Toric Varieties David Cox John Little Hal Schenck D EPARTMENT 01002
OF
M ATHEMATICS , A MHERST C OLLEGE , A MHERST, MA
E-mail address:
[email protected] D EPARTMENT OF M ATHEMATICS AND C OMPUTER S CIENCE , C OLLEGE THE H OLY C ROSS , W ORCESTER , MA 01610
OF
E-mail address:
[email protected] D EPARTMENT OF M ATHEMATICS , U NIVERSITY C HAMPAIGN , U RBANA , IL 61801 E-mail address:
[email protected]
OF I LLINOIS AT
U RBANA -
c 2009, David Cox, John Little and Hal Schenck
Preface
The study of toric varieties is a wonderful part of algebraic geometry that has deep connections with polyhedral geometry. Our book is an introduction to this rich subject that assumes only a modest knowledge of algebraic geometry. There are elegant theorems, unexpected applications, and, as noted by Fulton [30], “toric varieties have provided a remarkably fertile testing ground for general theories.” The Current Version. The January 2009 version consists of seven chapters: Chapter 1: Affine Toric Varieties Chapter 2: Projective Toric Varieties Chapter 3: Normal Toric Varieties Chapter 4: Divisors on Toric Varieties Chapter 5: Homogeneous Coordinates Chapter 6: Line Bundles on Toric Varieties Chapter 7: Projective Toric Morphisms These are the chapters included in the version you downloaded. The book also has a list of notation, a bibliography, and an index, all of which will appear in more polished form in the published version of the book. Two versions are available on-line. We recommend using postscript version since it has superior quality. Changes to the August 2008 Version. The new version fixes some typographical errors and has improved running heads. Other additions include: Chapter 4 includes more sheaf theory. Chapter 4 has an exercise about support functions and tropical polynomials. Chapter 5 now discusses sheaves associated to a graded modules. iii
Preface
iv
The Rest of the Book. Five chapters are in various stages of completion: Chapter 8: The Canonical Divisor of a Toric Variety Chapter 9: Sheaf Cohomology of Toric Varieties Chapter 10: Toric Surfaces Chapter 11: Toric Singularities Chapter 12: The Topology of Toric Varieties When the book is completed in August 2010, there will be three final chapters: Chapter 13: The Riemann-Roch Theorem Chapter 14: Geometric Invariant Theory Chapter 15: The Toric Minimal Model Program Prerequisites. The text assumes the material covered in basic graduate courses in algebra, topology, and complex analysis. In addition, we assume that the reader has had some previous experience with algebraic geometry, at the level of any of the following texts: Ideals, Varieties and Algorithms by Cox, Little and O’Shea [17] Introduction to Algebraic Geometry by Hassett [43] Elementary Algebraic Geometry by Hulek [52] Undergraduate Algebraic Geometry by Reid [84] Computational Algebraic Geometry by Schenck [90] An Invitation to Algebraic Geometry by Smith, Kahanp¨aa¨ , Kek¨al¨ainen and Traves [94] Readers who have studied more sophisticated texts such as Harris [40], Hartshorne [41] or Shafarevich [89] certainly have the background needed to read our book. We should also mention that Chapter 9 uses some basic facts from algebraic topology. The books by Hatcher [44] and Munkres [72] are useful references. Background Sections. Since we do not assume a complete knowledge of algebraic geometry, Chapters 1–9 each begin with a background section that introduces the definitions and theorems from algebraic geometry that are needed to understand the chapter. The remaining chapters do not have background sections. For some of the chapters, no further background is necessary, while for others, the material more sophisticated and the requisite background will be provided by careful references to the literature. The Structure of the Text. We number theorems, propositions and equations based on the chapter and the section. Thus §3.2 refers to section 2 of Chapter 3, and Theorem 3.2.6 and equation (3.2.6) appear in this section. The end (or absence) of a proof is indicated by , and the end of an example is indicated by .
Preface
v
For the Instructor. We do not yet have a clear idea of how many chapters can be covered in a given course. This will depend on both the length of the course and the level of the students. One reason for posting this preliminary version on the internet is our hope that you will teach from the book and give us feedback about what worked, what didn’t, how much you covered, and how much algebraic geometry your students knew at the beginning of the course. Also let us know if the book works for students who know very little algebraic geometry. We look forward to hearing from you! For the Student. The book assumes that you will be an active reader. This means in particular that you should do tons of exercises—this is the best way to learn about toric varieties. For students with a more modest background in algebraic geometry, reading the book requires a commitment to learn both toric varieties and algebraic geometry. It will be a lot of work, but it’s worth the effort. This is a great subject. What’s Missing. Right now, we do not discuss the history of toric varieties, nor do we give detailed notes about how results in the text relate to the literature. We would be interesting in hearing from readers about whether these items should be included. Please Give Us Feedback. We urge all readers to let us know about: Typographical and mathematical errors. Unclear proofs. Omitted references. Topics not in the book that should be covered. Places where we do not give proper credit. As we said above, we look forward to hearing from you! January 2009
David Cox John Little Hal Schenck
Contents
Preface
iii
Notation
xi
Part I: Basic Theory of Toric Varieties
1
Chapter 1.
3
Affine Toric Varieties
§1.0.
Background: Affine Varieties
§1.1.
Introduction to Affine Toric Varieties
10
§1.2.
Cones and Affine Toric Varieties
22
§1.3.
Properties of Affine Toric Varieties
34
Appendix: Tensor Products of Coordinate Rings Chapter 2.
Projective Toric Varieties
3
48 49
§2.0.
Background: Projective Varieties
49
§2.1.
Lattice Points and Projective Toric Varieties
55
§2.2.
Lattice Points and Polytopes
63
§2.3.
Polytopes and Projective Toric Varieties
75
§2.4.
Properties of Projective Toric Varieties
86
Chapter 3.
Normal Toric Varieties
93
§3.0.
Background: Abstract Varieties
93
§3.1.
Fans and Normal Toric Varieties
106
§3.2.
The Orbit-Cone Correspondence
115
§3.3.
Equivariant Maps of Toric Varieties
124
§3.4.
Complete and Proper
138 vii
Contents
viii
Appendix: Nonnormal Toric Varieties Chapter 4.
Divisors on Toric Varieties
148 153
§4.0.
Background: Valuations, Divisors and Sheaves
153
§4.1.
Weil Divisors on Toric Varieties
169
§4.2.
Cartier Divisors on Toric Varieties
174
§4.3.
The Sheaf of a Torus-Invariant Divisor
187
Chapter 5.
Homogeneous Coordinates
193
§5.0.
Background: Quotients in Algebraic Geometry
193
§5.1.
Quotient Constructions of Toric Varieties
202
§5.2.
The Total Coordinate Ring
216
§5.3.
Sheaves on Toric Varieties
224
§5.4.
Homogenization and Polytopes
229
Chapter 6.
Line Bundles on Toric Varieties
243
§6.0.
Background: Sheaves and Line Bundles
243
§6.1.
Ample Divisors on Complete Toric Varieties
260
§6.2.
The Nef and Mori Cones
279
§6.3.
The Simplicial Case
289
Appendix: Quasicoherent Sheaves on Toric Varieties Chapter 7.
Projective Toric Morphisms
299 303
§7.0.
Background: Quasiprojective Varieties and Projective Morphisms
303
§7.1.
Polyhedra and Toric Varieties
307
§7.2.
Projective Morphisms and Toric Varieties
315
§7.3.
Projective Bundles and Toric Varieties
321
Appendix: More on Projective Morphisms Chapter 8.
The Canonical Divisor of a Toric Variety
332 337
§8.0.
Background: Reflexive Sheaves and Differential Forms
337
§8.1.
One-Forms on Toric Varieties
347
§8.2.
p-Forms on Toric Varieties
352
§8.3.
Fano Toric Varieties
352
Chapter 9.
Sheaf Cohomology of Toric Varieties
353
§9.0.
Background: Cohomology
353
§9.1.
Cohomology of Toric Line Bundles
365
§9.2.
Serre Duality
380
Contents
ix
§9.3.
The Bott-Steenbrink-Danilov Vanishing Theorem
385
§9.4.
Local Cohomology and the Total Coordinate Ring
385
Appendix: Introduction to Spectral Sequences
395
Topics in Toric Geometry
401
Chapter 10.
403
Toric Surfaces
§10.1.
Singularities of Toric Surfaces and Their Resolutions
403
§10.2.
Continued Fractions and Toric Surfaces
412
§10.3.
Gr¨obner Fans and McKay Correspondences
423
§10.4.
Smooth Toric Surfaces
433
§10.5.
Riemann-Roch and Lattice Polygons
441
Chapter 11.
Toric Singularities
453
§11.1.
Existence of Resolutions
453
§11.2.
Projective Resolutions
461
§11.3.
Blowing Up an Ideal Sheaf
461
§11.4.
Some Important Toric Singularities
461
Chapter 12.
The Topology of Toric Varieties
463
Chapter 13.
The Riemann-Roch Theorem
465
Chapter 14.
Geometric Invariant Theory
467
Chapter 15.
The Toric Minimal Model Program
469
Bibliography
471
Index
477
Notation
Basic Notions
! "!# $ "
integers, rational numbers, real numbers, complex numbers semigroup of nonnegative integers image and kernel direct limit inverse limit
Rings and Varieties
&% ')(*',+.&%0% ' ( *' + -0&% '21( ( *' +1 ( 3547698 :.4<;=8 &%>;&%>;-? &4<;A8 B9C D.47EF8 GH JIK 4
polynomial ring in / variables formal power series ring in / variables ring of Laurent polynomials affine or projective variety of an ideal ideal of an affine or projective variety coordinate ring of an affine or projective variety ; graded piece in degree @ when is projective ; field of rational functions when is irreducible E affine variety of coordinate ring L projective variety of graded ring ; subset of an affine variety where OQR P E L localization of at O , a multiplicative set , a prime ideal Y E integral closure of the integral domain
xi
Notation
xii
E
4<;=8
! ; ! ;
L
B 4 8
affine cone of a projective variety diagonal map singular locus of a variety
E L H U
;
E
completion of local ring local ring of a variety at a point and its maximal ideal Zariski tangent space of a variety at a point dimension of a variety and dimension at a point L Zariski closure of in a variety product of varieties tensor product of rings over fiber product of varieties
Semigroups
6
%$
Z$ 1$ 2
203 2 34 5 &% 2 <
#"
lattice ideal of lattice ! $ lattice generated by "')( (+* ' ,-'/. %$ " (0* ' R elements & with & '( $ affine semigroup generated by affine semigroup affine semigroup 687:9; 2 semigroup algebra of 2+3 Hilbert basis of when 6 is strongly convex
Cones and Fans
= I> 4
B9C+B ) 4 6 8
! 6
67
XC D 4 6 8 EFD 4 6 8 H0I H0M OQP OVP U Y
L
convex cone generated by rational convex polyhedral cone in ?A@ subspace spanned by 6 dimension of 6 dual cone of 6 relative interior of 6 BC+B )4 6 8 R ?G@ interior of 6 when set of ,. ;J@ with K , H+L R face of 6N7 dual to H !6 , equal to 687G9 H I 4R L R , ,. hyperplane in ?G@ defined by K , 4R LXW , ,. half-space in ?G@ defined by K , fan in ? @
; @TS J ;J@TS
Notation
Y 4.8
Y
3
?
? 46 8 ; 46 8 B DB 46 8 Y M 46 8 46 8
xiii
Y
-dimensional cones of minimal generator of 9 ? , Y maximal cones of 4 8 R BC+B )4 6 sublattice 6:9 ? 3 quotient lattice ? ? 4 8 dual lattice of ? 6 , equal to 6 4 8 star of 6 , a fan in ? 6 Y star subdivision of along 6 index of a simplicial cone
. YS4 8
8 9-? I 9-;
Polyhedra
= I> 4
! O O U
+
4 A8
6"! Y%$ &
$
L
convex hull of polytope or polyhedron dimension of RS L R . hyperplane in ; @ defined by K , S R L%W . half-space in ; @ defined by K , dual or polar of a polytope standard / -simplex Minkowski sum multiple of a polytope or polyhedron cone over a polytope or polyhedron cone of a face # ! normal fan of a polytope or polyhedron support function of a polytope or polyhedron
? @QS G ?G@QS
Toric Varieties
P (' ; ? )
5 ;J@ ;0/ ? @ ?2/ K, L 3 8 .3 4 3 54 3 5 6 8
6 5
character lattice of a torus and character of , . ; lattice of one-parameter subgroups of a torus and one-parameter subgroup of . ? +* M R-, I..*4 ; J M 8 associated to ? and torus ? 1* ; +* vector spaces ; built from ; 1* +* vector spaces ? built from ? ? . , . ; or ; @ with ? or ?G@ pairing of $ !; affine and projective toric variety of affine toric variety of a cone 6J! ?:@ toric variety of a fan
;
Notation
xiv
$
@
3
46 8
; 4H 8 R 46 8 R 4 8
4
$ 4
projective toric variety of a lattice polytope or polyhedron toric variety of a basepoint free divisor 6 6 lattice homomorphism of a toric morphism and its real extension 4 3 distinguished point of Y orbit of 6 . Y , toric variety of B D B 4 6 8 closure of orbit of 6 . 6 of . YS4 8 torus-invariant prime divisor on $ torus-invariant prime divisor on of facet ! affine toric variety of recession cone of a polyhedron affine toric variety of a fan with convex support
6
Specific Varieties
+ + 4 + 8 M 47 M 8 + ?. ? 47 + 8 4 6 8 <
L!
affine and projective / -dimensional space weighted projective space multiplicative group of nonzero complex numbers standard / -dimensional torus rational normal + cone and curve blowup of at the origin 6 ; 4 8 Y blowup of along H , toric variety of Hirzebruch surface rational normal scroll
M 4H 8
Divisors
$
#"
V4 8 O
&%('
W ) 4 8
) )4 8 ) +*-, 4 8 = ) 4 8 = ) .*/,X4 8
6
= 4 8
6
local ring of a variety at a prime divisor discrete valuation of a prime divisor principal divisor of a rational function linear equivalence of divisors effective divisor group of principal divisors on group of Weil divisors on 6 group of torus-invariant Weil divisors on group of Cartier divisors on group of torus-invariant Cartier divisors on divisor class group of a normal variety 6
S
Notation
xv
G ! D.4 8 B C C 4 8
Picard group of a normal variety support of a divisor restriction of a divisor to an open set local data of a Cartier divisor on Cartier data of a torus-invariant Cartier divisor on polyhedron of a torus-invariant divisor fan associated to a basepoint free divisor Cartier divisor of a polytope or polyhedron support function of a Cartier divisor support functions integral with respect to ?
4 4 3 ' O 3' 86
,
Y
$ &
B 4 Y ? 8
6
Intersection Products
24 8
Z
( ? 4 T4 4 4 G !D. 4 4 A8
8 ? ( 4 8 8 8 8 8@
Z
degree of a divisor on a curve intersection product of Cartier divisor and complete curve numerically equivalent Cartier divisors and complete curves 4 = )4 8 8N * and 4 ( 4 8 F8N * ( 4 8 cone in ? generated by nef divisors ( 4 8 cone in ? generated by complete curves 4 8 Mori cone, equal to the closure of
)
G !D.4 8 *
primitive relation of a primitive collection
Sheaves and Bundles
"
"M
#" 4 8
"
44 8
;
BC D 47EF8
sheaf on
"! 4#" 8
%$&('
"
structure sheaf of a variety sheaf of invertible elements of sheaf of a Weil divisor on constant sheaf of rational functions when restriction of a sheaf to an open set sections of a sheaf over an open set ideal sheaf of a subvariety !
sheaf on
6
of an
E
L of a graded
-module
;
-module L L 4#" 8 sheaf on of the graded -module stalk of a sheaf at a point tensor product of sheaves of -modules
6
"
is irreducible
Notation
xvi
, $& 4 7 ; ; O M <
Y
4<; 8 4 S8
' 8
"
sheaf of homomorhisms < , 4 8 dual sheaf of , equal to vector bundle
rank vector bundle of a line bundle pullback of a line bundle ! 4 8 map to projective space determined by complete linear system of ; for R ! 4 8 fan that gives projective bundle of vector bundle or locally free sheaf
"
Quotients and Homogeneous Coordinates
E ; ; L
'
L
24 ' 8
'3
4Y 8
&4 Y 8
' P
' P
' ' P $ $ '
#4 "T8
E
ring of invariants of acting on good geometric quotient good categorical quotient 6 total coordinate ring of L YS4 8 variable in corresonding to . = 4 6 8 L graded piece of in degree . = 4 6 8 L degree in of a monomial in Y monomial generator of corresponding3 to 6 . L ' irrelevant ideal of , generated by the 354 4 Y 8*8 exceptional set, equal to I. *4 = 4 6 8 J M P 8 used in the quotient construction group , ' Laurent monomial P , ,. ; ' 9-; homogenization of , ,. facet variable of a facet ! -monomial associated to ,. 9-; 9 ; vertex monomial associated to vertex . L graded -module = 4 6)8 " shift of by .
Part I: Basic Theory of Toric Varieties
Chapters 1 to 9 introduce the theory of toric varieties. This part of the book assumes only a minimal amount of algebraic geometry, at the level of Ideals, Varieties and Algorithms [17]. Each chapter begins with a background section that develops the necessary algebraic geometry.
1
Chapter 1
Affine Toric Varieties
§1.0. Background: Affine Varieties We begin with the algebraic geometry needed for our study of affine toric varieties. Our discussion assumes Chapters 1–5 and 9 of [17].
6 ! L R F % ' ( *',+.- gives an affine variety 354768 R . + O 4 8 R for all O . 6 + ; and an affine variety ! gives the ideal :.4<;&8 R O . L O 4 8 R for all . ; . ; By the Hilbert Basis Theorem, an affine variety is defined by the vanishing of L 6 finitely many polynomials in , and for any ideal , the Nullstellensatz tells us that :.4 3547698*8 6 L 6 R R O . O . for some W ;since is algebraically Coordinate Rings. An ideal
closed. The most important algebraic object associated to
is its coordinate ring
A%>;S- R L :.4<;&8 &%>;- can be interpreted as the -valued polynomial functions on ; . Elements of A%>;S- is a -algebra, meaning that its vector space structure is compatible Note that with its ring structure. Here are some basic facts about coordinate rings:
A%>;S-
is an integral domain
:.4<;A8
is a prime ideal
;
; (A ; between affine variPolynomial maps (also called morphisms) F>% ; - &%>; ( - , where eties correspond to -algebra homomorphisms M M 4 98 R for . &%>; - . is irreducible.
Two affine varieties are isomorphic if and only if their coordinate rings are isomorphic -algebras. 3
Chapter 1. Affine Toric Varieties
4
A point
of an affine variety
;
gives the maximal ideal
%>; - O 4 8 R ! F%>; -<
O . F F%>; - arise this way. and all maximal ideals of
Coordinate rings of affine varieties can be characterized as follows (Exercise 1.0.1).
E
Lemma 1.0.1. A -algebra is isomorphic to the coordinate ring of an affine E variety if and only if isP a finitely generated -algebra with no nonzero nilpotents, E satisfies O R for some , W , then O R . i.e., if O .
; and &%>;- , we sometimes write ; R BC D47&%>; - 8 (1.0.1) ; with the set of maximal ideals of This can be made canonical by identifying F%>; - via the fourth bullet above. More generally, one can take any commutative BC D.47EF8 . The general definition of Spec uses E ring and define the affine scheme E all prime ideals of , not just the maximal ideals as we have done. Thus some ; BC D 47A%>;S- 8 , the maximal spectrum of A%>;S- . authors would write (1.0.1) as R To emphasize the close relation between
Readers wishing to learn more about affine schemes should consult [26] and [41].
+ ; An affine variety !
+ use. The Zariski Topology. has two topologies we will . The The first is the classical topology, induced from the usual topology on ; second is the Zariski topology,+ where the Zariski closed sets are subvarieties of contained in ; ) and the Zariski open sets are their (meaning affine varieties of complements. Since subvarieties are closed in the classical topology (polynomials are continuous), Zariski open subsets are open in the classical topology. L
;
L
Given a subset is the smallest ! L , its closureL in the Zariski topology ; L subvariety of containing . We call the Zariski closure of . It is easy to give examples where this differs from the closure in the classical topology.
Affine Open Subsets and Localization. Some Zariski open subsets of an affine ; &%>;- SX , let variety are themselves affine varieties. Given O .
;,N R . ; O 4 8 R P ! ;M ;,N is Zariski open in ; and is also an affine variety, as we now explain. Then + ; :.4<;&8 R K O ( O " L and pick . &% ' ( *' + - repreLet have ! 2 ; N R ; R 354 8 is Zariski open +in ; . Now consider a new senting O . Then 354 O ( O "
R 8 ! . Since the projection variable+ and let + R + maps bijectively onto ;VN , we can identify ;2N with the affine map . variety ! ; ;)N is easy to describe. Let F4<;=8 When is irreducible, the coordinate ring of F%>; - . Recall that elements of F4<;=8 be the field of fractions of the integral domain
§1.0. Background: Affine Varieties
give rational functions on
;
5
. Then let
A%>;S- N R O . A 4<;&8 . A %>;S-< W . (1.0.2) BC D.47A%>;S- N 8 is the affine variety ;2N In Exercise 1.0.3 you will prove that
.
Example 1.0.2. The / -dimensional + + torus is the affine open + subset
47 M 8 R S 354 ')( J',+98 ! with coordinate ring A% ' ( *' + - R A% ' 1( ( *'
+1 ( -<
Elements of this ring are called Laurent polynomials.
A%>;S- N
The ring from (1.0.2) is an example of localization. In Exercises 1.0.2 and 1.0.3 you will show how to construct this ring for all affine varieties, not just irreducible ones. The general concept of localization is discussed in standard texts in commutative algebra such as [2, Ch. 3] and [25, Ch. 2].
E
Normal Affine Varieties. Let be an integral domain with field of fractions . E Then is normal, or integrally closed, if every element of which is integral E E % ' - ) actually lies in E . over (meaning that it is a root of a monic polynomial in For example, any UFD is normal (Exercise 1.0.5). Definition 1.0.3. An irreducible affine variety F%>; - is normal.
;
is normal if its coordinate ring
+
&% ' ( *' +.-
For example, is normal since its coordinate ring and hence normal. Here is an example of a non-normal affine variety.
is a UFD
5 3 4 R 8 ' R ! Example 1.0.4. Let . This is an irreducible plane F% - R A% ' - K ' R L . curve with a cusp at the origin. It is easy to see that ' ' &% - respectively. This gives ' . FNow 4 8. let and be the cosets of and in ' &% - and that 4 'V 8 R ' . Consequently F% A computation shows that .
and hence
are not normal.
We will see below that
is an affine toric variety.
; has a normalization defined as follows. Let F%>; - Z R " . F 4<;=8 " is integral over &%>;- . A%>;S-!Z the integral closure of F%>;S- . One can show that A%>;S- Z is normal We call and (with more work) finitely generated as a -algebra (see [25, Cor. 13.13]). This An irreducible affine variety
gives the normal affine variety
; Z R BC D 47A%>;S- Z 8 ;AZ the normalization of ; . The natural inclusion &%>;- ! F%>;S-7Z R F%>;FZ We call ;A0Z ; . This is the normalization map. corresponds to a map
Chapter 1. Affine Toric Varieties
6
! Example We saw in Example 1.0.4 that the curve defined by ' R has1.0.5. ' &% - such that ' . F% - is integral F% - . In elements . over 8 A % ' F 4 F% Exercise 1.0.6 you will show that ! is the integral closure 4 8 of and that the normalization map is the map defined by .
At first glance, the definition of normal does not seem very intuitive. Once we enter the world of toric varieties, however, we will see that normality has a very nice combinatorial interpretation and that the nicest toric varieties are the normal ones. We will also see that normality leads to a nice theory of divisors. In Exercise 1.0.7 you will prove some properties of normal domains that will be used in §1.3 when we study normal affine toric varieties. Smooth Points of Affine Varieties. In order to define a smooth point of an affine ; ; variety , we first need to define local rings and Zariski tangent spaces. When ; is irreducible, the local ring of at is
Thus
R
O
. &4<;A8 O . F%>; -
consists of all rational functions on we have the maximal ideal
24 8 R P .
that are defined at . Inside of
. 4 8 R .
R
In fact, is the unique maximal ideal of Exercises 1.0.2 and 1.0.4 explain how to define The Zariski tangent space of
;
and
;
at
X , so that ; when
is a local ring. is not irreducible.
is defined to be
4<;&8 R-, I. T 4 J8 +
! 47 + 8 R In Exercise 1.0.8 you will verify that / for every .
. According to [41, p. 32], we can compute the Zariski tangent space of a point in an affine variety as follows.
+ ; ; Lemma 1.0.6. Let and let . :.4<;&8 R K O ( O " L ! ! F % ' (bean *' affine + - . Forvariety each , let ' 4 ' 8 R O 4 8 ')( O ' 4 8 ' + @ O ' ( '+
. Also assume that
+ 4<;&8 Then the Zariski tangent 4 ( 8 R R 4 " is8 R isomorphic to the subspace
! 4<;&of8
equations
@ O
@ O
. In particular,
;
/
defined by the .
Definition 1.0.7. A point of an affine variety is smooth or nonsingular if
4<;=8 R ! ;
! ; , where maximum of the dimensions of the ; containingis the. The irreducible components of point is singular if it is not ; ; smooth. Finally, is smooth if every point of is smooth.
§1.0. Background: Affine Varieties
7
Points lying in the intersection of two or more irreducible components of are always singular ([17, + Thm. 8 of Ch. 9, §6]).+ +
! 47 8
;
R / for every . + , we see that is smooth. For an Since ; ; and :.4<;&8 R ! irreducible affine variety of dimension @ , fix . write ( ; KO O " L . Using Lemma 1.0.6, it is straightforward to show that is smooth at if and only if the Jacobian matrix
(1.0.3) has rank /
R @
'O ' 4 8 ( ' " ( +
4 ( " 8 R O O
(Exercise 1.0.9). Here is a simple example.
Example 1.0.8. As noted 1.0.4, the plane curve defined by 8 R L Ain% ' Example . : 4 ' 4 * 8 . has Jacobian R has K ! . A point R
' R
R 4 * 4R 8
so the origin is the only singular point of
4<;&8
I. 4 J 8
.
;
R , see that is smooth at when
Since ; equals the dimension of ,aswea vector space over #. In ; . terms of commutative algebra, this means that is smooth if and only if is a regular local ring. See [2, p. 123] or [25, 10.3]. We can relate smoothness and normality as follows. Proposition 1.0.9. A smooth irreducible affine variety
F%>; -
. . ;
R Proof. In §3.0 we will see that X normal once we prove that is normal for all is normal whenever is smooth. that
;
is normal.
A%>;S-
By Exercise 1.0.7, is . Hence it suffices to show
This follows from some powerful results in commutative algebra: is a ; regular local ring when is a smooth point of (see above), and every regular local ring is a UFD (see [25, Thm. 19.19]). Then we are done since every UFD is X ; is sketched in normal. A direct proof that is normal at a smooth point . Exercise 1.0.10.
The converse of Propostion 1.0.9 can fail. We will see in §1.3 that the affine 354 ' R S8 ! is normal, yet 354 ' R 8 is singular at the origin. variety
; (
;
Products of Affine Varieties. Given affine varieties , there are several ; ( P ; is an affineand variety. The most direct ways to show that the cartesian product ; ( R B9C D.47A% ')( *' P - 8 ; way Let + R is BtoC proceed D 47A% ( as follows. + - 8 . Take :.4<; ( 8 ! R K O ( O " L and :.4<; 8 R K ( and ! L . Since the O ' and depend on separate sets of variables, it follows P + that
;( ; R 5 3 4 O ( O " ( 8 ! U
is an affine variety.
Chapter 1. Affine Toric Varieties
8
A fancier method is to use the mapping properties of the product. This will ; ( ; above, also give an intrinsic description of its coordinate ring. Given ;(# ; should be an affine variety with projections ' ;V( ; and ; ' as such that whenever we have a diagram
;$( ; ;
" /
;(
$
$
;
' are morphisms from an affine variety , there should be where ' a unique morphism (the dotted arrow) that makes the diagram commute, i.e., ' R ' . For the coordinate rings, this means that whenever we have a diagram
$
&%>; -
F%>;(*-
A%>;(
/
; -
%
&%>; - A% -
.
F% -
' with -algebra homomorphisms 'M , there should be a unique M algebra homomorphism (the dotted arrow) that makes the diagram commute. By A%>; ( - 1 F%>; the universal mapping property of the tensor product of -algebras, F > % ; * ( =&%>; has the mapping properties we want. Since is a finitely generated -algebra with no nilpotents (see the appendix to this chapter), it is the coordinate F > % ; (
; ring . For more on tensor products, see [2, pp. 24–27] or [25, A2.2].
;
$
Example 1.0.10. Let be an affine variety. Since ; + has product coordinate ring
+ R BC D.47A% ( + - 8 , the
A%>;S- &% ( + - R A%>;S- % ( + -< P ; :.4<;&8 R K O ( O " L ! A% ' ( *'+PX- , it follows that If is contained in with + :.4<; 8 R K O ( O " L ! A % ')( *'+PF ( +.-< ; 47 M 8 + is For later purposes, we also note that the coordinate ring of F%>; - VA% 1( ( + 1 ( - R &%>;- % (1 ( + 1 ( -< ; ( and ; , we note that the Zariski topology on ; ( ; Given affine varieties ;T( and ; . is usually not the product of the Zariski topologies on
§1.0. Background: Affine Varieties
9
R Example 1.0.11. Consider . By definition, a basis for the product 4 ( 4 4 of the Zariski topologies consists of sets where ' are Zariski open in . Such a set is the complement of a union of collections of “horizontal” and such “vertical” lines in . This makes it easy to see that Zariski closed sets in 354 R ' 8 cannot be closed in the product topology. as
Exercises for §1.0. 1.0.1. Prove Lemma 1.0.1. Hint: You will need the Nullstellensatz.
!"$#%&')(* for some + (a) Show that the usual formulas for adding and multiplying fractions induce well-defined binary operations that make into -algebra. (b) If has no nonzero nilpotents, then prove that the same is true for . For more on localization, see [2, Ch. 3] or [25, Ch. 2]. 1.0.3. Let be a finitely generated -algebra without nilpotents as in Lemma 1.0.1 and let ,-. be nonzero. Then (0/1123,23,54&26+6+7+98 is a multiplicative set. The localization : is denoted :; and is called the localization of at , . (a) Show that ; is a finitely generated -algebra without nilpotents. (b) Show that ; satisfies <>=?7@1 A ; 'B(C<>=D?E@1 F' ; . (c) Show that ; is given by (1.0.2) when is an integral domain. 1.0.4. Let G be an affine variety with coordinate ring IH G:J . Given a point KLMG , let %(N/7H GJPO&Q RKS'U(VT W8 . (a) Show that is a multiplicative set. The localization IH G:J is denoted XZYW[ \ and is called the local ring of G at K . (b) Show that every ] %X^YW[ \ has a well-defined value ]P RK' and that _ YW[ \:(N/]%XZY>[ \O]P RKS'B(V `8 is the unique maximal ideal of XaYW[ \ . (c) When G is irreducible, show that XaY>[ \ agrees with the definition given in the text. 1.0.5. Prove that a UFD is normal. 1.0.6. In the setting of Example 1.0.5, show that HAc b d b Jef Fg' is the integral closure of H gUJ and that the normalization -hig is defined by kjhl A 4m2npo7' . 1.0.7. In this exercise, you will prove some properties of normal domains needed for §1.3. (a) Let be a normal domain with field of fractions q and let .f be a multiplicative subset. Prove that the localization is normal. (b) Let :r , stu , be normal domains with the same field of fractions q . Prove that the intersection v r`wmx yr is normal. 1.0.8. Prove that zW{}|f~ \ A')(V for all K . 1.0.2. Let be a commutative -algebra. A subset is a multipliciative subset provided , , and is closed under multiplication. The localization consists of all formal expressions , , , modulo the equivalence relation
Chapter 1. Affine Toric Varieties
10
1.0.9. Use Lemma 1.0.6 to prove the claim made in the text that smoothness is determined by the rank of the Jacobian matrix (1.0.3).
G
K%G
X^Y>[ \
( d d zW{}|VG
H H 6 2 7 + 6 + 9 + 2 } J J _ ( d 26+7+6+92 d IH}H d 27+6+7+92 d J J Since K G is smooth, [70, §1C] proves the existence of a -algebra homomorphism XZYW[ \h IH}H d 26+6+7+62 d J}J that induces isomorphisms X YW[ \ _Y>[ \ IH H d 26+7+6+72 d J}JA _ for all f . This implies that the completion [2, Ch. 10] X YW[ \ ( {}| X Y>[ \ _ YW[ \
is isomorphic to a formal power series ring, i.e., X Y>[ \ H H d 26+7+6+62 d J}J . Such an isomorphism captures the intuitive idea that at a smooth point, functions should have power series expansions in “local coordinates” d 26+7+6+92 d . If NX YW[ \ is an ideal, then ( v _ YW[ \ '9+ This theorem of Krull holds for any ideal in a Noetherian local ring u and follows from [2, Cor. 10.19] with (Vu: . Now assume that KG is smooth. (a) Use the third bullet to show that XaY>[ \ h IH}H d 26+7+6+72 d J}J is injective. (b) Suppose that 2 X YW[ \ satisfy O in IH}H d 26+7+6+72 d J}J . Prove that O in X YW[ \ . Hint: Use the second bullet to show XaY>[ \ _ Y>[ \ and then use the third bullet. (c) Prove that X^Y>[ \ is normal. Hint: Use part (b) and the first bullet. This argument can be continued to show that XaYW[ \ is a UFD. See [70, (1.28)] 1.0.11. Let G and be affine varieties and let VCG be a subset. Prove that ! " ( . # G % is irreducible. Hint: 1.0.12. Let G and be irreducible affine varieties. Prove that $ G % (( 1) ' *) ' 4 , where ' 2 ' 4 are closed. Let ,G + ( ./ - G OW./ -D/8 % '0+ 8 . Suppose & G 4 and that ,G + is closed in G . Exercise 1.0.11 will be useful. Prove that GM(CG
1.0.10. Let be irreducible and suppose that is smooth. The goal of this exercise is to prove that is normal using standard results from commutative algebra. Set and consider the ring of formal power series . This is a local ring with maximal ideal . We will use three facts: is a UFD by [102, p. 148] and hence normal by Exercise 1.0.5.
§1.1. Introduction to Affine Toric Varieties We first discuss what we mean by “torus” and then explore various constructions of affine toric varieties.
47 8 +
M is a group under component-wise The Torus. The affine variety multipli+ 7 4 8 M cation. A torus is an affine variety isomorphic to , where inherits a group structure from the isomorphism. Associated to are its characters and oneparameter subgroups. We discuss each of these briefly.
§1.1. Introduction to Affine Toric Varieties
A character of a torus phism. For example, , R defined by
4 * ( * + 8 .
is a morphism
'
11
+ M
that is a group P 47 homomor+ ' M8 M gives a character
P 4 ( + 8 ( + R + 47 8 arise this+ way (see [53, §16]). Thus the One can show that+ all characters of M 7 4 8 form a group isomorphic to . characters of M For an arbitrary torus , its characters form a free abelian group ; of rank equal P to the dimension of . It is customary to say that ,. ; gives the character ' M. '
(1.1.1)
We will need the following results concerning tori (see [53, §16] for proofs).
(
Proposition 1.1.1.
(
(a) Let and be tori and let homomorphism. Then the image of
O
(b) Let be a torus and let ! O subgroup. Then is a torus.
be a morphism that is a group is a torus and is closed in .
be an irreducible subvariety of
that is a
Now assume that a torus acts linearly on a finite dimensional vector space on . . Given , . ; , over , where the action of . is denoted we get the eigenspace P
.
If
P R P
. R
, then every . P P S=
P R
48
'
, with eigenvalue given by
48.
'
for all
. .
is a simultaneous eigenvector for all
Proposition 1.1.2. In the above situation, we have
R P
P
.
This proposition is a sophisticated way of saying that a family of commuting diagonalizable linear maps can be simultaneously diagonalized. A one-parameter subgroup of a torus 4 group homomorphism. For+ example, R ) 7 4 8 M M defined by subgroup
is a morphism+
( +98 .
)
M
that is a gives a one-parameter
8 4 8 4 R (1.1.2) 47 M 8 + arise this way All one-parameter subgroups of + (see [53, §16]). It follows+ 7 4 8 M that the group of one-parameter subgroups of is naturally isomorphic to . )
For an arbitrary torus , the one-parameter subgroups form a free abelian group ? of rank equal to the dimension of . As with the character group, an element ) M . . ? gives the one-parameter subgroup There is a natural bilinear pairing P '
K L ;
?
defined as follows. )
(Intrinsic) P Given a character and a one-parameter subgroup , the com' ) M M is character for position of M , which is given by some . . Then K , L R .
Chapter 1. Affine Toric Varieties
12
47 8 +
M with (Concrete) If R + , then one computes that
K, L R
(1.1.3)
+
, R 4 * ( * + 8 .
+ ')( (
,
R 4 ( + 8 .
* ''
i.e., the pairing is the usual dot product.
It follows that the characters and one-parameter subgroups of a torus form ? free abelian groups ; and ? of finite rank with a pairing K L ; I. * 4 ; JX8 and ; with , I. * 4 ? JX8 . In terms of tensor that identifies ? with , 2*S M via ) 4 8 . products, one obtains a canonical isomorphism ? 5 . Hence it is customary to write a torus as +
47 8
5
+ M induces dual From this point of view, picking an isomorphism + bases of ; and ? , i.e., isomorphisms ; and ? that turn characters into Laurent monomials (1.1.1), one-parameter subgroups into monomial curves (1.1.2), and the pairing into dot product (1.1.3). The Definition of Affine Toric Variety. We now define the main object of study of this chapter.
;
Definition 1.1.3. An affine 5 47 M 8 + toric variety is an irreducible affine variety contain 5 on ing a torus as a Zariski open subset such that the action of 5 ; itself extends to an action of on . Obvious examples of affine toric varieties are less trivial examples.
R 354 ' #R
Example 1.1.4. The plane curve This is an affine toric variety with torus
R R
where the isomorphism is normal toric variety.
47 M 8 +
8 !
and
+
. Here are some
has a cusp at the origin.
47 M 8 R 4 8 . M M 4 8 . Example 1.0.4 shows that 9
is a non-
; R 354 ' R 8 ! is a toric variety with torus ; 9 47 M 8 R 4 ( ( " ( 8 ' . M 47 M 8 ( 4 ( ( " 894 ( 8 . We will see later that where the isomorphism is ; is normal. ? U ( parametrized by the map Example 1.1.6. Consider the surface in R+ ? U ( 4 8 4 ? ? " ( ? " ( ? 8 . Thus is defined using all degree defined by @ monomials in .
Example 1.1.5. The variety
§1.1. Introduction to Affine Toric Varieties
?U (
' *'2?
' ' ( )' ( '
13
Let the coordinates of be and let minors ideal generated by the of the matrix
6 ! &% ' *',?-
be the
'2? " ,' ? " ( '2? " ( ', ? 6 ' ' ( R ' ' U (J' @ R L . In Exercise 1.1.1 you will verify so R K ' U 47 8 R 354768 , so that ? R 47 8 is an affine variety. You will also prove that :.4 ? 8 R 6 , so that 6 is the ideal of all polynomials vanishing on ? . It follows that 6 3547698 is irreducible by Proposition 1.1.8 below. The affine that is prime since ? surface is called the rational normal cone of degree @ and is an example of a 6 determinantal variety. We will see below that is a toric ideal. ? It is straightforward to show that is a toric variety with torus 4*47 M 8 8 R ? 9 47 M 8 ? U ( 47 M 8 We will study this example from the projective point of view in Chapter 2.
We next explore three equivalent ways of constructing affine toric varieties. Lattice Points. In this book, a lattice + is a free abelian group of5 finite rank. Thus a . For example, a torus has lattices ; (of lattice of rank / is isomorphic to characters) and ? (of one-parameter subgroups).
; , a set $ R , ( , " 5 with M . Thenlattice 5 character consider the map
Given a torus P ' gives characters
defined by
5 R "
3
(1.1.4)
! ;
4 8 R ' P 4 8 (' P 4 8 . " 3
$
Definition 1.1.7. Given a finite set ! ; , the affine toric variety 3 to be the Zariski closure of the image of the map from (1.1.4).
3
is defined
This definition is justified by the following proposition.
$
%$
Proposition 1.1.8. Given ! ; be the sublattice $ "3 ! ; as above, let generated by . Then is an affine toric variety whose torus has character %$ . In particular, the dimension of 3 is the rank of %$ . lattice Proof. The map (1.1.4) can be regarded as a map 3
5 R 47 M 8 "
4 5 8
of tori. By Proposition 1.1.1, the image R that is closed in 47 M 8F" . The latter implies that 3 9 47 M 8F" R since is3 a torus Zariski closure 3 is. the of the image. It follows that the image is Zariski open in Furthermore, is 3 . irreducible (it is a torus), so the same is true for its Zariski closure 3
Chapter 1. Affine Toric Varieties
14
"
We next consider the action of . Since and takes varieties to varieties. Then
! 74 M 8F" , an element .
acts on
R ! 3 3 is a variety containing . Hence 3 ! 3 by the definition shows that ( 3 R 3 , so that the action of " to of Zariski closure. Replacing with 3 induces an action on " 3 . We concludeleads that is an affine toric variety. 4 8 of . Since R 34 5 8 , It remains to compute the character lattice ; 3
the map
gives the commutative diagram
47 8 "
5
/ M FF O FF FF FF F" " ?
where denotes a surjective map and an injective map. This diagram of tori induces a commutative diagram of character lattices
;
" o dHH HH HH HH H2 R
; 4 8
"
(
(
, " , the image Since ; takes the standard basis 4 8 %$ " to , 3 %$ is . By the diagram, we obtain ; . Then we are done since of the dimension of a torus equals the rank of its character lattice. 3
In concrete terms, fix+ a basis of ; , so that we may assume ; R $ ! the vectors in can be regarded as the columns of an / 3
with integer entries. In this case, the dimension of matrix .
+
. Then matrix is simply the rank of the
We will see below that every affine toric variety is isomorphic to $ of a lattice. finite subset
3
for some
" R 9B C D 47&% ')( *' " - 8 be the affine toric variety $ R , ( , " ! ; . We can describe the ideal :.4 3=8 ! A% ')( *' " - as follows. As in the proof of Proposition 1.1.8, (1.1.4)
Toric Ideals. Let ! coming from a finite set 3
induces a map of character lattices 3
(
" R ;
" to , that sends the standard basis map, so that we have an exact sequence
R+
R+
( , " . Let
" R+ ;
be the kernel of this
§1.1. Introduction to Affine Toric Varieties
15
4 ( " 8
In down to earth terms, elements R hence record the linear relations among the ,
R 4 ( " 8 . , set U R ' ' and R " and that U " . Note that R U ' R ' R Given
of
'.
satisfy
" R R
"
' ' R
R '
.
R
' '
' " '
3
' R '
3
since
Proposition 1.1.9. The ideal of the affine toric variety
and
. It follows easily that the binomial
vanishes on the image of (1.1.4) and hence on of the image.
:.4 3=8 R '
" & ' ( ( ' ,-' R
"H
3
. "
is the Zariski closure
" ! and
"
is
R
.
Proof. We leave it to the reader to prove equality of the two ideals on the right 6 denote this ideal and note that 6 ! :.4 3 8 . We prove (Exercise 1.1.2). Let the opposite inclusion following [97, Lem. 4.1]. Pick a monomial order on+ F% ' (*' " - and an isomorphism 5 47 M 8 + . Thus we may assume ; R + " P 7 4 8 M and the map is given by Laurent monomials in variables ( + . If 6 R P :.4 3 8 , then we . :.4 3 8 S 6 with minimal leading can pick O ' R '" ( ( ' ' . Rescaling if necessary, ' becomes the leading term monomial of O . P P
4 8
( +
Since O is identically zero as a polynomial in , there ' must be cancellation involving from . In other words, O must ' R '" ( ( the ' ' term' coming contain a monomial such that
"
4 P 8 R
'( ( "
This implies that
'( ( " R R & '" ( ( 4 * ' R ' 8 ' . so that 6 R ' ' of . It follows that O
* ' ,-' R
"
')( (
4P 8
" '( (
' ,-'
' R '
. . Then . : 4 3 8 also lies in leading term. This contradiction completes the proof.
6
the second description 6S byand has strictly smaller
$
finite set ! ; , there are several methods to compute the ideal :.4 3=Given 8 R 6 a of Proposition 1.1.9. For simple examples, the rational implicitization >6 using a algorithm of [17, Ch. 3,§3] can be used. It is also possible to compute basis of and ideal quotients (Exercise 1.1.3). Further comments on computing 6 can be found in [97, Ch. 12]. Inspired by Proposition 1.1.9, we make the following definition.
Chapter 1. Affine Toric Varieties
16
" be a sublattice. ' R ' "H . " and "-R
Definition 1.1.10. Let ! (a) The ideal
6 R
.
is called a lattice ideal.
(b) A prime lattice ideal is called a toric ideal. Since toric varieties are irreducible, the ideals appearing in Proposition 1.1.9 are toric ideals. Examples of toric ideals include:
K' R
Example 1.1.4 Example 1.1.5 Example 1.1.6
%' L ! & % ' XL ! A K ' GR K ' ' ' U ( R ' ' U (*'
@ R ? L ! ( A% ' *',? -< ? U
(The latter is the ideal of the rational normal cone ! .) In each example, we have a prime ideal generated by binomials. As we now show, such ideals are automatically toric.
Proposition 1.1.11. An ideal and generated by binomials.
6 ! F% ' ( *' " -
is toric if and only if it is prime
6
Proof. One direction is obvious. So suppose that is prime and generated by ' R ' . Then observe that 354768 9 47 M 8 " is nonempty (it contains binomials 4 8 ) and is a subgroup of 47 M 8F" (easy to check). Since 354768 ! " is irre354768 9 47 M 8 " is an irreducible subvariety of 47 M 8 " that is ducible, it follows that 3547698 9 47 M 8 " is a torus. also a subgroup. By Proposition 1.1.1, we see that R
47 8F"
47 8F"X
M M, Projecting on the th coordinate of M gives P a character ' 4 8 M which by our usual convention we write as for ,'G. ; . It 5 3 7 4 6 8 3 $ R , ( , " 6 R follows easily that for , and since is prime, we 6 :.4 3 8 by the Nullstellensatz. Then 6 is toric by Proposition 1.1.9. have R
We will later see that all affine toric varieties arise from toric ideals. For more on toric ideals and lattice ideals, the reader should consult [68, Ch. 7].
2
Affine Semigroups. A semigroup is a set with an associative binary operation and an identity element. To be an affine semigroup, we further require that 2 The binary operation on is commutative. We will write the operation as $ ! 2 gives and the identity element as . Thus a finite set
1$ R & P
3
*P ,
* P . ! 2
The semigroup is finitely generated, meaning that there is a finite set #$ R 2 . such that
;+ . ! The simplest example of an affine semigroup is $ The semigroup can be embedded in a lattice
+
$ !
2
. More generally, given a lattice ; and a finite set ! ; , we get the affine semigroup $ ! ; . Up to isomorphism, all affine semigroups are of this form.
§1.1. Introduction to Affine Toric Varieties
17
F% 2 -
2
Given an affine semigroup ! ; , the semigroup algebra is the vector 2 space over with as basis and multiplication induced by the semigroup structure 2 5 of . To make this precise, we think P of ; as the character lattice of a torus , so ' that ,. ; gives the character . Then
&% 2 - R P
P ' P
.
P
with multiplication induced by
2 $ If R
'
P
$ R , ( , " for
and
P
R P
R
for all but finitely many ,
P U P
2 - R A% ' P (' P F % , then . '
'
+ ! + gives the polynomial ring A% - R F % ' ( *' + -< ' ' and ( + is the standard basis of + . where ' R ( + is a basis of a lattice ; , then ; is generated by Example 1.1.13. If $ R ( + '
as an affine semigroup. Setting ' R gives the Laurent polynomial ring F% ; - R F% 1( ( +1 ( -< F% - is the coordinate ring of the torus 5 . Using Example 1.0.2, one sees that ; Here are two basic examples.
Example 1.1.12. The affine semigroup +
Affine semigroup rings give rise to affine toric varieties as follows. Proposition 1.1.14. Let
2
!;
be an affine semigroup.
A% 2 - is an integral domain and finitely generated as a -algebra. (a) BC D.47A% 2 - 8 is an affine toric variety whose torus has character lattice 2 , and (b) 2 # $ for a finite set $ !; , then BC D 47A% 2 - 8 R 3 . if R $ R , ( , " implies &% 2 - R F% ' P ( ' P - , so Proof. As noted above, F% 2 - is finitely generated. Since A% 2 - ! A% ; - follows from 2 ! ; , we see that F% 2 - is an integral domain by Example 1.1.13. $ R , ( , " , we get the -algebra homomorphism Using F% ' ( *' " N- R F% ; P ' ' . F% ; - . This corresponds to the morphism where ' 5 R " 3 4 3=8 M in the notation of §1.0. One checks that from (1.1.4), i.e., R kernel P the P - of & % ' ( ' F%is2 - the toric ideal :.4 3 8 (Exercise 1.1.4). R The image of is 3 , and then the coordinate ring of is &% 3A- R F% ' ( *' + - :.4 3 8 (1.1.5) R F% ' ( *',+.- 4 8 E 4 8 R A% 2 -<
Chapter 1. Affine Toric Varieties
18
BC D.47A% 2 - 8 R "3 . Since 2 R #$ implies 2 R %$ , the torus - 8 has the desired character lattice by Proposition 1.1.8.
This proves that 3 R BC D 47A% 2 of
Here is an example of this proposition.
2
Example 1.1.15. Consider the affine semigroup ! generated by and , so 2 . To study F% 2 - , we use (1.1.5). that R the semigroup algebra If $ R , then 3 4 8 R 4 8 and the toric ideal is :.4 3 8 R K ' R L by Example 1.1.4. Hence
F% 2 - R &% - A% ' - K ' R 3 is the curve ' R . and the affine toric variety
L
Equivalence of Constructions. We can now state the main result of this section, which asserts that our various approaches to affine toric varieties all give the same class of objects. Theorem 1.1.16. Let (a)
;
(b) (c) (d)
;
be an affine variety. The following are equivalent:
;
is an affine toric variety according to Definition 1.1.3.
; R 3
for a finite set
$
in a lattice.
is an affine variety defined by a toric ideal.
; R BC D 47A% 2 - 8
2
for an affine semigroup .
Proof. The implications (b) (c) (d) (a) follow from Propositions 1.1.8, ; 1.1.9 and 1.1.14. For (a) (d), let be an affine toric variety containing the torus 5 with character lattice ; . Since the coordinate ring of 5 is the semigroup A% 5 ! ; induces a map of coordinate rings algebra ; , the inclusion
5
This map is injective since A% as a subalgebra of ; . Let
2
&%>;-8R+ A % ; -<
is Zariski dense in
2 R ,.
;
'
P
;
, so that we can regard
A%>;S-
. A%>;S- 2
and note that is a semigroup. We will show that is finitely generated with F%>;- . This will complete the proof of the theorem. semigroup algebra equal to
F% 2 - ! A%>;S- is obvious. For the opposite inclusion, pick OQR P The inclusion A > % S ; & > % ; - ! F % ; - , we can write in . Using P O R P P ' where !; is finite and P R P for all ,. . Let P B C + B ) 4 ' 8 ,. R 'P ! F% ; F%>; - . If . 5 , then the action be the subspace spanned by the . Thus O . 9 F%>; - , so that we get O . A%>;S- . ; of on induces an action on the coordinate ring
§1.1. Introduction to Affine Toric Varieties
19
5
By Definition 1.1.3, this extends the usual P 4 8 of' P on , which in terms of the ' F% - is given by ' P R action coordinate ring ; that and hence 9 F%>; - are stable under the action of 5 . Since .9 ItAfollows %>;S- is finite-dimensional, A%>;S- is spanned by simultaneous eigenvectors 9 Proposition 1.1.2 implies that 5 . But this is taking place in F% ; - , where the simultaneous eigenvectors are of 9 A%>;S- is ' spanned characters. Then the above the characters! It follows that P &%>by A > % S ; . ;- for , . . It follows that expression for O . implies that 9 O . F% 2 - , proving that A%>;S- R A% 2 - .
A%>;S-
2
It remains to show that is finitely generated. Since is finitely generated, ( O " . &%>;- with &%>;- R &% O ( O " - . Expressing ' we can find O 2 each O 2 in terms of characters as above gives the desired finite generating set of . Hence is an affine semigroup. Here is one way to think about the above proof. When an irreducible affine ; 5 as a Zariski open subset, we have the inclusion variety contains a torus
A%>;S- ! F % ; -<
F%>; -
5
Thus consists of those functions on the torus that extend to polynomial ; ; functions on . Then the key insight is that is a toric variety precisely when the functions that extend are determined by the characters that extend.
;
8
354 ' R
R ! Example 1.1.17. We’ve seen that variety ' R A % ' X- . Also, the torus 47 isM 8 a toric L ( is via the map with toric ideal K ! 4 ( 8 4 ( ( " 8 . The lattice points used in this map can be represented as the columns of the matrix
(1.1.6)
2
R
! The corresponding semigroup consists of the -linear combinations of 2 the column vectors. Hence the elements of are lattice points lying in the poly pictured in Figure 1 on the next page. In this figure, the four hedral region in 2 vectors generating are shown in bold, and the boundary of the polyhedral region is partially shaded. In the terminology of §1.2, this polyhedral region is a rational 2 polyhedral cone. In Exercise 1.1.5 you will show that consists of all lattice points ; lying in the cone in Figure 1. We will use this in §1.3 to prove that is normal.
Exercises for §1.1.
$( d + d
# d + d Om #f -IH d 27+6+7+62 d J
1.1.1. As in Example 1.1.6, let
and let
g be the surface parametrized by F12nn'B(M F 23 26+7+6+92 6 2 'a +
Chapter 1. Affine Toric Varieties
20
(0,0,1)
(0,1,0)
(1,0,0)
(1,1,−1)
Figure 1. Cone containing the lattice points corresponding to
'Z( F 4 'a . Thus g ( ' . (b) Prove that g ' is homogeneous.
(c) Consider lex monomial order with d *d *d . Let ,% g ' be homogeneous of degree and let be the remainder of , on division by the generators of . Prove that can be written ( d 2 d ' . d 2 d 4 ' . d 2 d ' (a) Prove that
d # 27+6+7+6237 2n ')(C to show that I(V . (d) Use part (c) and W F 23 (e) Use parts (b), (c) and (d) to prove that $ ( g ' . Also explain why the generators of are a Gr¨obner basis for the above lex order. 1.1.2. Let - Prove that d"!$# be ad sublattice. & % # O U' ( d r # d)( OsB+2 * , ! 2 s -# * . + Note that when y. , the vectors 2 . , ! have disjoint! support (i.e., no coordinate is , with s -# *%' . positive in both), while this may fail for arbitrary sZ/2 * ' 1.1.3. Let 0 be a toric ideal and let 26+7+6+92 1 be a basis of the sublattice "! . Define 32 ( d 54# # d $4% O (L126+6+7+9/2 + Prove + 0 ( 3276 d 8pd ! . Hint: Given sZ/2 * . , ! with s-# *%. , write s # *( 9 +1 that + , + . This implies > < > < d r ( #f(;: dd $44#%? 4 : dd $ 44# % ? 4 #fm+ < 45= < 4$@ d d Show that multiplying this by ! +' A gives an element of 82 for B C . (By being where + is homogeneous of degree . Also explain why we may assume that the coefficient of + in ,+ is zero for .
more careful, one can show that this result holds for lattice ideals. See [68, Lem. 7.6].)
§1.1. Introduction to Affine Toric Varieties
21
G , 6 2 7 + 6 + 9 + 2 , f I H : G J . This gives a polynomial map 6 ! GMh ! 61H d 27+6+6+72 d ! Je#h H G:J"2 d + j#h , + + Let f ! be the Zariski closure of the image of . (a) Prove that 'B(:?7 '. (b) Explain how this applies to the proof of Proposition 1.1.14. 1.1.5. Let ( p12 `23 1'92 4 ( A 2612 1'92 o (L A 2 26E'92 (M nm26126#&' be the columns 1.1.4. Fix an affine variety and , which on coordinate rings is given by
of the matrix in Example 1.1.17 and let
g (
+ + O +
+
o
g Uo is a semigroup generated by 2 4 2 o 2 .
be the cone in Figure 1. Prove that
1.1.6. An interesting observation is that different sets of lattice points can parametrize the same affine toric variety, even though these parametrizations behave slightly differently. In this exercise you will consider the parametrizations
4 o and F 1 n 2 n Z ' L ( F 3 2 7 3 2 7 ' 4 F12nn'B(M F o 237 2nd o ' + c o (a) Prove that and both give the affine toric variety ( % # o7'^- . 4 (b) We can regard and as maps 4 61 4 #h and 4 6m 4 #h + Prove that is surjective and that is not. 4 6W ! " . The image of In general, a finite subset gives a rational map # in ! is called a toric set in the literature. Thus A 4 ' and A 4 ' are toric sets. The 4 papers [57] and [85] study when a toric set equals the corresponding affine toric variety.
1.1.7. In Example 1.1.6 and Exercise 1.1.1 we constructed the rational normal cone g using all monomials of degree in 12n . If we drop some of the monomials, things become more complicated. For example, consider the surface parametrized by 2 o 237 o 2n ' + F 1 n 2 n ) ' ( F This gives a toric variety - . Show that the toric ideal of is given by ')( %d $ # &c 2 'c $ 4 # o 2 "d 4 # c 4 $ 2 d 4 # c o fIH d 2 c 2 2 $ J + The toric ideal for g has quadratic generators; by removing the monomial 4 4 , we now
get cubic generators. In Chapter 2 we will use this example to construct a projective curve that is normal but not projectively normal.
B G B F12n 2 S')(M F 2n 2 23,+9 P2n 4 o 'a'B ) + (a) Find generators for the toric ideal ( G$'^ BQH d 2 d 2 d 2 d 2 d J . 4 o ) (b) Show that zW{}|-GM(.- . You may assume that Proposition 1.1.8 holds over B . (c) Show that $(0/ d d + d 2 d d 4 d o . o ) 4 o
1.1.8. Instead of working over , we will work over an algebraically closed field *) parametrized by characteristic ( . Consider the affine toric variety
of
Chapter 1. Affine Toric Varieties
22
G0 B
G
) has codimension two and can be defined by two equations, i.e., It follows that is a set-theoretic complete intersection. The paper [3] shows that if we replace with an algebrically closed field of characteristic ( , then the above parametrization is never a set-theoretic complete intersection.
B
K
80
f !
!
1.1.9. Prove that a lattice ideal for is a toric ideal if and only if is torsionfree. Hint: When is torsion-free, it can be regarded as the character lattice of a torus. The other direction of the proof is more challenging. If you get stuck, see [68, Thm. 7.4].
!
$( d 4 #fm2 dc # 12 &c # is the lattice ideal for the lattice t(N/> 2 2 'Z' o O 0 :|>z (W8: o +
1.1.10. Prove that
Also compute primary decomposition of to show that is not prime.
~
*~ gives an ] '( !n' ] is a group )jh ] ~ 1|S 23 ' + 1.1.12. Consider tori ~ and ~ with character lattices and . By Example 1.1.13, 4 H J and IH J . Let 6m~ 4 h ~ be a morphism the coordinate rings of ~ and ~ are I 4 4 4 that is a group homomorphism. Then induces maps
6 4 #h and 6mIH 4 Je#Sh IH J
1.1.11. Let be a torus with character lattice . Then every point
. Prove that evaluation map defined by induces a group isomorphism homomorphism and that the map
] 6 h
by composition. Prove that affine semigroups.
is the map of semigroup algebras induced by the map
of
N( ( %( - , :/E W8 t
1.1.13. A commutative semigroup is cancellative if # $ implies - $ for $ all and torsion-free if implies for all and . Prove that is affine if and only if it is finitely generated, cancellative, and torsion-free.
P2 2
Q %(
1.1.14. The requirement that an affine semigroup be finitely generated is important since lattices contain semigroups that are not finitely generated. For example, let be irrational and consider the semigroup
(N/> 2 9'Z', 4 O *8$ 4 +
Prove that is not finitely generated. (When satisfies a quadratic equation with integer coefficients, the generators of are related to continued fractions. For example, when and ( is the golden ratio, the minimal generators of are for , where is the th Fibonacci number. See [95] for further details.) (
(M n '
nm23 1' 4 2 4 ' (Mm2 `26+7+6+ 1.1.15. Suppose that ] 6, h is a group isomorphism. Fix a finite set and and #" are equivariantly isomorphic let ! ( ]P ' . Prove that the toric varieties (meaning that the isomorphism respects the torus action).
§1.2. Cones and Affine Toric Varieties We begin with a brief discussion of rational polyhedral cones and then explain how they relate to affine toric varieties.
§1.2. Cones and Affine Toric Varieties
23
Convex Polyhedral Cones. Fix a pair of dual vector spaces ; @ and ? @ . Our discussion of cones will omit most proofs—we refer the reader to [30] for more details and [76, App. A.1] for careful statements. See also [10, 38, 87].
@
Definition 1.2.1. A convex polyhedral cone in ?
= 6 R I> 4
where
L ! ? @
U
)
is a set of the form
W
)
! ?@
L
is finite. We say that 6 is generated by . Also set
= I> 4.8 R
.
One easily checks that a convex polyhedral cone 6 is in fact convex, meaning ' . 6 ) ' 4 1R)28 . 6 for all ) , and is a cone, meaning ' . 6 ) ' . 6 for all ) W . Since we will only consider convex cones, the cones satisfying Definition 1.2.1 will be called simply “polyhedral cones.”
that
of polyhedral cones include the first quadrant in = I> 4 ( ( 8 or first octant in . Examples For another example, the cone is pictured ! in Figure 2 below. It is also possible to have cones that contain entire lines. For = I> 4 (4R ( 8 = I> 4 (4R ( 8 ' 4 ' 8 ! is the -axis, is the closed example, while W . upper half-plane
. As we will see below, these last two examples are not strongly convex. We can also create cones using polytopes, which are defined as follows. Definition 1.2.2. A polytope in ?@ is a set of the form
where
L ! ? @
R = I>)4
U
)
is finite. We say that
R ! ?@ L is the convex hull of .
)
W
)
U
z
y
x
Figure 2. Cone in generated by
Chapter 1. Affine Toric Varieties
24
Polytopes include all polygons in and bounded polyhedra in . As we will see in later chapters, polytopes play a prominent role in the theory of toric varieties. Here, however, we simply observe that a polytope !? @ gives a polyhedral cone in ?@ by taking the cone
6 R ) 4 8 . ?@
W . R = I>)4
4
If shows what this looks when
.
)
is a pentagon in the plane.
P
Figure 3. Cone over a pentagon
! cone 6 is the dimension of the smallest B9C+B )4 6 8 the span of 6 . ) 4 6 8 6 ofof?Ga@ polyhedral containing 6 . We call Dual Cones and Faces. As usual, the pairing between ; @ and ? @ is denoted K L .
The dimension R B9C+B subspace
Definition 1.2.3. Given a polyhedral cone 6J!?
6 7 R ,. ;J@ K , LXW
@
, its dual cone is defined by
for all
. 6 .
Duality has the following important properties. Proposition 1.2.4. Let 6 4 8 cone in ;J@ and 67 7 R Given ,
RP
O P PU
! ?:@
be a polyhedral cone. Then
6 7
is a polyhedral
.
in ;J@ , we get the hyperplane
O P R
and the closed half-space
Then O and
6
. ? @
O PU R . ? @
K, L R ! ? @
L W ! ?@ K, X
O PU
is a supporting hyperplane of a polyhedral cone 6 ! ? @ if 6 ! , O P is a supporting half-space. Note that is a supporting hyperplane of 6
§1.2. Cones and Affine Toric Varieties
25
if and only if , . 6N7QS . Furthermore, if straightforward to check that
U 6 R O P 9
(1.2.1)
, ( , "
generate
6N7
, then it is
U 9 O P
Thus every polyhedral cone is an intersection of finitely many closed half-spaces. We can use supporting hyperplanes and half-spaces to define faces of a cone.
OP 9 6 H RP 6
Definition 1.2.5. A face of a cone of the polyhedral cone 6 is H R some , . 6 7 . Using , R shows that 6 is a face of itself. Faces called proper faces.
for are
The faces of a polyhedral cone have the following obvious properties. Lemma 1.2.6. Let 6
R = I> 4
be a polyhedral cone. Then:
(a) Every face of 6 is a polyhedral cone.
(b) An intersection of two faces of 6 is again a face of 6 . (c) A face of a face of 6 is again a face of 6 .
You will prove the following useful property of faces in Exercise 1.2.1.
Lemma 1.2.7. Let H be a face of a polyhedral cone 6 . If . H. then
. 6
!
and
R
. H
,
H R A facet of 6 is a face H of codimension 1, i.e., 6 . An edge of 6 is a face of dimension 1. In Figure 4 we illustrate a 3-dimensional cone with shaded facets and a supporting hyperplane (a plane in this case) that cuts out the vertical edge of the cone.
σ
supporting hyperplane
Figure 4. A cone
with shaded facets and a hyperplane supporting an edge
Here are some properties of facets.
Chapter 1. Affine Toric Varieties
26
+ ! ?:@ be a polyhedral cone. Then:
! R P If 6 / and the facets of 6 are H ' R O 96 for , ' . 6 7 ,
Proposition 1.2.8. Let 6 (a)
then
U 6 R O P 9
U 9 O P
= 6 7 R I> 4 , ( , " 8 (b) Every proper face H of 6 is the intersection of the facets of 6 containing H .
! R ! @ Note how part (a) of+ the proposition refines (1.2.1) when 6 +?
,
and
When working in
.
. From , dot product allows us to identify the dual with ( , " in part (a) of the proposition are facet this point of view, the vectors , normals, i.e., perpendicular to the facets. This makes it easy to compute examples. Example 1.2.9. It easy to see that the facet normals to the cone 6 ( R ( , R , R , R ( R . Hence are ,
= 6 7 R I> 4 ( (
!
in Figure 2
%R 8 !
This is the cone of Figure 1 at the end of §1.1 whose lattice points describe the 354 ' R S8 (see Example 1.1.17). As we semigroup of the affine toric variety will see, this is part of how cones describe normal affine toric varieties.
Now consider 6N7 , which is the cone in Figure 1. The reader can check that the ( ( . Using duality and part (b) of facet normals of this cone are Proposition 1.2.8, we obtain
= 6 R 4 6 7 8 7 R I> 4 ( ( Hence we recover our original description of 6 .
8
In this example, facets of the cone correspond to edges of its dual. More generally, given a face H of a polyhedral cone 6J! ? @ , we define
H I R , . V ; @ K , L R for all . H H M R , . 6 7 K , L R for all . H R 6 7 9 HI We call H M the dual face of H because of the following proposition. Proposition 1.2.10. If H is a face of a polyhedral cone 6 and H M R 6 7 9 H0I , then: (a) H0M is a face of 6N7 . H M is a bijective inclusion-reversing correspondence between (b) The map H the faces of 6 and the faces of 687 .
! H ! H M R (c) /.
!
Here is an example of Proposition 1.2.10 when 6 ! ? @ .
§1.2. Cones and Affine Toric Varieties
27
z
z
y
y
x
x σ
σ
Figure 5. A -dimensional cone
= I> 4 ( 8
and its dual
! Example 1.2.11. Let 6 R . Figure 5 shows 6 and 687 . You should check that the maximal face of 6 , namely 6 itself, gives the minimal face 6 M of 67 , namely the -axis. Note also that
even though
6
! 6
has dimension .
! M R 6
Relative Interiors. As already noted, the span of a cone 6 ! ? @ is the smallest C D 4 6 8 , is subspace of ?G@ containing 6 . Then the relative interior of 6 , denoted C D 4 8 6 as follows: the interior of 6 in its span. Exercise 1.2.2 will characterize
. C D 4 6 8 K , L for all ,. 6 7 S 6 I EF D4 6 8 When the span equals ?G@ , the relative interior is just the interior, denoted . H For an example of how relative interiors arise naturally, let be a face of a cone 6 . This gives the dual face H M R 67Q9 H I of 67 . Furthermore, if , . 687 ,
then one easily sees that
H ! O P 9 6 , . 6 7 , then In Exercise 1.2.2, you will show that if H R O P 9 6 ,. CX D 4 H M 8 C D 4 H M 8 Thus the relative interior tells us exactly which supporting hyperplanes H of 6 cut out the face . ,. H M
Strong Convexity. Of the cones shown in Figures 1–5, all but 6 7 in Figure 5 have the nice property that the origin is a face. Such cones are called strongly convex. This condition can be stated several ways.
Chapter 1. Affine Toric Varieties
28 Proposition 1.2.12. Let 6J!
6
is strongly convex
+ ?:@ 6
be a polyhedral cone. Then:
is a face of 6
contains no positive-dimensional subspace of ?
6T9 4FR 6 8 R
7 R 6 /
@
You will prove Proposition 1.2.12 in Exercise 1.2.3. One corollary is that if a polyhedral cone 6 is strongly convex of maximal dimension, then so is 6 7 . The cones pictured in Figures 1–4 satisfy this condition.
In general, a polyhedral cone 6 always has a minimal face that is the largest subspace contained in 6 . Furthermore:
R 6T9 F4 R 6 8 . R O P 9-6
6 R 6
whenever ,.
! ? @
C D 4 6 7 8 .
is a strongly convex polyhedral cone.
See Exercise 1.2.4. Separation. When two cones intersect in a face of each, we can separate the cones with the following result, often called the Separation Lemma.
(
Lemma 1.2.13 (Separation Lemma). Let 6 6 ( meet along a common face H R 6 9 6 . Then
be polyhedral cones in
H R O P 9 6 ( R O P 9 6 XC D 4 6 (7 9 4FR 6 8 7 8 . for any ,. R * R * . ! ?G@ , we set R Proof. Given
.
? @
that
. A standard
result from cone theory tells us that
6 (7 9 F4 R 6 8 7 R 4 6 ( R 6 8 7 C D4 6 (7 9 4FR 6 8 7 8 . The above equation and Exercise 1.2.4 imply Now fix ,. O P ( R 6 , i.e., that cuts out the minimal face of 6 O P 9 46 ( R 6 8 R 46 ( R 6 8 9 46 R 6 ( 8 However, we also have
4 6 ( R 6 8 9 4 6 %R 6 ( 8 R H R H ( One inclusion is obvious since H R 6 9V6 . For the other inclusion, write . 4 6 ( R 6 8 9 4 6 %R 6 ( 8 as * ( R * R XR ( * ( ( . 6 ( * . 6 R ( * ( ( R * Then that this element lies in H R 6 9 6 . Since (* ( . 6 ( , we have * ( ( implies . H by Lemma 1.2.7, and * . H follows similarly. ( R . H R H * * R Hence , as desired.
§1.2. Cones and Affine Toric Varieties
We conclude that
29
OTP 9 4 6 ( R 6 8 R H R H ( . Intersecting with 6 , we obtain O P 9 6 ( R 4H R H 8 9 6 ( R H
where the last equality again uses Lemma 1.2.7 (Exercise 1.2.5). If instead we R intersect with 6 , we obtain and
O P 9 F4 R 6 8 R H
O P 9 F4 R 6 8 R 4 H R H 8 9 F4 R 6 8 R R H follows.
O P
In the situation of Lemma 1.2.13 we call
a separating hyperplane.
and ; be dual+ lattices with associated vector * . For we usually use the lattice + . ; + = I> 4
Rational Polyhedral Cones. Let ? * and ;J@ R spaces ? @ R ?
The cones appearing in Figures 1, 2 and 5 are rational. We note without proof that faces and duals of rational polyhedral cones are rational. Furthermore, if 6 R = I> 4
6 9 ?
/
R &
U
)
)
W
in
One new feature is that a strongly convex rational polyedral cone 6 has a canonical generating set, constructed as follows. Let be an edge of 6 . Since 6 is strongly convex, is a ray, i.e., a half-line, and since is rational, the semigroup T9J? is generated by a unique element of the intersection. We call the ray generator of . Figure 6 shows the ray generator of a rational ray in the plane. The dots are the lattice ? R and the white ones are 9 ? . ←ρ
↑ ray generator
Figure 6. A rational ray
and its unique ray generator
Lemma 1.2.15. A strongly convex rational polyhedral cone is generated by the ray generators of its edges.
It is customary to call the ray generators of the edges the minimal generators of a strongly convex rational polyhedral cone. Figures 1 and 2 show -dimensional strongly convex rational polyhedral cones and their ray generators.
Chapter 1. Affine Toric Varieties
30
In a similar way, a rational polyhedral cone 6 of maximal dimension has unique facet normals, which are the ray generators of the dual 6 7 , which is strongly convex by Proposition 1.2.12. Here are some especially important strongly convex cones. Definition 1.2.16. Let 6J! (a) (b)
6
? @
be a strongly convex rational polyhedral cone.
is smooth or regular if its minimal generators form part of a -basis of ? ,
6
is simplicial if its minimal generators are linearly independent over .
The cones pictured in Figure 5 are smooth, while those in Figures 1 and 2 are not even simplicial. Note also that the dual of a smooth (resp. simplicial) cone is again smooth (resp. simplicial). Later in the section we will give examples of simplicial cones that are not smooth. Semigroup Algebras and Affine Toric Varieties. Given a rational polyhedral cone 6J! ? @ , the lattice points
2 3 R 7 6 9-;
!;
form a semigroup. A key fact is that this semigroup is finitely generated. Proposition 1.2.17 (Gordan’s Lemma). hence is an affine semigroup.
2 3 R 6 7 9J;
is finitely generated and
= I> 4 8
Proof. Since 6 7 is rational polyhedral, 6 7 R for a finite set + ! ; . P P
, so that P R ,
& + Then
is a bounded region of ; @ 2 3 4 8 . Note that .
9 ; is finite since ;
9-; !
*
4
9 ; 8 generates 2+3 as a semigroup. To prove this, take R & P * ) P , where )0P W . Then )+P R )0P P with . P , so that 0 ) P R P * )0P , P * P , The second sum is in 9X; (remember . ; ). It follows that is a nonnegative 4 9 ; 8. integer combination of elements of
We claim 2 3 and write . and
+ Theorem 1.2.18. Let 6 ! ?:@ 2 3
Since affine semigroups give affine toric varieties, we get the following.
R 67:9-;
. Then 4
be a rational polyhedral cone with semigroup
3 R BC D 47A% 2 3 - 8 R B9C D 47&% 7 6 9 ; -8
is an affine toric variety. Furthermore,
4
3 R /
the torus of
4
3
is
5 R ? *S M
6
is strongly convex
§1.2. Cones and Affine Toric Varieties
31 4
Proof. By Gordan’s Lemma and Proposition 1.1.14, 2 3 whose torus has character lattice !; . To study
3
2 3 is an affine toric variety , note that 203 R 203 R 2 3 R , ( R , , ( , . 2 3 .
+ 2 3
and ,. ; . Then , R , ( R , Now suppose that , . for some ( , . 2 3 R 6 7 9 ; . Since , ( and , lie in the convex set 6 7 , we have for , ( ( , , R , ( " , . 6 7 203 4 , 8 R , . 2 3 , so that ; It follows that , R , is torsion-free. Hence 5 203 R B 2 3 R 4 3 ; / (1.2.3) the torus of is
! 7 R Since 6 is strongly convex if and only if 6 / (Proposition 1.2.12), it remains to show that
!
4
3 R /
B 2 3 R /
7 R 6 /
The first equivalence follows since the dimension of an affine toric variety is the dimension of its torus, which is the rank of its character lattice. We leave the proof of the second equivalence to the reader (Exercise 1.2.6).
5
Since we want our affine toric varieties to contain the torus , we consider 4 3 only those affine toric varieties for which 6 !? @ is strongly convex. Our first example of Theorem 1.2.18 is an affine toric variety we know well.
= I> 4 ( (
8
Example 1.2.19. Let 6 R ! ?@ R with ? R . This is the cone pictured in Figure 2. By Example 1.2.9, 6 7 is the cone pictured in Figure 1, and by Example 1.1.17, the lattice points in this cone are generated 4 3 by columns of matrix (1.1.6). It follows from Example 1.1.17 that is the affine 5 3 4 ' R 8 . toric variety
+ / and set 6 R = >I 4 ( 8 ! = 6 7 R >I 4 ( U ( + 8
Here are two further examples of Theorem 1.2.18. Example 1.2.20. Fix
3 R BC D.47A% ')( *' * ' 1 U ( ( *' +1 ( - 8 R 5 47 M 8 + " + (Exercise 1.2.7). This implies the general fact that if 6 ! ? @ + 4 3 547 M 8 " . cone of dimension , then and the corresponding affine toric variety is 4
. Then
is a smooth
R and / R . = >I 4 % ( R 8 ! . @ Example 1.2.21. Fix a positive integer @ and let 6 R = I> 4 ( ( 8 This has dual cone 6 7 R 7 on the next page shows 6 7 L 3 @ . Figure R R when @ . The affine semigroup 6 7 9 is generated by the lattice points Figure 5 illustrates the cones in Example 1.2.20 when
Chapter 1. Affine Toric Varieties
32
Figure 7. The cone
4 8
when
"
@ . When @ R , these are the white dots in Figure 7. (You will for prove these assertions in Exercise 1.2.8.)
By §1.1, the variety 47 M 8 affine ? U ( toric map defined by
4
3
is the Zariski closure of the image of the
4 8 R 4 . ? 8
4 8 4 ? ? " ( ? " ( ? 8 This map has the same image as the map ?U ( ? 4 3 used ! in Example 1.1.6. Thus is isomorphic to the rational normal cone minors of the matrix whose ideal is generated by the ' ')( '2? " ',? " ( ')( ' '2? " ( ',? Note that the cones
6
and 6
7
are simplicial but not smooth.
We will return to this example often. One thing evident in Example 1.1.6 is the difference between cone generators and semigroup the cone 6 7 has L 3 R 6 7 9 has generators: two generators but the semigroup @ .
L 3
R 67Q9 ; When 6 ! ? @ has maximal dimension, the semigroup unique minimal generating set constructed as follows. Define an element , 2 3 Z , Z Z for , Z7 , Z Z . 2 3 implies , Z R or , to be irreducible if , R ,
Proposition 1.2.22. Let 6 2 3 R 6 7 9-; . Then
<
! ?:@
<
(a) (b) (c)
<
<
be strongly convex of maximal dimension and let
R , . 2 3 ,
has the following properties: is finite and generates
2+3
has a
R P of ZZ R .
.
is irreducible
contains the ray generators of the edges of 6
is the minimal generating set of
2 3
7
.
with respect to inclusion.
§1.2. Cones and Affine Toric Varieties
33
Proof. Proposition 1.2.12 implies that 6 element . 6 9-? S such that K , and only if , R .
7
is strongly convex, so we can find an L . for all ,. 2 3 and K , L R if
203
Now suppose that , . 3 irreducible. Then Z Z are nonzero elements ofis2 not and , . It follows that
Z K, L R K, L
K,
, R , Z
, ZZ
where
, Z
ZZ L
K , Z L K , Z Z L . SX , so that Z ZZ K , L K , L and K , L K , L 2 3 Using induction on K , L , we conclude that every element of is a sum of irre< 2 3 ducible elements, so that generates . Furthermore, using a finite generating < 203 with
set of , one easily sees that is finite. This proves part (a). The remaining parts of the proof are covered in Exercise 1.2.9.
<
203
2 3
! The set 2+is3 called the Hilbert basis of and its elements are the minimal generators of . Algorithms for computing Hilbert bases are discussed in [68, 7.3] and [97, Ch. 13]. Exercises for §1.2.
( for . 1.2.2. Here are some properties of relative interiors. Let be a cone. (a) Show that if , then ? }{ 5' if and only if 2 for all . (b) Let be a face of and fix . Prove that .? { E ' ( S+
1.2.1. Prove Lemma 1.2.7. Hint: Write
1.2.3. Prove Proposition 1.2.12. 1.2.4. Let
be a polyhedral cone.
(a) Use Proposition 1.2.10 to prove that has a unique minimal face with respect to inclusion. Let denote this minimal face.
( ' . # 5' . (d) Prove that ( p (e) Fix . Prove that ?. { & ' if and only if ( . (f) Prove that (e Q is a strongly convex polyhedral cone. 1.2.5. Let be a face of a polyhedral cone and let U# be defined as in the proof of Lemma 1.2.13. Prove that (L U# ' . Also show that # (<> = ' , i.e., y# is the smallest subspace of containing . 1.2.6. Fix a lattice and let <> = ! e FB' denote the span over of a subset .! . (a) Let ! be finite. Prove that " ! $# U (*zW{ |*<>%= 5 FB' . (b) Prove that (c) Prove that
is the largest subspace contained in .
Chapter 1. Affine Toric Varieties
34
z`{ | ! `? F)')(*zW{}|V<W= e )'
(b) Let ! be finite. Prove that . (c) Use parts (a) and (b) to complete the proof of Theorem 1.2.18. 1.2.7. Prove the assertions made in Example 1.2.20. 1.2.8. Prove the assertions made in Example 1.2.21. Hint: First show that when a cone is smooth, the ray generators of the cone also generate the corresponding semigroup. Then write the cone of Example 1.2.21 as a union of such cones.
/E `8 ( 1.2.10. Let V be a cone generated by a set of linearly independent vectors in .
1.2.9. Complete the proof of Proposition 1.2.22. Hint for part (b): Show that the ray generators of the edges of are irreducible in . Given an edge of , it will help to pick such that . Show that
is strongly convex and simplicial.
1.2.11. Explain the picture illustrated in Figure 8 in terms of Proposition 1.2.8.
σ ↑
σ ↑
→
↑ Figure 8. A cone
in the plane and its dual
1.2.12. Let be a polytope lying in an affine hyperplane (= translate of a hyperplane) not containing the origin. Generalize Figure 3 by showing that gives a convex polyhedral cone in . Draw a picture.
( ! `?1 - # ( 4 2 '^ 4 . (a) Describe and find generators of 4 . Draw a picture similar to Figure 7. (b) Compute the toric ideal of the affine toric variety and explain how this exercise relates to Exercise 1.1.6. 1.2.14. Consider the simplicial cone ( ! ?1 2 4 2 4 ( o '^Po . (a) Describe and find generators of o . (b) Compute the toric ideal of the affine toric variety . 1.2.15. Let be a strongly convex polyhedral cone of maximal dimension. Here is an example taken from [30, p. 132] to show that and need not have the same number of be the cone generated by ( + for all 2 , ( T . edges. Let (a) Show that has 12 edges. 9 + , and has 8 edges. (b) Show that is generated by + and # + ( 1.2.13. Consider the cone
§1.3. Properties of Affine Toric Varieties The final task of this chapter is to explore the properties of affine toric varieties. We will also study maps between affine toric varieties.
§1.3. Properties of Affine Toric Varieties
35
Points of Affine Toric Varieties. We first consider various ways to describe the points of an affine toric variety.
B9C D.47&% 2 - 8
;
R Proposition 1.3.1. Let be the affine toric variety of the affine 2 semigroup . Then there are bijective correspondences between the following: . ;
(a) Points
.
(b) Maximal ideals
! A% 2 .
2
(c) Semigroup homomorphisms under multiplication.
, where
is considered as a semigroup
Proof. The correspondence between (a) and (b) is standard (see [17, Thm. 5 of Ch. 5, §4]). The correspondence between (a) and (c) is special to the toric case. P
;
. P , define Given a point ' . A% 2 This makes sense since semigroup homomorphism.
by sending , . 2 to ' 4 8 . . A > % R ;S- . One easily checks that 2 is a
2
2
P Going 2 let the other way, F % P
is a basis of ,
be a semigroup homomorphism. Since F% 2 - which is induces a surjective linear map A% 2 - is a maximal ideal a -algebra homomorphism. The kernel of the map ; and thus gives a point . by the correspondence between (a) and (b). '
2 $ R , ( , " as follows. Let generate , so " concretely " 4 4 ( 8 4 8 * 8 , " . . Let us prove that . ; . . Let R , ! ' R ' vanishes at for all exponent By Proposition 1.1.9, it suffices to show that " R 4 * ( * " 8 4 ( * * " 8 satisfying vectors and R
; that R
We construct 3
"
This is easy, since
'( (
* ' ,-' R
"
' ,-' ( ' (
being a semigroup homomorphism implies that
" " " ( 4 ,-' 8 R ( * ' ,-' R ( ' ,-' R ( 4 ,-' 8 )' ( ' ( ' ( ' (
"
It is straightforward to show that this point of the previous paragraph (Exercise 1.3.1).
;
agrees with the one constructed in
;
As an application of this result, we describe the torus action on . In terms ; R "3 ! " , the proof of Proposition 1.1.8 shows that the of the embedding 5 3 is induced by the usual action of 47 M 8 " on " . But how do we action of on see the action instrinsically, without embedding into affine space? This is where 5 and . ; , and let semigroup homomorphisms prove their value. Fix . 4 , 8 . In ExerciseP 1.3.1 you will correspond to the semigroup homorphism , ' 4 8 4 , 8 . This show that is given by the semigroup homomorphism , abstract description will prove useful in Chapter 3 when we study the orbits of the torus action.
Chapter 1. Affine Toric Varieties
36
5
;
From the point of view of group actions, the action of on is given by a 5 ; ; . Since both sides are affine varieties, this should map be a morphism, meaning that it should come from a -algebra homomorphism
A% 2 - R A%>;S-NR A% 5 ; - R &% 5 - 1&A%>;S- R F % ; - 1=A% 2 -< P P P
This homomorphism is given by
'
'
'
for ,.
2
(Exercise 1.3.2).
We can also characterize those affine toric varieties for which the torus action 2 2 4FR 2 8 R , has a fixed point. We say that an affine semigroup is pointed if 9
2 i.e., if is the only element of with an inverse. This is the semigroup analog of being strongly convex. Proposition 1.3.2. Let
;
be an affine toric variety. Then:
; BC D.47F% 2 - 8 , then the torus action has a fixed point if and only (a) If we write R 2 if is pointed, in which case the unique fixed point is given by the semigroup 2 homomorphism defined by
, R R , (1.3.1) , RP ; "3 ! " for $ ! 2 S , then the torus action has a fixed (b) If we write R 3 point if and only if . , in which case the unique fixed point is . ; be represented by the semigroup Proof. For part (a), let . 2 8 4, 8 R 4, 8 . Then is fixed by the torus action if and only if ' P 4 homomorphism 2 5 4 8 for all , . and . . This equation P 4 8 is satisfied for , R4 8 since R , ' R P shows that , R . Thus, if a and if , R P , then picking with
fixed point exists, then it is unique and is given by (1.3.1). Then we are done since 2 (1.3.1) is a semigroup homomorphism if and only if is pointed.
; 3 ! " has a fixed point, in which case (b), first assume that R 2 R For $ part $ ! 2 S is pointed and the unique point is given by (1.3.1). Then " 3 . and the proof of Proposition 1.3.1 imply that is the origin in , so that . " 5 7 4 F 8 " 7 4 F 8 " M . ! The converse follows since . is fixed by M hence by ; R 4 3 When
, we can state Proposition 1.3.2 as follows (Exercise 1.3.3).
Corollary 1.3.3. Let 6J!?G@ be a strongly convex polyhedral cone. Then the torus
! R ! 4 3 action on has a fixed point if and only if 6 ?:@ , in which case the fixed point is unique and is given by the maximal ideal
K'
where as usual
203 R 67:9 ;
P
,. 2 3 S L ! &% 2 3 -<
.
We will see in Chapter 3 that this corollary is part of the correspondence be4 3 tween torus orbits of and faces of 6 .
§1.3. Properties of Affine Toric Varieties
37
Normality and Saturation. We next study the question of when an affine toric ; variety is normal. We need one definition before stating our normality criterion.
2
Definition 1.3.4. An affine semigroup 2 2 and ,. ; , ,. implies ,. .
! ;
is saturated if for all
. S
2 3 R For example, if 6 ! ?G@ is a strongly convex rational polyhedral cone, then 6 7 9-; is easily seen to be saturated (Exercise 1.3.4). ; 5
Theorem 1.3.5. Let are equivalent: (a)
;
be an affine toric variety with torus
. Then the following
is normal.
; R B C D 47A% 2 - 8 , where 2 !; is a saturated affine semigroup. (b) ; R BC D.47&%>L 3 - 8 4 R 4 3 8 , where 203 R 6 7 9 ; and 6 ! ? @ (c) convex rational polyhedral cone.
BC D 47A% 2 - 8
;
is a strongly
2
Proof. By Theorem 1.1.16, R for an affine semigroup contained ; has the character lattice in a lattice, and by Proposition 1.1.14, the torus of + 2
! ; . Also let / R , so that ; . We will use this to prove (a) ; R (b) (c).
;
F% 2 -
A%>;S-
R (a) (b): If is normal, then is integrally closed in its field 2 A < 4 & ; 8 S& and , . ; . , . of fractions . Suppose that for some . P' 5 ; Then is a polynomial function on and hence a rational function on since P 5P ! ; is Zariski open. We also have ' . P F% 2 - since , . 2 . It follows that ' R ' F% 2 - . By the is a root of the monic polynomial in P A% 2 - with2 coefficients 2 ' . definition of normal, we obtain , i.e., ,. . Thus is saturated. 2
$
2
2
(b) (c): Let rational = I> ! 4 $ 8 be a finite generating B %$ setR of . Then !lies = inI> the 4 $ 8 R / ! ; @ , and = I> 4 $ 8 / implies polyhedral cone 7 ! ? @ is a strongly convex by Exercise 1.2.6. It follows that 6 R 2 rational polyhedral cone such that ! 6 7 9; . In Exercise 1.3.4 you will prove 2 2 2 3 . that equality holds when is saturated. Hence R
F% 2 3 -
&%
-
R 6 7 9T; is normal when 6J! ?G@ (c) (a): We need to show that ( be the rays of 6 . Since is a strongly convex rational polyhedral cone. Let 6 is generated by its rays (Lemma 1.2.15), we have Intersecting with ;
gives
2 3 R
'7 ( ' (
6 7 R
(2 ')( , which easily implies
F% 2 3 - R F%>L 3 -
')( (
F% 2 7 -<
&% 2 -
By Exercise 1.0.7, is normal if each is normal, so it suffices to prove 2 A % that is normal when is a rational ray in ?:@ . Let . Q9V? be the ray
Chapter 1. Affine Toric Varieties
38
(
generator of . Since is primitive, i.e, . ? for all , we can find a ( + of ? with R ( (Exercise basis 1.3.5). Thus we can assume that R = I> 4 ( 8 , so that ( (
% 2 - R F% ')(*'V1 *' +1 F &% ' ( *' + - is normal (it is a UFD), so its localization by Example 1.2.20. But A% ')( *' +.- R A% ')(*' 1 ( *' +1 ( 354 ' R 8
is also normal by Exercise 1.0.7. This completes the proof.
;
Example 1.3.6. We saw in Example 1.2.19 that R is the affine toric = I> 4 ( . ( 8 4 3 R of the cone 6 variety pictured in Figure 1. Then ; Theorem 1.3.5 implies that is normal, as claimed in Example 1.1.5. ? (
? ! U
Example 1.3.7. By Example 1.2.21, the rational normal cone is the affine toric variety of a strongly convex rational polyhedral cone and hence is normal by Theorem 1.3.5. It is instructive to view this example ? ? using ? ( ? ( the parametrization
4 . 8 R 4 " " 8 3
$
from Example 1.1.6. Plotting the lattice points in for @ R white 2 gives #$ the squares in Figure 9 (a) below. These generate the semigroup R , and the = I> 4 ( 8 , which is the first quadproof of Theorem 1.3.5 gives the cone 6 7 R rant in the figure. At first glance, something seems wrong. The affine variety is normal, yet in Figure 9 (a) the semigroup generated by the white squares misses some lattice points in 6 7 . This semigroup does not look saturated. How can the affine toric variety be normal?
σ
(a)
σ
(b)
Figure 9. Lattice points for the rational normal cone
The problem is that we are using the wrong lattice! Proposition 1.1.8 tells us %$ to use the lattice , which gives the white dots and squares in Figure 9 (b). This figure shows that the white squares generate the semigroup of lattice points in 6 7 . 2 Hence is saturated and everything is fine.
§1.3. Properties of Affine Toric Varieties
39
This example points out the importance of working with the correct lattice. The Normalization of an Affine Toric Variety. The normalization of an affine toric ; BC D47&% 2 - 8 for an affine semigroup 2 , so that variety is easy to describe. Let R 2 = I> 4 2 8 ; the torus of has character lattice ; R . Let denote the cone of any 2 = I> 4 2 8 R 7 !? @ . In Exercise 1.3.6 you will finite generating set of and set 6 prove the following. Proposition 1.3.8. The above cone 6 is a strongly convex rational polyhedral cone F% 2 - ! A% 687A9 ; - induces a morphism 4 3 ; that is in ? @ and the inclusion ; the normalization map of .
3
The normalization of an affine toric variety of the form is constructed by 1$ %$ applying Proposition 1.3.8 to the affine semigroup and the lattice . Example 1.3.9.
. Then considered in Exercise 1.1.7. This is almost
$ R 4 8 4 8 4 8 4 8 ! Let 3 4 8 R 4 8 3 !
parametrizes the surface ( 4 rational the normal cone , except that we have omitted 8 4 8 , we see that 1$ is not saturated, so that
3
4 .8 R
. Using is not normal.
3
is . This Applying Proposition 1.3.8, one sees that the normalization of , and the normalization map is induced by the is an affine variety in obvious projection .
In Chapter 3 we will see that the normalization map Proposition 1.3.8 is onto but not necessarily one-to-one.
4
3
;
constructed in
Smooth Affine Toric Varieties. Our next goal is to characterize when an affine toric variety is smooth. Since smooth affine varieties are normal (Proposition 1.0.9), 4 3 we need only consider toric varieties coming from strongly convex rational polyhedral cones 6J! ?G@ .
3
We first study when 6 has maximal dimension. Then 6 7 is strongly convex, < 203 R so that 6 7 9 ; has a Hilbert basis . Furthermore, Corollary 1.3.3 tells us 3 . 4 3 4 3 that the torus action on has a unique fixed point, denoted here by . The < 3 point and the Hilbert basis are related as follows. 4
Lemma 1.3.10. Let 6 !?G@ be a strongly convex rational polyhedral cone of 4 4 3 8 be the Zariski tangent space to the affine toric maximal dimension and let 3
! 4 4 3 8 R < 4 3 variety at the above point . Then .
A%>L 3 -
Proof. 1.3.3, the maximal ideal of corresponding to P By Corollary P ,. 2 3 SX L . Since ' P is a basis of F%>L 3 - , we obtain K'
R ' P R P P(
%' P
irreducible
P
%' P R ' P P
reducible
3
is
R
Chapter 1. Affine Toric Varieties
40
It follows that local ring
!
R <
. To relate this to the maximal ideal , we use the natural map
in the
4 4 3 8 is the dual space which is always an isomorphism (Exercise 1.3.7). Since < 3 of . , we see that ! 4 4 8 R < L 3 gives 4 3 R ! " , where R < . This The Hilbert basis of 4 3 affine embedding is especially nice. Given any affine embedding , we
! 4 4 3 8
! 4 4 3 8 have by Lemma 1.0.6. In other words, is a lower
R
bound on the dimension of an affine embedding. Then Lemma 1.3.10 shows that 2 3 when 6 has maximal dimension, the Hilbert basis of gives the most efficient 4 3 affine embedding of .
?
Example In Example 1.2.21, we saw that the rational normal cone = I> 4 (1R 8 ? U ( is the1.3.11. and that! toric variety coming from 6 R @ ! 2 3 R 4 8 for @ . These generators form the 67T9 is generated by 2+3 ? Hilbert basis of? ( , so that the Zariski tangent space of . has dimension U ? @ . Hence in the smallest affine space in which we can embed .
We now come to our main result about smoothness. Recall from §1.2 that a rational polyhedral cone is smooth if it can be generated by a subset of a basis of the lattice. Theorem 1.3.12. Let 6 ! ?G@ be a strongly convex rational polyhedral cone. 4 3 is smooth if and only if 6 is smooth. Furthermore, all smooth affine toric Then varieties are of this form. Proof. If an affine toric variety is smooth, then it is normal by Proposition 1.0.9 4 3 and hence of the form . Also, Example 1.2.20 implies that if 6 is smooth as 4 3 a cone, then is smooth as a variety. It remains to prove the converse. So fix 3 3
6J! ? @ such that 4 is smooth. Let / R ! 4 R ! ? @ .
3
3
. First suppose that 6 has dimension / and let be the point studied in 3 4 3 4 4 3 8 has Lemma 1.3.10. Since is smooth in , the Zariski tangent space dimension / by Definition 1.0.7. On the other hand, Lemma 1.3.10 implies that < 2 3 R
4 4 3 8 67G9-; . Thus is the cardinality of the Hilbert basis of / R <
W
edges
4
! 6 7 W /
where the first inequality holds by Proposition 1.2.22 (each edge ! 6 7 con<
! 6 7 R / . It follows tributes an element of < ) and the second holds since that 6 has / edges and consists of the ray generators of these edges. Since+ 203 R ; by (1.2.3), the / edge generators of 6 7 generate the lattice ; 4 8 and hence form a basis of ; . Thus 6 7 is smooth, and then 6 R 6 7 7 is smooth since duality preserves smoothness.
§1.3. Properties of Affine Toric Varieties
( ?
? ? ?
!
41
6 R / . We reduce to the previous case as follows. Let Next suppose ! ( ? be the smallest saturated sublattice containing the generators of 6 . Then by Exercise is torsion-free, which B 1.3.5 implies the B existence of a sublattice ! ? with ? R ? ( ? . Note ? ( R and ? R / R0 .
4 (8
3 5
The cone 6 lies in both ? @ and ? @ . This gives affine toric varieties 4 3 5 ( R ? ? induces and and / respectively. Furthermore, ? ( of dimensions ( ;2 3 5 R ; ( ; 2 , 3 so 5 that 6 ! 4 ? 8 @ and 6 ! ? @ give the affine semigroups ! ; and !; respectively. It is straighforward to show that
4
2 34 5 R 2 3 5
;
which in terms of semigroup algebras can be written
F% 2034 5 - F% 2 3 5 - 1=F% ; <- 34 5 5 43 5 + 5
The right-hand side is the coordinate ring of 4
(1.3.2)
3 5
4
4
. Thus
3 5 5 47 M 8 + " ! 4 3 5 + " +" 4 34 5 4 34 5 & Since we are assuming that it follows that is smooth 4 .8 in 4 3 5 0 547 M is8 + smooth, " . In Exercise at any point 1.3.8 you will show that +" 4 3 5 0 4 .8 R 4 34 5 is smooth at is smooth at (1.3.3) 3 . 4 3 5 ( since Letting R , the previous case implies that 6 is smooth in ?
R ! 4 ( 8 6 ? @ . Hence 6 is clearly smooth in ? R ? ( ? . ; ( ; which in turn implies that 4
3 5
4
Equivariant Maps between Affine Toric Varieties. We next study maps ;T( and ; . between affine toric varieties that respect the torus actions on
BC D.47A% 2 - 8
;
' be the affine toric varieties coming from Definition 1.3.13. Let ' R 2 ' R ;)( ; is toric if the affine semigroups , . Then a morphism 2 A% -1 &% 2 ( - is induced by a the corresponding map of coordinate rings M 2 2 ( semigroup homomorphism .
Here is our first result concerning toric morphisms.
5
be the torus of the affine toric variety ;
;)( ; (a) A morphism
Proposition 1.3.14. Let
is toric if and only if
4 5 8 ! 5
-* , 5 5 is a group homomorphism. ; ( ; is equivariant, meaning that (b) A toric morphism 4 8 R 48 4 8 5 and . ;( . for all . and
' , R
.
Chapter 1. Affine Toric Varieties
42
BC D 47A% 2 - 8
;
5
2
' , so that the character lattice of ' . If Proof. Let ' R is ; ' R 2 2 ( comes from a semigroup homomorphism , then extends to a group ; ; ( and hence gives a commutative diagram homomorphism
A% 2 - R
&% ; - R BC D , we see that 4 5 8 ! Applying 5 R , I..*4 ; homomorphism since
A% 2 ( -
F% ; ( -<
, and -* , 5 Q 5 is a group ' J M 8 by Exercise 1.1.11. Conversely, if satisfies these conditions, then */, 5 : 5 induces a diagram as above ; ; ( . This, where the bottom map comes from a group homomorphism 47A% 2 - 8 ! F% 2 ( - , implies that induces a semigroup homomorcombined with M 2 2 ( phism . This proves part (a) of the proposition. ; ( ; . The action of 5 For part (b), suppose that we have a toric map 5 ; ' R+ ; ' , and equivariance means that ; on ' is given by a morphism ' we have a commutative diagram
;
5
5
5 ;V(
;V( /
5 ;
; /
* ,
/* , 5 is
If we replace ' with in the diagram, then it certainly commutes since 5 is a group homomorphism. Then the whole diagram commutes since 5 ;V( . Zariski dense in
We can also characterize toric morphisms between affine toric varieties coming from strongly convex rational polyhedral cones. First note that a homomorphism 5 5 of tori. ? ( ? of lattices a group homomorphism 5 R gives * This follows from one sees that is a morphism. Also, ? ' 4 ( 8 M , and 4 ? @ ? 8 @ . tensoring with gives @
Here is the result, whose proof we leave to the reader (Exercise 1.3.9).
Proposition 1.3.15. Suppose we have strongly convex rational polyhedral cones 5 5 6 ' ! 4 ? ' 8 @ and a homomorphism4 3 ? (X4 3 ? . Then 4 X ( 8 extends to a map of affine toric varieties if and only if @ 6 ! 6 .
In the remainder of the chapter we explore some interesting classes of toric morphisms. Faces and Affine Open Subsets. Let 6 ! ?A@ be a strongly convex rational polyhedral cone and let H ! 6 be a face. Then we can find , . 6 7 9V; such that H R OQP 9 6 . This allows us to relate semigroup algebras of 6 and H as follows.
§1.3. Properties of Affine Toric Varieties
43
H R O P 96 -
Proposition 1.3.16. Let H be a face of 6 and as above write F% 2 - R F% H 7 9 ; ,. 6 7 9Q; . Then the semigroup algebra P A% 2 3 - R F% 6 7 9-; - at ' . F% 2 3 - . In other words, of
, where is the localization
F% 2 - R F% 2 3 -
, and since K , L R F4 R , 8 ! 2
2 3 2
Proof. The inclusion H ! 6 implies we have , . H 7 . It follows that
2 3
. H
for all
This inclusion is actually an equality, as we now prove. Fix a finite set = I> 4
,
L ! ;
R B U K , Z L . F Z , . 2 3 . This proves that It is straightforward to show that , 2 3 F4 R , 8 R 2 F% 2 - R A% 2 3 - follows immediately. from which
This interprets nicely in terms of toric morphisms. By Proposition 1.3.15, the ? 4 4 3 identity map ? and the inclusion H ! 6 give the toric morphism 2 3 - F% 2 A % that corresponds to the inclusion . By Proposition 1.3.16, !
R BC D47&% 2 - 8 R BC D 47A% 2 3 - 8 R B9C D 47&% 2 3 - 8 R 4 3 8 ! 3 becomes an affine open subset of 3 when H is a face of 6 . In other words, (1.3.4)
4
4
4
4
4
This will be useful in Chapters 2 and 3 when we study the local structure of more general toric varieties. Sublattices of Finite Index and Rings of Invariants. Another interesting class of toric morphisms arises when we keep the same cone but change the lattice. Here is an example we have already seen.
= I> 4 ( 8
Example 1.3.17. In Example 1.3.7 the dual of 6 R interacts ! with the lattices shown in Figure 10 on the next page. To make this precise, let us name the lattices involved: the lattices Z
R ? ! ? R 4 * .8 * . * I Z have 6J! ? @ ! ? @ , and the dual lattices Z
R ; ; R 4 * 8 * . * I Z have 6 7 !; @ ! ;V@ . Note that duality reverses inclusions and that ; and ? are indeed dual under dot product. In Figure 10 (a), the black dots in the first quadrant 2 34 5 R Z , and in Figure 10 (b), the white dots in the 7 form the semigroup 6 9 ; 2+3 5 R first quadrant form 6 7 9-; .
Chapter 1. Affine Toric Varieties
44
σ
σ
(a)
(b)
Figure 10. Lattice points of
relative to two lattices
and
3 5
4
4
34 5
4
34 5 R
This gives the affine toric varieties . Clearly since Z 4 3 5 is smooth for ? , while Example 1.3.7 shows that is the rational normal Z !? gives a toric morphism cone . The inclusion ?
6
R
4
34 5 R
3 5 R
4
Our next task is to find a nice description of this map.
Z
Z
!? , where ? has finite index in ? , In general, suppose we have lattices ? Z and let 6 ! ? @ R ?@ be a strongly convex rational polyhedral cone. Then the Z ! ? gives the toric morphism inclusion ?
3 5 R Z ; , so that The dual lattices satisfy ; 4
4
34 5
A% 6 7 9 ; Z -
corresponds to the inclusion
&% 6 7 9 ; &% - as a ring of invariants of of semigroup algebras. The idea is to realize 6 7 9; A % Z a group action on 6 7 9-; . Z Z Proposition 1.3.18. Let ? have finite index in ? with quotient R ? ? and Z let 6 ! ? @ R ? @ be a strongly convex rational polyhedral cone. Then:
I. * 4 ; Z ; J M 8 R 4 5 5 8 A% Z - with ring of invariants acts on 6 7 9; A% 6 7 9-; Z - R A% 6 7 9 ; -<
(a) There are natural isomorphisms
(b)
(c)
4
34 5
,
acts on , and the morphism and induces a bijection 4
3 5
3 5
4
4
3 5
4
3 5
is constant on
-orbits
§1.3. Properties of Affine Toric Varieties
Proof. Since
45
5 R-, I. *V4 ; J M 8 by Exercise 1.1.11, applying , I. * 4FRSJ M 8 R+ ; R+ ; Z R ; Z ; R+
I.*4 ; Z ; J M 8XR 5 R+ 5 R I. *)F4 RSJ M 8 is left exact and M is divisible.
gives the sequence
R
,
This is exact since , ? ? Z into the picture, note that
Z ! ?! ?2/
?
;
and
!;
To bring
to
R
Z ! ;0/
; / 2 ? P/ , the map Z Z 4*% , Z -< % - 8 R+ ' ; ; ? ? R+ M I. *V4 ; Z ; J M 8 (Exercise 1.3.10). is well-defined and induces , 5 on 4 3 5 induces an action of on 4 3 5 since ! 5 . The action of 4 3 5 ( Using Exercise 1.3.1, one sees that if . and . , then is defined by 2 * 4 % 8 4 8 Z Z Z Z , , for , . 67A9-; Z . It follows the semigroup homomorphism , that the corresponding action on the coordinate ring is given by P ( P ' R V4*% , Z - 8 " ' , Z . 6 7 9-; Z Z . ; Z lies in ; if (Exercise 5.0.1 explains why we need the inverse.) Since , V4*% Z - 8 R and only if , for all . , the ring of invariants &% 6 7 9 ; Z - R O . A% 6 7 9 ; Z - O R O for all . . F% is precisely 6N7A9-; , i.e., A% 6 7 9-; Z - R A% 6 7 9 ; -< Since the pairing between
;
and ?
induces a pairing
+ When a finite group acts algebraically on Thm. 10 of Ch. 7, §4] &% ' (*',+.- ! E , [17, shows that the ring of invariants gives a morphism of affine
This proves part (b).
varieties
+ R BC D 47A% ' (*' + - 8XR+ BC D.47F% ' ( *',+.- 8
BC D.47&% ')( *',+.- 8
-orbits+ and induces a bijection
that is constant on
The proof extends without difficulty to the case when acts algebraically on BC D 47EF8 . Here, E ! E gives a morphism of affine varieties that is constant on
; R B C D.47EF8 R B C D 47E 8
-orbits and induces a bijection
;
BC D 47E
8
From here, part (c) follows immediately from part (b).
; R
Chapter 1. Affine Toric Varieties
46
We will give a careful treatment of these ideas in §5.0, where we will show BC D47EF8 BC D 47E M8 is a good geometric quotient. that the map Here are some examples of Proposition 1.3.18.
Example 1.3.19. In the situation of Example 1.3.17, one computes that R
acting on 4 34 5 R BC D 47A% - 8 by R 4 . 8 R 4FR group Thus the rational normal cone is the quotient
R
3 5
4
4
3 5 R
. is4R the 8.
&% . - R F% - R &% - F% ' *' ( *' - K ' ' R ' ( L
We can see this explicitly as follows. The invariant ring is easily seen to be
where the last isomorphism follows from Example 1.1.6. From the point of view of invariant theory, the generators of the ring of invariants give a morphism
R 4 8 4 8
that is constant on
-orbits. This map also separates orbits, so it induces
47 8 R
where the last equality is by Example 1.1.6. But we can also think about this in give the Hilbert basis terms of semigroups, where the exponent vectors of 2+3 5 of the semigroup pictured in Figure 10 (b). Everything fits together very nicely.
In Exercise 1.3.11 you will generalize Example 1.3.19 to the case of the ratio ? for arbitrary @ . nal normal cone
+
+ a simplicial cone of dimension / with Example 1.3.20. Let 6 ! ?G@ ( + . Then ? Z R be + of finite index in & '( Z ( ' is a4 sublattice ray generators 3 5 R . It follows that ? . Furthermore, 6 is+ smooth relative to ? , so that R ? ? Z acts on with quotient
+
R
4
3 5
4
3 5
Hence the affine toric variety of a simplicial cone is the quotient of affine space by 4 34 5 a finite abelian group. In the literature, varieties like are called orbifolds and are said to be -factorial.
Exercises for §1.3.
(<>=?7@1 AIH J!'
( / 27+6+6+72 ! 8
, where , and 1.3.1. Consider the affine toric variety . Let be a semigroup homomorphism. In the proof of Proposition 1.3.1 . we showed that lies in (a) Prove that the maximal ideal is the kernel of the -algebra homomorphism induced by .
( ,
6 h
K (M k '926+6+7+92 k ! ' ' /,-f H JUOe, RK'( W8 H J5h
§1.3. Properties of Affine Toric Varieties
~
47
(
% ~
and fix . As in the (b) The torus of has character lattice discussion following Proposition 1.3.1, this gives the semigroup homomorphism . Prove that this corresponds to the point
jh !n' ' 26+7+6+92 ' ' 27+6+7+92k ! 'n')( k ' 27+6+7+92 ! ' ' f ! . coming from the action of ^ ~ F '/! on K 1.3.2. Let G ( <W=D?E@1 FIH >J ' with the torus ~ ( <W=D?E@m A H J!' , ( . The action ~ G h G comes from a -algebra homomorphism H > a J h I H IH >J . Prove . Hint: First showJ that this that this homomorphism is given by jh IH J that gives theformula determines the -algebra homomorphism IH J h H J group operation ~ ~ h ~. 1.3.3. Prove Corollary 1.3.3.
&
1.3.4. Let
be a finite set.
, is saturated in `?1 C'^! .
(a) Prove that the semigroup Hint: Apply (1.2.2) to
(b) Complete the proof of (b)
if and only if
,
( `?1 C'%
.
(c) from Theorem 1.3.5.
is torsion-free. Prove that there is a 4 ( 4 . (b) Let be primitive as defined in the proof of Theorem 1.3.5. Prove that has a basis 26+6+7+92 such that (V .
1.3.5. Let be a lattice. (a) Let be a sublattice such that sublattice such that
1.3.6. Prove Proposition 1.3.8.
of an irreducible affine variety G . Then K gives the maximal ideal _ ( /&,t K beH GaJ^point O`, RKS'(M_ `8 _ as well as_ the _ maximal ideal _ YW[ \ X^Y>[ \ defined in §1.0. YW[ \m 4YW[ \ is an isomorphism of -vector spaces. Prove that the natural map 4y#h
1.3.7. Let
1.3.8. Prove (1.3.3). Hint: Use Lemma 1.0.6 and Example 1.0.10. 1.3.9. Prove Proposition 1.3.15.
Dh nH J"27H J!')jh &4 [ O ( 8 be the group of th roots of unity. Then acts on 1.3.11. Let ( / d c d c 4 by 2 '( 2 ' . Adapt Example 1.3.19 to show that 4 g . Hint: Use lattices D( U4Z (N/> ` D2 6 'aO2 .2 # :|>z 8 . 1.3.12. Let ' and 0 ' be strongly convex rational polyhedral cones. 4 44 This gives the cone ' . Prove that . Also explain 4 how this result applies to (1.3.2). 1.3.13. By Proposition 1.3.1, a point K of an affine toric variety G (C<>=?7@ AIH J!' is represented by a semigroup homomorphism 6 h . Prove that K lies in the torus of G if and only if never vanishes, i.e., k 'U(V T for all . 1.3.10. Prove the assertions made in the proof of Proposition 1.3.18 concerning the pairing
defined by
+ .
Chapter 1. Affine Toric Varieties
48
Appendix: Tensor Products of Coordinate Rings In this appendix, we will prove the following result used in §1.0 in our discussion of products of affine varieties.
and .
Proposition 1.A.1. If the same is true for
are finitely generated -algebras without nilpotents, then
H d 27+6+7+62 d J!
Proof. Since the tensor product is obviously a finitely generated -algebra, we need only has no nilpotents. If we write prove that , then is radical and hence has a primary decomposition + + , where each + is prime ([17, Ch. 4, §7]). This gives
( v !
IH d 26+7+6+62 d JA #Dh + ! I H d 27+6+6+72 d J!
+ where the second map is injective. Each quotient IH d 26+7+6+92 d JA + is an integral domain q + . This yields an injection and hence injects into its field of fractions 1 M#Dh + ! q + 2
and since tensoring over a field preserves exactness, we get an injection
q
h + ! q
+
+
q
q
Hence it suffices to prove that has no nilpotents when is a finitely generated field extension of . A similar argument using then reduces us to showing that has no nilpotents when and are finitely generated field extensions of . Since has characteristic , the extension has a separating transcendence basis ([56, p. 519]). This means that we can find such that are algebraically independent over and is a finite separable extension. Then
q
c c i c 26+7+6+72 c 26+7+6+92 c c ( 26+6+7+62 ' q Z' Aq I' + q N(Cq I c 62 +7+6+62 c p')(*q c 26+7+6+92 c p' is a field, so that we are reduced But gN(Cq to considering g where g and are extensions of and The latter and the is finite and separable. theorem of the primitive element imply that H JA , ' , where , ' has distinct roots in some extension of . Then g g H JA , ' g H J! , ' + Since , ' has distinct roots, this quotient ring has no nilpotents. Our result follows. A final remark is that we can replace with any perfect field since finitely generated
extensions of perfect fields have separating transcendence bases ([56, p. 519]).
Chapter 2
Projective Toric Varieties
§2.0. Background: Projective Varieties Our discussion assumes that the reader is familiar with the elementary theory of projective varieties, at the level of [17, Ch. 8].
In Chapter 1, we introduced affine toric varieties. In general, a+ toric variety over containing a torus 5 47 M 8 as a Zariski is an irreducible variety + 7 4 8 M open set such that the action of on itself extends to an action on . We will learn in Chapter 3 that the concept of “variety” is somewhat subtle. Hence we will defer the formal definition of toric variety+ until then and instead concentrate on toric varieties that live in projective space , defined by
+ R 47 + U ( S 8 M .4 * * +98 R 4 * * + 8 M + ( where M acts via homotheties, i.e., for . U 4 9 + 8 4 + 8 * + * . * * are homogeneous coordinates of a and . Thus and are well-defined up to homothety. point in
(2.0.1)
)
)
)
)
The goal of this chapter is to use lattice points and polytopes to create toric + varieties that lie in . We will use the affine semigroups and polyhedral cones introduced in Chapter 1 to describe the local structure of these varieties.
+ ; Homogeneous Coordinate Rings. A projective variety ! is defined by the vanishing of finitely many homogeneous polynomials in the polynomial ring L R F% ' *' +.- . The homogeneous coordinate ring of ; is the quotient ring A%>;S- R L :.4<;&8 :.4<;=8 is generated by all homogeneous polynomials that vanish on ; . where 49
Chapter 2. Projective Toric Varieties
50
L
,4 ' 8
' R . This gives the decomThe polynomial ring is graded by setting L V L ? L ? ? R ( position , where is the vector space of homogeneous polynomials of degree @ . Homogeneous ideals decompose similarly, and the above coordinate F%>; - inherits a grading where ring
%>; -? R VL ? :.4<;&8 ? F :.4<;&8 ! L R F% ' *' + - also defines an affine variety ; ! + U ( , The ideal ; ; called the affine cone of . The variety satisfies ; R 4 ; SX 8 M (2.0.2) ; and its coordinate ring is the homogeneous coordinate ring of , i.e., A% ;S- R F%>; -<
6 R K ' ' ' U ( R ' ' U (J' @ R L ! A% ' *',? minors of the matrix generated by the ' ')( '2? " ',? " ( ' ( ' '2? " ( ',? ? ? 6 Since is homogeneous, it defines a projective variety ! that is the image of the map ( R ? 4 . 8 4 ? ? " ( ? " ( ? 8 (see defined in homogeneous coordinates by ? Exercise 1.1.1). This shows that is a curve, called the rational normal curve of? degree @ . Furthermore, the affine cone of ? is the rational normal cone ? ! U ( discussed in Example 1.1.6. ? We know from Chapter 1 that is an affine toric surface; we will soon see ? that is a projective toric curve. 354 ' R S8 ! studied in Chapter 1 is Example 2.0.2. The affine toric variety the affine cone of the projective ( ( surface ; R 354 ' R 8 ! . Recall that this surface is isomorphic to via the Segre embedding ( ( R+ 4 8 4 8 ; ( ( Example 2.0.1. In Example 1.1.6 we encountered the ideal
given by . We will see below that projective toric variety coming from the unit square in the plane.
is the
; ! + has the classical topology, As in the affine case, a projective+ variety induced from the usual topology on , and the Zariski topology, where+ the Zariski ; contained closed sets are subvarieties of (meaning projective varieties of ; in ) and the Zariski open sets are their complements.
