Deformations of Algebraic Schemes

Deformations of Algebraic Schemes EDOARDO SERNESI 2 Preface In some sense deformation theory is as old as algebraic ...

Author: Edoardo Sernesi

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Deformations of Algebraic Schemes EDOARDO SERNESI

2

Preface In some sense deformation theory is as old as algebraic geometry itself: this is because all algebro-geometric objects can be “deformed” by suitably varying the coefficients of their defining equations, and this has of course always been known by the classical geometers. Nevertheless a correct understanding of what “deforming” means leads into the technically most difficult parts of our discipline. It is fair to say that such technical obstacles have had a vast impact on the crisis of the classical language and on the development of the modern one, based on the theory of schemes and on cohomological methods. The modern point of view originates from the seminal work of Kodaira and Spencer on small deformations of complex analytic manifolds and from its formalization and translation into the language of schemes given by Grothendieck. Here I will not recount the history of the subject since good surveys already exist (e.g. [189], [28], [154]). Today, while this area is rapidly developing, a self-contained text covering the basic results of what we can call “classical deformation theory” seems to be missing. Moreover a number of technicalities and of “well known” facts are scattered in a vast literature as folklore, sometimes with proofs available only in the complex analytic category. This book is an attempt to fill such a gap, at least partially. More precisely it aims at giving an account with complete proofs of the results and techniques which are needed to understand the local deformation theory of algebraic schemes over an algebraically closed field, thus providing the tools needed for example in the local study of Hilbert schemes and moduli problems. The existing monographs, like [15], [101], [122], [183], [205], all aim at goals different from the above. For these reasons my approach has been to work exclusively in the category of locally noetherian schemes over a fixed algebraically closed field k, avoiding to switch back and forth between the algebraic and the analytic category. I tried to make the text self-contained as much as possible, but i

ii without forgetting that all the technical ideas and prerequisites can be found in [3] and [2]: therefore the reader is advised to read this text keeping a copy of them on his table. In any case a good familiarity with [89] and with a standard text in commutative algebra like [49] or [139] will be generally sufficient; the classical [188] and [213] will be also useful. A good acquaintance with homological algebra is assumed throughout. One of the difficulties of writing about this subject is that it needs a great deal of technical results, which make it hard to maintain a proper balance between generality and understandability. In order to overcome this problem I tried to keep the technicalities to a minimum, and I introduced the main deformation problems in an elementary fashion in Chapter I; they are then reconsidered as functors of Artin rings in Chapter II, where the main results of the theory are proved. The first two chapters therefore give a self-contained treatment of formal deformation theory via the “classical” approach; cotangent complexes and functors are not introduced, nor the method of differential graded Lie algebras. Another chapter treats in more detail the most important deformation functors, with the single exception of vector bundles; this being motivated by reasons of space and because good monographs on the subject are already available (e.g. [98], [60]). Although they are not the central issue of the book, I considered necessary to include a chapter on Hilbert schemes and Quot schemes, since it would be impossible to give meaningful examples and applications without them, and because of the lack of an appropriate reference. Deformation theory is closely tied with classical algebraic geometry because some of the issues which had remained controversial and unclear in the old language have found a natural explanation using the methods discussed here. I have included a section on plane curves which gives a good illustration of this point. Unfortunately important topics and results have been omitted because of lack of space, energy and competence. In particular I did not include the construction of any global moduli spaces/stacks, which would have taken me too far from the main theme. The book is organized in the following way. Chapter I starts with a concise treatment of algebra extensions which are fundamental in deformation theory. It then discusses locally trivial infinitesimal deformations of algebraic schemes. Chapter II deals with “functors of Artin rings”, the abstract tool for the study of formal deformation theory. The main result of this theory is Schlessinger’s theorem. A section on obstruction theory, an elementary but

iii crucial technical point, is included. We discuss the relation between formal and algebraic deformations and the algebraization problem. This part is not entirely self contained since Artin’s algebraization theorem is not proved in general and the approximation theorem is only stated. The last section explains the role of automorphisms and the related notion of “isotriviality”. Chapter III is an introduction to the most important deformation problems. By applying Schlessinger’s theorem to them, we derive the existence of formal (semi)universal deformations. Many examples are discussed in detail so that all the basic principles of deformation theory become visible. This chapter can be used as a reference for several standard facts of deformation theory, and it can be also helpful in supplementing the study of the more abstract Chapter II. Chapter IV is devoted to the construction and general properties of Hilbert schemes, Quot schemes and their variants, the “flag Hilbert schemes”. It ends with a section on plane curves, where the main properties of Severi varieties are discussed. Our approach to the proof of existence of nodal curves with any number of nodes uses multiple point schemes and is apparently new. In the Appendices we have collected several topics which are well known and standard but we felt it would be convenient for the reader to have them available here. Acknowledgements. Firstly I would like to express my deepest gratitude to D. Lieberman, who initiated me to the study of deformations of complex manifolds long time ago. More recently R. Hartshorne, after a careful reading of a previous draft of this book, made me a number of comments and suggestions which significantly contributed in improving it. I warmly thank him for his generous help. For many extremely useful remarks on another draft of the book I am also indebted to M. Brion. I gratefully acknowledge comments and suggestions from other colleagues and students, in particular: L. Badescu, I. Bauer, A. Bruno, M. Gonzalez, A. Lopez, M. Manetti, A. Molina Rojas, D. Tossici, A. Verra, A. Vistoli. I would also like to thank L. Caporaso, F. Catanese, C. Ciliberto, L. Ein, D. Laksov and H. Lange for encouragement and support which have been of great help.

iv

Contents Terminology and Notation

vii

Introduction

1

1 Infinitesimal deformations 1.1 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 The module ExA (R, I) . . . . . . . . . . . . . . . . . . 1.1.3 Extensions of schemes . . . . . . . . . . . . . . . . . . 1.2 Locally trivial deformations . . . . . . . . . . . . . . . . . . . 1.2.1 Generalities on deformations . . . . . . . . . . . . . . . 1.2.2 Infinitesimal deformations of nonsingular affine schemes 1.2.3 Extending automorphisms of deformations . . . . . . . 1.2.4 First order locally trivial deformations . . . . . . . . . 1.2.5 Higher order deformations - Obstructions . . . . . . . . 2 Formal deformation theory 2.1 Obstructions . . . . . . . . . . . . . . . 2.2 Functors of Artin rings . . . . . . . . . 2.3 The theorem of Schlessinger . . . . . . 2.4 The local moduli functors . . . . . . . 2.4.1 Generalities . . . . . . . . . . . 2.4.2 Obstruction spaces . . . . . . . 2.4.3 Algebraic surfaces . . . . . . . . 2.5 Formal versus algebraic deformations . 2.6 Automorphisms and prorepresentability 2.6.1 The automorphism functor . . . 2.6.2 Isotriviality . . . . . . . . . . . v

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7 7 7 11 15 21 21 25 28 31 35 41 41 50 63 74 74 81 84 87 104 105 114

vi

CONTENTS

3 Examples of deformation functors 121 3.1 Affine schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.1.1 First order deformations . . . . . . . . . . . . . . . . . 122 3.1.2 The second cotangent module and obstructions . . . . 128 3.1.3 Comparison with deformations of the nonsingular locus 137 3.1.4 Quotient singularities . . . . . . . . . . . . . . . . . . . 142 3.2 Closed subschemes . . . . . . . . . . . . . . . . . . . . . . . . 144 3.2.1 The local Hilbert functor . . . . . . . . . . . . . . . . . 144 3.2.2 Obstructions . . . . . . . . . . . . . . . . . . . . . . . 153 3.2.3 The forgetful morphism . . . . . . . . . . . . . . . . . 156 3.2.4 The local relative Hilbert functor . . . . . . . . . . . . 161 3.3 Invertible sheaves . . . . . . . . . . . . . . . . . . . . . . . . . 163 3.3.1 The local Picard functors . . . . . . . . . . . . . . . . . 163 3.3.2 Deformations of sections, I . . . . . . . . . . . . . . . . 167 3.3.3 Deformations of pairs (X, L) . . . . . . . . . . . . . . . 172 3.3.4 Deformations of sections, II . . . . . . . . . . . . . . . 181 3.4 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 3.4.1 Deformations of a morphism leaving domain and target fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 3.4.2 Deformations of a morphism leaving the target fixed . 191 3.4.3 Morphisms from a nonsingular curve with fixed target . 205 3.4.4 Deformations of a closed embedding . . . . . . . . . . . 210 3.4.5 Stability and costability . . . . . . . . . . . . . . . . . 214 4 Hilbert and Quot schemes 4.1 Castelnuovo-Mumford regularity . . . . . 4.2 Flatness in the projective case . . . . . . 4.2.1 Flatness and Hilbert polynomials 4.2.2 Stratifications . . . . . . . . . . . 4.2.3 Flattening stratifications . . . . . 4.3 Hilbert schemes . . . . . . . . . . . . . . 4.3.1 Generalities . . . . . . . . . . . . 4.3.2 Linear systems . . . . . . . . . . 4.3.3 Grassmannians . . . . . . . . . . 4.3.4 Existence . . . . . . . . . . . . . 4.4 Quot schemes . . . . . . . . . . . . . . . 4.4.1 Existence . . . . . . . . . . . . . 4.4.2 Local properties . . . . . . . . . .