§2.0. Background: Projective Varieties
51
Rational Functions on Irreducible Projective Varieties. A homogeneous polynoL of degree @ does not give a function on + since mial O . ?
O 4 ) ' ) ' + 8 R
O 4 ' *' + 8 . L ? gives the well-defined However, the quotient of two such polynomials O function O + S 35 4 98 S + and say that O is a rational provided R P + . We write this as O function on . ; ! + is irreducible, and let O . F%>; - R More generally, suppose that F% ; - be homogeneous of the same degree with R P . Then O and give functions ; &4 ;A8 . By (2.0.2), this induces a on the affine cone and hence an element O . ; . Thus rational function O &4<;A8 R O . A4 ;&8 O . A%>;S- homogeneous of the same degree, R P ; is the field of rational functions on . It is customary to write the set on the left as F4 ;=8 since it consists of the degree elements of A4 ;&8 . ; ! + is a union of+ Affine Pieces of Projective Varieties. A projective variety + 4 S 354 ' ' 8 . Then 4 ' Zariski open sets that are affine. To see why, let ' R )
so that in the notation of Chapter 1, we have ' R BC D via the map
(2.0.3)
4 * * +98 R
4
;
Then+ 9 ' is a Zariski open subset of defined by the equations in 4
;
that maps via (2.0.3) to the affine variety
O R :.4<;&8 . as O varies over all homogeneous polynomials in ; 9 4 ' an affine piece of ; . These affine pieces cover ; We call+
(2.0.4)
since the cover . Using localization, we can describe the coordinate rings of the affine ' ' &%>;- , so that we get pieces as follows. The variable ' induces an element '#. the localization 4
'
F%>; - R O ' ' O . A%>;S-< W A%>;S- has a well-defined -grading as in Exercises 1.0.2 and 1.0.3. Note that
V4 ' ' 8 R V4 /8 R O O given by when O is homogeneous. Then 7 4 A > % S ; - 8 R ' & (2.0.6)
O ' . %>;- O is homogeneous of degree F%>; - consisting of all elements of degree . This gives an affine is the subring of ; piece of as follows.
(2.0.5)
Chapter 2. Projective Toric Varieties
52
Lemma 2.0.3. The affine piece
; 9
F%>; 9
'
4
of
;
has coordinate ring
' - 47A%>; - 8
4
Proof. We have an exact sequence
%>; -NR R :.4<;A8%R+ A% ' *' + -8R F ' If we localize at ' , we get the exact sequence %>;S- R (2.0.7) R :.4<;&8 R A% ' *',+.- R+ A since localization preserves exactness (Exercises 2.0.1 and 2.0.2). These sequences preserve degrees, so that taking elements of degree 0 gives the exact sequence
+R 47:.4<;&8 8 R 74 A% ' *' + - 8 R 74 &%>; - 8 R+ 47A% ' *' + - 8 R . If O . .: 4<;&8 Note that
is
47:.4<;=8 8 ' . R ' O O 8 8 maps :.4<; 9 4 ' 8 . To show that this 47:.4< ;A By (2.0.4), we conclude that to 4 . :.4<; ' 8 . For , ' ' R 9 map is onto, let ' O; 4 ' *' + 8 is homogeneous of degree . It follows easily that ' O vanishes on ; 4 ' 4 ' :.4<;A8 , since R on 9 ( ' and ' R on the complement of ' . Thus ' O . 4 ' 8 4 ' U 8 . 47:.4<;=8 8 maps to . The lemma follows immediately. and then ' O '
homogeneous of degree , then
;
; 9
' and One can also explore what happens when we intersect affine pieces 9 4 ; 4 4 ' 9 for XR P . By Exercise 2.0.3, 9 is affine with coordinate ring
(2.0.8)
F%>; 9
4
' 9
4
-
4
47&%>;- 8
We will apply this to projective toric varieties later in the chapter. We will use these results in §2.2 when we explore the structure of projective toric varieties. We will also see later in the book that Lemma 2.0.3 is related to the “Proj” construction, where Proj of a graded ring gives a projective variety, just as Spec of an ordinary ring gives an affine variety.
+ P Products of Projective Spaces. One can study the product of projective F% '! *',+ P - , where ' ' has spaces using the bigraded ring bidegree 4 8 and ' has bidegree 4 8 . Then a bihomogeneous polynomial of bidegree O 4 * 8 gives a+ well-defined + P . This allows us to define equation O R in P + using :.4<;&8 of varieties in bihomogeneous ideals. In particular, the ideal P ; is a bihomogeneous ideal. a variety ! + P Another way to study is via the Segre embedding + P R+ + P U + U P
§2.0. Background: Projective Varieties
53
4 * * + P8 to the point 4 * * ( * P * ( * ( P * + * + P 8 +P U+UP This map is studied in [17, Ex. 14 of Ch. 8, §4]. + If ' , P ! + PhasU + homogeneous U P / coordinates ' for , then 4 %8 54 , 8 matrix is defined minors of the / by the vanishing of the ' ' FP .. .. . ' /+ ' + . P This follows if it is a since an 4 / 18 4 , 8 matrix has rank if and only
product , where and are nonzero row matrices of lengths / and , . give the same notion of what it means to be a subvariety of + These P . Aapproaches 4 ' ' 8 of degree @ gives the bihomogeneous homogeneous polynomial + P U + U P lying of polynomial + P 4 ' ' 8 of bidegree 4 @ @ 8 . Hence any subvariety F% ' *' +, Pdefined by mapping
in can be defined by a bihomogeneous ideal in Going the other way takes more thought and is discussed in Exercise 2.0.5.
; ! +
We also have the following useful result proved in Exercise 2.0.6. Proposition 2.0.4. + GivenP subvarieties is a subvariety of .
and
!
P
, the product
;
.
+ is &% ' *',+.- , - + be positive
Weighted Projective Space. The graded ring associated to '
where each variable ' has degree . More generally, let .D 4 + 8 R and define integers with
4 + 8 R 47 + U ( XS 8 %
% where is the equivalence relation 4 * * + 8% 4 + 8 * ' R ' R / for some . M 4 +98 a weighted projective space. Note that + R 4 8 . We call 4 + 8 is the graded ring F% ' *' + - where The ring corresponding to ' ' now has degree ' . A polynomial O is weighted homogeneous of degree @ if ev' appearing in O satisfies " 4 +98 R @ . The O R is wellery monomial 4 + 8 when O is weighted homogeneous, so that one can define defined on 4 + 8 using weighted homogeneous ideals of F% ' *' +.- . varieties in 4 .8 in using Example 2.0.5. We can embed the weighted projective plane ( ' * ' ' * ' * ' ( of weighted degree . In other words, the map the monomials 4 .8#R
given by
)
4 * * ( * 8 R 4 * * * ( * ( * 8
)
Chapter 2. Projective Toric Varieties
54
is well-defined and injective. One can check that this map induces
where
4 .8 354 R ( 8 ! ( are homogeneous coordinates on .
Later in the book we will use toric methods to construct projective embeddings of arbitrary weighted projective spaces. Exercises for §2.0.
, t:/& W8 and an exact sequence of h 4h oh #Dh y; #h 4 y; #h o y; #Dh is also exact, where ; is the localization of at , defined in Exercises 1.0.2 and 1.0.3. has coordinate ring H GJ( FG$' , where ( 2.0.2. A projective variety G H d 26+7+6+62 d J . Let d b + be the image of d + in IH G:J . (a) Note the IH G:J is an -module. Prove that H GJ 4 IH GJ I 4 . h
2.0.1. Let be a commutative -algebra. Given -modules , prove that
(b) Use part (a) and the previous exercise to prove that (2.0.7) is exact.
2.0.3. Prove the claim made in (2.0.8).
G be a projective variety. Take , 26+7+6+723, such that the intersection G% F, 26+6+7+62 , ' is empty. Prove that the map 26+7+6+62 ')j#hl F, 26+7+6+92 '926+7+6+62 , 27+6+6+72 'n' . induces a well-defined map function 6 GMh be defined by , d +n2 c 'B(V , where , d +32 c ' is bihomogenous 2.0.5. Let G of bidegree 2 ' , t( m 26+7+6+723 . The goal of this exercise is to show that when we in via the Segre embedding described in the text, G becomes embed a subvariety of . (a) For each , pick an integer *| S/ 2 8 and consider the polynomials , [ r [ ( ( d r c ( , + d + 2 c ' where (* `27+6+6+623 and O saOm( # 2ZO *ZO ( "# . Note that , [ r [ ( is bihomogenous of bidegree 2 ' . Prove that G is defined by the vanishing of the , [ r [ ( . (b) Use part (a) to show that G is a subvariety of under the Segre embedding. 2.0.4. Let
2.0.6. Prove Proposition 2.0.4
h o d c % # 'aV
2.0.7. Consider the Segre embedding after relabeling coordi in is the variety . Show that featured in many $ nates, the affine cone of examples in Chapter 1.
o
§2.1. Lattice Points and Projective Toric Varieties
55
§2.1. Lattice Points and Projective Toric Varieties
+ We first observe that is a toric variety with torus R + S 354 ' ! ' +98 R 4 * * + 8 . + R 4 ( +98 . ( + .
+ * ! * + R P +
M 74 M 8
+ on itself clearly extends to an action on
The action of
+ , making
, we use the exact sequence aoftoric tori
R M R 47 M 8 + U ( R+ R + is coming from the definition (2.0.1) of . Hence the character lattice of + R 4 * * + 8 . + U ( +')( * ' R .
& (2.1.1) variety. To describe the lattices associated to
and the lattice of one-parameter subgroups
+ R
+U (
+ is the quotient 4 8 5
Lattice Points and Projective Toric Varieties. Let be a torus with lattices ; $ R and ? as usual. In Chapter 1, we used a finite set of lattice points of ( 3 , " ! ; to create the affine toric variety
, as the Zariski closure of the image of the map
4' P 4 8 (' P 4 8*8 3 47 M 8F" To get a projective toric variety, we regard as a map to " 47 M 8 to obtain with the homomorphism 5 R+ " R+ ! " " ( (2.1.2)
5 R+ "
3
R
$
and compose
Definition 2.1.1. Given toric variety " " ( a finite set !; , the projective 3 Zariski closure in of the image of the map from (2.1.2).
3
53
is the
Proposition 2.1.2. is a toric variety of dimension equal to the dimension of $ the smallest affine subspace of ; @ containing .
8" " (
! Proof. The proof that is a toric variety is similar to the proof given 3 ! " is a toric variety. The assertion in Propostion 1.1.8 of Chapter 1 that 3 concerning the dimension of will follow from Proposition 2.1.6 below. 3 5 R+
3
R 4' P 4 8 (' P 4 8*8 $ R , ( , " ! ; given by+ the characters coming from . In particular, if P' P and .3 is the Zariski , then is the Laurent monomial closure of ; R the image of 5 R+ " " ( R 4 P P 8 More concretely,
"" (
is the Zariski closure of the image of the map
Chapter 2. Projective Toric Varieties
56
+
$
! $ In the literature, so that the elements of
is often given as an / are the columns of .
matrix
with integer entries,
Here is a example where the lattice points themselves are matrices. be the lattice of integer matrices and let Example 2.1.3. Let ; R R permutation matrices !
Write
&% ; - R F % 1( ( 1 ( - , where the variables give the generic (
matrix
'
with nonzero entries. coordinates ' indexed by ( Also let have homogeneous L ! triples such that ' is a permutation in . Then is the Zariski closure given by the Laurent monomials ' ( of the Limage of the map 5 . . The ideal of for ' is
:.4 8 R K ' ( ' J( ' ( %R )' ( ' J( ' ( L ! &% ' ' -<
where the relation comes from the fact that the sum of three of the permutation matrices is equal to the sum of the other three (Exercise 2.1.1). Ideals of the toric varieties arising from permutation matrices have applications to sampling problems in statistics [97, p. 148].
8" " (
! The Affine Cone of a Projective Toric Variety. The projective variety 3 ! " 3 has an affine cone . How does relate to the affine toric variety 3 ! " constructed in Chapter 1? Recall from Chapter 1 that when induces an exact sequence (2.1.3)
R+
3
6 R ' R '
and that the ideal of
$ R , ( , " ! ;
, the map
' , '
" R ;
R+
is the toric ideal
"M
3
. "
and
R "
.
(Proposition 1.1.9). Then we have the following result. Proposition 2.1.4. Given (a) (b) (c)
3
,
3 3
" ! is the affine cone 6 R .: 4 .3 8 . 3
6
is homogeneous.
(d) There is
. ?
and
in
and of
64 3
as above, ( the following are equivalent:
! 8 " "
such that
.
K, ' L R
for
R .
§2.1. Lattice Points and Projective Toric Varieties
Proof. The equivalence (a) (b) follows from the equalities 6 R :.4 3 8 , and the implication (b) (c) is obvious. and
57
:.4
3
8 R :.4
3
8
6 ' R ' . assume that is homogeneous ' and . 6 R :.4 3 8 for *' take " R For. (c) ' (d), then .3 If and ' had different degrees, and ' would vanish ' on . This is impossible since 4 4 8 . 3 by8 R (2.1.2). Hence . for all have the same degree, which implies that . Now tensor (2.1.3) with and take duals to obtain an exact sequence " ?2/ R+ R , I. / 4 / JX8 R+ 4 8 . " maps to zero in , I. / 4 / JX8 The above argument shows that
for all . and hence comes from an element . ? / . In other words, K ,' L R Clearing denominators gives the desired . ? and in . 6 R :.4 3 8 , it suffices to show that Finally, we prove (d) (b). Since 3 9 47 M 8 " ! 3 .3 9 47 M F8 " 3 9 3 Let . . Since is the torus of , it follows that P P R 4 ' 4 8 ( ' 4 8*8 M and . 5 . Let . ? be as in part (d). This gives a onefor some . 5 , which we write as H ) 4 H 8 for H . M . Then parameter subgroup of ) 4 H 8 . 5 maps to the point . "3 given by R ' P 4 ) 4 H 8 8 ( ' P 4 ) 4 H 8 8 R H P ' P 4 8 H P ' P 4 8 P 4 ) 4 8*8 P ' H R H since by the description of K L given in §1.1. The hypothesis
4' P 4 8 (' P 4 8*8 . 3 . This completes the proof. Using , we can choose H so that R
, for some . ? and in The condition K , ' L R , R $ means that lies in an affine hyperplane of ; / not containing the origin. When + $ ; R and consists of the columns of an / integer matrix , this is 4 8 lying in the row space of (Exercise 2.1.2). equivalent to ? ? ! Example 2.1.5. We will examine the rational normal curve using two
of part (d) allows us to rewrite as
R H
different sets of lattice points. First let
$
consist of the columns of the
5 4 @ 8 matrix
@ R @
@ @ R ? The columns give the Laurent monomials defining the rational normal curve ? ? in Example 2.0.1. It follows that is a projective toric variety. The ideal of
R
is the homogeneous ideal given in Example 2.0.1, and the corresponding affine $ (= the columns of ) consists of all points 4 * 8 hyperplane of containing
Chapter 2. Projective Toric Varieties
58
*
R @ . It is equally easy to see that satisfying . In particular, we have
Now let
4 8
is in the row space of
.3 R ? and 3 R ? R @ R @ ! . This gives the map ? ? ( ? M R 4 " 8
?
R , but the The resulting projective variety is the rational normal curve, i.e., RP ? affine variety of is not the rational normal cone, i.e., is because :.4 8 ! A? % ' ( ? *',?? - is( not homogeneous. For example, ' ( R . This ' vanishes at 4 " 8 . U for all . M . Thus ' ( R ' . :.4 8 .
$
$
Given any way to modify so that the conditions !; , there is a $ standard
of Proposition 2.1.4 are met: use
! ; . This lattice corresponds to + M , and since the torus P 4 8 (' P 4 8 8 P 4 8 (' P 4 8*8 ' 4 4' 3 4 8 R ( R (2.1.4)
8" (
3 " . Since ( R ! it follows immediately that lies in an 3 has affine hyperplane missing the origin, + Proposition 2.1.4 implies that 3 ( R .3 . When ; R $ is represented by the columns affine cone and of an $
4
8 /
by adding the row integer matrix , we obtain to .
3
$
The Torus of a Projective Toric Variety. Our next task is to determine the torus of .3 . We will do so by identifying the character lattice. This will also tell us the 3 . Given $ R , ( , " !; , we set dimension of
Z$
Z $ R & '" ( ( * ' ,-' * ' . & '" ( ( * ' R
The rank of is the dimension of the smallest affine subspace of ; $ the set (Exercise 2.1.3). Proposition 2.1.6. Let
Z $ The lattice
3
be the projective toric variety of
$ !;
@
containing
. Then:
3 . (a) is the character lattice of the torus of 3 is the dimension of the smallest affine subspace of ; @ (b) The dimension of $ containing . In particular, B %$ R if $ satisfies the conditions of Proposition 2.1.4
! . 3 R B %$ otherwise.
4 " 8
Proof. To prove part (a), let ; be the character lattice of the torus .3 . By (2.1.2), we have the commutative diagram
5
/
EE O EE EE EE" " ?
"
/
"" (
"
of
§2.1. Lattice Points and Projective Toric Varieties
59
which induces the commutative diagram of character lattices
;
(
4 (
*" " R * by" (2.1.1). The map
since
' to ,-' . Thus
Z$
o
"
dHH HH HH HH H2 R
"(
" ; 4 8 8 . " & '" ( * ' ,-' R is the character lattice of " " ( ; is induced by the map " ; that sends " 8 " " (G ; ; 4
is the image of
and hence equals
by
the above diagram. assertion of part (b) follows from part (a) and the observation that B TheZ $ first $ is the dimension of the smallest affine subspace of ; @ containing . K ,-' " L ( R 8 4 * & '( '
3 R P . 3
Then
"3
3
equals the affine cone of , then there is " for all by Proposition 2.1.4. This implies that KF& ')( , which gives the exact sequence
Furthermore, if
. (0* ' ,-? ' with L R
Z R + R $ R %$ + R implies B %$6 R R B Z $ R 3 . On the other hand, if , then the ideal is not homogeneous, where is defined by + R R+ " R %$ R 6 , 4 8 R
by (2.1.3). Using the usual description of , it follows that some ,. . This implies there is such that we have a diagram
/
/
Z
$
"(
"
/
"(
9 "
/
/ "
/ /
/
/
/
/
%$ /
for
P
B %$ R B Z $ R ! .3 . $ R ( ( ( ! . One computes that Example 2.1.7. Let %$ R Z $ R 4 * 8 . * I . Thus Z $ has index in but %$ 3 R P .3 " . This means that and the map of tori with exact rows and columns. Hence
is two-to-one, i.e., its kernel has order
R
"
(Exercise 2.1.4).
Chapter 2. Projective Toric Varieties
60
Affine Pieces of a Projective Toric Variety. So far, our treatment of projective toric varieties has used lattice points and toric ideals. Where are the semigroups? " " ( There 3 are actually lots of semigroups, one ! . ( for each affine piece of
. Thus
3 4 ' ' 8 contains the torus ' R 8 " " S 5 3 9 ! .3 9 4 ' " R . .3 8" " ( , it follows that 3 9 4 ' is the " Since is the Zariski closure of ( in " in 4 ' " " . Thus 3 9 4 ' is an affine toric variety. Zariski closure of $ R , ( , " ! ;V@ (" affine semigroup associated to 3 9 4 ' Given , the " 4 is easy to determine. Recall that ' is given by 4 * ( * " 8 R 4 * ( * ' * ' " ( * ' * ' U ( * ' * " * ' 8 P ' P ' P " P ' 3 9 4 ' R Combining this and with the map (2.1.2), we see that The affine open set
4
5 R "" (
is the Zariski closure of the image of the map
given by (2.1.5) If we set
R+
$ '
P " P 4 8 P " P 4 8 P " P 4 8 P " P 4 8 R $ R , ' R , R , ' R P , it follows that 9 ' R R B9C D.47&% 2 ' - 8 '
('
.3
4
('
('
.3
$
2
' is the affine semigroup generated by where ' R the following result.
$ ' .
We have thus proved
3 ! 8" " ( $ R , ( , " ! ;J@ Proposition 2.1.8. Let for 4 3 affine piece 9 ' is the affine toric variety 3 9 4 ' R 3 R BC D 74 A% 2 ' - 8 . $ $ R , ' and 2 ' R #$ ' . where ' R
. Then the
"
We also note that the results of Chapter 1 imply that the torus of %$ ' . Yet the torus is , which has character lattice character lattice %$ ' R Z $ for all . ( Proposition 2.1.6. These are consistent since
3
XZ $
has by
8" "
9 ' of Besides describing the affine pieces , we can also describe how they patch together. In other words, we can give a completely toric description of the inclusions
' P ' ' P RP ' "
when
(2.1.6)
RP RP
.3 9
.
'
4
For instance, 3 . In terms of . Thus
3
9
4
' 9
4
4
9
' 9 4
.3 9 4
' R
4
4
3
' 9
4
3
! .3 9
4
9 ' where consists of those points of BC D.47A% 2 ' - 8 , this means those points where
R B9C D F% 2 ' -
R BC D A% 2 ' -
3
4
§2.1. Lattice Points and Projective Toric Varieties
R ,
so that if we set , be written
R ,-' 3 9 , then the inclusion
BC D 47A% 2 ' - 8
!
(2.1.7)
61 4
' 9 4
! .3 9
' 4
can
BC D.47F% 2 ' - 8
This looks very similar to the inclusion constructed in (1.3.4) using faces of cones. We will see in §2.3 that this is no accident.
= I> ) 4 $ 8
R ! ; @ We next say a few words about how the polytope 3 relates to . As we will learn in §2.2, the dimension of is the dimension of the smallest affine subspace of ; @ containing , which is the same as the smallest $ affine subspace of ; @ containing . It follows from Proposition 2.1.6 that
Furthermore, the vertices of
R !
3
give an especially efficient affine covering of
3
.
R , ( , " ! ; , let R = I> 4 $ 8 ! ;J@ Proposition 2.1.9. Given and set R . , is a vertex of . Then .3 R .3 9 4 $
.3 9
9 ' ! Proof. We will prove that if .
, then . . The discussion of polytopes from §2.2 below implies that
9;0/ R
&
3
4
, . 9 &
for some
4
R
Given .
, we have , ' . 9 ; , so that , ' is a convex , combination of the vertices . Clearing denominators, we get integers W such that
-linear and
,-' R & , & R R L 4, R , '8 R Thus & P' " P A% 2 ,- which implies , A' % 2 - , . ' when 2 ' - . Fix A % R . ' ' such a . Then so . By (2.1.6), 3 9 4 ' 9 4 R BC D 47A% 2 ' - 8 R is invertible, 3 9 4 ' , giving 3 9 4 ' ! 3 9 4 .
;
+
Projective Normality. An+ irreducible variety ! is called projectively normal U ( ; if its affine cone is normal. A projectively normal variety is always ! normal (Exercise 2.1.5). Here is an example to show that the converse can fail. Example 2.1.10. Let
$ !
consist of the columns of the matrix
. The polytope R = I> 4 $ 8 is the giving the Laurent monomials 4 8 and 4 8 , with vertices corresponding to and . line segment connecting 3 The affine piece of corresponding to has coordinate ring F% - R A% . 4 8 4 8 - R &% -<
Chapter 2. Projective Toric Varieties
62
which is normal since it is a polynomial ring. Similarly, one sees that the coordinate F% - , also normal. These affine pieces cover 3 by ring corresponding to is 3 is normal. Proposition 2.1.9, so that
4 8
3
3
Since is in the row space of the matrix, affine cone of 3 is not normalis the by Proposition 2.1.4. The affine variety by Example 1.3.9, so .3 is normal but not projectively normal. that
The notion of normality used in this example is a bit ad-hoc since we have not formally defined normality for projective varieties. Once we define normality for abstract varieties in Chapter 3, we will see that Example 2.1.10 is fully rigorous. We will say more about projective normality when we explore the connection with polytopes suggested by the above results. Exercises for §2.1.
o o of - - permutation matrices defined in Example 2.1.3. o (a) Prove the claim made in Example 2.1.3 that three of the permutation matrices sum to the other three and use this to explain why d d d # d d d . E' . 3 4 o 4 o o 4 3 o 4 o 4 4 o (b) Show that zW{}| ( by computing . o (c) Conclude that E'B( d d d # d d d . 4 o 4 o o 4 o 4 o34 4 o 2.1.2. Let consist of the columns of an matrix u with integer entries. Prove ! that the conditions of Proposition 2.1.4 are equivalent to the assertion that nm27+6+6+626E'^' lies in the row space of u over or . 2.1.3. Given a finite set ( , prove that the rank of equals the dimension of the smallest affine subspace (over or ) containing . ' and check that it is 2.1.4. Verify the claims made in Example 2.1.7. Also compute m not homogeneous. be projectively normal. Use (2.0.6) to prove that the affine pieces 2.1.5. Let G G + of G are normal. 2.1.6. Fix a finite subset . Given , let ( ,/ * & O B N8 . 2.1.1. Consider the set
This is the translate of by . (a) Prove that and its translate .
(
give the same projective toric variety, i.e.,
"
can differ. Hint: (b) Give an example to show that the affine toric varieties and Pick so that it lies in an affine hyperplane not containing the origin. Then translate to the origin.
2.1.7. In Proposition 2.1.4, give a direct proof that (d)
g
(c).
was parametrized using the 2.1.8. In Example 2.1.5, the rational normal curve + , . Here we will consider the curve parametrized homogeneous monomials
(
by a subset of these monomials corresponding to the exponent vectors
(N/> 2 '926+6+7+62E 2 ' 8
§2.2. Lattice Points and Polytopes
8 and + ! +^( for every . This gives the projective curve . We assume ( . , explain why we can obtain the same projective curve using (a) If or
where
63
(b) Assume ( and ( . Use Proposition 2.1.8 to show that g is smooth if and only if ( #- and ( . Hint: For one direction, it helps to remember that monomials of strictly smaller degree.
smooth varieties are normal.
§2.2. Lattice Points and Polytopes Before we can begin our exploration of the rich connections between toric varieties and polytopes, we first need to study polytopes and their lattice points.
! ; @ is the convex hull of Polytopes. Recall from Chapter 1 that a polytope L R = I> 4
The dimension of a polytope ! ; @ is the dimension of the smallest affine subspace of ;J@ containing . Given a nonzero vector in the dual space ? @ and O O U . , we get the affine hyperplane and closed half-space defined by
O. R R and O U R , . ;J@ K , XL W . , . ;J@ K , L such that A subset # ! is a face of if there are . ?:@ SX and . U O 9 R . # ! O and O
We say that is a supporting affine hyperplane in this situation. Figure 1 shows
a polygon with the supporting lines of its -dimensional faces. The arrows in the figure indicate the vectors .
↑ P
→
→
↑ Figure 1. A polygon
and four of its supporting lines
We also regard as a face of itself. Every face of is again a polytope, and if R = I>)4 4
Here are some properties of faces.
Chapter 2. Projective Toric Varieties
64
(a)
be a polytope.
is the convex hull of its vertices.
R = I> 4
(b) If (c) If
!; @
Proposition 2.2.1. Let
is a face of #
, then the faces of
(d) Every proper face
#
L
lies in . are precisely the faces of
is the intersection of the facets #
containing #
lying in # . .
A polytope ! ;J@ can also be written as a finite intersection of closed halfspaces. The converse is true provided the intersection is bounded. In other words, if an intersection "
O U
R
' ( ( is bounded, then is a polytope. Here is a famous example. ?? . @ matrix ; Example 2.2.2. A @ is doubly-stochastic if? it? has nonnega as the affine tive entries columns are all . If we regard ? and its row and sum ? * space with coordinates ' , then the set of all doubly-stochastic matrices
is defined by the inequalites
*' W ? ? & '? ( ( * ' W & ? ')( ( * ' & ( ( * ' W & ( ( * '
4 all 8 4 all 8 4 all 8
(We use two inequalities to get one equality.) These inequalities easily imply that * ' for all , so that ? is bounded and hence is a polytope.
?
Birkhoff and Von Neumann proved independently that
? R 4 R 8 the vertices of ? are @ the @ permutation matrices and that . In the literature, has various names, including the Birkhoff polytope and the transportation polytope. See [103, p. 20] for more on this interesting polytope.
!
R When is full dimensional, i.e., ; @ , its presentation as an intersection of closed half-spaces has an especially nice form because each facet has a unique supporting affine hyperplane. We write the supporting affine hyperplane and corresponding closed half-space as
U R , . J ; @ K , L R R * M and O R ,. ;V@ K , LXW R * H. 4 * M 8 . ? @ is unique up to multiplication by a positive real number. where an inward-pointing facet normal of the facet . It follows that We call O U R , . ; @ K , L%W R * for all facets of =. R (2.2.1) facet
O
In Figure 1, the supporting lines plus arrows determine the supporting half-planes whose intersection is the polygon . We write (2.2.1) with minus signs in order to simplify formulas in later chapters.
§2.2. Lattice Points and Polytopes
65
Here are some important classes of polytopes.
Definition 2.2.3. Let !;@ be a polytope of dimension @ .
vertices. (a) is a simplex or d-simplex if it has @
(b) (c)
is simplicial if every facet of
is a simplex.
is simple if every vertex is the intersection of precisely
Examples include the Platonic solids in
A tetrahedron is a -simplex.
@
facets.
:
The octahedron and icosahedron are simplicial since their facets are triangles. The cube and dodecahedron are simple since three facets meet at every vertex.
(
Polytopes
and
are combinatorially equivalent if there is a bijection
faces of ( faces of
that preserves dimensions, inclusions and intersections. For example, simplices of the same dimension are combinatorially equivalent, and in the plane, the same holds for polygons with the same number of vertices.
Sums, Multiples, and Duals. Given a polytope = I> )4 LM8 is again a polytope for any W . If
R = I>)4
R
is defined by the inequalities
K , ' L W * ' .
then
is given by
K, 'L W
* '
In particular, when is full dimensional, then pointing facet normals. The Minkowski sum of subsets
(
(
!;J@
and
have the same inward-
is
R * ( * * ( . ( * . . ( R = I> 4 4 I 4
! ;J@ is full dimensional and When the dual or polar polytope
R . ? @ K , %L W R
is an interior point of for all ,.
R , . ; @ K , L1W R * facet as in (2.2.1), we is in the interior. Then is 4 the convex hull of . since ? @ (Exercise 2.2.1). We also have 8 R in this
When we write * for all have 4 * 8 the vectors situation.
, we define
! ? @
Figure 2 on the next page shows an example of this in the plane.
Chapter 2. Projective Toric Varieties
66
P
P°
Figure 2. A polygon
Lattice Polytopes. Now let ; and ? ;J@ and ? @ . A lattice polytope ! It follows easily that a polytope in ; lie in ; (Exercise 2.2.2).
@
and its dual
in the plane
be dual lattices with associated vector spaces ; @ is the convex hull of a finite set L ! ; . is a lattice polytope if and only if its vertices
+ is ( + 8 ( (
Example 2.2.4. The standard / -simplex in
Another simplex in
is
+ R = I> R = I>)4
4
8 , shown in Figure 3.
e1 + e2 + 3e3
0
e2
e1
Figure 3. The simplex
The lattice polytopes and are combinatorially equivalent but will give very different projective toric varieties.
§2.2. Lattice Points and Polytopes
67
Example 2.2.5. The Birkhoff polytope defined? in? Example 2.2.2 is a lattice polytope relative to the lattice of integer matrices since its vertices are the permu tation matrices, whose entries are all or .
One can show that faces of lattice polytopes are again lattice polytopes and that Minkowski sums and integer multiples of lattice polytopes are lattice polytopes (Exercise 2.2.2). Furthermore, every lattice is an intersection of closed " ( ( O polytope U R ' half-spaces defined over ; , i.e., where ' . ? and ' . .
When a lattice polytope is full dimensional, the facet presentation given in (2.2.1) has an especially nice form. If is a facet of , then the inward-pointing facet normals of lie on a rational ray in ?A@ . Let denote the unique ray * L R R * when , is a , generator. The corresponding is integral since K vertex of . It follows that
(2.2.2)
R ,. ; @ K , LXW R *
for all facets
is the unique facet presentation of the lattice polytope
.
R = I>)4 (
Example 2.2.6. Consider the square ( normals of are and and the facet presentation of
* Since the
R L W F K, ( X R L W F K, X
are all equal to , it follows that
lattice polytope. The polytopes
and
.
8 !
=.
of
. The facet is given by
R = I> 4 ( 8
is also a
are pictured in Figure 2.
It is rare that the dual of a lattice polytope is a lattice polytope—this is related to the reflexive polytopes that will be studied later in the book.
= I> 4 ( (
Example 2.2.7. The -simplex R in Example 2.2.4 has facet presentation
K , 4R
K, K, (R
R
L W K, X ( R LXW R LXW LXW
8 !
pictured
R
R
(Exercise 2.2.3). However, if we replace in the last inequality, we get 4 .8 , whichwith integer inequalities that define is not a lattice polytope.
The combinatorial type of a polytope is an interesting object of study. This leads to the question “Is every polytope combinatorially equivalent to a lattice polytope?” If the given polytope is simplicial, the answer is “yes”—just wiggle the vertices to make them rational and clear denominators to get a lattice polytope. The same argument works for simple polytopes by wiggling the facet normals. This will enable us to prove results about arbitrary simplicial or simple polytopes
Chapter 2. Projective Toric Varieties
68
using toric varieties. But in general, the answer is “no”—there exist polytopes in every dimension W not combinatorially equivalent to any lattice polytope. An example is described in [103, Ex. 6.21]. Normal Polytopes. The connection between lattice polytopes and toric varieties comes from the lattice points of the polytope. Unfortunately, a lattice polytope might not have enough lattice points. The -simplex from Example 2.2.7 has only four lattice points (its vertices), which implies that the projective toric variety $ * is just (Exercise 2.2.3).
We will explore two notions of what it means for a lattice polytope to “have enough lattice points.” Here is the first.
.
for all
4 A8 9 ;
.
is normal if
4 A8 9-; R 4*4
4 A 8
!; @
Definition 2.2.8. A lattice polytope
4 A8
8 A8
4*4
9;
8 A8
9; 9-; 4 ! 8 9-; is automatic. Thus The inclusion come from lattice points of normality means that all lattice points of and . In particular, a lattice polytope is normal if and only if
9-;
9-; R 4 A8 9 ;
times
for all integers W . In other words, normality says that has enough lattice points to generate the lattice points in all integer multiples of .
Lattice polytopes of dimension class of normal polytopes. Definition 2.2.9. A simplex R , , for vertices! , differences ,
are normal (Exercise 2.2.4). Here is another
is basic if has a vertex , such that the , of , form a subset of a -basis of ; .
;@
RP
, . is chosen. The standard This definition + + is independent of which vertex simplex is basic, and any basic simplex is normal (Exercise 2.2.5). More ! general simplicies, however, need not be normal. = I> )4 ( (
Example 2.2.10. Let R that the only lattice points ( of
(
R
4 8
8 !
. We noted earlier are its vertices. It follows easily that ( (
( 4 (8
is not the sum of lattice points of is not basic.
4 8
4 (
. This shows that
8
.
is not normal. In particular,
Here is an important result on normality. Theorem 2.2.11. Let !; @ be a full dimensional lattice polytope of dimension / W . Then is normal for all W / R .
§2.2. Lattice Points and Polytopes
69
Proof. This result was first explicitly stated in [12], though (as noted in [12]), its essential content follows from [28] and [64]. We will use ideas from [64] and [78] to show that 4 A8
8 A8 9-; 9 ; 9-; R 4*4 R for all integers W / . In Exercise 2.2.6 you will prove that (2.2.3) implies that W is normal when / R . Note also that for (2.2.3), it suffices to show 4*4 8 A8 9 ; ! 4 A8 9-; 9;
(2.2.3)
since the other inclusion is obvious.
First consider the case where is a simplex with no interior lattice points. Let , + and take W / R . Then 4 8 has vertices the vertices of be , 4 8 , 4 8 , + , so that 4*4 8 A8 9 ; is a convex a point , . combination
+ 4 8 + ' , ' where ' W & '( ' R ) 4 8 ' , then If we set ' R + + , R & '( ) ' ,-' where ) ' W & ')( ) ' R )
for some , then one easily sees that , R , ' . 4 &8 9 ; . Hence If ' W , R 4 , R ,-' 8 ,-' is the desired decomposition. On the other hand, if ) ' , R & '(
for all , then
+ / R 4 / +R 8 R & '( ) ' / R and & ')( ) ' R / . Now consider the lattice point so that R / + , R , , + 8 R , R & ')( 4 R ) ' 8 ,-' + 4 R ) 8 )
' ' R The coefficients are positive since for all , and their sum is & '(
R
0) R R , ' / / for . Hence is a lattice point in the interior of since
all . This contradicts our assumption on and completes the proof when lattice simplex containing no interior lattice points.
is a
To prove (2.2.3) for the general case, it suffices to prove that is a finite union of / -dimensional lattice simplices with no interior lattice points (Exercise 2.2.7). For this, we use Carath´eodory’s theorem (see [103, Prop. 1.15]), which asserts that $ !; @ , we have for a finite set
= I> ) 4 $ 8 R
$
= I> ) 4 8
! = I>)4 $ 8
where the union is over all subsets of affinely = I>! 4 consisting 8 is a simplex. independent elements. Thus each This enables us to write our lattice polytope as a finite union of / -dimensional lattice simplices.
Chapter 2. Projective Toric Varieties
70 If an / -dimensional lattice simplex lattice point , then +
#
R
' ' ( #
#
R = I> 4 + 8 #
has an interior
' R = I> 4 ' +2 8
is a finite union of / -dimensional lattice simplices, each of which has fewer interior lattice points than # since becomes a vertex of each # ' . By repeating this process on those # ' that still have interior lattice points, we can eventually write # and hence our original polytope as a finite union of / -dimensional lattice simplices with no interior lattice points. You will verify the details in Exercise 2.2.7.
This theorem shows that for the non-normal -simplex of Example 2.2.10, is normal. Here is another consequence of Theorem 2.2.11. its multiple Corollary 2.2.12. Every lattice polygon
!
is normal.
We can also interpret normality in terms of the cone
4 A8 R = I> 4 8 !;J@
introduced in Figure 3 of Chapter 1. The key feature of this cone is that is the 4 A8 at height , as illustrated in Figure 4. It follows that lattice points “slice” of ,. correspond to points 4 , 8 . 4 A8 9 4 ; X8 .
2P
height = 2
C(P) P
Figure 4. The cone
height = 1
sliced at heights
In Exercise 2.2.8 you will show that the semigroup 4 &8 points in relates to normality as follows.
and
4 A8 9 4 ; X8
of lattice
§2.2. Lattice Points and Polytopes
71
Lemma 2.2.13. Let ! ; @ be a lattice polytope. Then 4 9-; 8X generates the semigroup 4 A8 9 4 ; if
X8is. normal if and only ! ; X8 @ is normal if and only if 4 9 ; 8 is This lemma tells us that 4 A8 4 the Hilbert basis of . 9 ; = >I )4 ( ( Example 2.2.14. In Example 2.2.10, the simplex R 8 gives the cone 4 A8 ! . The Hilbert basis of 4 A8 9 4 ; X8 is 4 8 4 ( 8 4 8 4 ( 8 4 ( .8 4 ( .8
(Exercise 2.2.3). Since the Hilbert basis has generators of height greater than ,
Lemma 2.2.13 gives another proof that
is not normal.
+
In Exercise 2.2.9, you will generalize Lemma 2.2.13 as follows.
be a lattice polytope of dimension / W and Lemma 2.2.15. Let ! ; @ 4 &8 let be the maximum height of an element of the Hilbert basis of . Then:
(a) (b)
/ R .
is normal for any
W
.
The Hilbert basis of the simplex of Example 2.2.14 has maximum height . is normal. The paper [64] gives a Then Lemma 2.2.15 gives another proof that version of Lemma 2.2.15 that applies to Hilbert bases of more general cones. Very Ample Polytopes. Here is a slightly different notion of what it means for a polytope to have enough lattice points.
! ;@ is very ample if for every vertex Definition 2.2.16. A lattice polytope , . , the semigroup 2 $ P R &4 9J; R , 8 generated by 9J; R , R
, Z R , , Z . 9-; is saturated in ; .
This definition relates to normal polytopes as follows. Proposition 2.2.17. A normal lattice polytope
. Proof. Fix a vertex ,
is very ample.
, . ; such that , . 2 $ P for some 2 $ P
, . 2 $ P as integer W . To prove that ,. , write , R & P $ * P 4 , Z R , 8 * P . F so that @ W & P $ * P . Then Pick @ . , @ , R & P $ * P , Z @ R & P $ * P , . @ Dividing by gives , @ , . @ , which by normality implies that
and take
@ , R ' ( ? 4 R , ' & ( ( ,-' We conclude that ,
?
, ' . 9; for all ( ,-' R , 8 . 2 $ P , as desired.
Chapter 2. Projective Toric Varieties
72
+ ! ; @
Combining this with Theorem 2.2.11 and Corollary 2.2.12 gives the following. Corollary 2.2.18. Let
! W (a) If , then
! R (b) If , then
be a full dimensional lattice polytope.
is very ample for all
W / R
.
is very ample.
Part (a) was first proved in [28]. We will soon see that very ampleness is precisely the property needed to define the toric variety of a lattice polytope. The following example taken from [11, Ex. 5.1] shows that very ample polytopes need not be normal, i.e., the converse of Proposition 2.2.17 is false.
% R
Example 2.2.19. Given
in positions and elsewhere. % , letThus $ R % -< % -< % - % -< % -< %> -< %
= I> 4 $ 8
-
vector in with 4 denote the 8 . Then let - % -< % -<%
!
The lattice polytope R lies in the affine hyperplane of where the coordinates sum to . As explained in [11], this configuration can be interpreted in terms of a triangulation of the real projective plane.
$
The points of of are the only lattice points of (Exercise 2.2.10), so that ( , ( . Then $ is the set of vertices of . Number the( points of ( as ,
$
4 8 R (
'( (
( ( 4 , '8 ( '(
,-' R
4 8 . . Since is not a sum of lattice points of R % - is not in $ ), we conclude that is not a
shows that R % - . $ , the vector (when normal polytope.
4 .8 is! very ample takes more work. The first step is to prove 4 A8 9 that , is a Hilbert basis of the semigroup 4 A8 where ! is the cone over . We used the software 4ti2 [47]. 2 $ P :R ,-' Now be the semigroup generated by the , . Take fix and let 2 $ P , . such that , . . As in the proof of Proposition 2.2.17, this implies that , @ ,-' . @ for some @ . . Thus 4 , @ , ' @ 8 . 4 A8 9 . Expressing $
Showing that
(
this in terms of the above Hilbert basis easily implies that
, R * 4 R5 ,-' 8
RQ
( ( * .4 ,
2 $ P
R ,-' 8
* * . F
2 $ P
,' . If we can show that , then , . follows immediately and 2 $ P
R proves that is saturated. When , one can check that % - R % - % - % -<
which implies that
R , ( R 4 , %R , ( 8
4, R , ( 8
4,
R , ( 8 . 2 $ P
§2.2. Lattice Points and Polytopes
73
R
One obtains similar formulas for the proof that is very ample.
(Exercise 2.2.10), which completes
The polytope has further interesting properties. For example, up to affine equivalence, can be described as the convex hull of the 10 points in given by
4 8 4 8 4 8 4 4 8 4 8 4 8 4
8 4 8
8 4 8 , this polytope ( has the Of all -dimensional polytopes whose vertices lie in
R , every pair most edges, namely (see [1]). Since it has vertices and of distinct vertices is joined by an edge. Such polytopes are -neighborly.
Exercises for §2.2. 2.2.1. Let point.
be a polytope of maximal dimension with the origin as an interior
( / O 2 M# for all facets 8 . Prove that U( ! p& 'p O a facet' . (b) Prove that the dual of a simplicial polytope is simple and vice versa. (c) Prove that I ':(L p ' for all . (a) Write and that
for all
(d) Use part (c) to construct an example of a lattice polytope whose dual is not a lattice polytope.
& be a polytope. L( L'
2.2.2. Let
(a) Prove that is a lattice polytope if and only if the vertices of lie in (b) Prove that is a lattice polytope if and only if is the convex hull of its lattice points, . i.e., (c) Prove that every face of a lattice polytope is a lattice polytope.
(d) Prove that Minkowski sums and integer multiples of lattice polytopes are again lattice polytopes.
( ! A 2 2 4 2 4 .- o ' o be the simplex studied in Exam given in Example 2.2.7. (b) Show that the only lattice points of are its vertices. (c) Show that the toric variety is o . (d) Show that the vectors given in Example 2.2.14 form the Hilbert basis of g I '
2.2.3. Let ples 2.2.4, 2.2.7, 2.2.10 and 2.2.14. (a) Verify the facet presentation of
' .
2.2.4. Prove that every one-dimensional lattice polytope is normal. 2.2.5. Recall the definition of basic simplex given in Definition 2.2.9. (a) Show that if a simplex satisfies Definition 2.2.9 for one vertex, then it satisfies the definition for all vertices.
is basic. (b) Show that the standard simplex (c) Prove that a basic simplex is normal.
Chapter 2. Projective Toric Varieties
74
2.2.6. Let & be an -dimensional lattice polytope. (a) Prove that (2.2.3) implies that
B I' * I' ( n B% 7' I' for all integers B . #f and f . Hint: When ( ( , we have BI' B I' C ( I' B% (m' I' + Apply (2.2.3) twice on the right. (b) Use part (a) to prove that B is normal when B . #- . 2.2.7. Let & be an -dimensional lattice polytope. (a) Follow the hints given in the text to give a careful proof that is a finite union of -dimensional lattice simplices with no interior lattice points. (b) In the text, we showed that (2.2.3) holds for an -dimensional lattice simplex with no interior lattice points. Use this and part (a) to show that (2.2.3) for
.
2.2.8. Prove Lemma 2.2.13.
B
2.2.9. In this exercise you will prove Lemma 2.2.15. As in the lemma, let be the maximum height of a generator of the Hilbert basis of . (a) Adapt the proof of Gordan’s Lemma (Proposition 1.2.17) to show that if is the Hilbert basis of the semigroup of lattice points in a strongly convex cone , then the lattice points in the cone can be written as the union
g I'
(b) Conclude that
, ) w
`?m C'
, m+
g I' .U'Q( ) +! + 23,+F' m2 where (,I n L' /1m8&' . (c) Use part (b) to show that (2.2.3) holds for B B . C' from Example 2.2.19. 2.2.10. Consider the polytope N( (a) Prove that is the set of lattice points of (b) Complete the proof begun in the text that
. is very ample.
2.2.11. Prove that every proper face of a simplicial polytope is a simplex.
B
B N #V
2.2.12. In Corollary 2.2.18 we proved that is very ample for using Theorem 2.2.11 and Proposition 2.2.17. Give a direct proof of the weaker result that is very ample for sufficiently large. Hint: A vertex gives the cone generated by the semigroup defined in Definition 2.2.16. The cone is strongly convex since for is a vertex and hence has a Hilbert basis. Furthermore, all . Now argue that when is large, ' contains the Hilbert basis of . A picture will help.
B
B
g [ g [ C( g A [ A
g[ g [ B , B B I' # B g [ ! A 2 2 4 2 o 'a o . 2.2.13. Fix an integer and consider the - -simplex L( (a) Work out the facet presentation of and verify that the facet normals can be labeled so that (C . 4 o (b) Show that is normal. Hint: Show that yo * B I' o)( n B%*E' I' o . We will see later that the toric variety of is the weighted projective space p126126m2 W' . [
§2.3. Polytopes and Projective Toric Varieties
75
§2.3. Polytopes and Projective Toric Varieties Our next task is to define the toric variety of a lattice polytope. As noted in §2.2, we need to make sure that the polytope has enough lattice points. Hence we begin with very ample polytopes. Strongly convex rational polyhedral cones will play an important role in our development. The Very Ample Case. Let very ample polytope ! ; !@ be a full dimensional R R / . If 9J ;5 8 " , " ( ( , " , then relative to the lattice ; , and let $ is the Zariski closure of P the image ofP the map given by (
4 8 4 8 . " " R ' (*' " for / " " ( . Fix homogeneous coordinates "" ( We examine the structure of using Propositions 2.1.8 and 2.1.9. ! '
('
$
For each ,'/.
9-;
consider the semigroup
2 ' R & 4 9-; R ,-' 8 1R ,-' , . 9 ; 8" " ( we have the affine open subset generated ( by , for . In 4 ' " " consisting of those points where ' ' R P . Proposition 2.1.8 showed that $ 9 4 ' of $ is the affine toric variety the affine open piece BC D.47F% 2 ' - 8 $ 9 4 '
9 ' R P . Here is our first major result about Theorem 2.3.1. Let be the projective toric variety of the very ample poly ! R /. tope !;J@ , and assume that is full dimensional with 9 ; , the affine piece 9 ' is the affine toric (a) For each vertex , ' . variety 9 ' R 3 R B9C D.47&% 6 ' 7 9 ; - 8 and Proposition 2.1.9 showed that
$
4
$
$
vertex of $
$
$
4
$
4
4
6 I>' ! 4 ? @ is R the strongly convex rational polyhedral cone dual to the
8 , ' 9 ; ! ;V@ . Furthermore, ! 6 ' R / . $ 5 . (b) The torus of has character lattice ; and hence is the torus = I> 4 , ' is a vertex, it has a supporting Proof. Let ' R 9V; O R U , ' 8 . Since O O R ,-' . It follows that O is hyperplane such that ! and 9 ' (Exercise 2.3.1), so that ' is strongly convex. a supporting hyperplane of .
! ' R ! It is also easy to see that (Exercise 2.3.1). It follows that ' and 6 ' R ' 7 are strongly convex rational polyhedral cones of dimension / . 2 ' 9V; R 6 ' 7 9V; . By hypothesis, is very ample, which We have ' ! 2 2 means that ' ! ; is saturated. Since ' and ' R 6 ' 7 are both generated by where = cone
Chapter 2. Projective Toric Varieties
76
2 R ,-' , part (a) of Exercise 1.3.4 implies ' R 6' 7 9 ;
9 ;
. (This exercise was part of the proof of the characterization of normal affine toric varieties given in Theorem 1.3.5.) Part (a) of the theorem follows immediately.
' and
For part (b), Theorem 1.2.18 implies that 4 5 ! 4 3 R $ strongly convex. Then 9 $ . the torus of
5 is the torus of 4 3 since 6 ' is 5 is also ' ! $ shows that
$
The affine pieces intersect in 9 9 9 ' 9 order to describe this intersection carefully, we need to study how the cones 6 6 fit together in ?G@ . This leads to our next topic.
$
4
$
4
4
4
'
. In and
The Normal Fan. The cones 6 ' ! ? @ appearing in Theorem 2.3.1 fit together in a remarkably nice way, giving a structure called the normal fan of . Let !; @ be a full dimensional lattice polytope, not necessarily very ample. Faces, facets and vertices of will be denoted by # , and respectively. Hence we write the facet presentation of as
R , . ;J@ K , LXW R *
(2.3.1)
A vertex
(When R containing
.
gives cones
R = I> 4 9 ; R 8 ! ;V@
and correspond bijectively to faces Q R+ Q R = I> 4 # #
(2.3.2)
Q
R+
R 4Q
#
6 R
and
, ' , these are the cones '
!
89
6 '
9-;
.
for all
R
7
! ? @
studied above.) Faces via the maps
! #
8
which are inverses of each other. These maps preserve dimensions, inclusions, and intersections (Exercise 2.3.2), as illustrated in Figure 5 on the next page. In particular, all facets of
come from facets of
R , . ;V@ K , LXW
for all
containing
By the duality results of Chapter 1, it follows that the dual cone 6
= 6 R I> 4
6"!
R = I> 4
8
contains
This construction generalizes to arbitrary faces
! #
containing , so that
.
is given by
by setting
contains
#
8
R since is the cone Thus the cone 6 is the ray generated by , and 6 generated by the empty set. Here is our main result about these cones. Theorem 2.3.2. Let
6"! # is a face of (a) For all 6
!
. Y%$
(b) The intersection
$
; @ be a full dimensional lattice polytope and set !. Then:
, each face of 6
6 ! 9 6
!
Y%$
of any two cones in Y . !
is also in
$
is a face of each.
Y
$
R
§2.3. Polytopes and Projective Toric Varieties
77
P Cv
v 0
Figure 5. The cone
of a vertex
A collection of strongly convex rational polyhedral cones satisfying conditions (a) and (b) of Theorem 2.3.2 is called a fan. General fans will be introduced in Y $ are built from the inward-pointing Chapter 3. Since the cones in the above fan % Y $ normal vectors , we call the normal fan or inner normal fan of .
The following easy lemma will be useful in the proof of Theorem 2.3.2.
O be a supporting affine hyperplane ! O 9 . ) , ) W . Then Proof. First suppose that . 6 ! and write R & ! R & ) * easily implies that ! O U and # ! .O 9 . setting R ! * come from the facet presentation (2.3.1). Recall that the integers O 9 . Take a vertex . # . Then Going the other way, suppose that # ! U U O O O . 7 R 6 , so that ! and imply that ! . Hence . R & ) ) W Let be a facet of containing but not # , and pick . # with . . Then . # ! . O imply that R K L R & ) K L R K L R & ) K L R R & ) * R * for . . These equations imply where the last equality uses K L R & ) K L R R & ) *
Lemma 2.3.3. Let # be a face of of . Then . 6"! if and only if #
and let
Chapter 2. Projective Toric Varieties
78
However, . gives K L ) R follows that whenever proof of the lemma. Corollary 2.3.4. If
#
is a face of #
, and since K AL W R * for all , it ! P . This gives . 6 and completes the . 6 if and only if ! . , then R *
!
"!
#
Proof. One direction is obvious by the definition of 6 ! , and the other direction O " follows from Lemma 2.3.3 since is a supporting affine hyperplane of " O with 9 R .
Theorem 2.3.2 is an immediate corollary of the the following proposition. Proposition 2.3.5. Let ! ; @ . Then:
!
Z
and #
if and only if 6
Z
#
!
be faces of a full dimensional lattice polytope
6"!
Z (b) If # ! # , then 6 ! is a face of 6"! , and all faces of 6 ! are of this form. ZZ Z (c) 6"!9 6 ! R 6 ! , where # is the smallest face of containing # and # . Z Z Proof. To prove part (a), note that if # ! # , then any facet containing # also (a)
#
#
!
.
contains # , which implies 6 ! ! 6! . The other direction follows easily from Corollary 2.3.4 since every face is the intersection of the facets containing it by Proposition 2.2.1.
For part (b), fix a vertex . # and note that by (2.3.2), # determines a face Q of . Using the duality of Proposition 1.2.10, Q gives the dual face
QM
of the cone obtains
6
. Then using 6
7
QI
9
R = I> 4
R = I> 4
QM
R
O
R 6 9
.
Q
.
QI
8
! O
and Q
Since . # , the inclusion Q ! is equivalent to turn is equivalent to # ! . It follows that
QM
(2.3.3)
Z
R = I> 4
#
! O
R 6 7
"
, one
, which in
8R 6
!
#
8
!
"!
arise in this way. ! 6 In particular, ! means that 6 is also a face of 6 , and since 6 by part (a), we see that 6 a face of 6 . Furthermore, every face of 6 a face for some face isZ . Using of 6 by Proposition 1.2.6 and hence is of the form 6 so that 6"! is a face of 6 , and all faces of 6 #
-#
!
!
!
part (a) again, we see that
ZZ
!
#
!
!
Z , and part (b) follows.
#
!
#
!
Z
For part (c), let # be the smallest face of containing # and # . This exists Z Z is the interbecause a face is the intersection of the facets containing it, so that # Z Z Z R ). section of all facets containing # and # (if there are no such facets, then # By part (b) 6 ! is a facet of both 6 ! and 6 ! . Thus 6 ! ! 6"!9 6 ! .
§2.3. Polytopes and Projective Toric Varieties
79
It remains to prove the opposite inclusion. If 6 ! 9V6 ! R R 6 , then Z Z R and we are done. If 6 ! 9 6 ! R P , any nonzero in the intersection # lies in both 6"! and 6 ! . The proof of Proposition 2.3.6 given below will show that O. . By Lemma 2.3.3, is a supporting affine hyperplane of for some . . 6"! and . 6 ! imply that # and # Z lie in O. 9 . The latter is a face of Z Z Z ! O. 9 since # Z Z is the smallest such face. containing # and # , so that # Applying Lemma 2.3.3 again, we see that . 6 ! .
$
Proposition 2.3.5 shows that there is a bijective correspondence between faces Y $ of and cones of the normal fan . Here are some further properties of this correspondence. Proposition 2.3.6. Let ! ; sion / and consider the cones
(a) (b)
!
? @ R
#
@
be a full dimensional lattice polytope of dimenY $ of . Then: in the normal fan
6 !
! R 6 ! / for all faces # " $ R 3 6 6! 6 vertex of
of
.
.
Proof. Let # be a face of and take a vertex of # . By (2.3.2) this gives a face Q of the cone , which has a dual face Q M of the dual cone 7 R 6 . Since QM R 6"! by (2.3.3), we have
!
#
! 6
!
R
Q
!
R /
QM
where the first equality uses Exercise 2.3.2 and the second follows from Proposition 1.2.10. This part (a). For part (b), U let . ? @ be nonzero and set K L proves = O and . O for at least one =R vertex of . Then ! vertex of , so that . 6 by Lemma 2.3.3. The final equality of part (b) follows immediately.
A fan satisfying the condition of part (b) of Proposition 2.3.6 is called complete. Thus the normal fan of a lattice polytope is always complete. We will learn more about complete fans in Chapter 3. An important observation is that the normal fan in Example 2.3.9 doesn’t de pend on the integer . In general, multiplying a polytope by a positive integer has no effect on its normal fan, and the same is true for translations by lattice points. We record these properties in the following proposition (Exercise 2.3.3). Proposition 2.3.7. Let ! ; @ be a full dimensional lattice polytope. Then for
any lattice point ,. ; and any integer W , the polytopes , and have the same normal fan as .
Examples of Normal Fans. Here are some examples of normal fans.
Example 2.3.8. Figure 6 on the next page shows a lattice hexagon in the plane ( , with corre together with its normal fan. The vertices of are labeled ( 6 in the normal fan. In the figure, is shown on the left, sponding cone 6
Chapter 2. Projective Toric Varieties
80
and at each vertex ' , we have drawn the normal vectors of the facets containing ' and shaded the cone 6 ' they generate. On the right, these cones are assembled at the origin to give the normal fan.
v5
v4 σ3
σ2
v6 v3
v1
σ4
ΣP
P
σ1 σ5
v2 Figure 6. A lattice hexagon
and its normal fan
σ6
Notice how one can read off the structure of from the normal fan. For Y $ example, two cones 6 ' and 6 share a ray in if and only if the vertices ' and lie on an edge of .
(
Example 2.3.9. The -simplex for ! has vertices Y . $Let R some positive integer . Figure 7 shows and its normal fan . At each vertex
v2 ΣP
P
v0
σ0 σ2
v1
Figure 7. The triangle
' of
σ1
and its normal fan
, we have drawn the normal vectors of the facets containing ' and shaded Y $ the cone 6 ' they generate. The reassembled cones appear on the left as .
Here is an example of the normal fan of a polytope in
.
4 8 .
Example 2.3.10. Consider the cube with vertices ( , and the ! facet presentation facet normals are of
K , ' LXW R
is
The
§2.3. Polytopes and Projective Toric Varieties
81
The origin is an interior point of . By Exercise 2.2.1, the facet normals are the vertices of the dual polytope , the octahedron pictured in Figure 8. z
z
y
y
P
P˚ x
x
Figure 8. A cube
and its dual octahedron
However, the facet normals also give the normal fan of , and one can check that in the above figure, the maximal cones of the normal fan are the octants of , which are just the cones over the facets of the dual polytope .
As noted earlier, it is rare that both and are lattice polytopes. However, !;J@ is a lattice polytope containing whenever an interior point, it is Y $ are theascones over the facets of still true that maximal cones of the normal fan ! ? @ (Exercise 2.3.4).
.
The special behavior of the polytopes and and 2.3.10 leads to the following definition.
discussed in Examples 2.2.6
Definition 2.3.11. A full dimensional lattice polytope facet presentation is
R , . ;J@ K , L W R
If is reflexive, then is a lattice point of * R for all point of (Exercise 2.3.5). Since
Thus
R = I> ) 4
! ;G@
for all facets
is reflexive if its
.
and is the only interior lattice , Exercise 2.2.1 implies that
facet of
A8
is a lattice polytope and is in fact reflexive (Exercise 2.3.5).
Chapter 2. Projective Toric Varieties
82
We will see later that reflexive polytopes lead to some very interesting toric varieties that are important for mirror symmetry.
$
The General Case. Earlier in the section, we studied the toric variety of a $ . This development very ample polytope and described the affine pieces of leaves one item of unfinished business: What is the intersection of two + of the affine pieces described in Theorem 2.3.1?
To study let !; R 9-this, ; . Then
@
be an / -dimensional very ample polytope and set
"" ( ! "" ( ' P Label the homogeneous as for ,. 9 ; . If is a vertex ( coordinates of 8 " ' " of , then ! is the affine open subset where R P , and Theorem 2.3.1
$
4
3 R B9C D 47&% 7 6 9 ; -8 $ 9 4 is the affine toric variety of the cone 6 in the normal fan Y $ of So Proposition 2.3.12. Let ! ; @ be full dimensional and very ample. If R P $
tells us that
9
4
R
4
9 9 R 3 R and the inclusions 9 9 9 are vertices of
and
#
is the smallest face of 4
$
3 4
(2.3.4)
4
4
$
can be written
4
4
$
4 4 3 8
R
containing
and
BC D 47A% 6 7 9; - 8 ! 4
!
3 R 4 4 3 8 4
!
, then
9 4
$
.
3
4
$
Proof. We analyzed the intersection of affine pieces of in §2.1. Translated to the notation being used here, (2.1.6) and (2.1.7) imply that
$
9 4
3 9 4 R 4 4 8
R 44 3 8
4 4 3 8 R 4 3 R . R 6 7 , so that H R O " 96 is a face of 6 . In However, we have this situation, Proposition 1.3.16 and equation (1.3.4) imply that 44 3 8 R 4 O " 9 6 R 6 ! . Thus the proposition will follow once we prove H R 6 ! , i.e., Since 6"! R 6 9 6 by Proposition 2.3.5, it suffices to prove that O " 9 6 R 6 9 6 O " 9V6 O Let . . If R P , there is . is a supporting affine O such O Thus all we need to show is that
hyperplane of . Then . 6 implies . by Lemma 2.3.3, so that . O " . Applying Lemma 2.3.3 again, we get . 6 . Going the other since .
§2.3. Polytopes and Projective Toric Varieties
83
R P , pick . as above. Then . 6 9V6 and . O , from which . O " follows easily. This
way, let . 6 9 6 . If Lemma 2.3.3 imply that completes the proof.
This proposition and Theorem 2.3.1 have the remarkable result that the normal Y $ $ $ fan completely determines the internal structure of : we build 4 3 from local pieces given by the affine toric varieties( , glued together via (2.3.4). "" for any of this—everything we We don’t need the ambient projective space need to know is contained in the normal fan.
As a consequence, we can now give the general definition of the toric variety of a polytope. Definition 2.3.13. Let define
where
! ;@
be a full dimensional lattice polytope. Then we
$ R
$
is any positive integer such that
is very ample.
Such integers exist by Corollary 2.2.18, and if and are two such integers, Y $ R then and have the same normal fan by Proposition 2.3.7, namely Y $ R Y $ $ $ . It follows that while and lie in different projective 4 3 glued together via (2.3.4). spaces, they are built from the affine toric varieties Once we develop the language of abstract varieties in Chapter 3, we will see that $ is well-defined as an abstract variety.
$
We will often speak of without regard to the projective embedding. When $ we want to use a specific embedding, we will say “ is embedded using ”, is very ample. In Chapter 6 we will use the language of where we assume that $ such that $ divisors and line bundles to restate this in terms of a divisor on $ is very ample precisely when is.
Here is a simple example to illustrate the difference between $ as sitting in a specific projective space. variety and
$
as an abstract
+
+
! Example 2.3.14. Consider the / -simplex . We can define using + + + W
for any integer since and hence very ample. The lattice U + R is normal F% ( +.- of total monomials points in correspond to the of ( + + " "
R R degree / . This gives an embedding . When , ! 9 ( , +
implies that +
R
The normal fan of ! can regard
+ in Exercise 2.3.6. 8" " is( described as the image of the map $ + R " " (
For an arbitrary
W , we
Chapter 2. Projective Toric Varieties
84
F% ' *' + -
defined using all monomials of total degree in (Exercise 2.3.6). It follows that this map is an embedding, usually called the Veronese embedding. But + when we forget the embedding, the underlying toric variety is just .
The Veronese embedding allows us to construct some interesting affine open + F% ' *' + - be nonzero and homogeneous of degree( subsets of . Let O . ' . We write the homogeneous coordinates of " " as and write O R & ( " R for . Then R & ( is a nonzero linear form in the variables " " ( /" " ( S 354 8 , so that is a copy of (Exercise 2.3.6). If follows that
+ S 5 3 4 O 8 $ 4 + 8 9 " " ( S 354 8 + has a richer supply of is an affine variety (usually not toric). This shows+ that S 354 ' ' 8 considered earlier in affine open subsets than just the open sets ' R 4
the chapter.
+ later in the + book, we will see the When we explain the construction of + Proj intrinsic reason why S 354 O 8 is an affine open subset of . + Example 2.3.15. The -dimensional analog of the rational normal curve is the L rational normal scroll , which is the toric variety of the polygon = >I 4 * ( ( 8 R !
* * and its normal fan are where . satisfy . The polygon R
pictured in Figure 9.
v2 = e2 P = P2,4 v1 = 0
σ4
v3 = 4e1 + e2
σ1
ΣP
σ3 σ2
v4 = 2e1
Figure 9. The polygon of a rational normal scroll and its normal fan
*
lattice points and gives the map 47 M 8 +R U U ( 4 8 4 8 !L R $ is the Zariski closure of the image. To describe the image, such that we rewrite the map as ( R+ U U ( 4 . )) 8 4 )) )) ) ) 8 In general, the polygon
has
§2.3. Polytopes and Projective Toric Varieties
4 ) 8
4 8
85
4 8
R , the map is , which When is the * coordinates of U U ( . In the rational normal curve mapped to the first 4 ) 8 R 4 8 gives same way, ( the rational normal curve mapped to the last
U U coordinates of . If we think of these two curves as the “edges” of a 4 )) 8 . ( vary gives scroll, then fixing gives a point on each edge, and letting the line of the scroll connecting the two points. So it really is a scroll!
important observation is that the normal fan depends only on the difference R An * , since this determines the slope of the slanted edge of . If we denote the , it follows that as abstract toric varieties, we have difference by . $ R $ R $ R
since they are all constructed from the same normal fan. In Chapter 3, we will see < that this is the Hirzebruch surface . (
L
U U
*
But if we think of the projective surface ! , then and have a unique meaning. For example, they have a strong influence defining equa U U ( ' onthe L * ' tions of . Let the homogeneous coordinates of be 4* 8 and consider the matrix
!
' ')( ' " ( ( " ( )' ( ' ' ( 624
Example 2.3.15 is another example of a determinantal variety, as is the rational + normal curve from Example 2.0.1. Note that the rational normal curve comes ( % ( from the polytope / R / , where the underlying toric variety is just .
Exercises for §2.3.
U
1[ <
2.3.1. This exercise will use the same notation as the proof of Theorem 2.3.1. (a) Let be a supporting hyperplane of a vertex "+ + porting hyperplane of
g (b) Prove that zW{}|-g + (Vz`{ | .
. Prove that
m[
is a sup-
2.3.2. Consider the maps defined in (2.3.2). (a) Show that these maps are inverses of each other and define a bijection between the faces of the cone and the faces of containing - .
g
(b) Prove that these maps preserve dimensions, inclusions, and intersections. (c) Explain how this exercise relates to Exercise 2.3.1. 2.3.3. Prove Proposition 2.3.7.
2.3.4. Let & be an -dimensional lattice polytope containing as an interior point, and let be its dual polytope. Prove that the normal fan consists of the cones over the faces of . Hint: Exercise 2.2.1 will be useful.
Chapter 2. Projective Toric Varieties
86
(b) Prove that
2.3.5. Let & be a reflexive polytope. (a) Prove that is the only interior lattice point of
.
is reflexive.
26+7+6+92 ! ` ? F ) ' C/ 2 26+7+6+92 8 (N# 9 L ( 12 `2 ! is given by the map (b) For an integer B i , show that the variety 6A #h ! defined using all monomials ofAtotal degree B in H d 26+6+7+62 d J . 2.3.7. Let be an -dimensional lattice polytope and let be a face. Prove the following intrinsic description of the cone f : (L/7 O 2 2 for all 2 8 +
2.3.6. This exercise is concerned with Example 2.3.14
be the standard basis of . Prove that the normal fan of the standard (a) Let -simplex consists of the cones for all proper subsets , where ( - . + + . Draw pictures of the normal fan for
2.3.8. Prove that all lattice rectangles in the plane with edges parallel to the coordinate axes have the same normal fan.
§2.4. Properties of Projective Toric Varieties We conclude this chapter by studying when the projective toric variety polytope is smooth or normal.
$
of a
Normality. Recall from §2.1 that a projective variety is projectively normal if its affine cone is normal. Theorem 2.4.1. Let (a) (b)
$ $
! ; @
be a full dimensional lattice polytope. Then:
is normal.
is projectively normal under the embedding given by is normal.
if and only if
$
4
3
Proof. Part (a) is immediate since is the union of affine pieces , a vertex 4 3 of , and is normal by Theorem 1.3.5. In Chapter 3 we will give an intrinsic definition of normality that will make this argument completely rigorous.
For part (b), the discussion following (2.1.4) shows that the projective em $ $ $ ( . By bedding of given by has affine cone given by 4*4 A8 9 ; 8X Theorem 1.3.5, this is normal if and only if the semigroup 4 A8 A8 * 4 4 8
is saturated in ; , and since , this is 9# ;
generates the cone 4 A8 &8 4 8
9 ;
generates the semigroup 9 4 ; MX8 . equivalent to saying that Then we are done by Lemma 2.2.13.
Smoothness. Given the results of Chapter 1, the smoothness of to determine. We need one definition.
$
is equally easy
§2.4. Properties of Projective Toric Varieties
Definition 2.4.2. Let
'
!;@
87
'
be a lattice polytope.
'
(a) Given a vertex of and an edge containing , let be the first lattice point of different from encountered as one tranverses starting at . In = I> 4 R 8 R other words, is the ray generator of the ray . (b)
! Theorem 2.4.3. Let following are equivalent:
; @
(b) (c)
R
'
is smooth if for every vertex , the vectors , where is an edge of
! R ! ; @ , containing , form a subset of a basis of ; . In particular, if R form a basis of ; . then the vectors We can now characterize when
(a)
'
$
Y
$
is smooth.
be a full dimensional lattice polytope. Then the
is a smooth projective variety.
$
is a smooth fan, meaning that every cone in Definition 1.2.16.
Y $
is smooth in the sense of
is a smooth polytope.
Proof. Smoothness is a local condition, so that a variety is smooth if and only if its $ is smooth if and only if 4 3 is smooth for every local pieces are smooth. Thus 4 3 vertex of , and is smooth if and only if 6 is smooth. Since faces of smooth Y $ consists of the 6 and their faces, the equivalence (a) cones are smooth and (b) follows immediately.
R 67 is For (b) (c), first observe that 6 is smooth if and only if its dual smooth. The discussion following (2.3.2) makes it easy to see that the ray genera R tors of are the vectors from Definition 2.4.2. It follows immediately that is smooth if and only if is smooth for every vertex , and we are done.
The theorem makes it easy to check the smoothness of simple examples such as the toric variety of the hexagon in Example 2.3.8 or the rational normal scroll L of Example 2.3.15 (Exercise 2.4.1).
We also note the following useful fact, which you will prove in Exercise 2.4.2. Proposition 2.4.4. Every smooth full dimensional lattice polytope ample.
!; @
is very
One can also ask whether every smooth lattice polytope is normal. This is an important open problem in the study of lattice polytopes. Here is an example of a smooth reflexive polytope whose dual is not smooth.
4
8 + R 4 8 ! + , where + is the standard + and have the same normal fan, so
Example 2.4.5. Let R / Proposition 2.3.7 implies that / -simplex. $ that and are smooth.
Chapter 2. Projective Toric Varieties
88
The polytope has the following interesting properties (Exercise 2.4.3). First, has the facet presentation
' ' W RF R / R' ( R R ',+ W F R
so that
is reflexive with dual
R = I> 4 ( + 8
R R ( R
The normal fan of consists of cones over the faces of Y $ corresponding to the vertex . is the cone of
6 Figure 10 shows
R = I>)4 ( +98
' R
and the cone 6 when /
R
4/
R +2
. In particular, the cone
8 '
.
v2 e2 σe0
P˚ e1
P e0
v1
Figure 10. The cone of the normal fan of
R
4
8 4 R 8
R / + ( ' + . This makes it easy For general / , observe that ' ( + 4
8 " in . Thus 6 is not smooth to see that has index / . It follows that the “dual” toric variety $ is singular for / W . when / W + $ Later we will construct as the quotient of under the action of a finite group + ( 47 4 / 8 X8 " .
(
$
is the Zariski closure of the image. If , then we have
R = I> 4 ( 8 ! . Since very ample, the ( ( give the map 47 M 8 is defined by 4 8 R+ 4 . 8
Example 2.4.6. Consider R lattice points 9 such that nates
$
R 354
R
has homogeneous coordi-
(8 ! $
Comparing this to Example 2.0.5, we see that the weighted projective space 4 .8 . Later we will learn the systematic reasonis why this is true.
§2.4. Properties of Projective Toric Varieties
89
$
The variety is not smooth. By working on the affine piece 4 8 is a singular point of $ . can check directly that
9 $
4
, one
We can also use Theorem 2.4.3 and the normal fan of , shown in Figure 11. ( Y $ One can check that the cones 6 and 6 are smooth, but 6 is not, so that is
v2 ΣP
P
v0
σ0
σ1
σ2
v1
Figure 11. The polygon giving
and its normal fan
not a smooth fan. In terms of , note that the vectors to the first lattice from points along the edges containing do not generate . Either way, Theorem 2.4.3 $ is not smooth. implies that
If you look carefully, you will see that 6 is the only nonsmooth cone of the Y $ . Once we study the correspondence between cones and orbits in normal fan Chapter 3, we will see that the non-smooth cone 6 corresponds to the singular 4
8 $ point of .
Products of Projective Toric Varieties. Our final task+ is to understand the toric 4 8 be lattice polytopes with variety of a product of polytopes. Let ' ! ; ' @
' R ' / ( for R . This gives a lattice polytope (X ! 4 ; (% ; 8 @ of / . dimension /
(
Replacing and with suitable multiples, we can assume that are very ample. This gives projective embeddings (
(
and
" " ' R ' 9-; ' is a subvariety of " " ( " " ( . Using so that by Proposition 2.0.4, the Segre embedding " " ( " " ( " " ( R (
$
$
we get an embedding (2.4.1)
$
$
- " " ( $
We can understand this projective variety as follows.
Chapter 2. Projective Toric Varieties
90
Theorem 2.4.7. If
(
and
are very ample, then
( (a) ! 4 ; ; 8 @ is a very ample polytope with lattice points 4 ( 8 9 4 ; ( ; 8 R 4 ( 9-; ( 8 54 9 ; 8 4 ( 8 9 4 ; ( ; 8 . Thus the integer defined above is R $ T $ " " ( coming from the very ample (b) The image of the embedding ( polytope equals the image of (2.4.1). $ $ $ $. (c)
(
Proof. For part (a), the assertions about lattice points are clear. The vertices of ( 2 4 ( 8 where ' is a vertex of ' (Exercise 2.4.4). consist of ordered pairs Given such a vertex, we have
4 H(
4 ( 8 R &4
8 9 4 ; ( ; 8 R 4 M( 9-; ( R ( 8X 54 9-; %R 8 ' 9 ; ' R ' 8 is saturated in ; ' . From Since ' is very ample, we know that ( here, it follows easily that is very ample.
5
$
5
For part (b), let 5 # 5 be the torus of . Since $ # it follows that is Zariski dense in combined with the Segre embedding, it follows that closure of the image of the map ( given by the characters Z and , ranges over the P 5 5 , the product ' coincides with the map
P ' P
'
'
R 5 5 1
$
"""
$
is Zariski dense in , (Exercise 2.4.4). When $ $ is the Zariski
(
(
(
, where , ranges over the elements of ; 5 8 5 9with elements of 9G; . When we identify P P P ' becomes the character , so that the above map
5 R "" (
5
(
coming from the product polytope part (c) is an immediate consequence.
( ! 4; ;
8@
. Part (b) follows, and
+ + Example 2.4.8. is the toric variety of the standard / -simplex , it + Since P + P is the toric variety of . ( follows that ( ( is the toric This also works for more than two factors. Thus Here is an obvious example.
variety of the cube pictured in Figure 8.
To have a complete theory of products, we need to know what happens to the normal fan. Here is the result, whose proof is left to the reader (Exercise 2.4.5). Proposition 2.4.9. Let
. Then
' ! 4; '8@ Y $
be full dimensional lattice polytopes for
R Y 0 1Y $
$
$
R
§2.4. Properties of Projective Toric Varieties
91
Here is an easy example. Example 2.4.10. The normal fan of an interval given by
6
(
%* - !
(
, where
*
in
, is
6
The corresponding toric variety is . The cartesian ( ( product of two such intervals by Theorem 2.4.7. If we set is a lattice rectangle whose toric variety is '6 R 6 ' 6 , then Proposition 2.4.9 gives the normal fan given in Figure 12.
σ10
σ00
σ11
σ01
Figure 12. The normal fan of a lattice rectangle giving
We will revisit this example in Chapter 3 when we construct toric varieties directly from fans.
Proposition 2.4.9 suggests a different way to think about the product. Let ' range over the vertices of ' . Then the 6 are the maximal cones in the normal fan Y$ , which implies that
R 3 R 3
(2.4.2) Thus
$
$
4
$
R
4
3 3 3 3
R
4
R
4
4
4
3
4
3
R $ $
In this sequence of equalities, the first follows from (2.4.2), the second is obvious, the third uses Exercise 1.3.12, the fourth uses Proposition 2.4.9, and the last follows 4 ( 8 ranges over all vertices of ( . since
This argument shows that we can construct cartesian products of varieties using affine open covers, which reduces to the cartesian product of affine varieties defined in Chapter 1. We will use this idea in Chapter 3 to define the cartesian product of abstract varieties.
Chapter 2. Projective Toric Varieties
92
Exercises for §2.4.
N( ! A `2 2 4 2 4 2 4 2 4 ' <[
( ( ( ( pictured 2.4.1. Show that the hexagon in Figure 6 and the trapezoid pictured in Figure 9 are smooth polygons. Also, of the polytopes shown in Figure 8, determine which ones are smooth. 2.4.2. Prove Proposition 2.4.4.
L(M A *E' #f nm27+6+6+626E' from Example 2.4.5. (a) Verify the facet presentation of given in the example. (b) What is the facet presentation of ? Hint: You know the vertices of . (c) Let - + ( A" &' + , where ( m27+6+7+62n and ( # # 8 # , and let .( - 8 - . Use the hint given in the text to prove y W ! $ E' )' . This shows that the index of in is A *E' , as claimed in the text. 2.4.4. Let +) +F' 4 be lattice polytopes with zW{}| + (* + for P(L12 ( . Also let + be the set of vertices of + . (a) Use supporting hyperplanes to prove that every element of 4 is a vertex of 4. (b) Prove that 4 ( ! F 4 ' and conclude that 4 is the set of vertices of 4. 2.4.5. The goal of this exercise is to prove Proposition 2.4.9. We know from Exercise 2.4.4 that the vertices of 4 are the ordered pairs - 2 - 4 ' where - + is a vertex of + . g . (a) Adapt the argument of part (a) of Theorem 2.4.7 to show that g [ (Ng [ ( . Taking duals, we see that the maximal cones of are 4 is a (b) Given rational polyhedral cones +) + ' and faces .+k + , prove that face of and that all faces of arise this way. 4 4 . (c) Prove that ( 8 with the property that 2.4.6. Consider positive integers N(9 9 4 + O for k(V 26+6+7+92n . Set B + ( E + for (Lm26+7+6+72n and let [ [ ( F `2 B 2&B 4 4 26+6+7+92&B 'P#- pm27+6+7+626&' + [ [ is reflexive and explain how it relates to Example 2.4.5. We will prove Prove that later that the toric variety of this polytope is the weighted projective space 27+6+6+62 ' . 2.4.7. The Sylvester sequence is defined by ( ( and (N 8 . It begins A fix a positive 4 integer A $ (`2-`2>2 -`271 >27+6+7+ and is described in [93, A000058]. Now and define 27+6+6+72 by ( (i and + ( (` #L&' + for ( (`26+7+6+92 . For ( - and this gives m27m2 D 2 and m27m26 (W2 ( `2 ( . Prove that 26+7+6+62 satisfies the conditions of Exercise 2.4.6 and hence gives a reflexive simplex, denoted in [75]. This paper proves that when , has the largest volume of all -dimensional reflexive 2.4.3. Consider the polytope
simplices and conjectures that it also has the largest number of lattice points.
Chapter 3
Normal Toric Varieties
§3.0. Background: Abstract Varieties The projective toric varieties studied in Chapter 2 are unions of Zariski open sets, each of which is an affine variety. We begin with a general construction of abstract varieties obtained by gluing together affine varieties in an analogous way. The resulting varieties will be abstract in the sense that they do not come with any given ambient affine or projective space. We will see that this is exactly the idea needed to construct a toric variety using the combinatorial data contained in a fan. Sheaf theory, while important for later chapters, will make only a modest appearance here. For a more general approach to the concept of abstract variety, we recommend standard books such as [26], [41] or [89].
;
B9C D 47EF8
R Regular Functions. Let be an affine variety. In §1.0, we defined V ; N R ; S 354 O 8 ! ; for O . E and showed that the Zariski open subset ;,N R B9C D.47ESN.8 , where ESN is the localization of E at O . The open sets ;)N form a ; 4 basis for the Zariski topology on in the sense that every open set is a (finite) 4 N U ; N for some L ! E (Exercise 3.0.1). union R ;
For an affine variety, a morphism is called a regular map, so that the ; ; coordinate ring of consists of all regular maps from to . We now define what ; it means to be regular on an open subset of .
;
Definition 3.0.1. Given an affine variety R 4 we say if for all ! is . regular , ; N 4 S E N . . Then define ! and
4 4S8 R
.
B9C D 47EF8
4
4
;
and a Zariski open ! E such that, , there exists O .
is regular
4
. 93
Chapter 3. Normal Toric Varieties
94
O 4 8 R P , and saying . EN means * O for some * . E and / W . that R X 4 4S8 Here are some cases where is easy to compute. B C ; D 7 4 F E 8 Proposition 3.0.2. Let R be an affine variety. 4<;=8 R E . (a) 4<; N.8 R EN E . (b) If O , then . E Proof. It is clear from Definition 3.0.1 that elements of define regular functions 4<;&8 4<;=8 ; ; on , hence elements of . Conversely, if + . , then for all . + . E 2 ; N S E N 6 * O . there is O such that . and R . The ideal R K O ; E 5 3 7 4 6 8 4 8 L R since O R P for all . ; . Hence the . ! satisfies 6 . : 4 5 3 7 4 9 6 * 8 8 E R R , so there exists a finite set L ! ; Nullstellensatz implies that L such that and polynomials for . +
R U O + U O R & U * . E , as desired. Hence R & 4 4 ; ! ; ;VN be4 Zariski open. For part (b), let Then is Zariski open in , and . E ; , ; N with coordinate ring satisfies ! , we have R whenever EN R 47ESN 8 N for all W . These observations easily imply that 4 4 8 R 4 4S8 (3.0.1) 4 ;,N gives Then setting R 4<; N 8 R 4<;,N 8 R ESN ;)N R B9C D.47ESN.8 . where the last equality follows by applying part (a) to ; BC D.47EF8 is an irreducible affine variety, we can describe Local Rings. When R The condition +
. ;2N
means that
%
regular functions using the local rings in §1.0. A rational func F4<;=8 is contained in the local ring %introduced tion in precisely when it is regular in a 4 ; is open, we have neighborhood of . It follows that whenever !
4
R 4 4S8
;
Thus regular functions on are rational functions on that are defined every4 4 ; , Proposition 3.0.2 implies where on . In particular, when R that (3.0.2)
R 4<;&8 R E R F%>; -<
§3.0. Background: Abstract Varieties
95
The Structure Sheaf . Given an affine variety 4
4 4S8
4
;
, the operation
! ;
open
has the following useful properties:
When
!
4
If
4
4 8 R R
defined by
, Definition 3.0.1 shows that there is an obvious restriction map
4 S8 4 8 4
. It follows that whenever ! ; ! 4 .
is an open cover of
R+ 4 4 8 R
!
4
;
is the identity map and that
, then the sequence
4 4 8 RR
44
9
4
8
is exact. Here, the second arrow and the is defined by the restrictions 44 8 double arrow is defined by and . Exactness at means that regular functions are determined locally (that is, two regular func4 4 tions on are equal if their restrictions to all are equal), and exactness at 4 4 8 4 means that regular functions on the agreeing on the overlaps 4 9 4 patch together to give a regular function on 4 .
sheaf of In the language of sheaf theory, these properties imply that ; 4<; 8 a ringedis aspace algebras, called the structure sheaf of . We call over . 4 ) ; N Also, since (3.0.1) holds for all open sets ! , we write
R 4<;N 8 R 4<; N9 8 . In terms of ringed spaces, this means
; ( ;
Morphisms. By §1.0, a polynomial mapping between affine varieties A > %
- ; A%>;( M corresponds to the -algebra homomorphism by 4M 98 R for . A%>; - . We now extend this to open sets of affinedefined varieties.
Definition 3.0.3. Let
. A function 4 (
(G 4
4
' ; ' Zariski open subsets of affine varieties for R defines a map ! 4 isbea morphism if M 4 4 8XR+ 4 4H( 8
4
Thus is a morphism if composing with regular functions on 4( gives regular functions on . Note also that M is a -algebra homomorphism since it comes from composition of functions. 4
; ( ;
Example 3.0.4. Suppose that a morphism according to DefiniBC D47E ' 8 , then the above ismap ; M gives the -algebra homotion 3.0.3. If ' R morphism
E R 4<; 8 R 4<;V( 8 R EA(
Chapter 3. Normal Toric Varieties
96
E
EA(
By Chapter 1, the -algebra homomorphism gives a map of affine ; ; ( . In Exercise 3.0.3 you will show that this varieties is the original map ;( ; we started with.
Example 3.0.4 shows that when we apply Definition 3.0.3 to maps between affine varieties, we get the same morphisms as in Chapter 1. In Exercise 3.0.3 you will verify the following properties of morphisms: If
4
is open in an affine variety
4 4S8 R
Hence regular functions on
;
, then
4
4
is a morphism
are just morphisms from
A composition of morphisms is a morphism. An inclusion of open sets
! 4
of an affine variety
;
4
.
to .
These observations enable us to understand what it means for 4 ; ' are arbitrary open sets. Let ; ' R BC D 47E a morphism when ' ! 4 ( R N U 4<; ( 8 N for L ! E ( finite. Then 4<; ( 8 N ! 4 ( and 4 ! morphism
4<;( 8 N ! H4 ( R
4
X(
is a morphism.
! ; E 74 EA( 8 N
4
;
'8
4
be and write gives the
4<;( 8 N
This is given by a -algebra homomorphism . Since the cover 4 ( , we see that locally, morphisms look like the polynomial maps introduced in Chapter 1. In particular, morphisms are continuous in the Zariski topology.
(
We say that a morphism ( 4 " its inverse function
4
H4 (
4
is an isomorphism if is also a morphism.
is bijective and
Gluing Together Affine Varieties. We now are ready to define abstract varieties by+ gluing together open subsets+ of affine varieties. The model is what happens for . Recall from §2.0 of that is covered by open sets
+ 4 ' R S 354 ' ' 8 R B9C D / . Each 4 ' is a copy of + that uses a different set of variables. For for R XR P , we “glue together” these copies as follows. We have open subsets 4 4 ' 8 ! 4 ' and 4 4 8 ! 4 (3.0.3)
' 4
and we also have the isomorphism
(3.0.4)
M '
4
' 8 R 4 4 8
+
since both give the same open set ' 9 in . The notation ' 4 '8 ' 4 ' . ' that means since the index is closest to , hence ' the level of coordinate comes from the isomorphism rings,
4
4
' was chosen so ' 4 'V8 . . At
4
§3.0. Background: Abstract Varieties
97
"
( R and + 4 ' We can turn this around and start from the affine varieties given above and glue together the open sets in (3.0.3) ( using the isomorphisms ' from (3.0.4). " ' and ' R ' wherever This gluing is consistent since ' R all three +
defined by
R+
4 R P 8
; To generalize this, suppose we have a finite collection of affine varieties "M ; ! ; and isomorphisms and for all pairs we have Zariski open sets ; ; " ( satisfying "Hthe following compatibility conditions: 4
.
%
%
the second shows that it is transitive. Hence is an equivalence relation and we with the quotient topology. For each " , let can form the quotient space R
4
Then ;
!
4
4
R %* - .
* . ; . 4 * 8 R %* -
is an open set and the map defines a homeomorphism ! . Thus locally looks like an affine variety.
Definition 3.0.5. We call
the abstract variety determined by the above data.
An abstract variety comes equipped with the Zariski topology whose open 4 sets are those sets that restrict to open sets in each Zariski closed subsets ! are called subvarieties of . We say that . The is irreducible if it is not the union of two proper subvarieties. One can show that is a finite union of T ( R " such that ' ! P for =R P . We call irreducible subvarieties the ' the irreducible components of .
Here are some examples of Definition + 3.0.5.
Example 3.0.6. We saw above that can be obtained by gluing together the+ open sets (3.0.3) using the isomorphisms ' from (3.0.4). This shows that + 4 is an abstract variety with affine open subset ' ! . More generally, given a + ; ; ; 4 projective variety ! , we can cover with affine open subset 9 ' , and the gluing implicit in equation (2.0.8). We conclude that projective varieties are also abstract varieties. P +
P 3.0.7. + P In a similar way, Example
can be viewed + P as gluing affine spaces P along suitable open subsets. Thus is an abstract variety, + ; and the same is true for subvarieties ! . 4
'
U
Chapter 3. Normal Toric Varieties
98
Example 3.0.8. Let with
; R
R BC D 47A% - 8
and
;( R
R BC D.47A% A - 8 ,
;V( R ; S 354 8 R B C D 47A% - 8 ; ( R ;V( S 5 3 4 8 R BC D 47A% A - 8
and gluing data
( ; ( ; ( coming from the -algebra homomorphism M ( A% A - % - defined by and F
and
( ; ( ; ( coming from the -algebra homomorphism M ( A% - ( F% A - defined by and ( R (" One checks that , and the other compatibility condition is satisfied since
; ' . It follows that we get an abstract variety . has another description. Consider the product ( 5 with The variety 4 '!*')( 8 on ( and coordinates( 4 ' 8 on . We will homogeneous coordinates with the subvariety R 35 4 !' R ' ( '8 ! , called the blowup identify 47 8 . First note that ( is covered by of at the origin, and denoted 4 R BC D 47A% ' ( ' *' - 8 and 4 ( R BC D 47A% ' ' (*' - 8 R 9 4 4 8 and ( R 9 4 4 ( 8 . Also, Then is covered by R 354 R 4 ')( ' 8 'V8 ! 4
there are only two
which gives the coordinate ring
A% ' ( ' *' - K R 4 ' ( ' 8 ' L &% ' * ' ( ' - via 4 ' ( ' 8 ' ( 3 5 4 ' R 4 ' ' ( 8 8 4 ( R ! Similarly, has coordinate ring &% ' ')(*' - K '-R 4 ' ')( 8 L A % *' ' ( - via ' 4 ' )' ( 8 ; You can check that these are glued together in in exactly the same way ;( are glued together in . We will generalize this example in Exercise 3.0.8. and
The Structure Sheaf of an Abstract( Variety. Let be an an open subset of an R " 4 4 9 4 8 ! ; . Then a function 4 abstract variety and set is regular if R
"
4
is regular for all . The compatibility conditions ensure that this is well-defined, so that one can define
" 4 8 R 4
4
is regular
.
§3.0. Background: Abstract Varieties
"
#"
99
#"
This gives the structure sheaf . Thus an abstract variety is really a ringed 4 8 with a finite open ofcovering 4 such that 4 4 8 is isospace
4<; 8 of the affine variety ; . (We leave the morphic to the ringed space definition of isomorphism of ringed spaces to the reader.)
4
4
Given an abstract variety and an open subset , we note that has a natural 4 , 4 9 4 is open ! structure of an abstract variety. For an affine open subset 4 4 4 4 4 8 N for a finite subset R NU in and hence can be written as a union 9 L ! F% 4 - . It follows that 4 is covered by finitely many affine open subsets and thus is an abstract The structure sheaf is simply the restriction of R variety. 4 to , i.e., .
"
"
also gives an abstract variety. For In a similar way, a closed subset !4 4 as above, 9 is closed in and hence is an affine variety. Thus is! covered by finitely many affine open subsets and thus is an abstract variety. This justifies the term “subvariety” for closed subsets of an abstract variety. The is related to as follows. Let inclusion structure sheaf 4 4S8 R 4 4 be 8 the and let M be the sheaf on defined by M 9 . Restricting M functions on to functions on gives a map of sheaves whose kernel is the subsheaf of functions vanishing on , meaning ! 4
"
"
"
4 4S8 R O . " 4 4S8 O 4 8 R
for all
4 . . 9 F
In the language of Chapter 6, we have an exact sequence of sheaves
R
R
" R+ M R
All of the types of “variety” introduced so far can be subsumed under the concept of “abstract variety.” From now on, we will usually be thinking of abstract varieties. Hence we will usually say “variety” rather than “abstract variety.”
;
Local Rings and Rational Functions. Let be a point of an affine variety . X Elements of the local ring are quotients O in a suitable localization with O . A%>;S- and V4 8 R P . It follows ; ; that is a neighborhood of in and O % ; is a regular function on . In this way, we can think of elements of as regular functions defined in a neighborhood of .
This idea extends to the abstract case. Given a point of an variety and 4 ( 4 4 ' are equivaneighborhoods of , we say that regular functions O ' ( O , if there is a neighbhorhood . 4 ! 4( 9 4 such that lent at , written O (O R O .
%
Definition 3.0.9. Let
" R
O is the local ring of at
be a point of a variety
4
.
4
. Then
is a neighborhood of
in
%
Chapter 3. Normal Toric Varieties
100
. "
#" Every
has a well-defined value is a local ring with unique maximal ideal
The local ring
"
4 8.
It is not difficult to see that
" R . " 4 8 R .
can also be defined as the direct limit
over all neighborhoods of
in
" R ! " #
" 4 8 4
(see Definition 6.0.1).
A4 8
When is irreducible, we can also define the field of rational functions 4 defined on a nonempty. A rational function on is a regular function O 4 ! , and two rational functions on are equivalent if they Zariski open set agree on a nonempty Zariski open subset. In Exercise 3.0.4 you will show that this relation is an equivalence relation and that the set of equivalence classes is a field, A4 8 . called the function field of , denoted
;
is an affine variety, the definition of a smooth point Smooth Varieties. When . ; (Definition 1.0.7) used 4<;&8 , the Zariski tangent space of ; at , and
; ; , the maximum dimension of an irreducible component of containing . 4 8
! You will show in Exercise 3.0.2 that and are well-defined for a of a general variety. point .
Definition 3.0.10. Let be a variety. A point . is smooth if
, and is smooth if every point of is smooth.
Morphisms of Varieties. Given varieties 4 4 Z and R , a function in the Zariski topology and the restrictions 4 9
! 4 8 R
R affine open covers and is awith morphism if it is continuous
" ( 4 4 Z 8 R 4
Z
are morphisms in the sense of Definition 3.0.3. One can check that this definition is independent of the affine open covers. Also, a function is a morphism 4 8 if and only if it gives an element of , i.e., is a regular function.
"
Normal Varieties. We return to the notion of normality introduced in Chapter 1.
"
Definition 3.0.11. An variety rings are normal for all
is called normal if it is irreducible and the local . .
At first glance, this appears to be different from the definition given for affine varieties in Definition 1.0.3. In fact the two notions are equivalent in the affine case.
;
Proposition 3.0.12. Let be an irreducible affine variety. Then ; . are normal for all . and only if the local rings
F%>; -
is normal if
§3.0. Background: Abstract Varieties
101
F%>; -
Proof. If is normal for all , then (3.0.2) shows that is an intersection of normal domains, all of which have the same field of fractions. Since such an F%>; - is normal. intersection is normal by Exercise 1.0.7, it follows that For the converse, suppose that (
F%>; -
4<;&8 . A * ' .
is normal and let
satisfy
' 8 R P . The product O R O ( O has * ' with O Write ' R &%>;- N and O 4 8 R P . The localization &%>;-7N is normal by * the properties that ' . % 4 8 R P . Hence " . &%>;-7N ! . since O Exercise 1.0.7 and is contained in
"
* (" " * R ' O ' . A%>;S- and O ' 4
"
This completes the proof. Here is a consequence of Proposition 3.0.12 and Definition 3.0.11.
Proposition 3.0.13. Let ; affine open sets . Then
be an irreducible variety with a cover consisting of ; is normal. is normal if and only if each
Products of Varieties. As another example of abstract varieties and gluing, we ( V of varieties ( and also has the structure indicate why the product of a variety. In §1.0 we constructed the product of affine varieties. From here, it ( is relatively routine to see that if is obtained by gluing together affine varieties 4 is obtained by gluing together 4 Z ( is obtained by and affines , then 4 4 Z ( - in the corresponding fashion. Furthermore, gluing together the has the correct universal mapping property. Namely, given a diagram
$ ( %
( #
/
' are morphisms, there is a unique morphism where ' (the dotted arrow) that makes the diagram commute. ( (
=
$
R Example 3.0.14. Let us construct the product ; R BC D.47F% - 8 and ;( R BC D 47A% - 8 , with the gluing. Write given by
( Then
A% -
is constructed from 4
A%
-
(
; ;V(
where
R BC D.47A% - &&% ' - 8
H( R BC D.47A% - 1A&% ' - 8 4
with gluing given by
44 8 44 ( 8
corresponding to the obvious isomorphism of coordinate rings.
Chapter 3. Normal Toric Varieties
102
Separated Varieties. From the point of view of the classical topology, arbitrary gluings can lead to varieties with some strange properties.
( Example 3.0.15. In Example 3.0.14 we saw how to construct affine vari; R BC D47&% - 8 and ;( R BC D.47F% - 8 with thefrom gluing given by eties on open sets M 4<; 8 ! ; and M 4<; ( 8 ! ; ( . This expresses ( as consisting of M plus two additional points. But now consider the abstract
variety arising from the gluing map
4<; 8 R+ 4<; ( 8
that corresponds to the map of -algebras defined by . As before, the glued M variety consists of together with two additional points. However here we ( 4* 8 " * M . have a morphism whose fiber over contains one point, ( corresponding ; and but whose fiber over consists of two points, to . ;)( . If 4H( 4 are classical open sets in with ( . 4 ( and corresponding to . . 4 4 ( 4 R , then 9 P . So the classical topology on is not Hausdorff.
Since varieties are rarely Hausdorff in the Zariski topology (Exercise 3.0.5), we need a different way to think about Example 3.0.15. Consider the product V and the diagonal mapping V defined by 4 8 R 4 8 for . . For from Example 3.0.15, there is a morphism whose 4 8 . Any Zariski closed subset of fiber over over consists of the four points ' containing one of these four points must contain all of them. The image of 4 ( ( 8 and 4 8 , but not the other two, so the the diagonal mapping contains diagonal is not Zariski closed. This example motivates the following definition.
Definition 3.0.16. We say a variety is separated if the image of the diagonal - - map is Zariski closed in .
+ is separated because the image of the diagonal in + + R B C D 47A% ' ( *' +, ( +.- 8 is the affine variety 354 ' (GR (*' + R +8 . For instance,
Similarly any affine variety is separated. The connection between failure of separatedness and failure of the Hausdorff property in the classical topology seen in Example 3.0.15 is a general phenomenon. Theorem 3.0.17. A variety is separated if and only if it is Hausdorff in the classical topology. Here are some additional properties of separated varieties (Exercise 3.0.6).
are morphisms, then . O 4 8 R 24 8 is a subvariety (a) If O of . 4 ; 4 ; is also affine. are affine open subsets of , then 9 (b) If
Proposition 3.0.18. Let
be a separated variety.
§3.0. Background: Abstract Varieties
103
The requirement that be separated is often included in the definition of an abstract variety. When this is done, what we have called a variety is sometimes called a pre-variety. Fibered Products. Finally in this section, we will discuss fibered products of varieties, a construction required for the discussion of proper morphisms in §3.4. First, L and L , then the fibered product if we have mappings of sets O HU is defined to be
U R 4' 8 .
(3.0.5)
O 4 '8 R 2 4 8.
The fibered product construction gives a very flexible language for describing ordinary products, intersections of subsets, fibers of mappings, the set where two mappings agree and so forth:
L
U
. L
U R 9 If are subsets of and O are the inclusions, then ( . ! L , then U R O " 4 .8 . If R
If
is a point, then
is the ordinary product
.
The third property is the reason for the name. All are easy exercises that we leave for the reader. In analogy with the universal mapping property of the product discussed above, the fibered product has the following universal property. Whenever we have map( and such that O ( R , there is a unique pings HU that makes the following diagram commute.
$
$ HU %
UG
" /
/
N L
U
Equation (3.0.5) defines as a set. To prove that is a variety, L L L we assume for simplicity that is separated. Then O and 4 98 L L , and one easily checks that give a morphism O (
4
LV L
HU R 4 O 8 " 4 4
LV L
L
where This is closed in is separated, ! isUAthe diagonal. V and hence hassince and it follows that is closed in a natural structure UG as a variety. From here, it is straightforward to show that has the desired universal mapping property.
UN
L
Proving that is a variety when is not separated takes more work. The R BC D 47E ( 8 , R B9C D 47E 8 , and idea is to begin with the affine case where L R BC D47EF8 . Then HU exists as an affine variety by the previous paragraph
Chapter 3. Normal Toric Varieties
104
L
since is affine and hence separated. When is constructed by a gluing procedure.
H L
are general varieties,
XU
XU . Let R also describe the coordinate ring of BC Dwe BC D In47EAthe( 8 ,affine R case, 47E can 8 , and L R BC D47EF8 . The morphisms O correspond E EA( M E E . Hence both E&(JE have theto ring homomorphisms O M E E ( E . This is also structure of -modules, and we have the tensor product a finitely generated -algebra, though it may have nilpotents (Exercise 3.0.9). To
get a coordinate ring, we need to take the quotient by the ideal ? Then one can prove that
HU R BC D 47EA(/ E
? 8
of all nilpotents.
U
We can avoid worrying about nilpotents by constructing as the affine B C ( 8 D 7 4 E E scheme . Interested readers can learn about the construction of fiber products as schemes in [26, I.3.1] and [41, pp. 87–89]. Exercises for §3.0.
GM(<>=?7@1 A'
C
3.0.1. Let be an affine variety. (a) Show that every ideal can be written in the form + . (This is the Hilbert Basis Theorem in .)
,
G
*( ,
(b) Let be a subvariety. Show that the complement of in a union of a finite collection of open affine sets of the form .
G
G;
G
27+6+6+623, ! , where can be written as
(c) Deduce that every open cover of (in the Zariski topology) has a finite subcover. (This says that affine varieties are quasicompact in the Zariski topology.)
K Wz {}| ~ \ 'I( G r %G (
zW{}| \
3.0.2. As in the affine case, we want to say a variety is smooth at if . In this exercise, you will show that this is a well-defined notion. , then the (a) Show that if is in the intersection of two affine open sets Zariski tangent spaces and are isomorphic as vector spaces over .
K* ~ Y>[ \ ~ Y [ \ (b) Show that zW{}| \ is a well-defined integer.
K
(c) Deduce that the proposed notion of smoothness at is well-defined.
K% G and set _ \ ( /&,%% O RK' .
3.0.3. This exercise explores some properties of the morphisms defined in Definition 3.0.3.
' _ \'
(a) Prove the claim made in Example 3.0.4. Hint: Take . Then describe in terms of
, RKS')(* `8
(b) Prove the properties of morphisms listed on page 96.
,2n
3.0.4. Let
be an irreducible abstract variety.
, - if , O (- O
M (b) Show that the set of equivalence classes of the relation in part (a) is a field. (c) Show that if M is a nonempty affine open subset of , then I ' I ' . (a) Let be rational functions on . Show that open set is an equivalence relation.
for some nonempty
3.0.5. Show that a variety is Hausdorff in the Zariski topology if and only if it consists of finitely many points.
§3.0. Background: Abstract Varieties
105
3.0.6. Consider Proposition 3.0.18. (a) Prove part (a) of the proposition. Hint: that if Show first by , then ' .
6 h
is defined c ')(M , c ' 2nS c 'n' *( 'n' (b) Prove (b) of the proposition. Hint: Show first that . G can be identified with ' part G$'^ . 6eG h G G 3.0.7. Let G ( <W=D?E@m A' be an affine variety. The diagonal mapping corresponds to a -algebra homomorphism h . Which one? Hint: Consider the universal mapping property of G G. $ , the blowup of 3.0.8. In this exercise, we will study an important variety in F 4 ' from Example 3.0.8. Write the at the origin, denoted F ' . This generalizes d d as 26+7+6+92 , and the affine coordinates on as homogeneous coordinates on c 26+7+6+92 c . Let ( A ')( d + c # d c +)O> fe'^ + (3.0.6) : Let 1+ , (N126+7+6+92 , be the standard affine opens in + ( d + ' 2 k (M 1 26+7+6 +92 . (note the slightly non-standard indexing). So the + form a cover of (a) Show that for each P(Mm26+7+6+92 , + ( + ' d <W=D?E@ d + 26+7+6+62 dd ++ 4 2 d d+ + 26+7+6+72 d d + 2 c + using the equations (3.0.6) defining
d ' and % d + ' . 3.0.9. Let G ( c 4$# d ' 4 and consider the morphism 6G h given by projection onto the d -axis. We will study the fibers of . A ' ( / F `23 1' 8 can be represented as the fibered (a) As noted in the text, the fiber d d .
(b) Give the gluing data for identifying the subsets
+
/E `8 G . In terms of coordinate rings, we have /E `8 ( <>=D?E@m FIH J! ' , *(C<W=D?E@m A H d J ' and GM(C<W=D?E@m A H d 2 c J! c 4# d . Prove that IH d J! d H d 2 c JA c 4 # d IH c J! c 4 + Thus, the coordinate rings IH d JA d , IH d J and H d 2 c JA c 4 # d lead to a tensor product that has nilpotents and hence cannot be a coordinate ring. T in , then W')(N/> 2 W' 8 . Show that the analogous tensor product is (b) If (* H d J! d # H d 2 c JA c 4 # d H c JA c 4 # H c JA c # IH c JA c + W' . This has no nilpotents and hence is the coordinate ring of What happens in part (a) is that the two square roots coincide, so that we get only one point with “multiplicity ( .” The multiplicity information is recorded in the affine scheme <>=?7@1 AIH c J! c 4 ' . This is an example of the power of schemes. product
Chapter 3. Normal Toric Varieties
106
§3.1. Fans and Normal Toric Varieties
Y
6
In this section we construct the toric variety corresponding to a fan . We will 6 to many of the examples encountered previously, and also relate the varieties we will see how properties of the fan correspond to properties such as smoothness 6 . and compactness of The Toric Variety Associated to a Fan. A toric variety continues to mean the same thing as in Chapters 1 and 2, although we now allow abstract varieties as in §3.0.
Definition + 3.1.1. A toric variety is an irreducible variety a torus 5 47 M 8 as a Zariski open subset such that the action of 5 containing on itself extends 5 on . to an action of The other ingredient in this section is a fan in the vector space
Y A fan
? @
.
in ? @ is a finite collection of cones 6 such that: Y . (a) Every 6 is a strongly convex rational polyhedral cone. Y Y . (b) For all 6 , each face of 6 is also in . ( Y , the intersection 6 ( 9 6 is a face of each (hence also in Y (c) For all 6 6 .
Definition 3.1.2.
).
It is also possible to consider infinite collections of cones with these properties, but we will not do so in this book. We have already seen some examples of fans. Theorem 2.3.2 shows that the Y $ of a full dimensional lattice polytope !; @ is a fan in the sense normal fan of Definition 3.1.2. However, there exist fans that are not equal to the normal fan of any lattice polytope. An example of such a fan will be given in Example 4.2.13. We now show how the cones in any fan give the combinatorial data necessary to glue a collection of affine toric varieties together to yield an abstract toric variety. Y By Theorem 1.2.18, each cone 6 in gives the affine toric variety
3 R BC D.47A% 2 3 - 8 R BC D 47A% 7 6 9; - 8 Moreover, if H is a face of 6 , then by Definition 1.2.5, there is , . 6 7 such that H R 6 9 O P , where O P R . ? @ K , L R is the hyperplane defined by , . In Chapter 1, we proved two useful facts: O P is the corresponding face of 6 , then First, if , . 6 7 9 ; and H R 6-9 4
Proposition 1.3.16 implies that
2 R 2 3
4FR , 8 % 2 - is the localization A% 2+3 - . Hence 4 R 4 4 3 8 F This shows that ( Second, if H R 6 9 6 , then Lemma 1.2.13 implies that (3.1.2) 6 ( 9 O P R H R 6 9 O PF (3.1.1)
.
§3.1. Fans and Normal Toric Varieties where ,. (3.1.3)
107
67( 9 F4 R 6 8 A 7 9-; . This shows that 4 3 4 4 3 8 R 4 R 4 4 3 8
!
4
3
These facts were also used to study projective toric varieties in Chapter 2. The 2 3 and their semigroup following proposition gives an additional property of the rings that we will need.
(6 . Y
Proposition 3.1.3. If 6
2 3
6 (7 R 4FR 4 6 ( 98 6 8 7 , . 67 9 4FR , 8 for 6 some79 ; . 203 203 that . . 6 (7
Proof. The inclusion
R 6 ( 9 6
and H
2 R 2 3
23
, then
2 3 2 follows directly from the general fact that ! R H 7 . For the reverse inclusion, take . 2 and let ( (3.1.2). Then (3.1.1) applied to 6 gives R 20satisfy 3 . But R , . 6 7 implies R ,. 203 , so and .
This result is sometimes called the Separation Lemma and is a key ingredient 6 are separated using Definition 3.0.16. in showing that the toric varieties
(
Example 3.1.4. Let 6 R = I> 4 ( ( 8 = inI> ? 4@ ( R dual cones 6N7 in Figure 1.
= I> 4 ( 8 1.2.11), and let 6 R ( (as in Exercise = > I 4 ( 8 H R 6 9A6 R R . We show the (4R ( . Then 8 8 R = I> 4 (+R H 7 R 6(7 6 7 67 ,
, and
τ σ1
σ1 τ
σ2
σ2
Figure 1. The cones
The darkest shaded region on the right is 6 6 9 O " P , where , R R ( Z . 6(7 and R , with , . 6 equals the set of all sums , 2 R 2 3 2 3 .
and their duals
( O P R (7 have H R 6 9 9 6 7 .( We 2 R , R Z . 67 . Note that (7 9 ; and , . 6 7 9 ; . Hence
Chapter 3. Normal Toric Varieties
108
4
6 H
BC D 47A% 2 3 - 8
3
R Now consider the collection of affine toric varieties , where ( Y runs over all cones in a fan . Let 6 and 6 be any two of these cones and let R 6 ( 9 6 . By (3.1.3), we have an isomorphism
4
3 3
4 38 4
44 38
which is the identity on . By Exercise 3.1.1, the compatibility conditions as in 4 3 4 4 3 8 are satisfied. §3.0 for gluing the affine varieties 6 along the subvarieties Y associated to the fan . Hence we obtain an abstract variety Theorem 3.1.5. Let variety.
Y
be a fan in ?
Y
@
. The variety
6
is a normal separated toric
Y
. . Proof. Since each cone in is strongly convex, 5 R BC D.47A% - 8 47 M 8 + 4 3 ! ? is a face of all 6 Hence we have for all 6 . These tori are all ; 5 6 ! identified by the gluing, so we have ! . We know from Chapter 1 that each 4 3 3 3 5 has an action of reduces to the identity 203 3 - . The gluing isomorphism & % mapping on . Hence the actions are compatible on the intersections of every pair of sets in the open affine cover, and patch together to give an action of 5 on 6 .
6
4
3
The variety is irreducible because all of the irreducible affine toric 5 . Furthermore, 4 3 is aarenormal varieties containing the torus affine variety by 6 is normal by Proposition 3.0.13. Theorem 1.3.5. Hence the variety
6
To see that is separated it will suffice to show that for each pair of cones 6 ( 6 in Y , the image of the diagonal map 4 4 3 4 3 H R ( 6 9-6 is Zariski closed (Exercise 3.1.2). But comes from the -algebra homomor-
M A% 2 3 - 1=F% 2 3 -8R+ &% 2 + ' ' ' P U + defined by . By Proposition 3.1.3, M is surjective, so that A% 2 - 74 A% 203 - &&% 2 3 - 8 4 M 8 ; 3 ; 3. Hence the image of is a Zariski closed subset of phism
P
6
Toric varieties were originally known as torus embeddings, and the variety 5 4 Y 8 in older references such as [76]. Other commonly would be written 4 Y 8 , or Q4 8 , if the fan is denoted by . When we want to used notations are 6 as 6 5 . emphasize the dependence on the lattice ? , we will write
Y
Every normal, separated toric variety is obtained as for some fan in ?G@ . 6 for For example, Theorem 1.3.5 implies that every normal affine toric variety is the fan consisting of a single cone 6 together with all of its faces. The construction of the projective toric variety associated to a lattice polytope in Chapter 2 shows 6 for suitable fans Y . that those varieties are also 6
§3.1. Fans and Normal Toric Varieties
Proposition 3.1.6. Let the projective toric variety
6
$
109
+ ! ; @ be an / -dimensional lattice polytope. Then Y $ is the normal fan of . , where
Proof. When is very ample, this follows immediately from the description of $ in Proposition 2.3.12 and the the intersections of the affine open pieces of Y $ definition of the normal fan . The general case follows since the normal fans of and are the same for all positive integers . The general statement is a consequence of a theorem of Sumihiro from [98].
5
Theorem 3.1.7 (Sumihiro). Let the torus act on a normal separated variety has a 5 -invariant affine open neighborhood. Then every point .
.
Corollary 3.1.8. Let be a normal separated toric variety containing the torus 5 as an affine open subset. Then there exists a fan Y in ? @ such that 6 . Proof. The proof will be sketched in Exercise 3.2.10 after we have developed some 5 -orbits on toric varieties. of the properties of Additional Examples. We now turn to some concrete examples. Many of these are toric varieties already encountered in previous chapters.
Y
Example 3.1.9. Consider the fan in ?A@ R in Figure 2, where ? R has ( standard basis . This is the same as the normal fan of each of the simplices R as in Example 2.3.9. Here we show all points in the cones inside a rectangular viewing box (all figures of fans in the plane in this chapter will be drawn using the same convention.)
σ1
σ0
σ2
Figure 2. The fan
for
From the discussion in Chapter 2, we expect , and we will show Y R = I> 4 ( 8 , this in detail. The fan has three two-dimensional cones 6 = = 6 ( R I> 4FR ( R 8 , and 6 R I> 4 (4R (R 8 , together with the three 6
Chapter 3. Normal Toric Varieties
110
rays H ' R 6 ' 9 6 for R P , and the origin. We see that the toric variety covered by the affine opens
3 R B C D47&% 2 3 - 8 BC D 47A% ' 3 R BC D47&% 2 3 - 8 B C D 47A% ' 3 R BC D47&% 2 3 - 8 B C D 47A% '
4 4 4
6
is
-8 " ( *' " ( - 8 " ( " (-8
%' - & % ' " ( * ' " ( - M( F % ' - & %' " ( " ( - M &
Moreover, by Proposition 3.1.3, the gluing data on the coordinate rings is given by
( ( - &% ' *' " ( - M ( A% ' " *' "
It is easy to see that if we use the usual homogeneous coordinates ' and identifies the standard affine open , then 6 6 4 3 ! . Hence we have recovered as the toric variety .
+ Example 3.1.10. Generalizing Example 3.1.9, let ? @ R ( +
standard basis
. Set
Y
R R ( R % R
4 '*')(*' 8 4
, where
' !
on with
? R
+
has
R +
and let be the fan in ?G@ consisting of the cones generated by all proper subsets + + +
6 . This is the normal fan of the / -simplex of and by Example 2.3.14 and Exercise 2.3.6. (You + will check the details to verify that this gives the usual affine open cover of in Exercise 3.1.3).
To relate this example to the discussion in Chapter 2, note that we have differ+ + ent embeddings of according to which is used to construct the Veronese + Y mapping. However, is the normal fan of for all , and the varieties obtained this way for different are isomorphic.
Example 3.1.11. We classify all -dimensional normal toric varieties as follows. . The only cones are the intervals 6 R % 8 We may assume ? R and ? @ R ( 4FR - and the trivial cone H R . It follows that there are only four and 6 R
possible fans, which gives the following list of toric varieties:
H . which gives M
6 H and 6 ( H . both of( which give
6 6 ( H . (which gives
Here is a picture of the fan for
:
6 ( This is the fan of Example 3.1.10 when /
R .
6
§3.1. Fans and Normal Toric Varieties
(
111
+
4 (8
P
4 8
? with ? @ Y andY ( ? Y@ Example 3.1.12. Let ? R ? . Let Y ' be the fan in 4 ? ' 8 @ as in Example R 3.1.10. Then is a fan in ? , consisting of all cones 6 ( 6 , where+ 6 ' isP in Y ' . By Example 2.4.8, the corresponding toric variety is the product .