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223 223 232 232 236 239 246 246 248 250 255 263 263 267

CONTENTS 4.5

Flag Hilbert schemes . . . . . . . . . . . . . . . . . . 4.5.1 Existence . . . . . . . . . . . . . . . . . . . . 4.5.2 Local properties . . . . . . . . . . . . . . . . . 4.6 Examples and applications . . . . . . . . . . . . . . . 4.6.1 Complete intersections . . . . . . . . . . . . . 4.6.2 An obstructed nonsingular curve in IP 3 . . . 4.6.3 An obstructed (non-reduced) scheme . . . . . 4.6.4 Relative grassmannians and projective bundles 4.6.5 Hilbert schemes of points . . . . . . . . . . . . 4.6.6 Schemes of morphisms . . . . . . . . . . . . . 4.6.7 Focal loci . . . . . . . . . . . . . . . . . . . . 4.7 Plane curves . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Equisingular infinitesimal deformations . . . . 4.7.2 The Severi varieties . . . . . . . . . . . . . . . 4.7.3 Non-emptiness of Severi varieties . . . . . . .

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271 271 275 281 281 283 285 287 296 299 301 304 304 307 314

A Flatness

323

B Differentials

335

C Smoothness

351

D Complete intersections 365 D.1 Regular embeddings . . . . . . . . . . . . . . . . . . . . . . . 365 D.2 Relative complete intersection morphisms . . . . . . . . . . . . 368 E Functorial language

375

List of symbols

401

viii

CONTENTS

Terminology and Notation All rings will be commutative with 1. A ring homomorphism A → B is called essentially of finite type (e.f.t.) if B is a localization of an A-algebra of finite type. We will also say that B is e.f.t. over A. We will always denote by k a fixed algebraically closed field. All schemes will be assumed to be defined over k, locally noetherian and separated, and all algebraic sheaves will be quasi-coherent unless otherwise specified. If X and Y are schemes we will write X × Y instead of X ×k Y . If S is a scheme and s ∈ S we denote by k(s) = OS,s /mS,s the residue field of S at s. As customary various categories will be denoted by indicating their objects within round parentheses when it will be clear what the morphisms in the category are. For instance (sets), (A-modules), etc. The class of objects of a category C will be denoted by ob(C). The dual of a category C will be denoted by C ◦ . Given categories C and D, a contravariant functor F from C to D will be always denoted as a covariant functor F : C ◦ → D. We will consider the following categories of k-algebras: A

= the category of local artinian k-algebras with residue field k ˆ A = the category of complete local noetherian k-algebras with residue field k ∗ A = the category of local noetherian k-algebras with residue field k (k-algebras) = the category of noetherian k-algebras

Morphisms are unitary k-homomorphisms, which are local in A, Aˆ and A∗ . ix

x

CONTENTS

For a given Λ in ob(A∗ ) we will consider the following: AΛ = the category of local artinian Λ-algebras with residue field k ∗ AΛ = the category of local noetherian Λ-algebras with residue field k ˆ then we They are subcategories of A and A∗ respectively. If Λ is in ob(A) will let AˆΛ = the category of complete local noetherian Λ-algebras with residue field k ˆ Moreover we will set: which is a subcategory of A. (schemes) = the category of schemes (i.e. of locally noetherian separated k-schemes) and (algschemes) = the category of algebraic schemes For a given scheme Z we set (schemes/Z) = the category of Z-schemes (algschemes/Z) = the category of algebraic Z-schemes hi (X, F) denotes dim[H i (X, F)] where F is a coherent sheaf on the complete scheme X. When no confusion is possible we will sometimes write H i (F) and hi (F) instead of H i (X, F) and hi (X, F) respectively. ` i Xi denotes the disjoint union of the schemes Xi . If E is a vector space or a locally free sheaf we will always denote its dual by E ∨ . If V is a k-vector space we will denote by IP (V ) the projective space Proj(Sym(V ∨ )) (where Sym(−) is the symmetric algebra of −): thus the closed points of IP (V ) are the 1-dimensional subspaces of V . Similarly, if E is a locally free sheaf on an algebraic scheme S, the projective bundle associated to E will be defined as IP (E) = Proj(Sym(E ∨ )) Note that this definition is dual to the one given in [89], p. 162. For all definitions not explicitly given we will refer to [89].

Introduction La m´ethode g´en´erale consiste toujours `a faire des constructions formelles, ce qui consiste essentiellement `a faire de la g´eom´etrie alg´ebrique sur un anneau artinien, et `a en tirer des conclusions de nature “alg´ebrique” en utilisant les trois th´eor`emes fondamentaux (Grothendieck [75], p. 11).

Deformation theory is a formalization of the Kodaira, Nirenberg, Spencer, Kuranishi (KNSK) approach to the study of small deformations of complex manifolds. Its main ideas are clearly outlined in the series of Bourbaki seminar exposes by Grothendieck which go under the name of “Fondements de la G´eom´etrie Alg´ebrique” [2]; in particular they are explained in detail in [76] (see especially page 17), while the technical foundations are laid in [75]. The quotation at the top of this page gives a concise description of the method employed. The first step of this formalization consists in studying infinitesimal deformations, and this is accomplished via the notion of “functor of Artin rings”; the study of such functors leads to the construction of “formal deformations”. This method enhances the analogies between the analytic and the algebraic cases, and at the same time hides some delicate phenomena typical of the algebraic geometrical world. These phenomena become visible when one tries to pass from formal to algebraic deformations. The techniques of deformation theory have a variety of applications which make them an extremely useful tool, especially in understanding the local structure of schemes defined by geometrical conditions or by functorial constructions. In this introduction we shall explain in outline the logical structure of deformation theory; for this purpose we will start by outlining the KNSK theory of small deformations of compact complex manifolds. Given a compact complex manifold X, a family of deformations of X is 1

2

CONTENTS

a commutative diagram of holomorphic maps between complex manifolds X ξ: ↓ ?

⊂

X ↓π to −→ B

with π proper and smooth (i.e. with everywhere surjective differential), B connected and where ? denotes the singleton space. We denote by Xt the fibre π −1 (t), t ∈ B. It is a standard fact that, locally on B, X is differentiably a product so that π can be viewed locally as a family of complex structures on the differentiable manifold Xdiff . The family ξ is trivial at to if there is a neighborhood U ⊂ B of to such that we have π −1 (U ) ∼ = X × U analytically. ∂ Kodaira and Spencer started by defining, for every tangent vector ∂t ∈ ∂ Tto B, the derivative of the family π along ∂t as an element ∂Xt ∈ H 1 (X, TX ) ∂t thus giving a linear map κ : Tto B → H 1 (X, TX ) called the Kodaira-Spencer map of the family π. They showed that if π is ∂ ∂ trivial at to then κ( ∂t ) = 0 for all ∂t ∈ Tto B. Then they investigated the problem of classifying all small deformations of X, by constructing a “complete family” of deformations of X. A family ξ as above is called complete if for every other family of deformations of X: X η: ↓ ?

⊂

Y ↓p mo −→ M

there is an open neighborhood V ⊂ M and a commutative diagram X . p−1 (V ) ↓ V

& → →

X ↓ B

inducing an isomorphism p−1 (V ) ∼ = V ×B X . The family is called universal if it is complete and moreover the morphism V → B is unique locally around

CONTENTS

3

mo for each family η as above. Kodaira and Spencer proved that if κ is surjective then the family ξ is complete. The following existence result was then proved: Theorem 0.0.1 (Kodaira-Nirenberg-Spencer[119]) If H 2 (X, TX ) = 0 then there exists a complete family of deformations of X whose KodairaSpencer map is an isomorphism. If moreover H 0 (X, TX ) = 0 then such complete family is universal. Later Kuranishi [124] generalized this result by showing that a complete family of deformations of X such that κ is an isomorphism exists without assumptions on H 2 (X, TX ) provided the base B is allowed to be an analytic space. We want to rephrase everything algebraically as far as possible. Let’s fix an algebraically closed field k and consider an algebraic k-scheme X. A local deformation, or a local family of deformations, of X is a cartesian diagram ξ:

X → X ↓ ↓π Spec(k) ⊂ S

where π is a flat morphism, S = Spec(A) where A is a local k-algebra with residue field k, and X is identified with the fibre over the closed point. If X is nonsingular and/or projective we will require π to be smooth and/or projective. We say that ξ is a deformation over Spec(A) or over A. If in particular A is an artinian local k-algebra then we speak of an infinitesimal deformation. The notion of local family has the fundamental property of being funtorial. Given two deformations of X: X → X X → Y ξ: ↓ ↓π and η: ↓ ↓ρ Spec(k) ⊂ Spec(A) Spec(k) ⊂ Spec(A) parametrized by the same Spec(A), an isomorphism ξ ∼ = η is defined to be a morphism f : X → Y of schemes over Spec(A) inducing the identity on the closed fibre, i.e. such that the following diagram X .

& f

X

−→ &

Y .