σ10
σ00
σ11
σ01
Figure 3. A fan
with
Y ! Z
BC B C B C B C
D.47A% 203 - 8 D.47A% 2 3 - 8 D.47A% 2 3 - 8 D.47A% 203 - 8
? @ pictured in Figure 3. as above. Then
When / R , R , we obtain the fan 6 Label the 2-dimensional cones 6 ' R 6 '
F% ' -
( F% ' " -
( ( F% ' " " ( F% ' " -<
( 3 We see that if and are the standard affine open sets in ( , then ( and it is easy to check that the gluing makes . 4
4
(
4
6
(
4
'
4
Y (
Example 3.1.13. Let ? R ? in ? ' as in the previous example. = I> 4 (Let S P 8 4 ? ( 8 @ be as above, but let Y be? the, with Y R fanYSconsisting GP Y of the cone?G@ ( together with all its faces. Then is a fan in + ( and the the cor 6 responding toric variety is . The case was studied in Example 3.0.14.
Examples 3.1.12 and 3.1.13 are special cases of the following general construction, whose proof will be left to the reader (Exercise 3.1.4). Proposition 3.1.14. Suppose we have fans is a fan in ?
(% ?
Y ( Y
and
6
Y (
R 6 ( 6
in
4? ( 8@
6 ' . Y '
- 6
6
6
and
Y
in
4 ? 8 @
. Then
Chapter 3. Normal Toric Varieties
112
(
Example 3.1.15. The two cones 6 and 6 in ? @ R from Example 3.1.4 (see Y Figure 1), together with their faces, form a fan . By comparing the descriptions ;3 it of the coordinate rings of ( what 6 given there with we did in Example 3.0.8, ! is easy to check that , where is the blowup of at the R 354 ' R ')+ (*'V8 (Exercise 3.1.5). origin, defined as
( +
and set R with standard basis Generalizing this, let ? R + Y fan in ? @ consisting of the cones generated by all .Let +, benotthecontaining +
( +, . Then the toric variety 6 is subsets of isomorphic to the blowup of at the origin (Exercise 3.0.8).
(
and consider the fan Y
.
in ? @ R
Example 3.1.16. Let consisting of the four cones 6 ' shown in Figure 4, together with all of their faces. The correσ4 (−1, r)
σ1
σ3
σ2
sponding toric variety
6
with
Figure 4. A fan
is covered by open affine subsets,
3 R BC D 47A% ' - 8 4
3 R BC D 47A% ' " ( - 8 4
4
3 R BC D 47A% ' " ( * ' 6
and glued according to (3.1.3). We call
" (-8
3 R BC D 47A% ' " ( * ' " 4
-8
the Hirzebruch surface
L
<
.
2.3.15 constructed the rational normal scroll using the polygon Example
Y " * W W with . The normal fan of fan above, so that < ( defined L < " . Note alsois the ( . We as an abstract variety, that will study smooth projective toric surfaces in Chapter 10.
+ . satisfy .D 4 ( 8 R . Consider Example 3.1.17. Let 4-+ 8 introduced in Chapter 2. Let ? R the+ weighted projective space (U 4 +98 + ( and let ' , R / be the images in ? of the standard
basis vectors in
U
, so the relation
+ + R
§3.1. Fans and Normal Toric Varieties
Y
113
holds in ? . Let be the fan made up of the cones generated by all+ the proper + . When ' R for all , we obtain 6 subsets of Exam 6 4 + 8 in general. This will be by ple 3.1.10. And indeed, + proved in Chapter 5 using a generalization of the homogeneous coordinates in .
4 .8
R ( R
R Here, we will consider the special case , where Y The fan in ? @ is pictured in Figure 5, using the plane spanned by
(
. . This
σ0 σ1
σ2
Figure 5. A fan
with
example has a different property from the others we have seen so far. Namely, = I> 4 ( 8 R = I> 4FR ( R ( 8 . We have 6 7 R consider the cone 6 R = I> 4FR ( R 8 ! ; , so the situation is similar to the case we studied in Example 1.2.21. Indeed, there is a change of coordinates defined by a matrix in 4 JX8 that takes 6 to the cone with @ R from that example. It follows that 4 3 354 ' TR 8 ! (Exercise 3.1.6). This is the there is an isomorphism rational normal cone , hence has a singular point at the origin. The toric variety 6 is singular because of the singular point in this affine open subset. We also saw $ 4 .8 a polytope such that the toric variety Y%$ coincides with the fan shown above. in Example 2.4.6, and the normal fan
Y
6
There is a dictionary between properties of and properties of that generalizes Theorem 1.3.12 and Example 1.3.20. We begin with some terminology. The first two items parallel Definition 1.2.16. Definition 3.1.18. Let (a) (b)
Y We say
Y We say Y
(c) We say
Y ! ?@
be a fan.
is smooth (or regular) if every cone 6 in
Y
is smooth (regular).
Y is simplicial if every cone 6 in is simplicial. Y R 3 6 6 is all of ? @ is complete if its support
.
Chapter 3. Normal Toric Varieties
114
Theorem 3.1.19. Let
6
be the toric variety defined by a fan
Y ! ?A@
.
Y (a) is a smooth variety if and only if the fan is smooth. 6 is an orbifold (that is, 6 has only finite quotient singularities) if and only (b) Y if the fan is simplicial. 6 is compact in the classical topology if and only if Y is complete. (c) 6
Proof. Part (a) follows from the corresponding statement for affine toric varieties, Theorem 1.3.12, because smoothness is a local property (Definition 3.0.10). In part (b), Example 1.3.20 gives one implication. The other will be proved later in the book. A proof of part (c) will be given in §3.4.
The blowup of at the origin (Example 3.1.15) is not compact, since the . The Hirzebruch support of the cones in the corresponding fan is not all of < surfaces from Example 3.1.16 are smooth and compact because every cone in . The variety the corresponding fan is smooth, and the union of the cones is 4 .8 from Example 3.1.17 is compact but not smooth. It is an orbifold (it has only finite quotient singularities) since the corresponding fan is simplicial.
Exercises for §3.1.
[ satisfy the compatibility G 3.1.2. Let be a variety obtained by gluing affine open subsets /&G r 8 along open subsets Gr ( *Gr by isomorphisms r ( 6>G r ( G ( r . Show that is separated when the image r G ( defined by RKS')(M Ke2p!r ( RKS'n' is Zariski closed for all sZ/2 * . of 6 GDr ( h GD" 3.1.3. Verify that if is the fan given in Example 3.1.10, then . 3.1.4. Prove Proposition 3.1.14. be the standard 3.1.5. Let , let 27+6+6+7 2 basis and let ( 8 . Let be the set of cones generated by all subsets of / 26+7+6+92 8 not containing / 26+7+6+9 2 8 . (a) Show that is a fan in .
3.1.1. Let be a fan in . Show that the isomorphisms conditions from §0 for gluing the together to create .
(b) Construct the affine open subsets covering the corresponding toric variety , and give the gluing isomorphisms. (c) Show that is isomorphic to the blowup of at the origin, earlier in described Exercise 3.0.8. Hint: The blowup is the subvariety of given by + + . Cover by affine open subsets + and compare those affines with your answer to part (b).
% d c # d 7c O5
Ne'
uM
t
(
4
(
`2+' 4 (b) Show that <>=?7@1 A H 9J ' % "d # c 4E'^f o . 3.1.7. In (.P4 , consider the fan with cones /E `8 , ! ?1 ' , and ! `?1 p# ' . Show that .
3.1.6. In this exercise, you will verify the claims made in Example 3.1.17. ( (a) Show that there is a matrix defining a change of coordinates that takes the cone in this example to the cone from Example 1.2.21, and find the mapping takes to the dual cone.
§3.2. The Orbit-Cone Correspondence
115
§3.2. The Orbit-Cone Correspondence
5
6
In this section, we will study the orbits for the action of on the toric variety . Our main result will show that there is a bijective correspondence between cones Y 5 -orbits in 6 . The connection comes ultimately from looking at limit in and 5 defined in §1.1. points of one-parameter subgroups of A First Example. We introduce the key features of the correspondence between orbits and cones by looking at a concrete example.
for the fan Y from Figure 2 of §3.1. We have Example 3.2.1. as the set of points 4 8 , 5 R 47 M 8 ! Consider with homogeneous coordinates R P . For each R 4 * 8 . ? R , we have the corresponding curve in :
6
)
4 8 R 4 8
)
We are abusing notation slightly; strictly speaking, the one-parameter subgroup 47 8 47 8 is a curve in M , but we view it as a curve in via the inclusion M ! .
48
We start by analyzing the limit of as . The limit point in 4 * 8 . It is easy to check that the pattern on R is as follows: )
depends
limit is (1,1,0) ↓
limit is (1,0,0) limit is (0,1,0) ← limit is (1,0,1) limit is (1,1,1)
←
a
limit is (0,0,1)
↑
b
limit is (0,1,1) Figure 6.
*
for
For instance, suppose in . These points lie in the first quadrant. ! # 4 8 R 4 8 . Next suppose that * R Here, it is obvious that in , corresponding to points on the diagonal in the third quadrant. Note that
4 8 R 4 8#% 4 " 8
R *
since we are using homogeneous coordinates in . Then implies that # 4 " 8 R 4 8 . You will check the remaining cases in Exercise 3.2.1.
Chapter 3. Normal Toric Varieties
116
Y
The regions of ? described in Figure 6 correspond to cones of the fan . C D4 6 8 , In each case, the set of giving one of the limit points equals ? 9 C D4 6 8 is the relative interior of a cone 6 . Y . In other words, we have where Y recovered the structure of the fan by considering these limits!
5
Now we relate this to the -orbits in . By considering the description 47 SF 8 M , you will see in Exercise 3.2.1 that there are exactly seven -orbits in :
5
( R 4 ' *')(*'
R 4 ' *')(*'
R 4 ' *')(*'
R 4 ' *')(*'
R 4 ' *')(*'
R 4 ' *' ( *'
R 4 ' *')(*'
' ' R P for all 54 8 ' R and ' *' ( R P 54 8 ')( R and ' *' R P 54 8 ' R and ')(*' R P 54 8 ')( R ' R and ' R P R 4 8 8 ' R ' R and ' ( R P R 4 8 8 ' R ')( R and ' R P R 4 8.
8 8 8 8 8
This list shows that each orbit contains a unique limit point. Hence we obtain a correspondence between cones 6 and orbits by
#
48 .
. CX D4 6 8 We will soon see that these observations generalize to all toric varieties 6
corresponds to
)
for all
6
.
Points and Semigroup Homomorphisms. It will be convenient to use the intrinsic 4 3 description of the points of an affine toric variety given in Proposition 1.3.1. We recall how this works and make some additional observations. 4
2 3
Points of
3
are in bijective correspondence with semigroup homomorphisms 2+3 R BC D 47A% 2 3 - 8 . 4 3 . Recall that 6 7 9-; and R
For each cone 6 we have a point of
4
3
defined by
,. 2 3 R
,. 2 3 9 6 I R 6 I 9 ; otherwise
This is a semigroup homomorphism since 6 7 9 6 I is a face of 6 7 . Thus, if , , Z . 203 and , , Z . 203 96 I , then , , Z . 2 3 96 I . We denote this 3 point by and call it the distinguished point corresponding to 6 .
3
The point is fixed under the (Corollary 1.3.3). If H
! 6
is a face, then
5
-action if and only if
!
6 R ! G ? @
. 3 . This follows since 6 I ! H I 4
.
Limits of One-Parameter Subgroups. In Example 3.2.1, the limit points of oneparameter subgroups are exactly the distinguished points for the cones in the fan of (Exercise 3.2.1). We now show that this is true for all affine toric varieties.
§3.2. The Orbit-Cone Correspondence Proposition 3.2.2. Let 6J! let . ? . Then
Proof. Given
#!
)
48
. CX . ?
be a strongly convex rational polyhedral cone and
#! D4 6 8 , then ! #
Moreover, if
?:@
117
. 6
, we have
exists in
4
3
) )
4 8 exists in 4 3 48 R 3 .
P 4 ) 4 8*8 203 exists in for all ,. ! P exists in for all ,. 2+3
#!
'
#
L W for all ,. 6 7 9 ; K, % 46 7 87 R 6 .
where the first equivalence is proved in Exercise 3.2.2 and the other equivalences are clear. This proves the first assertion of the proposition. In Exercise 3.2.2 you will also show that when . point corresponding to the semigroup homomorphism
! ) 4 8 2 6:3 9 ? , # defined by
is the
P ,. 6 7 9 ; R+ # 203 X C D 4 8 SG6 I (Exercise 1.2.2), and 6 , then K , L for all , . If . 0 2 3 I ,K L R if , . 96 . Hence the limit point is precisely the distinguished 3
point
.
Y
6
Using this proposition, we can recover the fan from cone by cone as in Example 3.2.1. This is also the key observation needed for the proof of Corollary 3.1.8 from the previous section.
354 ' R S8
Let us apply Proposition 3.2.2 to a familiar example.
R Example 3.2.3. Consider the affine toric variety studied in a number of examples from Chapter 1. For instance, in Example 1.1.17, we showed that is the normal toric variety corresponding to a cone 6 whose dual cone is = 6 7 R I> 4 ( ( B C D.47F% 6 7 9-; - 8 . and R (3.2.1)
In Example 1.1.17, we introduced the torus image of (3.2.2) Given (3.2.3)
%R 8
R 74 M 8
included in
4 ( 8 4 ( ( " ( 8 R 4 * 8 . ? R , we have the one-parameter subgroup ) 4 8 R 4 U " 8
as the
Chapter 3. Normal Toric Varieties
118
contained in , and proceed to examine limit points using Proposition 3.2.2. ! # ) 4 8 weexists * W and * W . These Clearly, in if and only if conditions determine the cone 6J! ?:@ given by
= 6 R I> 4 ( (
(3.2.4)
8
One easily checks that (3.2.1) is the dual of this cone (Exercise 3.2.3). Note also CX D4 8 * * , in which case the limit and that . 3 # ) 4 8 R 4 6 means 8 , which is the distinguished point .
The Torus Orbits. Now we turn to the Y has a distinguished point 3 cone 6 .
5 .
3
6
. We saw above that each
6 ! . This gives the torus orbit 3 6 !
-orbits in 4
46 8 R 5
4 8 In order to determine the structure of 6 , we need the following lemma, which you will prove in Exercise 3.2.4. Lemma 3.2.4. Let 6 be a strongly convex rational polyhedral cone in ?Q@ . Let ? 3 4 8 be the sublattice of ? spanned by the points in 6T9 ? , and let ? 6 R ? ? .
3
(a) There is a perfect pairing
9 ; ? 4 6 8 K L 6 I ? . induced by the dual pairing K L ;
(b) The pairing of part (a) induces a natural isomorphism
I.*4 6 I 9-; J M 8 5 3 5 3 R ? 4 6 8 * M is the torus associated to ? 4 6 8 . where
4 6 8 ! 4 3 , we recall how . 5 acts on semigroup homomorTo study 2+3 , then by Exercise 1.3.1, the point 4 3 phisms. If . is represented by ,
P 48 4 8 ,
is represented by the semigroup homomorphism
(3.2.5)
, R
'
Lemma 3.2.5. Let 6 be a strongly convex rational polyhedral cone in ?Q@ . Then
where ?
46 8
4 6 8 R 2 3 4 , 8 R P , . 6 I 9 ; I. )4 6 I 9-; J M 8 5 3 ,
.*
is the lattice defined in Lemma 3.2.4.
Z
2 3 4 , 8 R P 5
R
Proof. The set is invariant under the action of
by (3.2.5).
,. 6 I 9 ;
contains
3
and
Next observe that 6 I is the largest vector subspace of ; @ contained in 6N7 . 2 3 R
Z , then restricting 7 Hence 6 I 9 ; is a subgroup of . If . 6 9 ; 203 to , . 9 6 I R 6 I 9V; yields a group homomorphism 6 I 9V; M
§3.2. The Orbit-Cone Correspondence
119
I 9 ; M is a group homomorphism , we . Z by defining 4 8 , . 6 I 9-; 4 , 8 R , if
6 (Exercise 3.2.5). Conversely, if obtain a semigroup homomorphism
It follows that
Z
,
I..*4 6 I 9-; J M 8 .
otherwise.
Now consider the exact sequence (3.2.6) Tensoring with
M
R+ ? 3 R+ ? R+ ? 4 6 8XR
and using Lemma 3.2.4, we obtain a surjection
* M R * M 5 R ? 5 3 R ? 4 6 8
I..*4 6 I 9-; J M 8
,
5 3 , I..*)4 6 I 9; J M 8 Z 5 -action, so that 5 acts transtively on Z . Then 3 . are compatible with the 5
Z 3 R 4 6 8 , as desired. R implies that The bijections
Z
The Orbit-Cone Correspondence. Our next theorem is the major result of this section. Theorem 3.2.6 (Orbit-Cone Correspondence). Let Y fan in ? @ .
6
be the toric variety of the
(a) There is a bijective correspondence
Y Q 5 6 -orbits in 4 6 8 , I..*)4 6 I 6 T 9 ; J M 8
!
Let / R ? @ . For each cone 6 . Y , 4 6 8 R / R ! 6 4 3
cones 6
(b)
in
(c) The affine open subset
.
4
is the union of orbits
3 R
is a face of
3
4H 8
4 8 4 H 8 , and (d) H is a face of 6 if and only if 6 !
4H 8 R
4 8 3 6 is a face of
4 8 where H denotes the closure in both the classical and Zariski topologies. For instance, Example 3.2.1 tells us that for , there are three types of cones and torus orbits:
Chapter 3. Normal Toric Varieties
120
4 8
4 8
5
R ! The cone R , which
! 4 8 R R corresponds
! to the orbit R satisfies . This is a face of all the other cones Y in , and hence all the other orbits are contained in the closure of this one by part (d). According to part (c), since there are no cones properly contained in 4 R 4 8 47 M 8 . ,
The three one-dimensional cones H give the torus orbits of dimension 1. Each of these is isomorphic to M . The closure ( of one of these orbits is one of the 354 ' ' 8 ! , a copy of . Note that each H is contained in coordinate axes two maximal cones.
Y ' fan correspond 4 8 4 8 4 8 6 of inthethetorus to the three fixed points action on . There are two of these in
The three maximal cones
the closure of each of the 1-dimensional torus orbits.
5 -orbit in 6 . Since 6 is covered by the 6 and 4 3 9 4 3 R 4 3 3 , there is a !
! 4 3 . We claim that R 4 6 8 . Note that
and consider those ,. 2 3 satisfying 4 , 8 R P . To prove the claim, let . In Exercise 3.2.6, you will show that these , ’s lie on a face of 6 7 . But faces of 687 are all of the form 6 7 9 H0I for some face H of 6 by Proposition 1.2.10. In other words, there is a face H of 6 such that
, . 2 3 4 , 8 R P R 6 7 9 H I 9 ; 4 This easily implies . (Exercise 3.2.6), and then H R 6 by the minimality of 0 2 3 4 8 R 6 I 9 ; , and then . 4 6 8 by Lemma 3.2.5. R , . , 6 . Hence
P
4 6 8 since two orbits are either equal or disjoint. This implies R
Proof of Theorem 3.2.6. Let be a 5 -invariant affine open subsets 4 3 Y with unique minimal cone 6 . part (a) follows immediately.
Part (b) follows from Lemma 3.2.5 and (3.2.6).
3
Next consider part (c). We know that is a union of orbits. If H is a face of 6 , 4 8 4 4 3 4 8 4 3 H H ! ! then implies that is an orbit contained in . Furthermore, 4 3 the analysis of part (a) easily implies that any orbit contained in must equal 4 H 8 for some face H of 6 .
4
4 8
H in the classical We now turn to part (d). We begin with the closure of 5 4 8 H topology, which we denote . This is invariant under (Exercise 3.2.6) and 4 6 8 ! 4 H 8 . Then hence is a union of orbits. Suppose that have 4 H 8 ! 4 3 , since otherwise 4 H 8 9 4 3 R , which would imply we 4 H 8 must 4 3 R 4 3 4 H 8 ! 4 3 , 9 it follows since is open in the classical topology. Once we have that H is a face of 6 by part (c). Conversely, let H be a face of 6 . To prove that 4 6 8 ! 4 H 8 , it suffices to show that 4 H 8 9 4 6 8 R P . We will do this by using limits of one-parameter subgroups as in Proposition 3.2.2.
Let point of
4
homomorphism corresponding to the distinguished ,besothe 4 semigroup , 8 R if , . H+I 9; , and otherwise. Let . CX D4 6 8 ,
§3.2. The Orbit-Cone Correspondence
121
4 8 . This semigroup homomorphism is P P 4 8 , , R ' 4 ) 4 8*8 4 , 8 R 4 8 . 4 H 8 for all . M since the orbit of is 4 H 8 . Now let . Note that CX D4 6 8 , L Since . , K if , . 6 7 SG6 4 I 3, and R if , . 6 I . It 4 8 ! 4 8 R #8 follows that exists as a point in by Proposition 3.2.2, and
4 8 4 represents a point in 6 . But it is also in the closure of H by construction, so
4 6 8 9 4 H 8 R P . This establishes the first assertion of (d), and
4H 8 R
4 8 3 6 and for each
. M
consider
48 R
)
is a face of
follows immediately for the classical topology.
It remains to show that this set is also the Zariski closure. If we intersect 4 3 with an affine open subset , parts (c) and (d) imply that
4H 8 9 3 R 3 4
3
4H 8
46 8
354768 ! 4 3 for the ideal 6 R K ' ,. H I 9 4 6 Z 8 7 9-; L ! F%04 6 Z 8 7 9; - R 2 3 (3.2.7)
4 8 6 This easily implies that the classical closure H is a subvariety of and hence
4 8 is the Zariski closure of H .
4H 8 Orbit Closures as Toric Varieties. In the example of , the orbit closures is a face of
containing
In Exercise 3.2.6, you will P show that this is the subvariety
also have the structure of toric varieties. The same is true in general. The torus that 85 from Lemma 3.2.5. The corresponding fan can be acts is the quotient torus Y described as follows. Consider the set of cones 6 in containing H as a face. For 4 8 each such 6 , let 6 be the image cone in ? H @ under the quotient map
? @ R ? 4H 8@ R
obtained from (3.2.6). Then (3.2.8) is a fan in ?
4H 8@
B DB 4H 8 R 6 ! ? 4H 8@ H
(Exercise 3.2.7).
Proposition 3.2.7. For each cone . the toric variety
H
in
Y
is a face of 6
, the orbit closure
4H 8
is isomorphic to
Proof. This follows from parts (a) and (d) of Theorem 3.2.6 (Exercise 3.2.7).
Y
Example 3.2.8. Consider the fan in ?A@ R shown in Figure 7 on the next Y Y page. The support of is the cone in Figure 2 of Chapter 1, and is obtained from
this cone by adding a new -dimensional cone H in the center and subdividing. The 4 8 H has dimension by Theorem 3.2.6. By Proposition 3.2.7, the orbit orbit 4 8 Y closure H is constructed from the cones of containing H and then collapsing
Chapter 3. Normal Toric Varieties
122
z
y τ
x
Figure 7. The fan
H
to a point in 4 8 so that H
and its -dimensional cone
? 4( H 8 @ R ( 4 ? ? 8 @ .
. This clearly gives the fan for
( (,
Final Comments. The technique of using limit points of one-parameter subgroups to study a group action is also a major tool in Geometric Invariant Theory as in [71], where the main problem is to construct varieties (or possibly more general objects) representing orbit spaces for the actions of algebraic groups on varieties. We will apply ideas from group actions and orbit spaces to the study of toric varieties in Chapters 5 and 14. We also note the observation made in part (d) of Theorem 3.2.6 that torus orbits have the same closure in the classical and Zariski topologies. For arbitrary subsets of a variety, these closures may differ. A torus orbit is an example of a constructible subset, and we will see in §3.4 that constructible subsets have the same classical and Zariski closures since we are working over . Exercises for §3.2. 3.2.1. In this exercise, you will verify the claims made in Example 3.2.1 and the following discussion. are as claimed in the (a) Show that the remaining limits of one-parameter subgroups example.
4
(b) Show that the
F 'n4 orbits in 4 are as claimed in the example.
(c) Show that the limit point equals the distinguished point in each case.
}{ |
3.2.2. Let consider
, !n' , where ,'6m h ~
of the corresponding cone
be a strongly convex rational polyhedral cone. This exercise will is an arbitrary function.
§3.2. The Orbit-Cone Correspondence
}{ |
, An'
123
(a) Prove that exists in if and only if . Hint: Consider a finite set of characters
{}|
, !n'
}{ |
{}|
F, An'n' exists in ( , .
such that
for all
exists in , prove that the limit is given by the semigroup homo(b) When morphism that maps to .
F, !n' '
3.2.3. Consider the situation of Example 3.2.3. (a) Show that the cones in (3.2.1) and (3.2.4) are dual. (b) Identify the limits of all one-parameter subgroups in this example, and describe the Orbit-Cone Correspondence in this case.
# uC( # defines an automorphism of o and the corresponding linear map on
(c) Show that the matrix
the cone
to .
(d) Deduce that the affine toric varieties tion 1.3.15.
maps
and are isomorphic. Hint: Use Proposi-
3.2.4. Prove Lemma 3.2.4. 3.2.5. Let be as defined in the proof of Lemma 3.2.5. In this exercise, you will complete the proof that is a -orbit in .
(a) Show that if , then is a group homomorphism. (b) Deduce that
~
6
h
has the structure of a group.
(c) Verify carefully that we have an isomorphism of groups
m|
2 '.
6 h be a semigroup homomorphism giving a point of . Prove that / O k 'U(*T `8U( for some face of . (b) Show ' is invariant under the action of ~ . (c) Prove that ' is the variety of the ideal defined in (3.2.7). 3.2.7. Let be a cone in a fan , and let < E ' be as defined in (3.2.8). (a) Show that < & ' is a fan in . ' . (b) Prove Proposition 3.2.7. ~ on the affine toric variety . Use parts (c) and (d) of 3.2.8. Consider the action of ~ acting on . Theorem 3.2.6 to show that 5' is the unique closed orbit of 3.2.9. In Proposition 1.3.16, we saw that if is a face of the strongly convex rational polyhedral cone in then ( <>=?7@m FIH J!' is an affine open subset of ( <>=?7@1 AIH J!' . In this exercise, we will show that the converse is also true, i.e., that if % and the induced map of affine toric varieties ] 6 h is an open immersion, then is a face of . (a) Let P2n Q , and assume t Q . Show that }{}| !n' }{}| !n' W+ 3.2.6. This exercise is concerned with the proof of Theorem 3.2.6. (a) Let
Chapter 3. Normal Toric Varieties
124
}{ |
!n'
An'
{}|
(b) Show that and
of points as semigroup homomorphisms. (c) Deduce that
P2n S
are each in . Hint: Use the description
, so is a face of .
3.2.10. In this exercise, you will use Proposition 3.2.2 and Theorem 3.2.6 to deduce Corollary 3.1.8 from Theorem 3.1.7. (a) By Theorem 3.1.7, and the results of Chapter 1, a separated toric variety has an open cover consisting of affine toric varieties + for some collection of cones + . is also affine. Hint: Use the fact that is separated. Show that for all , +
(
( 4
2
(
+ . (b) Show that + is the affine toric variety corresponding to the cone Hint: Exercise 3.2.2 will be useful. ,+% , then show that is a face of both + and . Hint: Use Exercise 3.2.9. (c) If
(d) Deduce that
+ and all their faces.
for the fan consisting of the
§3.3. Equivariant Maps of Toric Varieties
Recall from §3.0 that if and are varieties with affine open covers R 4 Z is a Zariski-continuous and R , then a morphism of varieties mapping such that the restrictions (
4
9
" 4 Z 8 R 4
are regular in the sense of Definition 3.0.1 for all
When
and
"M
4
Z
4
.
are normal toric varieties, the results on mappings of affine toric varieties from Propositions 1.3.14 and 1.3.15 yield a class of morphisms whose construction comes directly from the combinatorics of the associated fans. The goal of this section is to study these special morphisms.
(
Y(
4 (8
Y
Definition 3.3.1. Let ? ? be two lattices a fan in ? @ and YS( a fan 4 8 @ . A -linear mapping ( ? with ? in ? is compatible with the fans Y if for every cone 6 ( in Y ( , there exists a cone 6 in Y such that 4 ( 8 and . @ 6 !6
(
(
Y
R Example 3.3.2. Let ? basis and let be the fan from Figure 4 < with 6 R in §3.1. By Example 3.1.16, ( is the Hirzebruch surface . Also let ? Y and consider the fan giving :
6 (
6
as in Example 3.1.11. The mapping
? (%R ? * ( R+ * Y and Y since each cone of Y is compatible with the fans
Y
. If
RP
, on the other hand, the mapping
? ( R+ ?
*( Y is not compatible with these fans since 6 .
R
maps onto a cone of
Y
does not map into a cone of .
§3.3. Equivariant Maps of Toric Varieties
125
Toric Morphisms. For our purposes, it will be convenient to take the result of part (a) of Proposition 1.3.14 as the definition of a toric morphism in this context. The discussion from §1.3 shows that this is consistent with what we did for affine toric varieties.
* ,
6
YS(
4 (8
@ Definition 3.3.3. Let , be normal toric varieties, with a fan in ? Y 4 8 6
6 and in ? is toric if maps the torus 5 ! a 6 faninto 5 @ ! . A 6morphism and is a group homomorphism.
6
The proof of part (b) of Proposition 1.3.14 generalizes easily to show that any 6 6 is an equivariant mapping for the 5 - and toric morphism 5 -actions. That is, we have a commutative diagram
5 0 6
(3.3.1)
where
(
and
/
*-,
6
/
6 5
6
give the torus actions.
( ?
Our first result in this section shows that toric morphisms are in bijective correspondence with -linear mappings ? Y( Y compatible with the fans and . Theorem 3.3.4. Let ?
(Q
( ?
6
6
that are
4 ? ' 8 @ , R . Y ( and Y , then -linear map that is compatible with Y '
be lattices, and let
(a) If ? ? is a there is a toric morphism
be a fan in
such that * , is the map ? (/ M R ? M is a toric morphism, then induces a -linear (b) Conversely, if ? ( ? that is compatible with the fans Y ( and Y . map ( Y ( YS( and Proof. To prove part (a), let 6 be a cone in . Since is compatible with 4 (8 Y Y , there is a cone 6 . with @ 6 3 3 ! 6 3 . Then Proposition 1.3.15 shows that induces a toric morphism . Using the general criterion for 3 gluing morphisms from Exercise 3.3.1, you will show in Exercise 3.3.2 that the . Moreover, is toric because glue together to give a morphism 5 5 , which is easily seen to be the group ( taking 6 R gives ? ( ? . homomorphism induced by the -linear map we show first that the toric morphism induces a -linear map ? For( part? (b), */, . This follows since is a group homomorphism. Hence, given . ? ( , the one-parameter subgroup M 5 can be composed with * , */, M 5 . This defines an to give the one-parameter subgroup 4 8 . ? . It is straightforward to show that ? ( ? is -linear. element 6
6
+*
6
+*
6
4
4
6
)
6
)
Chapter 3. Normal Toric Varieties
126
YF(
Y
It remains to show that is compatible with the fans 5 5 -orbit ( ! 6 is mappedandinto a. Because of the equivariance (3.3.1), each 5 -orbit
6 Orbit-Cone Correspondence (Theorem 3.2.6), each ( R! 4 6 (. 8 Byforthesome ( Y ( 4 6 is8 5 -orbit is R -orbit cone 6 in , and similarly each Y . Furthermore, if H ( ! 6 ( is a face, then by the same for some cone 6 in Y 4 4 H ( 8*8 ! 4 H 8 . such that reasoning, there is some cone H in
H
in this situation must be a face of 6 . This follows since
4 6 We
4 H 8 that 8 ! claim by part (d) of Theorem 3.2.6. Since is continuous in the Zariski 4 ( 8 4 8
H H . But the only orbits contained in the closure of ! topology, 4 H 8 are the orbits corresponding to cones which have H as a face. So H is a face of 6 . It follows from part (c) of Theorem 3.2.6 that also maps the affine open 3 4 3 ! 6 into 4 ! 6 , i.e., subset
4 38 ! 3 3 3 It follows that induces a toric morphism , which by Proposition 1.3.15 4 ( 8 ! 6 . Hence is compatible with the fans YS( and Y . implies that @ 6 4
(3.3.2)
4
4
4
First Examples. Here are some examples of toric morphisms defined by mappings compatible with the corresponding fans. Example 3.3.5. Let ?
( R
and ?
R
, and let
? ( R+ ? * ( R * is compatible with the fans be the first mapping in Example 3.3.2. We saw Y of the Hirzebruch surface < and Y of ( . that 3.3.4 implies that there < ( ( Theorem is a corresponding toric morphism . We will see ( later in this section < that this mapping gives the structure of a -bundle over . + , the multiplicaY Example 3.3.6. Let ? R and be a fan in ? @ . For .
tion map
Y is compatible with 6 6
? R+ ?
* R+
*
. By Theorem 3.3.4, there is a corresponding toric morphism 5 ! 6 is the group endomorphism whose restriction to
! *-, 4 ( +98 R 4 ( + 8
. R For a concrete example, let be the fan in ?A @ from Figure 2 and take R Then we obtain the morphism defined in homogeneous coordinates 4 ' *')(*' 8 R 4 ' *' ( *' 8 . by
Y
Sublattices of Finite Index. We get an interesting toric morphism when a lattice Y ? Z has finite index Zin a larger lattice ? . If Y is a fan in ? @ , then we can view Z as a fan either in ? @ or in ? @ , and the inclusion ? with the ? is compatible Y Z 6 5 6 5 fan in ? @ and ? @ . As in Chapter 1, we obtain toric varieties and
§3.3. Equivariant Maps of Toric Varieties
127
depending on which lattice we consider, and the inclusion ? morphism
5 R 5
Z
induced by the inclusion
Z
a sublattice of finite index in ? Z .beThen
and let
5 +R 5
? Z ?
induces a toric
6
6
Proposition 3.3.7. Let ? ? @ R ? @ Z . Let R ? ?
Z ?
Y
be a fan in
6
6
presents
6
5
as the quotient
6
5
.
Proof. Since ? has finite index in ? , Proposition 1.3.18 shows that the finite R ? ? Z is the kernel of 5 5 . It follows that acts on 6 5 . group 5 4 34 5 This action is compatible with the inclusion ! 4 6 34 5 for each cone 6 . Y . 4 34 5 Using Proposition 1.3.18 again, we see that , which easily implies 6 5 6 5 that .
We will revisit Proposition 3.3.7 in Chapter 5, where we will show that the 6 5 6 5 map is a good geometric quotient.
Y
5
Example 3.3.8. Let ? R , and be the fan shown in Figure 5, so is 4
. 8 Z . Let ? be the sublattice of isomorphic to the weighted projective space
? Z given by ? Z R 4 * ( 8 . ( ? I , so ? Z has index 2 in ? . Note that ? is generated by R , R and that
Z
R R ( R5 R R ( R
6
. ? Z
Let ? ? Z be6 the map. It is not difficult to see that with respect 5 inclusion to the lattice ? , (Exercise By Theorem 3.3.4, the -linear 3.3.3). 4
.8 , and by Proposition 3.3.7, it map induces a toric morphism 4
. 8 . Z follows that for R ? ?
σ2
Figure 8. The semigroups
and
The cone 6 from Figure 5 has the dual cone 6/7 shown in Figure 8. It is Z dual to ? Z . One checks instructive to consider how 6 7 interacts with the lattice ; Z 4 * .8 * . and 6 7 R = I> 4 (XR 4R 8 . In Figure 8, the that ;
Chapter 3. Normal Toric Varieties
128
Z
points in 67 9 ; are shown in white, and the points in 6/7 9 ; not in 6N7 9 ; are shown in black. Note that the picture in 6 7 9 ; is the same (up to a change 4 JX8 Figure 10 from Chapter 1. This shows again that of coordinates in 4 .8 contains the affine) as 4 3 5 open subset isomorphic to the rational normal 4 3 5 is smooth. The other affine open subsets cone . On the other hand ( in both 4 .8 . corresponding to 6 and 6 are isomorphic to and in
6
Torus Factors. A toric variety has a torus factor if it is equivariantly isomorphic to the product of a nontrivial torus and a toric variety of smaller dimension.
Proposition 3.3.9. Let equivalent: (a)
6
6
Y
be the toric variety of the fan . Then the following are
has a torus factor.
(b) There is a nonconstant morphism
M .
6
. YS4 8 , do not span ?G@ . 6 6 47 M 8 6 Proof. If for and some toric variety , then a non6 7 4 8 7 4 M8 M. trivial character of M gives a nonconstant morphism 6 M is a nonconstant If morphism, then Exercise 3.3.4 implies that 5 is ' P where . M , . ; . Multiplying by the( restriction of to P " , we may assume that *-, R ' . Then is aand toric morphism coming from M ? a surjective homomorphism . Since comes from the trivial fan, Y Y S 4
8 maps all cones of to the origin. Hence . for all . , so that the do not span ? @ . Y 4 8 S Finally, suppose that the , . span a proper subspace of ?:@ . Then B + C B Z ) 4 Y S 4
* 8 8 Z is . is proper sublattice of ? such that ? ? ? R 9 ? Y Z Z Z Z Z Z ? Y . Furthermore, with ? R ? can torsion-free, so ? has a complement ? Y Z Z Y Y Z Y Z Z R be regarded as a fan in ? @ , and then is the product fan , where Y ZZ ZZ is the trivial fan in ? @ . Then Proposition 3.1.14 gives an isomorphism 6 6 5 5 6 5 5
47 M 8 + "
!
? @ R / and ! ? @ Z R . where (c) The
,
In later chapters, torus varieties without torus factors will play an important role. Hence we state the following corollary of Proposition 3.3.9. Corollary 3.3.10. Let equivalent: (a)
6
6
Y
be the toric variety of the fan . Then the following are
has no torus factors.
6 M is constant, i.e., 4 . YS4 8 , span ? @ .
(b) Every morphism (c) The
,
6
" 8M R M.
We can also think about this from the point of view of sublattices.
§3.3. Equivariant Maps of Toric Varieties
129
? @ R / , ! ? @ Z R G @ .
Z
! ? be a sublattice with Proposition 3.3.11. Let ? Y Z Let be a fan in ? @ , which we can regard as a fan in ? (a) If ?
Z
.
is spanned by a subset of a basis of ? , then we have+ an isomorphism
5 5 5 5 5 5 47 M 8 "
6
6
(b) In general, a basis for ? 6 of finite index. Then
6
5
Z
5 6
6
? ZZ ! ?
can be extended to a basis of a sublattice is isomorphic to the quotient of+
5
5 5
ZZ by the finite abelian group ? ? .
5 547 M 8 "
6
Proof. Part (a) follows from the proof of Proposition 3.3.9, and part (b) follows from part (a) and Proposition 3.3.7.
Y Z
Y
Y
Refinements of Fans and Blowups. Given in ? @ , a fan Y Z R a Y fan Y Z Y Y refines if every cone of is contained in and . Hence every cone of is a union YZ YZ refines Y , the identity mapping on ? is automatically of cones of . When Y Z Y 6 6 . compatible with and . This yields a toric morphism
Y Z
Example 3.3.12. Consider the fan in ? in Figure 1 from §3.1. = pictured Y > I 4 ( 8 and This is a refinement of the fan consisting of its faces. The 6 6 R 354 ' RQ' ( 'V8 ! are and corresponding toric varieties ( at the origin (see Example 3.1.15). The identity map , the blowup of on( ? induces a toric morphism . This “blowdown” ( morphism maps ! to . and is injective outside of
in .
+
We can generalize this example and Example 3.1.5 as follows.
Y
= I> 4 ( + 8
Definition 3.3.13. Let be a fan in ?@ . Let 6 R Y ( + smooth cone in , so that is a basis for ? . Let R Y Z46 8 be the set of all cones generated by subsets of ( + . Then
( +
+
be a and let not containing
Y M 4 6 8 R 4 Y S 6 8 Y Z 4 6 8 Y is a fan in ? @ called the star subdivision of along 6 . = I> 4 ( 8 ! ?@ be a smooth cone. Example 3.3.14. Let 6 R Figure 9 on the next page shows the star subdivision of 6 into three cones = I> 4 ( 8 = I> 4 ( 8 = I> 4 8 Y 4 8 The fan M 6 consists of these cones, together with their faces. Y 4 8 Y Proposition 3.3.15. M 6 is a refinement of , and the induced toric morphism 6 3 R+ 6 3 6 -3 6 makes the blowup of at the distinguished point corresponding to the cone 6 .
Chapter 3. Normal Toric Varieties
130
u0
u2 u1
u3
Figure 9. The star subdivision
Y 4 8
Y
Proof. Since and M 6 are the same outside the cone 6 , without loss of generY + fan consisting of 6 and all of its faces, ality, we may reduce to the case that is the
6 4 3 and is the affine toric variety .
Under the Orbit-Cone Correspondence (Theorem 3.2.6), 6 corresponds to the 3 distinguished point , the origin (the unique fixed point of the torus action). By Theorem 3.3.4, the identity map on ? induces a toric morphism
3-
6
4
3 +
3
+ It is easy to check that the affine open sets covering are the same as for the blowup of at the origin from Exercise 3.0.8, and they are glued together in the same way by Exercise 3.1.5.
3
4 8 . In this notation, the
+ +8 The blowup at is sometimes denoted 7 4 blowup of at the origin is written . 6
6
6
The point blown up in Proposition 3.3.15 is a fixed point of the torus action. In some cases, torus-invariant subvarieties of larger dimension have equally nice+ ( + of blowups. We begin with the affine case. + The standard basis 3 = = > I 4 ( + 8 4 > I 4 ( 8 , R gives 6 R , and the face H R with / , gives the orbit closure
4 H 8 R A + "
4 8 R ( and consider the fan To construct the blowup of H , let Y M 4 H 8 R = I> 4 8 ! + . ( ! P . (3.3.3)
= I> 4 ( 8 = ( ! ? @ and H R I> 4 8 . 6 into the cones = >I 4 ( 8 = I> 4 8
Example 3.3.16. Let 6 R H The star subdivision relative to subdivides
§3.3. Equivariant Maps of Toric Varieties
as shown in Figure 10. The fan their faces.
Y M 4H 8
131
consists of these two cones, together with
e0
e2
τ e1
e3
Figure 10. The star subdivision
+" + 6 For the fan (3.3.3), the toric variety is the blowup+ of . Y 4 8 +" ! R To see why, observe that M H is a product fan. Namely, , and Y M 4 H 8 R Y ( Y Y ( .47 8 (coming from = I> 4 ( 8 ) and Y is the where + is the fan for " (coming from = I> 4 U ( + 8 ). It follows that fan for 6 R 47 8X + " " ( .47 + 8 is built by replacing . with Since follows ( +6 " R .47 8 " is built by replacing Q + " ! , + itwith " that . the 47 + 8 separates directions through the origin in , while +" intuitive idea is that 8 6 47 R the blowup normal directions to + . One can also study 47separates + 8 by working in on affine pieces given by Y 4 8 M H the maximal cones of —see [76, Prop. 1.26].
We generalize (3.3.3) as follows.
Y
+
Definition 3.3.17. Let be a fan in ?@ and assume H . Y H R & that all cones of containing are smooth. Let Y containing H , set cone 6 .
the property - ( has and for each
Y 3M 4 H 8 R = I> 4 8 ! 6 4 8 H 4 8 ! P Y Then the star subdivision of relative to H is the fan Y M 4 H 8 R 6 . Y H !P 6 Y 3 4 8 3 M H
Y
.
Chapter 3. Normal Toric Varieties
132
The fan
Y M 4H 8
4H 8.
Under the map , ; 4 8 closure H R
Y
is a refinement of 6
and hence induces a toric morphism
6
6
becomes the blowup
4 8
6
of
6
along the orbit
In Chapters 10 and 11 we will use toric morphisms coming from refinements of fans to resolve the singularities of toric varieties. Exact Sequences and Fibrations. Next, we consider some of the local structure of toric morphisms. To begin, consider a surjective -linear mapping
Z ? ? Y Y Z Z If in ? @ and in ? @ are compatible with , then we have a corresponding toric morphism R 4 8 , so that we have an exact sequence Now let ? R ? RR ? Z R (3.3.4) R+ ?
6
It is easy to check that
6
R 6 . Y 6 ! 4 ? 8 @ Y 4 8 @V! ? @ . By Proposition 3.3.11, is a subfan of whose cones lie in ? (3.3.5)
Y
Z
6
5 6 5 5
? . Furthermore, is compatible with since ? ? Z
in ? @ . This gives the toric morphism
"
,
" 4 5 8 R
6
Y
in ?@ and the trivial fan
5 5
In fact, by the reasoning to ( prove Proposition 3.3.4,
5 6 5 5 6 lying over 5 ! 6 is identified with the product In other words, the part of 5 6 5 . We say this subset of 6 is a fiber bundle of and the toric variety 5 with fiber 6 5 . over Y When the fan has a suitable structure relative to , we can make a similar 6 statement for every torus-invariant affine open subset of . Y Y Z Y Definition 3.3.18. In the situation of (3.3.4), we say is split by and if there Y Y exists a subfan ! such that: Y bijectively to a cone 6 Z . Y Z such that 6 6 Z defines (a) maps each cone 6 . Y Y Z
6
(3.3.6)
a bijection
(b)
. Y YGiven cones 6 arises this way.
.
and 6
. Y , the sum 6
6
Y
lies in , and every cone of
§3.3. Equivariant Maps of Toric Varieties
Y
Theorem 3.3.19. If is split by 6 locally trival fiber bundle over 4 open affine subsets satisfying
Y Z
133
Y
6
and as in Definition 3.3.18, then 6 5 . That is, 6 has a coverisbya with fiber
"(4 8 4
6
5
4
YS4 6 Z 8 R 6 . Y 4 6 8 ! 6 Z . Then " ( 4 4 3 8 R 6 3 6 3 6 5 .4 3 YS4 Z 8 Y 9 YS4 6 Z 8 It remains to show that . Since 6 is split by Y YS4 6 Z8 , we may assume 6 R 4 3 . In other words, we are reduced to the and 9 Y Z Z case when consists of 6 and its proper faces. ? Z ? splits the exact sequence (3.3.4) provided $ A -linear map $ Z is the identity on ? . A splitting induces an isomorphism Z ? ? ? Y such that maps 6 bijectively to 6 Z . By Definition 3.3.18, there is a cone 6 . Z Using 6 , one can find a splitting $ with the property that $ @ maps H to H for all Y H . (Exercise 3.3.5). Using Definition 3.3.18 again, we conclude that 4 ? 8 @ ? Z ?@ @ 4 Y 4 ? 8 @ 8 4 Y Z ? Z 8 4Y 8 carries the product fan @ to the fan ?G@ . By ProposiProof. Fix 6
Z
in
Y Z
and let
tion 3.1.14, we conclude that
6 6 5 - 6
6 5 4
3
and the theorem is proved. Example 3.3.20. To complete the discussion from Examples 3.3.2 and 3.3.5, con< ( sider the toric morphism induced by the mapping
Y
<
R
*(
(
Y
R+ *
R The fan of is split by the fan of and Y Y because of the subfan of consisting of the cones
= I> F4 R (
6 . Y @ 4 6 8 R
8 . = I> 94 ( 8 Y Z These cones are mapped bijectively to the cones in under @ consists of the cones = >I 4 8 . = >I F4 R 8
The fans
Y
and
Y
are shown in Figure 11.
Y
. and 6 . < As we vary over all 6 < . Hence Theorem 3.3.19 shows that with fibers isomorphic to
6 5
Y
, the sums 6
Y
. Note also that
give all cones of ( is a locally trivial fibration over ,
(
6
Chapter 3. Normal Toric Varieties
134
Σ0 ↓
〉
Σ ↓
σ4 (−1, r)
σ1
σ3
〉
←Σ σ2
↓
↓
↓
Figure 11. The Splitting of the Fan
4 8
R where ? gives the vertical axis in Figure 11. ( This( fibration is not < globally trivial when , i.e., it is not true that . There( is some " 44 3 8 “twisting” on the fibers involved when we try to glue together the ( < 4 3 to obtain .
We will give another, more precise, description of these fiber bundles and the “twisting” mentioned above using the language of sheaves in Chapter 6.
4 8
Images of Distinguished Points. Each orbit contains 6 4 in8 a toric variety 3 a distinguished point , and each orbit closure 6 is a toric variety in its own right. These structures are compatible with toric morphisms as follows.
6
6
6
Lemma 3.3.21. Let be the toric morphism coming from a map Z Y Y Z Y , let 6 Z . Y Z be the and . Given 6 . ? ? thatY Z is compatible with 4 8 minimal cone of containing @ 6 . Then:
4 3 8 R 3 , where 3 . 4 4 6 8*8 ! 4 6 Z8 and (b) 3 (c) The induced map
46 8
3 . 4 6 Z08
46 8 ! 46 Z8. 4 6 8 4 6 Z 8 is a toric morphism. Z( Z Y Z contain 4 6 8 , then so does their intersecProof. First observe that if 6 6 . @4 8 Y Z tion. Hence has a minimal cone containing @ 6 . (a)
and
are the distinguished points.
§3.3. Equivariant Maps of Toric Varieties
135
4 8 . C D4 6 Z 8 4 3 8 R ! # 4 8 R ! # 4 4 *8 8 R ! # 4 8 R 3
To prove part (a), pick Z the minimality of 6 . Then )
. C D 4 6 8
and observe that )
by
)
where the first and last equalities use Proposition 3.2.2. The first assertion of part (b) follows immediately from part (a) by the equivariance, and the second assertion follows by continuity (as usual, we get the same closure in the classical and Zariski topologies).
3- 4 8 4 Z08
For (c), observe that 6 6 is a morphism that is also a group homomorphism—this follows easily from equivariance. Since the orbit closures 3 4 6 8X 4 6 Z 8 is a toric are toric varieties by Proposition 3.2.7, the map morphism according to Definition 3.3.3.
Exercises for §3.3.
/ 8
3.3.1. Let be a variety with an affine open cover + , and let be a second variety. is + be a collection of morphisms. We say that a morphism Let + + for all . Show that there exists such a obtained by gluing the + if if and only if for every pair ,
] 6 h
] 6 h ] ] O 4 (N] ] 6 h 2 ],+ O 4 *( ] O 4 + 3.3.2. Let 2 be lattices, and let in ' , in ' be fans. Let ] 6! h 4 4 4 4 be a -linear mapping that is compatible with the corresponding fans. Using Exercise 3.3.1 above, show that the induced toric morphisms ] 6 h glue together to form a morphism ] 6 h . 3.3.3. This exercise asks you to verify some of the claims made in Example 3.3.8. 4. (a) Verify that with respect to the lattice , Q[
(b) Verify carefully that the affine open subset 6[ g , the rational normal cone of 4 degree 2. , gives a morphism ~ h . Here you will determine all 3.3.4. A character , " morphisms ~ h . (a) Explain why morphisms ~ h correspond to invertible elements in the coordinate ring of ~ . r (b) Let t and s . Prove that is invertible in IH 27+6+6+72n J and that all invertible elements of IH 26+6+7+62 J are of this form. ~ are of the form for (c) Use part (a) to show that all morphisms ~ 0h on and .
3.3.5. Let and let and be cones in ] 6 h be a surjective -linear mapping
and respectively with the property that ] maps
bijectively onto . Prove that ] has a splitting 6! Dh such that maps to . 3.3.6. Let be the fan obtained from the fan for 4 in Example 3.1.9 by the following process. Subdivide the cone into two new cones and 4 4 4 4 by inserting an edge `?m p # 4 '.
Chapter 3. Normal Toric Varieties
136
explicitly and give the gluing homomorphisms. (b) Show that the resulting toric variety is smooth. (c) Show that is isomorphic to the Hirzebruch surface . (a) Construct an affine open cover for
nm2612 1'
3.3.7. Let be the fan obtained from the fan for ( in Example 3.1.17 by the following process. Subdivide the cone into two new cones and by inserting an edge . (a) Construct an affine open cover for explicitly and give the gluing homomorphisms.
`?1 n# 4 '
4
(c) Construct a morphism of .
434
is smooth.
] 6 h
(b) Show that the resulting toric variety
4
and determine the fiber over the singular point
(d) One of our smooth examples is isomorphic to
. Which one is it?
( / W2 o&'UO I( m8$M F 'p4 4 4 and (M/ W W2 6 1'ZO2 ' 2 - |Wz >8 < . Also let ) ( &4 + ) . (a) Let D( Prove that the map h F 'n4 defined by ` `2 6 m':jh 2 ' induces an exact ) ) sequence #Dh #Dh #h #h `+ `?1 2 4 'L ' ( ( 4 . The inclusion Qh induces a toric (b) Let ( morphism [ h [ . Prove that this is the quotient map 4 h 4
for the above action of on 4 .
on . We will ) 3.3.8. Consider the action of the group study the quotient and its resolution of singularities using toric morphisms.
4
(c) Find the Hilbert basis (i.e., the set of irreducible elements) of the semigroup Hint: The Hilbert basis has four elements.
(d) Use the Hilbert basis from part (c) to subdivide . This gives a fan with Prove that is smooth relative to and that the resulting toric morphism
Q[ h [ C(C 4
.
O :O( .
is a resolution of singularities. This is a special case of the theory to be developed in Chapter 10.
A 'p4:- 4
m 'B( d # 12 c # d o '
(e) The group gives the finite set with ideal . ) Read about the Gr¨obner fan in [18, Ch. 8,§4] and compute the Gr¨obner fan of . The answer will be identical to the fan described in part (d). This is no accident, as shown in the paper [54]. There is a lot of interesting mathematics going on here, including the McKay correspondence and the -Hilbert scheme. See also [69] for the higher dimensional case.
ko F `2 26#&'
3.3.9. Consider the fan in shown in Figure 12 on the next page. This fan has five one-dimensional cones with four “upward” ray generators and one “downward” generator . There are also nine two-dimensional cones. Figure 12 shows five of the two-dimensional cones; the remaining four are generated by the combining the downward generator with the four upward generators.
$12 26E'927 F `2$126&'
c h . (b) Show that h is a locally trivial fiber bundle over with fiber nm27m2 (1' . Hint: Theorem 3.3.19 and pm23 `27E' C p#m23 `27E' ( A 2 `27#E' (V . See Example 3.1.17.
(a) Show that projection onto the -axis induces a toric morphism
§3.3. Equivariant Maps of Toric Varieties
137
z
y x
Figure 12. A fan
in
p126m2 m'
(c) Explain how you can see the splitting (in the sense of Definition 3.3.18) in Figure 12. Also explain why the figure makes it clear that the fiber is ( .
)4
3.3.10. Consider the fan in
with ray generators
( 4 2e ( 2e 4 ( 4 2 o (M# and two-dimensional cones `?1 A 2 '92 `?1 A 2n ' 2 ! ?1 ! 2 ' . 4 4 ato one point. (a) Draw a picture of and prove that is the blowup of h such that (b) Show the a toric morphism ] 6 map ! 4 jh induces ] As that ' for s * and ] A ' is a union of two copies of meeting at a F 1' gives point. Hint: Once you understand ] A 1 ' , show that the fan for ] . (c) To get a better picture of , consider the map 6 F 'n4Uh o defined by F12 n'B(L F o 2 4 237 2 4 ' 2 n' + n F ' 4 'y o be the closure of the image. Prove that and Let ( o Chi to gives the toric morphism ] of that the restriction of the projection part (b). o U is defined by the equations (d) Let d 2 c 2 2 $ be coordinates on o . Prove that 'c $ # 4 (* 2 d # c 4 (C `2 %d $ # &c (V `+ FsP' for sLV , and explain how Also use these equations to describe the fibers ] this relates to part (b). Hint: The twisted cubic is relevant.
This is a semi-stable degeneration of toric varieties. See [51] for more details.
Chapter 3. Normal Toric Varieties
138
§3.4. Complete and Proper The Compactness Criterion. We begin by proving part (c) of Theorem 3.1.19. Theorem 3.4.1. The following are equivalent for a toric variety (a)
6
! #
is compact in the classical topology.
(b) The limit (c)
Y
)
48
exists in
is complete (that is,
Y
6
3 6 6 R ?@
R
. ?
for all
6
.
.
).
6
Proof. First observe that since is separated (Theorem 3.1.5), it is Hausdorff as a topological space (Theorem 3.0.17). In fact, since the classical topology on 4 3 6 is a metric topology, is compact if and only if every each affine open set 6 sequence of points in has a convergent subsequence.
(b), assume that is compact and fix . ? . Given a sequence ) 4 8 . 6 . By compactness, . For M (a)converging to , we get the sequence this sequence has a convergent subsequence. Passing to this subsequence, we can ! ) 4 8
6 6 R . assume that . Because the union of the affine 34 # 3 Y , we may assume . 4 . Nowis take ,. 6 7 9 ; . The open subsetsP for 6 . ' 4 3 character is a regular function on and hence is continuous in the classical topology. Thus 6
'
P 4 8 R
#
P 4 ) 4 8*8 R
'
#
! P
, the exponent must be nonnegative, i.e., K , L-W for all , . , . 687 , so that . 4 6N7 8 7 R 6 . Then 67 9 ; . This implies K , L!W for) all 4 8 exists in 4 3 and hence in 6 . # Proposition 3.2.2 implies that
Since
!
48
# To prove (b) (c), take . ? and consider the limit . This lies 4 3 . in some affine open , which implies 6 Y 9? by Proposition Y 3.2.2. Thus every lattice point of ?G@ is contained in a cone of . It follows that is complete.
)
?:@ . In the case / R , the We will prove (c) (a) by induction on / R Y only complete fan is the fan The correspond 6 R ( in pictured in Example L 3.1.11. ing toric variety is . This is homeomorphic to , the two-dimensional sphere, and hence is compact.
Now assume the statement is true for all complete fans + of dimension strictly Y . 6 be a less than / , and consider a complete fan in ? @ . Let sequence. We will show that has a convergent subsequence.
6
4 8
H , we may assume the seSince is the union of finitely many orbits 4 8 4 8 H H R quence . If P , then the closure of H 6 lies entirely within an orbit R in is the toric variety of dimension / Proposition 3.2.7. Since B D B 4 H 8 by Y is complete, 4 8 it is easy to check that the fan is also complete in ? H @ (Exercise 3.4.1). Then the induction hypothesis implies that there is a convergent
§3.4. Complete and Proper
139
4 8
subsequence in H . Hence, without loss of generality again, we may assume that 5 ! 6 . our sequence lies entirely in the torus Recall from the discussion following Lemma 3.2.5 that
5 . 5
,
I. *V4 ; J M 8
Moreover, when we regard as a group homomorphism Y , restriction yields a semigroup homomorphism 6 for any 6 . 4 3 hence a point in .
; M , then 7Q9 ; and
5 ? @ ;
A key ingredient of the proof will be the logarithm map 5 ; M of , consider the map as follows. Given a point by the formula
, R
I 4 , 8
4 8
. , This is a homomorphism and hence gives an element For more properties of this mapping, see Exercise 3.4.2 below. the most important property of 4 8 R 6 for some 6 . Y 5 us, . For satisfies . of
defined defined
I. *4 ; * 8 ? @
.
is the following. Suppose that a point . If , . 687:9 ; , then the definition
implies that
I 4 , 8 R K , 4 8 L 4 8 R 6 . Hence 4 , 8 . Thus we have which is since , . 6 7 and . proved that 4 , 8 4 8 . R 6 R (3.4.2) for all ,. 6 7 9-; 4 8 . ? @ . Since Y Now apply to our sequence, which gives a sequence R Y (3.4.1)
is complete, the same is true for the fan consisting of the cones Hence, by passing to a subsequence, we may assume that there is 6
4 8 . R 6 4 , 8 . By (3.4.2), we conclude that
6 for 6 . . . Y such that
for all for all ,. 6 7 9 ; . It follows that the are a sequence of mappings to the closed unit disk in . Since the closed unit disk is compact, there is a subsequence which converges to a point 4 3 . . You will check the details of this final assertion in Exercise 3.4.3.
Proper Mappings. The property of compactness also has a relative version that is used most often in the theory of complex manifolds.
Definition 3.4.2. A continuous mapping O "O ( 4 8 is compact in for every compact subset
is proper if the inverse image ! .
It is immediate that is compact if and only if the constant mapping from to the space R pt consisting of a single point is proper. This relative version of compactness may also be reformulated, for reasonably nice topological spaces, in the following way.
Chapter 3. Normal Toric Varieties
140
Proposition 3.4.3. Let O be a continuous mapping of locally compact first countable Hausdorff spaces. Then the following are equivalent: (a) (b) (c)
O
is proper.
4 4S8
is closed in for all closed subsets . , are compact. ' . 4 ' 8 . converges has a subsequence Exery sequence ' that converges in . such that O 4
O
is a closed mapping (that ( is, O ! ), and all its fibers O " 4 8 ,
Proof. A proof of (a) (a) (c).
(b) can be found in [33, Ch. 9,§4]. See Exercise 3.4.4 for
Before we can give a definition of properness that works for morphisms, we L need another criterion for properness. Recall from §3.0 that morphisms O L U and give the fibered product . Fibered products can also be defined for continuous maps between topological spaces. In Exercise 3.4.4, you will prove that properness can formulated using fibered products.
Proposition 3.4.4. Let O be a continuous map between locally compact Hausdorff spaces. Then O is proper if and only if O is universally closed, meaning
that for all spaces and all continuous mappings , the projection defined by the commutative diagram
&
/
N
/
is a closed mapping. In algebraic geometry, it is customary to use the following definition of properness for morphisms of algebraic varieties.
(
Definition 3.4.5. A morphism of varieties is proper if it is univer
sally closed in the sense that for all varieties and morphisms , the projection defined by the commutative diagram
&
/
/
is a closed mapping in the Zariski topology. A variety pt is proper. the constant morphism
is said to be complete if
§3.4. Complete and Proper
141
Example 3.4.6. The + Projective Extension Theorem [17, Thm. 6 of Ch. 8, §5] shows that for R , the mapping + P P
P ; . It follows that if ! is any affine + variety, the projection ; ; is a closed mapping in the Zariski By the gluing+ construction, it follows + topology.
pt is proper, so is a complete variety that the constant morphism is closed in the Zariski topology for all ,
(the prototypical complete variety). Moreover, any projective variety is complete (Exercise 3.4.5).
that this is not On the other hand, consider the morphism
pt . We claim J R proper, so is not complete. To see this, consider and the diagram pt
.
The closed subset ( under . Hence
/
/ pt
354 ' R 8 ! does not map to a Zariski-closed subset of ( is not a closed mapping, and is not complete.
Completeness is the algebraic version of compactness, and it can be shown that a variety is complete if and only if it is compact in the classical topology. This is proved in Serre’s famous paper G´eom´etrie alg´ebrique et g´eom´etrie analytique, called GAGA for short. See [92, Prop. 6, p. 12]. The Properness Criterion. Theorem 3.4.1 can be understood as a special case of the following statement for toric morphisms.
6
6
Theorem 3.4.7. Let be the toric morphism corresponding to a ? ? Z that is compatible with fans Y in ?A@ and Y Z in ? @ Z . homomorphism Then the following are equivalent:
is proper in the classical topology (Definition 3.4.2). is a proper morphism (Definition 3.4.5). ! # 4 8 exists in , then # 4 8 exists in and
(b)
6
(a)
6
(d)
6
(" . 4 Y? Z 8 Y R @
(c) If
6
)
)
6
6
.
.
Proof. The proof of (a)
(b) uses two fundamental results in algebraic geometry.
First, given any morphism of varieties O and a Zariski closed subset ! , a theorem of Chevalley tells us that the image O 4 4<8 ; ! is8 constructible, 4 8 R ' ' S ' , where ; ' and meaning that it can be written as a finite union O ' are Zariski closed in . A proof appears in [41, Ex. II.3.19].
Chapter 3. Normal Toric Varieties
142
Second, given any constructible subset of a variety , its closure in in the classical topology equals its closure in the Zariski topology. When is a Zariski open, a proof is given in [70, Thm. (2.33)], and when is the image of a morphism, a proof can be found in GAGA [92, Prop. 7, p. 12].
6
6
that is proper in the classical topology and let
Now suppose 6 be a morphism. This gives the commutative diagram
"
6
"
6
/
/
6
4 8
Let be Zariski closed. We need to prove that is Zariski ! closed in . First observe that is also closed in the classical topology, so that 4 =8 is closed in in the classical topology by Proposition 3.4.4. However, 4 =8 is constructible by Chevalley’s Theorem, and then, being classically closed, it is also Zariski closed by GAGA. Hence is a closed map in the Zariski topol 6 ogy for any morphism . It follows that is a proper morphism. 6
48 . ? and assume that Z R ! # ) ) 4 8 exists in 6 under the extra assumptionexists that 6 is a nontrivial one-parameter subgroup in . ) 7 4 8
6 M be the closure of ! in the classical topology.
47 8
prove (b) (c), let To ! 6 in . We first prove 4 8 R P . This means that ) #
6
M ! Let Our earlier remarks imply that this equals the Zariski Since ) 47 M 8 closure. it is closed in the Zariski topology, so that is closed in topologies. It follows that )
6
is proper, in both
47 M 8 74 M 8 ! 47 M 8 Z Hence there is . to . Thus there is a sequence of points . M such that 4 8 mapping . Then 4 8 Z R 4 8 R 4 4 8*8 R # # 4 8 48 Z ! )
)
)
)
)
# This, together with R here, the arguments used to prove (a) ! # ) 4 8 exists in 6 . that
)
)
and (b)
RP
, imply that . From (c) of Theorem 3.4.1 easily imply
4 8 assume R P , consider the map 4 For 8 the 6 general case 6 when we. Thisno longer is proper since is proper (Exercise 3.4.6). 6 and 6 are toric varieties by Proposition 3.1.14, and Furthermore, 4 *,8 ? ? Z . Then applying the corresponding map on lattices is 4
8 ! # ) 4 8 exists in 6 . We . ? the above argument to shows that leave the details to the reader (Exercise 3.4.6).
§3.4. Complete and Proper
143
(d), first observe that the inclusion (
For (c)
Y
@ " 4 Y Z 8
!
is compatible with Y and YXZ . For the opposite inclusion, take " . @ 4 Y Z 8 9 ? . Then 4 8 . YZ , which by Proposition 3.2.2 implies that 6 # ) 4 8 exists in 6 . By assumption, ! # ) 4 8 exists in . Using Y . . Proposition 3.2.2, we conclude that . Because all the 6-9 ? " ( for some 6 8 Y 4 Y Z cones are rational, this immediately implies . ! ( since is automatic
@
Finally, we prove (d)
(a). We begin with two special cases.
6
5
Special Case 1. Suppose that a toric morphism (d) 5 ? Z is onto. The fan of satisfies consists and has the additional property that ? of the trivial cone , so that (d) implies (
R @" 4 8 R Y Y Z Z 4 8 When we think of as a fan in @ 6 6
4 8
Y
(3.4.3)
@
! ?@ , (3.3.5) implies that 5 6 5 5 . The fan YZ Z is complete in Then corresponds to the projection 4 8 by (3.4.3), so that 6 is compact by Theorem 3.4.1. Thus 6 pt @ 6 5 5
is proper, which easily implies that is proper. We conclude that is proper in the classical topology. 5 5 has the Special Case 2. Suppose that a homomorphism of tori
? ? Z is injective. Then (d) is obviously satisfied. is proper is given in Exercise 3.4.7. satisfying (d). We will Now consider a general toric morphism prove that is proper in the classical topology using part (c) of Proposition 3.4.3. is a sequence such that 4 8 converges in . We Thus assume that . need to prove that a subsequence of converges in .
additional property that An elementary proof that
6
6
6
6
6
5 -orbits, we may assume that the sequence Since has only finitely many
4 6 8 . As in Lemma 3.3.21, let 6 Z be the minimal cone of YZ lies in an orbit 4 8 containing 6 . The restriction -3 4 6 8 4 6 Z 8
6
4 8
4 Z8
6 and 6 are given is a toric morphism by Lemma 3.3.21, and the fans of B D B 4 6 8 in ? 4 6 8 @ and B D B 4 6 Z8 in ? Z<4 6 Z08 @ respectively. by Furthermore, one B D B 4 6 8 Y Y Z can check that since and satisfy (d), the same is true for the fans B D B 4 Z8 (Exercise 3.4.8). It follow that we may assume that . 5 and and 4 8 . 6 5 for all .
The limit the sequence
Z R ! # 4 8 lies in an orbit 4 H Z8 for some H Z . Y Z . Thus 4 8 and its limit Z all lie in . Note that 6 . Y 4 6 8 H Z is 4
Chapter 3. Normal Toric Varieties
144
" ( 4 8 . Since (d) implies that @" ( 4 H Z 8 R 3 6 4
the fan giving
, i.e., and " ( 4 H Z8 R Y . R X5 Z . If we write as the composition
6 R 4 we can assume that
4 Z8 Z If H R , then H R
6
4
4
Z ? 4 ? 8 ? 5 factors 6 5 5
6
then . Special Cases 1 and 2 imply that these maps are proper, and since the composition of proper maps is proper, we conclude that is proper.
Z R P . When we think of Z . 4 , Lemma 3.2.5 tells us that Z 8 R for all , Z . 4 H Z 8 7 9; Z S 4 H Z 8 I 9 ; Z M converge to Z in 4 , we see that 8 4 , Z 8 R for all , Z . 4 H Z 8 7 9-; Z S 4 H Z 8 I 9 ; Z
It remains to consider the case when H Z 4 H Z 8 7G9-; a semigroup homomorphism
Z4, 4 8 ; Since the ! 4 4 Z8
#
Z
as
Since H 7 91; is finitely generated, it follows that we may pass to a subsequence and assume that (3.4.4)
5
4 8 4, Z8
for all
and all ,
Z . 4 H Z 8 7 9 ; Z S 4 H Z 8 I 9-; Z
in the proof of Theorem 3.4.1 gives maps 5 map 5 introduced The? logarithm Z @ and ? @ linked by a commutative diagram:
*, M Z
Let ; ; be dual to implies that for all , we have (3.4.5)
5 5
,
?@ /
,
5
/
? @Z
? ? Z . Then , Z . 4 H Z8 7 9; Z S 4 H !Z 8 I 9 ; Z
Z 5 Z 5 K M 4 , 8 4 8 L R K , @ 4 4 8*8 L R K , Z 5 4 4 *8 8 L R I
4 8 4, Z8
where the first equality is standard, the second follows from the above commutative diagram, the third follows from (3.4.1), and the final inequality uses (3.4.4).
§3.4. Complete and Proper
@" ( 4 H Z 8
145
Now consider the following equivalences:
.
@4 8 . HZ Z K , @ 4 8 LXW Z K M 4 , 8 LXW
Z . 4H Z8 7 Z 9 ; Z 4 Z8 Z for all , . H 7 9-; Z 4 Z8 where the first and third equivalences are obvious and the second uses H R H 77 4 Z8 Z Z and the rationality of H . But we also know that H R P , which means that H 7 4 8 Z is a cone whose maximal subspace H I is a proper subset. This implies that " ( 4H Z8 Z Z Z Z Z Z . K M 4 , 8 L%W for all , . 4 H 8 7 9; S 4 H 8 I 9 ; @ " ( 4 H Z8 5 R 4 8 . (Exercise 3.4.9). Using (3.4.5), we( conclude that for all . @ " 4 H Z08 R Y . It follows that But, as noted above, (d) means R 5 4 8 . Y
for all ,
. Y
for all . Passing to a subsequence, we may assume that there is 6
5 4 8 . R 6
such that
for all . From here, the proof of (c) (a) in Theorem 3.4.1 implies that there is a 4 3 6 . This proves that is subsequence which converges to a point . ! proper in the classical topology. The proof of the theorem is now complete. An immediate corollary of Theorem 3.4.7 is the following more complete version of Theorem 3.4.1. Corollary 3.4.8. The following are equivalent for a toric variety (a)
6 6
(b) (d)
6
.
is compact in the classical topology.
! #
is complete.
(c) The limit
Y
)
48
is complete (that is,
exists in
Y
R
6
. ?
for all
3 6 6 R ?@
.
).
We noted earlier that a variety is complete if and only if it is compact. In a similar way, a morphism O of varieties is a proper morphism if and only if it is proper in the classical topology. This is proved in [36, Prop. 3.2 of Exp. XII]. Thus the equivalences (a) (b) of Theorem 3.4.7 and Corollary 3.4.8 are specific instances of this general phenomenon. Theorem 3.4.7 and Corollary 3.4.8 show that properness and completeness can be tested using one-parameter subgroups. In the case of completeness, we can formulate Given . ? , the one-parameter subgroup gives a map 5 ) S this G as follows.
6 ! ) 4 8 6 means that # ! , and saying that exists in
Chapter 3. Normal Toric Varieties
146 )
extends to a morphism diagram
)
6
S '
)
. In other words, whenever we have a
u
u u
u
pt .
/
/ 6 u:
)
6
the dashed arrow exists. The existence of tells us that is not missing any points, which is where the term “complete” comes from. In a similar way, the properness criterion given in part (c) of Theorem 3.4.7 can be formulated as saying that whenever . ? gives a diagram,
SX '
the dashed arrow
)
u
u u
u
/ 6 u:
/
6
exists.
For general varieties, there are similar completeness and properness ) S 6 and ) criteria 6 for that replace with maps coming from discrete valuation rings, to be discussed in Chapter 4. An example of a discrete valuation E F%0% -0- , whose field of fractions is the ring is the ring of formal power series R A * 4 4 * 8 8 . By replacing with BC D 47EF8 and field of formal Laurent series R SM with BC D4 8 in the above diagrams, where E is now an arbitrary discrete valuation ring, one gets the valuative criterion for properness ([41, Ex. II.4.11 and BC D47EF8 and Thm. II.4.7]). This requires the full power of scheme theory since BC D 4 8 are not varieties as defined in this book. Using the valuative criterion of properness, one gets purely algebraic proofs of (d) (b) in Theorem 3.4.7 and Corollary 3.4.8 (see [30, Sec. 2.4] or [76, Sec. 1.5]).
Example 3.4.9. An important class of proper morphisms are the toric morphisms 6 6 Y Z Y induced by a refinement of . Condition (d) of Theorem 3.4.7 is obviously fulfilled since ? ? is the identity and every cone of Y is a Y Z union of cones of . In particular, the blowups
(
6
3-
6
studied in Proposition 3.3.15 are always proper. Exercises for §3.4.
3.4.1. Let be a complete fan in and let be a cone in . Show that the fan defined in (3.2.8) is a complete fan in .
6~ h
t '
<" 7 '
3.4.2. In this exercise, you will develop some additional properties of the logarithm map ping defined in the proof of Theorem 3.4.1.
§3.4. Complete and Proper
147
(a) Let be the unit circle in the complex plane, a subgroup of the multiplicative group . Show that there is an isomorphism of groups
6m #h
j#h O O 2 O O ' 2 where the operation in the second factor on the right is addition. (b) Show that the compact real -dimensional torus ' can be viewed as a subgroup of ~ and that 6`~ h induces an isomorphism ~ : ` F ' . Hint: Use from part (a). (c) Let be a fan in . Show that the action of the compact real torus F ' C~ on ~ extends to an action on the toric variety and that the quotient space ' ` F ' ( t 5' P2 where ( denotes homeomorphism of topological spaces, and the union is over all cones in the fan. Hint: Use the Orbit-Cone Correspondence (Theorem 3.2.6). (d) Let in 4 be the fan from Example 3.1.9, so 4 . Show that under the that action of ' 4 N A ' 4 as in part (c), 4 ` F ' 4 ( , the 2-dimensional simplex. 4 We will say more about the topology of toric varieties in Chapter 12. 3.4.3. This exercise will complete the proof of Theorem 3.4.1. Let 1| 2 ' be the set of semigroup homomorphisms h . Assume that 1| 2 ' O N for all and all B . A We want to show that is a sequence such that O '6 A there is a subsequence that converges to a point m| 2 U' . A (a) The semigroup ( is generated by a finite set ,/ 27+6+6+72 8 . Use this fact ! and the compactness of / LO`O O 8 to show that there exists a subsequence A such that the sequences ' converge in for all . A (b) Deduce that the subsequence converges to a 1| 2 ' . A
3.4.4. In this exercise, you will prove some characterizations of properness stated in the text. (a) Prove (a)
(c) from Proposition 3.4.3.
, 6 h / 8
(b) Prove Proposition 3.4.4. Hint: Show first that compactness of is equivalent to the statement that the mapping pt is universally closed. Then use the easy fact that any composition of universally closed mappings is universally closed. 3.4.5. Show that any projective variety is complete according to Definition 3.4.5. 3.4.6. Complete the proof of (b)
(c) of Theorem 3.4.7 begun in the text.
] 6S~ h ~ be a map of tori corresponding to an injective homomorphism ] 6! h . Also let ] 6 h be the dual map. Finally, let A ~ be a sequence such that ]P ' converges to a point of ~ . A (a) Prove that { | ] 'I has finite index. Hence we can pick an integer such that f{ | ] ' . (b) Show that $ ' converges for all t{ | ] ' . Conclude that ' converges A is as in part (a). A for all " , where 3.4.7. Let
Chapter 3. Normal Toric Varieties
148
~
F '
M( A 26+7+6+92 A 'a% A ' . Show A 26+7+6+72 ^' % F ' .
(c) Pick a basis of so that and write that converges to a point
A 62 +7+6+92 A ' (d) Show that the th roots + can be chosen so that of the sequence a subsequence . A ( A 26+7+6+92 A ' converges to a point (L 27+6+6+62 ^ ' ~ (e) Explain why this implies that ~ h ~ is proper in the classical topology. 3.4.8. Here are some details from the proof of (d) (a) of Theorem 3.4.7. Given a toric ] 6 h and a cone t , let be the smallest cone containing morphism ] 5' . (a) Prove that ] induces a homomorphism ] 6!t 5'Zh t ' . < & '6O 'B( O < & 5'7O . (b) Assume further that ] O O 'B( O :O . Prove that ] ' 3O T /E `8 be a strongly convex polyhedral cone in . Prove that 3.4.9. Let k(L 2n f for all ' y ' and then apply this to a( ] A S' to complete the argument in the text. Hint: To prove , first show that the right hand side of the equivalence implies that 2n N for all B ' ' . Then show that U( T /& W8 implies that any element of ' is a limit of elements in ' y ' .
3.4.10. Give a second argument for the implication
compact
complete
from part (c) of Theorem 3.1.19 using induction on the dimension of . Hint: If is not complete and , then there is a one-dimensional cone in the boundary of the support of . Consider the fan and the corresponding toric subvariety of .
< E ' 3.4.11. Let 2 be fans in ] 6 h
toric morphism
h
compatible with the identity map . Prove that the is proper if and only if is a refinement of .
Appendix: Nonnormal Toric Varieties In this appendix, we discuss toric varieties that are not necessarily normal and relate them to the normal toric varieties studied in this chapter. We begin with an example to show that Sumihiro’s Theorem (Theorem 3.1.7) on the existence of a torus-invariant affine open cover can fail in the nonnormal case.
c 4 ( d 14 d ' g K.( A `23 `27E' g I/3K58 K K K g To see that g is a toric variety, we begin with the standard parametrization obtained by intersecting lines c (V d with the affine curve c 4 ( d 4 d -E' . This easily leads to the parametrization d (* 4 # 12 c (-9 A 4 #fE' + g g4
Example 3.A.1. Consider the nodal cubic defined by . The only singularity of is . We claim that is a toric variety with as torus. Assuming this for the moment, consider a torus-invariant neighborhood of . It contains and the torus and hence is the whole curve! We conclude that has no torusinvariant affine open neighborhood. Thus Sumihiro’s Theorem fails for .
Appendix: Nonnormal Toric Varieties
( $
149
K
The values map to the singular point . To get a parametrization that looks more like a torus, we replace with to obtain
d ( ! P# m &' 2 c ( m !9P !#f *E' E' + 4 o Then )(* 2 map to K and Z maps bijectively to g/3K58 . Using this parametrization, we get g , and the action of on itself given by multiplication extends to an action on g by making K a fixed point of the action. With
some work, one can show that this action is algebraic and hence gives a toric variety. (For readers familiar with elliptic curves, the basic idea is that the description of the group law in terms of lines connecting points on the curve reduces to multiplication in for our curve .)
g
g
In contrast, the projective toric varieties constructed in Chapter 2 satisfy Sumihiro’s Theorem by Proposition 2.1.8. Since these nonnormal toric varieties have a good local structure, it is reasonable to expect that they share some of the nice properties of normal toric varieties. In particular, they satisfy a version of the Orbit-Cone Correspondence (Theorem 3.2.6).
( / 26+7+6+62 ! 8 ! ( . `?1 C' #h induced by the inclusion of semigroup algebras H , VJe IH J"+ Recall that IH "J is the integral closure of IH , fJ in its field of fractions. We now apply standard results in commutative algebra and algebraic geometry: Since the integral closure H J is a finitely generated -algebra, it is a finitely generated module over IH , fJ (see [2, Cor. 5.8]). Thus the corresponding morphism h is finite (as defined in [41, p. 84]). A finite morphism is proper with finite fibers (see [41, Ex. II.3.5 and II.4.1]). is the identity on the torus, the image of the normalization is Zariski dense Since h
& , We begin with the affine case. Given and a finite subset we get the affine toric variety whose torus has character group (Proposi and tion 1.1.8). Assume isletthe map be dual to ( . By Proposition 1.3.8, the normalization of
in . But the image is also closed since the normalization map is proper. This proves that the normalization map is onto. Here is an example of how the normalization map can fail to be one-to-one.
( / 2 4 2 ( 4 8N 4 . This gives the parametrization F12 n'B(L m2 7 2n 4 ' , and one can check that ( % c 4 # d 4 'Z- o + 4 and ( `?1 C' ( `?1 2 4 ' . It follows easily that the Furthermore, ( normalization is given by 4 #h 12nn')j#hl 1237 2n 4 ' + #
Example 3.A.2. Let
Chapter 3. Normal Toric Varieties
150
A 2nn'hj F `2 2n346'
This map is one-to-one on the torus (the torus of is normal and hence is unchanged under normalization) but not on the -axis, since here the map is . We will soon see the intrinsic reason why this happens.
We now determine the orbit structure of Theorem 3.A.3. above. Then:
Let
.
be an affine toric variety with
(
(a) There is a bijective correspondence
$
such that a face of
"
/ faces of 8 h /7~ B
-orbits in
and let
8
of dimension corresponds to an orbit of dimension
is the orbit corresponding to a face (b) If character group .
zW{}|
be as
# B.
$ % C" ' (c) The normalization h induces a bijection /7~ -orbits in D 8 h /E~ -orbits in 8 such that if i and are the orbits corresponding to a face of , is the map of tori corresponding to the inclusion then the induced map h I *' of character groups. of , then
is the torus with
Proof. We will sketch the main ideas and leave the details for the reader. The proof uses the Orbit-Cone Correspondence (Theorem 3.2.6). We regard points of and as semigroup homomorphisms, so that in maps to in . Note also that is equivariant with respect to the action of . By Lemma 3.2.5, the orbit corresponding to a face of is the torus consisting of homomorphisms . Thus " "is the character group of . The normalization maps this orbit onto an orbit , where a point of maps to its restriction to . Since
h
' '
,
'
6
h
-6!
h
%
" '
O ~ 6, h
L' ( ( I C' 2 it follows that I *' is the character group of " ' . This proves part (b), and the final assertion of part (c) follows easily. Since is the saturation of , , it follows that there is an integer such that , . It follows easily that % C' has finite index in , so that zW{}| 'Z(*zW{} | 'Z(CzW{}| #%zW{}| (*zW{ | #%zW{}| 2 proving the final assertion of part (a). comes from an orbit in since h is onto. If orbits Finally, every orbit in , then '92 4 ' map to the same orbit of I C') ( I 4 C'9+ This easily implies % ( , so that ( . The bijections in parts (a) and (c) now 4 4 follow easily.
We leave it to the reader to work out other aspects of the Orbit-Cone Correspondence (specifically, the analogs of parts (c) and (d) of Theorem 3.2.6) for . Let us apply Theorem 3.A.3 to our previous example.
Appendix: Nonnormal Toric Varieties
151
( / 2 4 2 ( 4 8 4 as in Example A.3.A.2. The cone ( `?m C' ( `?1 2 4 ' has a face such that %t(<>= ! e 4 ' . Thus $ *'B( $ ( 4 ' ( 4+ It follows that I % C' has index ( in % , which explains why the normalization map is two-to-one on the orbit corresponding to . We now turn to the projective gives the projec case. Here, l(L/ 27+6+7+92 ! 8I! tive toric variety 9 ! whose torus9 has character group (Proposition 2.1.6). Recall that ( I2 +! + (V . +! + + O + One observation is that translating by leaves the corresponding projective ( (see part (a) of Exercise 2.1.6). Thus, variety unchanged. In other words, by translating an element of to the origin, we may assume . Note that the torus has character lattice ( when . of
Example 3.A.4. Let
We defined the normalization of an affine variety in §1.0. Using a gluing construction, one can define the normalization of any variety (see [41, Ex. II.3.8]). We can describe the as follows. normalization of a projective toric variety
%
(
( ! C' , then the normalization of is the toric variety normal fan of with respect to the lattice ( 1| +2 y' . Theorem 3.A.5. Let
be a projective toric variety where
and
. If
of the
Proof. Again, we sketch the proof and leave the details to the reader. We use the local given in Propositions 2.1.8 and 2.1.9. There, we saw that has an description of affine open covering given by the affine toric varieties , where - is a vertex of and - - .
(M<>=D?E@1 , ' L( ! C' ( # ( / # O N8 For the moment, assume that is very ample. Then Theorem 2.3.1 implies that has an affine open cover given by the affine toric varieties ( <>=D?E@m "L' , where - ( ! ?1 #!-W'
is a vertex of and . One can check that . The gluings are also is the saturation of , so that is the normalization of compatible by equations (2.1.6), (2.1.7) and Proposition 2.3.12. It follows that we get a that is the normalization of . natural map
,
h
B
B
In the general case, we note that is very ample for some integer and that and have the same normal fan. Since is a maximal cone of the normal fan, the
B
above argument now applies in general, and the theorem is proved.
Combining this result with the Orbit-Cone Correspondence and Theorem 3.A.3 gives the following immediate corollary. Corollary 3.A.6. With the same hypotheses as Theorem 3.A.5, we have: (a) There is a bijective correspondence
/ cones
of
such that a cone of dimension
^
B
8
h /7~
-orbits in
8
corresponds to an orbit of dimension
(b) If is the orbit corresponding to a cone character group .
$ % C'
of
zW{}| # B .
, then is the torus with
Chapter 3. Normal Toric Varieties
152
(c) The normalization
/7~
h
-orbits in
induces a bijection
8 h /7~
-orbits in
8
such that if are the orbits corresponding to a cone of and , then the induced map is the map of tori corresponding to the inclusion of character groups.
I *'
for
h
We leave it to the reader to work out other aspects of the Orbit-Cone Correspondence . A different approach to the study of appears in [31, Ch. 5].
Chapter 4
Divisors on Toric Varieties
§4.0. Background: Valuations, Divisors and Sheaves Divisors are defined in terms of irreducible codimension one subvarieties. In this chapter, we will consider Weil divisors and Cartier divisors. These classes coincide on a smooth variety, but for a normal variety, the situation is more complicated. We will also study divisor classes, which are defined using the order of vanishing of a rational function on an irreducible divisor. We will see that normal varieties are the natural setting to develop a theory of divisors and divisor classes. First, we give a simple motivational example.
4 '8
A4 '8
Example + 4.0.1. then there is a unique / . such that O 2. 4 'V8 4 'Vis8 nonzero, 4O '8 R ' , Ifwhere A % ' ' . , are not divisible by . This is possible F % ' 4 'V8 at : if / , because The integer / describes the behavior of O 4O '8 vanishesistoa UFD. 4 V ' 8 , O has order / at , and if / a pole of order / at . F4 'V8 M to the additive group Furthermore, the map from the multiplicative group 4 V ' 8 / is easily seen to be a group homomorphism. via O This construction works in the same way if we replace with any point of .
Discrete Valuation Rings. The simple construction given in Example 4.0.1 works in far greater generality. We begin by reviewing the algebric machinery we will need. Definition 4.0.2. A discrete valuation on a field
is a group homomorphism
$ M R+ 4 ' 8 W )4 $ 4 'V8 $ 4 8*8 that is surjective and satisfies $
whenever
M R S . The corresponding discrete valuation ring is the ring E R ' . M $ 4 '8 W .
' *'
.
153
Chapter 4. Divisors on Toric Varieties
154
One can check that a DVR is indeed a ring. Here are some properties of DVRs. Proposition 4.0.3. Let
E
be a DVR with valuation
$ M
' E is invertible in E if and only if $ 4 'V8 R . (a) . E ' E (b) is a local ring with maximal ideal R . E (c) is normal. E (d) is a principal ideal domain (PID). E (e) is Noetherian. E (f) The only proper prime ideals of are and
$
$ 4 'V8
. Then:
.
.
is a homomorphism, we have ( 8 R 4 'V8 4 ' " $ R $ (4.0.1) ' ' E is a unit, then $ 4 'V8 $ 4 ' " ( 8 W ( since ' *' " ( . E . Thus for all . M . If . 4 'V8 R , then $ 4 ' " 8 R by (4.0.1), so that $ 4 '8 R (4.0.1). Conversely, if $ ' " ( . E . by This proves part (a). ' E $ 4 'V8 is an ideal of E (this For part (b), note that R . Proof. First observe that since
follows directly from Definition 4.0.2). Then part (a) easily implies that with maximal ideal (Exercise 4.0.1).
'+
To prove part (c), suppose
E
'
E
' +
. M R S + ( " (*' "
E
is local
satisfies
R
' + . E . Then / and $ 4 '8 . " . E (' " + 4 ' + + (J' + " ( 8 and hence R "( ( + ' RF4 + " ( + " ' " ' " 8 . E . showing that R E satisfy $ 4 8 R and let 6 R P be an ideal of E . Pick ' . 6 S Let . 4 V ' 8 minimal. Then R ' " . satisfies $ 4 8 R $ 4 '8 R $ 4 8 R , with R $ E 6 so that is invertible in . From here, one proves without difficulty that R K L . with ' . . If . , we are done,( so suppose ( ' " . E . So ' Using (4.0.1) again, we see that
This proves part (d), and part (e) follows immediately.
For part (f), it is obvious that R K NL RP also that . Now let L R paragraph, K for some give a contradiction.
and the maximal ideal are prime. Note
be a proper prime ( ideal. By the previous ( . If , then " . and " .
This shows that every DVR is a Noetherian local domain of dimension one. In
! E E general, the dimension of a Noetherian ring is one less than the length of ? E the longest chain of proper prime ideals contained in . Among Noetherian local domains of dimension one, DVRs are characterized as follows.
§4.0. Background: Valuations, Divisors and Sheaves
155
47E= =8
Theorem 4.0.4. If is a Noetherian local domain of dimension one, then the following are equivalent.
E
(a) (b) (c) (d)
is a DVR.
E
is normal. is principal.
47E= =8
is a regular local ring.
Proof. (a) (b) and (a) (c) follow from Proposition 4.0.3, and equivalence (c) (d) is covered in Exercise 4.0.2. The remaining implications can be found in [2, Prop. 9.2].
DVRs and Prime Divisors. DVRs have a natural geometric interpretation. Let ! be an irreducible variety. A prime divisor irreducible subvariety of
!
! R is an. Recall R codimension one, meaning that from §3.0 that A4 8 has a field of rational functions with field . Our goal is to define a ring F 4 8 of fractions is a DVR when This will give a &4 such 8 M that such that A4 8 M is, normal. 4 O 8 gives valuation for O . the order of vanishing of O along .
$
"
"
$
Definition 4.0.5. For a variety X and prime divisor A4 8 defined by of
" R
! , "
is the subring
. F4 8 is defined on ! open with 9 R P . " We will see below that is a ring. Intuitively, this ring is built from rational functions on that are defined somewhere on (and hence defined on most of since is irreducible). 4
4
&4 8 R F4 4S8
4
!
is irreducible, Exercise 3.0.4 implies that Since is open and nonempty. If we further assume that
(4.0.2) follows easily (Exercise 4.0.3).
E
DI
#" R
4
9
whenever is nonempty, then
BC D 47EF8
R Hence we can reduce to the affine case for an integral domain . The codimension of a prime ideal Y , also called its height, is defined to be
! R ! E R 354 8 Y Y . It follows easily that Y 354 Y 8 induces a bijection
codimension one prime ideals of E& prime divisors of . 354 Y 8 , we can interpret " in terms of E as follows. Given a prime divisor R A4 8 is the field of fractions of E , and a rational The field of rational functions . E , is defined somewhere on R 354 Y 8 precisely function R O . , O :.4 8 R Y . It follows that when . " R
O . O . E= . Y .
Chapter 4. Divisors on Toric Varieties
156
EW
E
E
E
which is the localization of at the multiplicative subset STY (note that STY is closed under multiplication because Y is prime). This localization is a local ring EW (Exercise 4.0.3). It follows that with maximal ideal Y
" R E W
(4.0.3) when
R B9C D 47EF8
and Y is a codimension one prime ideal of
E
.
Example 4.0.6. In Example 4.0.1, we constructed a discrete valuation on 4 '8 . A4 '8 M to / . , provided sending O
A4 '8
by
4O 'V8 R ' + V44 'V'8 8 2 4 '8 4 '8 . A% ' -< 24 8 4 8 R P A% ' - . It follows that the prime divisor The corresponding DVR is the localization
R 354 'V8 ! R BC D.47F% ' - 8 has the local ring R A% ' - which is a DVR.
More generally, a normal ring or variety gives a DVR as follows. Proposition 4.0.7.
E
(a) Let be a normal domain and Y E W is a DVR. the localization
"
(b) Let
be a normal variety and is a DVR.
! E
be a codimension one prime ideal. Then
!
a prime divisor. Then the local ring
Proof. By Proposition 3.0.13, part (b) follows immediately from part (a) together with (4.0.2) and (4.0.3).
E W
W
E W
R Y It remains to prove part (a). The maximal ideal of is the ideal E W generated by Y in . The localization of a Noetherian ring is Noetherian (Exercise 4.0.4), and the same is true for normality by Exercise 1.0.7. It follows that the 47E W W8 is Noetherian and normal. local domain ESW
!
! E
R We compute the dimension of as follows. Since (see [17, 5 3 4 8 R Ex. 17 and 18 of Ch. 9, §4]), our hypothesis on Y implies that there are no E prime ideals strictly between and Y in . By [2, Prop. 3.11], the same is true FW E W EW has dimension one. Then EW is a DVR by for and in . It follows that Theorem 4.0.4.
When is a prime divisor on a normal variety we have a discrete valuation
A4 8
$ &4 8 M R+
, the DVR
$ 4 8
"
means that
M , we call O the order of vanishing of O along the divisor Given O . . Thus the local ring consists of those rational functions whose order of consists of those rational W vanishing along is , and its maximal ideal
"
"
§4.0. Background: Valuations, Divisors and Sheaves
functions that vanish on order / along .
. When
157
$ 4 O 8 R / , we say that O
has a pole of
Weil Divisors. Recall that a prime divisor on an irreducible variety ducible subvariety of codimension one.
) )4 8
is an irre-
)
is the free abelian group generated by the prime divisors Definition 4.0.8. 4 8 . on . A Weil divisor is an element of
) 4 8
*
) V4 8
. Thus a Weil divisor is a finite sum R & ' ' ' . of * W , ' ' . with for all . The divisor is effective, written prime divisors * if the ' are all nonnegative. The support of is the union of the prime divisors appearing in :
B C C 4 8 R
'
(
The Divisor of a Rational Function. An important class of Weil divisors comes from rational functions. If is normal, any prime divisor on corresponds to A 4 8 A 4 8 M , the integers M a DVR with valuation . Given O . 4 O 8 tell us how O behaves on the prime divisors of . Here is an important property of these integers.
"
$
$
Lemma 4.0.9. If is normal and ! finite number of prime divisors
O . F4 8 M , then $ 4 O 8 .
is zero for all but a
A4 8
M . It follows Proof. If O is constant, then it is a nonzero constant since O . 4 8 O R for all 4. On the that other hand, if O is nonconstant, then we can find such 4 is a nonconstant morphism. that O a nonempty open subset ! ( ; 7 4 8 ; M " M Then R O . The S ; is a nonempty open subset of such that O complement is Zariski closed and hence is a union of irreducible components of dimension ( " . / . Denote the irreducible components of codimension one by
$
;
;
Now let be prime divisor in . If 9 R ; , then ! S , so that is contained in an irreducible component of S since is irreducible. ' . On the other hand, Dimension considerations then imply that R for some ; R R , which implies if 9 , then O is an invertible element of P 4O 8 R . that
$
Definition 4.0.10. Let (a)
"
be a normal variety.
F4 8 M is The divisor of O .
4 8 R O
$ 4 O +8 ! . where the sum is over all prime divisors
.
Chapter 4. Divisors on Toric Varieties
158
(b)
4 8 O is) called a principal 4 8 denoted .
'
divisor, and the set of all principal divisors is
% ' if their difference ) 4 8 for, some O . F4 8 M .
'
(c) Divisors and are linearly equivalent, written R R 4 O 8 . is a principal divisor, i.e.,
) )4 8 . If O . F4 8 M , then 4 O 98 R 8 valuations are group homomor) O 8 issince ) 4 8 . a subgroup of P 4 ' R * 8 P ( 4 ' R 8 * R . &% ' - be a polynomial of Example 4.0.11. Let O ( * * . are distinct. Then: , where . M and degree , R V 4 8 R & ')( ( ,-' * ' O
. When , ( 4 8 ( * #R
V4 8
. implies that (
V4 Lemma
V4 " 8 R O R 4 8O 4.0.9 V4 8 O F4 8 M and .4 . It follows that phisms on
R
, ' ' , . O R & '( A4 8 M can be written 4 O 8 R 4 O 8R 4 O 8 , where The divisor of O .
4 8 R $ 4 O +8 O N
4 8 R R $ 4 O 8+ O N
4 8 We call O the divisor of zeros of O and 4 O 8 the divisor of poles of O . When
R
,
Note that these are effective divisors.
R & Cartier Divisors. If nonempty open subset, then
is a Weil divisor on
4
'*' '
R
is a Weil divisor on
. ( * '
called the restriction of
and
4
!
is a
' 9
4
to
4
We now define a special class of Weil divisors.
.
Definition 4.0.12. A Weil divisor on a normal variety is Cartier if it is locally has an open cover 4 ' ' such that is principal principal, meaning that R 4 ' 8 4 6 6 4 4 ' O ' 8 ' in ' for every . . If for . , then we call
O the local data for .
V4 8
A principal divisor is obviously locally principal. Thus O is Cartier for all A 4 8 M . O R . One can also show that if and are Cartier divisors, then and are Cartier (Exercise 4.0.5). It follows that the Cartier divisors on form = 4 8 a group satisfying
)
'
) 4 8 ! = ) 4 8 ! ) )4 8
'
§4.0. Background: Valuations, Divisors and Sheaves
159
Divisor Classes. For Weil and Cartier divisors, linear equivalence classes form the following important groups
Definition 4.0.13. Let
be a normal variety. Its class group is
= 4 8 R ) )4 8 ) 4 8
and its Picard group is
GH!D.4 8 R = ) )4 8 ) .4 8 G !D.4 8 in Chapter 6. Note that We will give a more sophisticated definition of = ) 4 8 ) ) 4 8 since is a subgroup of , we get a canonical injection GH!D. 4 8 = 4 8 In [41, II.6], Hartshorne writes “The divisor class group of a scheme is a very interesting invariant. In general it is not easy to calculate.” Fortunately, divisor class groups of normal toric varieties are easy to describe, as we will see in §4.1. More Algebra. Before we can derive further properties of divisors, we need to B9C D.47EF8 is learn more about normal domains. Equation (3.0.2) shows that if R irreducible, then
" E R "
E
If a point . corresponds to a maximal ideal ! , then the local ring E is the localization . Hence the above equality can be written
E R
E When
"
E
maximal
is normal, we get a similar result using codimension one prime ideals.
Theorem 4.0.14. If
E
is a Noetherian normal domain, then
E R
E W S W ( ( E E , * * Proof. Let be the field of fractions of and assume that . , . W E * lies in for all codimension one prime ideals Y . It suffices to prove that . K L . E This is obviously true when is invertible in , so we may assume that K L is a E proper ideal of . Then we have a primary decomposition (see [17, Ch. 4, §7]) ( (4.0.4) K L R 9 9 " E . In the and each prime ideal Y ' R ' is of the form Y ' R K L ' for some ' . terminology of [66, p. 38], the Y ' are the prime divisors of K L . E Since is Noetherian and normal, the Krull Principal Ideal Theorem states
that every prime divisor of K L has codimension one (see [66, Thm. 11.5] for a proof). This implies that in the primary decomposition (4.0.4), the prime divisors Y ' have codimension one and hence are distinct.
Chapter 4. Divisors on Toric Varieties
160
E W
*
4 8 W
EW
*
R It follows that . for all , which implies . . Since R ' for P (Exercise 4.0.6), localizing (4.0.4) at Y shows that for all , we have
Since
' E W 9 E R '
* . E W
R 'E W
(Exercise 4.0.6), we obtain
* .
'" ( ( ' R K L .
This result has the following useful corollary.
Corollary 4.0.15. Let be a normal variety and let O 4 . If S 4 has codimension defined on an open set ! to a morphism defined on all of .
E W
4
W
in
be a morphism , then O extends
B9C D 47EF8
R an affine open cover, we assume that , where Proof. Since E is a Noetherianhasnormal domain. If ! is an irreducible hypersurface, then 4 R R . 9 P for dimension reasons. It follows that O , so that
O .
(4.0.5)
" R
R
SW ( (
"
E W R E=
where the final equality is Theorem 4.0.14. These results enable us to determine when the divisor of a rational function is effective.
O . F4 8 M , then:
4 8 W (a) O if and only if O is a morphism, i.e., O . " 4 8 .
4 8 R O if and only if O M is a morphism, i.e., O . " M 4 8 . (b)
Proposition 4.0.16. Let
In general,
"M
be a normal variety. If
is the sheaf on
" M 4 4S8 R
defined by
invertible elements of
" 4 S8. 4
This is a sheaf of abelian groups under multiplication.
"
Proof. If O is a morphism, then O .
V4 8 W for every prime divisor 4 8 W O . Hence O . Going the other way, , which in turn implies
V4 8 W suppose that we restrict to an affine open O . This remains true when
4 8 W subset, so we may assume that is affine. Then O implies
$
O .
"
where the intersection is over all prime divisors. By (4.0.5), we conclude that O is defined everywhere. This proves part (a), and part follows immediately since
V4 8 R
4 8 W
4 " ( 8 (b) W O if and only if O and O .
§4.0. Background: Valuations, Divisors and Sheaves
161
Singularities and Normality. The set of singular points of a variety
B 0 4 8 !
B 0 4 8
is denoted
B 4 8
We call the singular locus of . One can show that is a proper closed subvariety of (see [41, Thm. I.5.3]). When is normal, things are even nicer. Proposition 4.0.17. Let
B 4 8
(a)
(b) If
be a normal variety. Then:
has codimension
is a curve, then
W
in
.
is smooth.
Proof. You will prove part (b) in Exercise 4.0.7. A proof of part (a) can be found in [89, Vol. 2, Thm. 3 of §II.5]. Computing Divisor Classes. There are two results, one algebraic and one geometric, that enable us to compute class groups in some cases. We begin with the algebraic result. Theorem 4.0.18. Let (a) (b)
E
E
be a UFD and set
R BC D 47EF8 . Then:
is normal and every codimension one prime ideal is principal.
= 4 8 R
.
Proof. For part (a), we know that a UFD is normal by Exercise 1.0.5. Let Y be a E E * codimension one prime ideal of and pick . Y1S . Since is a UFD,
* R with the ' prime and ' . Y , and since D I !
+
' )' ( (
E in . Because Y Y isR invertible
, this forces Y R K ' L .
is prime, this means some
:.4 8
! Turning to part (b), let be a prime divisor. Then Y R is a L R codimension one prime ideal and hence is principal, say Y O generates ESW , which implies 4 O 8 R K O . Then the maximal ideal of the DVR (see the proof of = 4 8 R
4 8 R Proposition 4.0.3). It follows easily that . Then O since all prime divisors are linearly equivalent to .
$
In fact, more is true: a normal Noetherian domain is a UFD if and only if every codimension one prime ideal is principal (Exercise 4.0.8). Example 4.0.19.
A% ' ( *' +.-
= 47 + 8 R is a UFD, so
by Theorem 4.0.18.
! Before stating the geometric result, we observe that if is a nonempty open subset, then the restriction map induces a well-defined map = 4 8 = 4 4 8 (Exercise 4.0.9).
4
Chapter 4. Divisors on Toric Varieties
162
4
Theorem 4.0.20. Let be a nonempty open subset of a normal variety and let ( " be the irreducible components of S 4 that are prime divisors. Then the sequence
"
( (
R = 4 8XR = 4 4 8XR+ " (0* = 4 8 is exact, where the first map sends & ( to its divisor class in 4
the second is induced by restriction to
Z
.
* ' 'Z . ) 4 4S8 with 'Z a prime divisor in 4 . Then the 'Z in is a prime divisor in , and R & ' * ' 'Z satisfies 8 = 4 4S8 is surjective.
R & ' Proof. Let Z Zariski closure ' of R Z . Hence = 4
and
) )4 8
Since each restricts to in , the composition of the two maps is % - . = 4 8 restricts to in trivial. To finish the proof of exactness, suppose that = 44 8 A4 4S8 M . Since F4 4 8 R . This means that is the divisor of some O . F4 8 and the divisor of O in )4 8 restricts to the divisor of O in 4 4 8 , it A4 8 M such that follows that we have O .
4
)
)
R V4 O 8 H R 4 O 8 is supported on S 4 , which means This implies that the difference
V4 8 . " ( ( R . that by the definition of the O ( R
and note that is a prime divisor on Example 4.0.21. Write (
(
( # R = 4 8XR+ = 478 R = 4 8 defined by * % * - is surjective. This map is Hence the map R V4 O 8 implies 4 O 8 R , so that O . 47S )8 M R * injective since
M by( Proposition 4.0.16. Hence O is constant, which forces * R . If follows that = 4 8 . . Then Theorem 4.0.20 and Example 4.0.21 give the exact sequence
Later in the chapter we will use similar methods to compute the class group of an arbitrary normal toric variety. Comparing Weil and Cartier Divisors. Once we understand Cartier divisors on normal toric varieties, it will be easy to give examples of Weil divisors that are not Cartier. On the other hand, there are varieties where every Weil divisor is Cartier. Theorem 4.0.22. Let (a) If the local ring is Cartier. (b) If
variety. Then: " beis aa normal UFD for every . , then every Weil divisor on
is smooth, then every Weil divisor on
is Cartier.
§4.0. Background: Valuations, Divisors and Sheaves
"
163
Proof. If is smooth, then is a regular local ring for all . . Since every regular local ring is a UFD (see §1.0), part (b) follows from part (a). For part (a), it suffices to show that prime divisors are locally principal. This B9C D 47EF8 is affine. condition is obviously local on , so we may assume that R 5 3 4 8 E one prime Let R Y be a prime divisor on , where Y ! is a codimension 4 R R S since . It remains ideal. Note that is obviously principal on to show that is locally principal in a neighborhood of a point . .
E
. implies Y ! The point E corresponds to the maximal ideal ! . Thus E ! E . Since Y ! has codimension one, it follows that the prime ideal Y also has codimension one (this follows from [2, Prop. 3.11]). Then Theorem 4.0.18 E is principal since E is a UFD by hypothesis. Thus Y E R implies that Y 4 * 8 E where * . E and . . Since is invertible in E , we in fact have YE R *E . * ( *
* . YE
E
Example 4.0.23. ( 8 = Since (8 H G ! D 4 4 R that
"
(
R *E
* , so that ' R R P . If we set R ( " , 8 is a neighborhood of , and * 8 on 4 .
" L ! . Then ' suppose Y R K 4 ' Now ' 8 * , where ' '#. E and ' . , i.e., ' 4 8 E R * E follows easily. Then 4 R BC D.47E then Y
V4 from here, it is straightforward to see that R
is smooth, Theorem 4.0.22 and Example 4.0.21 imply .
Sheaves of -modules. Weil and Cartier divisors on lead to some important sheaves on . Hence we need a brief excursion into sheaf theory (we will go was defined in §3.0. The deeper into the subject in Chapter 6). The sheaf 4 ! , definition of a sheaf of -modules is similar: for each open subset 44 8 4 4S8 -module with the following properties: there is an When
"
"
"
!
4
, there is a restriction map
4 8 4
4 8 R
"
such that when is the identity and 4 4S8 Furthermore, is compatible with the restriction map If
4
is an open cover of
R
4 4S8%R+
!
4
, then the sequence
4 4 8 RR
44 9
4
98
is exact, where the second arrow is defined by the restrictions and . double arrow is defined by When
4
4 4S8
#"
satisfies just the first bullet, we say that
4 4S8
!
; ! 4 . " 4 8.
and the
is a presheaf.
Given a sheaf of -modules , elements of are called sections of 4 4 over . In practice, the module of sections of over ! is expressed in several ways:
4 4S8 R
44 8 R O 44 8
Chapter 4. Divisors on Toric Varieties
164
O
We will in later chapters. Traditionally, 4 8 use in this section and switch to is called the module of global sections of .
be a morphism of varieties and let be a sheaf Example 4.0.24. Let O of -modules on . The direct image sheaf O M on is defined by
"
R+ 4 O " ( 4 4S8*8
4
for ! image 4
If
M
open. Then O
M
-modules. For
is a sheaf of was mentioned in §3.0.
'
, the direct
"
#"
are sheaves of -modules, then a homomorphism of sheaves 4 4S8 consists of -module homomorphisms
' and
4 8%R+ ' 4 S8 4
such that the diagram
44 8
4<;=8
4 4S8 '
/
;
4
/
'
4<;A8
commutes whenever ! . It should be what it means for sheaves clear ' . -modules to be isomorphic, written
"
4
Example 4.0.25. Let O be a morphism of varieties. If then composition with O induces a natural map
!
4
'
of
is open,
4 4 8XR " 4 O " ( 4 4S8*8 R O " 4 4S8 M " This defines a sheaf homomorphism OM . R BC D 47EF8 Over an affine variety , there is a standard way to get sheaves of " E E N such -modules. Recall that a nonzero element O . N R BC D.47ESN.8 S 354 O 8 gives the E localization; that is the open subset . Given an -module , we get " E N N =E S N R -module ; . Then there is a unique sheaf ; of -modules the ;
such that
O . E
for every nonzero
; 4 N 8 R ; N . This is proved in [41, Prop. II.5.1].
We globalize this construction as follows.
"
of on a variety is quasicoherent if Definition 4.0.26. A sheaf 4 , 4 -modules R BC D 47E 8 , such that for each " , there is has an affine open cover
E -module ; satisfying ; . Furthermore, if each ; is a finitely an E -module, then we say that is coherent. generated
§4.0. Background: Valuations, Divisors and Sheaves
"
165
#"
The Sheaf of a Weil Divisor. Let be a Weil divisor on a normal variety . We 4 8 will show that determines a sheaf of -modules on . Recall that if 44 8 4 4 ! is open, then consists of all morphisms . Proposition 4.0.16 F4 8 M is a morphism on 4 tells us that an arbitrary element O . if and only if
V4 8 W . It follows that the sheaf is defined by O
"
"
4
R+
" 4 4S8 R 4 8 M 4 O 8
O . F " 4 8
In a similar way, we define the sheaf
W .
by
" 4 8 4 S8 R O . F4 8 M 4 4 O 8 8 W . Proposition 4.0.27. Let be a Weil divisor on a normal variety . Then the sheaf " 4 8 " defined in (4.0.6) is a coherent sheaf of -modules on . " 4 8 is a sheaf of #" -modules. Proof. In Exercise 4.0.10 you will show that
R
4
(4.0.6)
4
" 4 8
The proof is a nice application of the properties of valuations.
BC D.47EF8
R To show that is coherent, we may assume that . Let E be the field of fractions of . It suffices to prove the following two assertions:
4 " 4 8*8 R O . V4 O 8 W is a finitely E generated -module. 4 N " 4 8*8 R ; N for all nonzero O . E . E S such that first bullet, we will prove the existence of . the " 4 For 4 * 8 8 E 4 " 4 8*8 E ! . This willE imply that is an ideal of and hence has a finite basis since is Noetherian. It will follow immediately that 4 " 4 8*8 E is a finitely generated -module. R & ')" ( +( * ' ' . Since C C 4 8 is a proper subvariety of , we can Write . E J 4 98 for every , so find on each ' . Then $ S that vanishes J 4 98 * ' for all . Since 4 98 W , it follows that there is , . with , $
V 4 98 R W . Now let O . 4 #" 4 8*8 . Then V4 O 8 W , so that ,
V 4 P 8 R , V 4 98 4 8 R , V 4 98 R 4 8 W O O O ;
R
since P a sum of effective divisors is effective. By Proposition 4.0.16, we conclude P that O . 4 8 R E . Hence R . E has the desired property.
"
To prove the second bullet, observe that ;
!
and O
. E S
imply that
4 #" 4 *8 8 , W O N 4 N " 4 8*8 For the opposite inclusion, let It is also " easy to see that ; R & '( ( * ' ' and write ! . R 6 .where ' 9 N R P for . 6 ! 354 O 8 for . . Given . 4 N9 " 4 8*8 , 4 4 8 8 " W and J4 8 W R * ' for . 6 . There is no constraint on $ 4 8 for . , implies that $ ; N R
P
.
Chapter 4. Divisors on Toric Varieties
166
but O vanishes on for sufficiently large such that
.
, so that
$ 4O 8
$ 4O 8 $ 4 8
. Hence we can pick
, .
P for .
V4 8 W
4 8 W on . Thus R Since O , it follows easily that P O P O . 4 " 4 8*8 , and then R O has the desired form. " 4 8 The sheaves are more than just coherent; they have the additional prop " 4 8 is invertible. The erty of being reflexive. Furthermore, when is Cartier, ,
"
definitions of invertible and reflexive will be given in Chapters 6 and 8 respectively. 4 8 For now, we record one property which will be crucial in proving that R BC D47EF8 E corre-is reflexive. Let , so that codimension one prime ideals Y ! W ! . Then a Weil divisor on is a sum spond to prime divisors
R
(4.0.7) where all but finitely many W will be written .
$
*W
SW ( (
* W &W
are zero. The discrete valuation coming from
W
SW ( * W W be an effective Weil divisor on 4 " 4FR 8*8 is the ideal of E given
R & ( Proposition 4.0.28. Let R BC D.47EF8 , where E is normal. Then by the intersection
4 " 4FR 8*8 R
E W Y S W ( (
V4 81R W 4 " 4FR 8*8 Proof. First note that O . implies O , so that
V4 8 W W E . O by Proposition 4.0.16, which " 4FR 8*8 is effective. Thus 4 #since E O
shows that follows from
O .
is an ideal of
4 " 4FR 8*8
. The final assertion of the proposition
V4 8 W O $ W 4O 8 W * W O . Y E W
D I ! Y R D I ! Y R for all R , Proposition 4.0.28 reduces to E R SW ( ( E W , which we When
for all
proved in Theorem 4.0.14. We thus have a nice circle of ideas: Proposition 4.0.16 M
r 5= rrrrr r r rrr rrrr Theorem 4.0.14 ks
MMM MMMM MMM M "*
Proposition 4.0.28
Linear equivalence of divisors tells us the following interesting fact about the associated sheaves.
§4.0. Background: Valuations, Divisors and Sheaves
% '
" '
167
"
Proposition 4.0.29. If are linearly equivalent Weil divisors, then 4 &8 and are isomorphic as sheaves of -modules. Proof. By assumption, we have
R '
V4 98
4 " 4 8*8
for some
. F4 8 M
" 4 8
. Then
W O .
4 8 ' 4 98 W O
4 98 ' W O . 4 " 4& ' 8*8 O 4 " 4 8*8 4 " 4'=8*8 Thus multiplication by induces an isomorphism 4 " 8 which is clearly an isomorphism of -modules.
4 8 O
4
The same argument works over any Zariski open set , and the isomorphisms are easily seen to be compatible with the restriction maps.
"
" ' "
% '
"
The converse of Proposition 4.0.29 is also true, i.e., an -module isomor 4 8 4 &8 phism implies that . The proof requires knowing more 4 8 about the sheaves and hence will be postponed until Chapter 8. Exercises for §4.0.
_
4.0.1. Complete the proof of part (b) of Proposition 4.0.3.
_ _ 4
4.0.2. Prove (c) (d) in Theorem 4.0.4. Hint: Let be the maximal ideal of . Since has dimension one, it is regular if and only if has dimension one as a vector space over . For (d) (c), use Nakayama’s Lemma (see [2, Props. 2.6 and 2.8]).
I _
4.0.3. This exercise will study the rings (a) Prove (4.0.2).
X^[
and .
(b) Let be a prime ideal of a ring and let denote the localization of with respect to the multiplicative subset . Prove that is a local ring and that its maximal ideal is the ideal generated by .
If
:
4.0.4. Let be a multiplicative subset of a Noetherian ring is Noetherian.
. Prove that the localization
4.0.5. Let and be Weil divisors on a normal variety. (a) If and are Cartier, show that and are also Cartier. (b) If
, show that
#
is Cartier if and only if
is Cartier.
4.0.6. Complete the proof of Theorem 4.0.14. 4.0.7. Prove that a normal curve is smooth. 4.0.8. Let be a Noetherian normal domain. Prove that the following are equivalent: (a) is a UFD. (b)
<>=?7@
A'n')(C .
(c) Every codimension one prime ideal of
is principal.
Chapter 4. Divisors on Toric Varieties
168
, > y(,S 26+7+6+72 !
i( `z { ,Q' ( ,
corresponds to . Use Theorem 4.0.14 to Hint: For (b) (c), assume that is primary in . Then show and use the Krull Principal Ideal Theorem to show and [2, Prop. 4.8] imply . For (c) (a), let be noninvertible and , + let be the codimension one irreducible components of . If + compare the divisors of and + + using Proposition 4.0.16.
!