Spec(A)

4

CONTENTS

is commutative. Consider the category A∗ = (noetherian local k-algebras with residue field k) and its full subcategory A = (artinian local k-algebras with residue field k) One defines a covariant functor Def X : A∗ → (sets) by Def X (A) = {local deformations of X over Spec(A)}/(isomorphism) This is the functor of local deformations of X; its restriction to A is the functor of infinitesimal deformations of X. One may now ask whether Def X is representable, namely if there is a noetherian local k-algebra O and a local deformation X → X◦ ↓ ↓p υ: Spec(k) ⊂ Spec(O) which is universal, i.e. such that any other local deformation ξ is obtained by pulling back υ under a unique Spec(A) → Spec(O). The approach of Grothendieck to this problem was to formalize the method of Kodaira and Spencer, which consists in a formal construction followed by a proof of convergence. In the search for the universal deformation υ the formal construction corresponds to the construction of the sequence of its restrictions to the truncations Spec(O/mn+1 O ): X → Xn◦ un : ↓ ↓ Spec(k) → Spec(O/mn+1 O )

n≥0

These are infinitesimal deformations of X because the rings O/mn+1 are in O A. The sequence uˆ = {un } can be considered as a formal approximation of υ. It is a special case of a formal deformation: more precisely, a formal deformation of X is given by a complete local k-algebra R with residue field k and by a sequence of infinitesimal deformations ξn :

X → Xn ↓ ↓ Spec(k) → Spec(R/mn+1 R )

n≥0

CONTENTS

5

such that ξn 7→ ξn−1 under the truncation R/mn+1 → R/mnR . In our case R ˆ The goal of the formal step in deformation theory is the construction R = O. of uˆ for a given X, i.e. of a formal deformation having a suitable universal property which is inherited from the corresponding property of υ, and which we do not need to specify now. Observe that in trying to perform the formal step we will at best succeed ˆ and not O. Since a formal deformation consists of infinitesin describing O imal deformations, for the construction of uˆ we will only need to work with the covariant functor Def X : A → (sets) A covariant functor F : A → (sets) is called a functor of Artin rings. To every complete local k-algebra R we can associate a functor of Artin rings hR by hR (A) = HomA (R, A) A functor of this form is called prorepresentable. By categorical general nonsense one shows that a formal deformation ξˆ defines a morphism of functors ( a natural transformation) hR → Def X and that this morphism is an isomorphism precisely when ξˆ is universal. Therefore we see that the search for uˆ is a problem of prorepresentability of Def X . More generally, to every local deformation problem there corresponds a functor of Artin rings F analogous to Def X ; the task of constructing a formal universal deformation for the given problem consists in showing that F is prorepresentable, producing the ring R prorepresenting F and the formal universal deformation defining the isomorphism hR → F . This is the scheme of approach to the formal part of every local deformation problem as it was outlined by Grothendieck. What one needs is to find criteria for the prorepresentability of a functor of Artin rings; we will also need to consider properties weaker than prorepresentability (semiuniversality) satisfied by more general classes of functors coming from interesting deformation theoretic problems. Necessary and sufficient conditions of prorepresentability and of semiuniversality are given by Schlessinger’s Theorem. After having solved the problem of existence of a formal universal (or semiuniversal) deformation by means of necessary and sufficient conditions for its existence one still has to decide whether O and υ exist and to find ˆ to O is the analogous of the convergence step in the them. To pass from O Kodaira-Spencer theory, and it is a very difficult problem, the algebraization problem. Under reasonably general assumptions one shows that there exists

6

CONTENTS

a deformation υ over an algebraic local ring (i.e. the henselization of a local k-algebra essentially of finite type) which does not quite represent the functor Def X but at least has a universal (or semiuniversal) associated formal deformation. The further property of representing Def X is not in general satisfied by (O, υ), being related with the existence of nontrivial automorphisms of X. This part of the theory is largely due to the work of M. Artin, and based on the notions of effectivity of a formal deformation and of local finite presentation of a functor, already introduced by Grothendieck. The main technical tool is Artin’s approximation theorem.

Chapter 1 Infinitesimal deformations The purpose of this chapter is to introduce the reader to deformation theory in an elementary and direct fashion. We will be especially interested in first order deformations and obstructions and in giving them appropriate ˇ interpretation mostly by elementary Cech cohomology computations. We will start by introducing some algebraic tools needed. For other notions used we refer the reader to the appendices.

1.1

Extensions

1.1.1

Generalities

Let A → R be a ring homomorphism. An A-extension of R (or of R by I) is an exact sequence (R0 , ϕ) :

ϕ

0 → I → R0 −→ R → 0

where R0 is an A-algebra and ϕ is a homomorphism of A-algebras whose kernel I is an ideal of R0 satisfying I 2 = (0). This condition implies that I has a structure of R-module. (R0 , ϕ) is also called an extension of A-algebras. If (R0 , ϕ) and (R00 , ψ) are A-extensions of R by I, an A-homomorphism ξ : R0 → R00 is called an isomorphism of extensions if the following diagram commutes: 0 → I → R0 → R → 0 k ↓ξ k 00 0→ I → R → R →0 7

8

CHAPTER 1. INFINITESIMAL DEFORMATIONS

Such a ξ is necessarily an isomorphism of A-algebras. More generally, given A-extensions (R0 , ϕ) and (R00 , ψ) of R, not necessarily having the same kernel, a homomorphism of A-algebras r : R0 → R00 such that ψr = ϕ is called a homomorphism of extensions. The following lemma is immediate. Lemma 1.1.1 Let (R0 , ϕ) be an extension as above. Given an A-algebra B and two A-homomorphisms f1 , f2 : B → R0 such that ϕf1 = ϕf2 the induced map f2 − f1 : B → I is an A-derivation. In particular, given two homomorphisms of extensions r1 , r2 : (R0 , ϕ) → (R00 , ψ) the induced map r2 − r1 : R0 → ker(ψ) is an A-derivation. The A-extension (R0 , ϕ) is called trivial if it has a section, that is if there exists a homomorphism of A-algebras σ : R → R0 such that ϕσ = 1R . We also say that (R0 , ϕ) splits, and we call σ a splitting. Given an R-module I, a trivial A-extension of R by I can be constructed ˜ whose underlying A-module is R ⊕ I and by considering the A-algebra R⊕I with multiplication defined by: (r, i)(s, j) = (rs, rj + si) The first projection ˜ →R p : R⊕I defines an A-extension of R by I which is trivial: a section q is given by q(r) = (r, 0). The sections of p can be identified with the A-derivations d : R → I. ˜ with σ(r) = (r, d(r)) then for all Indeed, if we have a section σ : R → R⊕I 0 r, r ∈ R: σ(rr0 ) = (rr0 , d(rr0 )) = σ(r)σ(r0 ) = (r, d(r))(r0 , d(r0 )) = (rr0 , rd(r0 ) + r0 d(r)) and if a ∈ A then: σ(ar) = (ar, d(ar)) = aσ(r) = a(r, d(r)) = (ar, ad(r)) hence d : R → I is an A-derivation. Conversely every A-derivation d : R → I ˜ by σd (r) = (r, d(r)). defines a section σd : R → R⊕I

1.1. EXTENSIONS

9

˜ p). If Every trivial A-extension (R0 , ϕ) of R by I is isomorphic to (R⊕I, 0 0 ˜ → R is given by: σ : R → R is a section an isomorphism ξ : R⊕I ξ((r, i)) = σ(r) + i and its inverse is ξ −1 (r0 ) = (ϕ(r0 ), r0 − σϕ(r0 )) An A-extension (P, f ) of R will be called versal if for every other A-extension (R0 , ϕ) of R there is a homomorphism of extensions r : (P, f ) → (R0 , ϕ). If R = P/I where P is a polynomial algebra over A then 0 → I/I 2 → P/I 2 → R → 0 is a versal A-extension of R. Therefore, since such a P always exists, we see that every A-algebra R has a versal extension. Examples 1.1.2 (i) Every A-extension of A is trivial because by definition ˜ for an A-module V . In it has a section. Therefore it is of the form A⊕V particular, if t is an indeterminate the A-extension A[t]/(t2 ) of A is trivial, and is denoted A[] (where = t mod (t2 ) satisfies 2 = 0). The corresponding exact sequence is: 0 → () → A[] → A → 0 A[] is called the algebra of dual numbers over A. (ii) Assume that K is a field. If R is a local K-algebra with residue field K a K-extension of R by K is called a small extension of R. Let (R0 , f ) :

f

0 → (t) → R0 −→ R → 0

be a small K-extension; in other words t ∈ mR0 is annihilated by mR0 so that (t) is a K-vector space of dimension one. (R0 , f ) is trivial if and only if the surjective linear map induced by f : f1 :

mR 0 mR → 2 2 mR 0 mR

is not bijective. Indeed for the trivial K-extension ˜ 0 → (t) → R⊕(t) →R→0

10

CHAPTER 1. INFINITESIMAL DEFORMATIONS

we have t ∈ mR⊕(t) \ m2R⊕(t) ˜ ˜ , hence the map f1 is not injective because f1 (t¯) = 0. Conversely, if f1 is not injective then f1 (t¯) = ¯0; choose a vector subspace U ⊂ mR0 /m2R0 such that mR0 /m2R0 = U ⊕ (t¯) and let V ⊂ R0 be the subring generated by U . Then V is a subring mapped isomorphically onto R by f . The inverse of f|V is a section of f , therefore (R0 , f ) is trivial. For example, it follows from this criterion that the extension of K-algebras 0→

(tn ) K[t] K[t] → n+1 → n → 0 n+1 (t ) (t ) (t )

n ≥ 2, is non-trivial. (iii) Let K be a field. The K-algebra K[, 0 ] := K[t, t0 ]/(t, t0 )2 is a K-extension of K[] by K in two different ways. The first p

0 → (0 ) → K[, 0 ] −→ K[] → 0 is a trivial extension, isomorphic to p∗ ((K[0 ], p0 )): 0 → (0 ) → K[] ×K K[0 ] k ↓ 0 → (0 ) →

→ p0

K[0 ]

−→

K[] → 0 ↓p K

→0

The isomorphism is given by K[, 0 ] a + b + b0 0

−→ K[] ×K K[0 ] 7−→ (a + b, a + b0 0 )

The second way is by “sum”: 0 → ( − 0 )

→ K[, 0 ]

−→

+

K[]

a + b + b0 0

7→

a + (b + b0 )

→0

We leave it as an exercise to show that (K[, 0 ], +) is isomorphic to (K[, 0 ], p ).

1.1. EXTENSIONS

1.1.2

11

The module ExA (R, I)

Let A → R be a ring homomorphism. In this subsection we will show how to give an R-module structure to the set of isomorphism classes of extensions of an A-algebra R by a module I, closely following the analogous theory of extensions in an abelian category as explained for example in Chapter III of [135]. Let (R0 , ϕ) be an A-extension of R by I and f : S → R a homomorphism of A-algebras. We can define an A-extension f ∗ (R0 , ϕ) of S by I, called the pullback of (R0 , ϕ) by f , in the following way: f ∗ (R0 , ϕ) :

0 → I → R0 ×R S k ↓ 0→ I → R0

(R0 , ϕ) :

→

S →0 ↓f → R →0

where R0 ×R S denotes the fibered product defined in the usual way. Let λ : I → J be a homomorphism of R-modules. The pushout of (R0 , ϕ) by λ is the A-extension λ∗ (R0 , ϕ) of R by J defined by the following commutative diagram: α

I −→ R0 ↓λ ↓ 0` 0→ J → R IJ 0→

where R0

a I

J=

ϕ

−→ R → 0 k → R →0

˜ R0 ⊕J {(−α(i), λ(i)), i ∈ I}

Definition 1.1.3 For every A-algebra R and for every R-module I we define ExA (R, I) to be the set of isomorphism classes of A-extensions of R by I. If (R0 , ϕ) is such an extension we will denote by [R0 , ϕ] ∈ ExA (R, I) its class. Using the operations of pullback and pushout it is possible to define a structure of R-module on ExA (R, I). If r ∈ R and [R0 , ϕ] ∈ ExA (R, I) we define r[R0 , ϕ] = [r∗ (R0 , ϕ)] where r : I → I is the multiplication by r.

12

CHAPTER 1. INFINITESIMAL DEFORMATIONS

Given [R0 , ϕ], [R00 , ψ] ∈ ExA (R, I), to define their sum we use the following diagram: 0 0 0 & ↓ ↓ I ⊕I I = I & ↓ ↓ 0 00 0 → I → R ×R R → R0 → 0 k ↓ & ↓ 0 → I → R00 → R → 0 ↓ ↓ & 0 0 0 which defines an A-extension: (R0 ×R R00 , ζ) :

ζ

0 → I ⊕ I → R0 ×R R00 −→ R → 0

We define [R0 , ϕ] + [R00 , ψ] := [δ∗ (R0 ×R R00 , ζ)] where δ : I ⊕ I → I is the “sum homomorphism”: δ(i ⊕ j) = i + j. Proposition 1.1.4 Let A → R be a ring homomorphism and I an Rmodule. With the operations defined above ExA (R, I) is an R-module whose ˜ p]. This construction defines a covariant functor: zero element is [R⊕I, ( R-modules) −→ I 7−→ (f : I → J) 7−→

( R-modules) ExA (R, I) (f∗ : ExA (R, I) → ExA (R, J))

Proof. Straightforward.

qed

It is likewise straightforward to check that if f : R → S is a homomorphism of A-algebras and I is an S-module, then the operation of pullback induces an application: f ∗ : ExA (S, I) → ExA (R, I) which is a homomorphism of S-modules. We have the following useful result.

1.1. EXTENSIONS

13

Proposition 1.1.5 Let A be a ring, f : S → R a homomorphism of Aalgebras and let I be an R-module. Then there is an exact sequence of Rmodules: ρ

0 → DerS (R, I) → DerA (R, I) → DerA (S, I) ⊗S R −→ f∗ v → ExS (R, I) −→ ExA (R, I) −→ ExA (S, I) ⊗S R Proof. v is the obvious application sending an S-extension to itself considered as an A-extension. An A-extension ϕ

0 → I → R0 −→ R → 0 is also an S-extension if and only if there exists f 0 : S → R such that the triangle R0 → R - ↑ S commutes, and this is equivalent to saying that f ∗ (R0 , ϕ) is trivial. This proves the exactness in ExA (R, I). ˜ p) where the The homomorphism ρ is defined by letting ρ(d) = (R⊕I, ˜ structure of S-algebra on R⊕I is given by the homomorphism s 7→ (f (s), d(s)) Clearly vρ = 0. On the other hand for (R0 , ϕ) : 0 → I

→ R0 ↑ S

ϕ

−→ R → 0

to define an element of ker(v) there must exist an isomorphism of A-algebras ˜ inducing the identity on I and on R. Hence the composition R0 → R⊕I 0 ˜ is of the form S → R → R⊕I s 7→ (f (s), d(s)) for some d ∈ DerA (S, I): therefore the sequence is exact at ExS (R, I). To ˜ → prove the exactness at DerA (S, I) note that ρ(d) = 0 if and only if p : R⊕I R has a section as a homomorphism of S-algebras, if and only if there exists an A-derivation R → I whose restriction to S is d: this proves the assertion. The exactness at DerS (R, I) and DerA (R, I) is straightforward. qed

14

CHAPTER 1. INFINITESIMAL DEFORMATIONS

Definition 1.1.6 The R-module ExA (R, R) is called the first cotangent mod1 ule of R over A and it is denoted TR/A . In case A = k we will write TR1 instead 1 of TR/k . Proposition 1.1.7 Let A → B be an e.f.t. ring homomorphism and let B = P/J where P is a smooth A-algebra. Then for every B-module M we have an exact sequence: DerA (P, M ) → HomB (J/J 2 , M ) → ExA (B, M ) → 0

(1.1)

If A → B is a smooth homomorphism then ExA (B, M ) = 0 for every Bmodule M . Proof. We have a natural surjective homomorphism HomB (J/J 2 , M ) → ExA (B, M ) λ 7→ λ∗ (η) where η : 0 → J/J 2 → P/J 2 → B → 0 The surjectivity follows from the fact that η is versal. The extension λ∗ (η) is trivial if and only if we have a commutative diagram 0 → J/J 2 ↓λ 0→ M

P/J 2 ↓ ˜ → B ⊕M →

→ B k → B

→0 →0

¯ : P/J 2 → M , equivalently to if and only if λ extends to an A-derivation D an A-derivation D : P → M . The last assertion is immediate (see Theorem C.53). qed Corollary 1.1.8 If A → B is an e.f.t. ring homomorphism and M is a finitely generated B-module then ExA (B, M ) is a finitely generated B-module. 1 In particular TB/A is a finitely generated B-module and we have an exact sequence: 0 → HomB (ΩB/A , M ) → HomB (ΩP/A ⊗P B, M ) → (1.2) → HomB (I/I 2 , M ) → T 1 (B/A, M ) → 0 if B = P/J for a smooth A-algebra P and an ideal J ⊂ P . Proof. It is a direct consequence of the exact sequence (1.1).

qed

1.1. EXTENSIONS

1.1.3

15

Extensions of schemes

Let X → S be a morphism of schemes. An extension of X/S is a closed immersion X ⊂ X 0 , where X 0 is an S-scheme, defined by a sheaf of ideals I ⊂ OX 0 such that I 2 = 0. It follows that I is, in a natural way, a sheaf of OX -modules, which coincides with the conormal sheaf of X ⊂ X 0 . To give an extension X ⊂ X 0 of X/S is equivalent to giving an exact sequence on X: ϕ

E : 0 → I → OX 0 −→ OX → 0 where I is an OX -module, ϕ is a homomorphism of OS -algebras and I 2 = 0 in OX 0 ; we call E an extension of X/S by I or with kernel I. Two such extensions OX 0 and OX 00 are called isomorphic if there is an OS -homomorphism α : OX 0 → OX 00 inducing the identity on both I and OX . It follows that α must necessarily be an S-isomorphism. We denote by Ex(X/S, I) the set of isomorphism classes of extensions of X/S with kernel I. In case Spec(B) → Spec(A) is a morphism of affine ˜ we have an obvious identification: schemes and I = M ExA (B, M ) = Ex(X/S, I) If S = Spec(A) is affine we will sometimes write ExA (X, I) instead of Ex(X/Spec(A), I). Exactly as in the affine case one proves that Ex(X/S, I) is a Γ(X, OX )-module with identity element the class of the trivial extension: ˜ → OX → 0 0 → I → OX ⊕I ˜ is defined as in the affine case (see Section 1.1). The correwhere OX ⊕I spondence I 7→ Ex(X/S, I) defines a covariant functor from OX -modules to Γ(X, OX )-modules. In deformation theory the case I = OX is the most important one, being related to first order deformations. If more generally I is a locally free sheaf we get the notions of ribbon, carpet etc. (see [18]). Using the fact that the exact sequence (1.2) of page 14 localizes, it is immediate to check that the cotangent module localizes. More specifically it is straightforward to show that given a morphism of finite type of schemes 1 f : X → S one can define a quasi-coherent sheaf TX/S on X with the following