, % W' 'B(
4<
ljh O induces a well-defined homomorphism " ')h ' 4.0.10. Let be a Weil divisor on a normal variety . Prove that (4.0.6) defines a sheaf X ' of X -modules. 4.0.11. For each of the following rings , give a careful description of the field of fractions q and show that the ring is a DVR by constructing an appropriate discrete valuation on q . (a) (N/ ` O 2 U. 2 (CT `2 @6zQ &2AKS')(N8 , where K is a fixed prime number. (b) (/m/ 8m8 , the ring consisting of all power series in with coefficient in that have a positive radius of convergence. 4.0.12. The plane curve % d o:# c 4E'M 4 has coordinate ring ( IH d 2 c JA d oy# c 4 . As noted in Example 1.1.15, this is the coordinate ring of the affine toric variety given by the affine semigroup (L/E 2 (W2 -`27+6+6+38 . This semigroup is not saturated, which means that H Jk(NIH p42npo9J is not normal by Theorem 1.3.5. It follows that is not a DVR by Theorem 4.0.4. Give a direct proof of this fact using only the definition of DVR. Hint: The field of fractions of H 4m2 po J is I !n' . If IH p42 po J comes from the discrete valuation , what is !n' ?
4.0.9. Prove that the restriction map .
4.0.13. Let sequence
be a normal variety. Use Proposition 4.0.16 to prove that there is an exact
:#hlX ' #h I ' #h U{ '^#h '^#h 2 where the map I ' hU{ ' is ,.jh zW{ F,Q' and { 'h ' is jh H J . Similarly, prove that there is an exact sequence y#Shl9 X ' #Dh ' #h U{ '^#Dh B{}@1 '^#h `+ be a Weil divisor on a normal affine variety ( 4.0.14. Let ( <>=?7@1 A' . As usual, let q be the field of fractions of . Proposition 4.0.28 describes Z 2EX $ n # 'n' when . Here we will explore the general situation. (a) Let be a codimension one prime of , so that is a DVR. Hence the maximal ideal is principal. Use this to define < q for all . . (b) For arbitrary , prove that < Z 2EX ' 'B( and explain how this relates to Proposition 4.0.28. (c) If is a prime divisor, use part (b) to prove that
, )(.-
"!$# % '& )('(+*
%/
4.0.15. Let be an integral domain with field of fractions 0 . A finitely generated submodule of 0 is called a fractional ideal. If is normal and is a Weil divisor on 1*32465$78 ( , explain why 9 :!# ; )((.-<0 is a fractional ideal.
§4.1. Weil Divisors on Toric Varieties
169
§4.1. Weil Divisors on Toric Varieties
Let be the toric variety of a fan in with . Then is normal of dimension . We will use torus-invariant prime divisors and characters to give a lovely description of the class group of .
The Divisor of a Character. The order of vanishing of a character along a torusinvariant prime divisor is determined by the polyhedral geometry of the fan.
!" #$% !"
By the Orbit-Cone Correspondence (Theorem 3.2.6), -dimensional cones correspond to -dimensional -orbits in . We let denote the set of one-dimensional cones (i.e., the rays) of . Then gives codimension . The orbit closure is a -invariant prime divisor . one orbit The DVR gives the discrete valuation
&#' ,.-0/21 354
(*) &#'
+
6 ) 6 3 487:9 !;=8@
#A !" BC)DE# FG 9 ; KML 7 C N< 9 ; + Proposition 4.1.1. Let be the toric variety of a fan . If the ray #O$ !" minimal generator B ) and K L is character corresponding to HIGJ , then 6 )PK L .RQHTSUBV)XWY@
Recall that the ray has a minimal generator . Also note that when , the character is a rational function in since is Zariski open in .
HIJ
B ) G
has
B ) to a basis Z:[\B ) S]Z"^_S`@`@`@"S]Z`a #Dcb\d:e2fghZ [ jilk a . By Example 1.2.20, m )ncoqprfts: 9vu w [ S wy^x [ S`@`@`@tS w aN x [ z . 9|{ 9}; aP~C[ m ) is defined by w [ N . It follows easily that the DVR is and ()F , - / 1 354 R,\ 41 4 354 9u w [tS`@`@`@"S w a zh
@ Similar to Example 4.0.6, E 9 w [ S`@`@`@tS w a ; has valuation 6 )gh+G> when w a[ S S 9vu w [ S`@`@`@tS w a zq Q w [ WY@ To relate this to 6 )PK L , note that w [ S`@`@`@"S w a are the characters of the dual basis of Z [ BV)S]Z ^ S`@`@`@S]Z a G . It follows that given any HIJ , we have U U K L w
[ L 1 w ^
L 1 `` w a
L 1 w
[ L 1 4 w ^
L 1 `` w a
L 1 @ Comparing this to the previous equation implies that 6 )gK5L8QHTSUB)XW . We next compute the divisor of a character. As above, a ray #$ !" gives: A minimal generator B ) #8FO . A prime + -invariant divisor (D)8 &*#2 on . I> a
Proof. Since is primitive, we can extend of , then we can assume and the corresponding affine toric variety is
Chapter 4. Divisors on Toric Varieties
170
We will use this notation for the remainder of the chapter.
H J
Proposition 4.1.2. For and its divisor is given by
, the character
K L
) [
KL
is a rational function on
,
QHTSUB)XWq():@
( ) are KL , it M KML ( ) ) [ 6 354:K L q():@ K L ) [ Then we are done since 6 3 4:K5L8.RQHTSUB)_W by Proposition 4.1.1. Computing the Class Group. Divisors of the form ) )+() are precisely the divisors invariant under the torus action on (Exercise 4.1.1). Thus >()}i ) [ is the group of -invariant Weil divisors on . Here is the main result of this Proof. The Orbit-Cone Correspondence (Theorem 3.2.6) implies that the . Since is defined and nonzero on the irreducible components of follows that is supported on . Hence
section.
Theorem 4.1.3. The following sequence is exact:
J V < \V< b 5\y
implies that we have an exact sequence
5\V< b 5\V< b \V
§4.1. Weil Divisors on Toric Varieties
171
9 + 8 ; , has zero divisor on 5 , so that 9u J z ; by Proposition 4.0.16. Thus yK L for some 9 ; and HIJ (Exercise 3.3.4). It follows that on O , ( N h+. yK L N K L YS which proves exactness at . Finally, suppose that HIJ with K0L8 ) [ QHTSUBV)Wq() is the zero divisor. Then QHTSUBV)_WN for all #$ !" , which forces H N when the Br) span . This gives the desired exact sequence. Conversely, if the sequence is exact, then one easily sees that the Br) span , which by Corollary 3.3.10 is equivalent to having no torus factors. b is a finitely generated abelian group. In particular, we see that Zg[tS`@`@`@S]Z`a J J R> a QHTSUBCW B0[tS`@`@`@tSUBON> a UQhZ [ SU B hWYS`@`@`@S"QhZ a SUB WU
Examples. It is easy to compute examples of class groups of toric varieties. In practice, one usually picks a basis of , so that and (via the dual basis) . Then the pairing becomes dot product. We list the rays of as with corresponding ray generators . We will think of as the column vector , where the superscript denotes transpose.
N> a #[tS`@`@`@tSU# B
With this setup, the map
J <
B [ S`@`@`@tSUB
in Theorem 4.1.3 is the map 7 > a < >
represented by the matrix whose columns are the ray generators. In other words, . By Theorem 4.1.3, the class group of is the cokernel of this map, which is easily computed from the Smith normal form of .
#$ !"
When we want to think in terms of divisors, we let . prime divisor corresponding to
(
be the
%bjd:e2fg PZ [ Z ^ S]Z ^ m O B[ PZ [ Z ^ St " B ^ Z ^ 'St" 7 > ^ ^ @ v
+
-invariant
Example 4.1.4. The affine toric surface described in Example 1.2.21 comes from the cone . For , is shown in Figure 1 on the next page. The resulting toric variety is the rational normal cone . Using the ray generators and , we get the map given by the matrix
This makes it easy to compute that
b N> g>}@
b
We can also see this in terms of divisors as follows. The class group is generated by the classes of the divisors corresponding to , subject to
( [S( ^
# [ SU# ^
Chapter 4. Divisors on Toric Varieties
172
ρ2
u2
u1 ρ1 Figure 1. The cone
when
RQhZ [ SUB [ Wq( [ RQhZ ^ SUB [ Wq( [ is generated by u ( [ z with u ( Thus b
the relations coming from the exact sequence of Theorem 4.1.3:
QhZ [ SUB ^ Wq( ^ ( [ QhZ ^ SUB ^ Wq( ^ R=( [ ( ^ @ [ z N , giving b N> P> . ^ Example 4.1.5. In Example 3.1.4, we saw that the blowup of 9 at the origin is ^ the toric variety : 9 given by the fan shown in Figure 2. N N
K K
ρ2
ρ0
u2
u0 ρ1
u1
Figure 2. The fan for the blowup of
at the origin
B [ Z [ SUB ^ Z ^ SUB Z [
Z^
The ray generators are corresponding to . By Theorem 4.1.3, the class group is generated by the classes divisors of the subject to the relations
(
( [S( ^S(
K K
( [ ( ( ^ ( _@
§4.1. Weil Divisors on Toric Varieties
b : 9 ^ U
>
173
u ( [ z u ( ^ z I u ( z
. This calculation Thus with generator can also be done using matrices as in the previous example. Example 4.1.6. The fan of has ray generators given by and . Thus the map can be written as
a
B+[ Z[tS`@`@`@"SUBVa Zta
J <
B v }Z [ `` Z a a
> a V a [ [ S`@`@`@tS a V< ! [ `` a S [ S`@`@`@tS a Y@
Using the map
> a [ V < > _S`@`@`@tS a V < `` a S
one gets the exact sequence
* a V a [ V , generalizing Example 4.0.21. It is easy to redo this
calculation using divisors as in the previous example. Example 4.1.7. The class group is isomorphic to
b a { L > ^ . More generally, b { cb lb Y @
You will prove this in Exercise 4.1.2.
Example 4.1.8. The Hirzebruch surfaces are described in Example 3.1.16. The fan for appears in Figure 3, along with the ray generators , , , .
B [ }Z [
B ^ cZ ^ B cZ [ B nR}Z ^
Z^
u1 = (−1, r) u2 u3 u4
Figure 3. A fan with
/
( [ S ( ^ S ( _S ( , with relations N K . = ( [ ( N K . ( [ ( ^ ( @
The class group is generated by the classes of
Chapter 4. Divisors on Toric Varieties
174
u ( [ z and u ( ^ z . Thus ^ b tN> @ N gives b "cb [ { [ N> ^ , which is a special case of
b
It follows that In particular, Example 4.1.7.
is the free abelian group generated by
Exercises for §4.1.
4.1.1. This exercise will determine which divisors are invariant under the -action on . Given and , the -action gives . If is a prime divisor, -action gives the prime divisor . For an arbitrary Weil divisor * , the * . Then is -invariant if * for all . (a) Show that
is
-invariant.
!
(b) Conversely, show that any -invariant Weil divisor can be written as in part (a). Hint: Consider 2#" 4 4 $)( and use the Orbit-Cone Correspondence. 4.1.2. Given fans % in '& ()( and %+* in $&,*()( , we get the product fan
% .- %+* *0/1 2- 13*5461 7% 9 8 ! ;: 3< which by Proposition 3.1.14 is the fan of the toric variety . Prove that = > ;: ?< = > ;: = > 3< / (A@ (!B ( Hint: The product fan has rays C - /D 8 and /D 8 - C * for C .7% 8 FE ( and C * G% * FE ( .
4.1.3. Redo the divisor class group calculation given in Example 4.1.5 using matrices, and redo the calculation given in Example 4.1.6 using divisors. 4.1.4. The blowup of HJI at the = > origin >ML is the toric variety K in Example 3.1.15. Prove that $K $H I ((A@ N .
>ML
7z $P L ! /// !QP I (
L ///
4.1.5. The weighted projective space O $P ! !QP ( , R 7 L /// I 9U fan in &1*N I T N 'P ! !QP ( . The dual lattice is
L //I /
$HJI( of the fan % described *SE , is built from a
L L X X (+7N I T 46 P YY I P I * D I L /// L /// !9Z [& denote the images of L the//standard !Q\ Let Z ! basis \ ! / I !P ( . Define maps are the ray generators of the fan giving O $P ! V A`badc a Lbg I /// !f a !9Z g ( &;^_N I T &e^ 9f !QZ ! L /// L L X I X N I T &e^_N ` ' ! ! I ( c&;^_ P Y I P I V
*W/ ' !
!Q
Show that these maps give an exact sequence
V
D 3& ^ &;^_N I T = > L /// and conclude that iO $P ! ! P ( (j@ N .
8 /
I ]N I T .
The
Z
/
&e^_N<&3^hD
I
§4.2. Cartier Divisors on Toric Varieties
Let be the toric variety of a fan . We will use the same notation as in §4.1, where each gives a minimal ray generator and a -invariant prime divisor . In what follows, we write for a summation over the rays when there is no danger of confusion.
D( )i #$ !"
#T !"
)
B+)
+
§4.2. Cartier Divisors on Toric Varieties
175
Computing the Picard Group. A Cartier divisor and hence
(
)( )S
)
(
on
is also a Weil divisor
) G>
) +) () is Cartier since ( is (Exercise 4.0.5). Let b \i denote the subgroup of consisting of 5 -invariant Cartier divisors. Since K L b for all H J , we get the following immediate by Theorem 4.1.3. Then
corollary of Theorem 4.1.3.
Theorem 4.2.1. The following sequence is exact:
J y< b V< s_ j
5\V< s5j< '@ Our next task is to determine the structure of b . In other words, which + -invariant divisors are Cartier? We begin with the affine case. Proposition 4.2.2. Let il be a strongly convex polyhedral cone. Then: m is the divisor of a character. (a) Every 5 -invariant Cartier divisor on m .N . (b) s: DV< J < b
I 9u FTJ
(
z
Proof. Let . First suppose that -invariant Cartier divisor. By Proposition 4.0.28,
+
m ) S",\ 4 !}(OU E N'S N i 9 K L L + 9 K L 3 9 K L
is an ideal
and
(
(4.2.1)
and
h+ l(
-invariant, we have
by argument used to prove (a)
(d) in Theorem 1.1.16 (Exercise 4.2.1).
#i gives an inclu#G.!" . Thus
Under the Orbit-Cone Correspondence (Theorem 3.2.6), sion of -orbits for all
+
is an effective
. Since
is
or
) 5) ()
&* 5 i &*#2ji *& #2N(D) & 5ji ()g@ )
176
Chapter 4. Divisors on Toric Varieties
*& 5 ( m m m m vcoqprfts g $ r N m m ; . Since ( is effective, $ by Thus ( N h+ for some 9 Proposition 4.0.16, and since is invertible on , we may assume . Then 6 3 4:h+q() 6 h+ 6 354:h+q()8N(@ h+ 3 4 ) ) Here, 3 4 denotes the sum over all prime divisors different from the (O) . The first equality is the definition of h+ , the second inequality follows since , m FT( ) for all and the final equality follows from ( I h+ since #G.!" . This shows that h+ l( , so that E . L with 9 ; and K L ( . Using (4.2.1), we can write $ XK m Restricting to , this becomes K L l h+ , which implies that K L _ m is a morphism on by Proposition 4.0.16. Then K5 L XK5L m imply that KM L _+ C for some . Hence KM L _ is nonvanishing and A m . It follows that in some open set with $ i K L h+ N( @ L and ( have support contained in ) () and every (D) meets Since K L .N( . (this follows from $NFO(*) , we have K finish the proof of (a), let ( be an arbitrary M -invariant Cartier divisor on m . To Since lJ ( is strongly convex), we can find H yF J such that QHTSUB ) W c for all #A.!" . Thus K L is a positive linear combination ( ( K L for sufficiently of the () , which implies that large. The above argument implies that ( is the divisor of a character, so that the same is true for ( . This completes the proof of part (a), and part (b) follows immediately using Theorem 4.2.1. Now fix a point . Since is Cartier, it is locally principal, and in particular is principal in a neighborhood of . Shrinking if necessary, we may , where satisfies . assume that
b\d:e fg PZ [ Z ^ S]Z ^ i|k ^ # [ SU# ^ u( [ z u( ^ z
b m > g> s: m
Example 4.2.3. The rational normal cone is the affine toric variety of the cone . We saw in Example 4.1.4 that . The edges of give prime divisors on , and the computations of Example 4.1.4 show that generates . Since by Proposition 4.2.2, it follows that the Weil divisors are not Cartier.
( [ S ( ^ m b
( [S( ^ Next consider the fan consisting of the cones #V[tSU#q^_S " . This is a subfan of the fan giving , and the corresponding toric variety is , where is the distinguished point that is the unique fixed point of the -action on . The variety is smooth since every cone in is smooth (Theorem 3.1.19).
§4.2. Cartier Divisors on Toric Varieties
177
Since and have the same one-dimensional cones, they have the same class group by Theorem 4.1.3. Thus
s: t. b t.cb cb N> P>}@ It follows that is a smooth toric surface whose Picard group has torsion. w 8 i 9 , which Example 4.2.4. One of our favorite examples is ?A is the toric variety of the cone A b\d:e2fghZ [ S]Z ^ S]Z
[ Z XS]Z ^ Z t ik . The ray generators are
B [ Z [ SMB ^ Z ^ S B} Z [ Z XSMB } Z ^ Z _@ Note that B5[ B BV^ B . Let ( i be the divisor corresponding to B
. In
[ ( [ ^ ( ^ r( +( is Cartier [ } ^ and that b T N> . Since s:T N , we see that the ( are not Cartier, and in fact no positive multiple of ( is Cartier. Exercise 4.2.2 you will verify that
Example 4.2.3 shows that the Picard group of a normal toric variety can have torsion. However, if we assume that has a cone of maximal dimension, then the torsion goes away. Here is the precise result.
? ka
Proposition 4.2.5. Let be the toric variety of a fan in contains a cone of dimension , then is a free abelian group.
s:
. If
(
Proof. By the exact sequence in Theorem 4.2.1, it suffices to show that if is a -invariant Cartier divisor and is the divisor of a character for some , then the same is true for . To prove this, write and assume that , .
( j( N K5L HIGJ Let have dimension
j(
(
) )5()
(
. Since is Cartier, its restriction to Using the Orbit-Cone Correspondence, we have
(
) [
m
is also Cartier.
)( )@
Proposition 4.2.2, so that there is H m . Thisby implies that
8cJ
This is principal on
( K L
such that
) QH SUBV)XW for all #.!"Y@ On the other hand, j( N KML implies that ) RQHTSUBV)W for all # !"Y@
Together, these equations imply
QhjH US B)XW N )nQHTSUB)XW for all #G.!"Y@ The B) span since . Then the above equation forces \HH ( N K L follows easily.
, and
Chapter 4. Divisors on Toric Varieties
178
This proposition does not contradict the torsion Picard group in Example 4.2.3 since the fan in that example has no maximal cone.
Comparing Weil and Cartier Divisors. Here is an application of Proposition 4.2.2. Proposition 4.2.6. Let equivalent:
be the toric variety of the fan . Then the following are
(a) Every Weil divisor on (b) (c)
s: cb . is smooth.
is Cartier.
m i b m < b m
Proof. (a) (b) is obvious, and (c) (a) follows from Theorem 4.0.22. For the converse, suppose that every Weil divisor on is Cartier and let be the affine open subset corresponding to . Since is onto by Theorem 4.0.20, it follows that every Weil divisor on is Cartier. Using from Proposition 4.2.2 and the exact sequence from Theorem 4.1.3, we conclude that induces a surjective map
s m }
H < K L J V< m .
c
) [
>()t@
.!" t # [ S`@`@`@tSU# , this map becomes J y (4.2.2) H y< UQHTSUB)WYS`@`@`@S"QHTSUB)`WUY@ Now define 7 > < by [ S`@`@`@tS [ XB) . The dual map
; 7 J nd: C S >}.< nd: r > S >}5 > Writing
>
is easily seen to be (4.2.2). In Exercise 4.2.3 you will show that is surjective is torsion-free. is injective and (4.2.3) can be extended to a basis of
;
B) `S @`@`@"SUB) $@ The above discussion shows that ; is surjective. Then (4.2.3) implies that the B+) for #G.!" can be extended to a basis of , which implies that is smooth. Then is smooth by Theorem 3.1.19. Proposition 4.2.6 has a simplicial analog. Recall that is simplicial when every is simplicial, meaning that the minimal generators of are linearly independent over k . You will prove the following result in Exercise 4.2.3. Proposition 4.2.7. Let be the toric variety of the fan . Then the following are equivalent:
(a) Every Weil divisor on (b) (c)
has a positive integer multiple that is Cartier.
s:5 has finite index in b + . is simplicial.
§4.2. Cartier Divisors on Toric Varieties
179
In the literature, a Weil divisor is called -Cartier if some positive integer multiple is Cartier. Thus Proposition 4.2.7 characterizes those normal toric varieties for which all Weil divisors are -Cartier.
) )+() .
Describing Cartier Divisors. We can use Proposition 4.2.2 to characterize be the subset of maximal invariant Cartier divisors as follows. Let cones of , meaning cones in that are not proper subsets of another cone in .
i?
Theorem 4.2.8. Let be the toric variety of the fan Then the following are equivalent:
(
and let
(
m for all T$ . J with QH SUBV)XW ) for all #OG.!" . (c) For each T$ , there is H J with QH SUBV)XW. ) for all #G.!" . (d) For each T$ , there is H Furthermore, if ( is Cartier and tH is as in part (c), then: is unique modulo J 5.N GFJ . (1) H H Dd'J . (2) If is a face of , then H Proof. Since ( ) [ )+() , the equivalences (a) (b) (c) follow immediately from Proposition 4.2.2. The implication (c) (d) is clear, and (d) (c) works follows because every cone in is a face of some and if H for , it also works for all faces of . GJ satisfies QHTSUBV)XWR ) for all #O.!" . Then, For (1), suppose that H given H GJ , we have QH SUB ) W ) for all #OG.!" QH TH SUB ) W for all #G.!" QH TH SUBrW. for all BEG H TH G FJ J 5Y@ is unique modulo J 5 . Since H works for any face of , It follows that H H 8 Ddq*J , and (2) follows. uniqueness implies that H of part (c) of the theorem satisfy ( K ~ L for all T . The H m ~ L is local data for ( in the sense of Definition 4.0.12. We Thus P U S K call tH the Cartier data of ( . (a)
(b)
(
is Cartier.
is principal on the affine open subset
The minus signs in parts (c) and (d) of the theorem are related to the minus signs in the facet presentation of a lattice polytope given in (2.2.2), namely
t H J 2 QHTSUBW c
for all facets
of
: @
We will say more about this below. The minus signs are also related to support functions, to be discussed later in the section.
Chapter 4. Divisors on Toric Varieties
180
Nk a n "l V ( ) )5() J with QHTSUB)_W. ) for all #G.!" . (d) For each , there is H Part (1) of Theorem 4.2.8 shows that these H ’s are uniquely determined. in Theorem 4.2.8 is only unique modulo J 5 . Hence In general, each H we can regard H as a uniquely determined element of J J 5 . Furthermore, if
is a face of , then the canonical map J J 5.< J J sends H to H .
When is a complete fan in , part (d) of Theorem 4.2.8 can be recast as follows. Let . In Exercise 4.2.4 you will show is Cartier if and only if: that a Weil divisor
b
There are two ways to turn these observations into a complete description of . For the first, write
5
" [ S`@`@`@S
and consider the map
J J F J J jy< H y< H H @
In Exercise 4.2.5 you will prove the following. Proposition 4.2.9. There is a natural isomorphism
b f
J J <
J J FO @
For readers who know about inverse limits (see [2, p. 103]), a more sophis ticated description of comes from the directed set , where
when is a face of . We get an inverse system where each gives , and the inverse limit gives an isomorphism
J J 5 < J J
b
8S n
~ J J 5Y@
b
ka
The Toric Variety of a Polytope. In Chapter 2, we constructed the toric variety of a full dimensional lattice polytope . If , this means that . As noted above, has a canonical presentation
i J
J
tHIJ 2QHTSUB.W c for all facets of :S where > and B G is the inward-pointing facet normal that is the minimal . The normal fan consists of cones generator of the ray # b\d:e2fgB . , where indexed by proper faces Ri (4.2.4)
cb\d:e2fgB contains vY@ Proposition 2.3.6 implies that the fan Furthermore, the vertices of correspond to the maximal cones in is complete. , and the facets of correspond to the rays in \!" .
§4.2. Cartier Divisors on Toric Varieties
181
B
The ray generators of the normal fan are the facet normals . The corresponding prime divisors in will be denoted . Everything is now indexed by the facets of . The normal fan tells us the facet normals in (4.2.4), but cannot give us the integers in (4.2.4). For these, we need the divisor
(
(4.2.5)
(
( @
B
As we will see in later chapters, this divisor plays a central role in the study of projective toric varieties. For now, we give the following useful result. Proposition 4.2.10.
(
is a Cartier divisor on
and
(
N
.
#
corresponds to a maximal cone , and a ray lies in Proof. A vertex if and only if . But implies that . Note also that since is a lattice polytope. Thus we have such that for all , so that is Cartier by Theorem 4.2.8. You will prove that in Exercise 4.2.6.
! " J Q SUB .W=
N
(
# N q!"
(
In the notation of Theorem 4.2.8, described by the collection
tH
(4.2.6)
a
H
Q SUB W J
is the vertex . Thus
is a vertex of
(
is completely
:@
This is very satisfying and explains why the minus signs in (4.2.4) correspond to the minus signs in Theorem 4.2.8.
u ( z s_ ( c( K L H J H L ( N( K L .N( ~ L (Exercise 4.2.6), so that the divisor class of ( gives all translates of . The divisor ( has many more wonderful properties. We will get a glimpse of this in §4.3 and learn the full power of ( in Chapter 6 when we study ample
, also has a nice interpretation. If The divisor class then for some . In Proposition 2.3.7 we saw that and its translate have the same normal fan and hence give the same toric variety, i.e., . We also have
( (
divisors on toric varieties.
tH
5
Support Functions. The collections that describe -invariant Cartier divisors can be cumbersome to work with. Here we introduce a more efficient computational tool. Recall that the support of a fan is .
Definition 4.2.11. Let
be a fan in
.
:
il
7 g
(a) A support function is a function set of all support functions is denoted
Chapter 4. Divisors on Toric Varieties
182
(b) A support function
is integral with respect to the lattice if \
FEji >8@
The set of all such support functions is denoted Let
(
) )( )
(4.2.7)
tH
be Cartier and let
QH US B ) W. )
for all
o S E . be as in Theorem 4.2.8. Thus
#G.!"Y@
We now describe Cartier divisors in terms of support functions.
be a fan in . ) )+() and tH as above, the function M3 7 qV< k B V< M38Br.RQH SUBrW when BE
Theorem 4.2.12. Let (a) Given
(
is a well-defined support function that is integral with respect to (b)
M3 B)_
(c) The map
)
for all
( < 0 3
#O$ !" , so that ( M3 B)Xq():@ )
induces an isomorphism
.
+co S TY@ is unique modulo FAJ and that Proof. Theorem 4.2.8 tells us that each H H H Dd' OF $F$J . It follows easily that 3 is well-defined. Also, M3 is linear on each since 03 Br\ QH SUBrW for BE , and it is integral with J . This proves part (a), and part (b) follows from the respect to since H definition of 03 and (4.2.7). It remains to prove part (c). First note that .3 Eo j 8S E by part (a). Since ($S $b 5 and OG> imply that M3 M3 3 M3 S the map b 5 < o S T is a homomorphism, and injectivity follows from part (b). To prove surjectivity, take |l o S T . Fix N . Since is integral with respect to , it defines a -linear map 7 OF?< > , which 7 , where coqp e+ 5+F . Since extends to -linear map S >} J J 5YS }d: r J such that BC* QH SUBrW for B c . Then it follows that there is H ( ) M38B)_q() is a Cartier divisor that maps to . b
§4.2. Cartier Divisors on Toric Varieties
183
In terms of support functions, the exact sequence of Theorem 4.2.1 becomes (4.2.8)
H J No S E
J V< o j 8S Ej<
where
s5j< 'S
B < vQHTSUBCW and ) \B ) q ( ) z s5 . Be sure
maps to the linear support function defined by maps to the divisor class you understand the minus signs.
u
Here is an example of how to compute with support functions.
Zg[ Zt^ Z Z ^ Z
k
Example 4.2.13. The eight points are the vertices of a cube in . Taking the cones over the six faces gives a complete fan in . Modify this fan by replacing with . The resulting fan has the surprising property that . In other words, is a complete toric variety whose Cartier divisors are all principal. We will prove by showing that all support functions for are linear. Label the ray generators as follows, using coordinates for compactness:
Z[ Z^ Z Z[ s \ |
k
s
B+[\R!_S S :YS.By^}R!_St _St"YSB R!_St_St "YS.B R! _St _St" B}R!_St _Stv"YSB}R! _Stv_St"YS.B R! _St_Stv"YS.B =R! _Stv_St "Y@
The ray generators are shown in Figure 4. The figure also includes three maximal cones of :
[ cb\d:e2fgB [ SUB ^ SUB _SUB " ^ cb\d:e2fgB [ SUB XSUBSUB t }cb\d:e2fgB [ SUB ^ SUBSUB "Y@ The shading in Figure 4 indicates [ F ^ S [ F _S ^ F . Besides [ S ^ S , the fan has three other maximal cones, which we call left, down, and back. Thus the cone left has ray generators B+^XSUB SUB SUB , and similarly for the other two. u1
σ3 u6
u4
σ2
u2 u8
u7 σ1
u5 Figure 4. A fan
u3
with
/
Chapter 4. Divisors on Toric Varieties
184
o S > . We show that is linear as follows. Since is linear, $ H [ > such that \BrRQH [ SUBrW for B$ [ . Hence the support function B < \BC0 QH [ SUBrW vanishes identically on 5[ . Replacing with this support function, we may assume that . Once we prove everywhere, it will follow that all support functions are linear, and then s N by (4.2.8). Our hypothesis implies that \B [ \B ^ \B " \ B " since B [ SUB ^ SUB _SU B G [ . It suffices to prove \B . \B " \ B t. \ B ". .
Take there is
To do this, we use the fact that each maximal cone has four generators, which must satisfy a linear relation. Here are the cones and the corresponding relations: cone
[ ^
left down back
relation
B [ B }B ^ XB B [ B = XB B B+[ XB BV^ B B ^ B } B B B B } B B B B } B B
\B [ \ B ^ \ B " \ B" , the
Since is linear on each cone and second, third, fourth and fifth relations imply
\B ". \B \B . \ B
B ". \B" B ". \ B tY@ The last two equation give \B . \ B , and substituting these into the first two shows that \BX \B B t. \ B " N .
Since the toric variety of a polytope has the non-principal Cartier divisor of Example 4.2.13 is not the normal fan of any three , its follows that the fan dimensional lattice polytope. As we will see later, this implies that is complete but not projective.
(
A full dimensional lattice polytope function on the normal fan .
i J
Proposition 4.2.14. Assume normal fan . Then the function
i J
leads to an interesting support
is a full dimensional lattice polytope with
7 < k defined by BC.N * eCUQHTSUBCW HI
has the following properties:
§4.2. Cartier Divisors on Toric Varieties
(a)
is a support function for
and is integral with respect to
(b) The divisor corresponding to
185
is the divisor
(
.
defined in (4.2.5).
Proof. First note that minimum used in the definition of compact. Now write
exists because
is
tH J 2QHTSUBW c for all facets of :@ Then ( ( is Cartier by Proposition 4.2.10, and Theorem 4.2.12 shows that the corresponding support function maps B to . It remains to show that BCD o j and B I . Recall that maximal cones of correspond to vertices of , where the vertex gives the . Take B B , where maximal cone G bjd:e2fgB . Then HI implies QHTSUB W c @ (4.2.9) QHTSUBCW. Thus Br | . Since equality occurs in (4.2.9) when H , we
obtain
jBC.
RQ SUBCWY@
o \S E . Furthermore, when
This shows that , as desired.
Q SUB W.
, we have
jB..
We will return to support functions in Chapter 6, where we will use them to give elegant criteria for a divisor to be ample or generated by its global sections. Exercises for §4.2. 4.2.1. Prove (4.2.1) using the argument used to prove (a)
(d) in Theorem 1.1.16.
4.2.2. Prove the assertions made in Example 4.2.4. 4.2.3. Prove (4.2.3) and Proposition 4.2.7. is Cartier if and only if it satisfies 4.2.4. When % is complete, prove that * condition (d) stated in the discussion following Theorem 4.2.8.
4.2.5. Prove Proposition 4.2.9.
zWD {. Hint: The normal fan of
4.2.6. A lattice polytope gives the toric variety X a (9* (a) Prove that for any (b) Prove that
and the divisor V .
from (4.2.5).
is complete.
4.2.7. Let be a -invariant Cartier divisor on . By Theorem 4.2.8, is determined a 8 satisfying condition (2) of the theorem. Given a V , show by a collection / X a & a 8 . Be sure to explain where the ( is given by the collection / that minus sign comes from.
z`{
Chapter 4. Divisors on Toric Varieties
186
4.2.8. Let be the toric variety of the fan % . Prove the following consequences of the Orbit-Cone Correspondence (Theorem 3.2.6). (a)
: // /
$1 ( *
(b) Rays Z
!
.
:
!9Z 7 % 9E$( lie in a cone of % if and only if
4.2.9. Let % be a fan in &
( @
I
and assume that
(a) Fix a cone 1GG% of dimension . Prove that
{ 7,
(A@
%
Y
*
.
has a cone of dimension .
/ / ] 2 )% !Q&:( 4 4 *D 8
(b) Explain how part (a) relates to Example 4.2.13. (c) Use part (a) to give a different proof of Proposition 4.2.5.
4.2.4, but instead of using the lattice generated by \ !Q\ * !\ , 4.2.10. Let 1 be as in Example X X X N '\ X \* X \ ( , where 6! are relatively instead use &1*N7 * \ N \b* N \ * X X X is Cartier. prime positive integers with E . Prove that no multiple of * Hint: The first step will be to find the minimal generators (relative to & ) of the edges of 1 .
= \ !! \*,! \ (.-" . 4.2.11. Let be the toric variety of the octahedron * = > (a) Show that (A@ N$# . (j@ N . (b) Use support functions and the strategy of Example 4.2.13 to show that { 78 L /// 4.2.12. In Exercise projective /// you showed that the weighted = > L 4.1.5, L /// space = O > 'P ! L /!Q/P / I ( has !P ((2@ N . Prove that iO 'P ! !P ('( class group $O $P ! > L{ 7 i/O / $/ P ! !QP I ('(%I I a a !QP ( . maps to the subgroup N - N , where * 7&% $P ! I be a smooth toric variety and let 7 ' G% be a cone of dimension (*) . This 4.2.13. Let gives the orbit closure + , ' ( * -' ( - . In §3.3 we defined the blowup K > . 0/ ( . Prove that { 78 $K >1. 0/ ((A@ { 7 (!BN / 43576 H;: ! /// ! < has Newton polytope 4.2.14. A nonzero polynomial 2 * 98 8 / 8 I 2 (+* = = a 4 6 * D 8?> I BA iZ ( * When dimension , Proposition 4.2.14 tells us that the function @2 g ( has CA a a % { CA 9f !QZ 4 2 ('( is the support function of a divisor on . Here we interpret
as the tropicalization of 2 . The tropical semiring @ ! B%!D ( has operations BE *F% X { $ !( tropical addition( / GDE * tropical multiplication( /// A tropical polynomial in real variables : ! tropical sum ' : : 8 : H * 6 D DYD 5 B Y! 8 I B is a6JI finite D 8 D YYKD 8 I 5 8 8 I D YYMD :Z ( times). For a more compact representation, define a where 6 L and Z *ZG] D YD 5 for a * $ ! /// !Q ( ONJI . Then, using tropical monomial to be * 8 8 8I HI the tropical analog of summation notation, the tropical polynomial is H *QP I 6 D SR ! a * ' ! /// ! ( / I X 8 X H I 6 (a) Show that *F % { !T T 8 Y X I 8 I ( .
§4.3. The Sheaf of a Torus-Invariant Divisor
187
2 is the tropical polynomial / HAQ L * P D9D 8
(b) The tropicalization of our original polynomial
HA
CA
* Prove that . (The D is explained as follows. In general, the coefficients of are Puiseux series, and the tropicalization uses the order of vanishing of the coefficients. Here, the coefficients are nonzero constants, with order of vanishing D .) (c) The tropical variety ofX a tropical polynomial is the set of points in where is X * & * X * , compute the tropical Ivariety of not linear. For :* and show that it consists of the rays in the normal fan of ( . A nice introduction to tropical algebraic geometry can be found in [86].
2
2 8 ) 8
H
8 8
@2
H
HA
§4.3. The Sheaf of a Torus-Invariant Divisor
(
+ ) ") () , - /5 (
If is a -invariant divisor on the normal toric variety , we get the sheaf defined in §4.0. We will study these sheaves in detail in Chapter 6. In this section we will focus primarily on global sections.
,\-/5 (
We begin with a classic example of the sheaf . Example 4.3.1. For , the divisors correpsond to the ray generators of the usual fan for . The computation from Example 4.1.6 shows that . These linear equivalences give isomorphisms
a ( `S @`@`@"S (a a a b > ( N( [ ` ` N( a
, (", ( [ ` ` R, ( a by Proposition 4.0.29. In the literature, these sheaves are denoted , !" . Simi larly, the sheaves , h=( h , > , are denoted , h . Global Sections. Let ( be a -invariant divisor on a toric variety O . We will give two descriptions of the global sections S",.-0/ (OU . Here is the first. Proposition 4.3.2. If ( is a -invariant Weil divisor on , then S", - /5 (U. 9 K L @ 3 S",-/5 (U , then h+ ( implies h+ Proof. If since ( . Since 9u J z is the coordinate ring of M , Proposition 4.0.16 implies E 9u J z . Thus S",-/5 (Uji 9u J z @ The action of 5 on itself induces an action on 9vu J z such that if Gc 0 and H J , then K LI K L PK5L , where the left-hand side of this equation is the action of on K L and the right-hand side is K L % 9 times K L . Then S",-/5 (U is invariant under this action since ( is M -invariant. Using the
argument used to prove (a)
(d) in Theorem 1.1.16, we see that
S",-/5 (U.
-/ 1 / 3
9 K L @
Chapter 4. Divisors on Toric Varieties
188
Since
K L S",.-0/M (U
KML8 ( l , we are done. ) )"() and HIJ , K L ( l
if and only if
The Polyhedron of a Divisor. For is equivalent to
(
QHTSUB ) W ) l
for all
#$ !"YS
for all
#$ !"Y@
which can be rewritten as
QHTSUBV)XW c )
(4.3.1)
This explains the minus signs! To emphasize the underlying geometry, we define (4.3.2)
We say that 3
3l t HIJ ' QHTSUB)XW
)
for all
#$ !" :@
is a polyhedron since it is an intersection of finitely many closed half spaces. This looks very similar to the canonical presentation of a polytope need not (see (4.2.4), for example). However, the reader should be aware that be a polytope, and even when it is a polytope, it need not be a lattice polytope. All of this will be explained in the examples given below.
3
For now, we simply note that (4.3.1) is equivalent to our second description of the global sections. Proposition 4.3.3. If
where 3Ni J
(
HI 3$FJ
-invariant Weil divisor on S",.-0/M (OU 9 K L S L
is a
is the polyhedron defined in (4.3.2).
. This gives
, then
As noted above, a polyhedron is an intersection of finitely many closed half spaces. A polytope is a bounded polyhedron. Examples. Here are some examples to illustrate the kinds of polyhedra that can occur in Proposition 4.3.3. Example 4.3.4. The fan erators For the divisor
: 9 ^ 9 ^ H R w S w cv w c _@
for the blowup of at the origin has ray genand correponding divisors if and only if . , a point lies in
B Z [ Z ^ S0B [ |Z [ SB ^ Z ^ ( _S.( [ S.( ^ 3 ( N( ( [ (D^ QHTSUB "W c QHTSUB5[W c QHTSUB ^ W c The fan and the polyhedron 3 are shown in Figure 5 on the next page. Note that 3 is not bounded. By Proposition 4.3.3, the lattice points of 3 (the dots in ^ , (OU . Figure 5) give characters that form a basis of : 9 YS"
§4.3. The Sheaf of a Torus-Invariant Divisor
189
u2
e2
u0
Σ
PD
u1
e1
Figure 5. The fan and the polyhedron
^ for the Hirzebruch surface ^ has ray generators B [ } Z [ Z ^ S B ^ Z ^ S B RZ [ SB }Z ^ . The corresponding divisors are ( [ S.( ^ S. ( _S.( , and^ Example 4.1.8 implies that the classes of ( [ and ( ^ are a b ^ N> . basis of i k ^ be the corresponding Consider the divisor ([ (D^ , G> , and let if and polyhedron, which is a polytope in this case. A point HI w S lies in
Example 4.3.5. The fan
only if
QHTSUB [ W c QHTSUB ^ W c QHTSUB W l QHTSUB W l
Figure 6 shows
^ , together with shaded areas marked SS
u1
a=1
0
u2
Σ2
[^ w ^ cv w l'@ '@
A B
u4
u3
C
− e2
Figure 6. The fan
e1 − e2
and the polyhedra
. These are related
a=2
a=3
Chapter 4. Divisors on Toric Varieties
190
to the polygons
R_S S
for
by the equations
[
@ } [ Notice that as we increase , the line ^ w ^ corresponding to B [ moves to the right and makes the polytope bigger. In fact, you can see that ^ is the normal for any . For , we get a lattice polytope fan of the lattice polytope ^
^
^
, but its normal fan is not —you can see how the “facet” with inward normal is not a lattice polytope since collapses to a point of . For , vector is a vertex.
^[ Z ^
B^
^
[
3
Chapter 6 will explain how the geometry of the polyhedron relates to the properties of the divisor . In particular, we will see that the divisor from Example 4.3.5 is ample if and only if since these are the only ’s for which is the normal fan .
(
^
( [
( ^
, h , h ( t ( B ( ( 3l a 'S a where DaOik is the standard -simplex. We can think of characters as Laurent monomials L [ `` a , where H R [ S`@`@`@tS a . It follows that a S", h U E 9u Y[ S`@`@`@S a z _2 f rh+ :@
Example 4.3.6. By Example 4.3.1, the sheaf can be written , where the divisor corresponds to the ray generator from Example 4.1.6. It is is straightforward to show that the polyhedron of
The homogenization of such a polynomial is
| w w [ w _S`@`@`@tS w a w tj 9vu w _S`@`@`@"S w a z @
In this way, we get an isomorphism
a S", h U
E 9u w XS`@`@`@"S w a z _
is homogeneous with
2f Vh+.c : @
The toric interpretation of homogenization will be discussed in Chapter 5.
ka
Example 4.3.7. Let be the toric variety of an -dimensional lattice polytope . The canonical presentation of gives the Cartier divisor is the polytope defined in (4.2.5), and one checks easily that the polyhedron that we began with (Exercise 4.3.1). It follows from Proposition 4.3.3 that
i J
3
\S",.- . ( .U.
L
9 K L @
(
§4.3. The Sheaf of a Torus-Invariant Divisor
K.L
Recall from Chapter 2 that the characters toric variety . We will see below that
\S",- . h ( .U.
191
j(
for
L
H F$J
give the projective , so that gives the polytope
9 K L @
KL j, - h ( .
In Chapter 2 we proved that is very ample for sufficiently large, in which is the toric variety . So the characters that realize as a case . In Chapter 6, we will projective variety come from global sections of pursue these ideas when we study ample and very ample Cartier divisors.
S",.- h ( .U
Note also that gives number of lattice points in mul tiples of . This will have important consequences in later chapters. operation sending a -invariant Weil divisor ( i to the polyhedron 3The i J defined in (4.3.2) has the following properties: 3 3 for l . 3 3 TH . 3 i 3. You will prove these in Exercise 4.3.2. The multiple 3 and Minkowski sum 3 are defined in §2.2, and H is translation.
Complete Fans. When the fan is complete, we have the following finiteness result that you will prove in Exercise 4.3.3.
Proposition 4.3.8. Let (a) (b) (c)
3
S",-/V. 9
be the toric variety of a complete fan
, so the only morphisms
5
is a polytope for any
S",-/5 (U .
< 9
-invariant Weil divisor
(
in
are the constant ones.
on
.
has finite dimension as a vector space over
divisor on
9
for any Weil
S",j-v
The assertions of parts (a) and (c) are true in greater generality: if complete variety and is a coherent sheaf on , then (see [89, Vol. 2, §VI.1.1 and §VI.3.4]).
S N
Exercises for §4.3.
4.3.1. Prove the assertion "* 4.3.2.
c Prove the properties of ^
. Then:
9
is any and
made in Example 4.3.7.
listed above.
a
* D 4.3.3. Prove g Proposition 4.3.8. Hint: For part (a), use completeness to show that V a D for all C . For part (b), assume ( * I and suppose a satisfy when f !9Z 4 4 a 4 4^ . Then consider the points on the sphere I - I .
(
S RR
4.3.4. Let % be a fan in & ( with convex support. Then V ( . cone with dual 4 %24 "-
E
4 %24 - & (
is a convex polyhedral
Chapter 4. Divisors on Toric Varieties
192
(a) Prove that 4 %24 is the polyhedron associated to the divisor
H (b) Conclude that 9 !# (9* .
P (c) Use part (b) to prove part (a) of Proposition 4.3.8.
4.3.5. Example 4.3.5 studied divisors on the X HirzebruchX surface consider the divisors * and * * * * . (a) Show that gives the same polygon as .
* D on
*
.
. This exercise will
= = a (c) Show that * 4 1GG% * 8) (( and that * = = a 461 G% * )8('( . Thus and give the same polygon but differ in how their < Cartier data relates to the # $)( is generated by global polygon. In Chapter 6 we will use this to prove that < sections while # ' ( has base points. (b) Since * is smooth, and are Cartier and hence correspond via Theorem 4.2.8 to a 8 3< * and / a 8 ?< * respectively. Compute these collections. collections /
Chapter 5
Homogeneous Coordinates
§5.0. Background: Quotients in Algebraic Geometry Projective space
where
9 ;
acts on
a
is usually defined as the quotient
9 a [
a R 9 a [ " X 9 ;S
by scalar multiplication, i.e.,
_S`@`@`@tS a R _S`@`@`@"S a Y@ a a The above representation defines as a set; making
into a variety requires the notion of abstract variety introduced in Chapter 3. The main goal of this chapter is to prove that every toric variety has a similar quotient construction as a variety.
w 7 w j? <
7 < ; 7 |<
Group Actions. Let be a group acting on a variety . We always assume that for every , the map defines a morphism . When is affine, comes from a homomorphism . We define the induced action of on by
I oqprfts: v
G ; h+ ~C[ w for all w $ . You will check for . In other words, + w \c in Exercise 5.0.1 this gives an action of on . Thus we have two objects: The set -orbits | w w . The ring of invariants T E for all
. To make into an affine variety, we need to define its coordinate ring, i.e., we $
need to determine the “polynomial” functions on if , then
. A key observation is that
0 w w
193
Chapter 5. Homogeneous Coordinates
194
7 < 9
gives a well-defined function . Hence elements of polynomial functions on , which suggests that as an affine variety
give obvious
SM co'pyfts Y@
As shown by the following examples, this works in some cases but fails in others.
^ v
9 ^ o'pyfts 9u S z
D ^
act on Example 5.0.1. Let , where acts . Note that every orbit consists of two elements, with the by multiplication by exception of the orbit of the origin, which is the unique fixed point of the action. The ring of invariants gives the affine toric variety . Hence we get a map
9vu S z
A w v ^
9u ^ S S ^ z
7g9 ^ +^ < oqprfts: 9u S z . A w ^ i 9 ^ ^ where the orbit ^ S` maps to S S . This is easily seen to be a bijection, 9 u S z is the perfect way to make 9 ^ ^ into an affine variety. so that oqprfts: v
This is actually an example of the toric morphism induced by changing the lattice—see Examples 1.3.17 and 1.3.19. Example 5.0.2. Let act on acts , where via In this case, the ring of invariants is
9 ; 9 oqprfts: 9vu w [ S w ^ S w _S w z [ S ^ S S .R [ S ^ S ~C[ _S ~C[ "Y@
9 ;
9vu w [ S w ^ S w S w z 9vu w [ w XS w ^ w XS w [ w S w ^ w z S
which gives the map
7P9 9 ;8V< o'pyfts 9u w [ S w ^ S w _S w z . A w 8ji 9 where the orbit 9 ; [ S ^ S _S maps to [ S ^ XS [ S ^ t . Then we have (Exercise 5.0.2): is surjective. If A w " , then ~C[ r consists of a single 9 ; -orbit which is closed in 9 . ~C[ g consists of all 9 ; -orbits contained in 9 ^ { P 'S g P 'S g {E9 ^ . ~C[ g is infinite. Thus
This looks bad until we notice one further fact (Exercise 5.0.2): The fixed point gives the unique closed orbit mapping to
* 9 'S g , then the 9 ; -orbit For example, if S` R 9 ; S S 'S g\ P S S 'S g 9 ; is not closed since
S S 'S gN . It follows that closed 9 ; -orbits A w 8Y@
under .
We will see that this is the best we can do for this group action.
§5.0. Background: Quotients in Algebraic Geometry
195
9 a [ coqprfts: 9u w S`@`@`@S w a z
9 ;
Example 5.0.3. Let act on by scalar multiplication. Then the ring of invariants consists of polynomials satsifying
w XS`@`@`@"S w a w XS`@`@`@tS w a
9 ; . Such polynomials must be constant, so that 9u w _S`@`@`@tS w a z 9 @ It follows that the “quotient” is o'pyfts 9 , which is just a point. The reason for this a [ . is that the only closed orbit is the orbit of the fixed point * 9 for all
Examples 5.0.2 and 5.0.3 show what happens when there are not enough invariant functions to separate -orbits.
co'pyftsv
The Ring of Invariants. When acts on an affine variety , a natural question concerns the structure of the ring of invariants. The coordinate ring is a ? It clearly finitely generated -algebra without nilpotents. Is the same true for has no nilpotents since . But is finitely generated as a -algebra? This is related to Hilbert’s Fourteenth Problem, which was settled by a famous example of Nagata that need not be a finitely generated -algebra! An exposition of Hilbert’s problem and Nagata’s example can be found in [23, Ch. 4].
9
i
9
9
oqprfts_
is finitely generated, then is an affine variety If we assume that that is the “best” candidate for a quotient in the following sense.
oqprftsv such that is a finitely generated 9 qo prfts_ . be the morphism of affine varieties < i . Then: (a) Given any diagram / ? AA AA AA A where is a morphism of affine varieties such that w . w for and w T , there is a unique morphism making the diagram commute, i.e., . (b) If is irreducible, then is irreducible. (c) If is normal, then is normal. oqprfts_ and that is induced by ; 7 c< . Then Proof. Suppose that ; .Di follows easily from w w for S w l . Thus ; factors uniquely as < h < @ The induced map 7 < clearly has the desired properties. 7
Lemma 5.0.4. Let act on -algebra, and let induced by the inclusion
Chapter 5. Homogeneous Coordinates
196
Part (b) is immediate since is a subring of . For part (c), let be the field of fractions of . If is integral over , then it is also integral over and hence lies in since is normal. It follows that , which obviously equals since acts trivially on . Thus is normal.
coqprfts
F
This shows that has some good properties when generated, but there are still some unanswered questions, such as:
is finitely
< surjective? Does have the right topology? Ideally, we would like m i to be open if ~C[ m }il is open. (Exercise 5.0.3 explores how this works for and only if
Is
7
group actions on topological spaces.)
While is the best affine approximation of the quotient a non-affine variety that is a better approximation?
, could there be
We will see that the answers to these questions are all “yes” once we work with the correct type of group action. Good Categorical Quotients. In order to get the best properties of a quotient map, we consider the general situation where is a group acting on a variety and is a morphism that is constant on -orbits. Then we have the following definition.
7 I<
<
7
Definition 5.0.5. Let act on and let be a morphism that is constant on -orbits. Then is a good categorical quotient if:
m i (a) If
, } m 8< ,- ~C[ m U induces an , m R, - ~C[ m U @ is closed. (b) If i is closed and -invariant, then \i (c) If [ S ^ are closed, disjoint, and -invariant in , then [ and ^ is open, then the natural map isomorphism
are disjoint in
.
We often write a good categorical quotient as properties of good categorical quotients. Theorem 5.0.6. Let
7 I<
7
<
. Here are some
be a good categorical quotient. Then:
(a) Given any diagram
EE EE EE EE "
<
/
§5.0. Background: Quotients in Algebraic Geometry
where and
197
w G
w
is a morphism of varieties such that for , there is a unique morphism making the diagram commute, i.e., .
w T (b) is surjective. m ~C[ m ji (c) A subset il is open if and only if (d) Given points w S , we have w . w F @
is open.
Proof. The proof of part (a) can be found in [23, Prop. 6.2]. The proofs of the remaining parts are left to the reader (Exercise 5.0.4).
Algebraic Actions. So far, we have allowed to be an arbitrary group acting on , assuming only that for every , the map is a morphism . We now make the further assumption that is an affine variety. To define this carefully, we first note that the group of invertible matrices with entries in is the affine variety
7
w < w ay 9 {
<
9 a 9 . i5ar 9
J aPa 9 . 9 a X2f N :@ a 9 o a 9 9 ; a
A subgroup is an affine algebraic group if it is also a subvariety of . Examples include , , , and finite groups.
a 9
If is an affine algebraic group, the connected component of the identity, denoted , has the following properties (see [53, 7.3]):
is a normal subgroup of finite index in
.
is an irreducible affine algebraic group.
An affine algebraic group acts algebraically on a variety defines a morphism
Sw < w
{
< $@
if the
-action
. RiN
Examples of algebraic actions include toric varieties since the torus acts algebraically on . Examples 5.0.1, 5.0.2 and 5.0.3 are also algebraic actions. Algebraic actions have the property that -orbits are constructible sets in This has the following nice consequence for good categorical quotients.
7
Proposition 5.0.7. Let an affine algebraic group , and assume that a good categorical quotient
C~ [ r (a) If O , then (b) induces a bijection
-orbits in
act algebraically on a variety exists. Then:
<
contains a unique closed
closed
.
-orbit.
@ C~ [ C
Proof. For part (a), note that uniqueness follows immediately from part (d) of Theorem 5.0.6. To prove the existence of a closed orbit in , let be
i
Chapter 5. Homogeneous Coordinates
198
C~ [ r is stable under , so we w w has minimal dimension. Note that w is irreducible since is irreducible, and since w is conm m structible, there is a nonempty Zariski open subset of w such that i w . If w is not closed, then w contains an orbit w . Thus i w w i w m @ However, w is irreducible, so that w m l w @ Hence has strictly smaller dimension, a contradiction. Thus w is closed. If [ S`@`@`@"S are coset representatives of , then w w [ shows that w is also closed. This proves part (a) of the proposition, and part (b) follows immediately from part (a) and the surjectivity of . C~ [ r
the connected component of the identity. Then can pick an orbit such that
For the rest of the section, we will always assume that group acting algebraically on a variety .
is an affine algebraic
Good Geometric Quotients. The best quotients are those where points are orbits. For good categorical quotients, this condition is captured by requiring that orbits be closed. Here is the precise result.
7
(a) All
<
Proposition 5.0.8. Let following are equivalent:
-orbits are closed in
be a good categorical quotient. Then the
.
w S , we have w . w and
(b) Given points
(c)
induces a bijection
-orbits in
(d) The image of the morphism is the fiber product
{ -
{
.
<
lie in the same
-orbit
@ {
defined by
@
S w < w S w
Proof. This follows easily from Theorem 5.0.6 and Proposition 5.0.7. We leave the details to the reader (Exercise 5.0.5).
In general, a good categorical quotient is called a good geometric quotient if it satisfies the condtions of Proposition 5.0.8. We write a good geometric quotient as since the points in the variety correspond bijectively to -orbits in .
7
<
§5.0. Background: Quotients in Algebraic Geometry
199
We have yet to give an example of a good categorical or geometric quotient. For instance, it is not clear that Examples 5.0.1, 5.0.2 and 5.0.3 satisfy Definition 5.0.5. Fortunately, once we restrict to the right kind of algebraic group, examples become abundant. Reductive Groups. An affine algebraic group is called reductive if its maximal connected solvable subgroup is a torus. Examples of reductive groups include finite groups, tori, and semisimple groups such as .
o Mar 9
For us, actions by reductive groups have the following key properties.
coqprfts:v
Proposition 5.0.9. Let be a reductive group acting algebraically on an affine variety . Then (a)
9 -algebra. < oqprfts_ . induced by i
is a finitely generated
(b) The morphism 7 quotient.
is a good categorical
In the situation of Proposition 5.0.9, we can write oqprfts_v oqprfts .
Proof. See [23, Prop. 3.1] for part (a) and [23, Thm. 6.1] for part (b).
Examples 5.0.1, 5.0.2 and 5.0.3 involve reductive groups acting on affine varieties. Hence these are good categorical quotients that have all of the properties listed in Theorem 5.0.6 and Proposition 5.0.7. Furthermore, Example 5.0.1 (the action of on ) is a good geometric quotient. This last example generalizes as follows.
^
9 ^
NiR
Example 5.0.10. Given a strongly convex rational polyhedral cone and a sublattice of finite index, part (b) of Proposition 1.3.18 implies that the acts on such that the induced map on coordinate finite group rings is
5i m 1 9vu FJ m z < 9vu F J z i 9vu FGJ z @ m It follows that 7 1 < 1 is a good categorical quotient. In fact,
is a good geometric quotient since the -orbits are finite and hence closed. This completes the proof of part (c) of Proposition 1.3.18. Constructing Quotients. Now that we can handle affine quotients in the reductive case, the next step is to handle more general quotients. Here is a useful result.
7 I< be a morphism of varieties such that 7 ~C[ j< is a good categorical quotient for every , then 7 < is a good categorical
Proposition 5.0.11. Let act on and let that is constant on -orbits. If has an open cover
quotient.
Proof. The key point is that the properties listed in Definition 5.0.5 can be checked locally. We leave the details to the reader (Exercise 5.0.6).
Chapter 5. Homogeneous Coordinates
200
ni
Example 5.0.12. Consider a lattice and a sublattice and let be a fan in . This gives a toric morphism
N
of finite index,
7 1 < 1 @ By Proposition 1.3.18, the finite group R| is the kernel of < + that acts on 1 . Since ~C[ m 1 m 1
, so
T$
for , Example 5.0.10 and Propostion 5.0.11 imply that is a good geometric quotient. This strengthens the result proved in Proposition 3.3.7. It is sometimes possible to construct the quotient of by locally by taking rings of invariants for a suitable affine open cover. If the local quotients patch together to form a separated variety , then the resulting morphism is a good categorical quotient by Proposition 5.0.11. Here are two examples that illustrate the strategy.
7
<
9 ; ^ act on 9 ^ m "m by scalar multiplication, where 9 ^ 9 " [ , where oqprfts 9 u w XS w [ z m 9 ^ A w .coqprfts 9u wrx [ S w [ z m [\ 9 ^ A w [.coqprfts 9u w S w x[ [ z m F m [\ 9 ^ A w w [Y.coqprfts 9u w x [ S w x[ [ z Y@
Example 5.0.13. Let . Then
The rings of invariants are
9u wyx [ S w [ z 9u w [ w z 9u w XS wyx[ [ z 9u w w [ z 9u w x [ S w x[ [ z 9u w [ w " x [ z @ m 9 ; glue together in the usual way to create [ . Since It follows that the M 9 ; -orbits are closed in 9 ^ " , it follows that [ 9 ^ "X 9 ;
is a good geometric quotient.
This example generalizes to show that
a 9 a [ "X 9 ; a [ is a good geometric quotient when 9 ; acts on 9
by scalar multiplication. At the beginning of the section, we wrote this quotient as a set-theoretic construction. It is now an algebro-geometric construction. Our second example shows the importance of being separated.
§5.0. Background: Quotients in Algebraic Geometry
201
^ ~C[ Then 9 ^ " act on 9 " by S`.R S m m 9 [ ; and m F m [ are as in Example 5.0.13.` .Here, the rings of 9vu w x [ S w [ z 9u w w [ z 9vu w _S w x[ [ z 9u w w [ z 9vu w x [ S w x[ [ z 9u w w [ x [ z @ m 9 ; along m \F m [ 9 ; gives the variety obtained from Gluing together two copies of 9 by identifying all nonzero points. This is the non-separated variety m
m [
Example 5.0.14. Let , where , invariants are
constructed in Example 3.0.15.
In Exercise 5.0.7 you will draw a picture of the quotient cannot be separated in this example.
9 ; -orbits that explains why the
<
7
In this book, we usually use the word “variety” to mean “separated variety”. For example, when we say that is a good categorical or geometric quotient, we always assume that and are separated. So Example 5.0.14 is not a good categorical quotient. In algebraic geometry, most operations on varieties preserve separatedness. Quotient constructions are one of the few exceptions where care has to be taken to check that the resulting variety is separated.
2 @2
Exercises for §5.0.
8 on 8 is. Bea morphism sure you
5.0.1. Let act on an affine variety such that : 2 4 57 for all . Show that defines an action of understand why the inverse is necessary. 5.0.2. Prove the claims made in Example 5.0.2.
8
U
5.0.3. Let be a group acting on a Hausdorff topological space, and let be the set U ` ^ U by of -orbits. Define . The quotient topology on is U defined by saying that is open if and only if is open. U (a) Prove that if is Hausdorff, then the -orbits are closed subsets of .
8
is closed and -invariant, then ! "# U (c) Prove that if disjoint, and -invariant in . ! * are disjoint$ in* areU closed,
(b) Prove that if
% '& )(
!
is closed. , then
% *+ )
and
,
5.0.4. Prove parts (b), (c) and (d) of Theorem 5.0.6. Hint for part (b): Part (a) of Defini UU tion 5.0.5 implies that injects into for all open sets . UU Use this to prove that is Zariski dense in . Then use part (b) of Definition 5.0.5. 5.0.5. Prove Proposition 5.0.8. 5.0.6. Prove Proposition 5.0.11.
.*
/D 8 described in Example 5.0.14. 5.0.7. Consider the H action on H (a) Show that with two exceptions, the H -orbits are the hyperbolas Also describe the two remaining H -orbits.
8 8 *
,
D
.
Chapter 5. Homogeneous Coordinates
202
(b) Give an intuitive explanation, with picture, to show that the “limit” of the orbits * as ^_D consists of two distinct orbits.
8 8 (c) Explain how part (b) relates to the non-separated quotient constructed in the example.
such that has 5.0.8. Give an example of a reductive -action on an affine variety a nonempty -invariant affine open set with the property that the induced map U U ^ U U is not an inclusion.
5.0.9. Let a finite group act on such that a good categorical quotient exists. Explain why is a good geometric quotient.
` ^ UU
§5.1. Quotient Constructions of Toric Varieties
Let
be the toric variety of a fan as a good categorical quotient
in
. The goal of this section is to construct
9 for an appropriate choice of affine space 9 , exceptional set i 9 group . We will use our standard notation, where each #$ !" generator By) #8F and a + -invariant prime divisor ()vi
, and reductive gives a minimal .
No Torus Factors. Toric varieties with no torus factors have the nicest quotient constructions. Recall from Proposition 3.3.9 that has no torus factors when is spanned by , , and when this happens, Theorem 4.1.3 gives the short exact sequence
By) #% !"
Dy< J y < ) >()ny< b jV< 'S L ) QHTSUBV)XWq() and b is the class group where H $J maps to KM8 defined in §4.0. We use the convention that in expressions such as ) , ) and , the index # ranges over all #O !" . ) We write the above sequence more compactly as *< J < > [ < b
< }d b YS 98; j< nd > S 9}; < }d hJ S 9n; \y< _S which remains a short exact sequence since nd +! S 9 ; is left exact and 9 ; is divisible. We have natural isomorphisms [ }d > S 9 ; R 9n; [ nd +hJcS 9 ; + S
and we define the group
by
nd C b YS 9 ; Y@
§5.1. Quotient Constructions of Toric Varieties
203
[ V< 9 ; V < + cV< _@
This gives the short exact sequence of affine algebraic groups
v<
(5.1.2)
The Group G. The group defined above will appear in the quotient construction of the toric variety . For the time being, we assume that has no torus factors.
[ 9 ;
The following result describes the structure of . for as a subgroup of the torus
b
Lemma 5.1.1. Let (a) (b)
Ri 9 ;` [
and gives explicit equations
be as in (5.1.2). Then:
is the character group of
.
is a torus, so that is isomorphic to a product of a torus and a finite Abelian group. In particular, is reductive.
Z:[tS`@`@`@tS]Zta of J , we have ) j 9 ;t [ ) )
L 1 4 for all HIJ )j 9 ; [ ) )
1 4 for @ is a finitely generated Abelian group, b Proof. Since b (c) Given a basis
where
is a finite Abelian group. Then
> {
,
{ S 9 ; R 9 ; { nd + S 9 ; Y@ }d r b YS 9 ; nd + > This proves part (b) since nd r S 9 ; is a finite Abelian group. For part (a), gives the map that sends nd C b YS 9 ; note that bh give characters b to E 9 ; . Thus elements of on , and the above isomorphisms make it easy to see that all characters of arise this way. [ For part (c), the first description of follows from (5.1.2) since J < > [ is defined by H lJ < UQHTSUBV)XWU > , and the second description follows immediately. a a Example 5.1.2. The ray generators of the fan for are B [ Z SUB [ a [ Z [ S`@`@`@SUB a Z a . By Lemma 5.1.1, _S`@`@`@"S a \A 9 ; lies in if and only if
L 1 ~ ~ ~
[ L 1 `` a
L 1 a for all HIJ N> . Taking H equal to Z [ S`@`@`@S]Z a , we see that is defined by ~C[ [ `` ~C[ a R_@ Thus
P S`@`@`@XS 9 ; 9 ;S a [ given by scalar multiplication. which is the action of 9 ; on 9
Chapter 5. Homogeneous Coordinates
204
[ { [
Example 5.1.3. The fan for has ray generators in . By Lemma 5.1.1, only if
B [ cZ [ SUB ^ }Z [ SUB [tS !^XS S \A 9 ; lies in if and
[ L 1 ^
L 1 ~
L 1
L 1 ~ R ^ for all HIJ N> . Taking H equal to Z [ S]Z ^ , we obtain [ ^~C[ ~C[ _@ Z"^SUB }Z"^
> ^ N
Thus
P S S S S 9 ; R 9 ; ^ @ ^ Example 5.1.4. gives the rational Let b\d:e2fg qZ [ Z ^ b ]S Z ^ % i Nk >, which normal cone . Example 4.1.4 shows that
P> , so that nd > g>nS 9 ; S i 9 ; is the group of th roots of unity. To see how acts on 9 ^ , one where uses the ray generators B [ qZ [ AZ ^ and B ^ cZ ^ to compute that P S P
(Exercise 5.1.1). This shows that
can have torsion.
9 [ [ 9
The Exceptional Set. In building the quotient representation of , we have the and the affine space . All that is missing is the exceptional set group that we remove from before taking the quotient by .
i 9 [ [ observation is that and 9 depend only on !" . In order to , useful One get we need something that encodes the rest of the fan . We will do this [ using a monomial ideal in the coordinate ring of 9 . Introduce a variable w ) for each # !" and let 9vu w ) X # !" z @ [ . Then oqprfts_ 9 . We call the total coordinate ring of For each cone , define the monomial w wV) @ ) [ Thus w is the product of the variables corresponding to rays not in . Then define
the irrelvant ideal
=
RQ w
A useful observation is that w is a multiple of w if is the set of maximal cones of , then
=
RQ w
} W\i M@
whenever is a face of . Hence,
XWY@
O }
Furthermore, one sees easily that the minimal generators of are precisely the , determines uniquely.
for . Hence, once we have
w
!" O }
§5.1. Quotient Constructions of Toric Varieties
205
= A O }U i 9 [ @
Now define
=
The variety of a monomial ideal is a union of coordinate subspaces. For , the coordinate subspaces can be described in terms of primitive collections, which are defined as follows.
i !" is a primitive collection if: i .!" for all . (a) , there is with ri .!" . (b) For every proper subset
Definition 5.1.5. A subset
Proposition 5.1.6. The = as a union of irreducible components is given by = A w )_# YS
where the union is over all primitive collections
=
i !" .
A w ) S`@`@`@"S w ) =i = w = Q wV) S`@`@`@S w) `W t# [ S`@`@`@"SU# w w) # .!" A w ) tS`@`@`@tS w ) Conversely, every primitive collection gives a maximal coordinate subspace A w )# contained in = , and the proposition follows.
Proof. It suffices to determine the maximal coordinate subspaces contained in . Suppose that is such a subspace and take . and Since vanishes on is prime, the Nullstellensatz im plies is divisible by some , i.e., . It follows that
satisfies condition (a) of Definition 5.1.5, and condition (b) follows easily from the maximality of . Hence is a primitive collection.
In Exercise 5.1.2 you will show that the algebraic analog of Proposition 5.1.6 is the primary decomposition
=
Q w) _ #
WY@
Here are some easy examples. Example 5.1.7. The fan for consists of cones generated by proper subsets of , where are as in Example 5.1.2. Let generate for , and let be the corresponding variable in the total coordinate ring. We compute in two ways:
tB _S`@`@`@tSUB a
a
}
w w
w
B _S`@`@`@tSUB a
B
#
b\d: e2fgB _S`@`@`@"S B S`@`@`@"SUB a .
The maximal cones of the fan are given by Then , so that . Hence
O } RQ w S`@`@`@tS w a2W = " . The only primitive collection is t# _S`@`@`@SU# a , so = A w S`@`@`@tS w a " by Proposition 5.1.6. [ [ { has ray generators B [ cZ [ SUB ^ }Z [ SUB Example 5.1.8. The fan for for a picture of this fan. Each B gives a ray # Z"^SUB }Zt^ . See Example 3.1.12 and a variable w . We compute = in two ways:
Chapter 5. Homogeneous Coordinates
206
w ^ w , and similarly the w [ w S w [ w S w ^ w . Thus = RQ w ^ w S w [ w S w [ w XS w ^ w tWYS ^ 9 ^ { " . and one checks that } " {G9 The only primitive collections are t# [ SU# ^ and t# _SU# , so that } A w [ S w ^ A w _S w " {G9 ^ 9 ^ { " by Proposition 5.1.6. Note also that = RQ w [ S w ^ WCFTQ w _S w W . [ [ A final observation is that 9 ; on 9 by diagonal matrices and = acts [ i 9 ; [ also acts . It follows that hence induces an action on 9 [ on 9 = . We are now ready to take the quotient. as a quotient, we first construct a The Quotient Construction. [ To represent !" be the standard basis toric morphism 9 . Let Zt) '# ==< [ of the lattice > . For each , define the cone cb\d:e2fghZt) X# .!"U i k [ @ b\d:e2fgB [ SUB t
gives the monomial The maximal cone other maximal cones give the monomials
It is easy to see that these cones form a fan
[ [ k . This fan has the following nice properties. in > Proposition [ 5.1.9. Let be the fan defined above. (a) 9 = is the toric variety of the fan . [ < that is compatible with (b) The map Zt) < B) defines a map of lattices > [ the fans in k and in .
7:9 [ }\V<
(c) The resulting toric morphism
is constant on
-orbits.
b\d:e2fPhZ) V# [ !"U 9
Proof. For part (a), let be the fan consisting of and its faces. Note that is a subfan of . Since is the fan of , we get the toric variety of by taking and then removing the orbits corresponding to all cones in . By the Orbit-Cone Correspondence (Theorem 3.2.6), this is equivalent to removing the orbit closures of the minimal elements of . But these minimal elements are precisely the primitive collections . Since the corresponding orbit closure is , removing these orbit closures means removing
[ 9
=
A w ) V#
A w )_#
Y@
i !"
§5.1. Quotient Constructions of Toric Varieties
207
7 > [ < hZt)X B) 5 B ) # .!" [ < C from the exact The map of tori induced by is the map 9 ; sequence i [ will check this in Exercise 5.1.3). Hence, if [ (5.1.2) (you 9 ; and w 9 = , then w w w YS where the first equality holds by equivariance and the second holds since is the [ kernel of 9 ;` < + . This proves part (c) of the proposition. . We can now give the quotient construction of be a toric variety without torus factors and consider the Theorem 5.1.10. Let [ toric morphism 7:9 =< from Proposition 5.1.9. [ Then: = , so that (a) is a good categorical quotient for the action of on 9 9 [ = @ (b) is a good geometric quotient if and only if is simplicial. given by m i , . By Proof. For part (a), we use the open cover of For part (b), define by . Since the minimal generators of are given by , , we have by the definition of . Hence is a compatible map of fans.
7 C~ [ m j < m
Proposition 5.0.11, it suffices to show that
(5.1.3)
is a good categorical quotient.
Ti m m of ~ C [ i b\d:e fghZ`) # .!"U 7 m V< m S where for simplicity we write instead of . The corresponding map ; on coordinate rings can be described as follows: For m , the cone gives the semigroup F> [ P ) G> [ ) l for all # .!" :@ m is the semigroup algebra Hence the coordinate ring of 9 ) w ) 4 ) l for all # .!"_@ This is the localization , where w ) [ wV) . For m , the coordinate ring is the usual semigroup algebra 9u FJ z . To study (5.1.3), first observe that if and , then . It follows that is equivalent to
is the toric variety . Hence (5.1.3) is the toric morphism
Chapter 5. Homogeneous Coordinates
208
[ [ < dualizes to the map J < > [ sending H EJ to . It follows that ; 7g9u FJ z < is given by ; K L ) w )
L 1 4 @ Note that QHTSUB ) W for all # , so that the expression on the right really lies in . m and hence As explained at the beginning of the chapter, the group acts on on its coordinate ring . Since is constant on -orbits, ; factors 9vu FJ z < i @ m < oqprfts is a good Since is reductive, Proposition 5.0.9 tells us that categorical quotient for the action of . Hence it suffices to prove that ; induces
7> QHTSUB ) W G>
The map
an isomorphism
9vu G F J z
@
m since 9 ; [ The map has Zariski dense image in by the exact sequence (5.1.2). It follows that ; is injective. To show that ; is surjective, take and write it as (5.1.4)
w ) w ) 4 satisfies ) N for all
. Then is -invariant where each w if and only if for all , we have w w @ Thus is -invariant if and only if for all whenever . is a character on and hence is an element of its character The map < group b (Lemma 5.1.1). This character is trivial when , so that by (5.1.1), the exponent vector J , i.e., must come from an element H w R Q T H U S B W
for all . But , which implies that
QHTSUB W l for all Y@ FEJ , and the desired surjectivity follows immediately. This shows that H
This completes the proof of part (a).
For part (b), first assume that is simplicial. By Proposition 5.0.8, it suffices to show that distinct -orbits map to distinct points of . Using the open cover , we are reduced to showing that distinct -orbits in map to distinct points of .
m i m
B
Since is simplicial, the ray generators , and by hypothesis, the collection of all ray generators Hence we can write as a disjoint union
B
m
[ k
, are linearly independent, , spans .
,
§5.1. Quotient Constructions of Toric Varieties
B
such that the for coordinates coming from
k [
209
form a basis of . Projection onto the gives an exact sequence
[ [ < > < > < '@ ? ? [ Note also that since the B ,
, form a basis, the map J < > given by H < QHTSUB W [ gives an exact sequence < J < > [ < <
where the cokernel
is finite.
Combining the two above exact sequences with (5.1.1), we get a commutative diagram with exact rows and columns:
(5.1.5)
<
J <
<
J
> [
>
>
b
<
> [ < <
}d S 9 ;
}d
. Then applying Now let
right gives the exact sequence
<
<
of affine algebraic groups. Note that
The group
<
9 ; <
<
<
S9 ;
to the column on the
is finite. m acts on , which we write as m 9 [ { 9n; { 9}; { 9 ; S
{ 9 ; m and , one easily checks that Given w S (5.1.6) S S as a subset of m via the where is the usual product in 9 ; . We regard S { 9 ; . Thus f acts on . Hence we have map < a commutative diagram m / m
O
(5.1.7)
|= || | || ||
Chapter 5. Homogeneous Coordinates
210
m < m
< m
where is constant on -orbits and is constant on -orbits. The induced maps on coordinate rings give a commutative diagram
9u m z II
/
9u m z
II II II I$
9u z
As shown earlier, the top line is an isomorphism. Recall that our proof used the exactness of (5.1.1), which reappears as the middle row of (5.1.5). The same argument, using the next row in (5.1.5), shows that (Exercise 5.1.4). It is a good categorical quotient for and follows that in (5.1.7), is a good categorical quotient for . But is finite, so that the -orbits are closed. Hence is a good geometric quotient, which by Proposition 5.0.8 implies that distinct -orbits map to distinct points in .
< m
m < m
{
9vu m z v9 u z m
m
< m
Consider two distinct -orbits in . Using (5.1.6), one sees that a -orbit intersects in an -orbit (Exercise 5.1.4). Since the -orbits map to distinct points in , the same is true for the -orbits by the commutativity of (5.1.7). This proves that is a good geometric quotient when is simplicial.
m
m
It remains to prove the converse. Suppose that has a non-simplicial cone . as follows. Since is non-simplicial, there We construct a non-closed orbit in is a relation where and for at least one . If we set for , then the one-parameter subgroup
N
[ B
>
. This follows easily from Lemma 5.1.1
CB
[ m i 9 The affine open subset
is nonzero for all
9n; [
is actually a one-parameter subgroup of (Exercise 5.1.4). and
consists of all points whose th coordinate . Hence the point
, where
B
B B
l
l'S
[ m B . The limit exists in since B | lies in . Now consider whenever . Furthermore, if , the th coordinate B is for B lies in m . By assumption, there is all , so that the limit B
with l . This has the following consequences: Since the th coordinate of B is nonzero, the same is true for every element in its -orbit B . Since l , the th coordinate of B B is zero.
§5.1. Quotient Constructions of Toric Varieties
m B
It follows that
B
m
211
is not closed in since its Zariski closure contains the point . This shows that is not a good geometric quotient and completes the proof of the theorem.
B
[
[ [ ;
One nice feature of the quotient is that it is compatible with the tori, meaning that we have a commutative diagram
where the isomorphism on the bottom comes from (5.1.2) and the vertical arrows are inclusions. Examples. Here are some examples of the quotient construction. has quotient representation Example 5.1.11. By Examples 5.1.2 and 5.1.7,
a
;
a
a [ " ; S
where acts by scalar multiplication. This is a good geometric quotient since is smooth and hence simplicial. Example 5.1.12. By Examples 5.1.3 and 5.1.8, has quotient representation
[ { [ " { ^ ; ^ acts via S S S XS
where geometric quotient.
[{ [ ^ { " n; ^ S S "S XS . This is again a good
:s
S`@`@`@tS `a
S`@`@`@tS `a > a [ `` `agBVa tB _S`@`@`@tSUB a _S`@`@`@"S a
Example 5.1.13. Fix positive integers and let with . The images of the standard basis in be the lattice give primitive elements satisfying . Let be the fan consisting of all cones generated by proper subsets of . . As in Example 3.1.17, the corresponding toric variety is denoted Using the quotient construction, we can now explain why this is called a weighted projective space. We have since has rays, and by the argument used in Example 5.1.7. It remains to compute . In Exercise 4.1.5, you computed the short exact sequence
> a [ X> `S`@`@`@tS a B S`@`@`@tSUBa
[
a [
B
" i ; a [
J > a [ > J QHTSUB "WYS`@`@`@S"QHTSUB a W > a [ and _S`@`@`@"S a l> a [ a [ where H ` ` a a > . This shows that the class group is > , and since Z G> maps to G> , it is easy to see that S`@`@`@S ; ; @
Chapter 5. Homogeneous Coordinates
212
a [ given by B S`@`@`@SUB a BXS`@`@`@"S B a Y @ Since is simplicial (every proper subset of tB _S`@`@`@SUB a is linearly independent
This is the action of
in
;
on
), we get the good geometric quotient
_S`@`@`@tS
a
a [ " ; @ XS`@`@`@S a from §2.0 and also gives its
This gives the set-theoretic definition of structure as a variety since we have a good geometric quotient.
m cbjd:e2f Z [ S]Z ^ S]Z [ Z _S]Z ^ Z ji k
Example 5.1.14. Consider the cone find the quotient representation of , we label the ray generators as
Then
;
. To
[ B5 [ Z_[` SMBV^ "Z ^ Z SMB Z"^X SMB Z_[ Z @ and since w . To determine the group i
, note that the exact sequence (5.1.1) becomes
[ S ^ S _ S
where to show that
G>
>
>
[ ^ S S ~C[ S C~ [
m
Hence we get the quotient presentation
> 'S > . This makes it straightforward
;
;@
;X@
w
In Example 5.0.2, we gave a naive argument that the quotient was . We now see that the intrinsic meaning of Example 5.0.2 is the quotient construction of given by Theorem 5.1.10. This example is not a good geometric quotient since is not simplicial.
m
^
Example 5.1.15. Let be the blowup of shown in Figure 1. By Example 4.1.5,
b
ρ2
u2
^
origin, whose fan is ^ at the> with generator u [ z
ρ0
u0 u1
Figure 1. The fan
ρ1
for the blowup of at the origin
§5.1. Quotient Constructions of Toric Varieties
u ^z
u z.
213
;
Hence and the irrelevant ideal is gives the good geometric quotient
^
This
{ " 'S n;S ~C[ S w S . where the ; -action is given by YS w S We also have u YS w S z u w S z . Then the inclusion { "'S }i
Qw S W.
induces the map on quotients
^
{ " 'S n;
where the final isomorphism uses
8; ^ S
; coqprfts u S w S z c oqprfts u w S z Y@ In terms of homogeneous coordinates, S w S w S . This map is the toric ^ ^
b\d:e2f B [ SUB ^ given by . ^ is the blowup of The quotient representation makes it easy to see why ^ at the origin. Given a point of ^ with homogeneous coordinates YS w S , there are two possibilities: N , in which case S w S S w S . This maps to w S ^ and^ is nonzero since w S cannot both be zero. It follows that the part of where N looks like ^ "'S . , in which case 'S w S maps to the origin in ^ . Since 'S w S ^ where w S and w S cannot both be zero, it follows that the part of [ N looks like ^ . ^ by replacing the origin with a copy of is a built from This shows that ~C[ [ which is called the exceptional locus . Since 'S , we see that , ^ induces an isomorphism
^ ^ 'S :@ morphism
induced by the refinement of
Note also that is the divisor corresponding to the ray . You should be able . to look at Figure 1 and see instantly that
[
S 'S
We can also check that lines through the origin behave properly. Consider the line defined by , where . When we pull this back to
, we get the subvariety defined by
w
^
' @ This is the total transform of . It factors as w
w
. Note that defines the exceptional locus, so that once we remove this, we get the curve in
defined by . This is the proper transform of , which meets
^
w
|
Chapter 5. Homogeneous Coordinates
214
'S "S
the exceptional locus at the point with homogeneous coordinates , cor responding to . In this way, we see how blowing up separates tangent directions through the origin.
[
S
The General Case. So far, we have assumed that has no torus factors. When still has a quotient construction, though it is no longer torus factors are present, canonical.
Let be a toric variety with a torus factor. By Proposition 3.3.9, the ray , span a proper subspace of generators , . Let be the intersection . of this subspace with , and pick a complement so that The cones of all lie in and hence give a fan in . As in the proof of Proposition 3.3.9. we obtain
B
{ ; > . Theorem 5.1.10 applies to since B
where span by construction. Note also that Hence
{
It follows that
[
@
,
and
, .
n; [ { n; (5.1.8) [ { ; { ; S In the last line, we use the trivial action of on ; . You will verify the last
isomorphism in Exercise 5.1.5.
; w [ `` w [ { ; [ YS { ; [ [ { and where [ { { w [ `` w Y@ as Hence we can write the quotient presentation of [ @ (5.1.9) We can rewrite (5.1.8) as follows. Using
, we obtain
This differs from Theorem 5.1.10 in two ways:
The representation (5.1.9) is non-canonical since it depends on the choice of the complement . contains and hence has codimension in . In constrast, (this follows from always has codimension in Proposition 5.1.6 since every primitive collection has at least two elements).
w [ `` w {
[
[
[
In practice, (5.1.9) is rarely used, while Theorem 5.1.10 is a common tool in toric geometry.
§5.1. Quotient Constructions of Toric Varieties
/ $
Exercises for §5.1.
%# f 8 % 9E .
5.1.1. In Example 5.1.4, verify carefully that 5.1.2. Prove that tive collections
g
4 C7
`
215
8 4
.
, where the intersection ranges over all primi-
^ & , and in the proof we use the map 5.1.3. In Proposition 5.1.9, we defined N ^ of tori 'H induced by . Show that this is the map appearing in (5.1.2).
: < ;: <
5.1.4. This exercise is concerned with the proof of part (b) of Theorem 5.1.10. @H (a) Prove that the map ^ in (5.1.7) induces an isomorphism H
(b) Prove that a -orbit intersects that the intersection is nonempty.
/ E8
in an
.
-orbit. In particular, you need to show
(c) Prove that $ when Z i D . Hint: Use Lemma 5.1.1. You can give a more conceptual proof by taking the dual of (5.1.1).
- U U @ - and / $ $ N * 4 X * 5.1.6. Consider the usual fan % for O but use the lattice & 8 , where is a positive integer. DG% (a) Prove that the ray generators are Z $ D , Z * $ D $ and L $ odd Z U U ) $ ) even 8 . V / $ U $ U +4 $ . N $ D?% (b) Prove that the dual lattice is = > N BN U N . (c) Prove that
5.1.5. Let be a variety with trivial action. Prove that use this to verify the final line of (5.1.8).
(d) Compute the quotient representation of
.
5.1.7. Find the quotient representation of the Hirzeburch surface
I
-"! ) %# when the fan % is smooth. Hint: Let 1 7% 5.1.8. Prove that acts freely on H i fixes Z i Z . Show that E for C GU 1 and then use and suppose that E for all C . Lemma 5.1.1 to show that -!
5.1.9. Prove that acts with finite isotropy subgroups on H simplicial. Hint: Use the proof of part (b) of Theorem 5.1.10.
)$#&% {C% ! %#)'# 4 % F EY4 . When % )#(% ) 0{ % ! # ) X #* {0!% ,+ X E , or (b) 4 % 9 E4 {0% E and )%# /bD 8 .
5.1.10. Prove that stronger result states that either (a) % *
in Example 3.1.16.
%# when the fan %
is
is a complete simplicial fan, a
This is proved in [6, Prop. 2.8]. See the next exercise for more on part (b).
Chapter 5. Homogeneous Coordinates
216
P X andE , where {C% a finite group $ $ I $ $ I L @O 'P $ $ P I U $ Also prove that the following are equivalent:
5.1.11. Let % be a complete fan such that thatL there is a weighted projective space O O $P P such that
4 % 9L E Y4 $P
. Prove acting on
is a weighted projective space. = > (b) A@ N . (c) & is generated by Z , C 7% 9E . L Z Z I . First show that % is simplicial Hint: Label the ray generators and that there L L L D E . Then are positive integers P P satisfying I P Z and R % 'P P I I consider the sublattice of & generated by the Z and use Example 5.1.13. You will also
(a)
$ $ $ $
$ $
need Proposition 3.3.7. If you get stuck, see [6, Lem. 2.11]. 5.1.12. In the proof of Theorem 5.1.10, we showed that a non-simplicial cone leads to a non-closed -orbit. Show that the non-closed -orbit exhibited in Example 5.0.2 is an example of this construction. See also Example 5.1.14. 5.1.13. Fill in the details omitted in Example 5.1.15.
*
5.1.14. Example 5.1.15 gave the quotient construction of the blowup of D H and used the quotient construction to describe the properties of the blowup. Give a similar treatment for the blowup of H [H I , using the star subdivision described in §3.3.
I
§5.2. The Total Coordinate Ring In this section we assume that coordinate ring
is a toric variety without torus factors. Its total
u w
z [ Roqprfts and contains the irrelevant was defined in §5.1. This ring gives ideal RQ w =W used in the quotient construction of . In this section we will explore how this . ring relates to the algebra and geometry of
is its grading by b [ J b . Given [ where maps to the divisor class b a monomial w , define its degree to be
w 2
f w b b , we let denote the corresponding graded piece of . For $ The grading on is closely related to the group }d b YS ; . Recall that b is the character group of , where as usual b gives The Grading. An important feature of the total coordinate ring . We have the exact sequence (5.1.1) the class group
§5.2. The Total Coordinate Ring
217
; . The action of on [ induces an action on with , we have K ~C[ ' for all
[ (5.2.1) w K ' w for all S w (Exercise 5.2.1). Thus the graded pieces of are the eigenspaces of the action of is homogeneous of degree . on . We say that a Example 5.2.1. The total coordinate ring of is u w XS "S w a z . By Exam a [ a ple 4.1.6, the map b is XS "S a `` a . This gives the grading on u w _S tS w a z where each variable w has degree , so that K
the character the property that given
“homogeneous polynomial” has the usual meaning.
In Exercise 5.2.2 you will generalize this by showing that the total coordinate , where the ring of the weighted projective space is variable now has degree . Here, “homogeneous polynomial” means weighted homogeneous polynomial. Example 5.2.2. The fan for is the product of the fans of and , and by Example 4.1.7, the class group is
w
u w S S w a z
S S `a
a { L
a
L
b a { a cb a { b L ^ The total coordinate ring is u w XS S w a S _S "S z , where L
f w S 2f ' S
(Exercise 5.2.3). For this ring, “homogeneous polynomial” means bihomogeneous polynomial.
^
^
^
Example 5.2.3. Example 5.1.15 gave the quotient representation of the blowup
of at the origin. The fan of is shown in Example 5.1.15 and has ray generators , , , corresponding to variables in the total . Since , one can check that the coordinate ring grading on is given by
[ ^ u YS w S z
f
^
b
and
S w S
2f w 2 f
(Exercise 5.2.4). Thus total coordinate rings can have some elements of positive degree and other elements of negative degree. The Toric Ideal-Variety Correspondence. For -dimensional projective space a homogeneous ideal defines a projective variety This generalizes to more general toric varieties as follows.
Ei
We first assume that
u w _S S w a z
[
vi
a
a
, .
is simplical, so that we have a good geometric quotient
Chapter 5. Homogeneous Coordinates
218
~C[
w
~C[
gives homogeneous by Theorem 5.1.10. Given , we say a point coordinates for . Since is a good geometric quotient, we have . for some . Thus all homogeneous coordinates for are of the form
w
w be homogeneous Now let be the total coordinate ring of and let for the b -grading on , say . Then w K ' w by (5.2.1), so that w N for one choice of homogeneous coordinates of if and only if w for all homogeneous coordinates of . It follows that the . We can use this to define subvarieties of equation is well-defined in
as follows.
Proposition 5.2.4. Let . Then: (a) If
i
be the total coordinate ring of the simplicial toric variety
is a homogeneous ideal, then
w w N . is a closed subvariety of arise this way. (b) All closed subvarieties of
i
[ w [ is a closed -invariant subset of categorical quotient (Definition 5.0.5), Conversely, given a closed subset ~C[ ji
Proof. Given
for all
as in part (a), notice that
w for all . By part (b) of the definition of good is closed in . i , its inverse image [ is closed and -invariant. Then the same is true for the Zariski closure ~C[ i [ ~C[ Ei is a homogeneous ideal It follows without difficulty that satisfying . Example 5.2.5. The equation w gives the -invariant closed subvariety w ji which is easily seen to be the prime divisor . This shows that
always has a global equation, though it fails to have local equations when Cartier (see Example 4.2.3).
a
a
is not
Classically, the Weak Nullstellensatz gives a necessary and sufficient condition for the variety of an ideal to be empty. This applies to and as follows:
For
a : Given an ideal i
u w [ S tS w a z , in a
§5.2. The Total Coordinate Ring
219
a : Given a homogeneous ideal i u w XS S w a z , in a Q w S S w a W i for some l
For
The toric version of the weak Nullstellensatz uses the irrelevant ideal .
Qw
}W\i
Proposition 5.2.6 (The Toric Weak Nullstellensatz). Let variety with total coordinate ring and irrelevant ideal homogeneous ideal, then
Proof. Let
ji
in
[
i
i
be a simplicial toric . If is a
i
l i
for some
the affine variety defined by denote in [
. Then:
ji i for some [ l'S where the last equivalence uses the Nullstellensatz in . a and a , the irrelevant ideal is Q W i u w [ S "S w a z and Q w _S S w a Wji For u w S tS w a z respectively. Furthermore, for a , the grading on u w [tS tS w a z is trivial, so that every ideal is homogeneous. Thus the toric weak Nullstellensatz a and a . implies the classical version of the weak Nullstellensatz for both
a
a
The relation between ideals and varieties is not perfect because different ideals can define the same subvariety. In and , we avoid this by using radical ideals:
a : There is a bijective correspondence a radical ideals i u w [ S "S w a z closed subvarieties of For a : There is a bijective correspondence a homogeneous ideals closed subvarieties of i radical Q w S tS w a W\i u w S "S w a z
For
Here is the toric version of this correspondence. Proposition 5.2.7 (The Toric Ideal-Variety Correspondence). Let cial toric variety. Then there is a bijective correspondence
closed subvarieties of
Ei
i
i
radical homogeneous ideals
\i
be a simpli-
Proof. Given a closed subvariety , we can find a homogeneous ideal with by Proposition 5.2.4. Then is also homogeneous and satisfies , so we may assume that is radical. Since
Y S
Chapter 5. Homogeneous Coordinates
220
$i is a radical homogeneous ideal satisfying . This proves surjectivity. To prove injectivity, suppose that S i are radical homogeneous ideals in . Then with [ [ i implies that is contained in and . However, S
we conclude that
Hence the above equality implies
so that
by the Nullstellensatz since
YS
and
are radical.
For general ideals, another way to recover injectivity is to work with closed subschemes rather than closed subvarieties. We will say more about this in the appendix to Chapter 6.
When is not simplicial, there is still a relation between ideals in the total coordinate ring and closed subvarieties of .
i
Proposition 5.2.8. Let (a) If
be the total coordinate ring of the toric variety
is a homogeneous ideal, then
there is
w
~C[
with
. arise this way. (b) All closed subvarieties of
wN
for all
. Then:
is a closed subvariety of
Proof. The proof is identical to the proof of Proposition 5.2.4.
~C[
The main difference between Propositions 5.2.4 and 5.2.8 is the phrase “there is ”. In the simplicial case, all such are related by the group , while this may fail in the non-simplicial case. One consequence is that the ideal-variety correspondence of Proposition 5.2.7 breaks down in the nonsimplicial case. Here is a simple example.
w
m
w
n;
Example 5.2.9. In Example 5.1.14 we described the quotient representation of for the cone , and in Example 5.0.2 we saw that the quotient map
b\d:e2f Z[tS]Z"^_S]Z_[
Z S]Zt^
Z i k
m w i is given by [tS ^_S S [ S ^ S [ S ^ . Note that the irrelevant u w [ S w ^ S w _S w z . ideal is
§5.2. The Total Coordinate Ring
[
The ideals contained in
221
^
Qw [ Sw ^ W
Q w _S w W m [ ^ { "^ " ^ " { "
are radical homogeneous ideals and that give the same subvariety in :
[ ^
m
m
Thus Proposition 5.2.7 fails to hold for this toric variety.
Local Coordinates. Let be an -dimensional toric variety. When smooth cone of dimension , we get an affine open set
contains a
m i with m a a are compatible with the homogeneous The usual coordinates for [ in the following sense. The cone gives the map [ coordinates for that sends [ to the point [ defined by
[
[
otherwise
Proposition 5.2.10. Let be a smooth cone of dimension and let be defined as above. Then we have a commutative diagram
[ / [ m / S
where the vertical maps are the quotient maps from Theorem 5.1.10. Furthermore, the vertical map to the left is an isomorphism.
~C[ m
m
m
Proof. We first show commutativity. In the proof of Theorem 5.1.10 we saw that . Since the image of lies in , we are reduced to the diagram
[ << << << <
m
/
m
Since everything is affine, we can consider the corresponding diagram of coordinate rings
uw
z
o
gNNN NNN NNN NN
uw
o7 ooo o o ooo ooo
u J z S
z
Chapter 5. Homogeneous Coordinates
222
w L and ; [ w
L for ; , and commutativity follows. For the final assertion, note that ; is an isomorphism since the
[ It is clear that ; ;
where ;
form a basis of
by our assumption on . This completes the proof.
[
m i m
i
vJ
,
i
.
,
i
It follows that if a closed subvariety is defined by an ideal , then the affine piece is defined by the dehomogenized ideal obtained by setting , in all polynomials of . We ,
will give examples of this below, and in §5.3, we will explore the corresponding notion of homogenization.
uw
Proposition 5.2.10 can be generalized to any cone (Exercise 5.2.5).
z
w
satisfying
^
^
Example 5.2.11. In Example 5.1.15 we described the quotient construction of the blowup of at the origin. This variety can be expressed as the union are as in Example 5.1.15. , where
m
m
[ S ^ ^ ^ The map
S w S
w S
in homogeneous cois given by ordinates. Combining this with the local coordinate maps from Proposition 5.2.10, we obtain
m
i
^ S w ^ S
w S i S S YS ^ , where w S w ^ . We Consider the curve w S in the plane ^ study this on the blowup using local coordinates as follows: On m , we get w S , i.e., w ^ ^ w . Since | . defines the exceptional locus, we get the proper transform w On m , we get Y S , i.e., ^ ^ ^ , with proper ^ N . transform ^
m
YS w S
Hence the proper transform is a smooth curve in . This method of studying the blowup of a curve is explained in many elementary texts on algebraic geometry, such as [83, p. 100].
^
We relate this to the homogeneous coordinates of as follows. Using the above map , we get the curve in defined by , i.e., . Hence the proper transform is . Then: Setting gives the proper transform on .
w
^ ^ w
^
^
w S R ^ w
w N m Setting w gives the proper transform ^ N on m .
Hence the “local” proper transforms computed above are obtained from the homo geneous proper transform by setting appropriate coordinates equal to .
§5.2. The Total Coordinate Ring
223
Exercises for §5.2. 5.2.1. Prove (5.2.1).
:8 $ $ 8 I <
8
L$ $PI
5.2.2. L Show that the total coordinate ring of the weighted projective space P . Hint: See Example 5.1.13. is H where R
O 'P
5.2.3. Prove the claims made about the total coordinate ring of the product O in Example 5.2.2.
I - O
made
5.2.4. Prove the claims made about the class group and the total coordinate ring of the * blowup of O at the origin made in Example 5.2.3.
5.2.5. Let be the toric variety of the fan % and assume as usual that has no torus 8 . Consider a full subfan % is full % / 1 0 % 4 1 9 E % 9 E if factors. A subfan % % % with the property that has no torus factors.
(a) Define the map given by
` H
^ H by sending $ to the point C % F E E
otherwise
-!
Prove that there is a commutative diagram H % /
H
-! %# /
$
where the vertical maps are the quotient maps from Theorem 5.1.10. (b) Explain how part (a) generalizes Proposition 5.2.10.
C{ % * # in H 5.2.6. The quintic 8 0{ %
(c) Use part (a) to give a version of Proposition 5.2.10 that applies to any cone 1 satisfying .
*
1 %
has a unique singular point at the origin. We will resolve the singularity using successive blowups.
>ML
*
(a) Show that the proper transform of this curve in K $H is defined by This uses the homogeneous coordinates from Exercise 5.2.3. : < (b) Show that the proper transform is smooth on but singular on .
$ 8 $
* 8 #
D
.
$ $8 $ , R iZ 'D E \ i E D R 9E E R FE ) 8 $ c^ $ iZ; Z * defines $ $ a toric $ $ ^_H *$ and use this (d) Show that $Z morphism 8$ $ $8 $ * * D. to show that the proper transform of the quintic in H is defined by Z3 8 # (e) Show that the proper transform is smooth by inspecting it in local coordinates. * * IT , (WE an integer. 5.2.7. Adapt the method Exercise 5.2.6 to desingularize 8 5.2.8. Given an ideal in a commutative ring , its Rees algebra is the graded ring L : < $ : <
Z (c) Subdivide 13* to obtain a smooh fan % . The toric variety = has > variables A@ N * with where Z corresponds to the ray that subdivides 1e* . Show that
Chapter 5. Homogeneous Coordinates
224
. There is also the extended Rees algebra : $ < 43 : $ < $ E , so that elements of where for #[D . These rings are graded by letting R $ have degree D . Seeg [13, 4.4] for more< about Rees algebras. (a) When f 8 $ H;: 8 $ < , prove that the extended Rees algebra : $ < is the polynomial ring H;: # 8 $ $ L
where is a new variable and
(b) Prove that the ring of part (a) is isomorphic to the total coordinate ring of the blowup * of H at the origin. (c) Generalize parts (a) and (b) to the case of
f 8 $ $ 8 g H;: 8 $ $ 8 < . I I
§5.3. Sheaves on Toric Varieties
Given a toric variety , we show that graded modules over the total coordinate . We continue to ring
give quasicoherent sheaves on assume that has no torus factors.
u w
z
Graded Modules. The grading on
such that
i
. for all
Definition 5.3.1. An -module
Ei
gives a direct sum decomposition
is graded if it has a decomposition
such that for all is the graded -module satisfying
for all
. Given
, the shift
.
The passage from a graded -module to a quasicoherent sheaf on requires some tools from the proof of Theorem 5.1.10. A cone gives the monomial
, and by (5.1.4), the map induces
an isomorphism where is the localization of at . Since monomials are homogeneous, is also graded by , and its elements of degree are precisely its -invariants (Exercise 5.3.1), i.e., . Hence the above isomorphism becomes
w w
(5.3.1)
;
w w
i
z w
u
; u z
These isomorphisms glue together just as we would hope.
§5.3. Sheaves on Toric Varieties
225
Lemma 5.3.2. Let
be a face of . Then and there is a commutative diagram of isomorphisms
u
/
z
/
u
z
,
, when , and is positive when
. This means that . Taking elements of degree zero commutes with localization, hence . The vertical maps in the diagram come from (5.3.1), and the horizontal maps are localization. In Exercise 5.3.2 you will chase the diagram to show that it commutes.
Proof. Since
From Modules to Sheaves. We now construct the sheaf of a graded module.
m i
Proposition 5.3.3. Let be a graded -module. Then there is a quasicoherent sheaf on such that for every , the sections of over are
m
Proof. Since is a graded -module, it is immediate that is a graded module. Hence is an -module, which induces a sheaf on . The argument of Lemma 5.3.2 applies verbatim to show that
m oqp ts u
Thus the sheaves construction.
z qo p ts
patch to give a sheaf
on
which is quasicoherent by
uw
w z
with the Example 5.3.4. The total coordinate ring of is standard grading where every variable has degree . The quasicoherent sheaf on associated to a graded -module was first described by Serre in his foundational paper Faisceaux alg´ebriques coh´erents [91], called FAC for short.
An important special case is when is a finitely generated graded -module. We will need the following finiteness result to understand the sheaf .
is a finitely generated -module for every . Proof. If " , there is nothing to prove. Otherwise, we can find a monomial w w . Consider rational polyhedral cone { k l for all i { k By Gordan’s Lemma, { is a finitely generated semigroup. Let the gener . Then the monomials ators with last coordinate equal to be w
for generate as a -module (Exercise 5.3.3).
Lemma 5.3.5.
Chapter 5. Homogeneous Coordinates
226
Here are some coherent sheaves on
Proposition 5.3.6. The sheaf graded -module.
on
.
is coherent when
is a finitely generated
Proof. Because is graded, we may assume its generators are homogeneous of degrees . Given , it follows immediately that is finitely generated over with generators in the same degrees. However, we need to be careful when taking elements of degree . Multiply a generator of degree by (finitely many by the previous lemma). Doing the -module generators of
this for all gives finitely many elements in that generate as an -module (Exercise 5.3.3). It follows that is coherent.
~
Given denoted
. Then:
.
u z , then
is a Weil divisor satisfying
m
Proof. The sections of
m are over
. Since the open cover m of satisfies m
for sheaf axiom gives the exact sequence
(a) There is a natural isomorphism (b) If
, the shifted -module gives a coherent sheaf on . This is a sheaf we already know.
Proposition 5.3.7. Fix
m m , the
w
The localization of all Laurent monomials has a basis consisting of degree such that for all . Then the exact sequence implies
of degree that basis consisting of all Laurent monomials
. These are precisely the monomials in such that for all of degree , which gives the desired isomorphism .
R
w
u z
We turn to part (b). Given a Weil divisor ,
with . By the above we need to construct a sheaf isomorphism description of the sections over , it suffices to prove that for every , we have isomorphisms
m
(5.3.2) compatible with inclusions
m
m Di m
Gi
induced by
in .
§5.3. Sheaves on Toric Varieties
m
227
m
To construct this isomorphism, we apply Proposition 4.3.3 to
A lattice point
w
When
w
for
since
w
w
gives the Laurent monomial
(5.3.3)
to obtain
w
, and in fact
, this monomial lies in
u z
We claim that map induces the desired isomorphism (5.3.2). If map to the same monomial, then
for all . This
since has no torus factors. Furthermore, if implies
such that , so that there is , then is a monomial in , for ,
for all . Since hence . . Then maps to
for This defines an isomorphism (5.3.2) which is easily seen to be compatible with the inclusion of faces. Example 5.3.8. For we have the standard grading by with . Then is the sheaf associated to for . The classes of the toric divisors correspond to , so that
u
z u
``
Thus
l
Also note that when
m
uw
, we have
w w
w z
w
is a canonical model for
z
w
`` . Compare Example 4.3.1.
Thus global sections of are homogeneous polynomials in degree . Compare Example 4.3.6.
w w
of
Sheaves versus Modules. An important result is that all quasicoherent sheaves on come from graded modules. Proposition 5.3.9. Let
be a quasicoherent sheaf on
(a) There is a graded -module (b) If
is coherent, then
. Then:
such that
.
can be chosen to be finitely generated over .
The proof will be given in the appendix to Chapter 6 since it involves tensor products of sheaves from §6.0.
Chapter 5. Homogeneous Coordinates
228
is surjective (up to isomorphism), it is far from Although the map injective. In particular, there are nontrivial graded modules that give the trivial sheaf. This phenomenon is well-known for , where a finitely generated graded module over gives the trivial sheaf on if and only if for (see [41, Ex. II.5.9]). This is equivalent to
N
uw
w z
w
w
for (Exercise 5.3.4). Since is the irrelevant ideal for , this suggests a toric generalization. In the smooth case, we have the following result.
w
w
be the irrelevant ideal of for a smooth toric Proposition 5.3.10. Let variety , and let be a finitely generated graded -module. Then if and only if for .
m
N
?
N
if and only if it vanishes on each affine open subset Proof. First observe that . But on an affine variety, the correspondence between quasicoherent if and only sheaves and modules is bijective (see [41, Cor. II.5.5]). Hence if for all .
N
w
First suppose that for some . Then , which easily implies that . Then follows from the previous paragraph. This part of the argument works for any toric variety.
u z ~ ~ w w
For the converse, we have for all . Given , we will show that for some , which will imply for , where since is finitely generated. Let . Since is smooth, there such that (this is part for all , we may assume that of the Cartier data for ). Replacing with and observe that . Now set
w
w w
Furthermore, w w
only w Since w involves
w w in , we can find w
for power of . Hence multiplying the above equation by some , as desired.
w l
w
w
w
has degree . Hence which by the definition of localization implies that there is
with
,
w w is a
such that implies
in
w c
for
Unfortunately, the situation is more complicated when is not smooth. Here is an example to show what can go wrong when is simplicial. Example 5.3.11. The weighted projective space has total coordinate ring , where have degree and has degree , and the irrelevant
uw
z
w
§5.4. Homogenization and Polytopes
229
w
. The graded -module ideal is has only elements of odd degree. Then since has degree , and it is clear that . It follows that , yet one easily checks that for all . Thus Proposition 5.3.10 fails for .
w
N l
Exercise 5.3.5 explores a version of Proposition 5.3.10 that applies to simpliis replaced with the weaker cial toric varieties. The condition that . condition that for all
s
N
We will say more about the relation between quasicoherent sheaves and graded -modules in the appendix to Chapter 6.
Exercises for §5.3.
: , prove 8
< , where 8
5.3.1. As described in §5.0, the action of on H induces an= action of on the total ` > j^_H . coordinate ring . Also recall that is a homomorphism (a) Given
8
(b) Show that
( 8 L
and
.
.
and that a similar result holds for the localization
5.3.2. Complete the proof of Lemma 5.3.2.
;: 8 L $ $ 8 < 8 E for all , and let be a finitely generated _D if and only if %# D for _D . 5.3.5. Let be a simplicial toric variety and generated graded D if and only if %# let D beforaallfinitely _D and { % . module. Prove that
5.3.3. Complete the proofs of Lemma 5.3.5 and Proposition 5.3.6. 5.3.4. Let H I whereD forR graded -module. Prove that
5.3.6. Let be a smooth toric variety. State and prove a version of Proposition 5.3.10 that applies to arbitrary graded -modules . Also explain what happens when is simplicial, as in Exercise 5.3.5.
§5.4. Homogenization and Polytopes
The final section of the chapter will explore the relation between torus-invariant divisors on a toric variety and its total coordinate ring. We will also see that when comes from a polytope , the quotient construction of relates nicely to the definition of projective toric variety given in Chapter 2.
Homogenization. When working with affine and projective space, one often needs to homogenize polynomials. This process generalizes nicely to the toric context. The full story involves characters, polyhedra, divisors, sheaves, and graded pieces of the total coordinate ring.
A Weil divisor
on gives the polyhedron for all
Chapter 5. Homogeneous Coordinates
230
by Proposition 4.3.3, which tells us that the This is linked to the sheaf global sections of are spanned by characters coming from lattice points , i.e., of
This relates to the total coordinate ring
Given , the -homogenization of is the monomial
w
uw
w
defined in (5.3.3). The inequalities defining guarantee that Here are the basic properties of these monomials.
u z
(a) For each
w
z
as follows.
w
Proposition 5.4.1. Assume that has no torus factors. If and is the divisor class of , then:
and
lies in
.
are as above
, the monomial
(b) The map sending the character of induces an isomorphism
lies in
.
to the monomial
w
Proof. Part (a) follows from the proof of Proposition 5.3.7. As for part (b), we use the same proposition to conclude that
One easily sees that this isomorphism is given by
w
.
Here are some examples of homogenization. Example 5.4.2. The fan for has ray generators
and . This gives variables . Since for and divisors for , the character of is the Laurent monomial
.
, has polyhedron For a positive integer , the divisor where is the standard -simplex. Given , its homogenization is
w
R
w `` w
w w ~ ~ ~ w T`` w w w w `` ww with respect to w . which is the usual way to homogenize w
§5.4. Homogenization and Polytopes
231
u z
This monomial has degree , in agreement with Proposition 5.4.1. The proposition also implies the standard fact that monomials . of degree in correspond to lattice points in Example 5.4.3. For , we have ray generators with corresponding variables and divisors . Given nonnegative is the . The polyhedron integers , we get the divisor , and given rectangle with vertices , the
w
w
{
Laurent monomial homogenizes to w w ~ w w ~ w
w
w w w
w
w
which is the usual way of turning a two-variable monomial into a bihomogeneous and monomial of degree (remember that ). Thus monomials of degree correspond to . See Exercise 5.4.1. lattice points in the rectangle
w
w w
w
Example 5.4.4. The fan for is shown in Example 5.1.15, and its total coordinate ring is described in Example 5.2.3. If we pick , is defined by the inequalities then the polyhedron
u w z
l
N
is the first quadrant in and , , the monomial homogenizes to
Since form a basis of . Given
w
w w
w w
correspond to the variables . For example, the singular cubic homogenizes to , which is the equation enountered in Example 5.2.11 when resolving the singularity of this curve.
where the ray generators
N
N
One thing to keep in mind when doing toric homogenization is that characters (in general) or Laurent monomials (in specific examples) are intrinsically defined on the torus or . The homogenization process produces a “global object” relative to a divisor that lives in the total coordinate ring or, via Proposition 5.4.1, in the global sections of . from Proposition 5.4.1 is compatible The isomorphism
w
;
with linear equivalence and multiplication. Let us explain each of these separately. First suppose that and are linearly equivalent torus-invariant divisors. This means that . Proposition 4.0.29 implies for some that induces an isomorphism (5.4.1)
Chapter 5. Homogeneous Coordinates
232
Turning to the associated polyhedra, we proved , then An easy calculation shows that if
w
w
in Exercise 4.3.2.
(Exercise 5.4.2). Hence (5.4.1) fits into a commutative diagram of isomorphisms
/
PPP PPP P(
(5.4.2)
u z u z
n nnn v nn n
Here, and the “diagonal” maps are the isomorphisms from Proposition 5.4.1. You will verify these claims in Exercise 5.4.2. It follows that gives a “canonical model” for , since the
latter depends on the particular choice of divisor to give a “canonical model” for the polyhedron
u z
Now consider multiplication. Let and . Then and set , in
u z
in the class . It is also possible (Exercise 5.4.3).
be torus-invariant divisors on induces a -linear map
such that the isomorphisms of Proposition 5.4.1 give a commutative diagram
(5.4.3)
/
/
where the bottom map is multiplication in the total coordinate ring (Exercise 5.4.4). Thus homogenization turns multiplication of sections into ordinary multiplication.
Polytopes. A full dimensional lattice polytope Recall that can be constructed in two ways:
As the toric variety
of the normal fan of
As the projective toric variety (Chapter 2).
gives a toric variety
(Chapter 3).
of the set of characters
.
for
We will see that both descriptions relate nicely to homogenenous coordinates and the total coordinate ring.
Given as above, set and let denote set of -dimensional consists of vertices the faces of . Thus and consists of facets. The facet presentation of given in equation (2.2.2) can be written as (5.4.4)
for all
§5.4. Homogenization and Polytopes
In terms of the normal fan
233
, we have bijections
(vertices
(facets
maximal cones)
rays)
When dealing with polytopes we index everything by facets rather than rays. Thus gives: each facet
The facet normal
, which is the ray generator of the corresponding cone.
The torus-invariant prime divisor
The variable
w
.
in the total coordinate ring . We call
We also have the divisor
w
a facet variable.
from (4.2.5). The polytope of this divisor is the polytope we began with (Exercise 4.3.1). Hence, if we set , then we get isomorphisms
u
z
In this situation, we write the homogenization of as
a
We call
w
w
-monomial.
The exponent of the variable in gives the lattice distance from to the facet . To see this, note that lies in the supporting hyperplane defined by . If the exponent of is , then to get from the sup
porting hyperplane to , we must pass through the parallel hyperplanes, namely . Here is an example. for
Example 5.4.5. Consider the toric variety
of the polygon
with vertices
Figure 2. A polygon with facets labeled by variables
, shown in Figure 2. In terms of (5.4.4),
Chapter 5. Homogeneous Coordinates
234
``
we have , where the indices correspond to the facet variables give -monomials indicated in Figure 2. The 8 points of
where the position of each -monomial corresponds to the position of the lattice point . The exponents are easy to understand if you think in terms of lattice distances to facets.
The lattice-distance interpretation of the exponents in shows that lattice of correspond to those -monomials points in the interior divisible by . For example, the only -monomial in Example 5.4.5 divisible by corresponds to the unique interior lattice point.
``
We next relate the constructions of toric varieties given in Chapter 2 and in §5.1. In Chapter 2, we wrote the lattice points of as and considered the map (5.4.5)
The projective (possibly non-normal) toric variety the image of .
is the Zariski closure of
On the other hand, we have the quotient construction of
. Also, the exceptional set
can be described where we write in terms of the -monomials coming from the vertices of the polytope.
Lemma 5.4.6. The vertex monomials properties:
(a)
irrelevant ideal of . (b)
.
,
, where
a vertex of
, have the following
is the
correspond bijectively to cones Proof. We saw above that vertices . Then the lattice-distance interpretation of shows the facet variables appearing in are precisely the variables appearing in . This implies part (a), and part (b) follows immediately.
as above, then the and give a map
If we set form a basis of (5.4.6)
where
-monomials
denotes the evaluation of the monomial . This map is well-defined since for each
,
,
at the point , Lemma 5.4.6
§5.4. Homogenization and Polytopes
implies that at least one be nonzero.
235
-monomial (in fact, at least one vertex monomial) must
The maps (5.4.5) and (5.4.6) fit into a diagram /
;
; ; ;;; ;; ;; / ;; VVVV VVVV ;; VV
VVVV ;; VVVV VVVV & ;; VV*
;
is described in (5.1.2) and
Here, the map is the quotient map. This diagram has the following properties. Proposition 5.4.7. There is a morphism represented by the dotted arrow in the above diagram that makes the entire diagram commute. Furthermore, the image of is precisely the projective toric variety .
Proof. When we regard the sequence (5.1.1) tells us that
as characters on
(5.4.7)
Multiplying each side by
;
;
, the exact
, we obtain
;
If we let , and apply this to a point in , we see that and give the same point in projective space since times vector for equals the vector for . It follows that, ignoring for the moment, the rest of the above diagram commutes.
We next show that is constant on -orbits. This holds since are homogeneous of the same degree. In more detail, fix points -monomial and a evaluating at gives
where the last equality follows from the description of Arguing as in the previous paragraph, it follows that
-monomials , . Then
given in Lemma 5.1.1. and give the
Chapter 5. Homogeneous Coordinates
236
same point in . This proves the existence of since is a good categorical quotient, and this choice of makes the entire diagram commute. is the Zariski The final step is to show that the image of closure of the image of . First observe that
since is continuous in the Zariski topology and by commutivity of the diagram. However, since is projective. You is Zariski closed in will give two proofs of this in Exercise 5.4.5, one topological (using constructible sets and compactness) and one algebraic (using completeness and properness). Once we know that implies is Zariski closed,
and
follows.
In Chapter 2, we used the map —construced from characters—to parametrize a big chunk of the projective toric variety . In contrast, Proposition 5.4.7 uses the map —constructed from -monomials—to parametrize all of .
If the lattice polytope is very ample, then the results of Chapter 2 imply that is the toric variety . So in the very ample case, the -monomials give an explicit construction of the quotient by mapping to projective space via the -monomials. It follows that we have two ways to take the quotient of by :
At the beginning of the chapter, we took of —to construct an affine quotient.