16

CHAPTER 1. INFINITESIMAL DEFORMATIONS

properties. If U = Spec(A) is an affine open subset of S and V = Spec(B) is an affine open subset of f −1 (U ), then 1 1 Γ(V, TX/S ) = TB/A 1 It follows from the properties of the cotangent modules that TX/S is coherent. 1 TX/S is called the first cotangent sheaf of X/S. We will write TX1 if S = Spec(k). For future reference it will be convenient to state the following:

Proposition 1.1.9 (i) If X is an algebraic scheme then TX1 is supported on the singular locus of X. More generally if X → S is a morphism 1 of finite type of algebraic schemes then TX/S is supported on the locus where X is not smooth over S. (ii) If we have a closed embedding X ⊂ Y with Y nonsingular, then we have an exact sequence of coherent sheaves on X: 0 → TX → TY |X → NX/Y → TX1 → 0

(1.3)

0 so that, letting NX/Y = ker[NX/Y → TX1 ], we have the short exact sequence 0 0 → TX → TY |X → NX/Y →0 (1.4) 0 NX/Y is called the equisingular normal sheaf of X in Y .

(iii) For every scheme S and morphism of S-schemes f : X → Y we have an exact sequence of sheaves 1 1 0 → TX/Y → TX/S → Hom(f ∗ Ω1Y /S , OX ) → TX/Y → TX/S → f ∗ TY1/S (1.5)

(iv) When S = Spec(k) and f is a closed embedding of algebraic schemes, 1 with Y nonsingular, we have TX/Y = 0 and NX/Y = TX/Y . Moreover (1.3) is a special case of (1.5) in this case. Proof. (i) use Proposition 1.1.7. (ii) (1.3) globalizes the exact sequence (1.2). (iii) (1.5) globalizes the exact sequence of Proposition 1.1.5. (iv) follows from (1.2) and 1.1.5).

qed

Note that the first half of the exact sequence (1.5) is the dual of the cotangent sequence of f . For more about (1.5) see also (3.43), page 193. The following is a basic result:

1.1. EXTENSIONS

17

Theorem 1.1.10 Let X → S be a morphism of finite type of algebraic schemes and I a coherent locally free sheaf on X. Assume that X is reduced and S-smooth on a dense open subset. Then there is a canonical identification Ex(X/S, I) = Ext1OX (Ω1X/S , I) which to the isomorphism class of an extension of X/S: E : 0 → I → OX 0 → OX → 0 associates the isomorphism class of the relative conormal sequence of X ⊂ X 0: δ cE : 0 → I → (Ω1X 0 /S )|X → Ω1X/S → 0 (which is exact also on the left). Proof. Suppose given an extension E. Since I is locally free and X is reduced in order to show that cE is exact on the left it suffices to prove that ker(δ) is torsion, equivalently that cE is exact at every general closed point x of any irreducible component of X. Since X is smooth over S at x it follows from 1.1.7 that there is an affine open neighborhood U of x such that E|U is trivial. From Theorem B.36 we deduce that the relative conormal sequence of E|U is split exact; since it coincides with the restriction of cX 0 to U we see that δ|U is injective; this shows that ker(δ) is torsion and cE is exact. Since isomorphic extensions have isomorphic relative cotangent sequences we have a well defined map c− : Ex(X/S, I) → Ext1OX (Ω1X/S , I) Let now

p

η : 0 → I → A → Ω1X/S → 0 define an element of Ext1OX (Ω1X/S , I). Letting d : OX → Ω1X/S be the canonical derivation, consider the sheaf of OS algebras O = A ×Ω1X/S OX : over an open subset U ⊂ X we have Γ(U, O) = {(a, f ) : p(a) = d(f )} and the multiplication rule is (a, f )(a0 , f 0 ) = (f a0 + f 0 a, f f 0 ) Then we have an exact commutative diagram 0→ I k 0→ I

→ O → OX ↓ d¯ ↓d → A → Ω1X/S

→0 →0

18

CHAPTER 1. INFINITESIMAL DEFORMATIONS

where one immediately checks that the projection d¯ is an OS -derivation and therefore it must factor as O → Ω1O/OS ⊗O OX → A and we have an exact commutative diagram 0→ I k I k 0→ I

→

O ↓ → Ω1O/OS ⊗O OX ↓ → A

→

OX ↓d → Ω1X/S k 1 → ΩX/S

→0 →0

(1.6)

→0

which implies Ω1O/OS ⊗O OX ∼ = A. Therefore, letting eη be the extension given by the firt row of (1.6), we see that ceη = η. Similarly one shows that ecE = E for any [E] ∈ Ex(X/S, I). Therefore c− and e− are inverse of each other and the conclusion follows. qed Corollary 1.1.11 Let X → S be a morphism of finite type of algebraic schemes, smooth on a dense open subset of X. Assume X reduced. Then there is a canonical isomorphism of coherent sheaves on X: 1 ∼ TX/S = Ext1OX (Ω1X/S , OX )

In particular, if X is a reduced algebraic scheme then TX1 ∼ = Ext1OX (Ω1X , OX ) and if moreover X = Spec(B0 ) then TB1 0 ∼ = Ext1k (ΩB0 /k , B0 ) Proof. An immediate consequence of the above theorem.

qed

A closer analysis of the proof of Theorem 1.1.10 shows that without assuming X reduced we only have an inclusion Ext1OX (Ω1X , OX ) ⊂ TX1 Notes and Comments

1.1. EXTENSIONS

19

1. An alternative approach to the topics treated in this section can be obtained by means of the so called “truncated cotangent complex”, first introduced in [80]. A more general version of the cotangent complex was introduced in [132] and later incorporated in general theories of Andr´e [4] and Quillen [163]. Here we will just recall the main facts about the truncated cotangent complex, without entering into any details, with the only purpose of showing the relation with the notions introduced in this section. For details we refer to [102]. Let A be a ring and R an A-algebra. To every A-extension ϕ

η : 0 → I → R0 −→ R → 0 we associate a complex c• (η) of R-modules (also denoted c• (ϕ)) defined as follows: c0 (η) = ΩR0 /A ⊗R0 R c1 (η) = I cn (η) = (0)

n 6= 0, 1

d1 : c1 (η) → c0 (η) is the map x 7→ d(x)⊗1. In other words c• (η) consists of the first map in the conormal sequence of ϕ. If r : (R0 , ϕ) → (R00 , ψ) is a homomorphism of A-extensions then r induces a homomorphism of complexes c• (r) : c• (ϕ) → c• (ψ) in an obvious way. The following is easy to establish: • Let r1 , r2 : (R0 , ϕ) → (R00 , ψ) be two homomorphisms of A-extensions of R. Then c• (r1 ) and c• (r2 ) are homotopic. As an immediate consequence we have: • if (E, p) and (F, q) are two versal A-extensions of R then the complexes c• (p) and c• (q) are homotopically equivalent. Definition 1.1.12 Let A be a ring, R an A-algebra and (E, p) a versal A-extension of R. The homotopy class of the complex c• (p) is called the (truncated) cotangent complex of R over A and denoted Tˇ(R/A). If R = P/I for a polynomial A-algebra P the A-extension 0 → I/I 2 → P/I 2 → R → 0

(1.7)

is versal and therefore the complex δ

I/I 2 −→ ΩP/A ⊗P R

(1.8)

20

CHAPTER 1. INFINITESIMAL DEFORMATIONS

where δ is the map appearing in the conormal sequence of A → P → R, represents Tˇ(R/A). If R is e.f.t. and P is replaced by a smooth A-algebra then (1.7) is again a versal A-extension and (1.8) again represents Tˇ(R/A). From the fact that every A-algebra R can be obtained as the quotient of a polynomial A-algebra P it follows that the cotangent complex Tˇ(R/A) exists for every A-algebra R. The cotangent complex can be used to define “upper and lower cotangent functors”, as follows. Definition 1.1.13 Let A → B be a ring homomorphism, M a B-module, and d1 C0 } represent Tˇ(B/A). Then for i = 0, 1 the lower cotangent let c• = {C1 −→ module of B over A relative to M is: Tˇi (B/A, M ) = Hi (c• ⊗B M ) and the upper cotangent module of B over A relative to M is: Tˇi (B/A, M ) = H i (Hom(c• , M )) Because of the definition of cotangent complex it follows that the cotangent modules are independent on the choice of the complex c• representing Tˇ(B/A), but only depend on A, B, M . Moreover one immediately checks that the definition is functorial in M and therefore we have covariant functors: Tˇi (B/A, −) : (B-modules) → (B-modules)

i = 0, 1

Tˇi (B/A, −) : (B-modules) → (B-modules)

i = 0, 1

and One immediately sees that for i = 0 the cotangent functors are: Tˇ0 (B/A, M ) = ΩB/A ⊗B M and Tˇ0 (B/A, M ) = DerA (B, M ) From the extension (1.7) we obtain the exact sequences: 0 → Tˇ1 (B/A, M ) → I/I 2 ⊗B M → ΩP/A ⊗P M → ΩB/A ⊗B M → 0 and

0 → HomB (ΩB/A , M ) → HomB (ΩP/A ⊗P B, M ) → → HomB (I/I 2 , M ) → Tˇ1 (B/A, M ) → 0

1.2. LOCALLY TRIVIAL DEFORMATIONS

21

If B is e.f.t. then in (1.7) P can be chosen to be a smooth A-algebra; in this case it follows that Tˇi (B/A, M ) and Tˇi (B/A, M ) are finitely generated B-modules if M is finitely generated. Moreover, recalling Corollary 1.1.8, we see that we have an identification Tˇ1 (B/A, M ) = T 1 (B/A, M ) 2. The topics of this section originate from [80]. See also [1], Ch. 0IV , §18. The proof of Theorem 1.1.10 has been taken from [18]; see also [70].