-invariant polynomials—elements
Here, we use -monomials—elements of —to construct a projective quo of “bad” points. tient, after removing a set
The -monomials are not -invariant but instead transform the same way under . This is why we map to projective space rather than affine space. We will explore these ideas further in Chapter 14 when we discuss geometric invariant theory. When is very ample, we have a projective embedding given by the -monomials in . If are homogeneous coordinates of , then the homogeneous coordinate ring of is
as in §2.0. We also have the cone ,affine ordinary coordinate ring of i.e.,
of
, and
Recall that
is an -graded ring since
Another -graded ring is
is a homogeneous ideal.
. This relates to
as follows.
is the
§5.4. Homogenization and Polytopes
Theorem 5.4.8. Let Then: (a)
be a very ample lattice polytope with
is normal.
(b) There is a natural inclusion normalization of .
237
such that
.
is the
(c) The following are equivalent: is projectively normal. (1) (2)
(3)
(4)
is normal.
.
is generated as a -algebra by its elements of degree .
Proof. Consider the cone
This cone is pictured in Figure 4 of §2.2. Recall that height . Since the divisor associated to is respect to induces an isomorphism
is the “slice” of at , homogenization with
is dual to , so that the semigroup algebra On the other hand,is the coordinate . Given ring of the affine toric variety , we write the corresponding character as . The algebra is graded using the last coordinate, the “height.” (this is the “slice” observation Since if and only if
made above), we have
Using (5.4.3), we obtain a graded
-algebra isomorphism
is normal. . For this, We next claim that is the normalization of the affine cone we let in the proof of Theorem 2.4.1, is . As noted . Since is very ample, one easily the affine cone of (Exercise 5.4.6). It is also checks that generates , i.e., . Hence clear that generates the cone is the normalization of This proves that
by Proposition 1.3.8. This immediately implies part (b).
Chapter 5. Homogeneous Coordinates
238
For part (c), we observe that (1) (2) follows from Theorem 2.4.1, and (1) (3) follows from parts (a) and (b) since the projective normality of is (4) is obvious since is equivalent to the normality of . Also (3) generated by the images of , which have degree . Finally, you will show (2), completing the proof. in Exercise 5.4.7 that (4)
Further Examples. We begin with an example of that illustrates how many different polytopes can give the same toric variety.
Example 5.4.9. The toric surface in Example 5.4.5 was defined using the polygon shown in Figure 2. In Figure 3 we see four polygons , all
Figure 3. Four polygons
with the same normal fan
of which have the same normal fan and hence give the same toric variety. Since we are in dimension 2, these polygons are very ample (in fact, normal), so that Theorem 5.4.8 applies. These four polygons give four different projective embeddings, each of which its own coordinate ring as a projective variety. By Theorem 5.4.8, these coordinate rings all live in the total coordinate ring . This explains the “total” in “total coordinate ring.”
Our next example involves torsion in the grading of the total coordinate ring. Example 5.4.10. The fan for has ray generators and
. in for and is the normal fan of the standard simplex Another polytope with the same normal fan is is reflexive in Example 2.4.5. One checks so that . We saw that is a translate that . Since has degree of , (5.4.2) implies that the -monomials for coincide with , which are homogeneous polynomials of the homogenizations coming from . degree in
``
§5.4. Homogenization and Polytopes
Since
is reflexive, its dual
239
is also a lattice polytope. Furthermore ,
since the ray generators of the normal fan of are the vertices of by duality for reflexive polytopes (be sure you understand this—Exercise 5.4.8). The vertices of are
(5.4.8)
The generate a sublattice the map defined by
. In Exercise 5.4.8 you will show that
induces an isomorphism
(5.4.9)
is the diagonal subgroup. It follows that
where is a lattice of index
in
.
, so that
The dual toric variety is determined by the normal fan of . The ray generators of are the vectors from (5.4.8). The only possible with these ray generators is the fan whose cones are generated complete fan in by all proper subsets of . Since and the generate , the toric variety of relative to is , i.e., . (Remember that is a fan in .) Since has index , Proposition 3.3.7 implies that
``
Hence the dual toric variety
is the quotient of by a group of order .
The total coordinate ring is the polynomial ring , graded by . The notation is challenging, since by duality is the character lattice of the torus of . Thus (5.1.1) becomes the short exact sequence
where
is dualizes to
of index
. Now consider the diagram
. If we let
, then
Chapter 5. Homogeneous Coordinates
240
/
/
/
/
/
/
/
/
with exact rows and columns. In the middle row, we use . By the snake lemma, we obtain the exact sequence
so
The polytope has only six lattice points in : the vertices the origin (Exercise 5.4.8). When we homogenize these, we get six
since
The equation
``
. Thus the class group has torsion.
and -monomials
(Exercise 5.4.8).
`` ` ` since it is built from -monomials. If we want defines a hypersurface is an irreducible hypersurface, we must have , in which case
isomorphic (via the torus action) to a hypersurface of the form
``
``
This is the quintic mirror family, which played a crucial role in the development of mirror symmetry. See [16] for an introduction to this astonishing subject. Exercises for §5.4.
{ X equivalent torus-invariant divisors with $ be V linearly , then prove that 8 8 T .
5.4.1. Verify the details of Example 5.4.3. 5.4.2. Let
a If
(a) (b) Prove (5.4.2).
.
§5.4. Homogenization and Polytopes
241
! 5.4.3. Fix a torus invariant divisor g V / a ( 4 fa Z for all C 8 . Define
and consider the polyhedron
$ ( ` V ( 3^ a Qf a Z g X +L . by $ embeds V ( as an affine subspace of . Hint: Remember that (a) Prove that
has no torus factors.
(b) Prove that induces a bijection `
4
@
V ( L
L { X . Prove that . Thus the polyhedron in
This realizes as the polyhedron obtained by intersecting the positive orthant of with an affine subspace.
(c) Let constructed in part (b) depends only on the divisor class of model” of .
5.4.4. Prove that the diagram (5.4.3) is commutative.
. This is the“canonical
`
^ O 5.4.5. In the proof of Proposition 5.4.7, we claimed that the image of ` ^ was Zariski closed. This follows from the general fact that if is a morphism of varieties and is projective, then is Zariski closed in . You will prove this two ways. (a) Give a topological proof that uses constructible sets and compactness. Hint: Remember that projective space is compact.
'
(b) Give an algebraic proof that uses completeness and properness from §3.4. Hint: Pro jective implies complete. Show that is proper and consider the graph of ^ .
V
( V be a very ample lattice polytope and letV 5.4.6. Let Prove that N - N . Hint: First show that N in the discussion preceding Proposition 2.1.6.
V - / E 8 V - N . - /D 8 , where N is defined
5.4.7. Prove of (4) (2) in part (c) of Theorem 5.4.8. Hint: (4) implies that is onto for all D.
T
(
5.4.8. This exercise is concerned with Example 5.4.10. (a) Prove that if I is reflexive, then the vertices of normal fan of .
^
are the ray generators of the
g E , where $ Z are defined in Example 5.4.10. Z $ % 3 = > $ H 'H # . Use Proposition 1.3.18 to prove / L $ $ 4 7 H $ # $ L Y E 8 @ H B V U V (e) Use part (e) and the quotient construction of to give another proof that V V V V O bU U . Also give an explicit description of the action of U on O . V 5.4.9. Here is another way to think about homogenization. Let \ \ be a basis of , $ $ I so that R , E $ $ , are coordinates for the torus . (b) Prove (5.4.9). (c) Prove f (d) Let
Chapter 5. Homogeneous Coordinates
242
8 R
8
(a) Adapt the proof of (5.4.7) to show that when we think of the as characters on $H . V a (b) Given , part (a) tells us that the Laurent monomial can be regarded as a Laurent monomial in the . Show that we can “clear denominators” by multiplying to obtain a monomial in the total coordinate ring . by
8
8
(c) Show that this monomial obtained in part (b) is the homogenization
8
.
5.4.10. Show that of Example 5.4.5 is the blowup of O - O at one = > the toric variety point. Compute and find the classes of the four polygons appearing in Figure 3. 9E E E 5.4.11. Consider the reflexive polytope . Work out the analog of Example 5.4.10 for .
$ $ Q = $D $ #\ $ #\b* $ \ #E . 5.4.12. Fix an integer ( E and consider the -simplex In Exercise 2.2.13, we claimed that the toric variety of is the weighted projective space O FE $ E $ E $ . Prove this.
Chapter 6
Line Bundles on Toric Varieties
§6.0. Background: Sheaves and Line Bundles
Sheaves of -modules on a variety were introduced in §4.0. Recall that for , an -module an affine variety gives a sheaf on such that
for all in . Globalizing this leads to quasicoherent sheaves on . These include coherent sheaves, which locally come from finitely generated modules. In this section we develop the language of sheaf theory and discuss vector bundles and line bundles.
The Stalk of a Sheaf at a Point. Since sheaves are local in nature, we need a . This is provided by the notion of method for inspecting a sheaf at a point direct limit over a directed set. Definition 6.0.1. A partially ordered set for all there exists If
is a directed set if
such that
is a family of rings indexed by a directed set
there is a homomorphism
and
such that whenever
satisfying and , then the form a directed system. Let be the submodule of generated by the relations , for and . Then the direct limit is defined as
243
Chapter 6. Line Bundles on Toric Varieties
244
For every
the elements and generally, two elements diagram
such that
, there is a natural map
and
.
For simplicity, we often write the direct limit as such as [2] write instead of .
. Note also that references
have the same image in are identified in
RRRR R R( m6 m m mmm
such that whenever
,
. More if there is a
Example 6.0.2. Given , the definition of sheaf shows that that the rings of , form a directed system under inclusion, , indexed by neighborhoods and in this case, the direct limit is the local ring . For a quasicoherent sheaf containing of -modules, take an affine open subset so that is the corresponding , where is an -module. If maximal ideal, then is the localization and
where is the localization of
at the maximal ideal .
The term sheaf has agrarian origins: farmers harvesting their wheat tied a rope around a big bundle, and left it standing to dry. Think of the footprint of the bundle as an open set, so that increasingly smaller neighborhoods around a point on the ground pick out smaller and smaller bits of the bundle, narrowing to a single stalk. Definition 6.0.3. The stalk of a sheaf
is
at a point
.
Injective and Surjective. A homomorphism of -modules was defined in §4.0. We can also define what it means for to be injective or surjective. The definition is a bit unexpected, since we need to take into account the fact that sheaves are built to convey local data.
Definition 6.0.4. A sheaf homomorphism
is injective if for any point and open subset containing , there exists an open subset containing , with injective. Also, is surjective if for any
point and open subset containing and any , there is an open subset . containing and such that
§6.0. Background: Sheaves and Line Bundles
245
In Exercise 6.0.1 you will prove that for a sheaf homomorphism
,
defines a sheaf denoted . You will also show that is injective exactly when
. On the other hand, surjectivity of a sheaf the “naive” idea works, i.e., homomorphism need not mean that the maps are surjective for all . Here is an example. , consider the Weil divisor . Example 6.0.5. On , then If we write of with and . Since
;
it follows easily that we have global sections For any
, multiplication by
. Doing this for
gives a sheaf homomorphism gives
In Exercise 6.0.2 you will check that this sheaf homomorpism is surjective. However, when we take global sections, we get
which is clearly not surjective. There is an additional point to make here. Given
,
need not define a sheaf. Fortunately, this can be rectified. Given a presheaf there is an associated sheaf known as the sheafification of , defined by
See [41, II.1] for a proof that
for all with and
and there is for all
is a sheaf with the same stalks as
has a natural sheaf associated to it, denoted
.
. Hence
,
Chapter 6. Line Bundles on Toric Varieties
246
Exactness. We define exact sequences of sheaves as follows.
Definition 6.0.6. A sequence of sheaves is exact at
if there is an equality of sheaves
The local nature of sheaves is again highlighted by the following result, whose proof may be found in [41, II.1].
Proposition 6.0.7. The sequence in Definition 6.0.6 is exact if and only if is exact for all
.
It follows from Example 6.0.5 that if
(6.0.1)
is a short exact sequence of sheaves, the corresponding sequence of global sections may fail to be exact. However, we always have the following partial exactness, which you will prove in Exercise 6.0.3. Proposition 6.0.8. Given a short exact sequence of sheaves (6.0.1), taking global sections gives the exact sequence
In Chapter 9 we will use sheaf cohomology to extend this exact sequence.
Quasicoherent Sheaves. For an affine variety , an -module gives a quasicoherent sheaf on . This operation preserves exactness, i.e., an exact sequence of -modules
gives an exact sequence of sheaves
(see [41, Prop. II.5.2]). There is a toric version of this result that uses the sheaf from §5.3 associated to a graded -module , where is the total coordinate ring of a toric variety .
Proposition 6.0.9. An exact sequence
-modules gives an exact sequence
of quasicoherent sheaves on
.
of graded
§6.0. Background: Sheaves and Line Bundles
247
Proof. For , the restriction of to is the sheaf associated to
, the elements of degree in the localization of at . Localization preserves exactness, as does taking elements of degree . This proves the desired exactness.
Another example is the following exact sequence of sheaves from §3.0.
Example 6.0.10. A closed subvariety
The sheaf
, defined by
; These are coherent sheaves on
The direct image sheaf
;
, defined by
gives two sheaves:
for
.
.
and are related by the exact sequence
;
Operations on Sheaves of -Modules. Operations on modules over a ring have natural analogs for sheaves of -modules. In particular, given quasicoherent
sheaves , it is easy to show that defines the quasicoherent
via sheaf . We can also define
On the other hand,
In Exercise 6.0.4 you will show that
is a quasicoherent sheaf.
is only a presheaf, so the tensor
is defined to be the sheaf associated to this presheaf. This sheaf product is again quasicoherent and satisfies
whenever
is an affine open set (see [41, Prop. II.5.2]).
Global Generation. For a module over a ring, there is always a surjection from a free module onto . This is true for a sheaf of -modules when is, in a certain sense, large enough.
Definition 6.0.11. A sheaf of -modules is generated by global sections if such that at any point there exists a set , the images of the generate the stalk .
Any global section follows that if is generated by
, there is a surjection of sheaves
gives a sheaf homomorphism
. It
In the next section we will see that when is toric, there is a particularly nice way of determining when the sheaves are generated by global sections.
Chapter 6. Line Bundles on Toric Varieties
248
Locally Free Sheaves and Vector Bundles. We begin with locally free sheaves.
Definition 6.0.12. A sheaf of exists an open cover
of -modules is locally free of rank r if there such that for all , .
Locally free sheaves are closely related to vector bundles. Definition 6.0.13. A variety there is a morphism
and an open cover
of
is a vector bundle of rank over a variety
if
such that:
(a) For every , there is an isomorphism
by projection onto is . followed (b) For every pair , there is such that the diagram 5 O kk such that
kkkk kkk
SSSS SSS SS )
commutes.
satisfying properties (a) and (b) is called a trivialization. The Data map gives a chart, where for . We call the fiber over . See Figure 1 on the next page.
Since the isomorphisms
given by and are related by the linear map , the fiber has a well-defined vector space structure. Hence a vector bundle really is a “bundle” of vector spaces.
On a vector bundle, the are called transition functions and can be regarded as a family of transition matrices that vary as varies. Just as there is no preferred basis for a vector space, there is no canonical choice of basis for a particular fiber. Note also that the transition functions satisfy (6.0.2)
on on
§6.0. Background: Sheaves and Line Bundles
249
φj: π −1(p) ≅ C r
φi : π−1 (p ) ≅ C r
g ij (p)
Uj × C r
Ui × C r
p
Ui
Uj
X
Figure 1. Visualizing a vector bundle
Definition 6.0.14. A section of a vector bundle over
such that A section
for all
picks out a point
. A section
in each fiber
is a global section. , as shown in Figure 2.
π −1( p) s(p)
s(x)
X p Figure 2. For a section ,
open is a morphism
Chapter 6. Line Bundles on Toric Varieties
250
We can describe a vector bundle and it global sections purely in terms of the transition functions as follows.
, and assume Proposition 6.0.15. Let be a variety with an affine open cover satisfying the compatibility that for every , we have conditions (6.0.2). Then:
(a) There is a vector bundle whose transition functions are the
of rank , unique up to isomorphism, .
is uniquely determined by a collection of -tuples (b) A global section , such that for all
Proof. One easily checks that the satisfy the gluing conditions from §3.0. It glue together to give a variety . Furfollows that the affine varieties glue together to give a morphism thermore, the projection maps . It follows easily that the open set of corresponding to , which gives an isomorphism . Hence is a is vector bundle with transition functions . . Thus Given a section , is a section of . By Definition 6.0.13, the satisfy the desired compatibility where
condition, and since every global section arises this way, we are done.
Let denote the set of all sections of over . One easily sees that is a sheaf on and in fact is a sheaf of -modules since the fibers are vector spaces. In fact, is an especially nice sheaf.
Proposition 6.0.16. The sheaf of sections of a vector bundle is locally free.
, the proof of Proposition 6.0.15 Proof. For a trivial vector bundle shows that a section is determined by a morphism , i.e., an element of . Thus the sheaf associated to a trivial vector bundle over is . , each For a general vector bundle with trivialization
:: :: :: :
gives an isomorphism of vector bundles
/
Since isomorphic vector bundles have isomorphic sheaves of sections, it follows that if is the sheaf of sections of , then .
§6.0. Background: Sheaves and Line Bundles
251
Line Bundles and Cartier Divisors. Since a vector space of dimension one is a line, a vector bundle of rank is called a line bundle. Despite the new terminology, line bundles are actually familiar objects when is normal.
Theorem 6.0.17. The sheaf of a line bundle.
of a Cartier divisor
is the sheaf of sections
;
; by Proposition 4.0.16. which implies
We use this data to construct a line bundle as follows. Since !; the quotients may be regarded as transition functions. These satisfy . the hypotheses of Proposition 6.0.15 and hence give a line bundle satisfies , so that on , A global section
we have
This shows that
. Then
Proof. Recall from Chapter 4 that a Cartier divisor is locally principal, so that has an affine open cover with , . Thus
is local data for . Note also that
which by part (b) of Proposition 6.0.15 gives a global section of . Conversely, the proposition shows that a global section of the vector bundle gives such that functions . It follows that is independent of . One easily checks that . The same argument works when we restrict to any open subset of . It follows that is the sheaf of sections of .
We will see shortly that this process is reversible, i.e., there is a one-to-one correspondence between line bundles and sheaves coming from Cartier divisors. First, we give an important examples. Example 6.0.18. When we regard as the set of lines through the origin in , each point corresponds to a line . We assemble these lines into a line bundle as follows. Let be homogeneous coordinates on and be coordinates on . Define
as the locus where the matrix
Chapter 6. Line Bundles on Toric Varieties
252
has rank one. Thus is defined by the vanishing of . Then define the map to be projection on the first factor of . To see that is a line bundle, consider the open subset where is invertible. On the equations defining become
Thus
for all
defines an isomorphism
In other words, is a local coordinate for the line over . Switching to the coordinate system over , we have the local coordinate , which over is related to via
Hence the the transition function from
to
!;
is given by
This bundle is called the tautological bundle on . In Example 6.0.20 below, we will describe the sheaf of sections of this bundle.
Projective spaces are the simplest type of Grassmannian, and just as in this example, the construction of the Grassmannian shows that it comes equipped with a tautological vector bundle. In Exercise 6.0.5 you will determine the transition functions for the Grassmannian .
Invertible Sheaves and the Picard Group. Propositions 6.0.16 and 6.0.17 imply that the sheaf of a Cartier divisor is locally free of rank . In general, a locally free sheaf of rank is called an invertible sheaf.
The relation between Cartier divisors, line bundles and invertible sheaves is described in the following theorem. Theorem 6.0.19. Let
on
be an invertible sheaf on a normal variety
(b) There is a line bundle on (a) There is a Cartier divisor
such that
. Then:
.
whose sheaf of sections is isomorphic to
.
Proof. The part (b) of the theorem follows from part (a) and Proposition 6.0.17. It remains to prove part (a).
Since is irreducible, any nonempty open gives a domain . By Exercise 3.0.4, , so that with field of fractions defines a constant sheaf on , denoted . This sheaf is relevant since defined as a subsheaf of .
is
§6.0. Background: Sheaves and Line Bundles
253
of First assume that is a subsheaf of . Pick an open cover that for every . Over , this gives homomorphisms
Let be the image of . Then that on satisfying .
;
such
. One can show without difficulty
is local data for a Cartier divisor
For the general case, observe that on an irreducible variety, every locally constant sheaf is globally constant (Exercise 6.0.6). Now let be any invertible sheaf on . On a small enough open set , , so that
Thus
is locally constant and hence constant. This easily implies that
, and composing this with the inclusion
expresses
as a subsheaf of .
We note without proof that the line bundle corresponding to an invertible sheaf is unique up to isomorphism. Because of this result, algebraic geometers tend to use the terms line bundle and invertible sheaf interchangeably, even though strictly speaking the latter is the sheaf of sections of the former.
We next discuss some properties of invertible sheaves coming from Cartier divisors. A first result is that if and are Cartier divisors on , then
(6.0.3) This follows because
induces a sheaf homomorpism
which is clearly an isomorphism on any open set where
is trivial.
By standard properties of tensor product, the isomorphism (6.0.3) induces an isomorphism In particular, when
where
, we obtain
and
is the dual of
.
More generally, the tensor of invertible sheaves is again invertible, and if invertible, then is invertible and
This explains why locally free sheaves of rank
are called invertible.
is
Chapter 6. Line Bundles on Toric Varieties
254
Example 6.0.20. There is a nice relation between the tautological bundle on and the invertible sheaf introduced in Example 4.3.1. Recall that the are all linearly equivalent, and so define isoinvariant divisors on . The local data for the Cartier divisor morphic sheaves, usually denoted
. is easily seen to be , where is the open set where
are given by Thus the transition functions for
These are the inverses of the transition functions for the tautological bundle from Example 6.0.18. It follows that the sheaf of sections of the tautological bundle is .
We can also explain when Cartier divisors give isomorphic invertible sheaves. Proposition 6.0.21. Two Cartier divisors give isomorphic invertible sheaves
if and only if
.
Proof. By Proposition 4.0.29, linearly equivalent Cartier divisors give isomorphic sheaves. For the converse, we begin with the case where . Then
. We show
by contradiction. , so
Assume and pick an irreducible divisor that appears in with positive coefficient. The local ring is a DVR, so we can find , where is the union of all irreducible divisors with . Set with . There are only finitely many such divisors, so that with . Then , and is a nonempty open subset of since vanishes on . However, . This contradiction proves . so that
Now suppose that Cartier divisors satisfy . Tensoring each side with and applying (6.0.3), we see that . If maps to via this isomorphism, then
as subsheaves of . Thus
where the last equality follows from the proof of Proposition 4.0.29. By the
, which clearly implies that previous paragraph, we have .
In Chapter 4, the Picard group was defined as the quotient
§6.0. Background: Sheaves and Line Bundles
255
We can interpret this in terms of invertible sheaves as follows. Given invertible, Theorem 6.0.19 tells us that , which is for some Cartier divisor unique up to linear equivalence by Proposition 6.0.21. Hence we have a bijection
isomorphism classes of invertible sheaves on
The right-hand side has a group structure coming from tensor product of invertible sheaves. By (6.0.3), the above bijection is a group isomorphism. In more sophisticated treatments of algebraic geometry, the Picard group of an arbitrary variety is defined using invertible sheaves. Also, Cartier divisors can be defined on an irreducible variety in terms of local data, without assuming normality (see [41, II.6]), though one loses the connection with Weil divisors. Since most of our applications involve toric varieties coming from fans, we will continue to assume normality when discussing Cartier divisors.
Stalks, Fibers, and Sections. From here on, we will think of a line bundle on as the sheaf of sections of a rank vector bundle . Given a section , we get the following: and . Then Since is a vector bundle of rank , we have the fiber gives . Since is a locally free sheaf of rank , we have the stalk . Then . gives
In Exercise 6.0.7 you will show that these are related via the equivalences
in (6.0.4) generates as an -module
A section
vanishes at
if
in
, i.e., if
.
Basepoints. It can happen that many (sometimes all) sections of a line bundle vanish at a point , called a basepoint. This leads to the following definition.
Definition 6.0.22. A subspace free if and only if for every
As noted earlier, a global section . Thus a subspace
defined by
has no basepoints or is basepoint
. with
gives
gives a sheaf homomorphism
. Then (6.0.4) and Proposition 6.0.7 imply the following.
Proposition 6.0.23. A subspace is surjective.
, there is
has no basepoints if and only if
Chapter 6. Line Bundles on Toric Varieties
256
of a Cartier divisor For a line bundle on a normal variety, the vanishing locus of a global section has an especially nice interpretation. The local data of gives the rank vector bundle with transition . Hence we can think of a nonzero global section of functions in two ways:
A rational function
;
satisfying
.
. The relation between and is given in the proof of Theorem 6.0.17: over , the . It follows that exactly for
section looks like when . Since
, the divisor of on is given by
A morphism
whose composition with
is the identity on
These patch together in the obvious way, so that the divisor of zeros of is
Thus the divisor of zeros of a global section is an effective divisor that is linearly equivalent to . It is also easy to see that any effective divisor linearly equivalent (Exercise 6.0.8). to is the divisor of zeros of a global section of
In terms of Cartier divisors, Proposition 6.0.23 has the following corollary.
Corollary 6.0.24. The following are equivalent for a Cartier divisor
(a)
is generated by global sections. is basepoint free, meaning that
(b)
(c) For every
there is
:
is basepoint free.
with
.
The Pullback of a Line Bundle. Let be a line bundle on and the associated rank vector bundle. A morphism gives the fibered product from §3.0 that fits into the commutative diagram
;
;
;
/
It is easy to see that
is a rank
;
/
vector bundle over
Definition 6.0.25. The pullback of the sheaf the rank vector bundle defined above.
;
.
is the sheaf of sections of
Thus the pullback of a line bundle is again a line bundle. Furthermore, there is a natural map on global sections
;
;
§6.0. Background: Sheaves and Line Bundles
defined as follows. A global section
gives the commutative diagram:
/
;
"
/
$
;
257
/
;
The universal property of fibered products guarantees the existence and uniqueness of the dotted arrow that makes the diagram commute. It . follows that Example 6.0.26. Let be a projective variety. If we write the inclusion as gives the line bundle , then the line bundle
;
;
;
Thus a projective variety always comes equipped with a line bundle. However, it is not unique, since the same variety may have many projective embeddings. You will work out an example of this in Exercise 6.0.9.
In general, given a sheaf of -modules on and a morphism , one gets a sheaf of -modules on . The definition is more complicated, so we refer the reader to [41, II.5] for the details.
;
Line Bundles and Maps to Projective Space. We now reverse Example 6.0.26 by using a line bundle on to create a map to projective space. with no basepoints and let Fix a finite-dimensional subspace be its dual. The projective space of is
If
is a basis of
(6.0.5)
, then
;
gives an isomorphism
, fix To define and pick with
. For . Then the map defined , there is such that by is linear and nonzero since has no base points. Thus , and since is unique up to an element of , the same is true for . Then
defines the desired map
Lemma 6.0.27. The map
;
.
is a morphism.
Chapter 6. Line Bundles on Toric Varieties
258
. These Proof. Let be a basis of and let open sets cover since has no basepoints. Furthermore, the natural map
is easily seen to be an isomorphism. Since all sections of , it follows that for all
for form , . write
are of the , we can
The definition of uses a nonzero vector , we can . Over . Then implies . It use follows that is the morphism
(6.0.6) When has no basepoints and span , is often written (6.0.7)
. with the understanding that this means (6.0.6) on
Furthermore, when rational functions such that (6.0.8)
, we can think of the global sections as
. Then can be written may be undefined, this needs explanation. The local data Since of . Then (6.0.8) means that implies that is This is a morphism on since the global sections corresponding to have no base points.
Exercises for §6.0. 6.0.1. For a sheaf homomorphism
`
^ , show that c;^
defines a sheaf. Also prove that the following are equivalent: (a) The kernel sheaf is identically zero. (b) (c)
is injective for every open subset
.
is injective as defined in Definition 6.0.4.
% : B % : A ^ % : is surjective.
6.0.2. In Example 6.0.5, prove that Prove that
%
6.0.3. Prove Proposition 6.0.8. c^ 6.0.4. Prove that show that it is quasicoherent when
$ ) are.
$
defines a sheaf
a
$ and
§6.0. Background: Sheaves and Line Bundles
259
$ + $ ) $ $ LL * * *'H . Then define up to left multiplication by elements of + F E $ - H to consist of all pairs $ such that $ . (a) A pair gives the matrix $ LLL ** 4 & * Prove that ) is a point of + if and only if the maximal minors of vanish. This $ shows that + 9E - H is a closed subvariety. $ ` + ^ F E . Explain why the (b) Projection onto the first factor gives a morphism $ fiber over 7 9 E is the 2-dimensional subspace of H corresponding to . $ (c) Given D # # , define / F E $ 4 D 8 Prove that @ H and that the give an affine open cover of 9 E . $ , let # %$ be the complementary indices, i.e., / $ 8 (d) Given D( # ! " # $$ $ /D $ E $ ) $ 8 . Prove that the map $ c^ $ $ '&! gives an isomorphism );( ^ - H * (e) By part (d), + is a vector bundle over F E . Determine its transition functions. $ 6.0.5. The Grassmannian 9E is defined as the space of lines in O , or equivalently, of 2-dimensional subspaces of H . This exercise will construct the tautological bundle on FE , which assembles these 2-dimensional subspaces into a rank vector bundle over FE . A point of FE corresponds to a full rank matrix
6.0.6. Prove that a locally constant sheaf on an irreducible variety is constant. 6.0.7. Prove (6.0.4).
%
6.0.8. Prove that any effective divisor linearly equivalent to a Cartier divisor divisor of zeros of a global section of ' .
` : O ^ * * % ,+ % .
6.0.9. Let
9E
2 `!
O
is the
be the Veronese mapping from Example 2.3.14. Prove that
- 2'.-
6.0.10. Let ^ be a morphism and let be a line bundle on that is generated by global sections. Prove that the pullback line bundle is generated by global sections.
2 $
. 8 give effective divisors X { @2 . % $ b / D , 7 H .$ $ Prove that these divisors are equal if and only if 2 (b) The complete linear system of is defined to be =0/ { 4 $ ( D 8 4 G4 /
6.0.11. Let (a)
be a Cartier divisor on a complete normal variety
X
{
on
.
Chapter 6. Line Bundles on Toric Varieties
260
Thus the complete linear system of consists of all effective Cartier divisors on linearly equivalent to . Use part (a) to show that 4 G4 can be identified with the projective space of
' , i.e., there is a natural bijection
. $ % 4 4 O . $ % ' ) . $ % $ - /bD 8 U H . % ' . Then O can (c) Assume that has no basepoints and set $ be identified 4 G4 . Prove that the morphism
` with the set hyperplanes in O ! " by ^hO is given / ]4 4 4 7 # ",,' 8 04 G4
§6.1. Ample Divisors on Complete Toric Varieties Our aim in this section is to determine when a Cartier divisor on a complete toric variety gives a projective embedding. We will use the key concept of ampleness.
Definition 6.1.1. Let we noted in §4.3,
be a Cartier divisor on a complete normal variety is finite-dimensional.
. As
has no (a) The divisor are very ample when and the line bundle basepoints and is a closed embedding. and is very ample for some integer . (b) are ample when
Our discussion will tie together concepts from earlier sections, including:
The very ample polytopes from Definition 2.2.16.
The polyhedra
from Proposition 4.3.3.
The support functions of Cartier divisors from Theorem 4.2.12.
We will see that support functions give a simple, elegant characterization of when is ample, as well as when is basepoint free.
of a complete fan in Basepoint Free Divisors. Consider the toric variety and let . By be a torus-invariant Cartier divisor on Propositions 4.3.3 and 4.3.8, we have the global sections
where
is the polytope defined by
We first study when being Cartier means that for every (6.1.1)
for all
is basepoint free. Recall from §4.2 that , there is with
Furthermore, is uniquely determined by the Cartier data complete. Then we have the following preliminary result.
since
is
§6.1. Ample Divisors on Complete Toric Varieties
261
Proposition 6.1.2. The following are equivalent: (a)
(b)
has no basepoints, i.e.,
for all
is generated by global sections.
.
. Proof. First suppose that is generated by global sections and take The -orbit corresponding to is a fixed point of the -action, and by the Orbit-Cone Correspondence,
By Corollary 6.0.24, there is a global section such that is not in the support of the divisor of zeros of . Since is spanned by for . The , we can assume that is given by for some discussion preceding Corollary 6.0.24 shows that the divisor of zeros of is
The point is not in the support of yet lies in for every . This
. Since is -dimensional, we conclude that forces for .
. Since For the converse, take global section whose divisor of zeros is one sees that the support of misses cover . Then we are done since the
, the character gives a . Using (6.1.1), , so that is nonvanishing on .
Later in the section we will improve this result by showing that (b) is equivalent , are the vertices of . This will to the stronger condition that the , imply in particular that is a lattice polytope when is basepoint free. We will also relate Proposition 6.1.2 to the convexity of the corresponding support function.
If is generated by global sections, we can write the corresponding map to projective space as follows. Suppose that The characters
span
(6.1.2)
, so that we can write
as
See (6.0.8) for a careful description of what this means. When we restrict to the torus , is the map (2.1.2). Hence our general theory relates nicely with the more concrete approach used in Chapter 2. Example 6.1.3. The fan for the Hirzebruch surface is shown in Figure 3 on the next page. Let be the divisor corresponding to . We will study the divisors
and
Chapter 6. Line Bundles on Toric Varieties
262
u1 = (−1,2) σ4 σ1
u2
σ3
u3 u4
Figure 3. A fan
For , let Figure 4 shows
1
σ2
< with
m2
m3
m′2
1
PD 2
2 m′4
Figure 4.
and
m′1
(left) and
Very Ample Polytopes. Let with facet presentation
and
This picture and Proposition 6.1.2 make it clear that is not.
(right)
is basepoint free while
(6.1.3)
corresponds to a maximal cone
Proposition 4.2.10 implies that when .
for all facets
a full dimensional lattice polytope
This gives the complete normal fan and the toric variety A vertex
m′3 PD′
m1 = m4
correspond to , and similarly for we have and (right) (see also Exercise 4.3.5).. and (left) and
. Write
is Cartier since
§6.1. Ample Divisors on Complete Toric Varieties
263
Recall from Definition 2.2.16 that is very ample if for every vertex , the semigroup is saturated in . The definition of given in Chapter 2 used very ample polytopes. This is no accident.
Proposition 6.1.4. Let (a)
be as above. Then:
is ample and basepoint free.
(b) If
, then
is very ample for every
is very ample if and only if
(c)
and
.
is a very ample polytope.
Proof. First observe that the polytope of the divisor is the polytope we began with. Thus is basepoint free by Proposition 6.1.2, which proves the final assertion of part (a). Furthemore, by (6.1.2), the map factors
where is the projective toric variety of understand when is an isomorphism. Fix coordinates of and let
from §2.3. We need to
be the set of in dices such that is a vertex of . Hence each gives a vertex and a corresponding maximal cone in the normal fan of . If , then for every facet containing . For all other facets , . Hence, if is the global section corresponding to , consists of those then the support of divisors missing the of . It follows that is the nonvanishing affine open toric variety locus of . Under the map maps to the affine open subset where , this . nonvanishing locus and , it Since suffices to study the maps
of affine toric varieties. By Proposition 2.1.8,
Since
by (6.1.3), we have an inclusion of semigroups
This is an equality precisely when is saturated in , we obtain the equivalences:
. Since
Chapter 6. Line Bundles on Toric Varieties
264
is an isomorphism
is an isomorphism for all is an isomorphism for all is saturated for all is very ample.
is very ample
This proves part (c) of the proposition. For part (b), recall that if and by Corollary 2.2.18. Hence is arbitrary, then is very ample when is very ample. This implies that is ample (the case is trivial), which completes the proof of part (a).
Example 6.1.5. In Example 2.2.10, we showed that
is not normal. We show that is not very ample as follows. From Chapter 2 we know that the only lattice points of are its vertices, so that . Since is singular (Exercise 6.1.1) of dimension 3, it follows that cannot and be a closed embedding. Hence and are not very ample. However, are very ample by Proposition 6.1.4.
Support Functions and Convexity. Let be a Cartier divisor on a is complete toric variety . As in Chapter 4, its support function determined by the following properties:
for all
This is where the for is given by
is linear on each cone
. .
from (6.1.1) appear naturally, since the explicit formula for all .
When is a rank two lattice, it is easy to visualize the graph of : imagine a tent, with centerpole extending from down the -axis, and tent stakes placed at positions . Here is an example. Example 6.1.6. Take and consider the divisor . This gives the support function where for the four ray generators . The graph of is shown in Figure 5 on the of the fan of next page. This should be visualized as an Egyptian pyramid, with base vertices at
and apex at the origin.
The first key concept of this section was ampleness. The second is convexity, which we now define.
§6.1. Ample Divisors on Complete Toric Varieties
265
u2 u1
u3 u4 (u2 ,−1)
(u 1,−1)
(u3 ,−1)
(u4 ,−1)
Figure 5. The graph of
Definition 6.1.7. Let
for all
and
be convex. A function
is convex if
.
is convex exactly if Continuing with the tent analogy, a support function there are unimpeded lines of sight inside the tent. It is clear that for Example 6.1.6, the support function is convex.
The following lemma will help us understand what it means for a support func , a cone is called a tion to be convex. Given a fan in , i.e, when wall when it is the intersection of two -dimensional cones
forms the wall separating and .
Lemma 6.1.8. The following are equivalent for a support function
(a)
(b) (c)
for all
is convex.
(d) For every wall
and
for all
, there is
.
. with
.
.
Chapter 6. Line Bundles on Toric Varieties
266
. Given Proof. First assume (a) and fix in the interior of
can find . By convexity, we have such that
This easily implies
, we
, proving (b). The implication (b) (c) is for , and (c) (a) follows because the immediate since minimum of a finite set of linear functions is always convex (Exercise 6.1.2).
Since (b) fix a wall
(d) is obvious, it remains to prove the converse. Assume (d) and . Then lies on one side of the wall. We claim that when are on the same side of
(6.1.4)
. If , the wall is defined by This is obvious if and then (d) implies that the side containing is defined by This proves (6.1.4).
Now take line segment in Figure 6.
,
.
. We can pick in the interior of so that the and intersects every wall of in a point. This gives the picture shown
wall
wall
wall
Figure 6. Crossing walls from to along
Using (6.1.4) repeatedly, we obtain
When we arrive at the cone containing . This proves (b).
, the pairing becomes
, so that
In terms of the tent analogy, part (b) of the lemma means that if we have a convex support function and extend one side of the tent in all directions, the rest of the tent lies below the resulting hyperplane, and part (d) means that it suffices to check this locally where two sides of the tent meet. The proof of our main result about convexity will use the following lemma that describes the polyhedron of a Cartier divisor in terms of its support function. Lemma 6.1.9. Let Then
be a fan and
be a Cartier divisor on
for all
.
§6.1. Ample Divisors on Complete Toric Varieties
Proof. Assume
for all
267
. Applying this with
so that by the definition of . For the opposite inclusion, take
. Then and . Thus , so that ,
where the inequality follows from the defining properties of .
gives
, and the last two equalities follow from
We now expand Proposition 6.1.2 to give a more complete characterization of when a divisor is basepoint free. Theorem 6.1.10. Assume Cartier divisor
(c)
(d) (e) (f) (g)
is complete and let be the support function of a . Then the following are equivalent:
on
free. is basepoint for all . . is the set of vertices of . for all . . for all
(a) (b)
is convex.
Proof. The equivalences (a) (b) and (f) (g) were proved in Proposition 6.1.2 and Lemma 6.1.8. Furthermore, Lemmas 6.1.8 and 6.1.9 imply that for all is convex
This proves (g)
for all
(b), so that (a), (b), (f) and (g) are equivalent.
and . Combining Assume (b). Then these with Lemma 6.1.9, we obtain proving (e). The implication (e) (g) follows since the minimum of a compact set of linear functions is convex (Exercise 6.1.2). So (a) (b) (e) (f) (g).
Consider (d). The implications (d) (c) (b) are clear. For (b) (d), take . Let be in the interior of and set . By Exercise 6.1.3, and is a supporting hyperplane of
(6.1.5)
Chapter 6. Line Bundles on Toric Varieties
268
This implies that is a vertex of . Conversely, let
be a supporting . This means , with hyperplane of a vertex for all . Since (b) holds, we also have (e) and (f). By (e), equality if and only if Combining this with (f), we obtain
Hence
must occur for some
.
, which forces
Example 6.1.11. In Example 6.1.3 we studied the Hirzebruch surface for and is basepoint free while showed that is not. Theorem 6.1.10 gives a different proof using support functions. Figure 7 shows the graph of the support function . In the figure, we have slightly distorted the position of
(u3 ,0) (u ,0)
(u 2,0)
4
(u 4,−1)
(u1 ,0)
Figure 7. The graph of
(which should really be closer to ) to make things easier to visualize. Notice that the portion of the “roof” containing the points and the origin lies
, and it is clear that for , there are unimpeded lines of sight in the plane within the tent. In other words, is convex.
The support function is shown in Figure 8 on the next page. Here, the line
, yet the ridgeline of the tent of sight from to lies in the plane
. Hence going from the apex at the origin to lies below the plane this line of sight does not lie inside the tent, so that is not convex.
When is basepoint free, Theorem 6.1.10 implies that the vertices of are . One caution is that in general, the need not the lattice points ,
§6.1. Ample Divisors on Complete Toric Varieties
269
(u3 ,0) (u2 ,0)
(u4 ,0)
(u2 ,−1) (u4 ,−1) (u1 ,0)
Figure 8. The graph of
<
can have be distinct, i.e., . An example is given by the divisor considered in Example 6.1.3—see Figure 4. As we will see later, this behavior illustrates the difference between basepoint free and ample.
also happen that has strictly smaller dimension than the dimension It .can of You will work out a simple example of this in Exercise 6.1.4.
The Normal Fan of a Basepoint Free Divisor. When has no base is a lattice polytope with the , , as vertices. points, Here we study the relation between and the normal fan of . Since may fail to be full dimensional, we need to explain what we mean by normal fan. Consider
struction of
. The inclusion since is saturated in .
induces a surjective
, we get a full dimensional lattice by a lattice point of is the normal fan of . The con . Then . Hereinis the key is independent of how we translate assertion.
Proposition 6.1.12. Let . If with polytope normal fan , then
with dual homomorphism
Translating polytope
be a basepoint free Cartier divisor on is a vertex and is the corresponding cone in the
Chapter 6. Line Bundles on Toric Varieties
270
Proof. We first give an alternate description of normal fans in §2.3, we saw that the vertex
whose dual in have whose dual in
is
. In the discussion of gives the cone
This has some nice consequences. First, since
(6.1.6)
and
. Since
is
, we also
is strongly convex, (6.1.6) implies that . It follows is a closed convex cone of dimension in that the proposition is equivalent to the assertion , (6.1.7) for all implies where “ ” denotes the interior (Exercise 6.1.5). satisfies A second consequence of (6.1.6) is that any
for all
for for all
We now prove (6.1.7). Assume element of the intersection. Since
In particular, basepoint free implies (6.1.8)
for some
, so that
and let , we have
be an
by convexity and part (b) of Lemma 6.1.8. Combining this with (6.1.8), we see that for all Since , proving (6.1.7). is open, this forces
This proposition has some nice geometric consequences, as we will see below.
Ampleness and Strict Convexity. We next determine when a Cartier divisor of is ample. The Cartier data satisfies on for all
Definition 6.1.13. The support function convex if it is convex and for every
of a Cartier divisor on satisfies
There are many ways to think about strict convexity.
is strictly
§6.1. Ample Divisors on Complete Toric Varieties
271
Lemma 6.1.14. The following are equivalent for the support function
(a)
is strictly convex. for all
(b)
(c) For every wall
(e) (f) (g)
, there is
and
when is convex and when is convex and for all
(d)
in
in
. with
and for all
and
.
.
is a wall.
.
.
not contained in the same
cone of .
Proof. You will prove this in Exercise 6.1.6. We now relate strict convexity to ampleness. is the support function of a Cartier divisor Theorem 6.1.15. Assume that . Then on a complete toric variety
Furthermore, if
is ample and
is strictly convex.
is ample, then
is very ample for all
.
Proof. First suppose that is very ample. Very ample divisors have no basepoints, so is convex by Theorem 6.1.10. If strict convexity fails, then Lemma 6.1.14 with
implies that has a wall
. Let . can be written Let , so that
for some . We will study as in (6.1.2). In this enumeration, . on the open subset . Theorem 6.1.10 implies that , so that the section First consider by the proof of Proposition 6.1.2. It corresponding to is nonvanishing on follows that on , is given by is the open subset where . where , the same argument works on . This gives a morphism Since q Since is a wall, are the only -dimensional cones of containing . Hence
by the Orbit-Cone Correspondence. Note also since is a wall. Since
is complete, Proposition 4.3.8 implies that all morphisms from
to affine
Chapter 6. Line Bundles on Toric Varieties
272
space are constant. Thus maps to a point, which is impossible since very ample. Hence is strictly convex when is very ample.
is
. Thus is very ample for If is ample, then strictly convex, which easily implies that is strictly convex.
must be
For the converse, assume is strictly convex. We claim that is the given fan full dimensional lattice polytope whose normal fan from Proposition 6.1.12 that the sublattice gives the dual such that is a fan in .
is a . Recall
Since is strictly convex, the are distinct by Lemma 6.1.14. Hence, given a vertex , the union of Proposition 6.1.12 reduces to when The right-hand side is strongly convex, while the left hand side contains . Hence the kernel is zero, so is full dimensional and , . Thus
when In other words, is the normal fan of , as claimed. Then , i.e., one easily sees that is the divisor associated to ample by Proposition 6.1.4.
, and . Hence is
The final assertion of the theorem also follows from Proposition 6.1.4. Here is a corollary of the proof of Theorem 6.1.15. Corollary 6.1.16. If is the normal fan of
is ample, then is a full dimensional lattice polytope, . , and is the Cartier divisor associated to
We have the following nice result in the smooth case.
Theorem 6.1.17. On a smooth complete toric variety and only if it is very ample.
, a divisor
is ample if
Proof. If is ample, then Corollary 6.1.16 shows that is the normal fan of . Since is very ample by Theoand is the divisor of is smooth, rem 2.4.3 and Proposition 2.4.4. Then we are done by Proposition 6.1.4.
Computing Ample Divisors. Given a wall
. Then a Cartier divisor (6.1.9)
, write
and pick
gives the wall inequality
Lemma 6.1.14 and Theorem 6.1.15 impy that the wall inequality (6.1.9) for every wall of .
is ample if and only if it satisfies
In terms of divisor classes, recall the map kernel consists of divisors of characters. If we fix
whose , then we have an
§6.1. Ample Divisors on Complete Toric Varieties
isomorphism
273
for all
(Exercise 6.1.7). Then (6.1.9) gives inequalities for determining when a divisor class is ample. Here is a classic example. Example 6.1.18. Let us determine the ample divisors on the Hirzebruch surface is shown in Figure 3 of Example 6.1.3, and this becomes the . The fan for . Hence we have ray generators fan for by redefining to be , then and maximal cones . If we set
(Note that every Weil divisor is Cartier since is smooth.) To determine when , we compute to be
Then (6.1.9) gives four wall inequalities which reduce to . Thus is ample (6.1.10)
For an arbitrary divisor
show that
is ample
, the relations
. Hence
Sometimes ampleness is easier to check if we think geometrically in terms of support functions. For , look back at Figure 6 and imagine moving the vertex at downwards. This gives the graph of , which is strictly convex when .
Here is an example of how to determine ampleness using support functions. as its has the eight orthants of Example 6.1.19. The fan for . Take the positive orthant maximal cones, and the ray generators are and subdivide further by adding the new ray generators
We obtain a complete fan by filling the first orthant with the cones in Figure 9 on with the plane the next page, which shows the intersection of . You will check that is smooth in Exercise 6.1.8. Let for
be a Cartier divisor on . Replacing with satisfies , we can assume that
Chapter 6. Line Bundles on Toric Varieties
274
e3
c d a
b
e1
e2
lying in
Figure 9. Cones of
Now observe that . Since and do not lie in a cone of , part (g) of Lemma 6.1.14 implies that
and generate a cone of , so that
However,
Together, these imply
. By symmetry, we obtain
an impossibility. Hence there are no strictly convex support functions. This proves that is a smooth complete nonprojective variety.
The Toric Variety of a Basepoint Free Divisor. We now return to the case of a basepoint free Cartier divisor . In on a complete toric variety the discussion leading up to Proposition 6.1.12, we defined the normal fan . The proposition explains how cones with of are combined to give the cones of . The normal fan gives the toric variety .
Before we can relate to , we need the following general fact about the pullback of a torus-invariant Cartier divisor by a toric morphism.
Proposition 6.1.20. Assume that is the toric morphism induced by , and let be a torus-invariant Cartier divisor with support . Then there is a unique torus-invariant Cartier divisor function with the following properties: on
;
(b) The support function
(a)
.
is the composition
§6.1. Ample Divisors on Complete Toric Varieties
275
now refers to an Proof. Let the local data of be , where arbitrary cone of . Recall that the minus sign comes from when
. Then the proof of Theorem 6.0.17 shows that is the sheaf of sections of a rank vector bundle with transition functions
Now take be the smallest cone satisfying and let , we set Using the dual map
;
; , one can show without difficulty that !; is the local data for a Cartier divisor
Since
.
Then straightforward to verify that
. It is on has the required properties (Exercise 6.1.9).
In the situation of Propositon 6.1.20, we say that . We now prove that a Cartier divisor on divisor pullback an ample divisor divisor on .
is the pullback of the without basepoints is the
Theorem 6.1.21. Let be a basepoint free Cartier divisor on a complete toric . Then there variety, and let be the toric variety of the polytope is a proper toric morphism such that is linearly equivalent to the pullback of an ample divisor on .
Proof. As in the discussion preceding Proposition 6.1.12, we have the sublattice so that it lies in , with dual . When we translate , the divisor changes by a linear equivalence. Then, given a vertex Proposition 6.1.12 implies
This implies that is compatible with the fans in and in and hence induces a toric morphism is proper since gives the ample, which and are complete. The polytope divisor on , and it follows that is the pullback of (Exercise 6.1.10).
, the cones and their Note that as we range over all vertices faces satisfy the conditions for being a fan, except that they may fail to be strongly convex. But they all contain the same maximal subspace, namely the kernel of . This is an example of what is called a degenerate fan. Once we mod out by the kernel, we get the genuine fan .
Here are two examples to illustrate what can happen in Theorem 6.1.21.
Chapter 6. Line Bundles on Toric Varieties
276
Example 6.1.22. While the toric variety of Example 6.1.19 has no ample divisors, it does have basepoint free divisors. The ray generators of are
with corresponding toric divisors
Then one can show that
is basepoint free (Exercise 6.1.8). Thus the support function is convex. is a union of three cones Figure 9 in Example 6.1.19 shows that
and
of . Using , one sees (Exercise 6.1.8). Hence we should combine that these three cones all have . these three cones. The same thing happens in and
e3
d
e1
e2
Figure 10. Combined cones of lying in
Thus, in the first orthant, the fan of looks like Figure 10 when intersected with . Hence is the blowup of at the point corresponding to the first orthant (Exercise 6.1.8). Note also that is a proper birational toric morphism since refines the fan of .
Example 6.1.23. Consider the divisor on the Hirzebruch surface , where we are using the notation of Example 6.1.3. For this divisor, the four cones of this fan give
is a line segment. When we combine and , Figure 3 from so that Example 6.1.3 shows that we get a degenerate fan. To get a genuine fan, we col lapse the vertical axis and obtain the fan for . Here, is the toric morphism from Example 3.3.5.
§6.1. Ample Divisors on Complete Toric Varieties
277
The Toric Chow Lemma. Recall from Chapter 3 that if is a refinement of , then there is a proper birational toric morphism . We close by proving is nonprojective (as in Example 6.1.19), then has the Toric Chow Lemma: if a refinement with projective.
Theorem 6.1.24. A complete fan
Proof. Suppose is a fan in the complete fan obtained from
has a refinement
. Let
such that
be obtained from
is projective.
by considering
So for each wall , we take the entire hyperplane spanned by the wall. This yields a subdivison with the property that
i.e., each hyperplane this way.
Choosing
by
Note that takes integer values on explains the minus sign).
is a union of walls of
so that
define the map
, and all walls of
arise
and is convex by the triangle inequality (this
Let us show that is piecewise linear with respect to . Fix and note that each cone of is contained in one of the closed half-spaces bounded by . This implies that is linear on each cone of . Hence the same is true for .
Finally, we prove that is strictly convex. Suppose that is a wall . Then . We label and so that ,
in
in The sum in is linear on , so is represented by different linear functions on each side of the wall . Since is convex, it is strictly convex by Lemma 6.1.14. Then is projective since
is ample by Theorem 6.1.15. of
Chapter 6. Line Bundles on Toric Varieties
278
In Chapter 11 we will improve this result by showing that to be smooth and projective. Exercises for §6.1. 6.1.1. Show that the toric variety
' % {
in Example 6.1.5 is singular. a ( be a compact set and define 6.1.2. Let by iZ f Explain carefully why the minimum exists and prove that is convex.
V
of the polytope
can be chosen
` & ( ^
$Z
g
.
a supporting hyperplane
is D . Then show a
6.1.3. Let be as in the proof of (b) (d). Prove that of that satisfies (6.1.5). Hint: Write Z Z , g g a a implies f Z f Z $Z .
$
$ ("
C{ %
6.1.4. As noted in the text, the polytope of a basepoint free Cartier divisor on a com plete toric variety can have dimension strictly less than . Here are some examples. (a) Let be one of the four torus-invariant prime divisors on O - O . Show that is a line segment. (b) Consider iO I and fix an integer with D . Find a basepoint free divisor on iO I such that . Hint: See Exercise 6.1.12 below.
C{ %
F
6.1.5. Prove (6.1.7). 6.1.6. This exercise is devoted to proving that the statements (a)–(g) of Lemma 6.1.14 are equivalent. Many of the implications use Lemma 6.1.8. (a) Prove (a) (b) and (c) (d). (b) Prove (b) (c) Prove (c)
(e) and (b) (f) (c). (b) by adapting the proof of (d)
(b) from Lemma 6.1.8.
(d) Prove (b) (g) and use the obvious implication (e) the lemma.
0/ { { % 0/ { .
(d) to complete the proof of
, . Prove that { % L
6.1.7. Let be = the toric % in & ( @ I and fix 1 variety of a fan the natural map induces an isomorphism j^ = L 46 D for all C 1 9E ,@
%
6.1.8. This exercise deals with Examples 6.1.19 and 6.1.22. (a) Prove that the toric variety of Example 6.1.19 is smooth.
) X ) X ) X X X X X \ $ \ * $ \ $ ) \ $ ) \ * $ ) \ $ ) \ )\ * $ )\ ) \ $ ) \ * ) \ $ ) \ )\ * )\ and conclude that is basepoint free.
* be the divisor defined in (b) Let Example 6.1.22. Prove that is the polytope with 10 vertices
(c) In Example 6.1.22, we asserted that certain maximal cones of % must be combined to get the maximal cones of % . Prove that this is correct. (d) Show that is the blowup of iO at the point corresponding to the first orthant.
6.1.9. Complete the proof of Proposition 6.1.20. 6.1.10. Complete the proof of Proposition 6.1.21.
§6.2. The Nef and Mori Cones
279
{ % and describe which torus* is the toric variety of the smooth complete fan % in with X % FE /4 \ $ \*$\ \*8 >
6.1.11. For the following toric varieties , compute invariant divisors are ample and which are basepoint free. (a)
(b)
is the blowup K
$O I of O I
at a fixed point of the torus action.
is the toric variety of the fan % from Exercise 3.3.9. See Figure 12 from Chapter 3. is the toric variety of the fan obtained from the fan of Figure 12 from Chapter 3 by combining the two upward pointing cones.
(c) (d)
$ $ \ I . Let X $ $ I denote I '#T T . X (a) Show that is basepoint free if and only if T ( D for all . X D for all . (b) Show that is ample if and only if T 6.1.13. Let be an ample divisor on a complete toric variety . Define = $Z 4bC G% FE &2( - 1
6.1.12. The toric variety iO I has ray generators \ the corresponding torus-invariant divisors. Consider
(a) Prove that 1 is strongly convex.
$
(b) Prove that the boundary of 1 is the graph of the support function . V (c) Prove that % is the set of cones obtained by projecting proper faces of 1 onto
(
.
§6.2. The Nef and Mori Cones In the last section, we saw that there are simple criteria which determine when a Cartier divisor is basepoint free or ample. We now study the structure of the set of basepoint free divisors and the set of ample divisors inside .
The main concept of this section is that of numerical effectivity. Roughly speaking, the goal is to define a pairing between divisors and curves, such that for a divisor and curve on a variety , the number counts the number of points of , with appropriate multiplicity. Example 6.2.1. Suppose with homogeneous coordinates , and let and and meet at the single point . Then
, where they share a common tangent. If we replace with the linearly equivalent divisor , then clearly and meet in two points. This suggests that the point should be counted twice, since it is a tangent point. Hence we should have .
Despite this encouraging example, there are several technical hurdles to overcome in order to make this precise in a general setting. Note that in , two lines may or may not meet, so to get a reasonable theory, we will work with complete curves on a normal variety . We also need to restrict to Cartier divisors on . With these assumptions, the intersection product should possess the following properties:
Chapter 6. Line Bundles on Toric Varieties
280
. when . is finite. Assume each Let be a prime divisor on that is smooth in , , andsuch and that the tangent spaces meet transversely. Then . from Note that these properties give a rigorous proof of the computation
Example 6.2.1.
The Degree of a Line Bundle. The key tool we will use is the notion of the degree of a divisor on an irreducible smooth complete curve . Such a divisor can be written as a finite sum where and .
be a divisor on an irreducible smooth complete
Definition 6.2.2. Let curve . Then the degree of
is the integer
Note the obvious property key result is proved in [41, Cor. II.6.10].
. The following
Theorem 6.2.3. Every principal divisor on an irreducible smooth complete curve has degree zero.
In other words, irreducible smooth complete curve
for all nonzero rational functions
. Thus
when
on
on an
and the degree map induces a surjective homomorphism
Note that all Weil divisors are Cartier since is smooth. is the set of isomorphism classes of line bundles In §6.0 we showed that on . Hence we can define the degree of a line bundle on . This leads immediately to the following result. Proposition 6.2.4. Let be an irreducible smooth complete curve. Then a line bundle has a degree such that has the following prop erties: . (a) (b) when . when . (c)
§6.2. The Nef and Mori Cones
281
The Normalization of a Curve. We defined the normalization of an affine variety in §1.0, and by gluing together the normalizations of affine pieces, one can define the normalization of any variety (see [41, Ex. II.3.8]). In particular, an irreducible curve has a normalization map
where
(a)
is an normal variety. Here are the key propeties of
Proposition 6.2.5. Let
(b)
.
be the normalization of an irreducible curve
is smooth. is complete whenever
is complete.
Proof. Since is a curve, Proposition 4.0.17 implies that covered by [41, Ex. II.5.8].
. Then:
is smooth. Part (b) is
One can prove that every irreducible smooth complete curve is projective. See [41, Ex. II.5.8].
The Intersection Product. We now have the tools needed to define the intersection product. Let be a normal variety. Given a Cartier divisor on and an irreducible complete curve , we have
The line bundle
The normalization
Then
;
on
. .
and
;
is a line bundle on the irreducible smooth complete curve
Definition 6.2.6. The intersection product of
is
.
.
Here are some properties of the intersection product.
Proposition 6.2.7. Let be an irreducible complete curve and sors on a normal variety . Then:
(a)
(b)
when
Cartier divi-
.
.
Proof. The pullback of line bundles is compatible with tensor product, so that part (a) follows from (6.0.3) and Proposition 6.2.4. Part (b) is an easy consequence of Propositions 6.0.21 and 6.2.4.
In Chapter 4, we defined a Weil divisor to be -Cartier if is Cartier for some integer . Given an irreducible complete curve , let
(6.2.1)
In Exercise 6.2.1 you will show that this intersection product is well-defined and satisfies Propostion 6.2.7.
Chapter 6. Line Bundles on Toric Varieties
282
Intersection Products on Toric Varieties. In the toric case, is easy to compute when and are torus-invariant in . In order for to be torus-invariant and
, where
is the wall complete, we must have , separating cones . We do not assume is complete.
In this situation, we have the sublattice and the
. Let and be the images of and in . Since quotient rays that correspond to the rays in the usual fan
is a wall, and , are is smooth, so no normalization is needed when for . In particular, computing the intersection product.
Proposition 6.2.8. Let
,
, be a complete torus-invariant corresponding to curve in . Let be a Cartier divisor with . Also pick that maps to the minimal generator of
, we can assume and is the fan Proof. Since consisting of and their faces. We also have The proof of Proposition 6.1.20 shows that the line bundle is determined . Thus by the transition function ; is the inclusion map. The pullback bundle is determined by where
the restriction of to
. Then
where
is the variety
and that
-orbit corresponding to . This is also the torus of the toric . In Lemma 3.2.5, we showed that is the dual of
;
Now comes the key observation: since the linear functions given by agree is the line bundle on
on , we have . Thus whose transition function is for .
It follows that by the data
;
;
, where
is the divisor on
Let be the torus fixed points corresponding to to the minimal generator
, we have
given
. Since
maps
§6.2. The Nef and Mori Cones
;
283
where the second equality follows from
in
Example 6.2.9. Consider the toric surface whose fan
. Hence
has ray generators
and maximal cones
The support of is the first quadrant and
complete torus-invariant curve .
are the divisors corresponding to If
is Cartier
When this condition is satisfied, we have
Also, basis of
gives the
, then
maps to the minimal generator of since . (You will check these assertions in Exercise 6.2.2.) Thus
form a
by Proposition 6.2.8. Since is -Cartier ( is simplicial), (6.2.1) shows that the formula for holds for arbitrary integers . In particular,
Later in the section we will see that these intersection products follow directly from and the fact that . the relation
Nef Divisors. We now define an important class of Cartier divisors.
Definition 6.2.10. Let be a normal variety. Then a Cartier divisor nef (short for numerically effective) if
for every irreducible complete curve
on
is
.
Any divisor linearly equivalent to a nef divisor is clearly nef. We also have the following useful result. Proposition 6.2.11. Every basepoint free divisor is nef.
Chapter 6. Line Bundles on Toric Varieties
284
Proof. The pullback of a line bundle generated by global sections is generated by global sections (Exercise 6.0.10). Thus, given and basepoint free, is generated by global sections. This allows us the line bundle for a basepoint free divisor on to write . A nonzero global
. Then gives an effective divisor section of
;
;
where the last inequality follows since
for every .
is effective if and only if
In the toric case, nef divisors are especially easy to understand.
Theorem 6.2.12. Let be a Cartier divisor on a complete toric variety following are equivalent.
(a)
is basepoint free, i.e.,
. The
is generated by global sections.
is nef.
(b) (c)
for all torus-invariant irreducible curves
.
Proof. The first item implies the second by Proposition 6.2.11, and the second
for all item implies the third by the definition of nef. So suppose that torus-invariant irreducible curves . We can replace with a linearly equivalent torus-invariant divisor. Then, by Theorem 6.1.10, it suffices to show that is convex.
Take a wall
Proposition 6.2.8, then
so that
of
and set
. If we pick
as in
Note that since the image of is nonzero in Then Lemma 6.1.8 implies that is convex.
.
A variant of the above proof leads to the following ampleness criterion, which you will prove in Exercise 6.2.3.
Theorem 6.2.13 (Toric Kleiman Criterion). Let be a Cartier divisor on a com for all torusplete toric variety . Then is ample if and only if invariant irreducible curves .
We also note that one direction of the proof follows from the general fact that
for all on any complete normal variety, an ample divisor satisfies irreducible curves (Exercise 6.2.4).
§6.2. The Nef and Mori Cones
285
Numerical Equivalence of Divisors. The intersection product leads to an important equivalence relation on Cartier divisors.
Definition 6.2.14. Let
be a normal variety.
(a) A Cartier divisor on is numerically equivalent to zero if all irreducible complete curves . (b) Cartier divisors and are numerically equivalent, written is numerically equivalent to zero.
for
What does this say in the toric case? be a Cartier divisor on a complete toric variety .
Proposition 6.2.15. Let Then if and only if
Proof. Clearly if is principal then and let converse, assume Proposition 6.2.8, then
, if
.
is numerically equivalent to zero. For the be a wall of . If we pick as in
since and . From here,
. This forces for for all , and it follows easily that is it is easy to see that
principal.
Numerical Equivalence of Curves. We also get an interesting equivalence relation on curves. Let be the free Abelian group generated by irreducible complete is called a proper -cycle. curves . An element of
be a normal variety.
(a) A proper -cycle on is numerically equivalent to zero if all Cartier divisors on . and are numerically equivalent, written (b) Proper -cycles is numerically equivalent to zero. extends naturally to a pairing The intersection product
Definition 6.2.16. Let
for
, if
between Cartier divisors and -cycles. In order to get a nondegenerate pairing, we work over and mod out by numerical equivalence.
Definition 6.2.17. For a normal variety , define and
It follows easily that we get a well-defined nondegenerate bilinear pairing A deeper fact is that and have finite dimension over . Thus and are dual vector spaces via intersection product.
Chapter 6. Line Bundles on Toric Varieties
286
The Nef and Mori Cones. The vector spaces interesting cones.
and
contain some
be a normal variety. is the cone in generated by classes of nef Cartier divisors. We (a) the nef cone. call is the cone in generated by classes of irreducible complete (b)
Definition 6.2.18. Let
curves.
(c)
is the closure of
in
. We call
the Mori cone.
Here are some easy observations about the nef and Mori cones.
are closed convex cones and are dual to each other, i.e., and
Lemma 6.2.19. (a)
(b) (c)
and
has maximal dimension in is strongly convex in
.
.
, and are convex cones, and Proof. It is obvious that
. In fact, is closed since it is defined by inequalities of the form
Then
by the definition of nef. follows easily. In general, need not be closed. However, since the closure of a convex cone is its double dual, we have
has maximal dimension since is spanned by the classes Note that . of irreducible complete curves. Hence the same is true for its closure is strongly convex since its the dual has maximal dimension. Then , then The toric case is especially nice. If we set since numerical and linear equivalence coincide by Proposition 6.2.15. Thus, in the instead of . When is complete, toric setting, we will write
the inclusion makes
a lattice in the vector space
Theorem 6.2.20. Let (a)
.
be a complete toric variety.
is a rational polyhedral cone in
.
§6.2. The Nef and Mori Cones
(b)
where
is a rational polyhedral cone in
is the class of .
287
. Furthermore,
Proof. Part (a) is an immediate consequence of part (b). For part (b), let and note that is a rational polyhedral cone contained in . Furthermore,Theorem 6.2.12 easily implies Then
where the third equality follows since
is polyhedral.
The formula from part (b) of Theorem 6.2.20
is called the Toric Cone Theorem. Although the Mori cone equals in this case, we will continue to write for consistency with the literature. Since , we get the following every irreducible curve gives a class in corollary of the Toric Cone Theorem.
Corollary 6.2.21. An irreducible curve in a complete toric variety is numeri cally equivalent to a non-negative combination of torus-invariant curves. When
is projective we can say more about the nef and Mori cones.
be a projective toric variety. Then: and are dual strongly convex rational polyhedral cones of (a) maximal dimension. lies in the (b) A Cartier divisor is ample if and only if its class in . interior of has an ample divisor . Then for every Proof. By hypothesis, irreducible curve in . This easily implies that the class of lies in the interior has maximal dimension and hence its dual of . Thus Theorem 6.2.22. Let
is strongly convex. When combined with Lemma 6.2.19, part (a) follows easily.
also shows that every irreducible curve gives The strict inequality a nonzero class in is in the interior of . Now suppose that the class of the nef cone. Then defines a supporting hyperplane of the origin of the dual
cone . Since for every irreducible curve ,
we have for all such . Hence is ample by Theorem 6.2.13.
Chapter 6. Line Bundles on Toric Varieties
288
When is not a projective variety, the ampleness criterion given in part (b) of Theorem 6.2.22 can fail. Here is an example due to Fujino [29]. with six minimal generators Example 6.2.23. Consider the complete fan in
and six maximal cones
You will draw a picture of this fan in Exercise 6.2.5 and show that the resulting complete toric variety satisfies
The maximal cones along the wall
and
meet
satisfies (Exercise 6.2.5), satisfies
However, any Cartier divisor so that the irreducible complete curve
by Proposition 6.2.8. Since this holds for all Cartier divisors on , we have . Then has no ample divisors by the Toric Kleiman Criterion, so that is nonprojective.
The nef cone of
is the half-line
gives a class in (Exercise 6.2.5). It follows that the Cartier divisor the interior of the nef cone, yet is not ample. Hence part (b) of Theorem 6.2.22 is false for . The failure is due to the existence of irreducible curves in that are numerically equivalent to zero. This shows that numerical equivalence can be badly behaved in the nonprojective case.
Exercises for §6.2.
6.2.1. Let be a normal variety. Prove that (6.2.1) gives a well-defined pairing between -Cartier divisors and irreducible complete curves. Also show that this pairing satisfies Propostion 6.2.7. 6.2.2. Derive the formulas for
a
and
a
given in Example 6.2.9.
6.2.3. Prove Theorem 6.2.13. 6.2.4. Prove that on a complete normal variety, an ample divisor all irreducible curves .
satisfies
D
for
§6.3. The Simplicial Case
289
6.2.5. Consider the fan % from Example 6.2.23. (a) Draw a picture of this fan in .
X (b) Prove that { % X j @0 /is $nef. (c) Prove that '
N
46
8
.
§6.3. The Simplicial Case
Henceforth we will assume that is a simplicial fan in . Then Proposition 4.2.7 implies that every Weil divisor is -Cartier. Since we will be working , it follows that we can drop the adjective “Cartier” when discussing in divisors.
Relations Among Minimal Generators. We begin our discussion of the simplicial case with another way to think of elements of . Recall from Theorem 4.1.3 that we have an exact sequence
(6.3.1) where
and sends the standard basis element to .
Proposition 6.3.1. Let exact sequences
where
Proof. Since
. Then there are dual
a standard basis vector of
In particular, we may interpret minimal generators of .
; ;
be a simplicial fan in
and
an irreducible complete curve
as the space of linear relations among the
is simplicial, all Weil divisors are -Cartier. Hence
Tensoring with preserves exactness, so exactness of the first sequence follows from (6.3.1). The dual of an exact sequence of finite dimensional vector spaces is still exact. Then the perfect pairings
Chapter 6. Line Bundles on Toric Varieties
290
easily imply that for
and
;
The map ducible complete curve
(6.3.2)
;
and
, we have
;
Taking the intersection product with
in Proposition 6.3.1 implies that an irregives the relation
in
This can be derived directly as follows. First observe that
. Writing this as
gives
By linearity, this holds for all we obtain
in
gives
,
Intersection Products. Our next task is to compute when is a torus invariant complete curve in . This means
, where is a wall, meaning that is the intersection of two cones in . Since we are not assuming that is complete, not every element of is a wall.
We begin with a case where
Since
is easy to compute. Fix a wall
is simplicial, we can label the minimal generators of so that
Thus is the side of “opposite” from . The intersection product of a simplicial cone is computed in terms of indices, where the index is the index of the sublattice
Lemma 6.3.2. If , and are as above, then
§6.3. The Simplicial Case
Proof. Since
291
is a basis of
, . On
such that
is the Cartier
. By (6.2.1) and Proposition 6.2.8,
. Recall that . where maps to a generator of When we combine with a basis of , we get a basis of . Thus there is a positive integer such that ,
. The minus sign is because and lie on opposite sides of . By considering the sublattices
one sees that . Thus
that such and
Pick a positive integer divisor determined by
, we can find
Since
, it follows that
Given a wall rays . Let
be a wall.
.
(6.3.3)
, our next task is to compute and write
and
Corollary 6.3.3. Let be a smooth fan in , then and generate a cone of If
for the other
This situation is pictured in Figure 11. Applying Lemma 6.3.2 to and , we obtain
(6.3.4)
To compute when
, note that the minimal generators are linearly dependent. Hence they satisfy a linear relation, which
we write as
(6.3.5)
Chapter 6. Line Bundles on Toric Varieties
292
uρ
n+1
← σ′ uρ
uρ
τ
2
n
uρ
1
← σ
Figure 11.
We may assume since and lie on opposite sides of the wall . Then (6.3.5) is unique up to multiplication by a positive constant since are linearly independent. We call (6.3.5) a wall relation.
On the other hand, setting
(6.3.6)
in (6.3.2) gives the linear relation
As we now prove, the two relations are the same up to a positive constant. Lemma 6.3.4. The relations given by (6.3.5) and (6.3.6) are equal after multiplication by a positive constant. In particular,
for all
and
for
.
Proof. First observe that if , then and never lie in the same cone of , so that by the Orbit-Cone Correspondence. This easily
(Exercise 6.3.1), which in turn implies that (6.3.6) reduces implies to the equation
§6.3. The Simplicial Case
293
The coefficients of and are positive by (6.3.4), so up to a positive constant, this must be the wall relation (6.3.5). The first assertion of the lemma follows.
for
Since the above relation equals (6.3.5) up to a nonzero constant, we obtain
. Then the desired formulas for
follow from (6.3.4).
For a simplicial toric variety, Lemmas 6.3.2 and 6.3.4 provide everything we need to compute when is a wall of .
Example 6.3.5. Consider the fan where
,
and the relation
implies
. Computing indices gives
Then Lemma 6.3.2 implies
from Example 6.2.9. We have the wall
and
in
by Lemma 6.3.4. Hence we recover the calculations of Example 6.2.9.
When is smooth, all indices are . Hence the wall relation (6.3.5) can be written uniquely as (6.3.7)
and then the intersection formula of Lemma 6.3.4 reduces to for
.
Example 6.3.6. For the Hirzebruch surface , the four curves coming from walls are also divisors. Recall that the minimal generators are
arranged clockwise around the origin (see Figure 3 from Example 6.1.3). Hence the wall generated by gives the relation
Chapter 6. Line Bundles on Toric Varieties
294
which implies
by Lemma 6.3.4. On the other hand, the wall generated by
Then the lemma implies
gives the relation
Although and are both smooth rational curves on , they intersect themselves very differently. In general, a divisor on a complete surface has . Self-intersections will play a prominent role in self-intersection Chapter 10 when we study toric surfaces.
Primitive Collections. In the projective case, there is a beautiful criterion for a Cartier divisor to be nef or ample in terms of the primitive collections introduced in Definition 5.1.5. Recall that is a primitive collection if is not contained in for some but any proper subset is. Since is simplicial, primitive means that does not generate a cone of but every proper subset does. This is the original definition given by Batyrev [5].
z
Example 6.3.7. Consider the complete fan in
shown in Figure 12.
ρ4
ρ1
ρ3 ρ2
y x ρ0
Figure 12. A fan in
One can check that
are the only primitive collections of .
§6.3. The Simplicial Case
295
Here is the promised characterization, due to Batyrev [5] in the smooth case. Theorem 6.3.8. Let
be a projective simplicial toric variety.
(a) A Cartier divisor
is nef if and only if its support function
(b) A Cartier divisor
for all primitive collections
satisfies
of .
is ample if and only if its support function
for all primitive collections
satisfies
of .
Before we discuss the proof of Theorem 6.3.8, we need to study the relations that come from primitive collections. be a primitive collection for the Definition 6.3.9. Let complete simplicial fan . Hence lies in the relative interior of a cone
. Thus there is a unique expression
Then is the primitive relation of .
" # # " " # $ " (6.3.8) ! # " #
otherwise % Then &' , so that gives an element of ( " when # ,$
In Exercise 6.3.2, you will prove that )+* The coefficient vector of this relation is
, where
" , so that
by Proposition 6.3.1. consists
of the positive coefficients in the primitive relation. The Mori cone for
has a nice description in terms of primitive relations.
Theorem 6.3.10. For a projective simplicial toric variety .- 0/
where the sum is over all primitive collections
of .
,
1 3 2 1 and a relation &4 9 8 and : ( is 1 : 2 1 ): 2
Proof. Given a Cartier divisor intersection pairing of 1 576 (6.3.9)
, the
296
Chapter 6. Line Bundles on Toric Varieties
(Exercise 6.3.3). In particular, when :
, we can rearrange terms to obtain
1 ) 2 2 ) % Since the support function of 1 satisfies &' 2 and is linear on , we can rewrite this as (6.3.10) 1 &4 &' &4 &4 % If 1 is nef, then it is basepoint free (Theorem 6.2.12), so that is convex. It
, which proves follows that 1 . Note also that is nonzero. To finish the proof, we need to show that is generated by the .
This cone is strongly convex ( is projective), which implies that it is generated by its 1-dimensional faces, which we call extremal rays. By Toric Cone Theorem,
each extremal ray is generated by the class coming from a wall. We call an extremal wall. It suffices to show that is a positive multiple of for some primitive collection . We first pause to make a useful observation about nef divisors. Any nef divisor
is linearly equivalent to a torus-invariant nef divisor 1 2 1 . Given , # " . As we have for noted above, 1 is with &4 2 basepoint free, so that # " & & 2
6 , we may hence assume that by Theorem 6.1.10. Replacing 1 with 1
# # " " (6.3.11) 1 2 1 2 and 2 %
. Consider the set Now assume we have an extremal wall and let # 1 % We will prove that is a primitive collection whose primitive relation is the class of , up to a positive constant. Our argument is taken from [19], which is based on ideas of Kresch [62]. We first prove by contradiction that " for all . Suppose " and take an ample divisor 1 (remember that is projective). Then in
particular 1 is nef, so we may assume that 1 is of the form (6.3.11). Since 2 # " for , we have
1
2 1 %
by (6.3.11), and " implies 1 for # " . It However, 2 follows that 1 , which is impossible since 1 is ample. Thus no cone of
contains all rays in
.
§6.3. The Simplicial Case
297
It follows that some subset is a primitive collection. This gives the ( , and we also have the primitive relation with coefficient vector . Let ( class
where
(
. We claim that if
# 1
(6.3.12)
is sufficiently small, then
*
# 1 *
To prove this, first observe that the definition of
%
implies
1 1 1 % . This forces 1 .
and 1 Now suppose that 1 * As noted above, 1 is the coefficent of & in the primitive relation of , # . Then implies 1
which by (6.3.8) is positive only when by the definition of . But we can clearly choose sufficiently small so that 1 1 whenever 1 % This inequality and the above equation imply 1 , which is a contradiction. We next claim that . By (6.3.12), we have # 1 #
" * 1 * where the second inclusion follows from , (6.3.4), and Lemma 6.3.4. Now let 1 be nef, and by (6.3.11) with , we may assume that # # " " 1 2 1 2 and 2 % Then
1 2 1 and 1 where the final inequality follows since 2 * can happen only # " when
. This proves that . . Thus the equation We showed earlier that
expresses as a sum of elements of . But is extremal, i.e., it lies in and a one-dimensional face of . By Lemma 1.2.7, this forces to is nonzero, it generates the face, so that is a positive lie in the face. Since multiple of . The relation corresponding to has coefficients 1 , and is the # set of ’s where 1 . But this relation is a positive multiple of , whose
positive entries correspond to
. Thus
and the proof is complete.
Chapter 6. Line Bundles on Toric Varieties
298
It is now straightforward to prove Theorem 6.3.8 using Theorem 6.3.10 and (6.3.10) (Exercise 6.3.4). We should also mention that these results hold more generally for any projective toric variety (see [19]). Example 6.3.11. Let # minimal generators of
be the fan shown in Figure 12 of Example 6.3.7. The % % % # are " " " " " " " " " & & & & & % The computations you did for part (c) of Exercise 6.1.11 imply that is # # # # # projective. Since the only primitive collections are and , The-
1 is nef if and only if
& & & &
& & & & & &
orem 6.3.8 implies that a Cartier divisor
2
(resp. and 2 1 1 is nef (resp. ample) if and only if 2 2
and ample if and only if these inequalities are strict. One can also check that
576 2 1 1
).
Exercise 6.3.5 will relate this example to the proof of Theorem 6.3.10.
These ideas will reappear in Chapter 15 when we study the toric minimal model program. Exercises for §6.3. 6.3.1. This exercise will describe a situation where is guaranteed to be zero.
(a) Let be a normal variety and assume that is a complete irreducible curve disjoint from the support of a Cartier divisor . Prove that . Hint: Consider .
,
(b) Let be a wall of a fan and pick !" #%$& such that and never lie in the same cone of . Use the Cone-Orbit Correspondence to prove that ('*),+- .0/ , and conclude that 1'2 +-3 4 .
6.3.2. In the primitive relation defined in Definition 6.3.9, prove 5 ' 6$ when (879);:<%$& . Hint: If >=?8:<%$ and 5@'@A?B6$ , then cancel CD'@A and show that CD'FEHGJIKIJI@GLCM'ON. : . 6.3.3. Let -P be a simplicial toric variety and fix a Cartier divisor QSR a relation R 'U ' C ' V . Prove that the intersection pairing of W YX [Z b c ' b U 'Jd Pfe =hg 8iY=j3 P is W YX kR ' T ' U ' .
' '
B{]\^'3 (T P_a`
and and
6.3.4. Prove Theorem 6.3.8 using Theorem 6.3.10 and (6.3.10). 6.3.5. Consider the fan from Examples 6.3.7 and 6.3.11. Every wall of is of the form Jlnm8poqsr>tsCMlLGhC mK for suitable u.vxw . Let y KlzmHY|{~} denote the wall relation of Klnm . Normalize by a positive constant so that the entries of y lnmj are integers with ^\KY$ . (a) Show the nine walls give the three distinct wall relations y3 F&@Ghy3 J
jOGLy3 JL& . (b) Show that y F
"y L 6y F and conclude that F
and h are extremal walls whose classes generate the Mori cone of P .
Appendix: Quasicoherent Sheaves on Toric Varieties
299
(c) For each extremal wall of part (b), determine the corresponding primitive collection. You should be able to read the primitive collection directly from the wall relation.
6.3.6. Let be the toric surface obtained by changing the ray C= in the fan of the Hirzebruch surface from $^GhyH to yjGJ$& . Assume y 6$ .
(a) Prove that is singular. How many singular points are there? (b) Determine which divisors T = = T JY? T
JY
# T K are Cartier and compute Yl J m for all uFGaw . (c) Determine the primitive relations and extremal walls of
.
Appendix: Quasicoherent Sheaves on Toric Varieties Now that we know more about sheaves (specifically, tensor products and exactness), we can complete the discussion of quasicoherent sheaves on toric varieties begun in §5.3. In this appendix, P will denote a toric variety with no torus factors. The total coordinate ring of P is W M' j -! #h$ X , which is graded by o a P . Recall from §5.3 that for xo 3 P , the shifted -module ; gives the sheaf ~ satisfying ~ ( for any Weil divisor with 6 W 1X . In §6.0 we constructed a sheaf homomorphism
! " #%$ In a similar way, one can define ;' )(*%$ (6.A.1) o 3 1P such that if W YX and ( xW " for G( h' ! " h ~ ! )(;
&" OI
8+(@I
X , then the diagram / c& " h /
, (;
commutes, where the vertical maps are isomorphisms.
-
From Sheaves to Modules. The main construction of §5.3 takes a graded -module and produces a quasicoherent sheaf on 1P . We now go in the reverse direction and show that every quasicoherent sheaf on (P arises in this way. We will use the following construction.
-.
0 -modules on P and !o 3 P , define /x;1/2 ~ 354 3 and then set /4 6 / ;LOI 7 d08!9 e g YP G:x 3 4 3 For example, 1 3 4 since 3 P G ~h#1 7 by Proposition 5.3.7. Using this and (6.A.1), we see that 4 / is a graded -module. 34 We want to show that / is isomorphic to the sheaf associated to 4 / when / is quasicoherent. We will need the following lemma due to Mustat¸aˇ [73]. Recall that for ; ! , we have the monomial = < ?> 'Ad @ = e =hg ' B . Let = 4tJD) = < oa P . Definition 6.A.1. For a sheaf
/
of
Chapter 6. Line Bundles on Toric Varieties
300
/ 3 = P G:/x = L such that C restricts / 3 =) < = = :/ 3 3 (b) If C! 3 P G:/4 restricts to in = G:/4 , then there is BS such that ) #= < FC 6 in P G:/x = L . 3 = G:/4 . Given c , let be the Proof. For part (a), fix ; [ and take x = ; ), ) restriction of to (3.1.3), we can find such that = ) "Q s ) .t \sBy W ) kX . In terms of the total coordinate ring , by (5.3.1). Hence the coordinate ring of = ) is the we have W ) kX localization G where < > ' ' since ) . This enables us to write = ) c ^ I 3 Since is quasicoherent, is determined by its sections / / G:4 / , and then
3 = ) >G 4 / is the localization . 3 3 It follows that , where 8B and C = G:4 / 3 equals C ) G:4 / . ( = G4 / . Since , ; , we see that Hence C restricts to ) x) = < ) B (6.A.2) 3 >G x / = h restricts for . This monomial has degree = . Then C - C 3 = = = ) G:x / h . By making sufficiently large, we can find one that to < Y . works for all / = , take = Gh J9! and To study whether the C patch to give a global section of x LA~) %E , and set :8| = ) J . Thus 3 (6.A.3) 44C A , C E G:x / = L 3 = ) G:x / = h . Arguing as above, this group of sections is the localrestricts3 to Y ization Gx / = h , where (: ) ; ) such that = ) c . Since 3 gives the zero element in this localization,= there is 4B with in / = 3 h . If we multiply by S < % for , we obtain = < G: x = in G:x / L h . Another way to think3 of this is that if we made in (6.A.2) big enough to begin with, then in fact c in G:x / = L for all >GL . Given 3 the definition (6.A.3) of , it follows that the C patch to give a global section C! P G:x / = L with the desired properties. 3
Lemma 6.A.2. Let be a quasicoherent sheaf on (P . (a) If 8 G 4 , then there are B4 and C, to G x L .
!$#(' ) * %
"!$#&%
+
"
-
. 0/
6 .
!$#&%
=
,?>
A "
;
,?>
B
;
C
5
HG
;
EFJ=
;
@;
. 7 6
=
;
6
6
<;
!1#2% 8 3
.
5
67
!1#2% 43
3
:9
-
,
6D
EF
HGI9
8 3
!$#&%
G
=
!1#2% 3=
;
=
;
The proof of part (b) is similar and is left to the reader. K
This lemma makes it easy to prove the following result.
/
34
Proposition 6.A.3. Let be a quasicoherent sheaf on (P . Then sheaf associated to the graded -module 4 .
-
34
/
/
Proof. Let 4 and recall from §5.3 that for every is the sheaf associated to the L -module L .
=
:-
;
/
is isomorphic to the
, the restriction of
-.
to
Appendix: Quasicoherent Sheaves on Toric Varieties
301
- . $ / = < = = /3 = ) < = = G = h' 3 = G:/x = L* $ 3 = G /4 induces a homomorphism of -modules 3 - % $ = G /4OI (6.A.4) $ / that patch to give a sheaf This defines compatible sheaf homomorphisms - . homomorphism - . $ / . / is quasicoherent, it= suffices to show that (6.A.4)3 is an isomorphism for every ; S Since . First suppose that3 C < kmaps to = G /4 . It follows easily / = h . By Lemma that C restricts to zero in = 3 G x 6.A.2 applied to x / = , there is / L > = h . Then BS such that ) = < LC84 in 3 P G:x C = < C 4 in - G ) = < = < 3 which shows that (6.A.4) is injective. To / and= apply 3 prove surjectivity, take 8 = G 4 = P Lemma 6.A.2 to find B and C| 3 G x / h such that C restricts to < . It = follows immediately that C ) < 2 maps to . 3
We first construct a sheaf homomorphism . Elements of L are C / for C 3 P G x h . Since 8 is a section of over , the map
L
L
, L
/
L
3
5
, L
5
E
5
3
L
3
/
L
K
This result proves part (a) of Proposition 5.3.9. We now turn our attention to part (b) of the proposition, which applies to coherent sheaves. Proposition 6.A.4. Every coherent sheaf to a finitely generated graded -module.
/
on (P is isomorphic to the sheaf associated
=
= 3 = G:/4 ) 0= < l = 3comes 4 - /4
Proof. On the affine open subset , we can find finitely many sections l ' which generate over . By Lemma 6.A.2, we can find BS such that . Now consider the graded -module from a global section ^l of x ' generated by the l . Proposition 6.A.3 gives an isomorphism
/
=
=
=
/ =
3 4
'
-
34
/
/41/ I
-. $ /
'
Hence 4 gives a sheaf homomorphism is injective by the , we have l , which l / exactness proved in Proposition 6.0.9. Over E(x L , ' ' over , it follows that . Then we are done and since these sections generate since is clearly finitely generated. K
=
/
-
=
34
=
-. /
= = <
/
-
The proof of Proposition 6.A.4 used a submodule of 4 because the full module need not be finitely generated when is coherent. Here is an easy example.
/
4 $ ! / |isu free in T
34 / 1/ 3 " /4 6 % G /x T h 6 I d$# d$# This module is not finitely generated over [ 9W h GJIKIJI@G: X since it is nonzero in all
negative degrees.
Example 6.A.5. A point ! gives a subvariety u . The sheaf can be thought of as a copy of sitting over the point . The line bundle a neighborhood of , so that x T for all T . Thus 7
7
Chapter 6. Line Bundles on Toric Varieties
302
Subschemes and Homogeneous Ideals. For readers who know about schemes, we can apply the above results to describe subschemes of a toric variety P with no torus factors. Let be a homogeneous ideal. By Proposition 6.0.9, this gives a sheaf of ideals ?whose quotient is the structure sheaf of closed subscheme of x P . This differs from the subvarieties considered in the rest of the book since the structure sheaf
may have nilpotents.
S
Proposition 6.A.6. Every subscheme Proof. Given an ideal sheaf
53 4
P
is defined by a homogeneous ideal
, we get a homomorphism of 34 # $ 13~ 4 I
.
-modules
$
If is the image of this map, then the map factors , where the first arrow is surjective and the second injective. Applying Propositions 6.0.9 and 6.A.3, the . .. K factors as . It follows immediately that inclusion
$
In the case of , it is well-known that different graded ideals can give the same ideal sheaf. The same happens in the toric situation, and as in §5.3, we get the best answer in the smooth case. Not surprisingly, the irrelevant ideal a plays a key role.
Proposition 6.A.7. Homogeneous ideals G in the total coordinate of a smooth toric variety 1P give the same ideal sheaf of if and only if ( a . K
a , and in the exact sequence $ $ a $ ( / $ >G the quotient (a / is annihilated by a power of ( since is finitely generated. The sheaf associated to this quotient is trivial by Proposition 5.3.10. Then and (a
Proof. Since is homogeneous, the same is true for
give the same ideal sheaf by Proposition 6.0.9. This proves one direction of the proposition. For the converse, suppose that
and
give the same ideal sheaf. This means that
for all ; . Take 7 for o 3 P and fix ; . Arguing as in the proof ; h$ such of Proposition 5.3.10, we can find a monomial involving only ' for = = that implies ) < Y , which in turn easily implies = ) < for . This that ) < . Thus ( , and from here the rest of the proof is
L
L
/
D
6D
/
3
straightforward.
L 9
6D
/
43
L
K
There is a less elegant version of this result that applies to simplicial toric varieties. See [15] for a proof and more details about the relation between graded modules and sheaves. See also [68] for more on multigraded commutative algebra.
Chapter 7
Projective Toric Morphisms
§7.0. Background: Quasiprojective Varieties and Projective Morphisms Many results of Chapter 6 can be generalized, but in order to do so, we need to learn about quasiprojective varieties and projective morphisms. Quasiprojective Varieties. Besides affine and projective varieties, we also have the following important class of varieties. Definition 7.0.1. A variety is quasiprojective if it is isomorphic to an open subset of a projective variety. Here are some easy properties of quasiprojective varieties. Proposition 7.0.2. (a) Affine varieties and projective varieties are quasiprojective. (b) Every closed subvariety of a quasiprojective variety is quasiprojective. (c) A product of quasiprojective varieties is quasiprojective. Proof. You will prove this in Exercise 7.0.1. Projective Morphisms. In algebraic geometry, concepts that apply to varieties sometimes have relative versions that apply to morphisms between varieties. For example, in §3.4, we defined completeness and properness, where the former applies to varieties and the latter applies to morphisms. Sometimes we say that the 303
Chapter 7. Projective Toric Morphisms
304
relative version of a complete variety is a proper morphism. In the same way, the relative version of a projective variety is a projective morphism.
a line
.
gives a
We begin with a special case. Let be a morphism and bundle on with a basepoint free finite-dimensional subspace with the morphism Then combining that fits into a commutative diagram morphism
"!$#&% '
(
/ PPP PPP PPP PPP PPP P'
(7.0.1)
)*
+ , -. /
If is a closed embedding (meaning that its image is closed and the induced map is an isomorphism), then you will show in Exercise 7.0.2 that has the following nice properties: is proper.
1
0 +
1
43 2
2 /
For every , the fiber and hence is projective.
5
is isomorphic to a closed subvariety of
The general definition of projective morphism must include this special case. In fact, going from the special case to the general case is not that hard.
6 7 98 "!;# % ' : 8 = 3 8 9 < is a closed embedding, where ;> @? and A B is projective with respect to .
Definition 7.0.3. A morphism is projective if there is a line bundle on and an affine open cover of with the property that for each , there is a basepoint free finite-dimensional subspace such that
3
8
; > @ ? . We say that
The general case has the properties noted above in the special case. Proposition 7.0.4. Let (a)
6 7
is proper.
2 .
(b) For every
, the fiber
be projective. Then:
3 2 is a projective variety.
Here are some further properties. Proposition 7.0.5. (a) The composition of projective morphisms is projective. (b) A closed embedding is a projective morphism. (c) A variety
is projective if and only if
7 DCFE
is a projective morphism.
§7.0. Background: Quasiprojective Varieties and Projective Morphisms
305
Proof. Parts (a) and (b) are proved in [37, (5.5.5)]. For part (c), one direction follows immediately from the previous proposition. Conversely, let be that is not contained projective, and assume that is nondegenerate, " meaning " . Then in any hyperplane of . Now let
is injective since
"
is nondegenerate. In Exercise 7.0.3 you will show that
" C % % % and that if - is the image of , then F is the embedding we began with. Hence Definition 7.0.3 is satisfied for 7 DCFE and . When the domain is quasiprojective, the relation between proper and projective is especially easy to understand. Proposition 7.0.6. Let Then:
0
is proper
be a morphism where
is quasiprojective.
is projective %
Proof. One direction is obvious since projective implies proper. For the opposite direction, our hypothesis implies that there is a morphism
+ + is open and via . It to a projective variety + such that follows that the product map B A+ * (7.0.2) * , where * is closed in . induces Since B is proper, 7 + is also proper (Exercise 7.0.4). Hence the image of is closed in , + since proper morphisms are universally = , where is closed in 5 + . This proves closed. Thus that (7.0.2) is a closed embedding. . Then we get a closed embedding Now take a closed embedding + of into . From here, it is straightforward to show that is projective
(Exercise 7.0.4).
In the literature, one finds two definitions of projective morphism. In [41, II.4], a projective morphism is defined as the special case considered in (7.0.1), while [37, (5.5.2)] and [100, 5.3] give a more general definition. Proposition 7.A.5 of the appendix to this chapter shows that the general definition is equivalent to ours. Projective Bundles. Vector bundles give rise to an interesting class of projective morphisms.
Chapter 7. Projective Toric Morphisms
306
8
"
Let be a vector . Recall from §6.0 that withbundleof rank has a trivialization . Furthermore, the $ that make the diagram transition functions
3 8
8
8 8 ! > ? ? k8 k5 $ 8 O kk kkkk k 3 8 $ 8 SSS ! > @? S ? S S SSS) $ 8 8
" induces an isomorphism " F8 $ 8 3 8 $ 8 3 % a variety . It is clear that induces a morphism This gives gluing data for the trivialization * and that induces
3 8 8 3 % is a projective morphism. We call the It follows easily that F commute. Note that
projective bundle of
.
Example 7.0.7. Let be a finite dimensional vector space over dimension. Then, for any variety , the trivial bundle trivial projective bundle .
of positive gives the
There is also a version of this construction for locally free sheaves. If locally free of rank , then is the sheaf of sections of a vector bundle " , we proved this in Theorem 6.0.19. Then define of rank . When
(7.0.3)
where
is the dual vector bundle of . Here are some properties of
is
.
" is locally free of rank . when when is locally free and is a line bundle. (b) . (c) If a homomorphism locally free sheaves is surjective, then the of ofprojective induced map bundles is injective.
Lemma 7.0.8. (a)
Proof. You will prove this in . Exercise 7.0.5. The dual in (7.0.3) explains why gives
,
F %
The appearance of the dual in (7.0.3) can be explained as follows. Let be a line bundle with basepoint free of finite dimension. As in §6.0, this gives a morphism
§7.1. Polyhedra and Toric Varieties Let
307
. The corresponding vector bundle is
(7.0.4)
%
%
A
By Proposition 6.0.23, the natural map is surjective since basepoints. By Lemma 7.0.8, we get an injection of projective bundles
The lemma also implies
has no
. Using this and (7.0.4), we get an injection
%
F
, so
Projection onto the second factor gives a morphism morphism from §6.0 (Exercise 7.0.6).
, which is the
We should mention that one can define the “projective bundle” coherent sheaf on . See [41, II.7].
for any
Exercises for §7.0. 7.0.1. Prove Proposition 7.0.2.
$
7.0.2. Prove Proposition 7.0.4. Hint: First prove the special case given by (7.0.1). Recall from §3.4 that is complete, so that is proper. 7.0.3. Complete the proof of Proposition 7.0.5.
$
(
$
(
$
and be morphisms such that is 7.0.4. Let proper. Prove that is also proper. Hint: As noted in the comments following Corollary 3.4.8, being proper is equivalent to being topologically proper (Definition 3.4.2). = = Also, implies 8 # k ~ 8 #h .
$
)( #
(
7.0.5. Prove Lemma 7.0.8. Hint: Work on an open cover of involved are trivial.
where all of the bundles
$
$
using 7.0.6. In the discussion following (7.0.4), we constructed a morphism . Prove that this coincides with the morphism the surjection 4 . 7.0.7. Show that
is quasiprojective but neither affine nor projective.
>G
'
§7.1. Polyhedra and Toric Varieties This section and the next will study quasiprojective toric varieties and projective toric morphisms. Our starting point is the observation that just as polytopes give projective toric varieties, polyhedra give projective toric morphisms. 8 is the intersection of finitely many Polyhedra. Recall that a polyhedron closed half-spaces 8 " % % % % & 2 A basic structure theorem tells us that is a Minkowski sum
Chapter 7. Projective Toric Morphisms
308
where is a polytope and is a polyhedral cone (see [103, Thm. 1.2]). If presented as above, then the cone part of is 8 " % % % % & (7.1.1) (Exercise 7.1.1). Following [103], we call the recession cone of .
is
Similar to polytopes, polyhedra have supporting hyperplanes, faces, facets, vertices, edges, etc. One difference is that some polyhedra have no vertices. 8 be a polyhedron with recession cone . Lemma 7.1.1. Let is a vertex is finite and is nonempty if and only if (a) The set is strongly convex. . (b) If is strongly convex, then
2 2 " % % % .
Proof. You will prove this in Exercises 7.1.2–7.1.5. Example 7.1.2. The polyhedron 2 % % % 2 has vertices % % % and recession cone
Lattice Polyhedra. We now generalize the notion of lattice polytope. 8 is a lattice polyhedron with respect to Definition 7.1.3. A polyhedron 8 the lattice if (a) The recession cone of is a strongly convex rational polyhedral cone. (b) The vertices of lie in the lattice .
A full dimensional lattice polyhedron has a unique facet presentation 8 (7.1.2) & 2 for all facets
( is a primitive inward pointing facet normal. This was defined in where & Chapter 2 for full dimensional lattice polytopes but applies equally well to full 8 dimensional lattice polyhedra. Then define by 8 % for all & 2 " considered When is a polytope, this reduces to the cone in §2.2.
at the origin is given by the fan in Example 7.1.4. The blowup of with minimal generators & and maximal cones & & & & , & & . For the divisor 1 1 1 1 , we computed in Figure 5 from Example 4.3.4 that the polyhedron is a -dimensional lattice polyhedron whose recession cone is the first quadrant. Figure 1 on the next page shows the -dimensional cone with at " height . Notice how the cone of appears naturally at height in Figure 1.
Some of the properties suggested by Figure 1 hold in general.
§7.1. Polyhedra and Toric Varieties
309
z
PD
1
y C x
Figure 1. The cone
Lemma 7.1.5. Let be a full dimensional lattice polyhedron in 8 cone . Then is a strongly convex cone in and $ 8
%
8
with recession
Proof. The final assertion of the lemma follows from (7.1.1) and the definition of 8 / implies . For strong convexity, note that 8 % $ Then we are done since
is strongly convex.
has height . Furthermore, when We say that a point the “slice” of at height is . If we write , where polytope, then for ,
, is a
since is a cone. It follows that as , the polytope shrinks to a point so that at height , only the cone remains, as in Lemma 7.1.5. You can see how this works in Figure 1. The Toric Variety of a Polyhedron. In Chapter 2, we constructed the normal fan of a full dimensional lattice polytope. We now do the same for a full dimensional lattice polyhedron . Given a vertex , we get the cone 8 % $
Chapter 7. Projective Toric Morphisms
310
Note that since is a lattice polyhedron. It follows easily that is a strongly convex rational polyhedral cone of maximal dimension, so that the same is true for its dual ( 8 % $
These cones fit together nicely. 8 be a full dimensional lattice polyhedron with reces Theorem 7.1.6. Let 8 sion cone . Then the cones , a vertex of , and their faces form a fan in ( whose support is .
Proof. The proof that we get a fan is similar to the proof for the polytope case (see §2.3) and hence is omitted. To complete the proof, we need to show $ . Then where is the set of vertices of . Take and . Taking $ $ , which easily implies . For and pick duals, we obtain the opposite inclusion, take & such that & & for all . We show & as follows. Any" $ where can be written , . Then and & & & & & % $ Thus for all & , which proves - & . - The fan of Theorem 7.1.6 is the normal fan of , denoted . We define to be the toric variety of the normal fan . Here is an example. " 2 % % % 2 2 Example 7.1.7. The polyhedron 2 " of Example 7.1.2 has vertices % % % . The facet of defined by 2 as inward normal. Then the vertex gives the cone has
These cones form - the fan of the blowup of
at the origin, so
-
% % % % % % %
.
complete in this example. In general, the normal fan has
is- . not We measure the deviation from completeness as follows.
- is a rational The support polyhedral cone but need not be strongly convex. $
Recall that in . Hence
gives the following: is the largest subspace contained Note that support
1
1 1
The sublattice
$ (
-(
The strongly convex cone The affine toric variety
8
- and the- quotient lattice ( - ( $ ( . - 9 ( 8 9 ( 9 8 .
of
.
§7.1. Polyhedra and Toric Varieties
-
-
311
-
A
( ( is compatible with the fans of The projection 8 map . Hence we get a toric - morphism
-
-
83 Since
8
-
%
-
and
(Exercise 7.1.6), Theorem 3.4.7 implies that
08
8
since
is proper.
The key result of this section is that is a projective morphism. gives the From a sophisticated point of view, this is easy to see. The cone semigroup algebra $ (7.1.3) $ where the character associated to is written . This algebra is graded by height, i.e, . The 5 construction in algebraic geometry associates a variety 5 to a graded ring . In the appendix to this chapter, we will discuss 5 and show that 5 $ - % -
Then standard properties of 5 easily imply that Proposition 7.A.1 in the appendix).
8
is projective (see
The Divisor of a Polyhedron. Let be a full dimensional lattice- polyhedron. As in the polytope case, facets of correspond to rays in the normal fan , so that each facet gives a prime torus-invariant divisor 1 . Thus the facet presentation (7.1.2) of gives the divisor
1
2 1
where the sum is over all facets of . Results from Chapter - 4 (Proposition 4.2.10 and Example 4.3.7) easily adapt to the polyhedral case to show that 1 is Cartier
. (with for every vertex) and the polyhedron of 1 is , i.e., - that Then Proposition 4.3.3 implies
(7.1.4)
1
-
%
The- definition of projective morphism given in §0 involves a line bundle and a finite-dimensional subspace of global sections. The line bundle will be (actually a multiple ) and will be determined by certain carefully
1 chosen lattice points of . The reason we need a multiple is that might not have enough lattice points.
Normal and Very Ample Polyhedra. In Chapter 2, we defined normal and very ample polytopes, which are different ways of saying that there are enough lattice points. For a lattice polyhedron , the definitions are the same.
Chapter 7. Projective Toric Morphisms
312
8 be a lattice polyhedron. Then: Definition 7.1.8. Let " , every lattice point of is a sum of (a) is normal if for all integers lattice points of . $ (b) is very ample if for every vertex , the semigroup $ generated by is saturated in .
We have the following result about normal and very ample polyhedra. 8 be a lattice polyhedron. Then: Proposition 7.1.9. Let (a) If is normal, then is very ample. (b) If 6 , then is normal and hence very ample for all
"
.
Proof. Part (a) follows from the proof of Proposition 2.2.17. For part (b), let be the convex hull of the vertices of , so that , where is the recession cone of . It is easy to see that is normal whenever is (Exercise 7.1.7). Note also that " % " Now suppose , then and hence are normal. If . If 6 6 , then is normal by Theorem 2.2.11, so that is normal. Then is very ample by part (a). 8, The Projective Morphism. Let be a full dimensional lattice polyhedron in $ and assume that is very ample. Then pick a finite set with the following properties: contains the vertices of . $ $ . $ For every vertex , generates We can always satsify the first condition, and the second is possible since is very ample. Using (7.1.4), we get the subspace
1 1
C
:
1 %
We claim that has no basepoints since contains the vertices of . To prove this, let be a vertex. Recall that 1 6 is the divisor of zeros of the global section given by . One computes that 1 % 2 1 6 & Since containing and & 2 for all other 2 for all facets -& facets, the support of 1 - 6 is the complement of the affine open subset , i.e., the nonvanishing set of the section is precisely . Then we are done since the cover .
8
8
It follows that we get a morphism for
1
-
F
-
. Here is our result.
8
§7.1. Polyhedra and Toric Varieties
313
Theorem 7.1.10. Let be a very ample full dimensional lattice polyhedron and define and as above. Then:
- is quasiprojective. (a) The toric variety (b) A 8 is a projective morphism.
Proof. The proof of part (a) is remarkably similar to the proof of Proposition 6.1.4. % % % and consider the projective toric variety We write
3 % " Let % % % be the set of indices corresponding to vertices of . So gives a vertex and a corresponding cone in . Also let 8 3 be the affine open subset where the th coordinate is nonzero. By our choice of , the proof of Proposition 6.1.4 shows that induces an isomorphism 8 $ 8 % that Since is the union of the 8 for- , it follows - $ (7.1.5)
F - 8 - % Since is projective, this shows that is quasiprojective. Part (b) now follows 8 is proper. immediately from Proposition 7.0.6 since A
Example 7.1.11. The polytope from Example 7.1.7 is very ample (in fact, it is normal), and the set used in the proof of Theorem 7.1.10 can be chosen to % % % (Exercise 7.1.8). This gives , be % % % where % % % % % % corresponding- to the elements has variables % % % % % % " of . Then is defined by the equations for by * (Exercise 7.1.8). Since Example 7.1.7, the isomorphism (7.1.5) implies % % % % % % % % % % % % for " * % and - % % % We get a better description of using the vertices of . This gives a map which, when combined with , gives a morphism
3 3 3
3 8 3 % Let 3 and have variables % % % Then and % % % respectively. is an embedding onto the subvariety of 3 be defined by for " (Exercise 7.1.8). Hence *
" * % % % % % % % 3 3
This description of the blowup can be found in many books on algebraic geometry and appeared earlier in this book as Exercise 3.0.8. Note also that the projective morphism of Theorem 7.1.10 is the blowdown map .
Chapter 7. Projective Toric Morphisms
314
When is not very ample, we know that a positive multiple is. Since and have the same normal fan and same recession cone, the maps and are identical. Hence we get the following corollary of Theorem 7.1.10. 8 Corollary 7.1.12. Let be a full dimensional lattice polyhedron in . Then is quasiprojective and is a projective morphism.
8
8
A 8
Exercises for §7.1.
!7 and let 7 be arbitrary. Show !7 ? T l . Then divide by and let $ . 7.1.2. Let 7x be a polyhedron in ` where is a polytope and is a polyhedral CD {]r d GLC for C ! . cone. Define 2
and conclude that ${ is (a) Show that 2CD {]r d GLC for C well-defined. 3C {]r d MGhC for C B , where + be the set of vertices of . (b) Show that ` CD ,GhC for all C44 . Hint: For the non(c) Show that 7
7.1.1. Prove (7.1.1). Hint: Fix for , so J 2,GhC l B
#
obvious direction, represent supporting hyperplanes.
J
J
7
#
as the intersection of closed half-spaces coming from
7.1.3. Let 7 be a polyhedron in ` with recession cone and let k ) * be the largest subspace contained in . Prove that every face of 7 contains a translate of and conclude that 7 has no vertices when is not strongly convex.
7.1.4. Let 7c 4S be a polyhedron in ` where is a polytope and is a strongly convex polyhedral cone. Let + be the set of vertices of . Assume that there is ! + and C in the interior of such that JMGhC v =9GhC for all = in + . Prove that is a vertex of 7 . Hint: Show that , T JMGhC , is a supporting hyperplane of 7 such that '7 )!7 . G Also show if )Aand C satisfy the hypothesis of the problem, then so do )?' 7 G and C for any C sufficiently close to C . Finally, Exercise 7.1.2 will be useful.
"
"
7.1.5. Let 7x cone such that
# `
!
is a polytope and is a polyhedral
`{$ 4 4W{$ |i ` . Let + be the set of vertices of and let C!&% r & ' GLC ( GhC whenever ) in + ^I
be a polyhedron in
where
J=
=
(a) Show that is open and dense in . , there is a vertex of 7 such that (b) Use Exercise 7.1.4 to show that for every C 23C JMGhC . Conclude that the set + of vertices of 7 is nonempty and finite. ]r d JMGhC for C! . (c) Show that *3C
( { * + (d) Conclude that 7 oqsr-, + 2x . Hint: Show that . , where 7 oqsr-, + _ . Then use part (c) of Exercise 7.1.2. 7.1.6. Let i ` be a polyhedral cone with maximal subspace 6) * . Let ;0/ i ` be the image of under the projection map : i9` $Qi ` . Prove that ; = ; . is strongly convex and that : 7.1.7. Let 7 be a lattice polytope and let be the convex hull of the vertices of 7 . Prove that if is normal then 7 is normal.
G
F
/
/
8
§7.2. Projective Morphisms and Toric Varieties
315
7.1.8. Prove the claims made in Example 7.1.11. 7.1.9. In this exercise, you will prove a stronger version of part (b) of Theorem 7.1.10. Let and be as in the proof of the theorem. Prove that there is a commutative diagram
(7.1.6)
such that ' Proposition 7.0.6.
;
H
/ QQQ QQQ QQQ QQQQQ A
$
Q(
is a closed embedding. Hint: See the proof of
7.1.10. Let i ` be a strongly convex rational polyhedral cone. This gives the semiA GKIJIKIJG # group algebra 9W X 9W B ) kX . Given a monomial ideal # W X , we get the polyhedron
=
;
=
7 2koq^r-,D Prove that 72koq^r, !=HGJIKIJI@G4O_ ;B .
H@G #
§7.2. Projective Morphisms and Toric Varieties We now study when a toric morphism
is projective.
Full Dimensional Convex Support. We first consider fans satisfy the following conditions:
1
( 8
is convex. 1 6 6 (
8
in
( 8
that
.
We say that has convex support of full dimension. Such fans satisfy (7.2.1) &' # "
(Exercise 7.2.1). In particular, the maximal cones of have dimension , so we can focus on , just as in the complete case considered in §6.1.
fail to be strongly convex. The largest $ . Hence we get the following:
may $ ( . 1 The sublattice $ ( ( and the quotient lattice ( ( 1 The strongly convex cone 9 ( 8 9 ( 98 . 1 The affine toric variety 8 8 . The projection
A ( ( is compatible with the fans of and 8 since 8 . map This gives a toric morphism
The rational polyhedral cone subspace contained in is
(7.2.2)
8
A 8
%
which as in §7.1 is easily seen to be proper. The difference between here and §7.1 is that may fail to be projective. Our first goal is to understand when
Chapter 7. Projective Toric Morphisms
316
is projective. As you might suspect, the answer involves support functions and convexity. The Polyhedron of a Divisor. A Weil divisor polyhedron 8 &4
1
21 # 2 for all %
on
gives the
When is complete, this is a polytope, but as we learned in §7.1, in general we have where
is a polytope and
is the recession cone of
.
Lemma 7.2.1. Assume is convex of full dimension and let 1 2 1 Weil divisor on . Then:
(a) The recession cone of is .
is a vertex is nonempty and finite. (b) The set (c) .
be a
Proof. Combining (7.1.1) with the definition of , we see that the recession cone of is 8 for all # &4
# " . This proves part (a). The recession cone is
has full dimension, so that parts (b) and (c) follow from
&' since strongly convex since Lemma 7.1.1.
Divisors and Convexity. The 8 definition of convex function given in §6.1 applies to any convex domain in ( . Thus, when is convex, we know what it means for the support function of a Cartier divisor to be convex. The convexity results of §6.1 adapt nicely to fans with full dimensional convex support.
Theorem 7.2.2. Assume is convex of full dimension and let be the support function of a Cartier divisor 1 on . Then the following are equivalent:
1
(a)
(b)
is basepoint free. for all
.
.
is the set of vertices of (d) .
6 & for all & . (e) & (f) & 6 & for all & .
(c)
(g)
is convex.
§7.2. Projective Morphisms and Toric Varieties
317
Proof. This theorem generalizes Theorem 6.1.10. We begin by noting that Proposition 6.1.2 and Lemma 6.1.8 remain valid when is convex of full dimension. (b) (e) Since Lemma 6.1.9 applies to arbitrary fans, the equivalences (a) (f) (g) follow exactly as in the proof of Theorem 6.1.10. The implication (d) (c) follows from Lemma 7.2.1, and (c) (b) is obvious. Finally, (b) (d) follows by the argument given the proof of Theorem 6.1.10.
As a corollary, we see that free Cartier divisor.
is a lattice polyhedron when
1
is a basepoint
8
Strict Convexity. Our next task is to show that is projective if has a Cartier divisor with strictly convex support function. We and only if continue to assume that has full dimensional convex support. As in §7.1, a support function is strictly convex if it is convex and for each , the equation & & holds only on . One can check that Lemma 6.1.14 remains valid in this situation.
Suppose that 1 function. Then 2 1 has a strictly convex support , , are distinct and Theorem 7.2.2 and Lemma 6.1.14 imply that the give the vertices of the polyhedron . This polyhedron has an especially nice relation to the fan .
Proposition 7.2.3. Assume that is convex of full dimension and 1 has a strictly convex support function. Then: (a) is a full dimensional lattice polyhedron. (b) is the normal fan of . Proof. As in §7.1, a vertex We claim that
This easily implies that Fix
gives the cone
%
# and let
32 1
$
.
has full dimension and that is the normal fan of $ implies " . Then
.
&4 &4 where the inequality the equality holds since & . $ and # " &4 holdsforbyallLemma 6.1.9 Thus , so that &' for all . Hence % (7.2.3)
Since
&
is the recession cone of
, the proof of Theorem 7.1.6 implies %
Chapter 7. Projective Toric Morphisms
318
E . Hence & for some . Then & and & imply % so & & & On the other if we apply (7.2.3) to the cone and , we obtain hand, &' . We conclude that &4 & & $ and the same equality holds for all elements of E . This easily implies that . Then by strict convexity, so that & . Now take
Here is the first major result of this section.
8
be the proper toric morphism where Theorem 7.2.4. Let affine. Then is convex and the following are equivalent: (a) (b) (c)
8
is
is quasiprojective. is a projective morphism. has a torus-invariant Cartier divisor with strictly convex support function.
8 3 . This shows
Proof. Since is proper, Theorem 3.4.7 implies that that is convex. First assume that has full dimension.
-
If (c) is true, then is the normal fan of the full dimensional lattice polyhedron by Proposition 7.2.3. It follows that , which is quasiprojective by Corollary 7.1.12, proving (a). Furthermore, (a) (b) by Proposition 7.0.6.
If (b) is true, we will use the theory of ampleness developed in [37]. The essential facts we need are summarized in the appendix to this chapter. Since is projective, there is a line bundle on that satisfies Definition 7.0.3. Then, is affine, Theorem 7.A.5 and Proposition 7.A.7 imply that since ( times) is generated by global sections for some .
8
1 1
1
The nonvanishing set of a global section of
is an affine open subset of
.
We know from §0 that for some Cartier divisor on , and since linearly equivalent Cartier divisors give isomorphic line bundles, we may assume that 1 is torus-invariant (this follows from Theorem 4.2.1). Then 1 is generated by global sections for some . This implies that is convex by Theorem 7.2.2, so that is convex as well. We will show that is strictly convex by contradiction.
convexity fails, then Lemma 6.1.14 implies that there is a wall $ If strict in with . Then corresponds to a global section
8
8 8 8
(since ) , which by the proof of Proposition 6.1.2 is nonvanishing on and on (since ). Thus the nonvanishing set contains , which contains the complete curve
. But being affine, the nonvanishing
8
8
§7.2. Projective Morphisms and Toric Varieties
319
set cannot contain a complete curve (Exercise 7.2.2). This completes the proof of the theorem when has full dimension.
fails to have full dimension. It remains to consider $ what happens when( such ( ( ( . The Let ( that
98 ( and pick ( 9 8 and hence give a fan in ( . If ( has rank , then cones of lie in ( Proposition 3.3.11 implies that
C
(7.2.4)
%
It follows that is the support function of a98 Cartier divisor 1 on . Note also that is convex of full dimension in ( . Since is quasiprojective, this allows us to reduce to the case of full dimensional support. See Exercise 7.2.3.
-Ample and -Very Ample Divisors. The definitions of ample and very ample from §6.1 generalize to the relative setting as follows. Recall from Definition 7.0.3 that a morphism is projective with respect to the line bundle when for a suitable open cover of , we can find global sections % % % of that give a closed embedding over
3 8
* 98
3 8
Then we have the following definition.
8 %
Definition 7.2.5. Suppose that 1 is a Cartier divisor on a normal variety is a proper morphism.
A B
1 are
(a) The divisor 1 and the line bundle with respect to the line bundle
1
(b)
1
and
A 7
Hence
Theorem 7.2.6. Let and let 1 21
1
is stricly convex.
.
has an -ample line bundle.
be a proper toric morphism where be a Cartier divisor on . Then:
and
is projective
is -very ample for some integer
A 8
-very ample if
.
is projective if and only if
is -ample if and only if 2 " and 1 (b) If 6 .
(a)
1
are -ample when
1
is -ample, then
1
is
8
is affine,
-very ample for all
Proof. This follows from Proposition 7.1.9 and Theorem 7.2.4. Here is an example to illustrate
" . The fan
for . 3 of 1 is
Example 7.2.7. Consider the blowdown morphism and & for has minimal generators & Let 1 be the divisor corresponding to & . The support function easily seen to be strictly convex (Exercise 7.2.4). Thus:
Chapter 7. Projective Toric Morphisms
320
1
1
1
is -ample by Theorem 7.2.6.
is projective by Theorem 7.2.4. is the polyhedron from Example 7.1.7. Note also that the polyhedron
3
Projective Toric Morphisms. Suppose we have fans in ( from §3.3 that a toric morphism
8
and
in (
8 . Recall
.
is induced from a map of lattices
(
(
that is compatible 8 . with the fans and , i.e., for every with
there is
is -ample. We first determine when a torus-invariant Cartier divisor on Since projective morphisms are proper, we can assume that is proper, which by Theorem 3.4.7 is equivalent to
83 %
(7.2.5) Here is our result.
5
Theorem 7.2.8. Let 32 1 be a Cartier divisor on
(b) If 6 " 1
be a proper toric morphism and let
.
is -ample if and only if for every
(a)
1
and
, is stricly convex on
is -ample, then
.
1
is
1
83 .
-very ample for all 8
Proof. The idea is to study what happens over the affine open subsets is the toric variety corresponding to the fan for . Observe that
3 8
8 %
Thus 3 8 . Let ! > ? and consider the commutative diagram
O
!
? !
3 8 ! Also let 1 be the restriction of 1 to 3 8
O /
8 ?
/ /
8 %
.
§7.3. Projective Bundles and Toric Varieties
321
! > @? is ! > ?
By Proposition 7.A.6, 1 is -ample if and only if 1 ample for all . Using the above notation, this becomes
1
is -ample
1
is
-ample for all
%
However, Theorem 7.2.6 implies that
1
is
-ample
is strictly convex %
This completes the proof of the theorem.
In Chapter 11 we will use Theorem 7.2.8 to construct interesting examples of projective toric morphisms. We can also characterize when a toric morphism is projective (Exercise 7.2.5). Theorem 7.2.9. Let equivalent: (a) (b)
is projective.
be a toric morphism. Then the following are
3
is proper and has a torus-invariant Cartier divisor whose support func8 for all . tion is strictly convex on
Exercises for §7.2. 7.2.1. Prove (7.2.1).
3
7.2.2. Prove that an affine variety cannot contain a complete variety of positive dimension. Hint: If is complete and irreducible, then 3 G .
+$
=
7.2.3. This exercise will complete the proof of Theorem 7.2.4. Let P satisfy the hypothesis of the theorem and write (P as in (7.2.4). We also have the Cartier divisors on P and -= on P A as in the proof of the theorem.
(a) Assume that is projective. Prove that P is quasiprojective and conclude that P A is quasiprojective. Now use the first part of the proof to show that is strictly convex. Hint: See Exercise 7.0.1.
(b) Assume that is strictly convex. Prove that P A is quasiprojective and conclude that P is quasiprojective. Then use Proposition 7.0.6.
7.2.4. Prove that the support function in Example 7.2.7 is strictly convex. We will 8 generalize this result considerably in Chapter 11. 7.2.5. Prove Theorem 7.2.9.
§7.3. Projective Bundles and Toric Varieties Given a vector bundle or projective bundle over a toric variety, the nicest case is when the bundle is also a toric variety. This will lead to some lovely examples of toric varieties.
Chapter 7. Projective Toric Morphisms
322
Toric Vector Bundles and Cartier Divisors. A Cartier divisor 1 2 1 on a 1 , which is the sheaf of sections toric variety " gives the line bundle . of the rank vector bundle
We will show that is a toric variety and is a toric morphism by constructing the fan of in terms of and 1 . To motivate our construction, recall that for , we have 1 for all & & & the graph of & & lies
.
%
above the graph of
The first equivalence follows from Proposition 4.3.3 and the second from 8 Propostion 6.1.9. The key word is ”above”: it tells us to focus on the part of ( that lies above the graph of .
,1
We define the fan
8
& & in (
as follows. Given , set
& "
# &' 2 " where the second equality follows since * &4 on . 2 and 8 is linear . Then let Note that is a strongly convex rational polyhedral cone in ( ,8 1 be the set consisting of the cones for and their faces. This is a fan , and the projection ) ( ( is clearly compatible with ,1 in (
and . Hence we get a toric morphism Proposition 7.3.1. sections is 1 .
*
" is a rank
% vector bundle whose sheaf of
Proof. We first show that is a toric fibration as in Theorem 3.3.19. The kernel98 of ( ( is " ( , and the fan ,1 ( has as its unique maximal cone. Also, for let
)
'& 2 # " % where * & consisting of points &
.
This is the face of . Thus 1 8 is a subfan of 1 . Since and and in fact maps bijectively to , we see that 1 is split by and in the sense of Definition 3.3.18. Since , Theorem 3.3.19 implies that
5
5
3 8 8
3 8 8 8 8
%
To see that this gives the desired vector bundle, we study the transition func tions. First note that , so that the above isomorphism is
%
§7.3. Projective Bundles and Toric Varieties
323
8 " $ follows directly from the definition of . Then, given another cone , the 8 to 8 8 is given by 8 transition map from & & & , where 3 (Exercise 7.3.1). which by projection induces a map . It is easy to check that this map is , where & & for & (Exercise 7.3.1). Note that
3
1 is We are now done, since the proof of Proposition 6.1.20 shows that the sheaf of sections of a rank 1 vector bundle over whose transition functions . are
3
This construction is easy but leads to some surprising rich examples.
Example 7.3.2. Consider with its usual fan and let 1 to the mini correspond 1 is denoted " . mal generator & . Recall that " This gives the rank vector bundle described in Proposition 7.3.1 whose fan in has minimal generators
% % %
%
You will check this in Exercise 7.3.2.
We can also describe this vector bundle geometrically as follows. Consider the lattice polyhedron in given by
% % % % % - % % The normal fan of is the fan (Exercise 7.3.2), so that is the above vector
. Note also that
- is dual to- the recession cone of . "
It is easy to see that is a smooth cone- of dimension
, so that the
bundle
8
projective toric morphism constructed in §7.1 becomes . When combined with the vector bundle map , we get a morphism
%
When the coordinates of and are ordered correctly, the image is precisely the variety defined by (Exercise 7.3.2). In this way, we recover the description of
given in Example 6.0.18.
Proposition 7.3.1 extends easily to decomposable toric vector bundles. Sup" % % % . This gives the pose we have Cartier divisors 1 2 1 , locally free sheaf
1
(7.3.1)
1
of8 rank . To construct the fan of the corresponding vector bundle, we work in 8 ( . Let % % % be the standard basis of and write elements of (
Chapter 7. Projective Toric Morphisms
324 as &
&
&4
& # " & for " % % % % % % % 2 2 . Then, given , we get the cone
One can show without difficulty that the set consisting of the cones for 8 and their faces is a fan in ( such that the toric variety of this fan is the vector bundle over whose sheaf of sections is (7.3.1) (Exercise 7.3.3).
Besides decomposable vector bundles, one can also define a toric vector bundle . Here, rather than assume that is a toric variety, one makes the weaker assumption the torus of acts on such that the action is linear on the fibers and is equivariant. Toric vector bundles have been classified by Klyachko [60] and others—see [79] for the historical background. Oda observed [77, p. 41] that if a toric vector bundle is a toric variety in its own right, then the bundle decomposes into a direct sum of line bundles, as above. This can be proved using Klyachko’s results.
Toric Projective Bundles. The decomposable toric vector bundles have associated toric projective bundles. Cartier divisors 1 % % % 1 give the locally free sheaf
1
1
" of rank . Then is a projective bundle whose fibers look like . To describe the fan of , we first give a new description of the fan of . In basis % % % . The “first orthant” % % %
, we" use the standard has
facets % % % % % % % % % % Now set (
. Then the8 images of sum to zero in ( and the images of give the fan for in ( . given in §0 involves taking the dual vector bundle. The construction of Thus , where is the vector bundle whose sheaf of sections is 1 1 % The fan of
&4 2 2 # " % % % and their faces, as ranges over the cones . To get the fan for , take and let be a facet of % % % . This gives the cone &' 8 2 8 2 # " ( 8 ( is the image of this cone under the natural projection and then ( ( 8 ( 8 ( 8 . is built from cones
Proposition8 7.3.3. 8 The cones fan in ( ( whose toric variety
% % %
and their faces form a is the projective bundle .
§7.3. Projective Bundles and Toric Varieties
325
8 Proof. Consider the fan in ( given by the and their faces. for ,
# " in Also, let be the image of & 2 2 ( 8 ( 8 . Then one easily adapts the proof of Proposition 7.3.1 to show that the toric variety of is a fibration over with fiber . Furthermore, working over an affine
, one sees that is obtained from open subset of in §0. We leave the details as Exercise 7.3.4.
by the process described
( with and we redefine as % % % % % %
In practice, one usually replaces % % % . Then set basis (7.3.2)
and the
# " (7.3.3) 4
2 2 & 2 2 8 . This way, is a fan in ( 8 . Here is a classic example. in ( Example fan for has minimal generators & and & & . Also let " 7.3.4. The 1 , where 1 be the divisor corresponding to & . Fix an integer and consider 2 2 % & & live As above, we get a fan in . The minimal generators in the first factor. In the second factor, the vectors % % %
in the above construction reduce to . Thus and . We will use & as the basis of . The maximal cones for the fan of are & and & . Then has four cones: & & &
& & 2
& 2 & 2 % & 2
This is the fan for the Hirzebruch surface . Thus 2 % and for a cone , redefine
as
=0
Note also that the toric morphism map for the projective bundle.
constructed earlier is the projection
This example generalizes as follows.
Example 7.3.5. Given integers projective bundle
"
and
2
2
2 , consider the
2 %
Chapter 7. Projective Toric Morphisms
326
has a nice description. We will work in . , where % % % & and has basis % % % . Also set & &
. As usual, & corresponds to the divisor 1 of such that 1 .
The fan of has basis & and
2
2
The description (7.3.3) of the cones in uses generators of the form
(7.3.4)
& 2 2 2 2
where the &' are minimal generators of the fan of the base of the projective bundle. % % % & . Since we are using the divisors 2 1 % % % 2 1 , Here, the & ’s are & the formula (7.3.4) simplifies dramatically, giving minimal generators
&' & &' &
& 2 2 & " % % % % Since the maximal cones of are & % % % & % % % & , (7.3.2) and (7.3.3)
% % % % % % % % % % % % for all % % % and % % % . It is also easy to see that the minimal generators % % % % % % have the following properties: 1 % % % % % % form a basis of . 1 . 1 2
2 . The first two bullets are clear, and the third follows from & and the definition of the . is smooth of dimension . Since has One also sees that " " minimal generators, the description of the Picard group given in §4.2 implies that 5 6 * %
imply that the maximal cones of are
% % % and % % % give primitive (Exercise 7.3.5). Also observe that collections of . We will see below that these are the only primitive collections of . Furthermore, they are extremal in the sense of §6.3 and their primitive relations generate the Mori cone of .
This is a very rich example!
A Classification Theorem. Kleinschmidt [59] classified projective toric 1 all. smooth varieties with Picard number , i.e., with 5 6 The rough idea is that they are the toric projective bundles described in Example 7.3.5. Following ideas of Batyrev [5], we will use primitive collections to obtain the classification.
We begin with some results from [5] about primitive collections. Recall from # # " §6.3 that a primitive collection % % % gives the primitive
§7.3. Projective Bundles and Toric Varieties
relation
&' &4
327
&4 is the minimal cone containing &
(7.3.5)
where &4 . When and projective, these primitive relations have some nice properties.
is smooth
be a smooth projective toric variety. Then: $ " (a) In the primitive relation (7.3.5), and ) for all # (b) There is a primitive collection with primitive relation & &4
Proposition 7.3.6. Let
Proof. The are integral since is smooth. Let the minimal generators of & % % % & , so the primitive relation becomes
" . .
be
& & & % & Suppose for example that & & . Then
&4 " & & & % &4 Note that &' % % % &4 generate a cone of since is a primitive collection. So the
above equation expresses an element of a cone of in terms of minimal generators in two different ways. Since is smooth, these must coincide. To see what this means, we consider two cases.
" , then & % % % & & & % % % & , so that & & for" some " . This is impossible since & & . On the other hand, if , then
" , which is &' % % % &4 & % % % & . Since &' & , we obtain If
impossible since
is a primitive collection.
Turning to part (b), let be the support function of an ample divisor on Thus is strictly convex. Since is complete, we can find an expression (7.3.6)
&4 &4
% % % are positive integers. Note that & % % % &4 such that cone of . By strict convexity and Lemma 6.1.14, it follows that (7.3.7)
4& &4 &4
cannot lie in a
&4 %
Pick a relation (7.3.6) so that the right-hand side is as big as possible.
.
The set & % % % & is not contained in a cone of and hence has a subset that is a primitive collection. By relabeling, we may assume that & % % % &4 , , is a primitive collection. Using (7.3.6) and the primitive relation (7.3.5), we obtain the nonnegative relation
" & & & %
Chapter 7. Projective Toric Morphisms
328
Since
is linear on and strictly convex,
&4
)&'
&4 &4
" &4
&4 &' " &'
4 & 4 & &' % This contradicts the maximality of the right-hand side of (7.3.7), unless " , in which case we get the desired primitive collection. which implies that
We now prove Kleinschmidt’s classification theorem.
Theorem 7.3.7. Let be a smooth projective toric variety with 576 ", 6 Then there are integers and 2
2 2
6 . Then 576
2 %
and
? 2 with "
.
Proof. Let and §4.2 imply that has elements. Following [59], we first use results about triangulations and polytopes to study the combinatorial structure of .
If we intersect the cones of the fan with a sphere centered at the origin, we " -sphere. This triangulation has " get a triangulation of the vertices, so by a result of Mani [65], the triangulation is combinatorially equivalent to the triangulation of the sphere induced by projecting from the interior point of an -dimensional simplicial polytope with vertices. The combinatorial type of such polytopes are classified in [38, 6.1]. The result states that the set of $ , such that the sets vertices can be divided into disjoint sets ,
2
2
are precisely the vertex sets of the facets of the polytope. Combining the combinatorial equivalences we see that
"
fan
triangulation
can be written as a disjoint union $ "
polytope
§7.3. Projective Bundles and Toric Varieties
329
# # where (7.3.8)
# " # # & % An immediate consequence of this description of is that such that
and are primitive collections. Be sure you understand why. It is also true that and are the only primitive collections of . To prove this, suppose that we had a third : , so there is # : , and similarly there is primitive collection : . Then # : since . By (7.3.8), the rays of : all lie in , which contradicts the definition of primitive collection.
Since is projective and smooth, Proposition 7.3.6 guarantees that has a primitive collection whose elements sum to zero. We may assume that is this " " " primitive collection. Let and , so since primitive collections have at least two elements, and since . % % % . Thus Now rename the minimal generators of the rays in as
%
The next step is to rename the minimal generators of the rays in as % % % . lies in a cone whose rays lie in the Proposition 7.3.6 implies that complement of , which is . Since is a primitive collection, must omit at least one element of , which we may assume to be the ray generated by . Then the primitive relation of can be written
2 2
and by further relabeling, we may assume 2 2 . Finally, observe % % % % % % generate a maximal cone of by (7.3.8). Since is that smooth, it follows that these vectors form a basis of ( . Comparing all of this to Example 7.3.5, we conclude that the toric variety of is the projective bundle 2 2 .
A proof of Theorem 7.3.7 that doesn’t use the combinatorial results of [38, 65] will be sketched in Exercises 7.3.6 and 7.3.7. The result proved in [59] is more general since Kleinschmidt does not assume projectivity. This can also be done using Batyrev’s approach (see [5, Thm. 4.3]). Exercises for §7.3. 7.3.1. Here you will supply some details needed to prove Theorem 7.3.1.
= ;
=
;
$
e
. Show that this map is 8 (a) In the proof we constructed a map 3C GLC for C . (b) Given cones Gh , the transition map from to L is given by 3C;G h CG 3C h . Prove that . # 8 #
= B
$
=
= B
=
7.3.2. In Example 7.3.2, we study the rank $ vector bundle sections is $ . Let be the fan of + in { = .
+
$
#
L
=
'
=%g , where
B
whose sheaf of
Chapter 7. Projective Toric Morphisms
330
= G = SJK0 = are the minimal generators of .
(b) Prove that is the normal fan of 7 koqsr-,f G = GJIKIJIKG f oq^r>t^ = G = = GJIKIJIJG = @I = . Prove that the image of this (c) The example constructs a morphism + $ (a) Prove that
= GJIJIKIJG
m sl and explain how this relates to Example 6.0.18. 7.3.3. Consider the locally free sheaf (7.3.1) and the cones ;. xi ` { defined in the discussion following (7.3.1). Prove that these cones and their faces give a fan in i ` !{ map is defined by l &m
whose toric variety is the vector bundle with (7.3.1) as sheaf of sections. 7.3.4. Complete the proof of Proposition 7.3.3.
){
$
7.3.5. Let 2 that Z \H ?h
" . be the toric projective bundle constructed in Example 7.3.5. Prove
be an extremal wall, which by the proof of Theorem 6.3.10 gives a primitive
7.3.6. This exercise and the next will use [83] to prove (7.3.8) without using [38, 65]. Let with generators Cf=jGJIKIJI@GLC . the - rays of ?h$ be >=&GKIJIJIKGh (a) Let collection 7 . Show that the wall relation of (after suitable relabeling) is
= 6>G U l_4 ' +-3 @I Hint: is smooth. See §6.3. (b) Explain why 7x >=jGh =
K l &u GJIKIJI@G and U l . Hint: See the proof of Theorem 6.3.10. (c) Let ; l 6oq^r>t^C =jGJIKIJI@G C l GJIJIKIJGLC =@ . We cite two results of Reid. By [83, p. 403],
; lG oqsr>tsC =jGJIKIJIJGLC =@ ' d C =
l =
U l3CDl C
and by [83, Cor. 2.10],
;
l ! when l !7 I Use this to prove that the cones of # _ not containing C
(d)
oqsr t^3Cf=&GKIJIKIJGC l GKIJIKI@GhC =jGC @G Prove that the walls in # k$ lying on the boundary of oqsr>t 3C=jGJIJIKI@GhC precisely the cones of the form
(e) (f)
l !7 areI the cones
=J are
oqsr>t 3Cf=&GKIJIJIJGC l GJIKIJIKGC m GKIJIKIJGhC =J where u v w and exactly one of lLGh sm is contained in 7 . are precisely the cones Use (d) to show that the cones of # _ containing C oq^r>t^C = GJIKIJIJG CM lhGJIKIJI@G C> msGKIJIJIJGhC = GLC K@G exactly one of l 4 Hm lies in 7 I Conclude that # _ oqsr>tsC =jGJIKIJIJGLC f l Gh m & & l 7 Gh m / 7 .
7.3.7. Here you will prove (7.3.8). We use the same notation as Exercise 7.3.6.
(a) Explain why the Mori cone !#"Y -P_ has exactly two extremal rays. Theorem 6.3.10 implies that each of these rays comes from a primitive collection. This gives primitive collections 7 and .
§7.3. Projective Bundles and Toric Varieties
331
#%$& and 7S) 6/ , and explain why this implies (7.3.8). Hint: (b) Prove that 7 Apply part (f) of Exercise 7.3.6 to both 7 and .
7.3.8. In Example 2.3.15, we defined the rational normal scroll of the polygon
7
7 'D
7 'D
to be the toric variety
6oqsr-, G T H=&G G U H=< { G T U , and in Example 3.1.16, we showed that
xJK
where T G U satisfy $ i.e., every rational normal scroll is a Hirzebruch surface. = There is an -dimensional analog of 7 . Take integers $ Then 7 A is the lattice polytope in { 7 ' D having the lattice points
'
'
7 'D
D 87
8
,
=.
G ^ = G &HG & = = G &
HG &
= GKIJIKIJG G 8 = = as vertices. The toric variety of 7 ' ' A is denoted ' ' A . (a) Explain why 7 ' ' A is a “truncated prism” whose base in S{ 8 = is the standard simplex = , and above the vertices of 8 = there are edges of lengths 8 means the = direction. Draw a picture when ! GKIJIJIJG 8 = . Here, “above” . A A (b) Prove that #A 4JK 8 =@L . ' '
A is smooth by part (b), so that 7 (c) A is very ample and hence gives ' ' ' '
a projective embedding of A . Explain= how this embedding consists of
' ' = embeddings of such that for each point ! , the resulting points in projective space are connected by an S$ -dimensional plane that lies in A . ' '
$ = = . in part (c) is the fiber
(d) Explain how part (c) relates to the scroll discussion in Example 2.3.15.
(e) Show that the |$ -dimensional plane associated to A ^& 4JK A = L of the projective bundle
8
7.3.9. Consider the toric variety 2 constructed in Example 7.3.5.
%$& T = |$& 4JK T 4$ h . Hint: Part (b) of (c) Find a lattice polytope in { ?{ whose toric variety is . Hint: In the polytope of = is*attached Exercise 7.3.8, each vertex of & to a line segment = { 8 {
in the normal direction. Also observe that a line segment is a multiple of = . Adapt this by using | { { as “base” and then, at each vertex of , attach a positive mutliple of in the normal direction.
(a) Prove that 2 is projective. Hint: Proposition 7.0.5.
(b) Show that ? Lemma 7.0.8 of §0.
8
8
7.3.10. Let -P be a projective toric variety and let 1sGJIKIJIKGL be torus-invariant ample divisors on 1P . Each Yl gives a lattice polytope 7;l~4 whose normal fan is . Prove that the projective bundle
|KJ KL is the toric variety of the polytope in i ` { oq^r, 7_ 7 = = JK 87 &OI Hint: If you get stuck, see [14, Sec. 3]. Do you see how this relates to Exercise 7.3.9? 7.3.11. Use primitive collections to show that is the only smooth projective toric variety with Picard number $ .
Chapter 7. Projective Toric Morphisms
332
Appendix: More on Projective Morphisms In this appendix, we discuss some technical details related to projective morphisms. Proj of a Graded Ring. As described in [26, III.2] and [41, II.2], a graded ring
6 f
, we have the affine that for every non-nilpotent Z q Z q such with t \^ e g OG where e g is the homogenous localization of at , i.e., e g B ^G- I Furthermore, if homogeneous elements =jGJIKIJIJG satisfy =&GJIKIJIKG 6 ^G then the affine open subsets = @GJIKIJIJG cover Z q . Thus we can construct Z q by gluing together the affine varieties l , just as we construct by gluing together copies of . The scheme Z q comes equipped induced with a morphism Z q t \s b $ is ant \saffine by the inclusions e g for all . For example, if variety, then we get the graded ring b W GJIKIJI@G: X where each Ml has degree $ , and Z q G where the map Z q $ MtK\^ t \s b is projection onto the first factor. ` be a full dimensional lattice polyhedron. As in Here is a toric example. Let 7 §7.1, this gives: . The toric morphism H $ ` 8{ . The cone Y79 Recall from (7.1.3) that Y 79 gives the semigroup algebra 1 9W Y79_) *" X where the character associated to G > Y79f), *" is written . We use the height grading given by setting tJD H . Then
gives the scheme open subset
Theorem 7.A.1. diagram
,4Z
5
#
3
q . Furthermore, if 7 7 /
is a closed embedding.
3
is normal, then there is a commutative
8 =
PPP PPP PPP PPP PP'
such that
#
5
A
Appendix: More on Projective Morphisms
333
Proof. We will sketch the argument and leave the details as an exercise. The slice of Y 79 at height is the recession cone of 7 . Recall that i |i / V)8i , where is the largest subspace contained in and that is the affine toric variety of , which is the image of in i ` . Then the inclusion
dual to i i gives
;
4 k
a`
$
`_I
;
"
, the degree part of the graded ring p 9W Y 79 )4 ? X , is W ) kX2 9W ;B ) kX . This implies MtK\s , so that we get a natural map Z q $ . Since 7 is normal, one sees easily that Y79_) ? is generated by its elements " + = , of height V$ . If is a Hilbert basis of Y79 )4 * , then J = GK$ @GJIJIKI@GK GJ$ , then where elements of "l have height u . If we write = A is generated as an -algebra by # OGKIJIKIJG # . In other words, we have a surjective It follows that
" "
homomorphism of graded rings
D W
l $ # O I = KG IJIJIKG X $ ~G = This surjection makes Z q a closed subvariety of Z q W j= GKIJIKIJG LX 8 by [41, Ex. III.3.12]. This gives the commutative diagram in the statement of the theorem, is projective. except that is replaced with Z q . Hence Z q $
It remains to prove one can prove:
Z
q . For this, let +
be the set of vertices of
7
. Then
- + 1 . If - + , then e$ g 1 9W ; ) kX , where ; koq^r>ts 7) . The first bullets implies that Z q is covered by the affine open subsets MtK\^ e$ g , and the second shows that MtK\s e$ g is the affine toric variety of the cone ; . These patch together in the correct way to give 4 Z q .
B
+
C
K
For an arbitrary full dimensional lattice polyhedron, some positive multiple is normal. Hence Theorem 7.A.1 implies Theorem 7.1.10. ´ ements Ampleness. A comprehensive treatment of ampleness appears in Volume II of El´ de g´ eom´ etrie alg´ ebrique (EGA) by Grothendieck and Dieudonn´e [37]. The results we need from EGA are spread out over several sections. Here we collect the definitions and theorems we will use in our discusion of ampleness.1 Definition 7.A.2. A line bundle on a variety sheaf on , there is an integer 5 such that for all 5 B 5 .
/
/ !
is absolutely ample if for every coherent 3 is generated by global sections
By [37, (4.5.5)], this is equivalent to what EGA calls “ample” in [37, (4.5.3)]. We use the name “absolutely ample” to prevent confusion with Definition 6.1.1, where “ample” is defined for line bundles on complete normal varieties. Here is another definition from EGA. 1The theory developed in EGA applies to very general schemes. The varieties and morphisms appearing in
this book are nicely behaved—the varieties are quasi-compact and noetherian, the morphisms are of finite type, and coherent is equivalent to quasicoherent of finite type. Hence most of the special hypotheses needed for some of the results in [37] are automatically true in our situation.
Chapter 7. Projective Toric Morphisms
334
$
be a morphism. A line bundle on is relatively Definition 7.A.3. Let f ample with respect to if has an affine open cover l such that for every u , A l = is absolutely ample on 8 la .
This is [37, (4.6.1)]. When mapping to an affine variety, relatively ample and absolutely ample coincide. More precisely, we have the following result.
H
Proposition 7.A.4. Let bundle on . Then:
$
be a morphism, where
is relatively ample with respect to
is affine, and let
be a line
is absolutely ample.
Proof. This is proved in [37, (4.6.6)].
K
The reader should be warned that in EGA, “relatively ample with respect to ” and “ -ample” are synonyms. In this text, they are slightly different, since “relatively ample with respect to ” refers to Definition 7.A.3 while “ -ample” refers to Definition 7.2.5. Fortunately, they coincide when the map is proper. $ be a proper morphism and a line bundle on . Then Theorem 7.A.5. Let s the following are equivalent: (a) is relatively ample with respect to in the sense of Definition 7.A.3. (b) is -ample in the sense of Definition 7.2.5. (c) There is an integer such that is projective with respect to in the sense of 5
3
Definition 7.0.3.
Proof. First observe that (b) and (c) are equivalent by Definition 7.2.5. Now suppose that l of such is projective with respect to 3 . Then there is an affine open covering that for each u , there is a finite dimensional subspace l G 3H that gives a closed = embedding of 8 l into l for each u . , which is the sheaf of From we get the locally free sheaf l . This gives sections of the trivial vector bundle l the projective bundle 2 l , so that we have a closed embedding
3
$
2OI By definition [37, (4.4.2)], g is very ample for A e g . Then [37, (4.6.9)] implies that A e g is relatively ample with respect to A e g , and hence absolutely ample by Proposition 7.A.4. Then is relatively ample with respect to by Definition 7.A.3. Finally, suppose that is relatively ample with respect to and let l be an affine open covering of . Then [37, (4.6.4)] implies that A e g is relatively ample with respect to A e g . Using [37, (4.6.9)] again, we see that A e g is very ample = la can be embedded in for A e g , which by definition [37, (4.4.2)] means that 2 for a coherent sheaf on l . Then of [100, Thm. 5.44] shows how to find l proof = the = l into finitely many sections of over give a suitable embedding of l . Ae 3
,
8
= l *
$
,
,
,
3
,
,
8
,
,
3
8
8
K
Appendix: More on Projective Morphisms
335
$
is projective In EGA [37, (5.5.2)], the definition of when a morphism ~ involves two equivalent conditions stated in [37, (5.5.1)]. The first condition uses the projective bundle 2 of a coherent sheaf on , and the second uses Z q , where is a quasicoherent graded -algebra such that != is coherent and generates . By [37, (5.5.3)], projective is equivalent to proper and quasiprojective, and by the defintion of quasiprojective [37, (5.5.1)], this means that has a line bundle relatively ample with respect to . Hence Theorem 7.A.5 shows that the definition of projective morphism given in EGA is equivalent to Definition 7.0.3.
$ be a proper morphism and a line bundle on . l of , the following are equivalent: (a) is -ample. (b) For every u , A e g is A e g -ample. = l $ l by the universal property of Proof. Since is ample, so is A e g properness. But for a proper morphism , being -ample is equivalent to being relatively ample with respect to . Then we are done by [37, (4.6.4)]. Here are some further properties of projective morphisms. Proposition 7.A.7. Let $ be a projective morphism with affine and let be an -ample line bundle on . Then: 3 3 G , let be the open subset where is (a) Given a global section nonvanishing. Then is an affine open subset of . Proposition 7.A.6. Let f Given an affine open cover
,
,
8
,
K
(b) There is an integer 5s such that
Proof. This is proved in [37, (5.5.7)].
3
is generated by global sections for all 5 B
^
.
5
K
Toric Varieties, Chapters 8–10 David Cox John Little Hal Schenck D EPARTMENT 01002
OF
M ATHEMATICS , A MHERST C OLLEGE , A MHERST, MA
E-mail address: [email protected] D EPARTMENT OF M ATHEMATICS AND C OMPUTER S CIENCE , C OLLEGE THE H OLY C ROSS , W ORCESTER , MA 01610
OF
E-mail address: [email protected] D EPARTMENT OF M ATHEMATICS , U NIVERSITY C HAMPAIGN , U RBANA , IL 61801 E-mail address: [email protected]
OF I LLINOIS AT
U RBANA -
Contents
Preface
iii
Notation
xi
Part I: Basic Theory of Toric Varieties
1
Chapter 1.
3
Affine Toric Varieties
§1.0.
Background: Affine Varieties
§1.1.
Introduction to Affine Toric Varieties
10
§1.2.
Cones and Affine Toric Varieties
22
§1.3.
Properties of Affine Toric Varieties
34
Appendix: Tensor Products of Coordinate Rings Chapter 2.
Projective Toric Varieties
3
48 49
§2.0.
Background: Projective Varieties
49
§2.1.
Lattice Points and Projective Toric Varieties
54
§2.2.
Lattice Points and Polytopes
62
§2.3.
Polytopes and Projective Toric Varieties
74
§2.4.
Properties of Projective Toric Varieties
86
Chapter 3.
Normal Toric Varieties
93
§3.0.
Background: Abstract Varieties
93
§3.1.
Fans and Normal Toric Varieties
105
§3.2.
The Orbit-Cone Correspondence
115
§3.3.
Equivariant Maps of Toric Varieties
124
§3.4.
Complete and Proper
138 ix
Contents
x
Appendix: Nonnormal Toric Varieties Chapter 4.
Divisors on Toric Varieties
148 153
§4.0.
Background: Valuations, Divisors and Sheaves
153
§4.1.
Weil Divisors on Toric Varieties
169
§4.2.
Cartier Divisors on Toric Varieties
174
§4.3.
The Sheaf of a Torus-Invariant Divisor
187
Chapter 5.
Homogeneous Coordinates
193
§5.0.
Background: Quotients in Algebraic Geometry
193
§5.1.
Quotient Constructions of Toric Varieties
202
§5.2.
The Total Coordinate Ring
216
§5.3.
Sheaves on Toric Varieties
223
§5.4.
Homogenization and Polytopes
229
Chapter 6.
Line Bundles on Toric Varieties
243
§6.0.
Background: Sheaves and Line Bundles
243
§6.1.
Ample Divisors on Complete Toric Varieties
260
§6.2.
The Nef and Mori Cones
279
§6.3.
The Simplicial Case
289
Appendix: Quasicoherent Sheaves on Toric Varieties Chapter 7.
Projective Toric Morphisms
299 305
§7.0.
Background: Quasiprojective Varieties and Projective Morphisms
305
§7.1.
Polyhedra and Toric Varieties
309
§7.2.
Projective Morphisms and Toric Varieties
317
§7.3.
Projective Bundles and Toric Varieties
323
Appendix: More on Projective Morphisms Chapter 8.
The Canonical Divisor of a Toric Variety
333 337
§8.0.
Background: Reflexive Sheaves and Differential Forms
337
§8.1.
One-Forms on Toric Varieties
348
§8.2.
Differential Forms on Toric Varieties
355
§8.3.
Fano Toric Varieties
369
Chapter 9.
Sheaf Cohomology of Toric Varieties
377
§9.0.
Background: Cohomology
377
§9.1.
Cohomology of Toric Divisors
388
§9.2.
Vanishing Theorems I
400
Contents
xi
§9.3.
Vanishing Theorems II
411
§9.4.
Applications to Lattice Polytopes
420
§9.5.
Local Cohomology and the Total Coordinate Ring
432
Appendix: Introduction to Spectral Sequences
445
Topics in Toric Geometry
449
Chapter 10.
451
Toric Surfaces
§10.1.
Singularities of Toric Surfaces and Their Resolutions
451
§10.2.
460
§10.3.
Continued Fractions and Toric Surfaces Gr¨obner Fans and McKay Correspondences
§10.4.
Smooth Toric Surfaces
479
§10.5.
Riemann-Roch and Lattice Polygons
487
Chapter 11.
470
Toric Singularities
499
§11.1.
Existence of Resolutions
499
§11.2.
Projective Resolutions
506
§11.3.
Blowing Up an Ideal Sheaf
507
§11.4.
Some Important Toric Singularities
507
Chapter 12.
The Topology of Toric Varieties
509
§12.1.
Homotopy of Toric Varieties
509
§12.2.
The Moment Map
516
§12.3.
Singular Cohomology of Toric Varieties
521
§12.4.
The Cohomology Ring
533
§12.5.
Complements
547
Chapter 13.
The Riemann-Roch Theorem
559
Chapter 13.
Geometric Invariant Theory
559
Chapter 14.
The Toric Minimal Model Program
561
Bibliography
563
Chapter 8
The Canonical Divisor of a Toric Variety
§8.0. Background: Reflexive Sheaves and Differential Forms This chapter will study the canonical divisor of a toric variety. The theory developed in Chapters 6 and 7 dealt with Cartier divisors. As we will see, the canonical divisor of a normal toric variety is a Weil divisor that is not necessarily Cartier. Reflexive Sheaves. A Weil divisor D on a normal variety X gives the sheaf defined by
U X
X
D
D f X div f D U 0 0
Our first task is to characterize these sheaves. Recall that the dual of a sheaf of X -modules We say that is reflexive if the natural map "!$#%
is
om
X
X
.
&
is an isomorphism. It is easy to see that locally free sheaves are reflexive. Here are some properties of reflexive sheaves. Proposition 8.0.1. Let be a coherent sheaf on a normal variety X and consider the inclusion j ' U0 ( # X where U0 is open with codim X ) U0 2. Then: (a)
and hence
&
are reflexive.
(b) If
is reflexive, then +*
(c) If
U0
is locally free, then
j
& *
U0
. j
U0
. 337
Chapter 8. The Canonical Divisor of a Toric Variety
338
Proof. Recall from §4.0 that the direct image j X open. U j U U0 for U
of a sheaf
on U0 is defined by
Parts (a) and (b) of the proposition are proved in [63, Cor. 1.2 and Prop. 1.6]. For part (c), we first observe that restriction is compatible with taking the dual, i.e., on X. Then U0 U0 for any coherent sheaf
& *
U 0
j
j
U0 & *
j
U0
where the first isomorphism follows from parts (a) and (b), and the last follows since U0 is locally free and hence reflexive.
Later in the section we will study the sheaf Xp of p-forms on X. This sheaf is locally free when X is smooth. For X normal, however, Xp may be badly behaved, though it is locally free on the smooth locus of X. Hence we can use part (c) of Proposition 8.0.1 to create a reflexive version of Xp .
For more on reflexive sheaves, the reader should consult [63] and [125].
Reflexive Sheaves of Rank One. We first define the rank of a coherent sheaf on an irreducible variety X. Recall that X is the constant sheaf on X given by X .
Definition 8.0.2. Given a coherent sheaf on irreducible variety X, the global sections of X X form a finite dimensional vector space over X whose dimension is the rank of . For a locally free sheaf, the rank is just the rank of the associated vector bundle. Other properties of the rank will be studied in Exercise 8.0.1. In the smooth case, reflexive sheaves of rank 1 are easy to understand.
Proposition 8.0.3. On a smooth variety, a coherent sheaf of rank 1 is reflexive if and only if it is a line bundle. This is proved in [63, Prop. 1.9]. We now have all of the tools needed to characterize which coherent sheaves on a normal variety come from Weil divisors.
Theorem 8.0.4. Let be a coherent sheaf on a normal variety X. Then the following are equivalent.
(a)
is reflexive of rank 1.
(b) There is an open subset j ' U0 ( # X such that codim X ) U0 2, * j line bundle on U0 , and U0 . (c) * X D for some Weil divisor D on X.
U0
is a
Proof. (a) (b) Since X is normal, its singular locus Y Sing X has codimension at least two in X by Proposition 4.0.17. Then U0 X ) Y is smooth, which * j implies that U0 is a line bundle by Proposition 8.0.3, and U0 by Proposition 8.0.1.
§8.0. Background: Reflexive Sheaves and Differential Forms
(b) (c) The line bundle U0 can be written as divisor E i ai Ei on U0 . Consider the Weil divisor D Zariski closure of Ei in X. Given f X , note that
339
E for some Cartier i ai Di , where Di is the
U0
div f & D 0 div f & D U0 0 since codim X ) U0 2, and the same holds over any open set of X. Combining this with E D U0 , we obtain X
D *
j
U0
E
j
U0 *
(c) (a) The proof of (b) (c) shows that X D * j X D U0 . But codim X ) U0 2, and X D U0 U0 D U0 is locally free since U0 is smooth. Thus j X D U0 * X D by Proposition 8.0.1, so X D * X D & is reflexive and has rank 1 since it is a line bundle on U0 .
Tensor Products and Duals. Given Weil divisors D E on a normal variety X, the map f g # f g defines a sheaf homomorphism (8.0.1)
X
D
X
X
E !$#+
X
D E
This is an isomorphism when D or E is Cartier but may fail to be an isomorphism in general.
Example 8.0.5. Consider the affine quadric cone X V y 2 ! xz 3 . From examples in previous chapters, we know that this is a normal toric surface. The line L V y z gives a Weil divisor that is not Cartier, though 2L is Cartier (this follows from Example 4.2.3). The coordinate ring of X is R x y z y 2 ! xz . Let x y z denote the images of the variables in R. In Exercise 8.0.2 you will show the following:
R. is the ideal z R (principal since !
X
X !
L is the ideal y z
X
X !
2L
2L is Cartier).
On global sections, the image of the map
X !
L
X !
X
L ! #+
X !
2L
z .
is y z 2 , which is a proper subset of X X ! 2L It follows that X ! L X X ! L * X ! 2L .
If we apply (8.0.1) when E X
D
!
D, we get a map X
X !
D ! #+
X
which in turn induces a map (8.0.2)
X !
D !$#
X
D
As noted in §6.0, this is easily seen to be an isomorphism when D is Cartier. Then we have the following general result about the maps (8.0.1) and (8.0.2).
Chapter 8. The Canonical Divisor of a Toric Variety
340
Proposition 8.0.6. Let D E be Weil divisors on a normal variety X. Then (8.0.1) induces an isomorphism
X
D
X
X
E *
X
D E
Furthermore, (8.0.2) already is an isomorphism, i.e., X !
D *
X
D
Proof. The first isomorphism follows from Proposition 8.0.1 since X D E is reflexive and (8.0.1) is an isomorphism on the smooth locus of X. The second isomorphism follows similarly since both sheaves are reflexive and (8.0.2) is an isomorphism on the smooth locus.
Divisor Classes. Recall that Weil divisors D and E on X are linearly equivalent (D E) if D E div f for some f X . Proposition 8.0.7. Let X be a normal variety. (a) If D and E are Weil divisors on X, then X
D *
E
X
E
D
(b) If D is a Weil divisor on X, then D is Cartier
+ X
D is a line bundle
Proof. Part (a) for Cartier divisors was proved in Proposition 6.0.21. The only place that we used Cartier in that proof was to show X
D *
X
E
X
D ! E *
When D E are Weil divisors, we prove this as follows:
X X
D *
X !
E
Taking the double dual and using Proposition 8.0.6, we get
X
X
D
X
X !
E *
X
E
X
X
E implies
D ! E *
X.
One direction of part (b) was proved in Chapter 6 (see Proposition 6.0.16 and Theorem 6.0.17). Conversely, if X D is a line bundle on X, then Theorem 6.0.19 shows that X D * E by part (a). X E for some Cartier divisor E. Thus D Then we are done since any divisor linearly equivalent to a Cartier divisor is Cartier by Exercise 4.0.5.
In Chapter 4 we defined the class group Cl X and Picard group Pic X in terms of Weil and Cartier divisors. Then, in Chapter 6, we reinterpreted Pic X in terms of isomorphism classes line bundles, where the group operation was tensor product and the inverse was the dual. We can now reinterpret Cl X in terms of isomorphism classes of reflexive sheaves of rank 1, where the group operation is the double dual of the tensor product and the inverse is the dual. This follows immediately from Propositions 8.0.6 and 8.0.7.
§8.0. Background: Reflexive Sheaves and Differential Forms
341
K¨ahler Differentials. In order to give an algebraic definition of differential form on a variety, we begin with the case of a -algebra.
Definition 8.0.8. Let R be a -algebra. The K a¨ hler differentials of R over , denoted R , is the R-module generated by the formal symbols d f for f R, modulo the relations (a) d c f g cd f dg for all c f g R. (b) d f g f dg gd f for all f g R.
Example 8.0.9. If R x1 xn , then
R
n
*
R dxi
i 1
This follows because the relations defining f R (Exercise 8.0.3). A -algebra homomorphism R # When we regard
S
R
R
n f i 1 xi dxi
imply d f
for all
S induces a natural homomorphism
!$#
S
as an R-module, we obtain a homomorphism of S-modules
S
R
R
! #
S
Here is a case when this map is easy to understand. See [99, Thm. 25.2] for a proof. Proposition 8.0.10. Let R # S be a surjection of -algebras with kernel I. Then there is an exact sequence of S-modules
where f I I maps to 1 d f S . I I2 ! #
S
2
R
R
R
!$#
R
S
! #
0
Example 8.0.11. Let R x1 xn and S R I, where I f 1 fs . The generators of I give a surjection Rs # I and hence a surjection S s # I I 2 . Combining this with Proposition 8.0.10 and Example 8.0.9, we obtain an exact sequence
Ss ! #
Sn ! #
S
! #
0
where is given by the reduction of the n s Jacobian matrix
(8.0.3)
f1 x1 .
f s x1
.. .
..
f1 xn
fs xn
modulo the ideal I (Exercise 8.0.4). This presentation of computing examples.
S
is very useful for
K¨ahler differentials also behave well under localization, as you will prove in Exercise 8.0.5.
Chapter 8. The Canonical Divisor of a Toric Variety
342
Proposition 8.0.12. Let R f be the localization of R at a non-zero divisor f R. Then R f * Rf. R Cotangent and Tangent Sheaves. Now we globalize Definition 8.0.8. Definition 8.0.13. Let X be a variety. The cotangent sheaf X -modules defined via 1 X U X U on affine open sets U. The tangent sheaf X is the dual sheaf
X
1 X
om
X
1 X
is the sheaf of
1 X X
The reason for the superscript in the notation for the cotangent sheaf will become clear later in this chapter. In Exercise 8.0.6 you will use Example 8.0.11 and Proposition 8.0.12 to show that 1X is a coherent sheaf. See [62, II.8] for a slightly different approach to defining the sheaf 1X , and [62, II.8, Comment 8.9.2] for the connection between these methods.
When U Spec R is an affine open of X, the definition of the tangent sheaf implies that X U HomR R R This can also be described in terms of derivations—see Exercises 8.0.7–8.0.8.
When X is smooth, these sheaves are nicely behaved, as shown by the following result from [62, Thm. II.8.15].
Theorem 8.0.14. A variety X is smooth if and only if 1X is locally free. When this happens, 1X and X are locally free sheaves of rank n, n X.
In the smooth toric case, it is easy to see that the cotangent sheaf is locally free.
n of dimension r gives the affine toric N * Example 8.0.15. A smooth cone variety U * r n r n Then Example 8.0.9 and Proposition 8.0.12 imply that U1 is locally free of rank n. It follows immediately that 1X is locally free for any smooth toric variety X .
We know from Chapter 6 that a locally free sheaf is the sheaf of sections of a vector bundle. When X is smooth, the vector bundles corresponding to 1X and X are called the cotangent bundle and tangent bundle respectively. Example 8.0.16. We construct the cotangent bundle for cover by the affine open sets
U * Spec yx x U * Spec xy y where are the maximal cones in the fan for U
0
*
Spec x y
1
1
1
1
1 2
0
1
2
2.
2.
Recall that
2
has a a
§8.0. Background: Reflexive Sheaves and Differential Forms
343
Let 2 Spec R for R x y . The module R is a free R-module of rank 2 with generators dx dy by Example 8.0.9. Thus a 1-form on 2 may be written uniquely as f 1 dx f2 dy, where f i R. To generalize this to 2 , we require that after changing coordinates, dx and dy transform via the Jacobian matrix described in Example 8.0.11. More precisely, the matrix for the transition function i j will be the Jacobian of the map U j # U i .
On U 2 , the coordinates a1 a2 are represented in terms of the x y coordinates on U 0 as xy 1 y 1 , yielding
1y
x y2 ! 1 y2 !
20
0
Next, we compute 12 . Things get messy if we keep everything in x y coordinates, so we first translate to coordinates a 1 a2 on U 2 , and then translate back. On U 2 we identify a1 a2 with xy 1 y 1 . Then U 1 has coordinates
yx x
1
1
So in a1 a2 terms, we have 12
1 a2 a1 a1
1 a21 0 2 ! a2 a1 1 a 1 !
Rewriting this in terms of x y yields 12
Finally, computing
10
y2 x2 0 ! y x2 y x !
directly, we obtain 10
y x2 1 x 0 ! 1 x2 !
A check shows that 10 12 20 . Similar computations show that the compatibility criteria are satisfied for all i j k, i.e.,
ik
jk
ij
On U i U j , the determinant of i j is invertible, so that the same is true for Hence we obtain a rank 2 vector bundle on 2 .
i j.
Relation with the Zariski Tangent Space. The definition of the tangent sheaf X seems far removed from the definition of the Zariski tangent space T p X Hom X p 2X p given in Chapter 1. Here we explain (without proof) the connection.
The stalk
stalk of
1 X
of the tangent sheaf at p X can be described as follows. The at p is the module of K¨ahler differentials
X p
1 X p
X p
Chapter 8. The Canonical Divisor of a Toric Variety
344
where X p is the local ring of X at p. Since X p X p * and to check), the exact sequence of Proposition 8.0.10 gives a surjection
2 X p ! #
X p
0 (easy
X p
X p
which is an isomorphism of vectors spaces over by [62, Prop. II.8.7]. Since is dual to 1X , taking the dual of the above isomorphism gives (8.0.4)
X p
X p
Hom *
2 X p
X p
X
Tp X
This omits many details but should help you understand why definition of tangent sheaf.
X
is the correct
x1 xn . Example 8.0.17. Let V n be defined by I I V f 1 fs The coordinate ring of V is S x1 xn I, so that Example 8.0.11 gives the exact sequence ! # Ss ! # Sn ! # 0 S
Now take p V and tensor with
to obtain the exact sequence 0
V p
s n V p !$#+ V p ! #
! #
V p
(Exercise 8.0.9). If we tensor this with and dualize, (8.0.4) and the isomorphism 2 give the exact sequence X p * X p X p X p
0! #
Tp V ! #
n
! #
s
where comes from the s n Jacobian matrix xfij p (Exercise 8.0.9). This explains the description of Tp V given in Lemma 1.0.6.
Conormal and Normal Sheaves. Given a closed subvariety i ' Y ( # X, it is natural to ask how their cotangent sheaves relate. We begin with the exact sequence 0 ! #
Y !$#+ X ! #
i
Y ! #
0
which we write more informally as 0 ! #
Y !$#
X !$#
Y ! #
0
The quotient sheaf Y Y2 has a natural structure as sheaf of Y -modules, as does X Y for any sheaf of X -modules. The following basic result is proved in [62, Prop. II.8.12 and Thm. II.8.17]. Theorem 8.0.18. Let Y be a closed subvariety of a variety X. Then: (a) There is an exact sequence of 2 Y Y ! #
Y -modules
1 X
X
Y ! #
1 Y ! #
0
(b) If X and Y are smooth, then this sequence is also exact on the left and is locally free of rank equal to the codimension of Y .
2 Y Y
§8.0. Background: Reflexive Sheaves and Differential Forms
call
345
Note that part (a) of this theorem is a global version of Proposition 8.0.10. We Y Y2 the conormal sheaf of Y in X and call its dual 2 Y Y
Y X
2
om
Y
Y Y Y
the normal sheaf of Y in X. When X and Y are smooth, we can dualize the sequence appearing in Theorem 8.0.18 to obtain the exact sequence
0! #
Y ! #
X
Y !$#
X
Y X !$#
0
The vector bundle associated to Y X is the normal bundle of Y in X. Then the above sequence says that when the tangent bundle of X is restricted to the subvariety Y , it contains the tangent bundle of Y with quotient given by the normal bundle. This is the algebraic analog of what happens in differential geometry, where the normal bundle is the orthogonal complement of the tangent bundle of Y .
Differential Forms. We call 1X the sheaf of 1-forms, and we define the sheaf of p-forms to be the wedge product
p X
p
1 X
For any sheaf of X -modules, the exterior power to the presheaf which to each open set U assigns the
p
is the sheaf associated p U -module U . X
Example 8.0.19. For n , 1 n is the sheaf associated to the free R-module p n xn . Then n is the sheaf associated to i 1 R dxi , R x1
Thus
p
n
p
is free of rank
R
1 i1 n p
R dxi1
ip n
.
R
dxin
More generally, Theorem 8.0.14 implies that Xp is locally free of rank np when X is smooth of dimension n. In particular, nX is a line bundle in this case.
Zariski p-Forms and the Canonical Sheaf . For a normal variety X, the sheaf of p-forms Xp may fail to be locally free. However, this sheaf is locally free on the smooth locus of X, and the complement of the smooth locus has codimension 2 since X is normal. Hence we can use Proposition 8.0.1 to define the sheaf of Zariski p-forms
(8.0.5)
p X
p X
j
p U0
where j ' U0 ( # X is the inclusion of the smooth locus of X. It follows that reflexive sheaf of rank np , where n X.
p X
is a
For later purposes, we note that by Proposition 8.0.1, (8.0.5) is valid for any smooth open subset U0 X whose complement has codimension 2. The case p n is especially important.
Chapter 8. The Canonical Divisor of a Toric Variety
346
Definition 8.0.20. The canonical sheaf of a normal variety X is
X
n X
where n is the dimension of X. This is a reflexive sheaf of rank 1, so that X *
D
X
for some Weil divisor D on X. We call this divisor a canonical divisor of X, often denoted KX . Proposition 8.0.7 shows that the canonical divisor KX is well-defined up to linear equivalence and hence gives a unique divisor class in Cl X , known as the canonical class of X. In the toric case, we will see later in the chapter that there is a natural choice for the canonical divisor. When X is smooth, we call X the canonical bundle since it is a line bundle. In this case, the canonical divisor is Cartier. There are also singular varieties whose canonical divisors are Cartier—these are the Gorenstein varieties to be studied later in the chapter.
While it often suffices to know X up to isomorphism, there are situations where a unique model of X is required. One such construction uses n X , where X is the field of rational functions on X. We can regard n X as the constant sheaf of rational n-forms on X, similar to way that X gives the constant sheaf X of rational functions on X. There is an obvious sheaf map
n X !$#
n
X
You will prove the following result in Exercise 8.0.10. Proposition 8.0.21. When X is normal, image of the map X
n
X
n # X
n
X
is
The canonical sheaf can be defined for any irreducible variety X as a subsheaf of n X , though the definition is more sophisticated (see [93, §9]). When X is projective, another approach is given in [62, III.7], where X is called the dualizing sheaf. We will see in Chapter 9 that X plays a key role in Serre duality.
Exercises for §8.0. 8.0.1. Here are some properties of the rank of a coherent sheaf on an irreducible variety X (Definition 8.0.2).
(a) Let R be an integral domain with field of fractions K, and let M be a finitely generated R-module. Show that M R K is a finite dimensional vector space over K whose dimension equals the rank of the coherent sheaf M on Spec R . (b) Let be a coherent sheaf on X let U X be an nonempty open subset. Prove that and U have the same rank.
(c) Let 0 rank rank .
be an exact sequence of sheaves on X. Prove that rank
§8.0. Background: Reflexive Sheaves and Differential Forms
8.0.2. Prove the claims made in Example 8.0.5.
8.0.3. Let R x1 xn . In Example 8.0.9 we claimed that d f for all f R. Prove this. 8.0.4. Prove that the map Jacobian matrix (8.0.3).
347
n i 1
f xi dxi
in
R
in the exact sequence from Example 8.0.11 comes from the
8.0.5. Prove Proposition 8.0.12. 8.0.6. Prove that the cotangent sheaf
1 X
defined in Definition 8.0.13 is a coherent sheaf.
-linear (a) Show that f d f defines a -derivation d R . (b) More generally, show that if M is an R-module homomorphism, then d R M is a -derivation. 8.0.8. Continuing Exercise 8.0.7, we let Der R M denote the set of all -derivations R M. This is an R-module where r f r f . (a) Use part (b) of Exercise 8.0.7 to construct an R-module isomorphism Der R M Hom M . Explain why d R is called the universal derivation. be the tangent sheaf of a variety X and let U Spec R be an affine open (b) Let U Der R R . subset of X. Prove that 8.0.7. Given a -algebra R and an R-module M, a -derivation R M is a map that satisfies the Leibniz rule, i.e., f g f g g f for all f g R. R
R
R
R
R
X
X
8.0.9. Fill in the details omitted in Example 8.0.17.
8.0.10. Prove Proposition 8.0.21. Hint: 1X is locally free when restricted to the smooth locus of X. Exercise 8.0.11 below will be useful.
!
8.0.11. Let j U X be the inclusion of a nonempty open subset of a variety X. (a) Show that there is a sheaf map j on X. U for any sheaf
"
(b) Show that the map of part (a) is an isomorphism when X is irreducible and constant sheaf.
is a
# $% M be the corre & & ( M . Show that f )# if and only if c 0 in and (a) Let f c ' c m 0 in M . Hint: Pick a basis e e of M and set t ' , so that f is a Laurent monomial in t * t . Then show that f +# if and only if f +# and 1 0 for all i. (b) Use part (a) to construct an isomorphism #%,-# M , and conclude that the Zariski tangent space of T at the identity is naturally isomorphic to N . 8.0.13. Let F be a free module of rank n over a ring R and fix 0 . p . n. Recall that wedge product induces an isomorphism / 0 F Hom / F / F . (a) If X is smooth of dimension n, then show that 0 om 1 2 . (b) Show 3 that if X is3 a normal variety of dimension n, then there is an isomorphism
0 om1 54 . Hint: If you get stuck, see [31, Prop. 4.7]. of a normal variety X satisfies (c) In a similar vein, show that the tangent sheaf om 1 3 *6 .
8.0.12. Let 1 TN be the identity element of the torus TN and let sponding maximal ideal. Set N N and M M . m
m m
2
m m
m m
1
1
n
i
ei
2
n
f ti
2
N
n p
p
n
R
n p X
n p X
X
p X
1 X
X
p X
n X
X
X
X
X
X
Chapter 8. The Canonical Divisor of a Toric Variety
348
§8.1. One-Forms on Toric Varieties
In this section we will describe two interesting exact sequences that involve the sheaves 1X and 1X on a normal toric variety X .
The Torus. The coordinate ring of the torus TN is the semigroup algebra M . Then the map defined by d
m #
m
m
M M
!$#
is easily seen to be an isomorphism. It follows that
(8.1.1) and dualizing, we obtain
M
M
1 TN *
N
TN *
TN
N
TN
TN
This makes intuitive sense since TN N as a complex Lie group. Thus its tangent space at the identity is N N via the exponential map. This is also true algebraically, as shown in Exercise 8.0.12. The group action transports the tangent space N over the whole torus, which explains the above trivialization of the tangent bundle TN .
As a consequence, the 1-form d m is a global section of 1TN that maps to m 1 in (8.1.1) and hence is invariant under the action of TN . See Exercise 8.1.1 for a description of invariant derivations on the torus. m
The First Exact Sequence. Now consider the toric variety X of the fan . For 1 , the inclusion i ' D # X gives the sheaf i on X , which following ( D §8.0 we write as D . Using the map M # given by m # m u , we obtain the composition
M
X
! #
This gives a natural map
(8.1.2)
M '
X
X
X
!$#
D
! #
D
where the direct sum is over all 1 . We also have a canonical map
1 X
'
(8.1.3)
! #
M
X
M , is defined by m M M d M patch These M -module homomorphisms M to give the desired map (8.1.3) (Exercise 8.1.2). Note that over the torus T , the
constructed as follows. On the affine piece U Spec m
M
!$#
m
M
#
N
map of (8.1.3) reduces to the isomorphism (8.1.1).
§8.1. One-Forms on Toric Varieties
349
Theorem 8.1.1. For a smooth toric variety X , the sequence
1 X
0! #
M
!$#
X
!$#
D ! #
0
formed using (8.1.2) and (8.1.3) is exact.
Proof. We first verify that is the zero map. On the affine piece U X , recall from Exercise 4.0.14 and (4.2.1) that D U U is defined by the ideal I
(8.1.4)
div
m
D
m
m
#
M m u 0
m
m, m Over U , the composition 1X # M X D takes a 1-form d M, to m u m M I . This is obviously zero if m u 0, and if m u 0, it vanishes since m I in this case.
We now verify that the sequence is exact over U . Since is smooth, we may assume Cone e1 er , where r n and e1 en is a basis of N. Then U r n r . Let x1 xn denote the characters of the corresponding dual basis of M, also denoted e1 en . The coordinate ring of U is R 1 x1 xr xr 1 xn 1 , and the 1-forms on U form the free R-module R n xi , i 1 R dxi by Example 8.0.9 and Proposition 8.0.12. Since takes dx i to ei we see that can be regarded as the map
that sends
R
n i 1 f i dxi
1
n
R dxi ! #
i 1
M
R
n
R i 1
to f1 x1 fn xn . This gives the exact sequence
0 !$# since xr
R
! #
n
R !$#
i 1
r
R xi !$#
i 1
0
xn are units in R, and the theorem follows.
Logarithmic Forms. The exact sequence of Theorem 8.1.1 has a lovely interpretation in terms of residues of logarithmic 1-forms. The idea is that M X can be thought of as the sheaf 1X D of 1-forms on X with logarithmic poles along D
D . We begin with an example.
Example 8.1.2. The coordinate ring of n is R x1 xn , and the divisor D D is the sum of the coordinate hyperplanes D i V xi . As above, R n has logarithi 1 R dxi . Now introduce some denominators: a rational 1-form mic poles along D if
n
fi i 1
dxi xi
fi R
Chapter 8. The Canonical Divisor of a Toric Variety
350
i These form the free R-module ni 1 R dx xi , and the corresponding sheaf is defined to be 1 n D . The formal calculation
d
m shows that the map d m #
m
n
m
m
m e dxx
i
i
i 1
i
1 induces an isomorphism of sheaves 1 n
such that the map ' 1 n # M 1 1-forms 1 n ( # n D .
D * M
n
n
n
n
of (8.1.3) is induced by the inclusion of
This construction works for any smooth affine toric variety U , and the sheaves of logarithmic 1-forms on U patch to give the sheaf 1X D for any smooth toric variety X . Furthermore, we have a canonical isomorphism
D * M
1 X
(8.1.5)
X
such that the map of (8.1.3) comes from the inclusion of 1-forms.
The construction of 1X D can be done more generally. Let X be a smooth variety. A divisor D i Di on X has smooth normal crossings if every Di is smooth and irreducible, and for every p X, the divisors containing p meet nicely. More precisely, if I p i p Di , we require that the tangent spaces Tp Di Tp X meet transversely, i.e.,
Tp Di
codim
Ip
i Ip
For example, the divisor D
D is a smooth normal crossing divisor on any smooth toric variety X . A nice discussion of 1X D for complex manifolds can be found in [56, p. 449].
The Poincar´e Residue Map. Let f z be an analytic (also called holomorphic) function defined in a punctured neighborhood of a point p . Take a counterclockwise loop C around p on X such that f z has no other poles inside C. Then the residue of the 1-form f z dz at p is the contour integral res p
1 2 i
C
In particular, if p 0 and f z g zz , with g z analytic at zero, then the residue theorem tells us that res p is the coefficient of 1z in the Laurent series for f z at 0, and is equal to g 0 . Note that the 1-form f z dz g z dzz has a logarithmic pole at p 0. When there are several variables, we can do the same construction by working one variable at a time. Here is an example.
§8.1. One-Forms on Toric Varieties
351
Example 8.1.3. Given f f x1 xn R x1 xn , we get the logarith1 mic 1-form f dx x1 . In terms of the above discussion of residues, we can regard f 0 x2 xn as the “residue” of at V x1 . Note also that f 0 x2 xn represents the class of f in R x1 . Doing this for every variable shows that the map
1 n
D * M
n
! #
n
Di i 1
can be interpreted as a sum of “residue” maps. More generally, if X is a smooth variety and D ing divisor, one can define the Poincar´e residue map
Pr '
1 X
i Di
a smooth normal cross-
D ! #
Di i
(see [122, p. 254]) such that we have an exact sequence
0 !$#
(8.1.6)
1 X ! #
1 X
P
D !$# r
Di ! # i
0
When applied to a smooth toric variety X and the divisor D
D , this gives the exact sequence of Theorem 8.1.1 via the isomorphism (8.1.5).
The Normal Case. When X is normal, we get an analog of Theorem 8.1.1 that uses the sheaf 1X of Zariski 1-forms in place of 1X . Since M X is reflexive, taking the double dual of (8.1.3) gives a map
1 X
M
!$#
X
Theorem 8.1.4. Let X be a normal toric variety. Then: (a) The sequence
1 X
0! #
! #
M
X
!$#
D
is exact.
(b) If X is simplicial, then the map on the right is surjective, so that
1 X
0! # is exact.
!$#
M
X
!$#
D ! #
0
Proof. Let j ' U0 X be the inclusion map for U0
U . Note that U0 is a smooth toric variety whose fan has the same 1-dimensional cones as , and codim X ) U0 2 by the Orbit-Cone Correspondence. By Theorem 8.1.1, we have an exact sequence
1 U0 ! #
0 !$#
M
U0 ! #
so that applying j gives the exact sequence
0 !$#
j
1 U0 !$#
j M
U0 ! #
D
U0 !$#
j
0 D
U0
Chapter 8. The Canonical Divisor of a Toric Variety
352
1 by the remarks since j is left exact (Exercise 8.1.3). However, j U1 0 X following (8.0.5), and j M U0 M X by Proposition 8.0.1. Hence we get an exact sequence
M j In Exercise 8.1.4 you will show that the maps M M j 1 X
0! #
! #
!$#
X
D
#
X
! #+ D ! #
X
where diately.
D #
j
D
U0
U0
j
D
U0
factor
U0
is injective. The exact sequence of part (a) follows imme-
It remains to show that M X # cial. Given , we need to show that
D
!$#
D
U
D
is surjective when X is simpli-
is surjective. Fix 1 and pick m M such that m u 0 and m u 0 for all in 1 . Such an m exists since is simplicial. Then m 1 maps to a nonzero constant function on and to the zero function on for . The desired surjectivity follows easily. When X has no torus factors, we learned in §5.3 that graded modules over the total coordinate ring S x 1 give quasicoherent sheaves on X . It is easy to describe a graded S-module that gives . We begin with M S !$# S S ! # S x where the first map comes from m # m u and the second map is obvious. This S x , and we define to be the kernel gives a homomorphism M S # M
U
D
1
U
D
U
1 X
1 S
of this map. Hence we have an exact sequence of graded S-modules
0! #
(8.1.7)
1 S !$#
M
S! #
S x
Using Proposition 6.0.9 and Theorem 8.1.4, we obtain the following result.
Corollary 8.1.5. When X has no torus factors, graded S-module 1S .
1 X
is the sheaf associated to the
The Euler Sequence. In [62, Thm. II.8.13], Hartshorne constructs an exact sequence (8.1.8)
0! #
1
n
! #+
n
!
1
n 1
! #+
n
! #
0
called the Euler sequence of n . He goes on to say “This is a fundamental result, upon which we will base all future calculations involving differentials on projective varieties.” Of course, n is toric, and there is a toric generalization of this result, due to Jaczewski [82] in the smooth case and Batyrev and Cox [8, Thm. 12.1] in the simplicial case.
§8.1. One-Forms on Toric Varieties
353
Theorem 8.1.6. Let X be a simplicial toric variety with no torus factors, i.e., 1 spans N . Then there is an exact sequence u
1 X
0 !$#
! #
X !
Cl X
D ! #
X
! #
0
Furthermore, if X is smooth, then the sequence can be written
1 X
0 !$#
! #
X !
Pic X
D ! #
X
! #
0
Proof. Consider the following diagram: 0
0
0
D
X !
0 /
0
/ Cl X
/ M
1 X
/
/
/ Cl X
X
X
X
/
/
0
D
/0
D
/0
/0
X
0
/0
0
0
The top row is from Theorem 8.1.4 and is exact since X is simplicial. The middle row is the direct sum of the exact sequences 0 !$#
X !
D !$#
X
D ! #
! #
0
and the third row is the obvious exact sequence that uses the identity map on Cl X X .
Since X has no torus factors, we have the exact sequence 0! #
M !$#
! #
Cl X ! #
0
from Theorem 4.1.3, and tensoring this with X gives the middle column. The column on the right is the another obvious exact sequence, and one can check without difficulty that the solid arrows in the diagram commute (Exercise 8.1.5). Then commutativity and exactness imply the existence of the dotted arrows in the diagram, which give an exact sequence by a standard diagram chase.
The exact sequence of sheaves in Theorem 8.1.6 is the (generalized) Euler sequence of the toric variety X . We will use it in the next section to determine the canonical sheaf of X . The Euler sequence also encodes relations generalizing the classical Euler relation for homogeneous polynomials (see Exercise 8.1.8).
Chapter 8. The Canonical Divisor of a Toric Variety
354
Exercises for §8.1.
2
5
8.1.1. We will study invariant derivations on the torus. Since the torus TN Spec M is affine, we know from Exercise 8.0.8 that the derivations Der M M give the global sections of the tangent sheaf TN . M M by (a) For u N, define u
' m u ' Prove that Der 5 M M (b) Let x * x be the characters corresponding to the elements of M dual to some particular basis e e of N. Thus M x x . Prove that x and is the free M -module generated by x x . that T generated is the free M -module (c) Dualizing, conclude that T by the T . invariant differentials 8.1.2. Consider the affine toric variety U Spec 2 M .
u
m
m
u
1
n
N
1
n
1
n
1
TN
1
ei
1 x1
N dxn xn
dx1 x1
i xi
n xn
1 TN
N
(a) Prove that the map
' m ' M & M defines a M -module homomorphism. (b) For a toric variety X , prove that these homomorphisms patch together to give the map M & 6 in (8.1.3). 8.1.3. Let 0 0 be an exact sequence of sheaves on X and let f X Y be a morphism. (a) Prove that 0 f " f " f " is exact on Y . (b) Suppose that Y pt and f X Y is the obvious map. Use part (a) to give a new m
d
1 X
m
M
X
proof of Proposition 6.0.8.
8.1.4. In the proof of Theorem 8.1.4, show that the map M where
M
6 j" 6 D
D
U0
& 6 6 j" 6 X
D
D
& 6 j" 6 X
D
U0
factors
U0
is injective.
6 6 /
8.1.5. The proof of Theorem 8.1.6 contains the square M
& 6
/
X
6 X
D
D
Describe the maps in this square carefully and prove that it commutes. 8.1.6. Show that the Euler sequence from Theorem 8.1.6 reduces to (8.1.8) when X
"!
n
.
8.1.7. Sometimes the name Euler sequence is used to refer to an exact sequence for the tangent sheaf X of a smooth toric variety.
(a) Show that for
!
n
6$# 6$# 1 % # 0
, we have an exact sequence 0
Hint: Use (8.1.8).
n
n
n 1
n
§8.2. Differential Forms on Toric Varieties
355
(b) What is the corresponding sequence for a general smooth toric variety X for Theorem 8.1.6? 8.1.8. Let f be a homogeneous polynomial of degree d in relation is the equation
(8.1.9)
n
f xi
xi
i 1
d f
as in
x x . The classical Euler 1
n
In this exercise, you will prove this relation and consider generalizations encoded by the generalized Euler sequence from a toric variety. (a) Prove (8.1.9). Hint: Differentiate the equation
tx
f tx1
d
* x
t d f x1
n
with respect to t.
graded by Cl X . The graded pieces S for Cl X consist of homogeneous polynomials as described by (5.2.1) from Chapter 5. If Hom & Cl X and f S show that we have a generalized Euler relation D x xf f Hint: Follow what you did for part a, which is the case X "! 0 . (c) When Cl X has rank greater than 1, there will be several distinct generalized Euler relations on homogeneous elements of S. For instance, what are the Euler relations on X "! ! ?
(b) To see how the classical Euler relation generalizes, recall from Chapter 5 that given a toric variety X with no torus factors (i.e., u 1 spans N ), we have the total coordinate ring S x 1
1
n 1
1
1
§8.2. Differential Forms on Toric Varieties
We now study the sheaves of p-forms Xp and Zariski p-forms Xp of a toric n ,n variety X . Recall from §8.0 that the canonical sheaf is X X . X
Properties of Wedge Products. We will need the following properties of wedge products of free R-modules. Proposition 8.2.1. Let F G H be free R-modules of finite rank. '
(a) An R-module homomorphism
p
'
F# p
G induces a homomorphism
F !$#
p
G
(b) An exact sequence 0 # F # G # H # 0 where rank F m and rank H n. Then rank G m n and there is a natural isomorphism
m n
G*
m
F
R
n
H
Chapter 8. The Canonical Divisor of a Toric Variety
356
Proof. Part (a) is straightforward (see Exercise 8.2.1 for an explicit description of p ), as is the rank assertion in part (b). For the isomorphism of part (b), m n G as follows. If the we assume n 0 and define a map m F R n H # maps in the exact sequence are ' F # G and ' G # H, then one checks that m m m n ' F # G is injective and n ' n G # H is surjective. Then m m n F R n H maps to m G, where n . This map is well-defined and gives the desired isomorphism (Exercise 8.2.1).
A corollary of this proposition is that if 0 # # # # 0 is an exact sequence of locally free sheaves on a variety X with rank m and rank m, then rank m n and there is a natural isomorphism
(8.2.1)
m n
*
m
n
X
Example 8.2.2. Suppose that Y X is a smooth subvariety of a smooth variety, and let n X, m Y . Then we have the exact sequence
0 !$#
from Theorem 8.0.18, where
1 X
1 Y !$#
Y ! #
X
0
is the ideal sheaf of Y . By (8.2.1), we obtain
One can check that . Now recall that and and that the normal sheaf of Y X is Hence the above isomorphism implies
n
2 Y Y ! #
1 X n
n
1 X
m
Y
1 Y
Y
X
n m
Y *
X
1 X
n
Y *
X
2
Y Y
1 X
m
1 Y
Y
Y
X
Y X
Y *
n m
X
Y X
X
X
2 Y Y
This isomorphism is called the adjunction formula.
.
The Canonical Sheaf of a Toric Variety. Our first major result gives a formula for the canonical sheaf of a toric variety.
Theorem 8.2.3. For a toric variety X , the canonical sheaf
Thus KX
X
*
X !
D
X
is given by
D is a torus-invariant canonical divisor on X . !
Proof. We assume first that X is smooth with no torus factors. Then we have the Euler sequence
1 X
0 !$#
!$#
X !
Pic X
D !$#
X
! #
0
from Theorem 8.1.6. Each X ! D is a line bundle since X is smooth, and if s n we set s 1 , then one sees easily that Pic X X * X . Hence we can apply part (b) of Proposition 8.2.1 to obtain (8.2.2)
n
1 X
s n
X
s n * X
s
X !
D
§8.2. Differential Forms on Toric Varieties
357
It follows by induction from Proposition 8.2.1 that the right-hand side of (8.2.2) is isomorphic to
X ! D * X !
D
s n
Turning to the left-hand side of (8.2.2), note that left-hand side is isomorphic to
n
1 X
n X
X
s n * X
X
, so that the
since X is smooth. This proves the result when X is smooth without torus factors. In Exercise 8.2.2, you will deduce the result for an arbitrary smooth toric variety.
Now suppose that X is normal but not necessarily smooth. Let j ' U0 X be the inclusion map for U0
U . We saw in the proof of Theorem 8.1.4 that U0 is a smooth toric variety satsifying codim X ) U0 2. Now consider X and
D . Since the fans for U0 and X have the same 1-dimensional cones, X ! these sheaves become isomorphic over U0 by the smooth case. Since these sheaves are reflexive and codim X ) U0 2, we conclude that X *
D by X ! Proposition 8.0.1.
Here are some examples. Example 8.2.4. Theorem 8.2.3 implies that the canonical bundle of
*
n
n
n
is
n ! 1 !
for all n 1 since Cl n * and D0 D1 Dn . In Exercise 8.2.3, you will see another way to understand and derive this isomorphism.
Example 8.2.5. The previous example shows that pute this directly using
2
2
2
2
1
2
*
2
!
3 . We will com-
2
and the description of 1 2 as a rank 2 vector bundle given in Example 8.0.16. Recall that the transition functions for this bundle are given by: 20
1y
0
x y2 ! 1 y2 !
12
!
y2 x2 0 ! y x2 y x
10
y x2 1 x ! 1 x2 0 !
By Exercise 8.2.1, the corresponding maps on 2 are given by the determinants of these 2 2 matrices: ! 1 ! y3 1 2 2 2 20 12 10 3 3 y x x3 Note that each is a cube. It is also evident that
2
10
2
12
2
20
so that these give the transition functions for a line bundle on
2.
2 is On the other hand, 2 1 2 * 2 ! 3 says the canonical bundle of the third tensor power of the tautological bundle described in Examples 6.0.18
Chapter 8. The Canonical Divisor of a Toric Variety
358
and 6.0.20. To see this directly, we first need to calibrate the coordinate systems. Example 8.0.16 used coordinates x y from U 0 , and Example 6.0.18 used homoge2) V x . neous coordinates x0 x1 x2 for 2 , with the standard open cover Ui i
Letting x x0 x2 and y x1 x2 gives an isomorphism U coordinates for U 1 , we have
1 x y x
1 x0 x2 x1 x2 x0 x2
x2
0
U2 . Translating
x0 x1 x0
hence U 1 U0 . A similar computation shows U 2 U1 . We are now set for the final calculation. Keep in mind that the i j are in the coordinate system with charts U i . We will use i j to denote the same transition function, but using the Ui charts. Thus we have 2 3 ! 1 y3 20 12 ! x2 x1 2 3 3 3 ! y x ! x x 1 0 12 01 2 3 1 x3 10 x2 x0 02
Up to a sign, these are indeed the cubes of the transition functions that we computed n ! 1 in Example 6.0.18. In Exercise 8.2.4, you will for the tautological bundle work through the definition of the canonical bundle directly to find the transition functions given in Example 8.0.16.
Example 8.2.6. When we computed the class group of the Hirzebruch surface in Example 4.1.8, we wrote the divisors D as D1 D2 D3 D4 and showed that D3
D1
D4
rD1 D2
r
Thus Cl r Pic r * 2 is freely generated by the classes of D 1 and D2 . It follows that the canonical bundle can be written
r
*
r
D1 ! D2 ! D3 ! D4 *
!
r
!
r 2 D1 ! 2D2
by Theorem 8.2.3.
The Canonical Module. For a toric variety X without torus factors, the canonical sheaf X comes from a graded S-module, where S x 1 is the total coordinate ring of X . This module is easy to describe explicitly.
Each variable x S has degree
0
x
D
x
D
Cl X . Define
Cl X
Then S ! 0 is the graded S-module where S ! 0 S in §5.3, the coherent sheaf associated to S ! 0 is denoted the following result. Proposition 8.2.7.
X !
0 *
X
.
0
for Cl X . As 0 . We have
X !
§8.2. Differential Forms on Toric Varieties
359
Proof. According to Proposition 5.3.7, X ! 0 * X D for any Weil divisor with ! 0 D Cl X . The definition of 0 allows us to pick D !
D . Then Theorem 8.2.3 implies
X !
We call S !
0
0 *
D
X !
the canonical module of S.
*
X
Corollary 8.2.8. For any normal toric variety X we have an exact sequence 0! #
X
! #+ X
!$#
D
Proof. First suppose that X has no torus factors. Multiplication by an exact sequence of graded S-modules
S !
S x
x . The sheaf associated to S x is 0! #
0 ! #
S! #
x induces
since 0 D (Exercise 8.2.5), and then we are done by Propositions 6.0.9 and 8.2.7. When X has a torus factor, the result follows using the strategy described in Exercise 8.2.2.
We can also describe X in terms of n-forms in the x . Fix a basis e1 en of M. For each n-element subset I 1 n
1 , we get the n n determinant
e u
uI
i
j
This depends on the ordering of the i , as does the n-form dx
the product
1
dx n , though
dx 1 dx n depends only on e1 en . It follows that the n-form uI
x dx dx is well-defined up to 1. If we set dx x , then is homogeneous of degree . This gives the submodule S which is isomorphic to S since . To see where comes from, let L x 1 be the field of fractions
(8.2.3)
0
I n
uI
I
1
n
0
n
S
0
n
0
!
S
0
0
of S. Then L is the function field
'
1
) 1
0
0
and the surjective toric morphism Z !$#
X
from Proposition 5.1.9 induces an injection on function fields
'
X !$#
L
Furthermore, the proof of Theorem 5.1.10 implies that for any m M, the character m X maps to m x m u
Chapter 8. The Canonical Divisor of a Toric Variety
360
We regard X as a subfield of L via , so that X This induces an inclusion of K¨ahler n-forms
A basis e1 from §8.1 that dtti i
dt
m u .
x
L
N
dt1 t1
n TN
ei
N
dtn tn
I n
L
x to clear denominators, we obtain
e u dxx
1
uI
n
i
e u dxx
x
dtn tn
which pulls back to a rational n-form in
dti ti
1 TN
0
n
ei u . When we mutiply by
x
1
x t 1
X
i
since ti
n
n
is a TN -invariant section of via (8.2.4). Note that
Hence
m
e of M gives coordinates t for the torus T , and we know . Then is a T -invariant section of
(8.2.4)
L and
dx
Ix
1
dx
e u dxx n
n
0
arises in a completely natural way.
This can be interpreted in terms of sheaves as follows. We regard (8.2.4) as an inclusion of constant sheaves on X . Then the inclusion
X
n
X
from Proposition 8.0.21 induces an inclusion X
L
S
On the other hand, it is easy to see that S
Using the above derviation of
n
n
0
0,
n
0
L
S
induces an inclusion
one can prove that S
0
X
as subsheaves of the constant sheaf n L C —see [93, Prop. 14.14]. Note that this is an equality of sheaves, not just an isomorphism. This shows that from the point of view of differential forms, S 0 deserves to be called the canonical module. It is isomorphic to the earlier version S ! 0 of the canonical module via the map that takes 0 S 0 to 1 S ! 0 .
§8.2. Differential Forms on Toric Varieties
361
The Affine Case. The canonical sheaf U of a normal affine toric variety U is determined by its module global sections, the canoncial module. When we think of U as the ideal sheaf U !
D , we get the ideal
U$
U
U$
U
M
Proposition 8.2.9. U U M is the ideal generated by the characters m for all m M in the interior of .
M
Proof. Corollary 8.2.8 gives the exact sequence 0 !$# where I I D
U
I m
U
U !$#
is the ideal of D
M m u 0
m
U U . Since
m
! #
m
M
M
I
M
by (8.1.4), it follows that U X is the direct sum of m for all m M such that m u 0 for all . Since the u generate , this is equivalent to saying that m is in the interior of .
The Projective Case. A full dimensional lattice polytope P M gives two interested graded rings, each with their own canonical module. For the first, we use an , a lattice point m k P M gives a auxiliary variable t so that for every k character mt k on TN . These characters span the polytope algebra
SP
(8.2.5) This ring graded by setting cone over P defined by
m
kP
mt k
m k
t
M
k. We can also describe SP in terms of the
C P Cone P 1
M
The “slice” of C P at height k is k P (see Figure 4 is §2.2), which implies that S P is the semigroup algebra
SP C P
M
(see the proof of Theorem 5.4.8). Then Proposition 8.2.9 tells us that the canonical sheaf of the associated toric variety comes from the ideal IP
m Int k P
M
m k
t
SP
This is the canoncial module of SP . The second ring associated to P uses the total coordinate ring S of the projective toric variety XP and the corresponding ample divisor D P . Let DP Cl XP
Chapter 8. The Canonical Divisor of a Toric Variety
362
be the divisor class of DP . The graded pieces Sk S Sk
S, k
, form the graded ring
k 0
The canonical module of S is S ! 0 , which is isomorphic to the ideal
x S via multiplication by x since 0
x . When restricted to S , this gives the ideal I Ik S
k 0
where Ik Sk consisting of monomials of degree pears to a positive power.
in which every variable ap-
It follows that the polytope P gives graded rings S P and S , each of which has an ideal representing the canonical module. These are related as follows. Theorem 8.2.10. The graded rings SP and S isomorphism that takes IP to I .
are naturally isomorphism via an
Proof. The proof of Theorem 5.4.8 used homogenization to construct an isomorphism SP * S . It is straightforward to see that this isomorphism carries the ideal IP SP to the ideal I S (Exercise 8.2.6).
For readers familiar with the Proj construction (see the appendix to Chapter 7), we note that XP * Proj SP by Theorem 7.A.1. Furthemore, graded S P -modules give quasicoherent sheaves on Proj S P , as described in [62, II.5] (this is similar to the construction given in §5.3). Then the following is true (Exercise 8.2.7). Proposition 8.2.11. The sheaf on XP associated to the ideal IP construction is the canonical sheaf of XP .
SP by the Proj
The situation becomes even nicer when P is a normal polytope. The ample divisor DP is very ample and hence embeds XP into a projective space, which gives the homogeneous coordinate ring XP . As we learned in §2.0, XP is also the ordinary coordinate ring of its affine cone XP , i.e.,
X X P
P
Furthermore, since P is normal, Theorem 5.4.8 gives isomorphisms
SP * S
*
X
P
and implies that XP is the normal affine toric variety given by
XP Spec SP Spec S
Then the canonical sheaf of the affine cone of XP comes from the ideal IP SP , and when we use the grading on SP and IP , they give XP and its canonical sheaf via XP * Proj SP . Everything fits together very nicely.
§8.2. Differential Forms on Toric Varieties
363
There is a more general notion of canonical module that applies to any graded S Cohen-Macaulay ring S k 0 k where S0 is a field—see [21, Sec. 3.6]. The canonical modules of SP * S constructed above are canonical in this sense.
When the Canonical Divisor is Cartier. For a toric variety X , the Weil divisor KX !
D is called the canonical divisor. Note that KX need not be a Cartier divisor if XX is not smooth. In fact, from Theorem 4.2.8, we have the following characterization of the cases when the canonical divisor is Cartier.
Proposition 8.2.12. Let X be a normal toric variety. Then KX is Cartier if and only if for each maximal cone in , there exists m M such that
m u
1 for all
1
Similarly, K is -Cartier if and only if for each maximal m M such that m u 1 for all 1 .
in , there exists
Example 8.2.13. Let Cone de ! e e . The affine toric variety U is the rational normal cone C . We computed Cl C * d in Example 4.1.4,
1
2
2
2
d
d
where the Weil divisors D1 D2 coming from the rays satisfy D2
D1 dD1
0.
The canonical divisor KCd ! D1 ! D2 has divisor class corresponding to ! 2 d . Since the Picard group of a normal affine toric variety is trivial (Proposition 4.2.2), it follows that KCd is Cartier if and only if d 2.
Another way to see this is via Proposition 8.2.12, where one easily computes that m 2 d e1 e2 satisfies m de1 ! e2 m e2 1. This lies in M 2 if and only if d 2.
We will use the following terminology. Definition 8.2.14. Let KX be a canonical divisor on a normal variety X. We say that X is Gorenstein if KX is a Cartier divisor. Since the canonical sheaf
X is Gorenstein
X
X
KX is reflexive, we have
KX is Cartier
X
is a line bundle
by Proposition 8.0.7. All smooth varieties are Gorenstein, of course. We will study further examples of singular Gorenstein varieties in the next section of the chapter. Refinements. A refinement of a fan induces a proper birational toric morphism ' X # X . The canonical sheaves of X and X are related as follows.
Theorem 8.2.15. Let ' X # X be the toric morphism induced by a refinement
of a fan in N * n . Then
In particular, if
X
*
X
is a smooth refinement of , then
n X
*
X
Chapter 8. The Canonical Divisor of a Toric Variety
364
D ,
Proof. First assume X U , so that refines . Since K !
the description of global sections given in Proposition 4.3.3 implies that
X
where
X
P
m
M
m
1
P m M m u 1 for all 1
1 . The opposite inclusion We clearly have P M Int M since 1 also holds, as we now prove. Given m Int M, we have m u 0 for all 0 for all 1 . This is an integer, so that u 0 in . In particular, m u m u 1, which implies m P M, as desired. Since
U$
U
m Int
by Proposition 8.2.9, we conclude that X * U since U is affine.
m
M X
X
U
U
. Then we have
In the general case, X is covered by affine open subsets U for , and 1 U induced by . The above is the toric variety of the refinement of paragraph gives isomorphisms
X
U when
*
U
X U
which are compatible with the inclusion U is a face of . Hence these isomorphisms patch to give the desired isomorphism X * X .
In Chapter 11, we will prove the existence of smooth refinements. It follows that for any n-dimensional toric variety X , there are two ways of constructing the canonical sheaf X from the sheaf of n-forms of a smooth toric variety:
(Internal) of X .
X
j
n U0 ,
where j ' U0
X
n , where (External) X ' X X from a smooth refinement of .
is the inclusion of the smooth locus
#
X is the toric morphism coming
X are also important (8.2.6) 0 M M as follows. The map M from (8.1.3) induces M
Sheaves of p-forms. The sheaves Xp for 1 p n for the geometry of X . We construct a sequence
p X
! #
! #
1 X
'
p
p
'
p
X
#
p X
!$#
p
p 1
X
p
1 X
! #
p
X
D
§8.2. Differential Forms on Toric Varieties
365
. To define , recall that for u N, the contraction map M has the property that i m m ! 1 m u m m m when m m M. Note also that i u M (Exercise 8.2.8). An element 1 gives u as usual. Using i ' M # M and # , we obtain the composition M ! # M !$# M
and then iu ' p M #
p p 1
p
p
p
u
1
1
i 1
p
i
1
p
i
i 1
p 1
u
p
u
X
p
p 1
D
p
p 1
X
This gives a natural map
p '
p
p 1
X
M
X
p 1
! #
D
M
D
Theorem 8.2.16. The sequence (8.2.6) is exact for any normal toric variety X .
Proof. We begin with the smooth case. Since exactness of a sheaf sequence is local, we can work over an affine toric variety U r n r , where is generated by the first r elements of a basis of e 1 en of N. By abuse of notation, e1 en will denote the corresponding dual basis of M. Setting x i ei , we get
U Spec R
1 1
R x1 xr xr
xn
1
Then (8.2.6) comes from the exact sequence of R-modules 0 !$#
p
R
where p maps dxi1 given in (8.1.3).
p
! #
p
M R !$#
dxi p to ei1
r
p
p 1
i 1
ei p
ei
M R x
i
xi1 xi p by the description of
It is thus obvious that p p 0. To prove exactness, we regard p M R as p F, where F is the free R module with basis e 1 en . Now suppose that p 0 for some p F. We can write uniquely as
i1
fi1 i p ei1 ip
ei p
Exactness will follow once we prove x i1 xi p divides f i1 i p for all i1
ip.
If an index i r appears in f i1 i p , then xi automatically divides f i1 i p since xi is invertible in R. Now suppose that an index i r appears in f i1 i p . We can write ei 1 2 , where ei does not appear in 2 and all f i1 i p involving the index i appear in 1 . Since ie1 1 , the only way for p to vanish is for f i1 i p to be zero in R xi , i.e., for xi to divide f i1 i p . This completes the proof of exactness in the smooth case.
The proof for the general case follows from the smooth case by an argument similar to what we did in the proof of Theorem 8.1.4.
Chapter 8. The Canonical Divisor of a Toric Variety
366
Note that when p 1, the exact sequence (8.2.6) reduces to the sequence appearing in Theorem 8.1.4, and when p n, we get the sequence in Corollary 8.2.8 n . since X X
When X has no torus factors, it is easy to find a graded S-module whose associated sheaf is . Adapting the definition of in (8.2.6) to the module case, we get a homomorphism M S ! # M S x
of graded S-modules whose kernel we denote . This gives an exact sequence of graded S-modules M S x 0 !$# ! # M S ! # p X
p
p
p 1
p S
p S
p
p 1
Using Proposition 6.0.9, we obtain the following corollary of Theorem 8.2.16.
Corollary 8.2.17. If X has no torus factors, then the graded S-module Sp .
p X
is the sheaf associated to
The Affine Case. For an affine toric variety U , the sheaf Up is determined by its global sections, which can be described using Theorem 8.2.16. We will need the following notation. If m M and , let
Span
V m
m0 M m0 the minimal face of containing m
where M M . Here is a result due to Danilov [31, Prop. 4.3].
M
Proposition 8.2.18. Let U be the toric variety of the cone . Then we have an isomorphism p p * V m m U U
m
M
M U M U 0 U of modules over U M . Also note that
Proof. Using the inclusion M M and the exact sequence of Theorem 8.2.16 over U , we obtain the exact sequence !$#
p U ! #
X
(8.2.7)
p
U ! #
U
m
M
M m u 0
p 1
m
U$ D
p
!$#
m
$ D
m
by the analysis given in the proof of Theorem 8.1.1. It follows that we have a direct p sum decomposition U Up m M U U m , where 0 !$#
(8.2.8)
is exact for m M and satisfying m u 0.
p U
p
m !$#
p
M
p
m u 0
p 1
M
is the sum of the contraction maps iu for all
§8.2. Differential Forms on Toric Varieties
367
p M is in the kernel of p if and only if iu 0 for all Thus with m u 0. You will show in Exercise 8.2.8 that i u 0 if and only if p . It follows that the kernel of p in (8.2.8) is the intersection
(8.2.9)
m u 0
p
p
However, the intersection F
m u 0
m u 0
is the minimal face of containing m, and one sees easily that
Span m0 M m0 F
m u 0
. This plus (8.2.9) imply that the kernel of It follows that V m m u 0 p p in (8.2.8) is V m , as claimed.
The Simplicial Case. When X is simplicial, the sequence (8.2.6) is exact on the right when p 1 by part (b) of Theorem 8.1.4. For p 1, similar though longer exact sequences exist in the simplicial case, as we now describe without proof.
Given a fan , consider the sheaf on X defined by
K0 p K p
p
1
M
p j
X
where V O
K j p
j
M
V
is the orbit closure corresponding to X
p 1
1
M
V
p 1
1
. Thus
M
D
and the exact sequence (8.2.6) can be written
p X
0 !$#
! #
K 0 p !$#
K1 p
When X is simplicial, one can extend this exact sequence as follows.
Theorem 8.2.19. When X is simiplicial, there is an exact sequence 0 !$#
p X
! #
K 0 p !$#
K 1 p !$#
!$#
K p p ! #
0
A discussion and proof of this result can be found in [115, Sec. 3.2]. We will use this exact sequence in Chapter 9 to prove a vanishing theorem for sheaf cohomology on simpicial toric varieties.
Chapter 8. The Canonical Divisor of a Toric Variety
368
Exercises for §8.2.
8.2.1. Given a map of free R-modules F G, we can pick bases of F G and represent by a n m matrix with entries in R, where m rank F n rank G . These bases p p induce bases of the wedge products F and G. Then prove that the induced map p p p F G is given by the p p minors of (with appropriate signs).
/
/
/ /
/
8.2.2. Complete the proof of Theorem 8.2.3 in the smooth case by showing how to reduce to the case when X is smooth with no torus factors. Hint: First prove that any smooth r toric variety is equivariantly isomorphic to a product X , where X is a smooth r . toric variety with no torus factors. Then consider X
!
,
* 4
5 "
8.2.3. Let TN be the torus of n. If x0 xn are the usual homogeneous coordinates, then yi xi x0 for i 1 n are affine coordinates for the open subset where x0 0. The n 1 dy1 dyn differential n-form TN nTN TN TN and has poles of y1 yn spans order 1 along each Di V xi for i 1 n. Show that if we write z j x j xi for the affine coordinates on the complement of V xi , and change coordinates to the z j , j i, then we can see also has a pole of order 1 along V x0 . Hence on n, defines a section n of n i 0 Di .
6 #
4
/ * , ! 4
!
#
2 8.2.4. Using the open subsets U i (see Figure 2 in Example 3.1.9), compute 1 2 U i , and write down the transition functions on U i U j . Compare to Example 8.0.16. If the result of your computation differs, describe how you can explain this via a change of basis.
,
8.2.5. Let S be the total coordinate ring of a toric variety X without torus factors. Prove that D is the sheaf associated to the graded S-module S x . Hint: Consider the exact sequence 0 x S S x 0. Propositions 5.3.7 and 6.0.9 will be helpful.
6
,
8.2.6. Prove Theorem 8.2.10.
8.2.7. (For readers familiar with the Proj construction) Prove that the sheaf associated to the ideal IP SP from Theorem 8.2.10 is the canonical sheaf of XP Proj SP . Hint: Let S be the total coordinate ring of XP . We saw in §5.3 that a graded S-module such as S 0 gives a sheaf on XP . Compare this to how I gives a sheaf on Proj S . See the proof of Theorem 7.A.1.
/ 0
8.2.8. Let F be a free module of finite rank over a domain R. Given u F , we get the p p 1 kernel u F and the contraction map iu F F. Assume u re1 where r R and e1 en is a basis of F .
*
-/
/ 0
p 1
(a) Prove that the image of iu is contained in u . p (b) Given F, prove that iu 0 if and only if
4 /
4
3
4 +/
3 3 3 to be the sheaf . Hint: If p
u .
8.2.9. Assume that X has no torus factors. For
Cl X , define p associated to the graded S-module Sp . Prove that X X
you get stuck, see [8, Prop. 8.5].
p X
p S
! , ! q * q and ! ! . is Gorenstein. 8.2.11. Prove that our favorite affine toric variety V V xy zw Hint: Example 1.2.19.
8.2.10. Compute the n-form
0
defined in (8.2.3) for
n
0
1
n
1
4
8.2.12. Prove that a product of two Gorenstein toric varieties is again Gorenstein. Hint: Use Propositions 3.1.14 and 8.2.12.
§8.3. Fano Toric Varieties
369
8.2.13. We will see in Chapter 10 that every 2-dimensional rational cone in 2 is lattice equivalent to a cone of the form Cone e2 de1 ke2 for integers d 0 and 0 k d with d k 1. Prove that U is Gorenstein if and only if d k 1.
.
8.2.14. A strongly convex rational polyhedral cone is Gorenstein if its associated affine toric variety is Gorenstein. In the discussion of the polytope algebra (8.2.5), we saw that a full dimensional lattice polytope P M gives the cone
Cone P 1 M (a) Prove that C P is Gorenstein. (b) Prove that the dual cone C P N is Gorenstein if and only if P is reflexive. C P
In general, a cone is reflexive Gorenstein is both the cone and its dual are Gorenstein. Reflexive Gorenstein cones play an important role in mirror symmetry—see [9, 104]. 8.2.15. Explain why S 0 k 0 Sk 0 gives another model for the canonical module of the graded ring S discussed in the text.
0
0
§8.3. Fano Toric Varieties We finish this chapter with a discussion of an interesting class of projective toric varieties and their corresponding polytopes. Definition 8.3.1. A complete normal variety X is said to be a Gorenstein Fano variety if the anticanonical divisor ! KX is Cartier and ample. Thus Gorenstein Fano varieties are projective. When X is smooth, we will simply say that X is Fano.
2 is Example 8.3.2. Example 8.2.5 shows that 2 ! K 2 * 2 3 is ample, so a Fano variety. We continue this example to introduce the key ideas that will lead to a classification of 2-dimensional Gorenstein Fano toric varieties.
The standard fan for 2 X has minimal generators u0 ! e1 ! e2 , u1 e1 and u2 e2 . The polytope corresponding to the anticanonical divisor of 2 is P m
m u
2
i
!
1 i 0 1 2
It is easy to check that P Conv ! e1 ! e2 2e1 ! e2 ! e1 e2 a lattice polygon with the origin as its unique interior lattice point, shown as the open circle in Figure 1 on the next page. The projective toric variety associated to P is the 3-tuple Veronese embedding of 2 , the image of 2 under the projective morphism defined by the global sections of 2 3 .
Note also that the dual polytope P u N m u ! 1 for all m P is given by P Conv e1 e2 ! e1 ! e2 which is the cone generated by the ray generators of the standard fan of 2 .
Chapter 8. The Canonical Divisor of a Toric Variety
370
P
P
Figure 1. The anticanonical polytope P and its dual P for
2
The special features of Example 8.3.2 involve an unexpected relation between Fano toric varieties and the reflexive polytopes introduced in Chapter 2. Fano Toric Varieties and Reflexive Polytopes. Recall from Definition 2.3.11 that a lattice polytope in M is reflexive if its facet presentation is
P m M m uF ! 1 for all facets F
(8.3.1)
It follows that if P is reflexive, the origin is the unique interior lattice point of P (Exercise 2.3.5). Since aF 1 for all facets F, the dual polytope is P Conv uF F is a facet of P
(8.3.2)
(Exercise 2.2.1). Finally, P is a lattice polytope and is reflexive (Exercise 2.3.5). Naturally, the polytopes pictured in Figure 1 are reflexive. The following result gives the connection between projective Gorenstein Fano toric varieties and reflexive polytopes generalizing what we saw above for 2 .
Theorem 8.3.3. Let X be a normal toric variety, and let KX !
D be the canonical divisor. If X is a projective Gorenstein Fano variety, then the polytope associated to the anticanonical divisor ! KX is reflexive. Conversely, if XP is the projective toric variety associated to a reflexive polytope P, then X P is a Gorenstein Fano variety.
Proof. If X is Gorenstein Fano, then ! KX is Cartier and ample. This implies that polytope associated to ! KX
D has facet presentation
P m M m u and hence is reflexive by (8.3.1).
!
1 for all 1
Conversely, let P be a reflexive polytope in M . The normal fan P of P defines the variety XP X P . The facet presentation (8.3.1) of P has a F 1 for every facet F of P. Hence the Cartier divisor corresponding to P is D P F DF ! KXP . We proved that DP is ample in Proposition 6.1.4, so that ! KXP is ample. Hence XP is Gorenstein Fano.
§8.3. Fano Toric Varieties
371
Classification. By Theorem 8.3.3, classifying toric Gorenstein Fano varieties is equivalent to classifying reflexive polytopes P in M . Since reflexive polytopes contain the origin as an interior point, “classify” means up to invertible linear maps of M induced by isomorphisms of M. This is called lattice equivalence. The first interesting case is in dimension two, where toric Gorenstein Fano surfaces correspond to 16 equivalences classes of reflexive polygons.
We begin with some general facts about a reflexive polytope P. First note that the lattice points of P are the origin and the lattice points lying on the boundary. Furthermore, any boundary lattice point is primitive. We give two less obvious results taken from [112]. Our first result concerns projections of reflexive polytopes. Given a reflexive polygon P and a primitive element m M, there is a projection map
m '
M ! #
M
m
whose image is a polytope whose vertices lie in M m.
n and let m be a lattice point Lemma 8.3.4. Let P be a reflexive polytope in M * m in the boundary of P. Then m P is a in lattice polytope in M m containing 0 as an interior lattice point, and
m
P
m
m F facet of P F
Proof. For each p m P , m 1 p is a line parallel to the line m. By taking the point in P that maps to p and is farthest along this line in the direction of m, we see that the points in m P are in bijective correspondence with the points in
m
U x P x
P for all
0
If F is an facet of P containing m, one easily sees that F U. Conversely, let x U. One can show without difficulty there exists some facet F of P, not parallel to m, containing x and such that uF m 0. Since P is reflexive and m M, uE m is an integer ! 1. Hence uE m ! 1, so that x m F. Thus U m F F, and from here the lemma follows easily.
We will also need the following fact about pairs of lattice points on the boundary of an reflexive polytope. The vertices are primitive vectors in M, but we can have other lattice points on the boundary of P as well. Lemma 8.3.5. Let m m be distinct lattice points on the boundary of a reflexive polytope P. Then exactly one of the following holds: (a) m and m lie in a common edge of P, (b) m m 0, or (c) m m is also on the boundary of P. The proof is left to the reader as Exercise 8.3.2.
Chapter 8. The Canonical Divisor of a Toric Variety
372
The Two-Dimensional Case. The following theorem classifies reflexive polygons 2 , up to lattice equivalence. in the plane M *
Theorem 8.3.6. There are exactly 16 equivalence classes of reflexive polygons in the plane, shown in Figure 2.
5a
6c
8a
6a
6d
6b
7a
8b
4c
4b
5b
4a
3
7b
9
8c
Figure 2. The 16 equivalence classes of reflexive lattice polygons in
2
In this figure the unfilled circle in the center of each polygon represents the origin, and the filled circles are the lattice points on the boundary. The numbers in the labels give the number of boundary lattice points. Note that polygons 3 and 9 are the dual pair that appeared in Example 8.3.2. Proof. We will sketch the proof following [113, Proposition 4.1], and leave the details for the reader to verify (Exercise 8.3.3). We consider several cases. Case A. First assume that P is a reflexive polygon such that each edge contains exactly two lattice points, and fix one such pair. These lattice points must form a basis for M since the triangle formed by these two vertices and the origin has lattice
§8.3. Fano Toric Varieties
373
points only at the vertices (part (a) of Exercise 8.3.3). Hence we can use a lattice equivalence to place the two of them at e 1 and e2 . Subcase A.1. If P has exactly three vertices, the third is located at ae 1 be2 for some a b . Since 0 is the only lattice point in the interior, we must have a b ! 1 (part (b) of Exercise 8.3.3). This gives the polygon of type 3. Subcase A.2. Next, still in Case A, assume that there are three distinct vertices m m m such that m m m . There must be edges of P containing the pairs m m and m m (part (c) of Exercise 8.3.3). Hence each pair must yield a basis for M and we can place m e1 m2 ! e1 e2 , and then m e2 . Project P from m e2 using Lemma 8.3.4. It follows that P can only contain points ! e 1 0 e1 on the line y 0 and ! e2 e1 ! e2 on the line y ! 1. Moreover, P cannot contain any points with y ! 2. It follows that the only other possible polygons in this case are 4b, 5a, 6a (part (d) of Exercise 8.3.3) up to lattice equivalence. Subcase A.3. Finally, in Case A, if we are not in subcases A.1 or A.2, then by Lemma 8.3.5, if m and m are not in the same edge, we must have m m 0. The only possibility here is clearly polygon 4a. Case B. Assume there are exactly three lattice points on some edge of P. By an isomorphism of M, we can place these at ! e 1 e2 e2 e1 e2 . Projecting from m e2 and using Lemma 8.3.4, we see that P must be contained in the strip ! 1 x 1. Moreover, P cannot contain any points with y ! 3, or else ! e 2 0 would be an interior lattice point. Up to lattice equivalence, the possibilities are 4c, 5b, 6b, 6c, 6d, 7a, 7b, 8a, 8b, and 8c (part (e) of Exercise 8.3.3).
Case C. Assume no edge of P has exactly three lattice points and some edge has four or more. Place the vertex of this edge at ! e 1 ! e2 so that ! e1 , ! e1 e2 and ! e1 2e2 also lie on this edge. Projecting from ! e 1 via Lemma 8.3.4 shows that P lies above the line y ! 1, and since the origin is an interior point, there must be a lattice point m ae1 be2 with a 0 and b ! 1. Subcase C.1 If b ! 1, then 2e1 ! e2 must lie in P. If ! e1 2e2 and 2e1 ! e2 don’t lie on an edge of P, their sum e1 e2 would lie in P by Lemma 8.3.5 and force e1 e2 to be interior points. This is impossible, so ! e 1 2e2 and 2e1 ! e2 lie on an edge of P. Hence P has type 9. Subcase C.2 If b 0, then m e1 , for otherwise e1 would be an interior point. Applying Lemma 8.3.5 to e1 and ! e1 e2 shows that e2 P. If no edge connects e1 e2 , then Lemma 8.3.5 would imply e1 e2 P. This point and ! e1 2e2 would force e2 to be interior, again impossible. Hence e 1 e2 lie on an edge, which forces P to be contained in the polygon of type 9. Then our hypotheses on P force equality.
Subcase C.3 If b 1, then e2 becomes an interior point of P (part (f) of Exercise 8.3.3). Hence this subcase can’t occur. There are many other proofs of classification given in Theorem 8.3.6, including [29, 112, 123]. References to further proofs are given in [29, Thm. 6.10].
Chapter 8. The Canonical Divisor of a Toric Variety
374
The collection of 2-dimensional reflexive polygons also exhibits some very interesting symmetries and regularities. For instance, we know that if P is reflexive, then its dual is also reflexive. In Exercise 8.3.4 you will determine which are the dual pairs in the list above. There is also a very interesting relation between the numbers of boundary lattice points in P and P : in all cases, we have
(8.3.3)
P M P
N 12
We will see in Chapter 10 that there is an explanation for this coincidence coming from the cohomology theory of sheaves on surfaces. Higher Dimensions. Reflexive polytopes of dimension greater than two and their associated toric varieties are currently being actively studied. See for instance [113] and the references therein. One reason for the interest in these varieties is the relation with mirror symmetry in physics. It is known that there are a finite number of equivalence classes of reflexive polytopes in all dimensions. Using their computer program PALP, Kreuzer and Skarke [92] determined that there are 4319 classes of 3-dimensional reflexive polytopes and 473800776 classes of 4-dimensional reflexive polytopes. Since these numbers grow so quickly, most more recent work has focused on subclasses, for instance the polytopes giving smooth Fano toric varieties. There are 5 types of such polytopes in dimension 2 (Exercise 8.3.5), and Batyrev [6] and Sato [132] have shown that there are 18 types in dimension 3 and 124 types in dimension 4. See [114] for further references. Exercises for §8.3. 8.3.1. Verify the claims about the polygons P and P defined in Example 8.3.2.
"
8.3.2. Prove Lemma 8.3.5. Hint: Assume that (a) and (b) do not hold. Show that for any facet F of P, uF m uF m 2, and use this to conclude that (c) must hold. 8.3.3. In this exercise, you will supply the details for the proof of Theorem 8.3.6.
(a) Let m m M 2 and assume that Conv 0 m m has no lattice points other than the vertices. Prove that m m form a basis of M.
(b) Show that if P is a reflexive triangle with vertices at e 1 e2 , then the third vertex must be e1 e2 . Hint: One way to show this succinctly is to use the projections from the vertices e1 and e2 as in Lemma 8.3.4.
(c) In the case that each facet contains exactly two vertices, show that if there are vertices m m m with m m m , then the pairs m m and m m must lie in edges.
(d) Complete the proof that the polygons of types 4b, 5a, 6a are the only possibilities in Subcase A.2. Hint: Show that Conv e2 e2 e1 e1 is lattice equivalent to 5a. (e) Show that every reflexive polygon in Case B is equivalent to one of type 4c, 5b, 6b, 6c, 6d, 7a, 7b, 8a, 8b, or 8c. (f) Prove the claim made in Subcase C.3. 8.3.4. Consider the 16 reflexive polygons in Figure 2. Since each polygon in the figure is reflexive, its dual must also appear in the figure, up to lattice equivalence.
§8.3. Fano Toric Varieties
375
(a) For each polygon in the Figure 2, determine its dual polygon and where the dual fits in the classification. In some cases, you will need to find an isomorphism of M that takes the dual to a polygon in the figure. Also, some of the polygon are self-dual, up to lattice equivalence. (b) Show that the relation (8.3.3) holds for each polar pair. 8.3.5. There are precisely five smooth toric varieties among the Gorenstein Fano varieties classified in Theorem 8.3.6. By determining the normal fans of these polygons, find the five smooth ones.
!
!
!
8.3.6. Some of the polygons in Figure 2 give well-known toric varieties. For example, type 9 gives 2 and type 8a gives 1 1. This follows easily by computing the normal fan. Here you will describe the toric surfaces coming from some of the other polygons in Figure 2. This exercise is based on [29, Rem. 6.11].
!
(a) Show that type 8b corresponds to the blow-up of 2 at one of the torus fixed points. Also show that this is the Hirzebruch surface 1 . Hint: Remember the description of blowing up given in §2.3.
!
(b) Similarly show that type 7a (resp. 6a) corresponds to the blow-up of three) torus fixed points.
! , ,
2
at two (resp.
(c) Show that type 1 corresponds to a quotient 2 3 and is isomorphic to the surface in 3 defined by w3 xyz. Hint: Compute the normal fan and show that its minimal generators span a sublattice of index 3 in 2. Proposition 3.3.7 will be helpful. For the final assertion, use the characters coming from the lattice points of the polygon.
!
! (e) Show that type 6d gives the weighted projective space ! 1 2 3 . (f) Show that type 7b (resp. 6b) corresponds to the blow-up of ! 1 1 2 at one (resp. two) smooth torus fixed points. (g) Show that type 5b gives the blow-up of ! 1 2 3 at its unique smooth torus fixed point. 8.3.7. In this exercise you will consider a toric surface that is not quite Fano. The lattice polygon P Conv 3e 3e 2e 2e M gives a toric surface X . Let D D be the anticanonical divisor of X . ! , ,
(d) Show that type 8c gives the weighted projective space 1 1 2 and type 4c gives the quotient 1 1 2 2 . Hint: See Example 3.1.17.
1
2
1
2
2
P
P
(a) Prove that the ample Cartier divisor DP associated to P is given by DP 6D. Conclude that D is not Cartier and that 6D is the smallest integer multiple of D that is Cartier.
(b) Show that the normal fan of P has minimal generators e1 2e2 2e1 e2 and that the minimal generators are the vertices of Q Conv e1 2e2 2e1 e2 N .
(c) Show that the dual of Q is Q of Q that is a lattice polytope.
1 6 P and conclude that 6 is
the smallest integer multiple
8.3.8. A complete toric surface X is log del Pezzo if some integer multiple of its anticanonical divisor KX is an ample Cartier divisor. The index of X is the smallest positive integer such that such that KX is Cartier. This exercise and the next will consider this interesting class of toric surfaces. (a) Prove that a complete toric surface is log del Pezzo of index 1 if and only if it is Gorenstein Fano.
Chapter 8. The Canonical Divisor of a Toric Variety
376
(b) Prove that the toric surface of Exercise 8.3.7 is log del Pezzo of index 6.
8.3.9. A lattice polygon Q N is called LDP if the origin is an interior point of Q and the vertices of Q are primitive vectors in N. The dual Q M of a LDP polygon contains the origin as an interior point but may fail to be a lattice polygon. The index of Q is the smallest positive integer such that such that Q is a lattice polygon. This exercise will explore the relation betweek LDP polygons and toric log del Pezzo surfaces. (a) Show that the polygon Q of Exercise 8.3.7 is LDP of index 6.
(b) A LDP polygon Q N gives a fan in N by taking the cones over the faces of Q. Show that the minimal generators of are the vertices of Q and that the toric surface X is log del Pezzo of index equal to the index of Q.
(c) Conversely, let X be a toric log del Pezzo surface and let Q be the convex hull of the minimal generators of . Note that Q is LDP and that is the fan obtained by taking the cones over the faces of Q. Hint: The key point is to show that every minimal generator of is a vertex of Q. This can be proved using the strict convexity of the support function of KX .
This exercise shows that classifying toric log del Pezzo surfaces up to isomorphism is equivalent to classifying LDP polygons up to lattice equivalence. This is an active area of research—see [84].
Chapter 9
Sheaf Cohomology of Toric Varieties
§9.0. Background: Cohomology By Theorem 6.0.8, a short exact sequence of sheaves on X 0 #%"! #
(9.0.1)
! #
! #
0
gives rise to an exact sequence of global sections 0 !$#
(9.0.2)
X ! #
X
! #
X
to be surjective is measured by a sheaf cohoThe failure of X # X mology group. The main goal of this chapter is to understand the sheaf cohomology of a toric variety.
Sheaves and Cohomology. A sheaf on a variety X has sheaf cohomology groups H p X . The abstract definition uses an exact sequence of sheaves 0 ! #%"!$#
0
d0
! ! #
1
d1
! ! #
2
d2
! !#
where 0 1 are injective. This term is from homological algebra: a sheaf !$# I and an injection ' # is injective if given a sheaf homomorphism there exists a sheaf homomorphism making the diagram below commute:
,
O aCC CC CC C / /
0
377
Chapter 9. Sheaf Cohomology of Toric Varieties
378
We say that is an injective resolution of . From global sections
X
0
d0
! ! #
1
X
d1
! !#
2
X
we get the complex of d2
! !#
The term complex refers to the fact that d p 1 d p 0 for all p 0. Then the pth sheaf cohomology group of is defined to be (9.0.3) H p X H p X d p im d p 1
0 . One can prove where for p 0, we define d 1 to be the zero map 0 # X that injective resolutions always exist and that two different injective resolutions of give the same sheaf cohomology groups.
This definition of H p X
H0
X
X
has some very nice properties, including:
.
A sheaf homomorphism # induces a homomorphism of cohomology i i groups H X # H X that is compatible with composition and takes the identity to the identity, i.e., # H i X is a functor.
A short exact sequence of sheaves (9.0.1) gives a long exact sequence 0! #
We call
H0 X
! #
H0 X
!$#
H 0 X + ! # 0
H1 X
! #
H1 X
!$#
H 1 X + ! # 1
Hp X
! #
Hp X
!$#
H p X + ! #
p p ' H
X
H p 1
#
p 1
!$#
p
X a connecting homomorphism.
You will prove the first bullet in Exercise 9.0.1. For the second bullet, the key step and injective resolutions is to show that given a sheaf homomorphism ' +# p # , # , there are sheaf homomorphisms ' p # p such that the diagram
0
0
/
d0
0
/ /
0
d0
1
/
d1
/
/
1
/
1
d1
commutes. We say that ' # is a map of complexes. Then p induces the p p desired map H X # H X . Finally, for the last bullet, an exact sequence (9.0.1) lifts to an exact sequence of injective resolutions / / / /0 0 O O O /
/
/ 0
and one can show that taking global sections gives an exact sequence of complexes
! # 0 (9.0.4) 0! # ! # !$# X X X
§9.0. Background: Cohomology
379
It is a general fact in homological algebra that any exact sequence of complexes indexed by 0 !$# A !$# B ! # C !$# 0
gives a long exact sequence 0 !$# H 0 A ! # (9.0.5) H1 A ! # Hp A ! #
H0 B ! # H1 B ! # Hp B ! #
0 H0 C $ ! # 1 H1 C $ ! # p Hp C $ ! #
p 1
! #
Applied to (9.0.4), we get the desired long exact sequence in sheaf cohomology.
In the language of homological algebra, is left exact since (9.0.2) is exact. Then the sheaf cohomology groups are the derived functors of . The texts [56, 62, 149] discuss sheaf cohomology and homological algebra in more detail. We especially recommend Appendix 3 of [37] and Chapters 2 and 3 of [80]. ˇ Cech Cohomology. While the abstract definition of sheaf cohomology has nice properties, it is not useful for explicit computations. Fortunately, there is a downˇ to-earth way of viewing sheaf cohomology, in terms of the Cech complex, which we now describe. Ui i 1 be an open cover of X. The definition of a sheaf Let shows that H 0 X X is the kernel of the map
(9.0.6)
d0
Ui ! #
1 i
on X
Ui U j
1 i j
where d 0 is defined as follows: if i 1 i Ui , then the i j component of d 0 is given by j Ui U j ! i Ui U j . Here, j # j Ui U j is the restriction map U j #% Ui U j , and similarly for i # i Ui U j . To get “higher” information about how sections of fit together, we extend ˇ (9.0.6) to the Cech complex. We will use the following notation. Let 1 be the index set for the open cover and let p denote the set of all p 1 -tuples i0 i p of elements of I satisfying i0 i p .