1.2 1.2.1

Locally trivial deformations Generalities on deformations

Let X be an algebraic scheme. A cartesian diagram of morphisms of schemes X → X η: ↓ ↓π s Spec(k) −→ S where π is flat and surjective, and S is connected, is called a family of deformations, or simply a deformation, of X parametrized by S, or over S; we call S and X respectively the parameter scheme and the total scheme of the deformation. If S is algebraic, for each k-rational point t ∈ S the schemetheoretic fibre X (t) is also called a deformation of X. When S = Spec(A) with A in ob(A∗ ) and s ∈ S is the closed point we have a local family of deformations (shortly a local deformation) of X over A. The deformation η will be also denoted (S, η) or (A, η) when S = Spec(A). The local deformation (A, η) is infinitesimal (resp. first order) if A ∈ ob(A) (resp. A = k[]). Given another deformation X → Y ξ: ↓ ↓ Spec(k) → S of X over S, an isomorphism of η with ξ is an S-isomorphism φ : X → Y inducing the identity on X, i.e. such that the following diagram is commutative: X . & φ X −→ Y & . S

22

CHAPTER 1. INFINITESIMAL DEFORMATIONS

By a pointed scheme we will mean a pair (S, s) where S is a scheme and s ∈ S. If K is a field we call (S, s) a K-pointed scheme if K ∼ = k(s). Observe that for every X and for every k-pointed scheme (S, s) there exists at least one family of deformation of X over S, namely the product family: X → X ×S ↓ ↓ s Spec(k) −→ S A deformation of X over S is called trivial if it is isomorphic to the product family. It will be also called a trivial family with fibre X. All fibres over k-rational points of a trivial deformation of X parametrized by an algebraic scheme are isomorphic to X. The converse is not true: there are deformations which are not trivial but have isomorphic fibres over all the k-rational points (see Example 1.2.2(ii) below). S The scheme X is called rigid if every infinitesimal deformation of X over A is trivial for every A in ob(A). Given a deformation η of X over S as above and a morphism (S 0 , s0 ) → (S, s) of k-pointed schemes there is induced a commutative diagram by base change X → X ×S S 0 ↓ ↓ Spec(k) → S0 which is clearly a deformation of X over S 0 . This operation is functorial, in the sense that it commutes with composition of morphisms and the identity morphism does not change η. Moreover it carries isomorphic deformations to isomorphic ones. An infinitesimal deformation η of X is called locally trivial if every point x ∈ X has an open neighborhood Ux ⊂ X such that Ux → X|Ux ↓ ↓ Spec(k) → S is a trivial deformation of Ux . Remark 1.2.1 Let

j

X −→ X η: ↓ ↓π s Spec(k) −→ S

1.2. LOCALLY TRIVIAL DEFORMATIONS

23

be a family of deformations of an algebraic scheme X parametrized by an algebraic scheme S and let Z ⊂ X be a proper closed subset. Then j

X\Z −→ X \j(Z) ↓ ↓ π0 s Spec(k) −→ S is a family of deformations of X\Z having the same fibres as π over t ∈ S for t 6= s: thus such fibres are deformations both of X and of X\Z. This shows that the definition of family of deformations given above is somehow ambiguous unless we assume that π is projective or that the deformation is infinitesimal. In what follows we will restrict to the consideration of deformations of projective schemes and/or of infinitesimal deformations when discussing the general theory, so that such ambiguity will be removed; only occasionally we will consider non-infinitesimal deformations of affine schemes. Examples 1.2.2 (i) The quadric Q ⊂ A3 of equation xy − t = 0 defines, via the projection A3 → A1 (x, y, t) 7→ t a flat family Q → A1 whose fibres are affine conics. This family is not trivial since the fibre Q(0) is singular, hence not isomorphic to the fibres Q(t), t 6= 0, which are nonsingular. (ii) Consider, for a given integer m ≥ 0, the rational ruled surface Fm = IP (OIP 1 (m) ⊕ OIP 1 ) The structural morphism π : Fm → IP 1 defines a flat family whose fibres are all isomorphic to IP 1 ; but if m > 0 then π is not a trivial family because Fm ∼ 6= F0 = IP 1 × IP 1 (see Example B.44(iii)). (iii) Let 0 ≤ n < m be two distinct nonnegative integers having the same parity and let k = 21 (m − n). Consider two copies of A2 × IP 1 given as Proj(k[t, z, ξ0 , ξ1 ]) =: W and Proj(k[t, z 0 , ξ00 , ξ10 ]) =: W 0 (here the rings are graded with respect to the variables ξi and ξi0 ). Letting ξ = ξ1 /ξ0 and ξ 0 = ξ10 /ξ00 consider the open subsets Spec(k[t, z, ξ]) ⊂ W,

Spec(k[t, z 0 , ξ 0 ]) ⊂ W 0

24

CHAPTER 1. INFINITESIMAL DEFORMATIONS

and glue them together along the open subsets Spec(k[t, z, z −1 , ξ]) ⊂ Spec(k[t, z, ξ]) and Spec(k[t, z 0 , z 0−1 , ξ 0 ]) ⊂ Spec(k[t, z 0 , ξ 0 ]) according to the following rules: z 0 = z −1 ,

ξ 0 = z m ξ + tz k

(1.9)

This induces a glueing of W and W 0 along Proj(k[t, z, z −1 , ξ0 , ξ1 ]) and Proj(k[t, z 0 , z 0−1 , ξ00 , ξ10 ]) Call the resulting scheme W and f : W → A1 = Spec(k[t]) the morphism induced by the projections. Then f is a flat morphism because it is locally a projection; moreover W(0) ∼ = Fm Let W ◦ = f −1 (A1 \{0}) and f ◦ : W ◦ → A1 \{0} the restriction of f . In k[t, t−1 , z, ξ] define zk ξ − t ζ= tξ and in k[t, t−1 , z 0 , ξ 0 ] ξ0 tz 0m−k ξ 0 + t2 It is straightforward to verify that the glueing (1.9) induces the relation ζ0 =

ζ 0 = znζ This means that we have an isomorphism W◦ ∼ = Fn × (A1 \{0}) compatible with the projections to A1 \{0}. Therefore the family f ◦ is trivial, in particular all its fibres are isomorphic to Fn , but the family f is not trivial because W(0) ∼ = Fm . (iv) Let f : X → Y be a surjective morphism of algebraic schemes, with X integral and Y an irreducible and nonsingular curve. Then f is flat. This is a special case of Prop. III.9.7 of [89]. Therefore f defines a family of deformations of any of its closed fibres.

1.2. LOCALLY TRIVIAL DEFORMATIONS

1.2.2

25

Infinitesimal deformations of nonsingular affine schemes

We will start by considering infinitesimal deformations of affine schemes. We need the following Lemma 1.2.3 Let Z0 be a closed subscheme of a scheme Z, defined by a sheaf of nilpotent ideals N ⊂ OZ . If Z0 is affine then Z is affine as well. Proof. Let r ≥ 2 be the smallest integer such that N r = (0). Since we have a chain of inclusions Z ⊃ V (N r−1 ) ⊃ V (N r−2 ) ⊃ · · · ⊃ V (N ) = Z0 it suffices to prove the assertion in the case r = 2. In this case N is a coherent OZ0 -module, and therefore H 1 (Z, N ) = H 1 (Z0 , N ) = 0 Let R0 be the k-algebra such that Z0 = Spec(R0 ). We have the exact sequence: 0 → H 0 (Z, N ) → H 0 (Z, OZ ) → R0 → 0 Put R = H 0 (Z, OZ ) and let Z 0 = Spec(R). We have a commutative diagram: Z