ˇ Definition 9.0.1. The group of pth Cech cochains is
Cp
i0
i p
p
Ui0
U
ip
One can think of an element of C p is as a function that assigns an element of Ui0 Ui p to each i0 i p p . Then we define a differential
Cp
dp
!$#
C p
1
Chapter 9. Sheaf Cohomology of Toric Varieties
380
by describing how d p operates on elements of p
d i0 i p As above, map
!
1
i0 ik i p 1 U i0
Ui0
to i0 ik i p
1
p 1 k
k 0
1
ik
i0 ik i p 1 U i0
Ui p
U U
p 1 :
ip
Ui p
1
is obtained by applying the restriction 1 1
!$#
Ui0
U
1
ip
. In Exercise 9.0.2 you will verify that d p d p
1
0.
Ui i I of X, the Definition 9.0.2. Given a sheaf on X and an open cover ˇ Cech complex is d0 d1 d2 C ' 0 !$# C0 ! # C1 !$# C2 ! # ˇ and the pth Cech cohomology group is H p H p C d p im d p 1
Notice that H 0 H 0 X is a sheaf. However, X since p H need not equal H p X for p 0. We will soon see that there is a nice case where equality occurs for all p.
Cohomology of a Quasicoherent Sheaf . To compute the cohomology of a quaˇ sicoherent sheaf on a variety X using Cech cohomology, we need to find open p covers of X such that H equals H p X for all p. The following vanishing theorem of Serre is very useful in this regard. Proofs can be found in [56, 62, 149].
Theorem 9.0.3 (Serre Vanishing for Affine Varieties). Let be a quasicoherent sheaf on an affine variety U. Then H p U 0 for all p 0.
By Theorem 9.0.3, we can compute the cohomology of a quasicoherent sheaf using any affine open cover. Here is the rough intuition: Consider an arbitrary open cover + Ui of X. Since we can construct X by “gluing together” the Ui , the cohomology of X should be obtained from the co homologies of the Ui and their various intersections. In other words, H X should be determined by the cohomology groups H Ui0 Ui p 1 as we vary over all p.
Ui is an affine open cover. Then Ui0 Now suppose that Ui p 1 is
affine and hence has vanishing higher cohomology by Serre’s theorem. So all ˇ that is left is H 0 Ui0 Ui p 1 . This gives the Cech complex, which thus computes the sheaf cohomology of .
This suggests the following result.
§9.0. Background: Cohomology
381
Ui be an affine open cover of a variety X and let Theorem 9.0.4. Let be a quasicoherent sheaf on X. Then there are natural isomorphisms
Hp
Hp X
*
for all p 0. The proof will be given later in the section after we introduce a spectral sequence that makes the above intuition rigorous. An more elementary proof that does not use spectral sequences can be found in [62, Theorem III.4.5]. Here is an application of Theorem 9.0.4. Example 9.0.5. We compute the cohomology of 1 , 1 ! 1 , and Consider the affine open cover U0 U1 of 1 , where
U0 Spec x
Note also that
U0
d0
U1 ! #
1
on
1
(9.0.7) where d 0 f x g x
U
1
U0
x !$# x x g x . Then
x
1
1.
1
It follows that H p 1 0 for p 2. Hence we need only consider H 0 1 and H 1 1 . For simplicity, we write these sheaf cohomology groups as H 0 and H 1 . ˇ For + complex becomes 1 , the Cech
1
U0 U1 Spec x x 1 ˇ complex is on , the Cech
For any sheaf
and U1 Spec x
1
f x ! H0
1
H
d0
1
1
d 0
1 1
0
coker d 0
The assertion for H 0 is clear since f x $! g x 1 0 implies that f and g are the same constant, and the assertion for H 1 follows since an element of x x 1 can be written m 1 n a 1 x a0 a1 x an x a m x
!
g x
1
f x
We can represent 1 ! 1 as the line bundle 1 ! D , where the divisor D is one of the fixed points of the torus action on 1 . It follows that we have an exact sequence of sheaves
0 !$#
1
!
1 !$#+
1
! #+ D ! #
0
The long exact sequence in sheaf cohomology gives 0#
H0
1
!
1 #
H0
1
#
H0
D #
H1
1
!
1 #
H1
1 #
Chapter 9. Sheaf Cohomology of Toric Varieties
382
The map H 0 1 # H 0 D is the isomorphism that sends a constant function of 1 to the same constant function on D. Since H 1 0, the long exact sequence 1 implies that
(9.0.8) Finally, for
H0 1
1
!
1 0
, we use the Euler sequence
1
1 H 1 !
1
1
0 !$#
1
!$#
!
1
1
1
!
1 ! #+
1
! #
0
* Hp X H p X (Exercise 9.0.3), the from (8.1.8). Since H p X vanishing (9.0.8) and the long exact sequence for cohomology imply that
H0 1
H
1 1
1
0
1 *
H0
1
Earlier, in Example 6.0.5, we showed that the surjective sheaf homomorphism
1
!
1
1
!
1 #+
1
is not surjective on global sections. This is what forces H 1
1 1
to be nonzero.
A key part of Example 9.0.5 was the surjectivity of (9.0.7). This is the algebraic analog of a Cousin problem in complex analysis. A discussion of Cousin problems and their relation to sheaves and cohomology can be found in [90, Ch. 13]. Serre Vanishing for Projective Varieties. The Serre vanishing theorem for affine varieties (Theorem 9.0.3) has a projective version. Theorem 9.0.6 (Serre Vanishing for Projective Varieties). Let bundle on a projective variety X. Then for any coherent sheaf
Hp X 0 and
for all p
X
be an ample line on X, we have
0
0.
A proof can be found in [62, Prop. II.5.3]. We will use this result later in the chapter to prove some interesting vanishing theorems for toric varieties. Higher Direct Images. Given a morphism f ' X # Y of varieties and a sheaf of X -modules on X, the direct image is the sheaf f on Y defined by for U
U! #
(9.0.9)
1
U
Y open. We noted in Example 4.0.24 that f
The definition of f since f
f
implies in particular that H0 Y f
1
Y
H0 X
is a sheaf of
X. More generally, there are homomorphisms Hp Y f
! #
Hp X
Y -modules.
§9.0. Background: Cohomology
383
which need not be isomorphisms for p
0.
In this situation, we also get the higher direct image R p f , which is the sheaf associated to the presheaf defined by
Hp f
U! # Proposition 9.0.7. Let f ' X # on X. Then:
1
U
Y be a morphism and
(a) The higher direct images R p f (b) If U Y is affine, then H p f 1 U .
Rp f
be a quasicoherent sheaf
are quasicoherent sheaves on Y . U
is the sheaf associated to the
Y
U -module
A proof can be found in [62, Propositions II.5.8 and III.8.5]. One especially nice case is when the higher direct images R p f vanish for p 0. We will see below that the maps (9.0.9) are isomorphisms when this happens. The proof involves our next topic, spectral sequences. Spectral Sequences. Readers not familar with spectral sequences should glance at the appendix to this chapter before proceeding further. Here we discuss two spectral sequences relevant to this section. First suppose that f ' X # Y is a morphism and is a quasicoherent sheaf on X. As above, we get the higher direct images R p f , which are sheaves on Y . Proposition 9.0.7 shows that these sheaves compute the cohomology of over certain open subsets of X. So when we “put these together,” i.e., compute H p Y Rq f , we should get the cohomology of on all of X. The precise form of this intuition is the Leray spectral sequence:
E2p q H p Y Rq f H p q X Furthermore, the map H p Y f # H p X from (9.0.9) is the edge homo morphism E2p 0 # H p X (see Definition 9.A.4). This spectral sequence is discussed in [50, II.4.17] and [56, p. 463]
Now assume Rq f + 0 for q
E2p q
0. Then H p Y f 0
q 0 q 0
Then Proposition 9.A.5 implies that (9.0.9) is an isomorphism. Thus we have proved the following. Proposition 9.0.8. Suppose f ' X # Y is a morphism and is a quasicoherent sheaf on X such that Rq f 0 for q 0. Then the map (9.0.9) is an isomorphism H p Y f * H p X .
For our second spectral sequence, let Ui be an open cover of X and
be a sheaf on X. In the discussion leading up to Theorem 9.0.4, we asserted that
Chapter 9. Sheaf Cohomology of Toric Varieties
384
H X is determined by H Ui0 Ui p 1 as we vary over all p. The precise meaning is given by the E1 spectral sequence
E1p q
(9.0.10)
i0
i p
H q Ui0
p
U
i p
H p
q
X
where the differential d1p q ' E1p q # E1p 1 q is induced by inclusion, with signs simˇ ilar to the differential in the Cech complex. This spectral sequence is constructed in [50, II.5.4]. We now have the tools needed to prove Theorem 9.0.4. Ui is an affine open cover Proof of Theorem 9.0.4. We are assuming that of X. First observe that the q 0 terms of (9.0.10) are given by
E1p 0
H 0 Ui0
U
i p
Cp
ˇ complex. Hence and the differentials d are the differentials in the Cech d E E im d E E E i0
i p
p
p0 1
p0 2
1 '
Hp C
p 1 0 1 p
p0 1 #
H
p 10 # 1
1 '
p0 1
Since is quasicoherent and the intersections Ui0 Ui p are affine for all p 0, it follows from Theorem 9.0.3 that E 1p q 0 for q 0. This implies that E2p q 0 for q 0. Using Proposition 9.A.5 again, we conclude that the edge homomorphism
E2p 0 H p
! #
Hp X
is an isomorphism for all p 0.
Cohen-Macaulay Varieties. We next discuss a class of varieties that play a crucial role in duality theory and are interesting in their own right.
We first define what it means for a local ring R to be Cohen-Macaulay. Elements f 1 fs form a regular sequence if f i is not a zero divisor in R x1 xi 1 for all i, and the depth of R is the maximal length of a regular sequence. Then R is Cohen-Macaulay if its depth equals its dimension. Examples include regular local rings. A nice discussion of what Cohen-Macaulay means can be found in [134, 10.2].
A variety X is Cohen-Macaulay if its local rings X p X p are CohenMacaulay for all p X. Thus smooth varieties are Cohen-Macaulay. Later in the chapter we will prove that normal toric varieties are Cohen-Macaulay. Serre Duality. Cohen-Macaulay varieties provide the natural setting for a basic duality theorem of Serre. Here is the simplest version.
§9.0. Background: Cohomology
385
Theorem 9.0.9 (Serre Duality I). Let X be the canonical sheaf of a complete normal Cohen-Macaulay variety X of dimension n. Then for every locally free sheaf of finite rank on X, there are natural isomorphisms
H p X * H n
p
X
X
X
In particular, when D is a Cartier divisor on X and KX is a canonical divisor, we have isomorphisms H p X
X
D * H n
p
X
X
KX ! D
A proof for the projective case can be found in [62, Thm. III.7.6]. The assertion for divisors follows from X
X
X
D *
X
KX
X !
X
D *
X
KX ! D
where the last isomorphism holds since D is Cartier. There is also a more general version of Serre duality that applies when is coherent but not necessarily locally free. The cohomology groups H p X are the derived functors of the global section functor # X , and in the same way, the ext groups Ext p X are the derived functors of the hom functor # Hom X for fixed . Then we have the following result.
Theorem 9.0.10 (Serre Duality II). Let X be the canonical sheaf of a complete normal Cohen-Macaulay variety X of dimension n. Then for every coherent sheaf on X, there are natural isomorphisms
H p X * Extn X p
X
When X is projective, this is proved in [62, Thm. III.7.6]. We will give a version of Serre duality especially adapted to the toric case in §9.2. When X fails to be Cohen-Macaulay, there is a more general duality theorem where canonical sheaf X is replaced with the dualizing complex X . A discussion of this version of duality can be found in [115, §3.2]. Singular Cohomology. Our discussion of the sheaf cohomology of a toric variety in §9.1 will use some algebraic topology. Here we review the topological invariants we will need, beginning with the singular cohomology groups H p Z R where Z is a topological space Z and R is a commutative ring, usually , These are defined using continuous maps ' n ! #
or .
Z
where n is the standard n-simplex. There are several good introductions to singular cohomology, including [55], [65] and [109]. Here are some important properties of singular cohomology:
Chapter 9. Sheaf Cohomology of Toric Varieties
386
A continuous map f ' Z # W induces f ' H p W R # H p Z R such that homotopic maps induce the same map on cohomology and the identity map induces the identity on cohomology. If i ' A ( # Z is a deformation retract (i.e., there is a continuous map r ' Z # A such that r i 1A and i r is homotopic to 1Z ), then i ' H p Z R # H p A R is an isomorphism.
If Z is contractible (i.e, there is z Z such that z ( # retract), then R p 0 H p Z R 0 otherwise For n
Z is a deformation
1, the singular cohomology of the n ! 1 -sphere S n H p Sn
1
R
1
n
is
p 0 n ! 1 otherwise
R 0
We will always assume that Z is a locally contractible metric space. This allows us to interpret the singular cohomology of Z in terms of sheaf cohomology as follows. The ring R defines a presheaf on Z where R is the group of sections over every nonempty open U Z. The corresponding sheaf is the constant sheaf of R. By [15, III.1], the sheaf cohomology of the constant sheaf of R on Z is the singular cohomology H p Z R . The Cohomology Ring. For a commutative ring R, H Z R is a graded R-algebra with multiplication given by cup product. A continuous map induces a ring homomorphism on cohomology, so in particular, a deformation retract i ' A ( # Z induces a ring isomorphism i ' H Z R * H A R .
Here are some examples. Example 9.0.11. The real torus S 1 n has cohomology ring H S1 n R R 1 n
where the i lie in H 1 S1 n R and satisfy the relations (see Examples 3.11 and 3.15 of [65]). Thus n H S1 n R * R
i j !
j i
i.e., the cohomology ring is the exterior algebra of a free R-module. Example 9.0.12. The torus n contains the real torus S 1 retract via t1 tn # t1 t1 tn tn . Hence n H n R * H S1 n R * R
More canonically, the torus TN N has cohomology ring H TN R * M R
n
and
2 i
0
as a deformation
§9.0. Background: Cohomology
387
where M is the dual of N. Thus a lattice homomorphism N # N gives a map TN # TN of tori, and the induced map H TN R # H TN R is the map M R ! # M R
determined by the dual homomorphism M #
Example 9.0.13. The cohomology ring of H n * H2
M. n
over is
n 1
n
(see [65, Theorem 3.12]). Later in the book we will give where a similar quotient description of the cohomology ring of any complete simplicial toric variety.
Reduced Cohomology. Given a topological space Z, we have the canonical map Z # pt that sends elements of Z to pt. This induces the map H Z M whose cokernel is denoted H Z M . Thus H Z M 0 Z is path connected and M R H 0 pt M ! # 0
0
0
0
The reduced cohomology of Z with coefficients in M is defined for p 0
H
and for p
!
(9.0.11)
1 by
p
Z M
H
1
H0 Z M H p Z M 0 M
Z M
Z Z
p 0 p 0
The definition of H 1 will be used in the next section when we compute sheaf cohomology on a toric variety. Exercises for §9.0.
" X .
9.0.1. Use (9.0.3) to show that H 0 X
0 0. X H X .
ˇ 9.0.2. Check that the map d p defined on the Cech cochains satisfies dp d p 9.0.3. Let
be sheaves on X. Prove H p X
Hp
p
1
0 "
9.0.4. A morphism f X Y is affine if Y has an affine open cover Ui such that f 1 Ui is affine for all i. Now assume that f X Y is affine and let be a quasicoherent sheaf on X. Use Theorem 9.0.3, Proposition 9.0.7 and Theorem 9.0.8 to prove that H p Y f Hp X for all p 0.
9.0.5. Use Exercise 9.0.4 to prove the following isomorphisms in cohomology: (a) H X i " H Y when i Y ! X is closed in X and is quasicoherent. (b) H X " H V when V X is a vector bundle and is quasicoherent. p
p
p
p
Chapter 9. Sheaf Cohomology of Toric Varieties
388
1 -sphere S 0 H S 0 R n 1
9.0.6. Consider the n
n
. Show that
R p n 1 0 otherwise
n 1
p
0
Hint: Remember that S consists of two points.
§9.1. Cohomology of Toric Divisors
For a toric variety X , there is a concrete description of the sheaf cohomology of a the sheaf X D of a torus-invariant Cartier divisor D due to Demazure [34]. We generalize this to torus-invariant Weil divisors, inspired by papers of Eisenbud, Mustat¸aˇ and Stillman [39], Hering, K¨uronya and Payne [69], and Perling [121].
ˇ ˇ The Toric Cech Complex. When we compute sheaf cohomology using Cech cohomology, the obvious choice of open cover is
U
max
where max is the set of maximal cones in . We write these as according to their indices.
X
If we set
i0
D
i0
i p
for
Cp
X
D
U
i0
i0 i p
H0 U
p
i p
(9.1.1)
p
and order them
i
ˇ complex
a D on X , the Cech
Given a torus-invariant Cartier divisor D is given by Cp
H 0 U
p,
ip
X
D
then we rewrite this as D
X
The Grading on Cohomology. For an affine open subset U , Proposition 4.3.3 implies that the sections of X D over U can be written
H 0 U$ where
X
P m M m u We can write this as
H 0 U
X
H 0 U
X
D
m
m P M
!
D
m M
0
m
m
a for all
H 0 U$
where for m M, (9.1.2)
D
m u
X
otherwise
!
1
D m
a for all
1
§9.1. Cohomology of Toric Divisors
389
ˇ ˇ This induces a grading of the Cech complex (9.1.1). The Cech differential is built from the restriction maps and hence respects the grading of the complex. Since H p X X D H p X D , we obtain a natural decomposition of sheaf cohomology
H p X
X
D
H p X
m M
X
D m
When decomposing cohomology this way, we often refer to the m M as weights. Here is an example of how weights can be used to compute sheaf cohomology.
Example 9.1.1. On 2 , label the rays as usual: u0 ! e1 ! e2 , u1 e1 , u2 e2 , and maximal cones i , starting with 0 in the first quadrant and going counterclockwise. We will compute H p 2 2 a for a , where 2 a 2 aD0 for the divisor D0 corresponding to u0 .
Let Ui be the affine open corresponding to i . Then Ui j is Ui U j and U012 is ˇ the triple intersection. This allows us to write the Cech complex as
(9.1.3)
0 !$#
1 ! 1 0 1 0 ! 1 0 1 ! 1
C0 a ! ! ! ! ! ! ! ! !$! !#
1 ! 1 1 C1 a !! ! ! ! ! ! ! ! # C2 a !$#
0
where
3
0
C a
C1 a
H 0 Ui
2
aD0
i 1
H 0 Ui j
2
aD0
i j 2
C a H 0 U012
2
aD0
We will compute the cohomology of this complex using the graded pieces for 2 defined by m M 2. For this purpose, let l0 l1 l2 be the lines in M m u0 ! a m u1 0 m u2 0. These lines divide the plane into various regions called chambers. To get a disjoint decomposition, we define
m M m u0 ! a m u1 0 m u2 0 Note that a minus sign corresponds to strict inequality ( 0) while a plus sign corresponds to a weak inequality ( 0). The other chambers C , C , etc. are defined similarly. C
We first consider the case when a 0. The corresponding chamber decomposition of M is shown in Figure 1 on the next page. The labels l i are placed on the plus side of the lines, i.e., where m u 0 ! a m u1 0 m u2 0. Each chamber is labeled with its sign pattern, and the shading indicates which chamber the points on the lines belong to.
Chapter 9. Sheaf Cohomology of Toric Varieties
390
l1
−−+
−++
l0 +−+
0
l2
−−−
a
+−−
−++
a ++−
0
Figure 1. The chamber decomposition for a
The cone
i
is generated by u j uk , where i j k 0 1 2 . Thus
(9.1.4)
H 0 U0
H 0 U1
0
H U2
2
a
m
2
a
m
2
a
m
0
m C
0
m C
M
m C
0
M
M
This follows easily from (9.1.2) (Exercise 9.1.1). Hence we can determine when C0 a m 0.
The cone 1 2 is generated by u0 , so its dual is the half-plane of M where a, i.e., on the plus side of l0 . Using Figure 1 and (9.1.2) again, we get 0
m u
H 0 U12
2
a
m
0
m C
C
C
Similar results hold for U01 and U02 , so we know when C1 a U012 is the torus of 2 , we have
C2 a
m
H 0 U012
2
a
m
m
M
0. Finally, since
0 for all m M
Putting everything together, we get Table 1, which shows the dimension of C p a for m M and a 0. For example, C1 a m 2 when m C since both U01 and U02 contribute 1-dimensional subspaces. Be sure you understand this.
C C
m M is in C C C C C
C0 a 1 0 0
m
C1 a 2 1 0
Table 1. Dimension of C p a
m
for a
0
m
C2 a 1 1 1
m
§9.1. Cohomology of Toric Divisors
391
ˇ When a 0, it is now easy to understand the Cech complex (9.1.5)
0! #
C0 a
C1 a
m !$#
C2 a
m ! #
m ! #
0
The first line of Table 1 corresponds to m C C C . One can check without difficulty that for these m’s, the complex (9.1.5) exact, and the same is true for the second row as well. These remarks imply that if m is a lattice point which is not in the interior of the triangle a 2 Conv 0 0 a 0 0 a then H p 2 2 a m 0 for all p However, if m is a lattice point in the interior of a 2 , then H 2 2 2 a m is nonzero. Summing up, we have:
(9.1.6)
0 Int a
h p a
when a 0. Here, h p a
Hp
2
M
2
2
p 2 p 2
a 1 2
a .
In Exercise 9.1.2, you will will adapt the above methods to show that (9.1.7)
h p a
a 0
2
M
a 2 2
p 0 p 0
when a 0. Thus, for any line bundle on 2 , we have completely determined the dimensions of the cohomology groups H p 2 2 a . The values for ! 7 a 4 are depicted in Table 2. Note the symmetry in the table.
! ! ! ! ! ! !
a h0 a 7 0 6 0 5 0 4 0 3 0 2 0 1 0 0 1 1 3 2 6 3 10 4 15
h1 a 0 0 0 0 0 0 0 0 0 0 0 0
Table 2. Sheaf cohomology of
h2 a 15 10 6 3 1 0 0 0 0 0 0 0
2
a on
2
In Exercise 9.1.3 you will explore how this symmetry relates to Serre duality.
Chapter 9. Sheaf Cohomology of Toric Varieties
392
D m
Then define three subsets of :
a D on X and m M, define m u ! a for all 1
The General Case. Given a divisor D
) 0 . (When is simplicial) V Conv u 1 . (When D is -Cartier) V u m u u , where support function of D (see Exercise 9.1.4).
(General) VD m
Dm
simp Dm supp Dm
Dm
D
is the
D
Figure 2 in Example 9.1.3 below gives a picture of these sets.
(a) H p X
X
D
a D be a Weil divisor on X . Then for any m M
Theorem 9.1.2. Let D and p 0, we have:
Hp
m *
1
VD m
(b) If is simplicial, then H p X Hp
(c) If D is -Cartier, then
X
X
D
H
VD m
.
supp VD m
.
p 1
m *
simp
1
Hp
m *
D
X
.
Proof. When is simplicial, it is easy to see that VD m VD m is a deformation retract (Exercise 9.1.5). Hence part (b) follows from part (a). simp
For part (a), we begin by showing
H 0 U
(9.1.8)
X
D
m
m u a for all V
0
1
Dm
!
Also note that V is convex and hence connected when it is nonempty. Thus H V when V . Since H U D
for all . The first equivalence uses (9.1.2). For the second, note that for , if and only if there is 1 such the definition of VD m implies that VD m that m u ! a . Dm
0
Dm
0
Dm
X
m
when it is nonzero, the equivalences (9.1.8) give a canonical exact sequence (9.1.9) for all 0 !$#
H 0 U
0 !$#
D
X
m !$#%
H 0 VD m
! #
! #
m
0
. It follows that for every p 0, we get an exact sequence p
X
0! #
Cp
H 0 U
which we write as
D
m !$#
X
D
p
! #
m !$#
B p !$#
p
H 0 VD m
Cp ! #
!$#
0
0
ˇ The formula for the differential in the Cech complex can be used to define differ 1 1 p p p p entials B # B and C # C . Then we get an exact sequence of complexes 0 !$# C X D m ! # B ! # C !$# 0
§9.1. Cohomology of Toric Divisors
Since H p comes 0#
X
H 1 X
H p X
D *
X
D
393
D , the long exact sequence (9.0.5) be-
X
# H0 B
m #
# H0 C
H 1 X
X
D
m #
In Exercises 9.1.6 and 9.1.7, you will use the theory of Koszul complexes to show that the complex B has very simple cohomology: Hp B
(9.1.10)
p 0 p 0
0
Thus our long exact sequence breaks up into an exact sequence 0 # H 0 X X D m # # H 0 C # H 1 X X D
and isomorphisms Hp
1
H p X
C *
X
We will show below that
D m
H p C * H p VD m
(9.1.11)
0#
H 0 X
Since H 0 X
X
X
D
D
m
m #
#
0
p 0
When p 1, we obtain the exact sequence
m #
p 2
For p 2, this gives the desired isomorphism H p 1 Vm H p 1 VD m * H p X
X
D m
H 0 VD m #
H 1 X
X
D
m #
V (Exercise 9.1.8), we obtain H 0Vimplies H X D
H
(9.1.12)
Dm
0
1
D m *
1
VD m *
0
X
H X
where the last line uses the definition of H
1
0
m
D m
X
given in (9.0.11).
It remains to prove (9.1.11). Since is the union of the maximal cones and VD m , we get the closed cover VD m i of VD m , where the i are the ˇ maximal cones of . Furthermore, C is the “Cech” complex C for the constant sheaf on VD m , where we use quotation marks since is a closed cover rather than an open cover.
There is still an E1 spectral sequence
E1p q
p
H q VD m
H p
q
VD m
ˇ that applies to this situation, were the differential d 1p q is determined by the Cech differential (see [50, Theorem II.5.4.2]). In particular,
E1p 0 C p C p
Chapter 9. Sheaf Cohomology of Toric Varieties
394
Also, we noted earlier in the proof that each VD m is contractible, so that VD m
E1p q
p
H q VD m
is convex. It follows that
0
0
q
As in the proof of Theorem 9.0.4, this implies that H p C H p C q 0 pq E2 0 q 0
Following the proof of Theorem 9.0.4, we invoke Proposition 9.A.3 to conclude that the edge homomorphisms are isomorphisms, i.e., H p C * H p VD m
The proves (9.1.11) and completes the proof of part (a). For part (c), we assume that D is -Cartier with support function one can modify the proof of (9.1.8) to obtain the equivalence
H 0 U
X
D
m
0
D.
supp
Then
VD m
(Exercise 9.1.9). This allows us to replace (9.1.9) with the exact sequence
H 0 U$
0! #
X
D
m ! #%
H 0 VDsupp m
! #
! #
0
From here, the proof is identical to what we did in part (a) (Exercise 9.1.9).
Example 9.1.3. Let X be the toric surface obtained by blowing up 2 at the three fixed points of the torus action. The minimal generators u 1 u6 of the fan
give divisors D1 D6 on X . Consider the divisor D ! D3 2D5 ! D6 . Let us compute H 1 X X D m for m e1 . Since
m u
1
u3
→
u4 u5
0 m u2 0 m u3 1 m u4 0 m u5
we get the sets
simp VD,m
simp VD m ,
VD m and
supp VD m
u2
VD,m
u1
u4
simp u6 ← VD,m
u2
supp VD,m
u1
u4
u6
supp Figure 2. VDsimp m , VD m and VD m for D
u3
D 3
2D5
D6 and m
u2 u1
u6
u5
← VD,m
2 m u6 1
shown in Figure 2 (Exercise 9.1.13). The
u3
u5
!
supp VD,m
e1
§9.1. Cohomology of Toric Divisors
395
supp
open circle at the origin is a reminder that VD m and VD m do not contain the origin. Then Theorem 9.1.2 implies
H 1 X
X
D
simp
supp
H 0 VD m * H 0 VD m * H 0 VD m *
m *
since all three sets have two connected components.
Each representation H p X X D m in Theorem 9.1.2 has its advantages. supp For example, the vanishing theorems in §9.2 use VD m because of its relation to
simp
convexity, while for surfaces, we will see below that VD m is easier to work with. We note that Theorem 9.1.2 can be stated using relative cohomology intead of reduced cohomology. Consider, for example, the case when D is -Cartier. Then supp the inclusion VD m gives the long exact sequence
! #
H p VD m !$#
H p !$#
supp
supp
H p VD m !$#
Since is contractible (it is a cone), this sequence and Theorem 9.1.2 imply
H p X
(9.1.13)
D
X
m *
H p VD m supp
p 0
(Exercise 9.1.10). Since VDsupp m is open in , algebraic geometers often write this relative cohomology group as HZpD m , where ZD m is the closed set
ZD m
)
VD m u m u
D
u
Then part (c) of Theorem 9.1.2 can be restated as
H p X
X
D
HZpD m
m *
This is the version that appears in [45, p. 74].
Other formulas for H p X X D m can be found in [39], [69] and [121]. Exercise 9.1.11 will explore one of these alternative formulations. The Surface Case. A toric surface is simplicial, which means that we can compute simp simp H p X X D m using VD m . The topology of VD m is especially simple and can be encoded by a sign sequence. We illustrate this with an example.
Example 9.1.4. We return to the situation of Example 9.1.3. Write the divisor D ! D3 2D5 ! D6 as D i ai Di , and label the ray generated by u i with if simp m ui ! ai and with ! if m ui ! ai . Then the picture of VD m from Figure 2 gives the sign pattern shown in Figure 3 on the next page.
If we start at u1 and go counterclockwise around the origin, we get the sign pattern ! ! ! , which we regard as cyclic. The positions of the ! ’s determine simp simp VD m , so that connected components of VD m correspond to strings of consecutive ! ’s in the sign pattern.
Chapter 9. Sheaf Cohomology of Toric Varieties
396
−
+ u3
simp VD,m
−
u2
→
+
u1
u4
simp u6 ← VD,m
u5
+
−
Figure 3. The sign pattern for D
D 3
2D5
D6 and m
e1
This observations holds in general. Let X be a complete toric surface with minimal generators u1 ur arranged counterclockwise around the origin. Given 2 a torus-invariant Weil divisor D i ai Di on X and m M * , the sign pattern signD m is the string of length r whose ith entry is if m u i ! ai and ! if m ui ! ai . We regard signD m as cyclic. Thus, for example, ! ! ! has only one string of consecutive ! ’s.
We have the following result from [70] and [77] (Exercise 9.1.12). Proposition 9.1.5. Given a Weil divisor D i ai Di on a complete toric surface X , the dimension of H p X X D m is given by
X
D
m
X
D
m
D
0
H X H 1 X
H 2 X
X
m
1 0
if signD m otherwise
0
0 1 0
Vsimp Dm
simp
connected components of VD m ! 1 strings of consecutive ! ’s in sign D m $! 1
if signD m otherwise
!
!
VDsimp m is a cycle
Example 9.1.6. Let us revisit Example 9.1.1. For each m M 2, the minimal generators u0 u1 u2 give a sign pattern for the line bundle 2 a 2 aD0 . When a 0, all possible sign patterns are recorded in Figure 1 of Example 9.1.1. Using Proposition 9.1.5, we obtain:
2
2
H0
H1 H2
2
2
a 0 since does not appear.
2
a 0 since all patterns have one string of consecutive ! ’s.
2
a
m Int a
2
m
since ! ! ! labels the interior of a
Thus the computation of Example 9.1.1 follows immediately from Figure 1.
2.
§9.1. Cohomology of Toric Divisors
397
Example 9.1.7. Consider the surface X and divisor D ! D3 2D5 ! D6 from Example 9.1.3. The sign patterns sign D m for all m M * 2 give the chamber decomposition shown in Figure 4 (Exercise 9.1.13). −+++−−
←++++−−
−++++− +++++− −−+++−
+++−−− →
l3
+++−+−
−+−++−
←++−++−
−−−++− +−−++−
l6
→
2e1
e1
0
++−−+−
++−−−−
+−−−+−
++−−−+
−−−+++
+−−−++ ←+−−+++
++−−++ l2
l4 = l1
l5
D
Figure 4. The chamber decompostion for D
3
2D5
D6
As in Figure 1, we have lines li defined by m ui ! ai , where D i ai Di . We put the label li on the side where m ui ! ai , and the shading indicates where the boundary points lie. Each chamber is labeled with its sign pattern (some labels are rather small), and we get five 1-dimensional chambers since l 1 l4 .
We now compute cohomology using Proposition 9.1.5. First,
H 0 X
X
D H 2 X
X
D 0
since neither nor ! ! ! ! ! ! appear anywhere. Furthermore, of the six chambers with sign patterns having more than one string of consecutive ! ’s, four of them ( ! ! , ! ! ! , ! ! ! ! , plus the 1-dimensional ! ! ! ) have no lattice points. Of the remaining two, the lattice points are:
m 0 from ! ! (1-dimensional). m e1 2e1 from ! ! ! .
Chapter 9. Sheaf Cohomology of Toric Varieties
398
These lattice points are indicated with white dots in Figure 3. Hence
In particular,
H 1 X
X
D *
H 1 X
X
D 3.
0
e1
2e1
The paper [70] applies these methods to classify line bundles with vanishing higher cohomology on the toric surface coming from three consecutive blowups of the Hirzebruch surface 2 . Exercises for §9.1. 9.1.1. In Example 9.1.1, verify the sign patterns in Figure 1. 9.1.2. Use the methods of Example 9.1.1 to prove (9.1.7).
4 # 6 # 1 . Use this and Serre duality to
!
n 9.1.3. The canonical class of n is n n explain the symmetry in Table 2 in Example 9.1.1.
9.1.4. Recall that a Weil divisor D a D on X is -Cartier if some positive integer multiple of D is Cartier. Let D be the support function of D as in Theorem 4.2.12.
,
(a) Show that 1 D is a support function (Definition 4.2.11) and depends only on D. We define the support function of D to be D 1 D .
,
(
, there is m M such that (b) Show that D is -Cartier if and only if for every m u a for all 1 . When this is satisfied, show that D u m u for . all u
9.1.5. Show that VDsimp m
VD m is a deformation retract when
9.1.6. Given a ring R and elements f 1
5
dp
/
* f p
by d p , where e1 i 1 f i ei p R , we get the complex
/
K
d0
K0
R
p 1
* e
d1
K1
is simplicial.
R, define
/ %
R
are the standard basis of R . Setting K p
2
d
K
0
1
1 d
K
and for an R-module M, we get the Koszul complex
M Let K K f f R be the Koszul complex defined in the text. (a) Given R R, show that there are maps s K K 0 such that s %s 5 s 5 1 s for all K and ( K . K
f * f M 1
K
R
1
p
p q
p
(b) Show that for all
p
p
q
p 1
p
and
q
K p , the maps s p satisfy
0 s 2 s % d 2 f e (c) Now assume that f f R. Prove that K f f M module M. Hint: Define e g where g f 1. 1
dp
1
p 1
p
p
i 1 i i
1
1
i
i
i 1 i i
is exact for every R-
§9.1. Cohomology of Toric Divisors
M
1 M
9.1.7. When f 1 f of Exercise 9.1.6 becomes (9.1.14)
399
1 and M is any R-module, the Koszul complex K 1 M
1 i
M M 1 i
j
1 i
j k
where the entries of the matrices representing the differentials are 0 or 1. (a) Use the previous exercise to prove that (9.1.14) is exact.
(b) Prove that the complex B defined in the proof of Theorem 9.1.2 satisfies (9.1.10). Hint: Show B is the Koszul complex (9.1.14) with M , minus its first term.
6 D
9.1.8. Prove that H 0 X
X
0 if and only if Vm
m
. Then prove (9.1.12).
9.1.9. Complete the proof of part (c) of Theorem 9.1.2.
9.1.10. Prove (9.1.13).
6 ( . (a) For , show that H U 6 D 0 if and only if . (b) Adapt the proof of Theorem 9.1.2 to show that H X 6 D H 0 has a natural structure Readers familiar with simplicial complexes should note that can be computed using simplicial cohomology. as a simplicial complex, so H 0
a D be a Weil divisor on X . Inspired by [121], we describe a 9.1.11. Let D “simplicial” computation of H p X is not simplicial. X D m that applies even when 1 be a real vector space with basis e , 1 . Each gives the simplex Let Conv e 1 . For m M, let D m mu a 1 as in the text, and set D m . D m
0
X
m
Dm
p
X
p 1
m
Dm
Dm
p 1
Dm
9.1.12. Prove Proposition 9.1.5. 9.1.13. Verify Figure 2 in Example 9.1.3 and Figure 4 in Example 9.1.7.
*6
9.1.14. Prove that H 0 X X D Hint: Part (a) of Exercise 9.1.8.
6 D
0 implies that H p X
m
X
m
0 for all p 0.
6 D 0 0 N X 6 (a) Deduce an exact sequence 0 N X H X 6 D H X 0. (b) Let S be the total coordinate of X and let x Pic X . Prove that ring S 1 n. H X (c) Let x x be a monomial in S . Use homogeneous coordinates from Chapter 5 to construct an automorphism of X that takes x to x x and x to x for . Since X is smooth, H the Lie algebra of the automorphism group Aut X . X S 1is automorphisms Part (c) constructs of X , which when combined with the automorphisms coming from the torus, generate the connected component of the identity of Aut X . This is explains the dimension formula in part (b). See [24] for more details.
9.1.15. Let X be the tangent sheaf of a smooth toric variety X of dimension n. Recall from §6.3 that the real vector space N1 X is dual to Pic X via intersection product. Since Theorem 8.1.6 gives an exact sequence of vector bundles, we can dualize to obtain 1
X
X
1
0
X
0
0
X
X
0
X
X
Chapter 9. Sheaf Cohomology of Toric Varieties
400
§9.2. Vanishing Theorems I In this section we prove two basic vanishing theorems for -Cartier divisors on a toric variety X . We then apply these results to show that normal toric varieties are Cohen-Macaulay and hence satisfy Serre duality. We also give a version of Serre duality that is special to the toric case.
Nef -Cartier Divisors. A Weil divisor D on a normal variety X is -Cartier if some positive integer multiple is Cartier. Also recall that the intersection product D C is defined whenever D is -Cartier and C is a complete irreducible curve in X. Then D is nef if D C 0 for all complete irreducible curves C X. This generalizes the definition of nef Cartier divisor given in §6.2.
In the toric case, we characterize
-Cartier nef divisors as follows.
Lemma 9.2.1. Let D
a D be a -Cartier divisor on X . If has convex support, then the following are equivalent: (a) D is nef. (b) For some integer 0, D is Cartier and basepoint free, i.e., line bundle generated by its global sections. (c) The support function
D '
#
X
D is a
is convex.
Proof. The proof combines ideas from Theorems 6.1.10, 6.2.12 and 7.2.2. We leave the details to the reader (Exercise 9.2.1).
In the Cartier case, we know that D is nef if and only if to X D is generated by global sections (Theorems 6.1.10 and6.2.12). This fails in the -Cartier case, as shown by the following example.
Example 9.2.2. Let X 2 3 5 and let D0 D1 D2 be the divisors given by the minimal generators u0 u1 u2 . Note that 2u0 3u1 5u2 0. Then Cl X * , where the classes of D0 D1 D2 map to 2 3 5 respectively. Let D D1 ! D0 . Then D0 2D D1 3D D2 5D, and in particular, D is a nef Q-Cartier divisor. However, you will check in Exercise 9.2.2 that H 0 X X D 0. Thus X D is not generated by its global sections.
Demazure Vanishing. The most basic vanishing theorem for toric varieties is the following result, first proved by Demazure [34] for Cartier divisors.
Theorem 9.2.3 (Demazure Vanishing). Let D be a -Cartier divisor on X . If is convex and D is nef, then
H p X
X
D 0 for all p
0
' # Proof. The support function D is convex by Lemma 9.2.1. Fix m M supp supp and let VD m u & m u u as in Theorem 9.1.2. Let u v VD m , D
§9.2. Vanishing Theorems I
so that m u
401
u and m v
D
m 1 !
u . Then, for 0
D
t u tv
1, we have
1 ! t m u t m v
t
1 ! t
u t
D
D
v
1 ! t u tv
D
supp
Here the last line follows since D is convex. This implies that VD m is convex and hence contractible. Combining this with Theorem 9.1.2, we obtain
H p X
X
D
Hp
m *
supp
1
VD m
0 for all p
0
Example 9.2.4. For n , the sheaf n a n aD0 is basepoint free when a 0. Hence the higher cohomology vanishes by Theorem 9.2.3, which agrees with what we found in Example 9.1.1 when n 2.
Vanishing of Higher Direct Images. Here is an easy application of Demazure vanishing. Theorem 9.2.5. Let ' X bundle on X 2 Then: (a) R p
(b) If the map then
#
1
X
0 for all p
'
N1 #
*
be a proper toric morphism and let
2
be a line
0.
N2 on the lattices of one-parameter subgroups is surjective, .
Proof. First assume X 2 be the affine X 2 , so that X 1 . Let U open corresponding to 2 . By Proposition 9.0.7, R p X 1 U is the sheaf p 1 is proper, Theorem 3.4.7 associated to the module H U X 1 . Since 1 tells us that U is a toric variety whose fan is supported on the convex set 1 p 1 U 0 by Theorem 9.2.3. It follows that . Thus H X 1 0 for p p R 0 for p 0 by Proposition 9.0.7. X 1
The morphism induces a homomorphism of sheaves X 2 # on X 2 X 1 (Example 4.0.25). It suffices to show that this map is an isomorphism on each affine 1 U open U X 2 , 2 . Note also that the restriction # U is again a 1 proper toric morphism. Hence we may assume that X 2 U . Then 1 is a convex cone, so that
H 0 X 1 by Exercise 4.3.4. We analyze
X
1
1
m
1
1
M1
M 1
m
M1 as follows.
M1
The surjection N1 # N2 gives an injection M2 N2 , 1 N1 have duals M2 1 1 implies 1 . It follows that
M2
M1 on character lattices, and M1 , 1 M1 . Then
M 1
M2
Chapter 9. Sheaf Cohomology of Toric Varieties
402
where the last equality follows since M1 is saturated in M2 by the surjectivity of N1 # N2 . Hence
H 0 X 1
X
1
m
M2
m
H 0 U
U
This shows that U # X 1 induces an isomorphism on global sections, which gives the desired sheaf isomorphism since we are working with quasicoherent sheaves over an affine variety.
For a line bundle R
p
on X 2 , take U
U *
R
'
0. Furthermore, when
for p
p
X
X
N1 #
*
2
*
U
2
U *
. Then R
p
X
1
U
X
, so that U
2
0
N2 is surjective, we have X
1
X
*
2
where the first isomorphism is part (a) of Exercise 9.2.3 and the second follows from X 1 * X 2.
The Injectivity Lemma. We introduce a method that will be used several times in this section and the next. The general framework uses a morphism f ' Y # X and coherent sheaves on Y and on X such that:
f is an affine morphism, meaning f
1
There is a split injection i ' # f #% r' f such that r i 1 . # By functorality, a split injection i ' +
of
Hp X
U
f
X -modules.
Y is affine.
This means that there is
induces a split injection
Hp X f
! #
X is affine when U
However, since f is affine, we also have Hp X f
*
Hp Y
by Exercise 9.0.4. Hence we get an injection Hp X
(#
Hp Y
In particular, H p Y 0 implies H p X 0. This method is used in the proofs of many vanishing theorems—see [95, Ch. 4]. In the toric context, Fujino [42] identified the best choice for the morphism f . Given a fan in N and a positive integer , let ' N # N be multiplication by . This maps to itself and hence induces a toric morphism ' X # X . The dual of is multiplication by on M, and the restriction of to TN M is the group homomorphism given by raising to the th power. Furthermore, given any , we have 1 U U since 1 . This shows that is an affine morphism.
Here is our first injectivity lemma.
§9.2. Vanishing Theorems I
403
Lemma 9.2.6. Let D be a Weil divisor on X and let be a positive integer. Then there is an injection
H p X
H p X
D ( #
X
X
D for all p 0
Proof. We will construct a split injection X D ( # then follows immediately from the above discussion.
X
D . The lemma
Write D
a D . Given , let P m M m u ! a for all M 1 . Over U , the sheaves X D and X D come from the modules m and m
m P M
1 Since U U , module structure given by
P M
m
m P M
m
!$# m
P M
that sends m to m is a homomorphism of Furthermore, one can show that the map given by
r
(9.2.2)
defines a
m
0
is determined by m P M ma. This implies that the map
X D m a
(9.2.1)
m
m
with the
m
M -modules (Exercise 9.2.4).
m M m m m M
M -module splitting of (9.2.1) (Exercise 9.2.4).
The map (9.2.1) and the splitting r are easily seen to be compatible with the inclusion U U when is a face of . It follows that the maps (9.2.1) patch to give the split injection X D ( # X D .
Batyrev-Borisov Vanishing. In Example 9.1.1, we saw that 2 a has nontrivial H 2 when a 0. If we write a ! b for b 0, we can rewrite (9.1.6) as
Hp
2
2
!
0 Int b
b
2
p 2 p 2
M
This has been generalized by Batyrev and Borisov [9]. Recall that a Weil divisor D
a D gives the polyhedron
PD m M m u
!
a for all 1
which is a polytope when is complete (Proposition 4.3.8).
a D be a -Cartier
Theorem 9.2.7 (Batyrev-Borisov Vanishing). Let D divisor on a complete toric variety X . If D is nef, then
(a) H p X X ! D 0 for all p PD . p PD , H X X ! D * (b) When p
m Relint PD
M
m.
Chapter 9. Sheaf Cohomology of Toric Varieties
404
Proof. First assume that D is Cartier and PD N X . Then D is basepoint free since it is nef (Theorem 6.2.12). By Proposition 6.1.12 and Theorem 6.1.21, combining cones of that share the same linear functional relative to D gives a fan D such that refines D and D is strictly convex relative to D .
For m M, H p X Z
D !
D
N ) V
V
Hp
m *
supp D m
1
supp D m
by Theorem 9.1.2. Set
u N m u D u Since D is strictly convex relative to D , the proof of Theorem 9.2.3 implies that Z D m is convex. In fact, it is strongly convex. To see why, note that for any nonzero u N , the strict convexity of D implies 0 D u ! u D u D ! u since u and ! u do not lie in the same cone of D . Thus
Dm
X !
Now assume u Z
u
!
Dm)
D !
u
D
u
for all u 0 in N
0 . Then m u D u , so that m ! u D u
Combining the displayed inequalities gives m ! u , which implies D ! u supp that ! u V D m N ) Z D m . Hence Z D m is strongly convex.
We next prove that (9.2.3) If m ! Int PD
Z
D m
0
m ! Int PD
M
M, then ! m Int PD , so that
!
m u
!
a
D
for all 1
u
Thus
m u u for all 1 which easily implies that Z 0 (Exercise 9.2.5). On the other hand, if we have m ! Int P M, then one of the inequalities of (9.2.4) must fail, i.e., m u u for some 1 Then u Z , so that Z
0 . This proves (9.2.3). We are now ready to compute H X ! D . If m ! Int P M, then V N ) 0 is homotopic to S . Hence 0 p n H X ! D * H V * H S p n If m ! Int P M, then V ) Z , where Z is a closed strongly convex cone of positive dimension. If u 0 in Z , we showed above that contracts to ! u (Exercise 9.2.6). Hence ! u V . It is easy to see that V H X ! D * H V 0 (9.2.4)
D
Dm
D
D
0
0
Dm
0
0
Dm
p
supp Dm
X
D
m
n 1
p
X
p 1
m
supp Dm
D
supp Dm
supp Dm n
p 1
Dm
X
m
Dm
Dm
supp Dm
p
n 1
p 1
supp Dm
This completes the proof when D is Cartier and PD has dimension n.
§9.2. Vanishing Theorems I
405
Now suppose D is Cartier but PD n. We will use the Proposition 6.1.12 and Theorem 6.1.21. After translation, PD spans MD M , where MD M is dual to a surjection ' N # ND , and the normal fan of PD lies in ND . If XD is the toric variety of PD MD , then Theorem 6.1.21 shows that induces a toric morphism ' X # XD such that D is the pullback of the ample divisor D on XD determined by PD . In particular, X ! D * XD ! D .
'
is proper and
Since
R
N#
ND is surjective, we have
p
X !
D 0 p
0
X !
D
D
XD !
by Theorem 9.2.5. Then Proposition 9.0.8 implies that H p XD
(9.2.5)
D * H p X
XD !
X !
D
This proves the theorem, as we now explain. Since MD is saturated in M by the surjectivity of N # ND , we have
PD M PD
MD
M PD
MD
Since PD PD is full dimensional in MD , we are done by (9.2.5) and the full dimensional case considered above.
Finally, we need to consider what happens when D is -Cartier. Pick an integer 0 such that D is Cartier. By Lemma 9.2.6, we have an injection
H p X
X !
H p X
D ( #
X !
D
Since D is Cartier and nef with P D PD , the Cartier case proved above implies that H p X X ! D 0 for p PD . Furthermore, when p PD ,
H p X
X !
H V
D
p 1
m M
V
supp D m
m
m Relint PD
from which we conclude Hp
1
supp Dm
0
m ! Relint PD otherwise
V and m ! Relint P M H X ! D H V
Since V
p
supp Dm
supp D m
D
p 1
X
supp Dm
m M
m
M
M
m ! Relint PD
m
m Relint PD
M, m
M
n a Example 9.2.8. We saw in Example 9.2.4 that the higher cohomology of n a n ! a D0 . Since a D0 is ample with vanishes a 0. When a 0, polytope a n , Theorem 9.2.7 implies
Hp
n
n
a
0 Int a
n
M
when a 0. This generalizes (9.1.6) from Example 9.1.1.
p n p n
Chapter 9. Sheaf Cohomology of Toric Varieties
406
The Cohen-Macaulay Property. Here is a surprising consequence of BatyrevBorisov vanishing. Theorem 9.2.9. A normal toric variety is Cohen-Macaulay.
Proof. First suppose that X is projective and let D be a torus-invariant very ample divisor on X . In [62, Thm. III.7.6], Hartshorne shows that a variety satisfies Serre duality if and only if it is Cohen-Macaulay. When applied to X , his proof that Serre duality implies Cohen-Macaulay has two main steps. The first step is to show that Serre duality implies
H p X
(9.2.6)
X !
D 0 for all
0 p n
and then the second step is to show that (9.2.6) implies Cohen-Macaulay. 0. Since D is ample, the divisor D is generated Now take any integer by global sections and its polytope P D has dimension n X . Thus (9.2.6) follows from Theorem 9.2.7, and then Hartshorne’s argument implies that X is Cohen-Macaulay.
Next consider an affine toric variety U . We can find a projective toric variety that contains U as an affine open subset (Exercise 9.2.7). Since being CohenMacaulay is a local property, it follows that U is Cohen-Macaulay, and then any normal toric variety X is Cohen-Macaulay.
This result was originally proved by Hochster [73]. Another proof can be found in [31, Thm. 3.4]. A projective variety is called arithmetically Cohen-Macaulay (aCM for short) if its affine cone is Cohen-Macaulay. In Exercise 9.2.8 you will show that normal lattice polytopes give aCM toric varieties. Serre Duality. Theorem 9.2.9 shows that Serre duality holds for any normal toric variety. However, version stated in Theorem 9.0.9 only applies to locally free sheaves, while the more general version Theorem 9.0.10 uses ext groups. Many of the sheaves we deal with, such as X D and Xp , fail to be locally free when X is not smooth. Fortunately, there is an “ext-free” version of Serre duality that holds for these sheaves.
Theorem 9.2.10 (Toric Serre Duality). Assume that X is a complete toric variety of dimension n.
(a) If D is a -Cartier divisor and KX is the canonical divisor, then we have natural isomorphisms
H p X
X
(b) If X is simplicial and p
H X
q X
D * H n
p
X
X
KX ! D
is locally free, then there are natural isomorphisms
X
*
Hn
p
X
n q X
X
§9.2. Vanishing Theorems I
407
Remark 9.2.11. Here are two interesting aspects of Theorem 9.2.10: (a)
X
KX ! D need not be isomorphic to X
D
X
X
X
KX
X !
X
D
when D is -Cartier. So Theorem 9.2.10 does not follow from Theorem 9.0.9. (b) Every Weil divisor is -Cartier on a simplicial toric variety. Theorem 9.2.10 holds for all Weil divisors in this case. The proof of Theorem 9.2.10 requires some tools from commutative algebra and will be given in later. This is typical of algebraic geometry—once a variety ceases to be smooth, the theory becomes more technically demanding.
In §9.0 we defined the depth of a local ring R . More generally, the depth of a finitely generated R-module F is the maximal length of a sequence f 1 fs fi is not a zero divisor in F x1 xi 1 F for all i, and the dimension of F is R Ann F , where Ann F f R f F 0 .
Then we define a coherent sheaf on a normal Cohen-Macaulay variety X to be maximal Cohen-Macaulay (MCM for short) if for every point p X, the stalk p is an X p -module whose depth and dimension both equal X. For example, every locally free sheaf on X is MCM.
The following result from [121, Prop. 4.24] shows that MCM sheaves satisfy a nice version of Serre duality. Theorem 9.2.12 (Serre Duality III). Let be a MCM sheaf on a complete normal Cohen-Macaulay variety X of dimension n. There there are natural isomorphisms Hp X
Hn
*
p
X
om
X
X
Proof. We will use the sheaf version of ext, where the sheaves xt q X the derived functors of # om X X . The stalk at p X is
xt q X
(9.2.7) and since
Ext q X p
X p
p
X
are
X p
Hom X ! X om X ! we can apply the Grothendieck spectral sequence for the composition of functors [150, Thm. 5.8.3] to obtain the spectral sequence
pq
E2 q
Hp X
q
xt
X
X
Ext
p q X
X
Since is MCM, [21, Cor. 3.5.11] implies that Ext q X p p X p 0 for 0 and p X. By (9.2.7), we obtain xt q X X 0 for q 0. Hence Hp X
om
X
X *
Ext p X
by Proposition 9.A.5. Since H p X * Ext n X p duality given in Theorem 9.0.10, we are done.
X
X
by the version of Serre
Chapter 9. Sheaf Cohomology of Toric Varieties
408
We can now prove the toric version of Serre duality.
Proof of Theorem 9.2.10. We first show that X D is MCM when D is 0 such that D is Cartier. We will use splitting methods, though Cartier. Pick for clarity we replace N with the sublattice N N. The dual lattice ' N # 1 M 1M . M M gives semigroup algebras N N
Over an affine open subset U , we can pick m M such that u m u for u . Now consider the S -module A U D . If we set P m , then one easily sees that B (9.2.8) A where the last equality uses m M. Note that B is free over and hence is MCM over . However, is finitely generated as a module (Exercise 9.2.9), so that B is MCM over S by [21, Ex. 1.2.26].
1
D
N
X
m
m P M
m
m P
m
1M
1
m
N
N
N
N
N
N
N
Another property of MCM modules is that any nontrivial direct summand of an MCM module is again MCM. This follows from [21, Thm. 3.5.7]. Hence it suffices to split (9.2.8). In this situation, we use the map r ' B # A that sends m B to r
m
1 0
if m A otherwise
Similar to what we did in Proposition 9.2.6, r is a homomorphism of S N -modules. From here, it follows easily that X D is MCM when D is -Cartier. Now we prove duality. Since
p
H X
X
D *
D is MCM, Theorem 9.2.12 implies that
X
H
n p
H
n p
X
om
X
X
om
X
X
D
X
X
D
X
KX
However, om X X D X E * X E ! D for any Weil divisors D E on a normal variety X (Exercise 9.2.10). This gives the desired duality for X D .
For part(b), we first show that is MCM when X is simplicial. Since is complete, we can work locally over U , n. The minimal generators of form a basis of a sublattice N N of finite index such that is smooth relative to N . Let M be the dual of N . As above, we get semigroup algebras N N . q X
By Proposition 8.2.18, the restriction of qX to U U the N -module q V m m A
m
M
N
is determined by
where V m Span m0 M m0 the minimal face of containing m . Now consider the larger N -module defined by
B m
V
q
M
m
m
§9.2. Vanishing Theorems I
409
Since V m is unaffected when M is replaced by M , we see that as a
-module,
B U However, is locally free since U is smooth by our choice of N . Hence B is MCM over . Arguing as in part (a), B is MCM over and we have a
q U
q U
N
N
N
N
N
N
splitting map r defined by
r
m
if m M otherwise
1 0
is MCM since is locally free. The proof is now easy to finish, since Theorem 9.2.12 implies H X H X om
Thus
N
q X
is MCM, and then
p
q X
X
q X
*
n p
However, using the local freeness of
om
q X
X
X
X *
X
om
X
q X
X
X
and Exercise 8.0.13, we have X
q X
X
X
*
n q X
X
From here, the theorem follows easily.
The proof of part (a) of the theorem was inspired by [31, Lem. 3.4.2]. Other proofs that -Cartier divisors give MCM sheaves can be found in [19, Cor. 4.2.2] and [121, Prop. 4.22]. In part (b) we followed [31, Prop. 4.8]. We should also mention that besides -Cartier divisors, there can be other Weil divisors that give MCM sheaves. For example, X is MCM on any normal Cohen-Macaulay variety (see [21, Def. 3.3.1]), so that X is MCM on any toric variety X , even when the canonical class KX fails to be -Cartier. You will give a toric proof of this in Exercise 9.2.11.
In Exercise 9.2.12 you will use Alexander duality to give a purely toric proof of Serre duality for simplicial toric varieties. Exercises for §9.2. 9.2.1. Prove Proposition 9.2.1. 9.2.2. Verify the claims made in Example 9.2.2.
Y induces a homomor6 "6 6 " f " 6 %1 (a) Let be a line bundle on Y . Construct an isomorphism f " f of 6 -modules. Hint: Construct a homomorphism and then study the homomorphism over open subsets of Y where is trivial. (b) Generalize part (b) by showing that for any sheaf of 6 -modules on X and any line " on Y , there is an isomorphism f " 1 f f " 1 . This is the bundle
9.2.3. We saw in Example 4.0.25 a morphism of varieties f X phism Y f X of Y -modules.
X
Y
X
X
projection formula.
Y
Y
Chapter 9. Sheaf Cohomology of Toric Varieties
410
'
'
9.2.4. Consider the map m P M m m P M
m M -module structure on m P M is given by (9.2.1) and (9.2.2) are M -module homomorphisms.
9.2.5. Prove that (9.2.4) implies Z
0
Dm
'
'
m
from (9.2.1), where the m a. Prove that a
'
m
0 , as claimed in the proof of Theorem 9.2.7. 0
9.2.6. As in the proof of Theorem 9.2.7, assume that Z D m is strongly convex and that u Z D m is nonzero. Thus u V supp V supp D m . Prove that for every v D m , the line segment uv supp is contained in V supp (this means that V is star shaped with respect to u). Conclude D m D m u is a contraction. that the constant map V supp D m
0
0 0 0
0
0
9.2.7. Prove that every affine toric variety U is contained in a projective toric variety X . 9.2.8. The lattice points of a normal lattice polytope P give a projective embedding of the toric variety XP . Prove that XP is aCM in this embedding.
9.2.9. Assume that N1 N2 has finite index, with dual M2 M1 , and let be a cone in N1 N2 . This gives semigroup algebras N2 M2 M1 . N1 Prove that N1 is finitely generated as a module over N2 . Hint: Let m1 mr M2 r generate and consider M1 . i 1 i 1 i mi 0
* .$
9.2.10. Let D E be Weil divisors on a normal variety X and let U X be the smooth locus. Then X 6 D " U 6 D , and the same holds for E. (a) Construct a natural map X 6 E D Hom1 6 D *6 E and prove that is an isomorphism when X is smooth. (b) Prove that om 1 6 D 6 E is reflexive. Hint: Let U is the smooth locus and study the restriction map Hom 1 6 D 6 E Hom 1 6 D 6 D . (c) Show that is an isomorphism and that 6 E D om 1 6 D 6 E . 9.2.11. Following [31, Cor. 3.5], you will show that 4 is MCM on a normal toric variety. Fix a cone N and pick a basis e * e of M such that e is in the interior of . Let lcm e u 1 and let M be the lattice with basis e e 0 0 e , with dual N N. By Proposition 8.2.9, 4 comes from the ideal ' M . 0 (a) Prove that u , e u u lies in N. Then use e u 1 to show that u is the minimal generator of with respect to N. (b) Conclude that the canonical divisor of U is Cartier. ' ' and conclude that (c) Construct a splitting of the canonical sheaf of U is MCM. 9.2.12. Alexander duality [109, §71] states that if A S 0 is a closed subset such that the pair S 0 A is triangulable (see [109, p. 150]), then H 0 A H 0 0 S 0 A a D on a complete simplicial toric variety X You will prove Serre duality when D of dimension n. Let K K be the canonical divisor of X . Theorem 9.1.2 implies H X *6 D H 0 V m M Set A V and Conv u 1 . Your goal is to prove that (9.2.9) H X 6 D H 0 X *6 K D 0 X
X
U
X
X
X
X
X
X
X
X
X
X
X
X
X
U
X
X
X
X
U
X
X
1
n
n
n
1
m Int 1
n
M
n
N m
m Int
M
m
m Int
M
N
n 1
n 1
p 1
n p 1
n 1
X
p
Dm
X
m
p 1
simp Dm
simp Dm
p
X
1
n 1
m
U
n p
m
X
m
U
X
n
§9.3. Vanishing Theorems II
(a) Explain why S
411
is homeomorphic to Sn
0
1
0 0
. Note that AD m AK
S.
0 0 0 0 0 be the face of generated by u ’s with (c) Fix an intermediate cone . Let m u a and % be the face generated by u ’s with% m 0u a % . Show % , can be written uniquely as u 1 t u tu where u that every u u0 0 and 0 . t . 1. Then show that % 0 is a deformation retract. (d) Prove that A 0 0 S A is a deformation retract. D
m
(b) Prove that AD m AK D m and that for , is contained in neither AD m 1 such that m u 1 a 1 and nor AK D m if and only if there are 1 2 mu 2 a 2 . We say that is intermediate when this happens.
K D
m
Dm
(e) Finally, prove (9.2.9) by applying Alexander duality to A D m
S.
This exercise was inspired by [31, Prop. 7.7.1] and [45, Sec. 4.4].
§9.3. Vanishing Theorems II Vanishing theorems play an important role in algebraic geometry. A glance at the index of Lazarsfeld’s two-volume treatise [95] lists vanishing theorems due to Bogomolov, Demailly, Fujita, Grauert-Riemenschneider, Griffiths, Kawamata-Viehweg, Kodaira, Koll´ar, Le Potier, Manivel, Miyoka, Nadel, Nakano, and Serre. We will explore toric versions of several of these results. In some cases, the toric version is much stronger, which is to be expected since toric varieties are so special. Twisting. If D is a Cartier divisor and then the twist of by D is the sheaf
D
a coherent sheaf on a normal variety X,
X
X
D
For example, if KX is a canonical divisor on X, then X
D
X
X
X
D *
X
KX
X
X
D *
X
KX D
This notation will be used some of the vanishing theorems stated below. However, when we start working with non-Cartier divisors, we will drop the twist notation. Kodaira and Nakano. Two of the earliest vanishing theorems are due to Kodaira and Nakano. For an ample divisor D on a smooth projective variety X of dimension n, Kodaira vanishing asserts that Hp X
X
D 0 p
0
and Nakano vanishing states that Hp X
q X
D 0 p q
Nakano’s theorem generalizes Kodaira’s since
X
n n X
in the smooth case.
In the toric case, we get more vanishing, in what has become known as the Bott-Steenbrink-Danilov vanishing theorem.
Chapter 9. Sheaf Cohomology of Toric Varieties
412
Theorem 9.3.1 (Bott-Steenbrink-Danilov Vanishing). Let D be an ample divisor on a projective toric variety X . Then
q X
Hp X for all p
D 0
0 and q 0.
Proof. We give a proof due to Fujino [42] that uses the morphism ' X # X from Lemma 9.2.6. We can assume that D is torus-invariant with support function . Let D ' N # X D be the associated line bundle.
The morphism has two key properties. The first concerns the pullback X D . Proposition 6.1.20 implies that X D comes from a divisor whose support function is D . Since is multiplication by , D is the support function of D. Hence
X
The second key property of
X
D *
D *
X
D
is that there is a split injection
q X
(9.3.1)
(#
q X
We will assume this for now.
Given these properties of , the theorem follows easily. Tensoring (9.3.1) with gives a split injection
Since (this is the projection formula from Exercise 9.2.3) and , we obtain a split injection q X
q X
*
X
q X
*
q X
(#
X
X
(#
X
q X
X
q X
X
As in the discussion leading up to Lemma 9.2.6, this gives the injection
q X
Hp X
X
(#
Hp X
q X
X
When p 0, the right-hand side vanishes for sufficiently large by Serre vanishing (Theorem 9.0.6). Hence the left-hand side also vanishes for p 0, which is what we want. It remains to prove (9.3.1). If we set D 0 in the proof of Lemma 9.2.6, we get a split injection i '& X # X . Recall that locally,
X
i sends
looks like m
X
to m .
The splitting r '
X
r
, with module structure is given by
#+ X m
is defined by 0
m
m M m m m M
m
a m a.
§9.3. Vanishing Theorems II
If we tensor i '
X
#
X
(9.3.2)
413
q
with
M
X
together with a splitting map (9.3.3)
q
q
M, we get a sheaf homomorphism
(#
q
M
X ! #
Thus (9.3.2) is a split injection.
q
X
M
M
X
q By Theorem 8.2.16, qX sits inside q M X , so that is a subsheaf X q M X . These subsheaves relate to (9.3.2) and (9.3.3) as follows. of Over U , Theorem 8.2.18 implies
V m M U M
*
where V m M is spanned by the lattice points lying in the same minimal face of as m. Then over U , we have: looks like with module structure a a. The map (9.3.2) takes V m to V m V m U$ since V m V m . Thus (9.3.2) induces a map (9.3.1). The map (9.3.2) takes V m to 0 or, when m m , to V m V m U$ # . since V m V m . Thus (9.3.3) induces a map
U
q X
q
m
m
M
q
m
q
m
M
X
q X
q X
m
q
q
m
q X
m
q
m
q
m
m
q
q
m
q X
m
q X
This gives the desired split injection (9.3.1).
q X
Although Theorem 9.3.1 generalizes the vanishing theorems of Kodaira and Nakano, its name “Bott-Steenbrink-Danilov” reflects the more special vanishing that happens for projective spaces (Bott [13]), weighted projective spaces (Steenbrink [143]), and projective toric varieties (Danilov [31], stated without proof). Theorem 9.3.1 was first proved by Batyrev and Cox [8] in the simplicial case and by Buch, Thomsen, Lauritzen and Mehta [22] in general. Further proofs have been given by Fujino [42] (noted above) and Mustat¸aˇ [111].
The Simplicial Case. When X is simplicial, there is another vanishing theorem q involving the sheaves X .
Theorem 9.3.2. If X is a complete simplicial toric variety, then
Hp X
q X
0 for all p q
Chapter 9. Sheaf Cohomology of Toric Varieties
414
Proof. Since X is simplicial, we have the exact sequence
q X
0! #
(9.3.4)
K 0 q !$#
!$#
from Theorem 8.2.19, where
K j q
K 1 q !$#
Kq q ! #
! #
0
M
q j
j
V
and V O X is the orbit closure corresponding to . Since V is a toric variety, its structure sheaf has vanishing higher cohomology by Demazure vanishing, so that H p X K j q 0 for p 0. Since the above sequence has q 1 terms with vanishing higher cohomology, an easy argument using the long exact sequence in cohomology implies H p X qX 0 for p q (Exercise 9.3.1).
Now suppose p q. Since X is simplicial, the version of Serre duality given in Theorem 9.2.10 implies
q X
Hp X
Hn *
The right-hand side vanishes since n ! p
p
X
n q X
n ! q, and the result follows.
When X complete but not necessarily simplicial, Danilov [31, Cor. 12.7] proves that H p X qX 0 when q p. In [100], Mavlyutov discusses a vanishing theorem for nef Cartier divisor. His result goes as follows.
q X
D , where D is a
Theorem 9.3.3. Let X be a complete simplicial toric variety. If D is a nef Cartier divisor on X , then H p X qX D 0 whenever p q or q p PD .
The proof for p q is relatively easy (Exercise 9.3.2). Note that Theorem 9.3.2 is the case D 0 of Theorem 9.3.3. The paper [100] computes H p X qX D explicitly for all p q. -Weil Divisors. A -Weil divisor or D on a normal variety X is a formal -linear combination of prime divisors. Thus a positive integer multiple of D is an ordinary Weil divisor, often called integral in this context. A -Weil divisor is -Cartier is some positive multiple is integral and Cartier. In the literature (see [95, 1.1.4]), -Cartier -Weil divisors are often called -divisors. These divisors and their close cousins, -divisors, are essential tools in modern algebraic geometry.
For x , x is the greatest integer x and x is the least integer x. Then, given a -Weil divisor D i ai Di , we get integral Weil divisors
D
i
D
i
ai Di
the “round down” of D
ai Di
the “round up” of D
§9.3. Vanishing Theorems II
415
We now prove an injectivity lemma for -Weil divisors due to Fujino [42].
Lemma 9.3.4. Let D be a -Weil divisor on a toric variety X and let integer such that D is integral. Then there is an injection
H p X
H p X
D ( #
X
X
0 be an
D for all p 0
Proof. We adapt the proof of Lemma 9.2.6 to our situation. Write D
a D , as where a b for b and 0 1. Thus D b D . With
above, we claim that ' X # X from Lemma 9.2.6 gives a split injection
(9.3.5)
D (#
X
X
D
Assuming (9.3.5), lemma follows from the discussion leading up to Lemma 9.2.6.
It remains to prove the existence of a split injection (9.3.5). Take an affine open subset U X and consider
Then
U
P m M m u
D
X
m P1 M
m
m P1 M
m
X
U
where the first equality uses a b , 0 the map m # m induces an inclusion
a for all
!
(#
1
D
M
m P
1. Since P
m P
m
M
M
m
P1
M ,
This is a M -module homomorphism, provided that the right-hand side has module structure m a m a, and the usual formula for the splitting map r is M -module homomorphism. From here, we get the required split also a injection (9.3.5) without difficulty.
Here are -divisor versions of Demazure and Batyrev-Borisov vanishing.
Theorem 9.3.5. Let D be a -Cartier
-Weil divisor on a toric variety X .
(a) If is convex and D is nef, then
H p X
D 0 for all p
X
(b) If is complete and D is nef, then
H p X
X !
0
P X D
D 0 for all p
D
Proof. Pick 0 with D Cartier. For part (a), H p 0 for p 0 X by Theorem 9.2.3. The desired vanishing follows immediately from Lemma 9.3.4. For part (b), replacing D with ! D in Lemma 9.3.4 gives
H p X
X !
D H p X
and then we are done by Theorem 9.2.7.
X
!
D ( #
H p X
X !
D
Chapter 9. Sheaf Cohomology of Toric Varieties
416
Here is an example that show how part (a) of Theorem 9.3.5 extends Demazure vanishing beyond nef -Cartier divisors. Example 9.3.6. The fan for the Hirzebruch surface r has minimal generators u1 ! 1 r u2 0 1 u3 1 0 u4 0 ! 1 , giving divisors D1 D4 . By Example 6.1.18, Pic r * aD3 bD4 a b
and by Theorem 6.3.8, the nef cone is generated by D 3 and D4 . Since D1 D3 and D2 D4 ! rD3 , it follows easily that a -Weil divisor D a 1 D1 a4 D4 is nef if and only a1 a3 ra2
(9.3.6) We will assume r
a 2 a4 0
and
0.
We will show that the divisors aD3 bD4 , a b ! 1, have vanishing higher cohomology. Given such a divisor, pick a rational number 0 12 and consider the -Weil divisor D 2 D1
r D2
a 1!
D3 b 1 !
r
D4
This satisfies (9.3.6) and hence is nef. Then D aD 3 bD4 has vanishing higher cohomology by Theorem 9.3.5. Taking a ! 1 or b ! 1, we get non-nef divisors whose higher cohomology vanishes.
Lemma 9.3.4 also leads to the following result due to Mustat¸aˇ [111].
Theorem 9.3.7. Let X be a projective toric variety and let 1 distinct. Then for any ample Cartier divisor D on X , we have
H p X
X
D ! D 1 !
D r 0 for all p !
r
1 be
0
Proof. Let B D 1 D r . Since D is ample, Serre vanishing (Theorem 9.0.6) 0 such that H p X X X D ! B 0 for p 0. implies that we can find 1 Now let E D ! B. Since E D ! B, the inclusion
H p X
X
X
D ! B ( #
H p X
E ( #
from Lemma 9.3.4 implies H p X
H p X
X
X
E
D ! B 0 p
0
Here is a non- -Cartier divisor with vanishing higher cohomology.
Example 9.3.8. The complete fan in 3 shown in Figure 5 on the next page has divisors D0 D4 corresponding to 0 4 . In Exercise 9.3.3, you will show that for a -Weil divisor D a0 D0 a4 D4 ,
D is -Cartier if and only a1 a3 a2 a4 . D is -Cartier and nef if and only a1 a3 a2 a4 0.
§9.3. Vanishing Theorems II
417
z
ρ4
ρ1
ρ3 ρ2
y x ρ0
Figure 5. A complete fan in
3
In particular, D D3 D4 is -Cartier and nef, so that D4 D ! D3 has vanishing higher cohomology by Theorem 9.3.7. Yet D 4 is not -Cartier.
Iitaka Dimension and Big Divisors. Given a nef Cartier divisor D on a complete toric variety X and an integer 0, the global sections W H 0 X X D give a morphism W ' X # W as in Lemma 6.0.27. Also recall that since D is nef, its polytope PD is a lattice polytope. The map W and the polytope PD are related as follows.
Lemma 9.3.9. For PD . In particular,
0, the image of W is isomorphic to the toric variety of X PD for 0.
W
Proof. This follows from Proposition 6.1.21 (Exercise 9.3.4).
This situation is a special case of the definition of the Iitaka dimension X D (see [95, Def. 2.1.3]) of a Cartier divisor D on a complete irreducible variety X. In this terminology, Lemma 9.3.9 implies that a nef Cartier divisor D on a complete toric variety X has Iitaka dimension
X D
PD
In general, a Cartier divisor is big if it has maximal Iitaka dimension, which for a nef Cartier divisor D on a complete toric variety X means D is big
PD
X
It should be clear what it means for a -Cartier
-Weil divisor to be big and nef.
Chapter 9. Sheaf Cohomology of Toric Varieties
418
Kawamata-Viehweg. The classic version of Kodaira vanishing can be stated as H p X
X
KX D 0 for all p
0
when D is an ample line bundle on a smooth projective variety X. In the 1982, Kawamata and Viehweg independently weakened the hypotheses on D. Here is the toric version of their result.
Theorem 9.3.10 (Toric Kawamata-Viehweg). Let X be a complete toric variety and let D be a -Cartier -Weil divisor on X that is big and nef. Then
H p X
X
KX
D 0 for all p
0
X . When D is Cartier, Serre duality implies H X K D * H X ! D
P by Theorem 9.2.7, and since Since D is nef, the latter vanishes for n ! p D is big, the condition on p becomes n ! p n, i.e, p 0. In general, write D
a D where a b ! for b and 0 1. Pick such that D is Cartier and 0 1 for all . Let E D K . b ! 1, we have E Since b ! ! D K . Thus the inclusion H X E ( # H X E Proof. Let n
p
X
n p
X
X
D
1
1
1
p
H p X
X
KX
X
H p X
D ( #
X
X
p
X
from Lemma 9.3.4 implies
KX D
X
Then we are done by the Cartier case already proved.
Another approach to Theorem 9.3.10 is to prove a version of Serre duality that implies H p X X KX D * H n p X X ! D (Exercise 9.3.5).
Grauert-Riemenschneider. We begin with the following preliminary result. Proposition 9.3.11. Let be a simplicial fan whose support is strongly convex. Then H p X X 0 for all p 0.
Proof. Fix p 1 and m M. Then Theorem 9.1.2 with D KX implies H p X X m H p X X D m * H p 1 VD m
where VD m Consider the set
)
Dm
0 and
W u ) 0 m u
D m
)
0
m u 1 for
0 u N m u
1 .
0
and note that W is convex since is strongly convex. Also, we can rewrite
as D m m u 0 for 1 since m u . Thus
(9.3.7)
VD m
W
The proposition will follow once we prove that (9.3.7) is a deformation retract.
Dm
§9.3. Vanishing Theorems II
Given with W The assumption W
419
, we construct r W
'
implies that
A
1 m u
#
VD m
as follows.
0
is nonempty. Since is simplicial, every u can be uniquely written u , define r u
1 u , 0, and when u W
A u . This is nonzero since u W , so that r u VD m since Cone u A VD m . It is also easy to see that r is compatible with r whenever is a face of . Thus we get a map r ' W # VD m , which is a deformation retraction by Exercise 9.3.6.
Here is a toric version of Grauert-Riemenschneider vanishing.
Theorem 9.3.12 (Toric Grauert-Riemenschneider Vanishing). Let ' X # X be a surjective proper toric morphism between toric varieties of the same dimension. If X is simplicial, then
Rp If in addition
X
is birational, then
X
0 for all p *
X
.
0
Proof. Our hypothesis on implies that the associated lattice map ' N # N induces an isomorphism ' N * N such that is the inverse image of . 1 Thus, given , we see that is strongly convex. This is the support of the fan of 1 U , so that
H U 0 0 by Proposition 9.3.11. Then R 0 for p
p
for p
1
X
p
X
0 by Proposition 9.0.7.
The final assertion of the theorem is an easy consequence of Theorem 8.2.15 (Exercise 9.3.7).
Most versions of Theorem 9.3.12 in the literature assume that X is smooth. Other Vanishing Theorems. There are many more toric vanishing theorems. Both Fujino [42] and Mustat¸aˇ [111] state general vanishing theorems that imply versions of Theorems 9.3.1, 9.3.5, 9.3.7 and 9.3.10. Further vanishing results can be found in [44] and [121]. Exercises for §9.3.
0 for p 0. Now suppose that 0 0 where are acyclic. Prove H X 0 for all p r by induction on r.
9.3.1. A sheaf on a variety X is acyclic if H p X we have an exact sequence of sheaves 0
0
r
1
r
p
9.3.2. Use (9.3.4) and the previous exercise to prove Theorem 9.3.3 when p q.
Chapter 9. Sheaf Cohomology of Toric Varieties
420
D be the divisors from Example 9.3.8 and consider a *a D . (a) Show that D 2D 2D D D D D. (b) Show that D is -Cartier if and only a a a a . (c) Show that D is -Cartier and nef if and only a a a a 0.
9.3.3. Let D0 D a 0 D0
4
4
-Weil divisor
4
0
3
4
1
3
2
4
1
3
2
4
1
3
2
4
9.3.4. Complete the proof of Lemma 9.3.9.
(a) Pick 0 such D is Cartier and consider 6 X D ! " 6 X D from (9.3.5). Use this to prove that 6 X D is MCM. Hint: Replace N N with N N as
9.3.5. Let D be a -Cartier
-Weil divisor on a complete toric variety X .
in the proof of Theorem 9.2.10.
(b) Adapt the proof of Theorem 9.2.10 to show that
6 D H 0 X 6 K D (c) Prove H X 6 D H 0 X *6 K D . Hint: D D . (d) Prove Theorem 9.3.10 using part (c) and Theorem 9.3.5. 9.3.6. Consider the map r W V defined in the proof of Proposition 9.3.11. (a) Prove that these maps patch to give a retraction r W V . Hp X
p
n p
X
n p
X
X
X
X
X
Dm
Dm
to itself, r is homotopic to the identity. (b) Prove that when regraded as a map from W Then formulate and prove a similar result for r.
X X be a proper birational toric morphism. (a) Prove that N N is an isomorphism. 0 (b) Let be the fan in N consisting of the cones for )
9.3.7. Let
1
0
refinement of
. Prove that
is a
0.
(c) Complete the proof of Theorem 9.3.12.
. .
a D a -Weil divisor 9.3.8. Given a toric variety X , let D be a Weil divisor and E such that 0 a 1 for all and E is integral for some integer 0. Prove that there is an injection
*6 D D E for all p 0 This is a strengthened version of Theorem 0.1 from [111]. Hint: As suggested by [42], apply Lemma 9.3.4 to the -Weil divisor D % E. 6 D !
Hp X
X
Hp X
X
1
§9.4. Applications to Lattice Polytopes In this section we use vanishing theorems to study lattice polytopes. The Euler Characteristic of a Sheaf . Let be a coherent sheaf on a complete variety X. Its Euler characteristic is defined to be the alternating sum
! p 0
1
p
H p X
§9.4. Applications to Lattice Polytopes
421
Our hypotheses on X and guarantee that Hp X for all p, and p H X 0 for p X by [62, Thm. III.2.7]. Hence is a well-defined integer. The Euler characteristic satisfies
(9.4.1)
whenever we have an exact sequence 0 # # # # 0 of coherent sheaves. This follows from the long exact sequence in cohomology (Exercise 9.4.1). Given a line bundle
where as usual
on X, define
om
X
X
times
0
0
X
X
X
0
. A basic result [87] states that
is a polynomial in , called the Hilbert polynomial. Exercises 9.4.2 and 9.4.3 sketch a proof in the ample case. When D for a Cartier divisor D, the Hilbert
X
polynomial is written
D via the twisting convention introduced in §9.2.
n . When Example 9.4.1. We will compute 0, Example 9.2.4 implies p n that H n 0 for p 0, so that
(9.4.2)
n
H0
n
n
n
n
n n
The second equality follows from Proposition 4.3.3 since the polytope associated to D0 is n , where n is the standard n-simplex from Example 4.3.6. You will prove the last equality in Exercise 9.4.4. When 0, Example 9.2.8 implies (9.4.3)
n
n
1
!
Hn
n
n
!
1 n Int
!
1
n
n
1
n
n
where the last equality uses Exercise 9.4.4. Now consider the polynomial p x and observe that p
x n x n ! 1 n
n n
when
x 1
x
.
n for all . For We claim that p 0 is follows immediately from (9.4.2). When 0, note that
p (Exercise 9.4.4). Then p
! n
1
n
for
n
1
0 by (9.4.3).
Chapter 9. Sheaf Cohomology of Toric Varieties
422
Note also that when
0, (9.4.2) and (9.4.3) imply that p p !
n !
M
n
1 Int
n
M
We will see below that this is a special case of Ehrhart reciprocity. The Ehrhart Polynomial. Given a full dimensional lattice polytope P functions l P l P
P
M
Int
P
M , the
M
count lattice points in P or in its interior. For example, if (9.4.2) and (9.4.3) imply that l n n n and l n
0 is an integer, then 1 . n
n be a full dimensional Theorem 9.4.2 (Ehrhart Reciprocity). Let P M * lattice polytope. There there is a polynomial L x x such that if , then
L l P Furthermore, if
is positive, then L !
!
1 nl P
Proof. Let XP be the toric variety of P and DP the associated ample divisor. We will show that the Hilbert polynomial L
XP
DP
has the desired properties. The case when 0 is easy since D P is basepoint free by Proposition 6.1.4. Thus
XP
H 0 XP
DP
XP
DP
P M l P
by Demazure vanishing (Theorem 9.2.3) and Example 4.3.7. Now assume
XP !
DP
0. Then P DP P is full dimensional, hence
!
1
n
H n XP
XP !
DP
!
1 n Int P
l P
M
by Batyrev-Borisov vanishing (Theorem 9.2.7).
The polynomial L x in Theorem 9.4.2 is called the Ehrhart polynomial of P. A more elementary approach to the Ehrhart polynomial and Ehrhart Reciprocity Theorem is described in [10].
When P is very ample, the Ehrhart polynomial has a nice interpretation. The s 1 , where very ample divisor DP on XP gives a projective embedding i ' XP ( #
§9.4. Applications to Lattice Polytopes
423
x1 xs gives the homogeneous s P M . Its homogeneous ideal I XP coordinate ring XP x1 xs I XP
This graded ring has a Hilbert function
X
!$#
P
0
which is a polynomial (the Hilbert polynomial of XP Ch. 6, §4]).
s 1)
for
0 (see [27,
Proposition 9.4.3. If P is a very ample, then the Ehrhart polynomial of P equals the Hilbert polynomial of the toric variety XP under the projective embedding given by the very ample divisor DP . Proof. Consider the exact sequence 0 !$#
Since XP DP is the restriction of an exact sequence XP ! #+
0 !$#
XP !$#+
s 1
s 1
!$#
XP !$#
0
! #+ XP DP !$#
s 1
1 to XP , tensoring with
s 1
gives
0
In Exercise 9.4.5 you will show that the resulting long exact sequence begins 0! #
I XP
! #
x x
1
s
H 0 XP
!$#
XP
DP !$#
H1
s 1
XP
The H 1 term vanishes for 0 by Serre vanishing (Theorem 9.0.6). Hence for large, we get an isomorphism
X P
*
H 0 XP
XP
DP
This implies that the Hilbert polynomial of XP is the Ehrhart polynomial of P.
We can describe the degree and leading coefficient of the Ehrhart polynomial L x of P. For the leading coefficient, we use the normalized volume in M . Let n e1 en be a basis of M and consider the “unit cube” i 1 . i 1 i ei 0 Then the normalized volume is the usual n-dimensional Lebesgue measure, scaled so that the unit cube has volume 1.
Given a full dimensional lattice polytope P denoted Vol P and is computed by the limit (9.4.4)
Vol P
M , its normalized volume is
l P n
This is proved in many places, including [10, Lem. 3.19]. Since l P is given by the polynomial L , it follows easily that L x has degree n and leading coefficient Vol P (Exercise 9.4.6). Here is a classic application of these ideas.
Chapter 9. Sheaf Cohomology of Toric Varieties
424
2 be a lattice polygon with Ehrhart polynomial L x . The Example 9.4.4. Let P leading coefficient of L x is Area P , and the constant term is easy to compute, since L 0 l 0 P 1. Thus we can write
L x Area P x2
1 2 Bx
where B is yet to be determined. The reason for the
1 1 2
will soon become clear.
By Ehrhart reciprocity, we have (9.4.5)
A A!
1 2B 1 2B
1 L 1 l P 1 L ! 1
Solving for B gives B l P ! l P Thus the Ehrhart polynomial of P is
!
1 2 l P l P
P M , where P is the boundary of P.
L x Area P x2
1 2
P M x 1
Furthermore, solving the bottom equation of (9.4.5) for the area gives 1 2
Area P l P &
P M ! 1
This equation, called Pick’s formula, shows how to compute the area of a lattice polygon in terms of its lattice points.
The p-Ehrhart Polynomials. Following Materov [98], we use the sheaves XpP to generalize the Ehrhart polynomial when lattice polytope P is simple. Recall n is simple if every vertex that a full dimensional lattice polytope P M * is the intersection of exactly n facets. Hence each vertex gives n facet normals that generated the corresponding cone in the normal fan P . It follows that P is simplicial whenever P is simple. Hence the toric variety XP is simplicial.
We proved earlier that the Ehrhart polynomial L x of P satisfies L
DP
XP
where DP is the ample divisor coming from P. More generally, given an integer 0 p n, the Euler characteristic XpP DP is a polynomial function of . Then the p-Ehrhart polynomial L p x x is the unique polynomial that satisfies L p
p XP
DP
Note that L0 x is the ordinary Ehrhart polynomial L x . To state the properties of L p x , we will use the following notation: P i Q Q is an i-dimensional face of P fi hp
P i n
i p
!
1
i-dimensional faces of P
i p
i fi p
§9.4. Applications to Lattice Polytopes
425
The fi are the face numbers of P. Furthermore, if Q is a face of P, we define l Q
Q
l Q
M
Relint
Q
M
These invariants are related to the p-Ehrhart polynomials by the following result of Materov [98].
Theorem 9.4.5. Let XP be the toric variety of a full dimensional simple lattice n. polytope P M * (a) If
0 is an integer, then n
L p
i p
i p
(b) If 0 is an integer, then L p
p
1
!
i
i 0
(c) If 0 p
Q P i
n! i n! p
!
L p 0
l Q
Q P n i
1 n Ln
n, then
l Q
n, then
L p ! x (d) If 0 p
p
x
1 php
!
We will defer the proof for now. Theorem 9.4.5 has some nice consequences. First, setting x 0 in part (c) and using part (d), we see that 1 php
!
1 n ! 1
!
This proves that h p hn
(9.4.6) for 0
p
h
n p
n p
p
n. These are called the Dehn-Sommerville equations.
Also, if we write L p using part (b) and Ln gives the following formulas for 0: L p
p
!
1
i
i 0
L p ! For p 0 and
!
n
1 Ln
n! i n! p p
p
using part (a), then part (c)
l Q
i n ! p
Q P n i
!
1
n
n
i n p
Q P i Q P i
0, these equations reduce to Ehrhart reciprocity: L0 l P L0 ! ! 1 n l P
l Q
Chapter 9. Sheaf Cohomology of Toric Varieties
426
Thus Theorem 9.4.5 simultaneously generalizes the Dehn-Sommerville equations and Ehrhart reciprocity. The proof of the theorem will use the following lemma.
n be a full dimensional lattice polytope with toric Lemma 9.4.6. Let P M * variety XP and ample divisor XP . Given an integer 0 and m P M, let V P m be the subspace of M such that V P m m is the smallest affine subspace of M containing the minimal face of P containing m. Then
0
H XP
p XP
p
DP m
P
V P m
M
m
Proof. We adapt the stragegy used in the proof of Proposition 8.2.18. To simplify notation, set D DP . Using M M and tensoring (8.2.6) with XP D , we obtain the exact sequence
0! #
p XP
D !$#
p
M
p 1
D !$#
XP
M
D
D
Take global sections over XP and consider the graded piece for m M. Since m , we get the exact sequence H 0 XP XP D m P M H 0 XP
0 !$#
when m P
p XP
D
0 !$#
XP
p
M ! #
p 1
D
M. To determine H 0 XP 0 ! #+
with
m !$#
D ! #+
XP !
M H 0 XP
D
D
m
D m , tensor D ! #
XP ! #
0
D and take global sections to obtain
H 0 XP
XP
D ! D ! #
H 0 XP
XP
D
H 0 XP
D !$#
D ! #
0
where the exactness on right follows from H 1 XP XP D ! D 0 courtesy of Theorem 9.3.7. Comparing the polytopes of D ! D and D
a D shows that H 0 XP
D
D
m
m u
0
otherwise
a
!
Let F be the facet of P corresponding to . For m P, m u only if m F . Hence we get an exact sequence 0 !$#
H 0 XP
p XP
D
m !$#
p
M ! #
p
m F
p 1
!
a if and
M
where p is a sum of contraction maps iu defined in the discussion leading up to Theorem 8.2.16.
p Thus M is in the kernel of p if and only if iu 0 for all with p m F . We know from Exercise 8.2.8 that iu 0 if and only if . It follows that the kernel of p in (8.2.8) is the intersection
(9.4.7)
m F
p
p
m F
§9.4. Applications to Lattice Polytopes
F is the minimal face of P containing m, one sees easily that
m F
Since F
427
m F
Span m0 ! m M m0 F m F
Thus V P m
. This and (9.4.7) imply
p
p
V P m .
We are now to prove the properties of the p-Ehrhart polynomials.
Proof of Theorem 9.4.5. We begin with part (a). Lemma 9.4.6 implies that
when
0
H XP
p XP
DP
M
H XP
p XP
We also have H q XP L p
p XP
DP
i p
i 0 p XP
V P m p
M, note that
Q is the minimal face of P containing m
When this happens, we have Q of P are Q for Q P i , we obtain 0
P M
m
0. Given a face Q of P and m P m Relint Q
V P m . Since the i-dimensional faces
Q P i
DP 0 for q
0
DP
H XP
l Q
n
i p
i p
Q P i
l Q
0 by Theorem 9.3.1. Hence
p XP
DP
n
i p
i p
Q P i
l Q
which proves part (a). We next compute L p 0 . Let LQ x be the Ehrhart polynomial of Q. If 0 and Q P i , then LQ ! ! 1 i l Q by Ehrhart reciprocity. Then the above equation gives the polynomial identity L p x
(9.4.8)
n
i p
i p
since it holds for all
Q P i
!
1 i LQ ! x
0. Setting x 0 gives n
L p 0
!
1
i
i p
i p
Q P i
LQ 0
1 php
!
by the definition of h p , and part (d) follows. For part (c), we use Serre duality as given in Theorem 9.2.10, which implies
! D since D is Cartier for any . This easily implies D ! 1 ! D (Exercise 9.4.7). Thus L ! 1 L ! for , proving part (d).
p XP
Hq X
(9.4.9)
DP * H n
q
X
n p XP
P
P
p XP
p
n
P
n
n p
n p XP
P
Chapter 9. Sheaf Cohomology of Toric Varieties
428
Finally, for part (b), take 0 and consider L p
!
n
1 Ln
p !
!
1
n
n
p
i
n!
i n p
Q P i
!
1 i LQ
where the first equality uses part (d) and the second uses (9.4.8) with p replaced by n ! p. Since 0 implies LQ Q l Q , we obtain L p
n
!
1
n i
i n!
i n p
p
Q P i
p
l Q
!
1
j
n
j 0
!
j n! p
Q P i
l Q
where the last equality follows by setting j n ! i. This completes the proof.
Identities equivalent to Theorem 9.4.5 were discovered by McMullen in 1977 in a context that had nothing to do with toric varieties. See [10, Ch. 5] for details and references, including a non-toric version of Theorem 9.4.5 in [10, Ex. 5.8– 5.10]. Examples. We first give a classic example of the Dehn-Sommerville equations. Example 9.4.7. Let P be a simple 3-dimensional lattice polytope in Sommerville equations can be written h3 h0 so f3
f0 ! f1
h2 h1 so f2 ! 3 f3
3.
The Dehn-
f2 ! f3
f1 ! 2 f2 3 f3
Since f3 1 (P is the only 3-dimensional face of itself), we obtain f0 ! f1
f2 2 f1 3 f2 ! 6
The first equation Euler’s celebrated formula V ! E F 2, which holds for any 3-dimensional polytope. The second seems more mysterious, but when combined with the first reduces to 2 f 1 3 f0 . This holds because is P is simple—every vertex meets three edges and every edge meets two vertices.
There is a version of the Dehn-Sommerville equations for the dual polytope P
N , which is simplicial since P is simple (Exercise 9.4.8). More on the Dehn-Sommerville equations and toric varieties can be found in [45, Sec. 5.6]. We also recommend [10, Ch. 5] and [152, Sec. 8.3]. We next give an application of Theorem 9.4.5. Example 9.4.8. The standard n-simplex Given 0, note that
Q
because
i n
l
n
n
has
n
1 n! i
n
as its associated toric variety. !
1 i
§9.4. Applications to Lattice Polytopes
n 1 i 1
An n-simplex has
n 1 n i
429
faces of dimension i.
Each Q n i is lattice isomorphic to the standard i-simplex and hence has 1 i interior lattice points by (9.4.3).
n
Then part (a) of Theorem 9.4.5 gives the formula n
L p Since
i 1 p i
L p
i p
1 p 1 p i p
!
1 p
i p
n
1 n! i
!
1 i
, this becomes
1 n! i
i 0
!
p! 1 i! p
!
1 p
n
!
p
where the last equality uses the Vandermonde identity discussed in Exercise 9.4.9. Hence n ! p ! 1 H 0 n p n p This formula was first proved by Bott [13] in 1957.
Cohomology of p-Forms. Given a simple polytope P as above, the sheaf Euler characteristic
(9.4.10)
p XP
L p 0
by Theorem 9.4.5. The factor of ! 1
p
p XP
has
1 php
!
is explained as follows.
Theorem 9.4.9. Let XP be the toric variety of a full dimensional simple lattice n . Then polytope in P M *
H q XP
q q
hp 0
p XP
p p
Proof. The toric variety XP is simplicial, so that Theorem 9.3.2 applies to XP . This shows that H q XP XpP 0 for q p. It follows that
p XP
!
p
1
H p XP
The theorem follows by comparing this with (9.4.10). A different proof of In Chapter 12, the sum
H p XP
hp
n
i p
p XP
!
1
p XP
h p will be sketched in Exercise 9.4.10.
i p
i fi p
will arise naturally when we compute the singular cohomology of a simplicial toric variety. More precisely, we will show that
hp
H 2p XP
Chapter 9. Sheaf Cohomology of Toric Varieties
430
The link between this and Theorem 9.4.9 is the Hodge decomposition
H 2p XP *
H i XP
i j 2p
j XP
When XP is smooth, this is a classical fact—see, for example, [56, p. 116]. In the simplicial case, this is covered in [144]. Exercises for §9.4. 9.4.1. Prove (9.4.1). 9.4.2. Here are some cases where it is easy to show that Euler charactersitics give Hilbert polynomials. (a) Let F be a finitely generated graded module over the polynomial ring S x0 xn . By [27, Ch. 6, Thm. (3.8)], there is an exact sequence 0
!
Fr
F1
F0
F
*
0
where each Fi is a finite direct sum of modules for the form S a . Now let coherent sheaf on n . Prove that there is an exact sequence of sheaves
0
r
1
0
0
such that each i is a finite direct sum of sheaves of the form Proposition 6.0.9.
1 6#
(b) Use part (a) together with (9.4.1) and Example 9.4.1 to show that n n . nomial in , where
6 # a . n
'
be a
Hint: Use is a poly-
'
(c) Let be a coherent sheaf on a complete variety X and let D be a very ample divisor D is a polynomial in on X. Use part (b) and Exercise 9.0.5 to show that .
9.4.3. Let be a coherent sheaf on a projective variety X, and let D be an ample divisor on X. Thus there is k0 0 such that kD is very ample for all k k0 . (a) Let a k be integers with k k0 . Use Exercise 9.4.2 to show that there is a polynomial pa k x x such that pa k a k D for all . (b) Show that the polynomials pa k x k and pa m x m are equal when k m k0 . Conclude that there is a polynomial pa x such that pa a D for all k0 . (c) Show that the polynomials pa x a and pb x b are equal for any a b and conclude that there is a polynomial p x x such that p D for all .
'
, ' ' ,
%
9.4.4. In this exercise will always denote an integer. n n (a) Prove that n n give n for nonnegative. Hint: Lattice points in xn by Exercise 4.3.6. Here is a combinatorial proof. monomials of degree in x0 Begin with a list of 1’s of length n. In n of the positions, convert the 1 to a 0. This divides the remaining 1’s into n 1 groups. The number of elements in each group gives the exponents of x0 xn . See also [26, Ex. 13 of Ch. 9, §3]. 1 n (b) Prove that Int n n for positive. Hint: Show that shifting the interior lattice points by 1 1 gives the lattice points in n 1 n. x n x n 1 x 1 n satisfies p 1 n n 1 for (c) Prove that p x negative.
*
* 0
*, 0 0
§9.4. Applications to Lattice Polytopes
431
0. After tensoring with 6 # 6$ # 6 6 0 whose long exact H ! 6$# H ! 6 H ! 0 H ! We know from Example 4.3.6 that H ! 6 # S , where S % x * x with the standard grading. Also recall that as a sheaf on ! , 6 stands for i " 6 , where i X ! ! is the inclusion map. It follows that H ! 6 H X 6 . I X , where I X S is the ideal of X. (a) Show that H ! (b) Show that the Hilbert polynomial of the coordinate ring X S , I X is the Euler characteristic ' 6 . 9.4.6. Let p x be a polynomial such that p , exists and is nonzero. (a) Prove that n p x and that the above limit is the leading coefficient of p x . !
6#
n 9.4.5. A projective variety X gives 0 n , we have an exact sequence 0 X sequence in cohomology begins 0
n
0
X
n
n
X
X
0
n
0
X
n
n
n
n
n
0
n
n
X
0
n
0
1
X
n
X
n
X
0
X
X
X
X
n
(b) Use part (a) to prove the assertion made following (9.4.4) concerning the degree and leading coefficient of the Ehrhart polynomial.
3 1 ' 3 0 ' D p XP
9.4.7. Show that (9.4.9) to prove that 9.4.8. Let P
M
n
i
n p XP
n
P
DP .
be a full dimensional simplicial lattice polytope and define hsimp p
n
1 0
i p
i p
p
fn
0 0 i 1
Rescaling and translating P does not affect the face numbers f i . So we may that assume the origin is an interior point of P. Let P N be the dual polytope of P. Use Exercise 2.3.4 and Proposition 2.3.6 to prove the following: (a) P is simple.
0 0
(b) The face numbers f iP of P and fiP are related by f nP simp
(c) h p
h0
simp n p.
i 1
fiP .
% % % %0 0 0 0 % 0 9.4.10. For a full dimensional simple polytope P M , Theorem 8.2.19 gives 3 0 K p K p K p 0 where K p / 0 M & 6 and V X is the 3 closure of the orbit corresponding to + 3 . You will use this to compute H X . (a) Show that ' 1 ' K p . (b) Use Theorem 9.2.3 to show that ' K p H X K0 p . (c) Use Proposition 2.3.6 to show that H X K p 0 f 0 . Hint: M has rank n j for j . 3 (d) Use Theorem 9.3.2 to conclude that H X h 0 . Hint: Set j n i.
9.4.9. The goal of this exercise is to prove the identity used in Example 9.4.8. a b (a) Prove the Vandermonde identity, which states that i j k ai bj . k for a b Hint: 1 x a 1 x b 1 x a b. n p (b) Use part (a) to show that i 0 nn 1i i p p 1 , as claimed in Example 9.4.8. n
p XP
i
0
1
P
p i
P
P
p
P
i
P
V
p
P
p XP
p j 0
P
j
j
j
P
p XP
P
0
P 0
j
P
P
n j n p
P
P
p
P
p XP
j
n p
P
n
j
Chapter 9. Sheaf Cohomology of Toric Varieties
432
9.4.11. When a polytope P has rational but not integral coordinates, the counting function 2 l P is almost a polynomial. Here you will study P Conv 0 e1 12 e2 . (a) Given
, prove that
1 2 4 1 2 4
l P
1
even odd
3 4
This is an example of a quasipolynomial. See [10, Sec. 3.7] for a discussion of the Ehrhart quasipolynomial of a rational polytope.
!
!
(b) The weighted projective plane 1 1 2 is given by the fan in 2 with minimal generators u0 e1 2e2 u1 e1 u2 e2 . Show that X2P 1 1 2 with D2P 2D0 , where D0 is the divisor corresponding to u0 . Also show that
' 6 #
1 1 2
D l P 0
The Euler characteristic is not a polynomial in . The reason is that D0 is not Cartier.
' 3
n 9.4.12. Let P M be a full dimensional lattice poltyope, not necessarily simple. p for all Let L p x be the unique polynomial satisfying L p . XP DP (a) Prove that if 0 in an integer, then
Lp
i
n
i p
p
Q P i
l
" Q
Hint: Look at the hypotheses of Theorem 9.3.1 and Lemma 9.4.6.
1 L . Hint: Serre duality. (c) Prove that L 0 1 h . Hint: Follow the proof of Theorem 9.4.5. n
(b) Prove that L0
p
p
n
p
This shows that some parts of Theorem 9.4.5 hold for arbitrary lattice polytopes. In the next exercise you will see that other parts can fail.
9.4.13. This exercise will show how things can go wrong for a non-simple polytope. Let 3 P Conv e1 e2 e3 . Note that P is not simple, so that XP is not simplicial. (a) Show that h1 h2 .
(b) Conclude that for this polytope, (9.4.9) cannot hold for all p q . So the version of Serre duality stated in part (b) of Theorem 9.2.10 can fail for non-simplicial toric varieties. Hint: The Dehn-Sommerville equations follow from Theorem 9.4.5.
§9.5. Local Cohomology and the Total Coordinate Ring In this section, we use local cohomology and ext to compute the cohomology of a coherent sheaf on a toric variety X . Our treatment is based on [39].
We first review the basics of local cohomology. Local Cohomology. Let R be a finitely generated -algebra, I an ideal and R-module. First define
I
a
I k a 0 for some k
an
§9.5. Local Cohomology and the Total Coordinate Ring
433
is left This is the I-torsion submodule of . One checks easily that # I exact. Hence, just as we did for global sections of sheaves in §9.0, we get the derived functors # HIi such that
HI0
I
0#
HI0
#
.
A short exact sequence 0 #
HI0
#
#
#
HI0 #
#
HI1
0 gives a long exact sequence HI1
#
HI1 #
#
ˇ ˇ The Local Cech Complex. As with sheaf cohomology, there is a Cech complex for local cohomology. The sheaf case used an affine open cover; here we use generators of the ideal. More precisely, if I f 1 f , let 1 be the index set and as in §9.0, let p denote the set of all p 1 -tuples i0 i p of elements of I satisfying i0 i p . Also set f f 1 f
Given an R-module
, define
Cp f
i0
1
ip
f i0
p
fi p
1
1
where fi0 fi p 1 is the localization of at f i0 fi p 1 . An element of C p f a function that assigns an element of fi0 fi p 1 to each i0 i p 1 Then define a differential
d p ' Cp f
C p
!$#
1
f
is 1.
p
by d p i0 i p
p
!
k
1
i0 ik i p
k 0
where we regard i0 ik i p as an element of
f i0
f ik
fi p
#
f i0
f i0
fi p
fi p
via the map
given by localization at f ik . Similar to Exercise 9.0.2, we have d p d p
1
0.
Definition 9.5.1. Given an R-module and generators f f 1 f of I, the ˇ local Cech complex is d0 d1 d2 C f ' 0 ! # C0 f ! # C1 f !$# C2 f !$#
ˇ ˇ Just as the Cech complex computes sheaf cohomology, the local Cech complex computes local cohomology. See [80, Thm. 7.13] for a proof of the following.
Theorem 9.5.2. Given an R-module are natural isomorphisms HIp for all p 0.
*
and generators f H p C f
f 1 f
of I, there
Chapter 9. Sheaf Cohomology of Toric Varieties
434
x y
Example 9.5.3. For I
0! #
ˇ S x y , the local Cech complex is
S! #
Sy ! #
Sx
Sxy ! #
0
Theorem 9.5.2 implies that HI2 S is the cokernel of Sx Sy # Sxy . Consider x 1 y 1 x 1 y 1 Span x a y b a b 0 with S-module structure given by
y x y i a j b 0 otherwise Then the map S # x y x y defined by f x y xy # gives an exact sequence (9.5.1) S S ! # S !$# x y x y $! # 0 (Exercise 9.5.1). Thus H S * x y x y . xi y j x
a
1
xy
x
a i
b
1
1
y
b j
1
1
xy
1
2 I
k
1
1
1
1
f x y x
y k
k
1
1
It follows then x and y have degree 1, then HI2 S is graded with
and
HI2 S HI2
S
HI2 S
! 1 for
HI2 S
6
HI2 S
4
2
0
0 (Exercise 9.5.1).
The number ! 1 just computed is also the dimension of H 1 1 Example 9.4.1). This is no accident, since we will prove below that HI2 S
H1
a *
1
1
!
(see
a
1
for all a . Relation with Ext. For us, an especially useful aspect of local cohomology is its relation to ext. In §9.0 we introduced ext in the context of sheaves. There is also a module version, where for a fixed R-module , the derived functors of HomR are denoted ExtRp . They have the expected properties, # and more importantly, are easy to compute by computer algebra systems such as Macaulay 2.
To see the relation between ext and local cohomology, we begin with the observation that an R-module homomorphism ' R I # is determined by 1 , and choosing any a gives a homomorphism, provided that I a 0. It follows that HomR R I k consists of those elements of annihilated by I k . Comparing this to the definition of I M , we obtain
M
I
HomR R I k M
k
Since direct limit is an exact functor (Exercise 9.5.2), it follows without difficultly that the derived functors are related the same way (see [80, Thm. 7.8] for a proof). Theorem 9.5.4. HIp
k
ExtRp R I k
.
§9.5. Local Cohomology and the Total Coordinate Ring
435
f f , let
We will need a variant of this result. For an ideal I
f
I k
k 1
1
fk
By [80, Rem. 7.9], we have the following result. Theorem 9.5.5. HIp
ExtiR R I k
.
k
Here is a simple example.
Example 9.5.6. Let I x y S x y . We know by Example 9.5.3 that HI2 S 5 has dimension 4. To compute this using ext, let A S I k S x k y k . Since A and S are graded S-modules, the ext group Ext 2S A S is also graded.
Let us compute the graded piece Ext2S A S 5 for various values of k. In Macaulay 2, we set the ring S x y by the command
S = QQ[x,y] and then A for k 3 is given by A = coker matrix{{xˆ3,yˆ3}} The dimension of Ext2S A S
5
is computed by the command
hilbertFunction(-5,Extˆ2(A,S))
which gives the answer 2. Since HI2 S 5 4, we see that k 3 is not big enough. If we switch to k 4, then the answer stabilizes at the number 4.
In this example, Ext2S S I k S 5 HI2 S prove below that for any degree a , we have Ext2S S I k S
a
for k sufficiently large. We will
5
HI2 S
a
provided that k is sufficiently large. The problem is that Ext 2S S I k S is finitely generated over S but HI2 S is not (Exercise 9.5.4). So we can’t use the same k for all degrees a. Explicit bounds on k in terms of a are needed in order to turn the method of Example 9.5.6 into an algorithm. This concludes our overview of local cohomology; for more, see [37, App. 4] or [80]. Our next task to is apply local cohomology to toric varieties.
The Toric Case. The total coordinate ring S x 1 of a toric variety a X is graded by the class group Cl X , where a monomial x has degree
a D $ Cl X . We also have the irrelevant ideal
x
B
max
x
x 1
introduced in Chapter 5. We assume that X has no torus factors.
We proved in Chapter 5 that a finitely generated graded S-module divisor class Cl X give the following:
and a
Chapter 9. Sheaf Cohomology of Toric Varieties
436
The coherent sheaf way (Proposition 5.3.9).
The shifted module
on X . Every coherent sheaf on X arises in this
, where
for Cl X .
The sheaf associated to S is denoted X , and Proposition 5.3.7 tells us that X * X D when D is a Weil divisor with divisor class . More generally, if " , then will denote the sheaf associated to . When is the class of a Cartier divisor, one can prove that
(9.5.2)
*
X
X
(Exercise 9.5.3). Our first main result is that the local cohomology for the irrelevant ideal B computes the sheaf cohomology of all twists a coherent sheaf on X . Theorem 9.5.7. Let coherent sheaf +
be a finitely generated graded S-module with associated on X . If p 2, then HBp
*
Cl X
Hp
X
Furthermore, we have an exact sequence HB0
0! #
! #
! #
Cl X
1
H0 X
HB1
! #
0
! #
Proof. Let max 1 , so that B is generated by the monomials fi ˇ x i , i 1 . Then the terms of the local Cech complex C f are direct sums of localizations at products of various f i ’s. These localizations are Cl X -graded since is graded and the f i are monomials. The differentials also preserve the grading, which by Theorem 9.5.2 implies that HBp has a natural Cl X -grading such that for all Cl X , we have HBp H p C f
Cp
To compute
Cp f
We will relate C p f
f
i0
1
ip
p
f i0
fi p
1
i0
1
fi0 fi p
1
x
i0
x ip
given in (9.1.1).
1
1
p
f i0
1
i 0 i p 1
ip
since shifting commutes with localization. For (9.5.3)
, first note that
ˇ to the Cech complex for
a
x
fi p
p 1,
1
0
note that
1
where a i j i j 0. If we set gives i0 i p 1 , then x
the same localization as (9.5.3). Since is the sheaf associated to , Proposition 5.3.3 implies that
f i0
fi p
1
0
x 0*
U
§9.5. Local Cohomology and the Total Coordinate Ring
437
ˇ Comparing this with the Cech complex (9.1.1) for the open cover we obtain
Cp f
p
U
Cp
*
1
U i
,
i
1
ˇ This isomorphism is compatible with the differentials. It follows that the local Cech ˇ complex C f is obtained from the Cech complex C by deleting the first term and shifting the remaining terms. When p 2, this implies
HBp
Hp
*
X 1
and with a little work (Exercise 9.5.5) we also get an exact sequence 0 !$#
HB0
! #
H0 X
! #
!$#
HB1
! #
0
Theorems 9.5.5 and 9.5.7 imply that when p 1,
Hp X
*
ExtSp
1
S B k
k
when and Cl X . We can compute ExtSp 1 S B k by the methods of Example 9.5.6. The problem is the direct limit. We tackle this next.
Stabilization of Ext. In the toric case, ext has a nice relation to local cohomology.
Lemma 9.5.8. Let M be a finitely generated Cl X -graded S-module and fix Cl X . If is a complete fan, then there exists k 0 such that for all k k0 , the natural map S B k 1 # S B k induces an isomorphism
ExtiS S B k
k 1
ExtiS S B
*
In particular, k k0 implies
ExtiS S B k
*
HBi
Proof. We give a proof only for S following Mustat¸aˇ [110, Thm. 1.1]. For the general case, one replaces with a free resolution and uses a spectral sequence argument. See [39, Prop. 4.1] for the details.
Earlier we described ext using the derived functors of # Hom R for fixed . Ext also comes from the derived functors of # Hom R for fixed , where one uses a projective resolution of instead of an injective resolution of (see, for example, [37, A3.11]). In particular, we can compute Ext iS S B k S using any free resolution of S B k . An especially nice resolution is given by the Taylor resolution, which is described as follows.
We begin with S B . The minimal generators of B are f i x i for i . Let Fs be a free S module with basis eI for all I with I s. To define the
Chapter 9. Sheaf Cohomology of Toric Varieties
438
with I
differential ds ' Fs # Fs 1 , take I J of I as i1 is . Then define
0
cIJ
r ! 1 fI
J
1 s and list the elements
J I I J ir
fJ
where fI lcm fi i I and similarly for f J . Then ds eI
J cIJ eJ
S B . One can prove that
Since F0 S, we have an obvious map F0 # 0! #
F !$#
! #
F1 ! #
S B ! #
F0 !$#
0
is exact (see [37, Ex. 17.11]). This is the Taylor resolution of S B . This construction applies to any monomial ideal. In particular, it works for B k f1k f k . Here, the Taylor resolution has the same modules Fsk Fs , and since the f i are square-free, we have f Ik lcm fik i I . Thus the differentials dsk in the Taylor resolution of S B k are given by
dsk eI
k J cIJ eJ
cIJ as above
We now compare the Taylor resolutions of S B k and S B k 1 . The maps s ' Fsk 1 # Fsk defined by s eI fI eI induces a commutative diagram Fsk ds
k 1
s
1
/ Fk s ds
k
Fsk 11
s 1
/ Fk s 1
These maps are compatible with the natural surjection S B It follows that the map H i HomS F k
(9.5.4) induced by the
s
1
k 1 #
S B k .
H i HomS F k S
S #
can be identified with the map ExtiS S B k S #
ExtiS S B
k 1
S
The next key idea involves the use of a finer grading than the Cl X -grading used so far. Recall that this grading is induced by the map (9.5.5)
a
!#
1
Cl X
where x a
x has degree a D . The ring S also has a 1 -grading where x a a. Since B k is a monomial ideal, the quotient ring S B k 1 -graded. Then Exti S B k S inherits a natural 1 -grading. is S Now grade the Taylor resolution F k d k by setting eI k fI . This guarantees two things:
§9.5. Local Cohomology and the Total Coordinate Ring
439
The differential dsk has degree 0, so the isomorphism Ext iS S B k S * 1 -graded. H i HomS F k S is
The map
s
-graded.
has degree 0, so (9.5.4) is
1
It follows that Fsk HomS Fsk S has dual basis eI with 1 , define Given a b
eI !
k
fI
.
a b if and only if a b for all 1 We claim that if a ! k ! k , then (9.5.6)
Fsk a ! #
s a '
Fsk
1 a
is an isomorphism for all s.
1
To prove this, first observe that s eI $ fI eI . It follows that s a is injective. Now take x b eI Fsk 1 a . Then b ! k 1 f I a, so b
k 1
a k f I & ! k ! k Since fI is square-free, k 1 f I is a vector with whose th entry is k 1 if x divides f I and 0 otherwise. Then the above inequality implies that f I divides x b , so that x b eI is in the image of s a . It follows that (9.5.6) is an isomorphism.
f I
1 -grading which is compatible its The local cohomology HBp S has a Cl X -grading via (9.5.5). This follows easily from Theorem 9.5.5 and the Taylor resolution. If Cl X and p 2, then
HBp
S
*
Hp
1
X
X
by Theorem 9.5.7. The right-hand side is finite dimensional since is complete. 1 -graded This implies that when we decompose HBp S into its nonzero pieces HBp S a , only finitely many can appear in HBp S . For these finitely many a’s, pick k0 such that they all satisfy a ! k0 ! k0 . This k0 has the required properties. The argument for p 0 1 is covered in Exercise 9.5.6.
Theorem 9.5.7 and Lemma 9.5.8 imply that for p 1 and k 0, H p X X * HBp 1 S * ExtSp 1 S B k S
Notice how these isomorphisms generalize Examples 9.5.3 and 9.5.6. Our final task is give an explicit method for deciding when k is big enough.
Bounds. For a complete fan , graded S-module , and divisor class Cl X , the paper [39] gives an explict value for the number k 0 appearing in Lemma 9.5.8. We will state the results of [39] without proof, though the following example suggests some of the ideas involved.
2 with minimal generators Example 9.5.9. Let be a complete fan in M * u1 ur arranged counterclockwise around the origin. Suppose we have a divisor 1 D i ai Di on X and m M such that H X X D m 0. It follows from
Chapter 9. Sheaf Cohomology of Toric Varieties
440
Proposition 9.1.5 that the sign pattern of m (determined by m u i ai 0 or 0) has at least two strings of consecutive ! ’s. The set I i r m u i ai 0 records the locations of the ! ’s in the sign pattern of m.