θ

−→ Z 0 -% Z0

The sheaf homomorphism θ−1 OZ 0 → OZ is clearly injective and θ is a homeomorphism. It will therefore suffice to prove that θ−1 OZ 0 → OZ is surjective. Let z ∈ Z and f ∈ Γ(U, OZ ) for some affine open neighborhood U of z. Let f0 = f|U ∩Z0 . It is possible to find ϕ0 , ψ0 ∈ R0 such that f0 = ψϕ00 , ψ0 (z) 6= 0 and ψ0 = 0 on Z0 \U , because Z0 is affine. Let ψ ∈ R be such that ψ|Z0 = ψ0 (it exists by the surjectivity of R → R0 ). Then ψ(z) 6= 0 and ψ = 0 on Z\U . There exists n 0 such that ψ n f =: g ∈ R (it suffices to cover Z with affines). Then f = ψgn ∈ θ−1 OZ 0 . qed Let B0 be a k-algebra, and let X0 = Spec(B0 ). Consider an infinitesimal deformation of X0 parametrized by Spec(A), where A is in ob(A). By

26

CHAPTER 1. INFINITESIMAL DEFORMATIONS

definition this is a cartesian diagram X0 → X ↓ ↓ Spec(k) → Spec(A) where X is a scheme flat over Spec(A). By Lemma 1.2.3 X is necessarily affine. Therefore, equivalently, we can talk about an infinitesimal deformation of B0 over A as a cartesian diagram of k-algebras: B ↑ A

→ B0 ↑ → k

(1.10)

with A → B flat. Note that to give this diagram is the same as to give A → B flat and a k-isomorphism B ⊗A k → B0 . We will sometimes abbreviate by calling A → B the deformation. Given another deformation A → B 0 of B0 over A, an isomorphism of deformations of A → B to A → B 0 is a homomorphism ϕ : B → B 0 of A-algebras inducing a commutative diagram: B0 %

ϕ

B0

−→

B -

% A

It follows from Lemma A.25 that such a ϕ is an isomorphism. An infinitesimal deformation of B0 over A is trivial if it is isomorphic to the product deformation B0 ⊗k A → B0 ↑ ↑ A → k The k-algebra B0 is called rigid if Spec(B0 ) is rigid. Theorem 1.2.4 Every smooth k-algebra is rigid. In particular every affine nonsingular algebraic variety is rigid.

1.2. LOCALLY TRIVIAL DEFORMATIONS

27

Proof. Suppose k → B0 is smooth, and suppose given a first order deformation of B0 : B → B0 η0 : ↑ f ↑ k[] → k Consider the commutative diagram: B → B0 ↑f ↑ k[] → B0 [] where B0 [] = B0 ⊗ k[]. Since f is smooth (because flat with smooth fibre, see [89], ch. III, Th. 10.2) and the right vertical morphism is a k[]-extension, by Theorem C.53 there exists a k[]-homomorphism φ : B → B0 [] making the diagram B → B0 ↑f & ↑ k[] → B0 [] commutative. Therefore φ is an isomorphism of deformations and η0 is trivial. Consider more generally a deformation of B0 B → B0 η: ↑f ↑ A → k parametrized by A in ob(A). To show that η is trivial we proceed by induction on d = dimk (A). The case d = 2 has been already proved; assume d ≥ 3 and let 0 → (t) → A → A0 → 0 be a small extension. Consider the commutative diagram: B → B ⊗A A0 ∼ = B0 ⊗k A0 ↑f ↑ A → B0 ⊗k A f is smooth, the upper right isomorphism is by the inductive hypothesis, and the right vertical homomorphism is an A-extension. By the smoothness of f and by Theorem C.53 we deduce the existence of an A-homomorphism B → B0 ⊗k A which is an isomorphism of deformations. qed

28

CHAPTER 1. INFINITESIMAL DEFORMATIONS

Example 1.2.5 Let λ ∈ k and B0 = k[X, Y ]/(Y 2 − X(X − 1)(X − λ)). If λ 6= 0, 1 then B0 is a smooth k-algebra, being the coordinate ring of a nonsingular plane cubic curve. By Theorem 1.2.4 B0 is rigid. On the other hand the elementary theory of elliptic curves (see [89]) shows that the following flat family of affine curves Speck[X, Y ]/(Y 2 − X(X − 1)(X − (λ + t))) ↓ Spec(k[t]) is not trivial around the origin t = 0 so that it defines a non-trivial (noninfinitesimal) deformation of B0 . This example shows that by studying infinitesimal deformations of affine schemes we are loosing some information. In this specific case we will see that this information is ricovered by considering the infinitesimal deformations of the projective closure of Spec(B0 ) (see Corollary 2.6.6 page 112).

1.2.3

Extending automorphisms of deformations

In deformation theory it is very important to have a good control of automorphisms of deformations and of their extendability properties. We will now begin to introduce such matters and to recall some terminology. In Section 2.6 we will consider these problems again in general, and we will relate them with the property of “prorepresentability”. Let’s start with a basic lemma. Lemma 1.2.6 Let B0 be a k-algebra, and e : 0 → (t) → A˜ → A → 0 a small extension in A. Then there is a canonical isomorphism of groups:

automorphisms of the trivial deformation B0 ⊗k A˜ inducing the identity on B0 ⊗k A

→ Derk (B0 , B0 )

In particular the group on the left is abelian. Proof. Every automorphism θ : B0 ⊗k A˜ → B0 ⊗k A˜ belonging to the first ˜ group must be A-linear and induce the identity mod t. Therefore: θ(x) = x + tdx

1.2. LOCALLY TRIVIAL DEFORMATIONS

29

˜ where d : B0 ⊗k A˜ → B0 is a A-derivation (Lemma 1.1.1). But ˜ B0 ) = Hom DerA˜ (B0 ⊗k A, ˜ (ΩB0 ⊗k A/ ˜ A ˜ , B0 ) = B0 ⊗k A = HomB0 (ΩB0 /k , B0 ) = Derk (B0 , B0 ) By sending θ 7→ d we define the correspondence of the statement. Since θ is determined by d the correspondence is one to one. Clearly the identity corresponds to the zero derivation. If we compose two automorphisms: θ σ B0 ⊗k A˜ −→ B0 ⊗k A˜ −→ B0 ⊗k A˜

where θ(x) = x + tdx, σ(x) = x + tδx, we obtain: σ(θ(x)) = θ(x) + tδ(θ(x)) = x + tdx + t(δx + tδ(dx)) = x + t(dx + δx) therefore the correspondence is a group isomorphism.

qed

Recall the following well known definition. Definition 1.2.7 Let G be a group acting on a set T and let π :G×T →T be the map defining the action. T is called a homogeneous space under (the action of ) G if π is transitive, i.e. if π(G × {t}) = T for some (equivalently for any) t ∈ T (i.e. if there is only one orbit). The action is called free if for every point t ∈ T the stabilizer Gt = {g ∈ G : gt = t} is trivial, i.e. gt = t implies g = 1G for all t ∈ T . If the action is both transitive and free then T is called a principal homogeneous space (or a torsor) under (the action of ) G. To an action π : G × T → T we can associate the map: p: G×T (g, t)

→ T ×T 7→

(gt, t)

30

CHAPTER 1. INFINITESIMAL DEFORMATIONS

The condition that the action is transitive (resp. free) is equivalent to p being surjective (resp. injective); therefore T is a torsor under G if and only if p is bijective. Note that π = pr1 p is determined by p. More generally suppose that we have a map of sets f : T → T 0 . Then the map p factors through T ×T 0 T ⊂ T × T if and only if the action π is compatible with f , i.e. if f (t) = f (gt) for all t ∈ T , g ∈ G. As before the map p : G × T → T ×T 0 T is surjective (resp. injective) if and only if the action of G is transitive (resp. free) on all the non-empty fibres of f . In particular p is bijective if and only if all the non-empty fibres of f are torsors under G. In this case one also says, according to [3], p. 114, that T over T 0 is a formal principal homogeneous space (or a pseudo-torsor) under G. Now we come back to deformations and we prove a generalization of Lemma 1.2.6. Lemma 1.2.8 Let B0 be a k-algebra, e : 0 → (t) → A˜ → A → 0 ˜ a deformation of B0 and A → B = B ˜ ⊗˜ A a small extension in A, A˜ → B A the induced deformation of B0 over A. Let σ : B → B be an automorphism of the deformation. Then: (i) If o

n

˜ := automorphisms τ : B ˜→B ˜ such that τ ⊗ ˜ A = σ 6= ∅ Autσ (B) A then there is a free and transitive action ˜ → Autσ (B) ˜ Derk (B0 , B0 ) × Autσ (B) defined by (d, τ ) 7→ τ + td ˜ 6= ∅ for any σ. (ii) If B0 is a smooth k-algebra then Autσ (B) Proof. (i) Recall that we have a chain of natural identifications ˜ B0 ) = Hom ˜ (Ω ˜ ˜ , B0 ) = HomB0 (Ω ˜ ˜ ⊗ ˜ k, B0 ) = DerA˜ (B, B B/A B/A A = HomB0 (ΩB0 /k , B0 ) = Derk (B0 , B0 )

1.2. LOCALLY TRIVIAL DEFORMATIONS

31

Therefore the action in the statement is well defined once we consider a ˜ ˜ into B0 . Given any two elements d ∈ Derk (B0 , B0 ) as an A-derivation of B ˜ we have by definition: τ, η ∈ Autσ (B) qτ = σq = qη ˜ → B is the projection: hence by Lemma 1.1.1, where q : B ˜ → tB0 = ker(q) η−τ :B ˜ is an A-derivation which is 0 if and only if η = τ . This implies that the action is free and transitive. ˜ is trivial (Theorem 1.2.4), (ii) Since B0 is smooth the deformation A˜ → B ∼ ˜ ˜ ˜ so that we have an A-isomorphism B = B0 ⊗k A; moreover B0 ⊗k A˜ is a smooth ˜ ˜ is A-smooth. ˜ A-algebra (Proposition C.46(iii)) hence B Let σ : B → B ˜ be any automorphism of the deformation and consider the diagram of Aalgebras: q σ ˜→ B B→ B ↑q ˜ B ˜ is A-smooth, ˜ Since ker(q) = tB0 is a square-zero ideal, and since B we deduce ˜→B ˜ such that q˜ that there is σ ˜:B σ = σq. It is immediate to check that σ ˜ ˜ is an isomorphism and therefore σ ˜ ∈ Autσ (B). qed In (ii) the condition that B0 is smooth cannot be removed. A simple example is given in 2.6.8(i). The extendability of automorphism of deformations of not necessarily affine schemes will be studied in §2.6.