Since H 1 X X D m 0 is finite dimensional, there are only finitely many m’s with the same sign pattern. In other words, the inequalities
m u m u i
ai 0 i I
i
ai 0 i
r )
I
have only finitely many integer solutions, which means that the region of the plane defined by these inequalities is bounded. You can see and example of this in Figure 4 from Example 9.1.7. Since this region determined by the u i and ai , it should be possible to bound the size of the m’s that appear in terms of the u i and ai .
m u a m u a . Then it H X D * H S Thus bounding the m’s with H X D 0 is equivalent to bounding the b’s with H S 0. And once we find this bound, we know how to pick k so that b ! k ! k for all such b’s. The proof of Lemma 9.5.8 shows that this k works. To relate this to Lemma 9.5.8, let b is easy to see that 1
X
1
2 B
1
m
2 B
X
1
r
r
b
m
0
b
0
0
0
In the general case, we proceed as follows. Pick a basis e 1 en of M and let A be the matrix whose rows give the coefficients of the u ’s with respect to the chosen basis. Then A is an r n matrix, where r 1 . For this matrix, define
nonzero n n minors of A Q entries of A Q n ! 1 n ! 1 minors of A qn
(9.5.7)
1
n 1
Here is [39, Cor. 3.3]. Theorem 9.5.10. Given a complete fan
a D $ Cl X , we have
ExtiS S B k S
for all k k0 , where k0 n2
in NR * HBi *
S
a Q Qq 1
n
n 1
and a divisor class
n
For a finitely generated graded S-module , the formula for k 0 involves the minimal free resolution of . Write the resolution as
!$#
F2 ! #
F1 ! #
F0 !$#
! #
0
§9.5. Local Cohomology and the Total Coordinate Ring
and write
Fj
Cl X
S !
Finally, for each with r j 0, write 4.1] gives the following bound.
rj
441
a D
Cl X . Then [39, Prop.
Theorem 9.5.11. Let and be as in Theorem 9.5.10, and let generated graded S-module. Then ExtiS S B k M for all k k0 , where k0 n2
M
HBi *
j r j 0 a !
be a finitely
Q1 Qn qn
a
1
The Cotangent Bundle. We end this section by using our methods to calculate the cohomology of the cotangent bundle of a smooth complete toric variety X . Our first task is to find a graded S-module with 1X * .
In (8.1.7) we constructed an exact sequence
M S #
1 S !$#
0! #
f #
m u f , and in CorolS x
where M S #
S x is defined by m lary 8.1.5 we showed that 1X is the sheaf of 1S . Our second task is to find a description of 1S that is easier to implement on a computer.
Pick bases of M and Pic X and order the elements of 1 as 1 r . Then the basic exact sequence
!#
M ! #
0! #
1
Pic X ! #
can be written (9.5.8)
0 !$#
!$# A
n
r
!$# B
r n
!$#
0
0
where A is an r n matrix whose ith row consists of the coefficients of u i in the basis of M. This is the same matrix A that appears in Theorem 9.5.10. The r ! n r matrix B is called the Gale dual of A. Lemma 9.5.12. The r ! n
r matrix
B
x 1 0 .. .
0 x 2 .. .
0
'
0
r i 1 S !
induces a graded homomorphism that 1X is the sheaf associated to & .
..
.
0 0 .. .
x
r
x i #
Sr
n
of degree 0 such
Chapter 9. Sheaf Cohomology of Toric Varieties
442
Proof. Let x be the r r diagonal matrix of variables appearing in the statement of the lemma. Similar to the proof of Theorem 8.1.6, we have a commutative diagram 0 0 0
/
/
1 S
/
/
r i 1 S !
0
0
x i
r n
x
S /
r
S
/
A
S
/
r i 1 S
r i 1 S
i
x
i
/0
S
0
x
/0
B
r n
n
/
0
/0
0
0
By the diagram chase from the proof of Theorem 8.1.6, the dotted arrows are exact, r r n S is and by commutivity, the dotted arrow from x i to i 1 S ! given by B x.
Example 9.5.13. For
2,
the matrices A and B of (9.5.8) are given by
1 0 0 1 ! 1 ! 1
A To create the module
1 S
B
1 1 1
in Macaulay 2, we use the commands
S = QQ[x,y,z] OM = ker matrix{{x,y,z}} We also need a free resolution of
1, S
which can be computed by Macaulay 2:
F = res OM F.dd The first command computes the resolution; the second prints out the differentials. The result is
x y z
0 !$#
S ! 3 !! ! #
3
S ! 2
! #
1 S !$#
0
The numbers from (9.5.7) are q2 Q1 1, so that for a , the formula for k0 from Theorem 9.5.11 is k0 4 For a in the range ! 4 Hp
a !
2 a ! 3
a 4, we can use k0 28. This implies that for p 1, 2
1
2
a * ExtSp S x28 y28 z28
1 S a
§9.5. Local Cohomology and the Total Coordinate Ring
Table 3.
Hp
2
443
1 2 a for 4 a 4
a) p 0 ! 4 0 0 ! 3 ! 2 0 0 ! 1 0 0 1 0 2 3 3 8 4 15
0 0 0 15 0 8 0 3 0 0 1 0 0 0 0 0 0 0 0 0
when ! 4 a 4. This can be computed by the methods of Example 9.5.6. We can also compute H 0 2 1 2 a directly from 1S (Exercise 9.5.7). The results are shown in Table 3.
In Exercise 9.5.8 you will calculate this table by hand. Notice also that the symmetry of the table comes from Serre duality.
Here is a slightly more complicated example. Example 9.5.14. Let us compute the first cohomology of various twists of the cotangent sheaf of the Hirzebruch surface 2 . We will use the notation of Example 9.3.6. Using the basis of Pic 2 given by the classes of D3 D4 , the matrices of (9.5.8) are
!
A
1 0 1 0 !
1
2 1 0 1
B
!
2 1 0 1 0 1
0
The total coordinate ring is S x1 x2 x3 x4 , where the degrees of the variables are given by the columns of B, i.e.,
x1
Then we create
1 0 1 S
x2
!
2 1
x3
1 0
x4
0 1
within Macaulay 2 by the commands
S = QQ[x1,x2,x3,x4,Degrees=>{{1,0},{-2,1},{1,0},{0,1}}] OM = ker matrix{{x1,-2*x2,x3,0},{0,x2,0,x4}}
Be sure you understand how this uses the matrix
from Lemma 9.5.12.
Using Macaulay 2, we also compute the free resolution 0 !$#
S 0 ! 2 ! #
S 1 ! 2
2
S ! 2 0 ! #
1 S !$#
0
Chapter 9. Sheaf Cohomology of Toric Varieties
444
The numbers from (9.5.7) are q2 1 and Q1 2, so that for Pic 2 , the formula for k0 from Theorem 9.5.11 is k0 16
1
a!
aD
2 b a 1 b ! 2 a b ! 2
3
bD4 in
We can use our methods to calculate the cohomology groups of 1 2 a b 1 aD3 bD4 for all a b. Table 4 records some computations of H . There are 2 Table 4.
H1
2
1
a b
2
for
2 a b 2
a ) b –2 –1 0 1 2 –2 0 0 3 4 3 –1 0 0 2 2 2 0 1 1 2 1 1 1 2 2 2 0 0 2 3 4 3 0 0
many things we can see from this table, including:
Theorem 9.4.9 implies that H 1 2 1 2 h1 f1 ! 2 f2 4 ! 2 1 2 since 2 comes from a quadrilateral (see Examples 2.3.15 and 3.1.16). This explains the 2 when a b 0 0 .
Serre duality implies that H 1 2 explains the symmetry in the table.
1 2
a b * H 1
2
1
2
!
a ! b . This
Bott-Danilov-Steenbrink vanishing implies that H 1 2 1 2 a b a b 0. This explains the 0s in the lower right corner of the table.
0 for
You will verify the details of this example in Exercise 9.5.9. One suggestion is that after computing an ext group E Ext Sp A 1S , use the command rank source basis({a,b},E) to compute
E
rather than
ab
hilbertFunction({a,b},E) since the former is faster than the latter and gives the same answer.
An algorithmic approach to these calculations is described in [39, Sec. 5] Exercises for §9.5. 9.5.1. Prove the exactness of (9.5.1).
0 A A
9.5.2. Suppose that Ai i , Ai i , Ai i are directed systems, and that for each i, there exist di and i commuting with the ’s, such that
di
i
i
i
Ai
0
Appendix: Introduction to Spectral Sequences
445
Prove the exactness of the sequence
A A A 0 9.5.3. Prove (9.5.2) when ) Cl X is the class of a Cartier divisor. Hint: Look carefully at Proposition 5.3.3 and remember that the restriction of 6 5 to U is trivial. 9.5.4. For I x y S x y , prove that H S is not finitely generated as an S-module. Hint: Use Example 9.5.3 to show that H S 0 for all a . 2. 0
i
i
i
i
i
i
X
2 I
2 I
a
9.5.5. Complete the proof of Theorem 9.5.7 by proving that there is an exact sequence 0
for all
HB0
Cl X .
9.5.6. Prove that HB0 S
5
H0 X
HB1 S
0
0. Hint: Proposition 5.3.7.
9.5.7. Here are some details to check from Example 9.5.13 (a) Use the exact sequence from Lemma 9.5.12 to show that for all Pic X when X is smooth and complete.
HB1
2 1 S
5
H0 X
1 X
(b) Verify first column of Table 3. (c) Show that the first column agrees with the Bott formula from Example 9.4.8. 9.5.8. We can check Table 3 of Example 9.5.13 with a “barehanded” approach. Observe that we have exact sequences involving 1 2:
# 0 6 # 3 1
0
#
2
6 # 1 6 # 0 6 # 2 # 0 3
2
2
3
2
1
2
2
The first comes from Lemma 9.5.12, and the second comes by sheafifying the free resolution of 1S computed in Example 9.5.13. Twist these sequences by a and consider the resulting long exact sequences in cohomology. Using the vanishing theorems we know, conclude that the nonzero values for H p 2 1 2 a are:
H0 H1 H
2
! #
1
2
! #
1
2
1
2
! #
! # a a 1 if a 2 a 1 if a 1 a a 1 if a . 2
2
2 2
2
2
These formulas give the numbers in Table 3. 9.5.9. This exercise concerns Example 9.5.14. (a) Use Macaulay 2 to compute the free resolution of
1 S.
(b) Show how the formula given for k0 follows from Theorem 9.5.11. (c) Compute the entries in Table 4.
Appendix: Introduction to Spectral Sequences Spectral sequences are used several times in this chapter. Here we give the necessary background, though our discussion is far from complete. We refer the reader to [56] or [150] for a more complete treatment of spectral sequences.
Chapter 9. Sheaf Cohomology of Toric Varieties
446
Definition and Intuition. At first glance, spectral sequences may seem rather complicated. Fortunately, most of the ones encountered in this book will be fairly simple to understand. Definition 9.A.1. A spectral sequence over consists of vector spaces Erp q and linear maps Erp r q r 1 drp q Erp q
% 0 % d
such that (a) drp (b)
%
rq r 1
Erp q1
pq r
0 for all p q r.
d , i.e., E % 0 % im d 0 % 0 E 0 % 0 Erp q
is the cohomology of
% d E
Erp q1
pq r
% 0 %
pq r
pq r
p rq r 1 r
p rq r 1 r
p rq r 1 r
Erp q
We will always work with first quadrant spectral sequences, which means that Erp q 0 when p 0 or q 0. Thus the nonvanishing terms lie in the quadrant where p q 0. The minimum value of r is usually 1 or 2, in which case we say that Erp q drp q is an E1 or E2 spectral sequence respectively. When working with an E1 spectral sequence, we often know the differentials d1p q explicitly. In general, however, the differentials drp q for r 2 are much more difficult to describe. For fixed p q, the differentials mapping to and from Erp q vanish when r is large by our first quadrant assumption. It follows that
E % E %
Erp q
pq r 1
pq r 2
for r 0. This common value is defined to be E Definition 9.A.2. A spectral sequence Erp q drp q
pq
if there are subspaces 0 such that
F%
k 1
Hk
.
converges to vector spaces H , k F0 H FH FH H k 1
FkHk
E p q
k
k
F pH p
1
% ,F% q
p 1
For an E1 or E2 spectral sequence, we write this as E1p q
Hp
%
or E2p q
q
Hp
%
k
0
k
0,
k
q
Hp
%
q
respectively. The intuition behind a convergent E2 spectral sequence is that we often know something about the E2 terms and are interested in what the spectral sequence converges to. One can think of E2p q as a first approximation to the convergent, and then E3p q E4p q as better and better approximations. A nice introduction to this way of thinking can be found in the first chapter of [102]. Here is an example of how to work with a spectral sequence.
Proposition 9.A.3. Suppose E2p q that E2p q 0 for all q 0. Then
E2p 0
Hp
%
Hp
q
is an E2 spectral sequence with the property for all p
0
Appendix: Introduction to Spectral Sequences
447
Proof. First observe that all differentials drp q vanish for r follows that E2p q E p q for all p q. Then we have
E2p 0
, %
F pH p F p
1
Hp
E 0 E 0 E implies 0 F 0 H , F H F 0 H , F 0 so that F H F 0 H F H F H p 11 2
and 0
p
p 22 2 p 1 p
p
p 1
0 p 2
p
p 2
p
p
1
p
p
0
p 1
p
Hp
2 since E2p q
0 for q 0. It
F pH p
H p . Thus
,
F 0H p F 1H p
E2p 0
H p , as claimed.
The hypothesis of Proposition 9.A.3 implies that the differentials d 2p q vanish for all p q. In general, we say that a spectral sequence degenerates at E r if the differentials drp q vanish for all p q. Note that degeneration at Er implies Erp q E p q .
Edge Homomorphisms. Another feature of a convergent E2 spectral sequence is that all differentials starting from Erp 0 vanish, so that the spectral sequence gives maps E2p 0
E3p 0
Definition 9.A.4. The map
E2p 0
E p 0
, %
F pH p F p
1
Hp
F pH p
Hp
p
H is called an edge homomorphism.
This leads to a more precise version of Proposition 9.A.3.
%
Proposition 9.A.5. Suppose E2p q H p q is an E2 spectral sequence with the property p q 0 for all q 0. Then the edge homomorphism that E2
is an isomorphism for all p
E2p 0
Hp
0.
For many E2 spectral sequences, we often know the edge homomorphisms E 2p 0 explicitly. This is true in the two applications of Proposition 9.A.5 given in §9.0.
Hp
Chapter 10
Toric Surfaces
§10.1. Singularities of Toric Surfaces and Their Resolutions In this chapter, we will apply the theory developed so far to give a detailed treatment of the structure of normal toric varieties of dimension two (toric surfaces).
2, Singular Points of Toric Surfaces. If X is the toric surface of a fan in N * then minimal generators of the rays 1 are primitive vectors (i.e., can be extended to a basis of N). Then Theorem 3.1.19 implies that the toric surface obtained by removing the fixed points of the torus action (i.e., the points corresponding to the 2-dimensional cones under the Orbit-Cone Correspondence) is smooth. There are only finitely many such points, so X has at most finitely many singular points. Moreover, 2-dimensional cones are always simplicial, so from Example 1.3.20, each of these singular points is a finite abelian quotient singularity (isomorphic to the image of the origin in the quotient 2 G where G is a finite abelian group).
All cones are assumed to be rational and polyhedral. A 2-dimensional strongly 2 has the following normal form that will facilitate our study convex cone in N * of the singularities of toric surfaces.
2 be a 2-dimensional strongly convex cone. N * Proposition 10.1.1. Let Then there exists a basis e1 e2 for N such that
where d
0, 0
k d, and
Cone e2 de1 ! ke2 d k 1.
Proof. We will need the following modified division algorithm here and at several other points in this chapter (Exercise 10.1.1). (10.1.1)
Given integers l and d
0, there are unique integers
s and k such that l sd ! k and 0
k d 451
Chapter 10. Toric Surfaces
452
Say Cone u1 u2 , where ui are primitive vectors. Since u1 is primitive, we can take it as part of a basis of N, and we let e 2 u1 . Since is strongly convex, for any basis e1 e2 for N, it will be true that
u2 de1 le2
for some d 0. By replacing e1 by ! e1 if necessary, we can assume d 0. By (10.1.1), there are integers s k such that l sd ! k, where 0 k d. Using this integer s, let e1 e1 se2 . Then e1 e2 is also a basis for N and u2 de1 l ! sd e2 de1 ! ke2
Hence Cone e2 de1 ! ke2 as claimed, and d k 1 follows since u2 is primitive. The uniqueness of d k is left to the reader (Exercise 10.1.2).
We will call the integers d k in this statement the parameters of the cone , and e1 e2 is called a normalized basis for N relative to . Using the normal form, we will next describe the local structure of the point p in the affine toric variety U . Recall from Example 1.3.20 that if N N is the sublattice generated 2 by the ray generators of , then U * G, where G N N . In our situation, N e1 e2 and
N e2
de1 ! ke2 d e1 e2
so it follows easily that G N N *
(10.1.2)
d
In particular, for singularities of toric surfaces, the finite group G is always cyclic. 2
The action of G on
is determined by the integers m k as follows. We write
d
d
1
the group of dth roots of unity in . Then a choice of a primitive dth root of unity defines an isomorphism of groups d * d . Proposition 10.1.2. Let M be the dual lattice of N and let m1 m2 M be dual to u1 u2 in N . Using the coordinates x m1 and y m2 of 2 , the action of 2 is given by d * N N on
Furthermore, U *
2
d
x y
x
k
y
with respect to this action.
Proof. The general discussion in §1.3 shows that N N acts on the coordinate ring of 2 via (10.1.3)
where m 0 j d ! 1.
u N
m
e2
i m u
m
M and each element of N N is given by u
je1 for some j,
§10.1. Singularities of Toric Surfaces and Their Resolutions
453
An easy calculation shows m1 e1 1 d and m2 e1 k d. Hence if we set up the isomorphism d * N N by mapping e2 i j d # je1 N , then for all e2 i j d d , from (10.1.3)
x y
e2
ij d
x e2
i jk d
y
k
x
y
which is what we wanted to show.
The isomorphism in (10.1.2) and the result of Proposition 10.1.2 also indicate that there is a slight but manageable ambiguity in the normal form for 2dimensional cones. Two cones are lattice equivalent if there is a bijective -linear mapping ' N # N taking one cone to the other. After choice of basis for N, such mappings are defined by matrices in GL 2 .
Proposition 10.1.3. Let Cone e2 de1 ! ke2 and Cone e2 de1 ! ke2 be cones in normal form that are lattice equivalent. Then d d and either k k or kk 1 d.
Proof. Since the cones are lattice equivalent, writing N and N for the sublattices as in (10.1.2), there is a bijective -linear mapping ' N # N such that N N . Hence N N * N N , so d d. The statement about k and k is left to the reader in Exercise 10.1.2.
We will next consider two examples to illustrate Proposition 10.1.2 and we will identify the resulting toric surface singularities. Example 10.1.4. First consider a cone
Cone e2 de1 ! e2
with parameters d 1 (so the cone is not smooth) and k 1. This is precisely the cone considered in Example 1.2.21. The corresponding toric surface U is rational normal cone Cd d 1 . The quotient Cd * 2 d was studied in the special case d 2 in Example 1.3.19, and the general case was described in Exercise 1.3.11. With the notation of Proposition 10.1.2, d acts on x y 2 via x y x y and the ring of invariants is
x y
so
U *
2
d
d *
x x d
d 1
y xyd
Spec xd xd
1
1
yd
y xyd
On the other hand, from the Hilbert basis of the semigroup the standard toric description
1
yd
M we obtain
U * Spec u uv uv2 uvd Exercise 10.1.3 studies the relation between these representations of the coordinate ring of U .
Chapter 10. Toric Surfaces
454
Example 10.1.5. Next consider a cone with parameters d and k d ! 1, so d k 1. We will express everything in terms of the parameter k in the following. Unlike the previous example, this is a case we have not encountered previously. Note that k ! 1 d. Hence by Proposition 10.1.2, the action of G N N on 2 is given by
x y
x
1
y
It is easy to check that the ring of invariants here is
x y
k 1
x
k 1
yk
1
xy
Moreover we have an isomorphism of rings '
X Y Z Z
k 1
!
XY * xk
x
Y#
yk
Z#
xy
yk
1
xy
k 1
X#
1
1
so we may identify the toric surface U with the variety V Z k
1!
XY
3.
The origin is unique singular point of the affine variety of Example 10.1.5 and is called a rational double point (or Du Val singularity) of type A k . Another standard form of these singularities is given in Exercise 10.1.4. They are called double points because the lowest degree nonzero term in the defining equation has degree two (that is, the multiplicity of the singularity is two). The rational double points are the simplest singularities from a certain point of view. The exact definition, which we will give in §10.3, depends on the notion of a resolution of singularities, which will be introduced shortly. All rational double points appear as singularities of quotient surfaces 2 G where G is a finite subgroup of SU 2 . There is a complete classification of such points in terms of the Dynkin diagrams of types Ak Dk E6 E7 , and E8 . The groups corresponding to the diagrams D k E6 E7 E8 are not abelian, so by the comment after (10.1.2) above such points do not appear on toric surfaces. We will see one way that the Dynkin diagram A k appears from the geometry of the toric surface U in Exercise 10.1.6, and we will return to this example in §10.4. More details on these singularities can be found in [133, Ch. VI] and in the article [36].
Here is another interesting aspect of Example 10.1.5. Recall that a normal variety X is Gorenstein if its canonical divisor is Cartier (Definition 8.2.14). You proved the following in Exercise 8.2.13.
Proposition 10.1.6. For a cone Cone e 2 de1 ! ke2 in normal form, the affine toric surface U is Gorenstein if and only if k d ! 1.
§10.1. Singularities of Toric Surfaces and Their Resolutions
455
Toric Resolutions of Singularities. Let X be a normal toric surface, and denote by Xsing the finite set of singular points of X (possibly empty). Definition 10.1.7. A proper morphism ' Y # X is a resolution of singularities of X provided Y is a smooth surface and induces an isomorphism of varieties
Y )
(10.1.4)
1
Xsing * X ) Xsing
Such a mapping modifies X to produce a smooth variety without changing the smooth locus X ) Xsing . One of the most appealing aspects of toric varieties is the way that many questions that are difficult for general varieties admit simple and concrete solutions in the toric case. The problem of finding resolutions of singularities is a perfect example. We illustrate this by constructing explicit resolutions of the toric surface singularities from Examples 10.1.4 and 10.1.5. Example 10.1.8. Consider the rational normal cone of degree d, the affine toric surface U for Cone e2 de1 ! e2 studied in Example 10.1.4. Let be the fan in Figure 1 obtained by inserting a new ray Cone e 1 subdividing into two 2-dimensional cones:
1
Cone e2 e1
2
Cone e1 de1 ! e2
σ1 ←τ σ2
Figure 1. The cones
1
2
in Example 10.1.8
We now use some results from Chapter 3. The identity mapping on the lattice N is compatible with the fans and as in Definition 3.3.1. By Theorem 3.3.4, we have a corresponding toric blowup morphism
(10.1.5)
'
X ! #
U$
Note that both 1 and 2 (as well as all of their faces) are smooth cones. Hence Theorem 3.1.19 implies that X is a smooth surface. In addition, the toric morphism is proper by Theorem 3.4.7 since is a refinement of . Finally, we claim that satisfies (10.1.4). This follows from the Orbit-Cone Correspondence on the
Chapter 10. Toric Surfaces
456
two surfaces: if p is the distinguished point corresponding to the 2-dimensional cone in (the singular point of U at the origin), then restricts to an isomorphism
p * U ) p $ U p is the curve on X given by the closure of the The inverse image E T -orbit O corresponding to the ray . That is, the singular point “blows up” to E * on the smooth surface. It follows that X and the morphism (10.1.5)
1
X )
smooth
1
N
1
give a toric resolution of singularities of the rational normal cone. E is called the exceptional divisor on the smooth surface. We will study more of the geometry of how E sits inside the surface X in §10.4.
Example 10.1.9. We consider the case d 4 of Example 10.1.5, for which the surface U has a rational double point of type A 3 . We will leave the details, as well as the generalization to all d 2, to the reader (Exercise 10.1.6). It is easy to find subdivisions of
Cone e2 4e1 ! 3e2
yielding collections of smooth cones. The most economical way to do this is to insert three new rays 1 Cone e1 , 2 Cone 2e1 ! e2 , 3 Cone 3e1 ! 2e2 to obtain a fan consisting of four 2-dimensional cones and their faces. The fan produced by this subdivision is somewhat easier to visualize if we draw the cones relative to a different basis u 1 u2 for N. Taking u1 e2 and u2 e1 ! e2 , the fan consisting of
(10.1.6)
1
Cone u1 u1 u2
2
Cone u1 u2 u1 2u2
3
Cone u1 2u2 u1 3u2
4
Cone u1 3u2 u1 4u2
and their faces appears in Figure 2. σ4 σ3
σ2
σ1
Figure 2. The cone
and the refinement
in Example 10.1.9.
§10.1. Singularities of Toric Surfaces and Their Resolutions
457
You will check that each of these cones is smooth. Hence X is a smooth surface. Since is a refinement of , we have a proper toric morphism
'
X
# U
1 p to As in the previous example, restricts to an isomorphism from X ) 1 p is the union X ) p . In this case, the exceptional divisor E
E O 1 & O 2 & O 3
on X . Each of the curves O i is isomorphic to 1 . The first two intersect transversely at the fixed point of the TN -action on X corresponding to the cone 2 , while the second two intersect transversely at the fixed point corresponding to 3 .
In these examples, we constructed toric resolutions of affine toric surfaces with just one singular point. The same techniques can be applied to any normal toric surface X .
Theorem 10.1.10. Let X be a normal toric surface. There exists a smooth fan
refining such that the associated toric morphism ' X # X is a toric resolution of singularities.
Proof. It suffices to show the existence of the smooth fan refining . The reasoning given in Example 10.1.8 applies to show that the corresponding toric morphism is proper and birational, hence a resolution of singularities of X .
We will prove this by induction on an integer invariant of fans that measures the complexity of the singularities on the corresponding surfaces. Let 1 n denote the 2-dimensional cones in a fan . For each i, we will write N i for the sublattice of N generated by the ray generators of i . Then we define
n
s
i 1
N ' N !
1 &
i
If s 0, then n 0 or N ' Ni 1 for all i. It is easy to see that this implies that is a smooth fan. Hence X is already smooth and we take this as the base case for our induction.
For the induction step, we assume that the existence of smooth refinements has been established for all fans with s s, and consider a fan with s s. If s 1, then there exists some nonsmooth cone i in . By Proposition 10.1.1, there is a basis e1 e2 for N such that i Cone e2 de1 ! ke2 with parameters d 0, 0 k d, and d k 1. Consider the refinement of obtained by subdividing the cone i into two new cones
i i
Cone e2 e1 Cone e1 de1 ! ke2
Chapter 10. Toric Surfaces
458
with a new 1-dimensional cone Cone e 1 . We must show that s
invoke the induction hypothesis and conclude the proof.
s to
In s , the terms corresponding to the other cones j for j i are unchanged. The cone i is smooth since e1 e2 is the normalized basis of N relative to i . So it contributes a zero term in s . Now consider the cone i . In order to compute its contribution to s , we must determine the parameters of i .
In terms of the basis e1 e2 for N, the -linear mapping defined by the matrix A
0
!
1 0
1
(a “90-degree rotation”) takes i to Cone e2 ke1 de2 . Since A GL 2 , it defines an automorphism of N, and hence i will have the same parameters as Cone e2 ke1 de2 . But now we apply (10.1.1) to write (10.1.7)
d sk ! l
where 0 l k. Since d k 1, we have k l 1 as well. Hence the cone i has parameters k and l obtained from (10.1.7). Since k d, if Ni is the sublattice generated by the ray generators of i , then by (10.1.2),
N ' N i
k
N ' N i
d
It follows that s s , and the proof is complete by induction.
We will see in the next section that the 1-dimensional cones in the refinement yielding the resolution of singularities of U have a very nice description. Al
though to this point we have usually only considered the semigroups M deN for the themselves. For fined by the dual cones, we can also consider each 2-dimensional cone in , the new cones in are produced by subdividing along the rays through the Hilbert basis (the irreducible elements) of the semigroup N. If you look back at Examples 10.1.8 and 10.1.9, you will see how this works in two special cases.
A resolution of a non-normal toric surface singularity can be constructed by first saturating the associated semigroup as in Theorem 1.3.5, then applying the results of this section. Toric resolutions of singularities for toric varieties of dimension three and larger also exist. However, we will postpone discussing the higher-dimensional case until Chapter 11. Exercises for §10.1. 10.1.1. Adapt the usual proof of the integer division algorithm to prove (10.1.1).
10.1.2. In this exercise, you will develop further properties of the parameters d k in the normal form for cones from Proposition 10.1.1 and prove part of Proposition 10.1.3. (a) Show that if is obtained from a cone by parameters d k by a -linear mapping of N defined by a matrix in GL 2 , then the parameter k of satisfies either k k, or
§10.1. Singularities of Toric Surfaces and Their Resolutions
459
d. Hint: There is a choice of orientation to be made in the normalization kk 1 process. Recall that d k 1, so there are integers d k such that d d kk 1.
(b) Show directly using Proposition 10.1.2 that the singularities at the origin of the toric varieties U and U are isomorphic if has parameters d k and has parameters d k d. with kk 1
$ $
(c) Show that if is a cone with parameters d k, then the dual cone M has parameters d d k. Hint: Use the normal form for , write down in the corresponding dual coordinates in M, then change coordinates in M to normalize .
10.1.3. With the notation in Example 10.1.4, show that
u uv uv * uv x x 0 y * xy 0 y under u x and v y , x, and use Proposition 10.1.2 to explain where these identifications come from in terms of the semigroup . Hint: We have u ' and v ' where e e is the normalized basis for N and m m is the dual basis for M. 2
d
d
d 1
d 1
d
d
m1
1
m2
1
2
2
%
%
10.1.4. In Example 10.1.5, we gave one form of the rational double point of type A k , 3 . Another namely the singular point at 0 0 0 on the surface V V Z k 1 XY commonly used normal form for this type of singularity is the singular point at 0 0 0 on the surface W V X k 1 Y 2 Z 2 . Show that V and W are isomorphic as affine varieties, hence the singularities at the origin are analytically equivalent. Hint: There is a linear change of coordinates in 3 that shows this.
10.1.5. Let X Y be irreducible varieties. Consider the class of morphisms F U Y where U is a nonempty Zariski-open subset of X. We will say F U Y and F V Y are equivalent if F U V F U V . A rational map F X Y is by definition an equivalence class of such morphisms under this notion of equivalence. A rational mapping F X Y is birational if it has a rational inverse G Y X (that is F G id and G F id where the compositions are defined). (a) Show that a birational mapping F induces an isomorphism of function fields F Y X .
"
(b) Show that if F is birational, then there are nonempty Zariski-open subsets U X and V Y such that F restricts to an isomorphism from U to V . (This applies in particular when F is a birational morphism, hence a mapping with domain equal to X.)
10.1.6. In this exercise, you will check the claims made in Example 10.1.9 and show how to extend the results there to the case Cone e2 de1 d 1 e2 for general d. (a) Check that each of the four cones in (10.1.6) is smooth, so that the toric surface X is smooth by Theorem 3.1.19. (b) For general d, show how to insert new rays i to subdivide and obtain a fan whose associated toric surface is smooth. Try to do this with as few new rays as possible. Hence we obtain toric resolutions of singularities X U for all d. 1 (c) Identify the inverse image C p in general. For instance, how many irreducible components does C have? How are they connected? Hint: One way to represent the structure is to draw a graph with vertices corresponding to the components and connect two vertices by an edge if and only if the components intersect on X . Do you notice a relation between this graph and the Dynkin diagram Ak Ad 1 mentioned before? We will discuss the relation in detail in §10.4.
0
0
Chapter 10. Toric Surfaces
460
§10.2. Continued Fractions and Toric Surfaces Hirzebruch-Jung Continued Fractions. What do continued fractions have to do with toric surfaces? To begin to answer this, let us consider what happens when we start with the singular point at the origin on the affine toric surface U for a cone in normal form with parameters d k, and construct a resolution of singularities using a process following the proof of Theorem 10.1.10. The first step in the resolution is to refine the cone to a fan containing the 2-dimensional cones
Cone e2 e1
and
Cone e1 de1 ! ke2
The first is smooth, but the second may not be. However, we saw in the proof of Theorem 10.1.10 that the cone has parameters k k1 where
d b1 k ! k1 b1 2, 0 k1 k as in (10.1.1). We used slightly different notation before, writing s rather than b1 and l rather than k1 ; the new notation will help us keep track of what happens if we now continue the process and refine the cone .
Using the normalized basis for N relative to , we insert a new ray and obtain a new smooth cone and a second, possibly nonsmooth cone with parameters k 1 k2 , where k b 2 k1 ! k2 using (10.1.1). Doing this repeatedly yields a modified Euclidean algorithm d b1 k ! k1
(10.2.1)
k b 2 k1 ! k2 .. .
kr kr
k bk
3
br
2
1 r 2!
kr
1
r r 1
that computes the parameters of the new cones produced as we successively subdivide to produce the fan giving the resolution of singularities. The process terminates with kr 0 for some r as shown, since as in the usual Euclidean algorithm, the ki are a strictly decreasing sequence of nonnegative numbers. Also, by (10.1.1), we have bi 2 for all i. The equations (10.2.1) can be rearranged: d k b 1 ! k1 k
(10.2.2)
k k 1 b2 ! k2 k1 .. .
kr kr
3
kr
2
kr
2
br
1
br
1!
kr
1
kr
2
§10.2. Continued Fractions and Toric Surfaces
461
and spliced together to give a type of continued fraction expansion for the rational number d k, with minus signs: 1
d k b1 !
(10.2.3)
1
b2 !
1 br !
This is the Hirzebruch-Jung continued fraction expansion of d k. For obvious typographical reasons, it is desirable to have a more compact way to represent these expressions. We will use the notation
b b b
d k
1
2
r
The integers bi are the partial quotients of the Hirzebruch-Jung continued fraction, and the truncated Hirzebruch-Jung continued fractions
b b b 1 i r are the convergents. Note that d b b b d k. di
1
2
r
i
1
2
r
Example 10.2.1. Consider the rational number 17 11. The Hirzebruch-Jung continued fraction expansion is
2 3 2 2 2 2
17 11
as may be verified directly using the modified Euclidean algorithm (10.2.1).
Proposition 10.2.2. Let d k 0 be integers with d k 1 and let d k b1 br . Define sequences Pi and Qi recursively as follows. Set
(10.2.4) and for all 2
i
P0 1
Q0 0
P1 b1
Q1 1
r, let
Pi bi Pi
(10.2.5)
Qi bi Qi
1!
Pi
1!
Qi
2 2
Then the Pi Qi satisfy: (a) The Pi and Qi are increasing sequences of integers.
b b (c) P Q ! P Q
(b) di
i 1
1
i
i
i
i 1
Pi Qi for all 1 1 for all 1
i
r.
i r.
(d) The convergents form a strictly decreasing sequence: d k
Pr Qr
Pr Qr
1 1
P1 Q1
Chapter 10. Toric Surfaces
462
Proof. The proof of part (a) is left to the reader (Exercise 10.2.1). To prove part (b), we will make use of the observation that the expression on the right side of (10.2.3) also makes sense when the b j are any complex numbers (not just integers). We will show that the sequences defined by (10.2.5) satisfy
b b 1
Ps Qs
s
for all lists of numbers b1 bs . The proof is by induction on the length s of the list. When s 1, we have b1 b1 QP11 by (10.2.4). Now assume that the result has been proved for all lists of of length t and consider the expression
b b b b ! t 1
1
1
1
t
bt
1
where the right side comes from a list of length t. By the induction hypothesis, this equals
1 bt 1
bt !
Pt
1
1
bt !
bt
Qt
1!
Pt
1!
Qt
2
2
By the recurrences (10.2.5), this equals Pt ! Qt !
1 bt 1 Pt 1 1 bt 1 Qt 1
bt 1 Pt ! Pt bt 1 Qt ! Qt
1 1
Pt Qt
1
1
which is what we wanted to show. Part (c) will be proved by induction on i. The base case i 1 follows directly from (10.2.4). Now assume that the result has been proved for i s, and consider i s 1. Using the recurrences (10.2.5), we have Ps Qs
1!
Ps 1 Qs Ps bs Qs ! Qs 1 ! bs Ps ! Ps Ps 1 Qs ! Ps Qs 1
1
Qs
1
by the induction hypothesis. Finally, from part (b), for each 1
Pi Qi
1 1
i
Pi Qi 1 Qi
r ! 1, we have
Pi 1 Qi Qi 1 Qi
Hence
since Qi
Pi Qi
1 1
0 by part (a). Hence part (d) follows.
§10.2. Continued Fractions and Toric Surfaces
463
Hirzebruch-Jung Continued Fractions and Resolutions. When is a cone with parameters d k, the process of computing the Hirzebruch-Jung continued fraction of d k by the modified Euclidean algorithm (10.2.1) yields a convenient method for finding the fan such that ' X # U is a toric resolution of singularities.
Theorem 10.2.3. Let Cone e2 de1 ! ke2 be in normal form. Let u0 e2 and use the integers Pi and Qi from Proposition 10.2.2 to construct vectors ui Pi Then the cones
i
1 e1 !
Cone ui
have the following properties: (a) Each
is a smooth cone and ui
i
(b) For each i, 1
i 1
i
1 e2
1
1
ui
1
1
ui are its ray generators.
i
r 1
i r 1
Cone ui .
, so the fan
refinement of . (d) The toric morphism ' X # U
(c)
Qi
r 1
consisting of the
i
and their faces gives a
is a resolution of singularities.
Proof. Both statements in part (a) follow easily from part (c) of Proposition 10.2.2.
For part (b), we note that the ratio ! Q i 1 Pi 1 represents the slope of the line through ui in the coordinate system relative to the normalized basis e 1 e2 for . By part (d) of Proposition 10.2.2, these slopes form a strictly decreasing sequence for i 0, which implies the statement in part (b). ur
Part (c) follows from part (b) by noting that u 0 1 de1 ! ke2 . Hence the cones i fill out .
e2 and Pr Qr
d k, so
Part (d) now follows by the reasoning used in Examples 10.1.8 and 10.1.5.
Example 10.2.4. Consider the cone Cone e 2 7e1 ! 5e2 in normal form. To construct the resolution of singularities of the affine toric surface U , we simply compute the Hirzebruch-Jung continued fraction expansion of the rational number d k 7 5 using the modified Euclidean algorithm:
7 2 5! 3 5 2 3! 1 3 3 1 Hence b0 b1 2 b2 3, and (10.2.6)
2 2 3
7 5
Chapter 10. Toric Surfaces
464
Then from Proposition 10.2.2 we have P0 P1 P2 P3
1 2 b2 P1 ! P0 3 b3 P2 ! P1 7
Q0 Q1 Q2 Q3
0 1 b2 Q1 ! Q0 2 b3 Q2 ! Q1 5
By Theorem 10.2.3, the cones in the refinement
singularities are
(10.2.7)
of
1
Cone e2 e1
2
Cone e1 2e1 ! e2
3
Cone 2e1 ! e2 3e1 ! 2e2
4
Cone 3e1 ! 2e2 7e1 ! 5e2
giving the resolution of
We can also see an interesting connection between this Hirzebruch-Jung continued fraction and the corresponding continued fraction expansion for 7 3, where 5 3 1 7 as in Proposition 10.1.3. We leave it to the reader to check that
3 2 2
7 3
in which the partial quotients are the same as those in (10.2.6), but listed in reverse order. We will return to this point shortly.
You will see several other explicit examples of this process in Exercises 10.2.4 and 10.2.9. Our next result gives an alternative, more intrinsic, way to understand the vectors ui in Theorem 10.2.3. Note that the set N has the structure of an additive semigroup. We define its irreducible elements and Hilbert basis as in Proposition 1.2.22. See Figure 2 in Example 10.1.9 for an example.
Theorem 10.2.5. Let
Cone e2 de1 ! ke2 be in normal form. Then the set S u0 u1 ur
1
constructed in Theorem 10.2.3 is the Hilbert basis of the semigroup
N.
Proof. Since each cone i is smooth, i N is generated by its ray generators ui 1 ui . Since N. 1 r 1 , it follows that S generates
We claim next that all the ui are irreducible elements of N. This is clear for u0 e2 and ur 1 de1 ! ke2 since they are the ray generators for . If 1 i r and ui is not irreducible, then ui would have to be a linear combination of the vectors in S ) ui with nonnegative integer coefficients, i.e., ui Pi
1 e1 ! Qi
1 e2
c ju j j i
c j Pj j i
1
e1 !
c jQ j j i
1
e2
§10.2. Continued Fractions and Toric Surfaces with c j 0 in . Hence Pi
1
c j Pj j i
1
Qi
465
1
c jQ j j i
1
Since the Pi and Qi are strictly increasing by part (a) of Proposition 10.2.2, we must have c j 0 for all j i. But this would imply that u i is a linear combination with nonnegative integer coefficients of the vectors in u 0 ui 1 . This contradicts the observation made in the proof of Theorem 10.2.3 that the slopes of the u i are strictly decreasing. It follows that the u i are irreducible elements of N.
Finally, we must show that there are no other irreducible elements in N. But this follows from what we have already said. Since 1 r 1 , if u is irreducible, then u i N for some i. But then u ci 1 ui 1 ci ui for some ci 1 ci 0 in . Thus u is irreducible only if u u i 1 or ui .
Next, we will show how the irreducible elements u i in N determine the partial quotients in the Hirzebruch-Jung continued fraction expansion of d k. Theorem 10.2.6. Let
Cone e2 de1 ! ke2 be in normal form, and let
b b b
d k The vectors u0 u1 ur
1
ui
i
2
r
constructed in Theorem 10.2.3 satisfy
(10.2.8) for all i, 1
1
1
ui
1
bi ui
2 e2 &
r.
Proof. By the recurrences (10.2.5), ui
1
ui
1
Pi 2 e1 !
Qi
Pi 2
Pi e1 ! Qi bi Pi 1 e1 ! Qi 1 e2 bi ui
Pi e1 ! 2
Qi e2
Qi e2
Later in this chapter we will see several important consequences of the equations (10.2.8) connected with the geometry of the smooth toric surface X . We will next show that the pattern noted at the end of Example 10.2.4 holds for all Hirzebruch-Jung continued fractions. We give a proof that uses the properties of the associated toric surfaces.
Proposition 10.2.7. Let 0 k k d and assume k k Jung continued fraction expansion of d k is d k
1
b b b 1
2
b b
r
d. If the Hirzebruch-
then the Hirzebruch-Jung continued fraction expansion of d k is d k
r
r 1
b1
Chapter 10. Toric Surfaces
466
Proof. Let Cone e2 de1 ! ke2 and Cone e2 de1 ! ke2 be the corresponding cones in normal form. Since k k 1 m, there is an integer d such that d d kk 1. The -linear mapping ' N # N defined with respect to the basis e1 e2 by the matrix
A
k d
d ! k
is bijective, maps to , and is orientation-reversing. Thus
de 1 ! ke2 e2 and e de ke . If we apply Theorem 10.2.3 to , then we obtain vectors u i ! 1 2 2 satisfying the equations ui 1 ui 1 bi ui
1 defined by the for all 1 i r. We claim that when we apply the mapping inverse of the matrix A above, then the vectors u i are taken to corresponding vec1 are orientation-reversing, the partial tors ui for the cone . But since and quotients in the Hirzebruch-Jung continued fraction will be listed in the opposite order. You will complete the proof of this assertion in Exercise 10.2.3.
Ordinary Continued Fractions. The Hirzebruch-Jung continued fractions studied above are less familiar than ordinary continued fraction expansions in which the minus signs are replaced by plus signs. If d k 0 are integers, then the ordinary continued fraction expansion of d k may be obtained by performing the same sequence of integer divisions used in the usual Euclidean algorithm for the gcd. Starting with k 1 d and k0 k, we write ai for the quotient and ki for the remainder at each step, so that the ith division is given by
(10.2.9) where 0
k i ki
ki
2
ai ki
1
ki
1.
Let kn 1 be the final nonzero remainder (which equals equations splice together to form the continued fraction 1
d k a1
d k ). The resulting
1
a2
1 an To distinguish these from Hirzebruch-Jung continued fractions, we will use the notation
a a a
d k
(10.2.10)
1
2
n
for the ordinary continued fraction. The a i are the (ordinary) partial quotients of d k, and the truncated continued fractions
a a a
ci
1
2
are the (ordinary) convergents of d k.
i
1 i
n
§10.2. Continued Fractions and Toric Surfaces
467
Example 10.2.8. Consider the rational number 17 11 from Example 10.2.1 above, where we saw that the Hirzebruch-Jung continued fraction expansion is
2 3 2 2 2 2
17 11
The ordinary continued fraction expansion is
1 1 1 5
17 11
Note that the partial quotients and the lengths are different. Each of these expansions determines the other and there are methods for computing the HirzebruchJung partial quotients b j in terms of the ordinary partial quotients a i and vice versa. See [72, p. 257], [30, Proposition 3.6] and Exercise 10.2.7 below.
The following result is mostly parallel to Proposition 10.2.2, but shows that ordinary continued fractions are slightly more complicated than Hirzebruch-Jung continued fractions. The proof is left to the reader (Exercise 10.2.5).
Proposition 10.2.9. Let d k 0 be integers with d k 1, and let d k a1 an . Define sequences pi and qi recursively as follows. First set
(10.2.11) and for all 2
i
p0 1
q0 0
p1 a1
q1 1
n, let
pi ai pi
(10.2.12)
qi ai qi
1
pi
1
qi
2 2
Then the pi qi satisfy:
a a (b) p q ! p q (a) ci
1
pi qi for all 1
i
i 1 i
i i 1
i ! 1
for all 1
i
i
n.
n.
(c) The convergents converge to m k, but in an oscillating fashion: p1 q1
p3 q3
d k
p4 q4
p2 q2
Ordinary Continued Fractions and the Dual Cone. Ordinary continued fraction expansions also have applications in the geometry of toric surfaces.
2 be a cone with parameters d k. Let e e be a Theorem 10.2.10. Let N * 1 2 normalized basis of N, and let m1 m2 be the dual basis of M. Let d k a1 an , and define a sequence of vectors in M by m 1 m1 , m0 m2 , and
m i ai mi
(10.2.13) for all 1
i
1
mi
2
n. Then the Hilbert basis of the semigroup n m2 j 1 0 j mn 2
M is
Chapter 10. Toric Surfaces
468
Proof. As usual, if Cone e2 de1 ! ke2 , then Cone m1 km1 dm2 . As in the proof of Theorem 10.2.3, we will consider the slopes of the m i with respect 2 induced by the basis m m . Write m to the coordinate system in M * 2 i 1 ri m1 si m2 for ri si . Then from (10.2.13), we see that r0 0, s0 1, r1 1, s1 a1 , and for i 2,
ri ai ri
1
ri
and
2
s i ai si
1
si
2
Comparing with (10.2.12), because the sequences generated by these recurrences are uniquely determined by the two initial terms, it follows that for i 1, r i qi and si pi , so that mi qi m1 pi m2 (Note the parallel with the definition of the vectors u i in Theorem 10.2.3.) Hence in this coordinate system, the slope of the line spanned by m i is qpii
By part (c) of Proposition 10.2.9, the odd-numbered vectors m 2 j 1 lie within the cone , while the even-numbered m2 j lie outside this cone, on the other side of the boundary ray Cone km1 dm2 . Hence each of the vectors in lies in the semigroup M. Moreover, the slopes of the elements of are strictly increasing from 0 to d k. By the reasoning used in the proof of Theorem 10.2.5, generates and is precisely the set of irreducible elements of this semigroup. You will check the details in Exercise 10.2.8.
Example 10.2.11. Consider the cone Cone 17e 1 ! 11e2 . We computed the Hirzebruch-Jung and ordinary continued fraction expansions of d k 17 11 in Examples 10.2.1 and 10.2.8. The smooth fan that refines has 7 cones i by Theorem 10.2.3. On the other hand, 17 11 1 1 1 5 implies that the vectors (10.2.13) in Theorem 10.2.10 are
It follows that the semigroup
1
m1
m0
m2
m1
m1 m2
m2
m1 2m2
m3
2m1 3m2
m4
11m1 17m2
m
m1 m1 m2 2m1 m3 11m1 17m2 is the Hilbert basis of M.
The connection between Hirzebruch-Jung continued fractions and resolutions of singularities of toric surfaces is standard and stems from work of Hirzebruch. The connection with ordinary continued fractions goes back to work of Felix Klein on lattice polygons. Our presentation is based on [30, Sec. 3].
§10.2. Continued Fractions and Toric Surfaces
469
Exercises for §10.2.
10.2.1. Prove part (a) of Proposition 10.2.2: Show that the sequences Pi and Qi from (10.2.5) are increasing sequences of nonnegative numbers. Hint: Use b i 2 for all i. 10.2.2. Verify that equations (10.2.8) hold for the vectors u i computed in Example 10.2.4. (See (10.2.7).) 10.2.3. Verify the last claim in the proof of Proposition 10.2.7. 10.2.4. In §10.1, we constructed several resolutions of singularities in a rather ad hoc way. In this exercise, we will see that the resolutions given by Theorem 10.2.3 are the same as what we saw before.
(a) When has parameters d 1, show that the Hirzebruch-Jung continued fraction method gives the same resolution of U as the one given in Example 10.1.8. (b) Do the same for the resolutions of the singular points for with parameters d k d 1 (the rational double points from Example 10.1.9). Hint: First show that d 22 2 d 1 where there are d 1 2’s.
10.2.5. In this problem we will consider Proposition 10.2.9.
(a) Prove the proposition. Hint: For part (a), argue by induction on the length n 1 of the expansion. The formal expression a1 a2 ai is well-defined even if the al are not integers, so we can write
* a a a 0 a a a * a 0 1, a 1
i 1
2
i
1
i 1
2
i
Then use the recurrences in (10.2.12). Then follow the reasoning from the proof of Proposition 10.2.2. (b) Suppose we modify the initialization and the recurrences (10.2.12) as follows. Let
s0 0 0 s 1 Then, for all 1 . i . n, compute r r0 ar0 s s0 as0 (note the change in sign!) What is true about r d s k for all i? What do we get with i n? Hint: This fact is the basis for the extended Euclidean algorithm. r
0 1 1
1
r0
0
i
i 2
i i 1
i
i 2
i i 1 i
,
i
10.2.6. Let d k be a rational number in lowest terms with 0 k d, and assume
, b * b a a
d k
1
r
1
n
are its continued fraction expansions. The recurrences defining the Pi Qi and pi qi considered in this section can also be expressed in matrix forms. b 1 (a) Let M b . Show that for all 1 i r, 1 0
0
P P0 i
Qi
. . 0 0 0 Q0 M b M b M b i 1
i 1
1
2
i
Chapter 10. Toric Surfaces
470
(b) Let M
% a
a
1 . Show that for all 1 1 0 pi qi
pi qi
0
. i.
n,
0 M% a M% a M% a 1
1
1
2
i
10.2.7. A key ingredient for a method for converting ordinary continued fraction expansions to Hirzebruch-Jung continued fractions is the following observation. (a) Given integers a1 a2 0 and a variable x, show that, as rational expressions,
a a x a 1 2 0 x 1
1
2
a2 1
1
where for any l 0, 2 l denotes a string of l 2’s. Hint: Argue by induction on a1 . (b) Show how this observation yields the equality
1 1 1 5 2 3 2 2 2 2
from Example 10.2.8. 10.2.8. Verify the claims in the proof of Theorem 10.2.10 by following the reasoning from the proof of Theorem 10.2.5.
!
10.2.9. In this exercise, you will apply the results of this section to the weighted projective plane q0 q1 q2 from §2.0 and Example 3.1.17.
!
(a) Construct a resolution of singularities for any 1 1 q2 , where q2 2. What smooth toric surface is obtained in this way? A complete classification of the smooth complete toric surfaces will be developed in §10.4. (b) Do the same for
! 1 q q in general. 1
2
§10.3. Gr¨obner Fans and McKay Correspondences
The fans obtained by resolving the singularities of the affine toric toric surfaces U have unexpected descriptions that involve Gr¨obner bases and representation theory. In this section we will present these ideas, following [78] and [79]. An example appeared earlier in Exercise 3.3.8. We will assume that the reader is familiar with “Gr¨obner basics” (see, e.g. [26]), and with the beginnings of representation theory for finite abelian groups. Gr¨obner Bases and Gr¨obner Fans. For more background on the material covered in this section, consult [27, Ch. 8, §4], or [146]. An nonzero ideal I x 1 xn has a unique reduced Gr¨obner basis with respect to each monomial order on the polynomial ring. However, the set of distinct reduced marked Gr o¨ bner bases for I (that is, reduced Gr¨obner bases with specified leading terms in each polynomial) is finite. This implies that every nonzero ideal I has a finite universal Gr o¨ bner basis, that is, a subset I such that is a Gr¨obner basis for all monomial orders simultaneously.
Let w n be a weight vector in the positive orthant (so w could be taken as the first row of a weight matrix defining a monomial order). Let
g1 gt
§10.3. Gr¨obner Fans and McKay Correspondences
471
be one of the reduced marked Gr¨obner bases for I, where gi x
i
ci x
and x i is marked as the leading term of gi . If w i w whenever ci 0, then I will have Gr¨obner basis with respect to any monomial order defined by a weight matrix with first row w. The set C w
(10.3.1)
n
w whenever ci
i
w
0
is the intersection of a finite collection of half-spaces, hence has the structure of a closed convex polyhedral cone in n . The cones C as runs over all distinct marked Gr¨obner bases of I, together with all of their faces, have the structure of a fan in n called the Gr¨obner fan of I. In particular, for each pair of marked Gr¨obner bases, the cones C and C intersect along a common face where the w-weights of terms in some polynomials in (and in ) coincide.
A First Example. Let be a cone in normal form with parameters d k, and recall d k 1 by hypothesis. By Proposition 10.1.2, the group
Gd k acts on
2
k
2
d
1 0
k
d ! 1 *
d
by componentwise multiplication
(10.3.2) with quotient
2
k
x y
Gd k * U .
k
x
y
Let I Gd k be the ideal defining Gd k as a variety in 2 . In the next extended example, we will introduce the first main result of this section.
Example 10.3.1. Let d 7 k 5, G G7 5 , and I I G . It is easy to check that
x
I
7
1 y ! x5 !
Moreover, for the lexicographic order with y 1
x
7
x, the set
1 y ! x5 !
is the reduced marked Gr¨obner basis for I, where the underlines indicate the leading terms. The corresponding cone in the Gr¨obner fan of I is (10.3.3)
C 1 w
a b
2
b 5a Cone e2 e1 5e2
There are three other marked reduced Gr¨obner bases of I as well: 2 3 4
x x y
5
!
y x 2 y ! 1 y2 ! x 3
3
!
y2 x 2 y ! 1 y3 ! x
7
!
1 x ! y3
Chapter 10. Toric Surfaces
472
It is easy to check that each of these sets is a Gr¨obner basis for I using Buchberger’s criterion. The corresponding cones are C
2
a b
C
3
a b
C
4
a b
2
b
2
5a 2b 3a Cone e1 5e2 2e1 3e2 2b 3a 3b a Cone 2e1 3e2 3e1 e2
2
3b
a Cone 3e1 e2 e1
Since these three cones fill out the first quadrant in 2 , the Gr¨obner fan of I consists of these three cones and their faces. We will denote this fan by in the following. Next, let us consider the resolution of singularities of
X ! #
U
for with m 7 k 5 computed in Example 10.2.4 in the last section. We leave it to the reader to check that the linear transformation T ' N # N with matrix relative to the basis e1 e2 given by A
(10.3.4)
7 0 5 1 !
maps the cones C i in the Gr¨obner fan to the corresponding cones i in the fan
. The matrix in (10.3.4) is invertible, but its inverse is not an integer matrix. The image of the lattice N 2 under T is the proper sublattice 7 e1 e2, and T 1 maps N to the lattice N a 7 b 7 a b b
5a
1
7 N
7
5 e2 7
e1
There is an exact sequence 0 !$#
N !$#
N ! #
G !$#
0
induced by the map N # 2 defined by a 7 b 7 # Cone e e , the corresponding toric morphism U 1 2 mapping 2 !$# 2 G.
e2
N ! #
. Letting U is the quotient e2 ib 7
ia 7
N
With respect to the lattice N , the cones in the Gr¨obner fan are smooth cones, and it follows from the discussion of toric morphisms in §3.3 that the toric surfaces X N and X are isomorphic, In other words, the Gr¨obner fan of the ideal I encodes the structure of the resolution of singularities of U .
A Tale of Two Fans. We next show that the last observation in Example 10.3.1 holds in general. As in the example, consider the ideal I I G d k and the action of Gd k given in (10.3.2). Each monomial in x y is equivalent modulo I to one of the monomials x j , j 0 d ! 1. This may be seen, for instance, from the remainders on division by the lexicographic Gr¨obner basis x d ! 1 y ! x k . As
§10.3. Gr¨obner Fans and McKay Correspondences
473
a result, we have a direct sum decomposition of the coordinate ring of the variety Gd k as -vector spaces
x y
(10.3.5)
d 1
I*
j 0
where V j is the subspace spanned by x j
Vj
I (hence
V j 1 for all j).
The following result establishes a first connection between the ideal I and the resolution of singularities of U described in Theorem 10.2.3.
Proposition 10.3.2. Let I I Gd k where 0 k d and u0 e2 ui Pi
1 e1 !
Qi
from Theorem 10.2.3. Let T ' N #
1
ui
1 e2
d 0 ! k 1
g i x bi ! y ai (b) The set
i 1 r 1
1 d
for i 0 r 1. Write wi
(a) The polynomials g0 gr
1
0
i
ai e1 bi e2 and define
r 1
are contained in the ideal I.
ae1 be2 N a b 0 and x b ! y a I
d k 1. Consider
N be the linear transformation with matrix A
and let wi T
N
is an additive semigroup. (c) The set
dwi i 0 r 1
is the Hilbert basis of the semigroup
of part (b).
Proof. It is an easy calculation to show T 1 maps Cone e2 de1 ! ke2 to the first quadrant 2 . Moreover, w0 e2 , and for i 1 r 1,
wi
1 Pi d
1 e1
Therefore, g0 x d ! 1 and gi x kPi
1
kPi
dQi
1!
1
!
dQi
y Pi
1
e2
1
for i 1 r 1. Since d 1, these polynomials clearly vanish at Therefore gi I Gd k I for all i. The proof of part (b) is left to the reader as Exercise 10.3.3.
k
Gd k .
Chapter 10. Toric Surfaces
474
For part (c), it follows from parts (a) and (b) that dw i is contained in . On the other hand, let ae1 be2 . Then x b ! y a I, which implies that b ak d. Hence 1 b ! ak T e2 ae1 be2 ae1 d d N. Since the ui are the Hilbert basis for the semigroup must be an element of N by Theorem 10.2.5, this vector is a nonnegative integer combination of the ui . Hence ae1 be2 is a nonnegative integer combination of the dw i . It follows that is generated by the dwi . The dwi are irreducible in because the corresponding ui T wi are irreducible in the semigroup N.
Let
denote the fan in the first quadrant consisting of the cones i
(10.3.6)
Cone ai
1 e1
bi
1 e2
ai e1 bi e2
for i 1 r 1, together with their faces. In Exercise 10.3.4, you will verify several of the following claims.
Lemma 10.3.3. The ai are increasing and the bi are decreasing with i.
For any monomial orders, the leading terms of g 0 x d ! 1 and gr 1 y d ! 1 are xd and yd respectively. It follows that for each monomial order , there is some index i i0 (depending on ) such that LT gi
(10.3.7)
x bi for all i
i0 and LT gi y ai for all i
i0
Lemma 10.3.4. Let be a monomial order and let i i 0 be the index as in (10.3.7). Then gi0 and gi0 1 are elements of the reduced Gro¨ bner basis for the ideal I I Gd k with respect to .
Proof. We know gi0 is an element of the ideal by part (a) of Proposition 10.3.2. If gi0 is not an element of the reduced Gr¨obner basis, then there exists some g I whose leading term divides LT gi0 x bI0 . This means that g x b f x y , where 0 b bi0 and LT g x b . Hence all the monomials appearing in f x y have the form x c y a for 0 c b and 0 a. Consider where each of these monomials lives in the direct sum decomposition (10.3.5). There must be some monomial x c y a appearing in g that lies in the same subspace Vb as the monomial x b , or else g could not be an element of the ideal. But then x b ! x c y a x c x b c ! y a I, so x b c ! y a I as well. Writing b b ! c, this gives w ae1 b e2 , where is the semigroup from part (b) of Proposition 10.3.2. Then part (c) of the same proposition implies that w must be a nonnegative integer combination of the mwi ai e1 bi e2 . Since b b bi0 and the bi are decreasing with i, this linear combination can include only the mw i with i i0 . But this is a contradiction because y ai x bi for all i i0 by hypothesis. Therefore gi0 must appear in the reduced Gr¨obner basis of I with respect to .
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475
The statement for gi0 1 now follows by an argument parallel to the one above, considering elements of the ideal whose leading terms are powers of y. The details are left to the reader in part (d) of Exercise 10.3.4.
We are now ready for the first major result of this section.
Theorem 10.3.5. Let be the fan consisting of the cones i from (10.3.6), together with their faces. Then equals the Gr o¨ bner fan of the ideal I Gd k .
Proof. Lemma 10.3.4 and the definition of the Gr¨obner cone C for a marked Gr¨obner basis imply that the rays Cone a i e1 bi e2 all occur as 1-dimensional cones in the Gr¨obner fan. To complete the proof, we will show that there are no other 1-dimensional cones. Suppose on the contrary that Cone ae 1 be2 appears in the Gr¨obner fan for some w ae1 be2 with a b 0. Then w lies in the interior of i for some i, 1 i r 1. But then any monomial order defined by a weight matrix with this vector as first row must have i0 i ! 1 in (10.3.7). It follows from Lemma 10.3.4 that gi 1 and gi are elements of the reduced Gr¨obner basis. All other elements will have the form x s y t ! 1 for some s t 0. Therefore, the corresponding Gr¨obner cone is exactly i and we have a contradiction.
As in Example 10.3.1, the following statement is an immediate consequence. Corollary 10.3.6. Let N be the lattice N a d b d a b b
k
d
1 k e1 e2 d d
e2
are isomorphic. In other words, the Gr¨obner fan of I G can be used to construct the minimal resolution of singularities of the affine toric surface U when has parameter d k.
The toric surfaces X and X
N
d k
Connections with Representation Theory for d . We now consider the above results in a slightly different context. The group G G d k from (10.3.2) acts on V 2 according to the 2-dimensional linear representation of the group d of dth roots of unity defined by
(10.3.8)
'
d !$#
GL V GL 2
!$#
0 k
0
Since d is abelian, its irreducible representations are 1-dimensional over , and hence each is defined by a character
j '
d ! #
! #
j
Chapter 10. Toric Surfaces
476
for j 0 d ! 1. The reason for the minus sign will soon become clear. Via (10.3.2), we get the induced action of
x
1
x
d
on the polynomial ring x y by
y
j
y
as explained in §5.0. Each monomial x a y b spans an invariant subspace where the action of d is given by the irreducible representation with character j for j a kb d. We call a kb d the weight of the monomial x a y b with respect to this action of d . Since the ideal of the group G 2 is invariant, the action descends to the quotient x y I G , and we have a representation of d on x y I G . The direct sum decomposition (10.3.5) shows that the irreducible representation with character j appears exactly once in this representation, as the subspace V j in (10.3.5). This means that the representation on x y I G is isomorphic to the regular representation of d (Exercise 10.3.5).
A 2-dimensional McKay Correspondence. In 1979, McKay pointed out that there is a one-to-one correspondence between the irreducible representations of d and the components of the exceptional divisor in the resolution ' X !$# U when was a cone in normal form with parameters k d ! 1, as in Example 10.1.5. In this case, the singular point of U is a rational double point, and the image of the representation from (10.3.8) lies in SL 2 . A great deal of research was devoted to explaining the original McKay correspondence in representation-theoretic and geometric terms (work of Gonzalez-Sprinberg, Artin and Verdier). However, for 1 k d ! 1, d is a subgroup of GL 2 , not SL 2 , and there are more irreducible representations of d than components in the exceptional divisor. The McKay correspondence can be extended to these cases by identifying certain sets of special representations that correspond to the components of the exceptional divisor (work of Wunram, Esnault, Ito and Nakamura, Kidoh, and others). To conclude this section, we describe a generalized McKay correspondence that applies for all d k. Writing G Gd k as before, consider the ring of invariants x y G . You will prove the following in Exercise 10.3.6.
Lemma 10.3.7. Let V j be an irreducible representations of d with character j , and consider the action of G Gd k * V j . Then the subspace of d on x y G invariants x y V j has the structure of a module over the ring x y G .
Example 10.3.8. Let G G7 5 as in Example 10.3.1. The ring of invariants is x y G x7 x 2 y xy4 y7 in this case. If v j is the basis of the representation V j , then it is easy to check that x a y b v j is invariant under G if and only if a kb ! j 0 7, or in other words if and only if x a y b has weight j under this action of 7 .
First consider the case j 1 in Lemma 10.3.7. The monomials in the complement of the monomial ideal
x x y xy y
7
2
4
7
§10.3. Gr¨obner Fans and McKay Correspondences
477
that have weight 1 are x and y 3 . Then x v1 and y 3 v1 generate the module V1 G . Since x and y3 have the same weight with respect to this action x y of 7 , the difference x ! y 3 is an element of the ideal I G , and this is one of the polynomials gi as in the proof of Proposition 10.3.2.
On the other hand, if j 2, thenthere are three monomials with weight 2 in the complement of , and these give three generators of x y V2 G , namely x 2 v2 xy 3 v2 , and y6 v2 . It is still true that x 2 ! xy 3 , x 2 ! y6 , and xy 3 ! y6 are elements of I G , but these polynomials cannot appear in a reduced Gr¨obner basis for I G . Moreover, no subset of the three generators generates the whole module V2 G . x y
Definition 10.3.9. Let G Gd k * d as above. We say that the representation V j is special with respect to k if x y V j G is minimally generated as a module over the invariant ring x y G by two elements.
Hence, in Example 10.3.8, V1 is special while V2 is not. According to our definition, the trivial representation V0 is never special, since x y V0 G is generated by the single monomial 1 over the invariant ring. Our next theorem gives a rudimentary form of a McKay correspondence for the groups G d k .
Theorem 10.3.10 (McKay Correspondence). Let be a cone with parameters d k, where 0 k d and d k 1. Then there is a one-to-one correspondence between the representations of d that are special with respect to k and the components of the exceptional divisor for the minimal resolution ' X ! # U .
Proof. Write G Gd k as above and consider the set B of monomials in the complement of the ideal generated by the G-invariant monomials. This set contains
L 1 x x 2 x d
1
y y 2 y d
1
Since d k 1, for each 1 j d ! 1, there is an integer 1 a j d ! 1 such that x j and y a j have equal weight (equal to j) for the action of G. The representation V j is special with respect to k if and only if these are the only two monomials of weight j in the set B, and nonspecial if and only if there is some monomial x a y b with a b 0 in B which also has weight j. Since x j ! y a j I G , saying V j is special with respect to k is in turn equivalent to saying that the corresponding vector a j e1 je2 is an irreducible element in the semigroup from part (b) of Proposition 10.3.2 (Exercise 10.3.8). By Theorem 10.3.5, Cone a j e1 je2 is one of the 1-dimensional cones of the Gr¨obner fan of I G and V j corresponds to one of the 1-dimensional cones in the fan , hence to one of the irreducible components of the exceptional divisor.
The original McKay correspondence is the following special case.
Chapter 10. Toric Surfaces
478
Corollary 10.3.11. When k d ! 1, there is a one-to-one correspondence between the set of all irreducible representations of d and the components of the exceptional divisor of the minimal resolution ' X ! # U .
Proof. In this case, the invariant ring is x d xy y d , so the sets L and B in the proof of the theorem coincide. There has also been much work devoted to extend the McKay correspondence to finite abelian subgroups G GL n for n 3, and several other ways to understand these constructions have also been developed, including the theory of G-Hilbert schemes. See Exercise 10.3.10 for the beginnings of this. Exercises for §10.3. 10.3.1. In this exercise, you will verify the claims made in Example 10.3.1, and extend some of the observations there.
(a) Show that each of the
(b) Show that IG.
i
is a Gr¨obner basis of I G .
x 1 y x x y 1 x y y 1 5
3
2
2
5
is a universal Gr¨obner basis for
(c) Determine the cones C i using (10.3.1).
(d) Verify the final claim that linear transformation defined by the matrix A from (10.3.4) maps the Gr¨obner cones C i to the i for i 1 2 3.
10.3.2. Verify the conclusions of Proposition 10.3.2 and Theorem 10.3.5 for the case d 17 k 11.
10.3.3. Show that the set defined in part (b) of Proposition 10.3.2 is an additive semigroup. Hint: A direct proof starts from two general elements ae1 be2 and a e1 b e2 in . Consider x b y a x b y a and x b y a x b y a .
10.3.4. In this exercise you will complete the proof of Lemma 10.3.4. (a) Show that for each i, kPi mQi ki , where the ki are produced by the modified Euclidean algorithm from (10.2.1).
(b) Prove Lemma 10.3.3. (c) Verify that there is an index i0 as in (10.3.7). (d) Verify that if i0 is as in part (a), then gi0 basis.
%
1
is also an element of the reduced Gr¨obner
10.3.5. If G is any finite group, the (left) regular representation of G is defined as follows. Let W be a vector space over of dimension G with a basis eh h G indexed by the elements of G. For each g G let g W W be defined by g eh egh .
(a) Show that g g is a group homomorphism from G to GL W . (b) Now let G be the cyclic group of order d. Show that W is the direct sum of 1dimensional invariant subspaces W , j 0 * d 1 on which G acts by the character 0
d
'
j
j
defined in the text, so that W decomposes as W
d 1 j 0Vj.
10.3.6. In this exercise you will consider the module structures from Lemma 10.3.7.
§10.4. Smooth Toric Surfaces
479
(a) Prove Lemma 10.3.7. (b) Verify the claims made in Example 10.3.8.
10.3.7. In this exercise you will prove an alternate characterization of the special representations with respect to k from Definition 10.3.9. We write G Gd k d as usual.
% x y . Show that x y dx dy % % % x y dx dy
. (b) Show that the spaces of G-invariants and 2 V have the structure of
2
(a) Let 2 f dx dy f defines an action of G on
a b
2
a b 1 k a b
2
modules over the invariant ring
% x y
2
G
2
2
G
j
2
G
.
(c) Show that V j is special with respect to k if and only if the “multiplication map”
2 x y V V 2
G
j
2
G
2
j
2
is surjective.
G
10.3.8. In the proof of the McKay correspondence, show that a j e1 je2 is irreducible in the semigroup from Proposition 10.3.2 if and only if the representation V j is special with respect to k.
10.3.9. Let G Gd k , and let gi , i tion 10.3.2. Show that
0 r
1, be the binomials constructed in Proposi-
g * g x y 1 x y is G-invariant is a universal Gr¨obner basis for I G (not always minimal, however). as in (10.3.2). As a point set, G can be viewed as the 10.3.10. Let G G act on orbit of the point 1 1 under this action. The ideal I G is invariant under the action of G on % x y and as we have seen, the corresponding representation on x y , I G is isomorphic to the regular representation of G. (a) Show that if p is any point in other than the origin, the ideal I of the orbit of 1
a b
r
a b
2
dk
2
x y , I (b) The G-Hilbert scheme can be defined as the set of all G-invariant ideals in x y such that the representation of G on x y , I is isomorphic to the regular representation of G. Show that every such ideal has a set of generators of the form x y y x x 0 y 0 for some + and where x and y (respectively y and x ) have equal weights p is another G-invariant ideal and the corresponding representation of G on is also isomorphic to the regular representation of G.
a
c
b
a
d
c
a d b c
b
d
for the action of G. It can be seen from this result that the G-Hilbert scheme is also isomorphic to the minimal resolution of singularities of U .
§10.4. Smooth Toric Surfaces Classification of Smooth Toric Surfaces. We begin by using the results of §10.1 and §10.2 to give a complete classification of the smooth complete toric surfaces. Our theorem will say that these are all obtained by toric blowups from either 2 1 1 , or one of the Hirzebruch surfaces r with r 2 from Example 3.1.16. The proof will be based on the following facts.
Chapter 10. Toric Surfaces
480
First, Proposition 3.3.15 implies that if Cone u 1 u2 is a smooth cone and we refine by inserting the new 1-dimensional cone Cone u 1 u2 , then on the resulting toric surface, the smooth point p is blown up to a copy of 1 .
For the second ingredient of the proof, we introduce the following notation. If
is a smooth complete fan, then list the ray generators of the 2-dimensional cones in as u0 u1 ur 1 in clockwise order around the origin in N , and we will
consider the indices as integers modulo d, so u r u0 . Then we have the following statement parallel to (10.2.8). Lemma 10.4.1. Let u0 ur be the ray generators for a smooth complete fan
2 . For all i, there exist integers b , i in N * 0 r ! 1, such that i
(10.4.1)
ui
1
1
ui
bi ui
Proof. This is a special case of the wall relation (6.3.5). We also have the following result.
Lemma 10.4.2. Let be a smooth fan that refines a smooth cone . Then is obtained from by a sequence of star subdivisions as in Definition 3.3.13.
Proof. Say there are r 2-dimensional cones in and the ray generators in are u0 u1 ur , listed clockwise starting from u0 . We argue by induction on r. If r 1, there is only one cone in and there is nothing to prove. Assume the result has been proved for all with r 2-dimensional cones, and consider the case of r 1 cones. The proof of Lemma 10.4.1 works locally in the fan, so under the hypotheses here, for all i, 1 i r, there exist integers b i such that (10.4.1) holds. Note that is strongly convex so bi 0 for all i. We claim that there exists some i such that bi 1. If not, that is, if bi 2 for all i, then as in §10.2, the HirzebruchJung continued fraction b1 b2 bs represents a rational number d k and the cone has parameters d k with d k 0. But then d 2, which would contradict the assumption that is a smooth cone.
Hence there exists an i, 1
i r, such that ui
1
ui
1
ui
In this situation, Cone ui 1 ui 1 is also smooth (Exercise 10.4.1). Moreover, Cone ui 1 ui and Cone ui ui 1 are precisely the cones in the star subdivision of Cone ui 1 ui 1 . Then we are done by induction.
We are now ready to state our classification theorem.
Theorem 10.4.3. Every smooth complete toric surface X is obtained from either
2
1
1
or
r
r 2 by a finite sequence of blowups at fixed points of the torus action.
§10.4. Smooth Toric Surfaces
481
Proof. We label the ray generators of the fan as in Lemma 10.4.1. As in the proof of Lemma 10.4.2, if bi 1 in (10.4.1) for some i, then our surface is a blowup of the smooth surface corresponding to the fan where u i is removed. Hence we only need to consider the case in (10.4.1) where b i 1 for all i.
Suppose first that u j ! ui for some i j. We relabel the vertices to make u j ! u1 for some j. Note that j 2 since the cones must be strongly convex. Then from (10.4.1), u0
!
u2 b1 u1
Using the basis u1 u2 of N, we get the picture shown in Figure 3. Comparing this
u0 u1 u2 uj
Figure 3. The ray generators u0 u1 u2 u j when u j
u1
with the fans for 2 , 1 1 and a (Figures 2, 3 and 4 from §3.1), we see that
is a refinement of the fan of r if r b1 2. The same follows if b1 ! 2 and r b1 (see Exercise 10.4.2). Since b1 1, the remaining possibilities are b 1 0 or ! 1, where we get a refinement of the fan of 1 1 or 2 respectively. Then the theorem follows from Lemma 10.4.2 in this case.
2 since is When 1 has only three elements, it is easy to see that X smooth and complete (Exercise 10.4.3). So assume 1 3. We will prove that u j ! ui for some i j using primitive collections (Definition 5.1.5). This will complete the proof of the theorem.
We begin with two easy facts (Exercise 10.4.3):
X
1
3, every primitive collection of has exactly two elements.
is projective.
The second bullet allows us to use Proposition 7.3.6, which implies that has a primitive collection whose minimal generators sum to 0. Then the first bullet implies that this primitive collection has two elements, whose minimal generators ui u j satisfy ui u j 0.
Chapter 10. Toric Surfaces
482
Since there is also a Hirzebruch surface 1 , the statement of this theorem might seem puzzling. The reason that 1 is not included is that this surface is actually a blowup of 2 (Exercise 10.4.4).
The problem of classifying smooth compete toric varieties of higher dimension is much more difficult. We did this when rank Pic X 2 in Theorem 7.3.7. See [5] for the case when rank Pic X 3.
Intersection Products on Smooth Surfaces. A fundamental feature of the theory of smooth surfaces is the intersection product on divisors. In §6.2, we defined D C when D is a Cartier divisor and C is complete irreducible curve. On a smooth surface, C is a divisor, and taking D C gives the self-intersection D D D 2 . Here is a useful result about intersection numbers on a smooth toric surface.
Theorem 10.4.4. Let D be the divisor on a smooth toric surface X corresponding to Cone u which is the intersection of 2-dimensional cones Cone u u 1 and Cone u u2 in . Then: (a) D D
!
b, where u1 u2 bu as in (10.4.1).
(b) For a divisor D D , we have D
D
Cone ui i 1 2 otherwise
1 0
Proof. Since X is smoooth, part (a) follows from Lemma 6.3.4 once you compare (10.4.1) to (6.3.5). Part (b) follows from Corollary 6.3.3 and Lemma 6.3.4.
Example 10.4.5. Let Cone u1 u2 be a cone in a smooth fan , and consider the star subdivision, in which Cone u 1 u2 is inserted to subdivide into two cones. Call the refined fan . Then the exceptional divisor E of the blowup satisfies E E
!
1 on X .
'
X
# X
Complete curves with self-intersection number ! 1 on a smooth surface are called exceptional curves of the first kind. They can always be contracted to a smooth point on a birationally equivalent surface, as in the above example. One of the foundational results in the theory of general algebraic surfaces is that every smooth complete surface S has at least one relatively minimal model. This means that there is a birational morphism S # S, where S is a smooth surface with the property that if ' S # S is a birational morphism to another smooth surface S , then is necessarily an isomorphism. This is proved in [62, V 5.8]. Interestingly, the possible relatively minimal models for rational surfaces are precisely the surfaces 2 , 1 1 , and r for r 2 from Theorem 10.4.3. The proof essentially shows that after contracting all possible exceptional curves of the first kind, every smooth toric surface will be taken to one of these.
§10.4. Smooth Toric Surfaces
483
On a smooth complete toric surface X , the intersection product can be extended by linearity to all of Pic X . This yields a -valued symmetric bilinear form. Here is an example.
Example 10.4.6. Consider the Hirzebruch surface r . Using the fan shown in Figure 3 of Example 4.1.8, we get divisors D 1 D4 corresponding to minimal generators u1 ! e1 re2 u2 e2 u3 e1 u4 ! e2 . By Theorem 10.4.4, we have the self-intersections D1 D1 D3 D3 0 D 2 D2 The Picard group Pic
r
!
r D 4 D4 r
is generated by the classes of D3 and D4 . Note also that D3 D4 D4 D3 1
D
0
by Theorem 10.4.4. The intersection product is described by the matrix 3 D4
If D
aD3 bD4 and E D E
(10.4.2)
D3 D3 D4 D3 D4 D4
0
1 1 r
pD3 qD4 are any two divisors on the surface, then a b
1 1 r
For instance, with D E D2
!
p q
bp aq rbq
rD3 D4 , we obtain
D2 D2 1 ! r & ! r 1 r 1 1
!
r
The self-intersection numbers D1 D1 D3 D3 0 reflect the fibration structure on r studied in Example 3.3.20. The divisors D 1 and D3 are fibers of the map1 . Such curves always have self-intersection equal to zero. You will ping r # compute several other intersection products on r in Exercise 10.4.5.
Resolution of Singularities Reconsidered. Another interesting class of smooth toric surfaces are those that arise from a resolution of singularities of the affine toric surface U of a 2-dimensional cone . Here is a simple example.
Example 10.4.7. Let Cone e2 de1 ! e2 have parameters d 1 where d 1. The resolution of singularities X # U constructed in Example 10.1.8 uses the smooth refinement of obtained by adding Cone e 1 . This gives the exceptional divisor E on X . Since
e2 de1 ! e2 de1 we see that E E is the self-intersection number of E.
!
d
Chapter 10. Toric Surfaces
484
More generally, suppose that the smooth toric surface X is obtained via a resolution of singularities of U , where the 2-dimensional cone has parameters d k with d 1. Let the Hirzebruch-Jung continued fraction expansion of d k be
b b b
d k
1
2
r
Recall from Theorem 10.2.3 that is obtained from Cone u 0 ur rays generated by u1 ur , and by Theorem 10.2.6, we have ui
1
1
ui
bi ui
1
i
1
by adding
r
It follows that D1 Dr complete curves in X . They are the irreducible components of the exceptional fiber, with self-intersection Di Di
!
bi
1
i r
by Theorem 10.4.4. Then the intersection matrix D i D j Di D j
(10.4.3)
!
bi
1 0
1 i j r
is given by
if j i if i ! j 1 otherwise
In Exercise 10.4.6 you will show that the associated quadratic form is negative definite. This condition is necessary for the contractibility of a complete curve C on a smooth surface S, i.e., the existence of a proper birational morphism ' S # S, where C is a (possibly singular) point on S.
Rational Double Points Reconsidered. If has parameters d d ! 1, then from Exercise 10.2.4, the Hirzebruch-Jung continued fraction expansion of d d ! 1 is given by m m ! 1 2 2 2 with d ! 1 terms. Hence bi 2 for all i, and (10.4.3) gives the d ! 1 d ! 1 matrix
! 2 1 0 0 1 ! 2 1 0 .. .. .. . . . 0 0
0 1 ! 2 1 0 0 1 ! 2 representing the intersection product on the subgroup of Pic X generated by the components of the exceptional divisor for the resolution of a rational double point of type Ad 1 . We can now fully explain the terminology for these singularities.
The problem of classifying lattices
e1
es
with negative definite bilinear forms B satisfying B e i ei ! 2 for all i arises in many areas within mathematics, most notably in the classification of complex simple Lie algebras via root systems. The matrix above is the (negative of) the
§10.4. Smooth Toric Surfaces
485
Cartan matrix for the root system type A d Dynkin diagram:
1,
which is often represented by the
with d ! 1 vertices. The vertices represent the lattice basis vectors. The edges connect the pairs with B ei e j 0 and B ei ei ! 2 for all i as above. In our case, the vertices represent the components D i of the exceptional divisor, and the bilinear form is the intersection product.
A precise definition of a surface rational double point follows. We say that a resolution of singularities ' Y # X is minimal if no component of the exceptional divisor of the map is a curve with self-intersection number ! 1 (see Example 10.4.5 and the following comments). Definition 10.4.8. A singular point p of a surface X is a rational double point or Du Val singularity if there is a minimal resolution of singularities ' Y # X such that every irreducible component E i of the exceptional divisor E over p, 1 i r, satisfies KY Ei 0 where KY is a canonical divisor on Y . We can relate these concepts to the toric case as follows.
Proposition 10.4.9. Assume has parameters d k with d 1 and let ' X # be the resolution of singularities constructed in Theorem 10.2.3. Then: (a)
is a minimal resolution.
U
(b) The singular point p U is a rational double point if and only if k d ! 1. Proof. We showed above that the components D i of the exceptional fiber satisfy Di Di ! bi , where d k b1 br . Since bi 1 always holds in a HirzebruchJung continued fractions, we see that is minimal.
The canonical divisor of X is K ! D , and one computes that K D b ! 2 1 i r Thus p is a rational double point if and only if b 2 for all i. This easily implies r 1 i 0
X
X
i
i
i
i
k d ! 1. You will verify these claims in Exercise 10.4.7.
There is much more to say about rational double points. For example, one can show that E E1 Er satisfies E E ! 2 (you will prove this in the toric case in Exercise 10.4.7). From a more sophisticated point of view, E E ! 2 implies that the canonical sheaf on Y is the pullback of the canonical sheaf on X under , and this is one way to define rational singularities more generally. See [36] for more on rational double points.
Chapter 10. Toric Surfaces
486
Exercises for §10.4. 10.4.1. Here you will verify several statements made in the proof of Lemma 10.4.2. (a) Show that if the cone is strictly convex, then the integers ai in (10.4.1) must be strictly positive. (b) Show that if ui 1 ui 1 ui , then Cone ui 1 ui 1 must also be smooth.
0 %
0 %
u and u u , where r a .
10.4.2. In the proof of Theorem 10.4.3, verify that if u j a1 2, then is a refinement of a fan with X
1
0
r
a1 u1 with
2
1
10.4.3. In this exercise, you will prove some facts used in the proof of Theorem 10.4.3. 2 . Let be a complete fan in N
(a) If is smooth with 1 3, prove that X ! (b) If 1 3, prove that every primitive collection has two elements. 2
(c) Prove that X is projective. Hint: Find a convex polygon whose vertices are rational points lying on the rays of . For example, intersect the rays with a circle and move the points of intersection to a nearby rational points on each ray.
!
10.4.4. In the statement of Theorem 10.4.3, you might have noticed the absence of the Hirzebruch surface 1 . Show that this surface is actually isomorphic to the blowup of 2 at one torus-fixed point. 10.4.5. This exercise studies several further examples of the intersection product on (a) Compute D1 D1 using (10.4.2) and also directly from Theorem 10.4.4. (b) Compute K 2
K K on
r,
where K
K
r
r.
is the canonical divisor.
10.4.6. Show that the matrix defined by (10.4.3) has a negative-definite associated quadratic form. Hint: Recall that if B x y is a bilinear form, the associated quadratic form is Qx Bxx.
10.4.7. This exercise deals with the proof of Proposition 10.4.9. (a) Show that KX Di bi 2 for 1 i r.
. . (b) Show that d , k 2 * 2 if and only if k d 1. Hint: Exercise 10.2.4. (c) Show that E D D satisfies E E 2.
1
r
10.4.8. Let have invariants d d 1, so that the singular point of U is a rational double point. By Proposition 10.1.6, U is Gorenstein. This implies that its canonical sheaf U is a line bundle. Let X U be the resolution constructed in Theorem 10.2.3. Prove that U is the canonical sheaf of X .
"4
+
4
10.4.9. Another interesting numerical fact about the integers b i from (10.4.1) is the following. Suppose a smooth fan has 1-dimensional cones labeled as in Lemma 10.4.1. Then (10.4.4)
b0
b b 0 3r 12 1
r 1
This exercise will sketch a proof of this fact. (a) Show that (10.4.4) holds for the standard fans of
! ,! 2
1
!
1
and
a,
a
2.
(b) Show that if (10.4.4) holds for a smooth , then it holds for the fan obtained by performing a star subdivision on one of the 2-dimensional cones of . (c) Deduce that (10.4.4) holds for all smooth fans using Theorem 10.4.3.
§10.5. Riemann-Roch and Lattice Polygons
487
§10.5. Riemann-Roch and Lattice Polygons Riemann-Roch theorems are a class of results about the dimensions of sheaf cohomology groups. The original statement along these lines was the theorem of Riemann and Roch concerning sections of line bundles on algebraic curves. This result and its generalizations to higher-dimensional varieties can be formulated most conveniently in terms of the following notion of the Euler characteristic of a sheaf, defined in §9.4. Riemann-Roch for Curves. A modern form of the Riemann-Roch theorem for curves states that if D is a divisor on a smooth projective curve C, then (10.5.1)
D
C
D &
C
where the degree D is defined in Definition 6.2.2. This equality can be rewritten using Serre duality as follows. Namely, if KC is a canonical divisor on C, then we have H 1 C
C
D * H 0 C
C
KC ! D
H 0 C
C
KC
H 1 C
C *
The integer g H 0 C C KC is the genus of the curve C. Then (10.5.1) can be rewritten in the form commonly used in the theory of curves: (10.5.2)
H 0 C
C
D $!
H 0 C
C
KC ! D
D 1 ! g
A proof of this theorem and a number of its applications are given in [62, Ch. IV]. Also see Exercise 10.5.1 below. As a first consequence, note that if D KC is a canonical divisor, then
(10.5.3)
KC 2g ! 2
1 . Then g 0 We will need to use (10.5.2) most often in the simple case X * and the Riemann-Roch theorem for 1 is the statement for all divisors D on 1 ,
(10.5.4)
1
D
D 1
The Adjunction Formula. For a smooth curve C contained in a smooth surface X, the canonical sheaves C of the curve and X of the surface are related by (10.5.5)
C *
X
C
X
C
This follows without difficulty from Theorem 8.2.2 (Exercise 10.5.2) and has the following consequence for the intersection product on X. Theorem 10.5.1 (Adjunction Formula). Let C be a smooth curve contained in a smooth complete surface X. Then KX C C C 2g ! 2 where g is the genus of the curve C.
Chapter 10. Toric Surfaces
488
Proof. Let i ' C ( #
X be the inclusion map. Then C *
X
so that 2g ! 2
C
C
i
C
X
i
X
X
C i
KX C
X
KX C
KX C C
where the first equality is (10.5.3) and the last is the definition of K X C C given in §6.2. Riemann-Roch for Surfaces. The statement for surfaces corresponding to (10.5.1) is given next. Theorem 10.5.2 (Riemann-Roch for Surfaces). Let D be a divisor on a smooth projective surface X with canonical divisor KX . Then
X
D D ! D KX 2
D
X
We will only prove this for X a smooth complete toric surface; there is a simple and concrete proof in this case. Proof. Note that the theorem certainly holds if D 0, so X D * X . Our proof will use the special properties of smooth complete toric surfaces. Recall that if X X , then Pic X is generated by the classes of the divisors D i , i 1 r, corresponding to the 1-dimensional cones in . Hence, to prove the theorem, it suffices to show that if the theorem holds for a divisor D, then it also holds for D Di and D ! Di for all i.
Assume first that the theorem holds for D. From the exact sequence 0! # we tensor with
X
X !
Di ! #+
X !$#
Di !$#
0
D Di to obtain an exact sequence of sheaves
0 ! #+
D !$#+
X
X
D Di ! #+
Di
D Di !$#
0
By (9.4.1), it follows that
X
D Di
X
D
Di
D Di
By the induction hypothesis, the first term on the right is
(10.5.6)
X
D
D D ! D KX 2
X
The second term can be computed using the Riemann-Roch theorem for curves and 1 . By (10.5.4), the fact that Di *
(10.5.7)
Di
D Di D Di Di Di 1
§10.5. Riemann-Roch and Lattice Polygons
489
Combining (10.5.6) and (10.5.7), we have
X
D D ! D KX D Di Di Di 1 X
2 D Di D Di Di Di ! D KX 2 2 ! D1 Dr and Theorem 10.4.4, we obtain
D Di
However, using KX
X
Di KX ! Di Di ! 2
(10.5.8)
Substituting this into the above expression for X D Di and simplifying, we obtain D Di D Di ! D Di KX X D Di X 2 which shows that the theorem holds for D D i The proof for D ! Di is similar and is left to the reader (Exercise 10.5.3).
The following statement is sometimes considered as the topological part of the Riemann-Roch theorem for surfaces. Theorem 10.5.3 (Noether’s Theorem). Let X be a smooth projective surface with canonical divisor KX . Then KX KX e X 12 where e X is the topological Euler characteristic of X defined by
X
e X
4
!
1
i 0
i
Hi X
As before, we will give a proof only for a smooth complete toric surface. We will also use the Hodge decomposition
Hp X *
Hi X
i j p
j X
mentioned at the end of §9.4. Proof. Demazure vanishing (Theorem 9.2.3) implies that for a smooth complete toric surface X X , we have (10.5.9)
X
H 0 X
X $!
H 1 X
X
H 2 X
X
1 ! 0 0 1
Thus the Noether theorem for a smooth complete toric surface is equivalent to (10.5.10)
KX KX e X 12
Chapter 10. Toric Surfaces
490
We prove this as follows. Set r 1 and let the minimal generators of the rays r 1 be u0 ur 1 as in Lemma 10.4.1. Since KX ! i 0 Di , (10.5.8) implies
KX KX If ui
1
uu
i
r 1
!
i 0
Di KX
!
r 1
!
i 0
bi ui as in (10.4.1), then Di Di
r 1
KX KX 2r !
r 1
Di Di ! 2 2r !
i 0
Di Di
bi by Theorem 10.4.4. Hence
bi 2r ! 3r ! 12 12 ! r
i 0
where the second equality uses Exercise 10.4.9. We next compute e X . Exercise 10.4.3 shows that
a polygon with r 1 sides. Then the formula for Theorem 9.4.9 implies
is the normal fan of H p X qX given in
H 0 X
H 0 X
X
H1 X
H0 X
1 X & 2 X &
H2 X
H 3 X 4
H X
0
H0 X
1
H 1 X H1 X
X 1 X
H2 X
1 X
0 0 0
f1 ! 2 f2 r ! 2 H1 X 2
H X
2 X & 2 X
0 0 0 H 2 X
X
1
where f1 r and f2 1 are the face numbers of a polygon with r sides. It follows that e X 1 r ! 2 $ 1 r. Then (10.5.10) follows easily from the above computation of KX KX .
We will give a topological proof of e X r in Chapter 12, and in Chapter 13, we will interpret e X in terms of the Chern classes of the tangent bundle. The Riemann-Roch theorems for curves and surfaces have been vastly generalized by results of Hirzebruch and Grothendieck, and the precise relation of Noether’s theorem to the Riemann-Roch theorem for surfaces is a special case of their approach. We will discuss Riemann-Roch theorems higher-dimensional toric varieties in Chapter 13. Lattice Polygons. For the remainder of this section, we will explore the relation between toric surfaces and the geometry and combinatorics of lattice polygons. We will see that the results from §9.4 for lattice polytopes have an especially nice form for lattice polygons.
Let X X be a smooth complete toric surface. Since Riemann-Roch for a divisor D on X becomes D D ! D KX X D 1 (10.5.11) 2
X
1 by (10.5.9),
§10.5. Riemann-Roch and Lattice Polygons
491
Thus, for any ,
D D ! D KX
2
1 12 D D ! 12 D KX 1
2 The theory developed in §9.4 guarantees that X D is a polynomial in ; the above computation gives explicit formulas for the coefficients in terms of intersection products.
(10.5.12)
X
D
Here is an example of how this formula works. Example 10.5.4. Let X be the Hirzebruch surface 2 . We will use the notation from Example 10.4.6. We will study D D 1 D2 . Using KX ! D1 ! ! D4 and Example 10.4.6, it is easy to compute that D D 0
D KX
!
2
Then (10.5.12) implies
X
D 1
Now consider H p X X D for 0. An easy application of Serre duality 2 shows that H X X D 0 (Exercise 10.5.4). Thus
H 0 X
X
D !
H 1 X
X
D 1
Things get more surprising when we compute H 0 X X D . Using the ray generators u1 u4 from Example 10.4.6, the polygon PD corresponding to D D1 D2 is defined by the inequalities
m u
1
!
1 m u2
!
1 m u3 0 m u4 0
This gives the polygon shown in Figure 4. Even though it is not a lattice polytope, Proposition 4.3.3 still applies. Thus
H 0 X
X
D
PD
M
PD M
1 2 4 1 2 4
1
even
3 4
odd
u1 u2 u3
PD u4
Figure 4. The polygon of the divisor D and the fan of
2
Chapter 10. Toric Surfaces
492
where the final equality follows from Exercise 9.4.11. Combining this with the above computation of X D , we obtain
H 1 X
1 2 4 1 2 4 !
X D
even 1 4
odd
This is a vivid example how the Euler characteristic smooths out the complicated behavior of the individual cohomology groups.
On the other hand, if D is nef, the higher cohomology is trivial by Demazure vanishing, so that the Euler characteristic reduces to H 0 . We exploit this as follows. 2 be any lattice polygon (2-dimensional lattice polytope). Recall Let P M * from Theorem 9.4.2 and Example 9.4.4 that the Ehrhart polynomial L x x of P satisfies L P M Area P 2 12 P M 1 for . We next describe this polynomial in terms of intersection products.
By the results of §2.3, we get the projective toric surface XP coming from the normal fan P of P. In general XP will not be smooth, so we compute a minimal resolution of singularities ' X ! # XP using the methods of this chapter. Recall that XP has the ample divisor DP whose associated polygon is P.
Proposition 10.5.5. There is unique torus-invariant nef divisor D on X such that (a) The support function of D equals the support function of D P .
(b) X D is the Ehrhart polynomial of P. We call D the pullback of DP .
Proof. Proposition 6.1.20 implies that there is a divisor D on X such that satisfies part (a). Since DP has a convex support function, the same is true for D, so that D is nef. Furthermore, P is the polytope associated to D P and hence is the polytope associated to D since the polytope of a nef divisor is determined by its support function (Theorem 6.1.10).
H 0 X X D P D M P M when 0, It follows that 0 so that H X X D equals the Ehrhart polynomial of P when 0. However, D is nef when 0 and hence has trivial higher cohomology by Demazure vanishing (Theorem 9.2.3). Thus 0 implies
Since
X
X
D
H 0 X
X
D
P
M
D is a polynomial in , it must be the Ehrhart polynomial of P.
Theorem 10.5.5 and (10.5.12) imply that the Ehrhart polynomial of P is L
1 2
D D
2
!
1 2
D KX 1
§10.5. Riemann-Roch and Lattice Polygons
493
Comparing this to the above formula for L , we have proved the following result. Proposition 10.5.6. Let P be a lattice polygon and let D be the pullback of D P constructed in Proposition 10.5.5. Then D D 2 Area P !
D KX
P M
Example 10.5.7. Take the fan of 2 shown in Figure 4 from Example 10.5.4 and combine the two 2-dimensional cones containing u 2 into a single cone. The resulting fan has minimal generators u 1 u3 u4 that satisfy u1 u3 2u4 0, so the resulting toric variety is 1 1 2 .
Let P Conv 0 2e1 ! e2 M , which is the double of the polytope shown in Figure 4. The normal fan of P is the fan of 1 1 2 . The minimal generators u1 u3 u4 of this fan give divisors D1 D3 D4 on 1 1 2 , and the divisor DP is easily seen to be the ample divisor 2D 1 .
Since the fan of 2 refines the fan of 1 1 2 , the resulting toric morphism 1 1 2 is a resolution of singularities. By considering the support function of DP , we find that the pullback of DP 2D1 is D 2D1 2D2 . We leave it as Exercise 10.5.5 to compute D D and D K 2 and verify that they give the numbers predicted by Proposition 10.5.6.
2#
Sectional Genus. The divisor DP on XP is very ample since P 2. Hence it s s 1 restricts to such that gives a projective embedding XP ( # X DP . In s gives curves X geometric terms, this means that hyperplanes H XP P H that are linearly equivalent to DP . For some hyperplanes, the intersection XP H can be complicated. Since XP has only finitely many singular points, the Bertini theorem [62, II 8.18 and III 7.9.1] guarantees that when H is generic, C X P H is a smooth connected curve contained in the smooth locus of XP . The genus g of C is called the sectional genus of the surface XP .
We will compute g in terms of the geometry of P using the adjunction formula. Since we need a smooth surface for this, we use the resolution ' X # XP and note that C can be regarded as a curve in X since is an isomorphism away from the singular points of XP . Since C DP on XP , we have C D on X , where D is the pullback of DP . Then the adjunction formula (Theorem 10.5.1) implies
2g ! 2 KX
C C C KX
1 2
D KX D 1
so that (10.5.13)
g
D D D
Then we have the following result. Proposition 10.5.8. The sectional genus of XP is g
Int
P
M.
Chapter 10. Toric Surfaces
494
Proof. Pick’s formula from Example 9.4.4 can be written as Int
P
1 2
M Area P !
P M 1
which by Proposition 10.5.6 becomes Int
P
M
1 2D
1 2D
D
KX 1
The right-hand side is g by (10.5.13), completing the proof. Example 10.5.9. Let P Veronese embedding and DP are the curves of degree n in
2 in its nth Conv ne1 ne2 0 . Then XP 2 nL, where L is a line. The hyperplane sections 2 , and the smooth ones have genus n ! 1 n ! 2 g Int n M 2 You will check this assertion and another example in Exercise 10.5.6. n
The curves C X studied here can be generalized to the study of hypersurfaces in projective toric varieties coming from sections of a nef line bundle. The geometry and topology of these hypersurfaces have been studied in many papers, including [7, 8, 32, 101]. Reflexive Polygons and The Number 12. The final topic gives a way to understand a somewhat mysterious formula we noted in the last section of Chapter 8. Recall from Theorem 8.3.6 that there are exactly 16 equivalence classes of reflexive lattice polytopes in 2 , shown in Figure 3 of §8.3. The article [123] gives four different proofs of the following result.
Theorem 10.5.10. Let P be a reflexive lattice polygon in M *
P M P
2.
Then
N 12
One proof consists of a case-by-case verification of the statement for each of 16 equivalence classes. You proved the theorem this way in Exercise 8.3.4. The argument was straightforward but not very enlightening! Here we will give another proof using Noether’s theorem.
Proof. Since Noether’s theorem requires a smooth surface, we need to refine the normal fan P of P M . Since P is reflexive, we can do this using the dual polgyon P
N . We know from (8.3.2) that the vertices of P are the minimal generators of P . Let be the refinement of P whose 1-dimensional cones are generated by the rays through the lattice points on the boundary of P . This is illustrated in Figure 5 on the next page. The fan has the following properties:
For each cone of , its minimal generators and the origin form a triangle whose only lattice points are the vertices. Thus is smooth by Exercise 8.3.3.
The minimal generators of are the lattice points of lying on the boundary. Thus 1 P N .
§10.5. Riemann-Roch and Lattice Polygons
495
P
P
Figure 5. A reflexive polygon P and its dual P
From the first bullet, we get a resolution ' X # XP . Recall that DP ! KXP since P is reflexive. The wonderful fact is that its pullback via is again anticanonical, i.e., D ! KX . To prove this, recall that D and DP have the same support function , which takes the value 1 at the vertices of P since DP ! KXP . It follows that 1 on the boundary of P . Then D ! KX because the minimal generators of
all lie on the boundary.
Now apply Noether’s theorem to the toric surface X . By (10.5.10), we have KX
KX e X 12
We analyze each term on the left as follows. First, D KX
KX
!
D KX
!
KX implies
P M
where the last equality follows from Proposition 10.5.6. Second, e X is the number of minimal generators of by the proof of Theorem 10.5.3. In other words,
e X
1
P
N
where the second equality follows from the above analysis of . Hence the theorem is an immediate consequence of Noether’s theorem.
A key step in the above proof was showing that the pullback of the canonical divisor on XP was the canonical divisor on XP . This may fail for a general resolution of singularities. We will say more about this when we study crepant resolutions in Chapter 11. Exercises for §10.5. 10.5.1. The Riemann-Roch theorem for curves, in the form (10.5.1), can be proved by much the same method as used in the proof of Theorem 10.5.2. Namely, show that if (10.5.1) holds for a divisor D then it also holds for the divisors D P and D P, where P is an arbitrary point on the curve.
10.5.2. Prove the adjunction formula (Theorem 10.5.1) using (10.5.3) and (10.5.5). 10.5.3. Complete the proof of Theorem 10.5.2 by showing that if the theorem holds for D, then it also holds for D Di where Di is any one of the divisors corresponding to the 1-dimensional cones in .
Chapter 10. Toric Surfaces
496
6
10.5.4. Let D a D be an effective -Cartier Weil divisor on a complete toric variety X of dimension n. Use Serre duality (Theorem 9.2.10) to prove that H n X 0. X D
10.5.5. In Example 10.5.7, compute D D and D K numbers given by Proposition 10.5.6.
2
and check that they agree with the
10.5.6. This exercise studies the sectional genus of toric surfaces. (a) Verify the formula given in Example 10.5.9 for the sectional genus of Veronese embedding.
!
2
in its nth
(b) Let P Conv 0 ae1 be2 ae1 be2 . What is the smooth toric surface XP in this case? Show that its sectional genus is a 1 b 1 . 10.5.7. Prove that the singularities (if any) of the toric surface of a reflexive polygon are rational double points. Hint: Proposition 10.1.6.
! !
!
10.5.8. According to Theorem 10.4.3, every smooth toric surface is a blow-up of either 2 , 1 1 , or r for r 2. For each of the 16 reflexive polygons in Figure 3 of §8.3, the process described in the proof of Theorem 10.5.10 produces a smooth toric surface X . Where does X fit in this classification in each case? (This gives a classification of toric Del Pezzo surfaces.)
3
10.5.9. From §9.3, the p-Erhart polynomials of a lattice polygon P
M are given by
' D p 0 1 2 We know that L x is the usual Ehrhart polynomial L x , and then L x L x Theorem 9.4.5. The remaining case is L x . Prove that L x 2Area P x f 2 Lp
p XP
P
0
2
by
1
1
2
1
where f1 is the number of edges of P. Hint: Use Theorem 9.4.9 for the constant term and part (c) of Theorem 9.4.5 for the coefficient of x. For the leading coefficient, tensor the exact sequence of Theorem 8.1.6 with XP DP , take the Euler charactersitic, and then let .
6
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Index
absolutely ample, 333 abstract variety, 97, 99 affine piece of a variety, 51, 52 scheme, 4, 104 toric variety, 12, 18 of a cone, 30 variety, 3 affine cone of a projective toric variety, 56 of a projective variety, 50 affine hyperplane, 63 affine semigroup, 16 ample, see Cartier divisor, ample absolutely, 333 relatively with respect to , 334
basepoint free, see also sheaf, generated by global sections Cartier divisor, see Cartier divisor, basepoint free subspace of global sections, 255 basic simplex, 68 binomial, 16 Birkhoff polytope, 64 blowup, 98, 105, 112, 129, 130, 172, 174, 186, 188, 278, 279, 308 cancellative semigroup, 22 Carath´eodory’s theorem, 69 Cartier data, 179, 192 Cartier divisor, 158 ample, 260, 263, 271, 295 basepoint free, 256, 261, 263, 267, 283, 284, 316 -ample, 319 -very ample, 319 local data of, 158
nef, 283, 284, 295 numerical equivalence of, 285 numerically equivalent to zero, 285 of a polyhedron, 311 of a polytope, 181 on an irreducible variety, 255 torus-invariant, see torus-invariant, Cartier divisor very ample, 260, 263 character, 11 Chevalley’s Theorem, 142 class group, 159 classical topology, 4, 50, 102, 114, 138, 145 closed half-space, 24, 63 codimension of a prime ideal, 155 coherent sheaf, 164 combinatorially equivalent polytopes, 65 compact, 114, 138, 145 compatibility conditions, 97, 250 compatible map of lattices, 124 complete fan, 113, 138, 145, 191 complete linear system, 259 complete variety, 140, 145 cone convex polyhedral, see polyhedral cone of a polyhedron, 332 of a polyhedron, 308 of a polytope, 24, 70, 86 polyhedral, see polyhedral cone rational polyhedral, see rational polyhedral cone simplicial, see simplicial cone smooth, see smooth cone strongly convex, see strongly convex cone constructible set, 122 convex function, 265
477
Index
478
hull, 23 polyhedral cone, see polyhedral cone convex function, 316 convex support of full dimension, 315 coordinate ring, 3, 4 degenerate fan, 275 degree of a divisor on a curve, 280 of a line bundle on a curve, 280 determinantal variety, 13 diagonal map, 102 dimension at a point, 6, 100 dimension of a cone, 24 dimension of a polytope, 63 dimension of a ring, 154 direct image, 164 direct limit, 100, 243 directed set, 180, 243 directed system, 243 discrete valuation, 153 discrete valuation ring (DVR), 153 distinguished point, 116, 118, 129, 134 divisor Cartier, see Cartier divisor linear equivalence of, 158 locally principal, see locally principal divisor nef, see Cartier divisor, nef numerically effective, see Cartier divisor, nef of a character, 170 of a rational function, 157 of poles, 158 of zeros, 158 principal, 158 Weil, see Weil divisor doubly-stochastic matrix, 64 dual cone, 24 dual face, 26 dual polytope, 65 dual sheaf, 253 edge, 25, 63 effective divisor, 157 equivariant map, 41, 125 exact sequence of sheaves, 246 extremal ray, 296 extremal wall, see wall, extremal
-ample, see Cartier divisor, -ample -very ample, see Cartier divisor, -very ample 4ti2, 72 face, 25, 63 facet, 25, 63 normal, 26, 64 presentation, 67 fan, 77, 106 complete, see complete fan
normal, see normal fan refinement of, see refinement of a fan simplicial, see simplicial fan smooth, see smooth fan fiber bundle, 132 fibered product of varieties, 103 Fibonacci number, 22 finite quotient singularities, 114 finitely generated semigroup, 16 formal power series, 10 fractional ideal, 168 full dimensional polytope, 64 full dimesional convex support, 315 polyhedron, 309 function field, 100 GAGA, 141, 142 generated by global sections, see sheaf, generated by global sections global sections, 187 of a sheaf, 164, 246 of a toric sheaf, 188, 191 of a vector bundle, 249 gluing data, 97, 250 Gordan’s Lemma, 30 graded module, 224, 246, 299 shift of, 224 Grassmannian, 252, 259
sheaf, 247, 253 Hausdorff topological space, 102 height of a prime ideal, 155 Hilbert basis, 32, 33 Hilbert Basis Theorem, 3 Hirzebruch surface, 112, 173, 189, 261, 268, 273, 293, 325 homogeneous coordinate ring, 49 homogeneous coordinates, 49 homogenization, 190 homomorphism of sheaves, 164, 244 injective, 244 sujective, 244 image sheaf, 245 index of a simplicial cone, 290 inner normal fan, see normal fan integral closure, 5 integrally closed ring, 5 intersection product, 281 on a toric variety, 282, 290 inverse limit, 180 inverse system, 180 invertible sheaf, 166, 252, 253 inward-pointing facet normal, see facet, normal irreducible components, 97
Index
irreducible variety, 97 irrelevant ideal, 204 isomorphism of varieties, 96 Jacobian matrix, 7 kernel sheaf, 245 Kleinschmidt’s classification theorem, 328 Krull Principal Ideal Theorem, 159, 168 lattice, 13 lattice ideal, 16 lattice polyhedron, 308 lattice polytope, 66 Laurent polynomial, 5 limit of one-parameter subgroup, 115, 117, 138, 141, 145 line bundle, 251, 253 ample, 260 -ample, 319 -very ample, 319 pullback of, 256 very ample, 260 linear equivalence, see divisor, linear equivalence of linearly equivalent divisors, 254 local data, 158, 251, 255 toric, 179 local ring, 6, 154 at a point, 6, 9, 94, 99 at a prime divisor, 155 localization, 5, 9 homogeneous, 332 locally principal divisor, 158 locally trival fiber bundle, 133
maximal spectrum, 4 minimal generator, 29 Minkowski sum, 65, 191, 307 Mori cone, 286 of a toric variety, 287, 295 morphism of varieties, 3, 95, 100 projective, see projective morphism proper, see proper morphism multipliciative subset, 9 Nakayama’s Lemma, 167 nef cone, 286 of a toric variety, 286 nef divisor, see Cartier divisor, nef Newton polytope, 186 nilpotents, 4, 8, 9, 48, 104, 105 Noetherian, 154 nonnormal toric variety, 148 nonsingular point, 6 normal affine toric variety, 37
479
polyhedron, 312 polytope, 68, 86 toric variety, 86, 108 variety, 5, 100 normal fan of a polyhedron, 310 of a polytope, 76, 77, 109, 269 normal ring, 5, 154, 155 normalization, 5, 149 of a projective toric variety, 151 of an affine toric variety, 39, 150 of an irreducible curve, 281 Nullstellensatz, 3 numerically effective divisor, see Cartier divisor, nef numerically equivalent divisors, see Cartier divisor, numerical equivalence of proper -cycles, see proper -cycle, numerical equivalence of to zero, see Cartier divisor, numerically equivalent to zero and proper -cycle, numerically equivalent to zero
one-parameter subgroup, 11 orbifold, 46, 114 orbit closure, 121, 134 Orbit-Cone Correspondence, 119 nonnormal case, 152 order of vanishing, 156 perfect field, 48 permutation matrix, 56 Picard group, 159, 254 pointed affine semigroup, 36 polar polytope, 65 pole, 157 polyhedral cone, 23 polyhedron, 188, 307 full dimensional, see full dimensional, polyhedron lattice, see lattice polyhedron normal, see normal, polyhedron of a torus-invariant divisor, 188, 266, 316 very ample, see very ample, polyhedron polytope, 23, 63 combinatorially equivalent, see combinatorially equivalent polytopes full dimensional, see full dimensional, polytope lattice, see lattice polytope normal, see normal, polytope simple, see simple polytope simplicial, see simplicial polytope smooth, see smooth polytope very ample, see very ample, polytope pre-variety, 103 presheaf, 163, 245
Index
480
primary decomposition, 159 prime divisor, 155 primitive collection, 294, 326 primitive relation, 295, 327 principal divisor, see divisor, principal principal ideal domain (PID), 154, 155 product variety, 7, 53, 101, 303 class group of, 173 toric, 47, 90, 111 Proj, 311, 332 projective bundle of a coherent sheaf, 307 of a locally free sheaf, 306 of a vector bundle, 306 toric, 324 projective morphism, 304, 305, 335 projective space, 49 projective toric variety, 55 projective variety, 49 projective with respect to a line bundle, 304, 334 projectively normal variety, 61, 86 proper -cycle, 285 numerical equivalence of, 285 numerically equivalent to zero, 285 proper continuous map, 139, 141 proper face, 25 proper morphism, 140, 141, 305 Puiseux series, 187 pullback, see line bundle, pullback of pullback of a torus-invariant Cartier divisor, 274
-Cartier divisor, 179 -factorial, 46 quasicoherent sheaf, 164 quasicompact, 104 quasiprojective toric variety, 314, 318 quasiprojective variety, 303, 305 rational function, 51, 100 rational normal cone, 13, 32, 38, 40, 46, 50, 176 rational normal curve, 50, 57 rational normal scroll, 84, 112 rational polyhedral cone, 29 ray generator, 29 real projective plane, 72 recession cone, 308 refinement of a fan, 129 reflexive polytope, 81, 87 reflexive sheaf, 166 regular cone, see smooth cone regular fan, 113 regular local ring, 7, 155 regular map, 93, 98 relative interior, 27 relatively ample with respect to , 334 restriction of a divisor, 158 ring of invariants, 44
ringed space, 95, 99 saturated affine semigroup, 37 section of a sheaf, 163 of a vector bundle, 249 Segre embedding, 52 self-intersection, 294 semigroup, 16 semigroup algebra, 17 of a cone, 30 semigroup homomorphism, 35, 116 separated variety, 102 separating transcendence basis, 48 Separation Lemma, 28, 107 set-theoretic complete intersection, 22 sheaf, 95 constant, 253 generated by global sections, 247, 256, 335 locally constant, 253 locally free, 248 -modules, 163, 243 of of a graded module, 225, 246, 299 of a torus-invariant divisor, 187 global sections of, 187, 188, 191 of sections of a vector bundle, 250 sheafification, 245 simple polytope, 65 simplex, 65 simplicial cone, 30 index of, 290 simplicial fan, 113 simplicial polytope, 65 simplicial toric variety, 178 singular point, 6 singular locus, 161 of a normal variety, 161 Smith normal form, 171 smooth cone, 30, 40 smooth fan, 113 smooth point, 6, 100 smooth polytope, 87 smooth toric variety, 40, 87, 114, 178 smooth variety, 6, 100 span of a cone, 24 Spec, 4, 332 splitting fan, 132 stalk of a sheaf, 244, 255 , 66 standard -simplex star subdivision, 129, 131 strictly convex, see support function, of a Cartier divisor, strictly convex strongly convex cone, 28 structure sheaf, 95, 99 sublattice of finite index, 44 subvariety, 97 Sumihiro’s Theorem, 109, 148
Index
support function, 181 integral with respect to a lattice, 182 of a Cartier divisor, 182, 264 convex, 265, 267, 316 strictly convex, 270, 271, 317–319, 321 of a polytope, 185 support of a divisor, 157 support of a fan, 113 supporting affine hyperplane, 63 supporting half-space, 24 supporting hyperplane, 24 Sylvester sequence, 92
-neighborly polytope, 73 tautological bundle, 252, 254, 259 tensor product, 8, 48 of sheaves, 247, 253 tent analogy, 264, 265, 268 Toric Chow Lemma, 277 Toric Cone Theorem, 287, 296 toric fibration, 133 toric ideal, 16 Toric Kleiman Criterion, 284 toric morphism, 41, 42, 125, 134 projective, 314, 318, 321 proper, 141 toric set, 21 toric variety, 106 affine, see affine, toric variety normal, see normal, toric variety of a basepoint free divisor, 274 of a fan, 108 of a polyhedron, 310, 314 of a polytope, 83, 109 projective, see projective toric variety quasiprojective, see quasiprojective toric variety torsion-free semigroup, 22 torus, 5, 10 embedding, 108 of a projective toric variety, 58 of an affine toric variety, 13 orbit, 118 torus-invariant Cartier divisor, 175 prime divisor, 169 Weil divisor, 170, 174 total coordinate ring, 246, 299 transportation polytope, 64 tropical polynomial, 186 tropical variety, 187 tropicalization, 186, 187 unique factorization domain (UFD), 7 universally closed, 140 valuative criterion for properness, 146 variety
481
abstract, see abstract variety affine, see affine, variety complete, see complete variety irreducible, see irreducible variety normal, see normal, variety projective, see projective variety projectively normal, see projectively normal variety quasiprojective, see quasiprojective variety separated, see separated variety toric, see toric variety vector bundle, 248 chart of, 248 decomposable, 323, 324 fiber above a point, 248, 255 sheaf of sections of, 250 toric, 322, 324 transition functions of, 248, 250 trivialization of, 248 vertex, 63 very ample divisor, see Cartier divisor, very ample polyhedron, 312 polytope, 71, 75, 87, 263 wall, 265, 291 extremal, 296 wall relation, 292 weighted homogeneous polynomial, 53 weighted projective space, 53, 112, 174, 186 Weil divisor, 157 torus-invariant, see torus-invariant, Weil divisor Zariski closure, 4 Zariski tangent space, 6, 100 Zariski topology, 4, 50, 97