1.2.4

First order locally trivial deformations

We will now apply 1.2.6 to first order deformations of any algebraic variety. Proposition 1.2.9 Let X be an algebraic variety. There is a 1-1 correspondence:

κ:

isomorphism classes of first order locally trivial deformations of X

→ H 1 (X, TX )

called the Kodaira-Spencer correspondence, where TX = Hom(Ω1X , OX ) = Derk (OX , OX ), such that κ(ξ) = 0 if and only if ξ is the trivial deformation

32

CHAPTER 1. INFINITESIMAL DEFORMATIONS

class. In particular if X is nonsingular then κ is a 1-1 correspondence

κ:

isomorphism classes of first order deformations of X

→ H 1 (X, TX )

Proof. Given a first order locally trivial deformation X → X ↓ ↓ Spec(k) → Spec(k[]) choose an affine open cover U = {Ui }i∈I of X such that X|Ui is trivial for all i. For each index i we therefore have an isomorphism of deformations: θi : Ui × Spec(k[]) → X|Ui by 1.2.4. Then for each i, j ∈ I θij := θi−1 θj : Uij × Spec(k[]) → Uij × Spec(k[]) is an automorphism of the trivial deformation Uij × Spec(k[]). By Lemma 1.2.6 θij corresponds to a dij ∈ Γ(Uij , TX ). Since on each Uijk we have −1 θij θjk θik = 1Uijk ×Spec(k[])

(1.11)

it follows that dij + djk − dik = 0 ˇ i.e. {dij } is a Cech 1-cocycle and therefore defines an element of H 1 (X, TX ). It is easy to check that this element does not depend on the choice of the open cover U. If we have another deformation X → X0 ↓ ↓ Spec(k) → Spec(k[]) and Φ : X → X 0 is an isomorphism of deformations then for each i ∈ I there is induced an automorphism: θi

Φ|Ui

αi : Ui × Spec(k[]) −→ X|Ui −→

0

θi −1 0 X|U −→ i

Ui × Spec(k[])

1.2. LOCALLY TRIVIAL DEFORMATIONS

33

and therefore a corresponding ai ∈ Γ(Ui , TX ). We have θi0 αi = Φ|Ui θi and therefore −1 (θi0 αi )−1 (θj0 αj ) = θi−1 Φ−1 |Uij Φ|Uij θj = θi θj thus 0 αi−1 θij αj = θij

equivalently: d0ij + aj − ai = dij namely {dij } and {d0ij } are cohomologous, and therefore define the same element of H 1 (X, TX ). ˇ Conversely, given θ ∈ H 1 (X, TX ) we can represent it by a Cech 1-cocycle 1 {dij } ∈ Z (U, TX ) with respect to some affine open cover U. To each dij we can associate an automorphism θij of the trivial deformation Uij × Spec(k[]) by Lemma 1.2.6. They satisfy the identities (1.11). We can therefore use these automorphisms to patch the schemes Ui × Spec(k[]) by the well known procedure (see [89], p. 69). We obtain a Spec(k[])-scheme X which by construction defines a locally trivial first order deformation of X. The equivalence between κ(ξ) = 0 and the triviality of ξ is easily proved. The last assertion follows from the first one because all deformations of a nonsingular variety are locally trivial by Theorem 1.2.4. qed Definition 1.2.10 For every locally trivial first order deformation ξ of a variety X the cohomology class κ(ξ) ∈ H 1 (X, TX ) is called the KodairaSpencer class of ξ. Let ξ:

X → X ↓ ↓f s Spec(k) −→ S

(1.12)

be a family of deformations of a nonsingular variety X. By pulling back this family by morphisms Spec(k[]) → S with image s and applying the Kodaira-Spencer correspondence (Proposition 1.2.9) we define a linear map κξ : TS,s → H 1 (X, TX ) also denoted κf,s or κX /S,s , which is called the Kodaira-Spencer map of the family ξ.

34

CHAPTER 1. INFINITESIMAL DEFORMATIONS

Examples 1.2.11 (i) Let m ≥ 1 and let π : Fm → IP 1 be the structural morphism of the rational ruled surface Fm (see B.44(iii)). Then π is not a trivial family but has a trivial restriction around each closed point s ∈ IP 1 , thus κπ,s = 0. (ii) Consider an unramified covering π : X → S of degree n ≥ 2 where X and S are projective nonsingular and irreducible algebraic curves. All fibres of π over the closed points consist of n distinct points, hence they are all isomorphic. Moreover each such fibre is rigid and unobstructed as an abstract variety. In particular the Kodaira-Spencer map is zero at each closed point s ∈ S. On the other hand π −1 (U ) is irreducible for each open subset U ⊂ S and therefore the restriction πU : π −1 (U ) → U is a nontrivial family; this follows also from the fact that π does not have rational sections. This example exhibits a phenomenon which is not detected by infinitesimal considerations and in some sense opposite to the one described in Example 1.2.5: we can have a flat projective family of deformations all of whose geometric fibres are isomorphic but which is nevertheless non trivial over every Zariski open subset of the base. Note that this is different from what happens with the projections Fm → IP 1 , m ≥ 1 of Example (i), which are non trivial but have trivial restriction to a Zariski open neighborhood of every point of IP 1 . See Subsection 2.6.2 for more about this. (iii) Let 0 ≤ n < m be integers having the same parity, and let k = − n). Consider the smooth proper morphism f : W → A1 introduced in Example 1.2.2(iii), whose fibres are W(0) ∼ = Fm , and W(t) ∼ = Fn for t 6= 0. Recall that the family f is given as the glueing of two copies of A2 × IP 1 : 1 (m 2

W = Proj(k[t, z, ξ0 , ξ1 ]),

W 0 = Proj(k[t, z 0 , ξ00 , ξ10 ])

along Proj(k[t, z, z −1 , ξ0 , ξ1 ]) and Proj(k[t, z 0 , z 0−1 , ξ00 , ξ10 ]) according to the rules: z 0 = z −1 , ξ 0 = z m ξ + tz k where ξ = ξ1 /ξ0 and ξ 0 = ξ10 /ξ00 are non-homogeneous coordinates on the corresponding copies of IP 1 . Let’s compute the local Kodaira-Spencer map κf,0 of f at 0. The image κf,0 ( dtd ) is the element of H 1 (Fm , TFm ) corresponding to the first order deformation of Fm obtained by glueing W0 := Proj(k[, z, ξ0 , ξ1 ])

W00 := Proj(k[, z 0 , ξ00 , ξ10 ])

1.2. LOCALLY TRIVIAL DEFORMATIONS

35

along Proj(k[, z, z −1 , ξ0 , ξ1 ]) and Proj(k[, z 0 , z 0−1 , ξ00 , ξ10 ]) according to the rules z 0 = z −1 , ξ 0 = z m ξ + z k By definition we have that κf,0 ( dtd ) is the element of H 1 (U, TFm ), where U = {W0 , W00 }, defined by the 1-cocycle corresponding to the vector field on W0 ∩ W00 ∂ {z k } ∂ξ According to Example B.44(iii) this element is non zero; therefore κf,0 is injective. Similarly we can consider a smooth proper family F : Y → Am−1 defined as follows. Y is the glueing of Y := Proj(k[t1 , . . . , tm−1 , z, ξ0 , ξ1 ]) and Y 0 := Proj(k[t1 , . . . , tm−1 , z 0 , ξ00 , ξ10 ]) along Proj(k[t1 , . . . , tm−1 , z, z −1 , ξ0 , ξ1 ]) and Proj(k[t1 , . . . , tm−1 , z 0 , z 0−1 , ξ00 , ξ10 ]) according to the rules: z 0 = z −1 , ξ 0 = z m ξ +

m−1 X

tj z j

j=1

The morphism F is defined by the projections onto Spec(k[t1 , . . . , tm−1 ]); the fibre of F over 0 is Y(0) ∼ = Fm . The computation we just did immediately implies that the local Kodaira-Spencer map κF,0 : T0 Am−1 → H 1 (Fm , TFm ) is an isomorphism.

1.2.5

Higher order deformations - Obstructions

Let X be a nonsingular algebraic variety. Consider a small extension e : 0 → (t) → A˜ → A → 0 in A and let ξ:

X → X ↓ ↓ Spec(k) → Spec(A)

36

CHAPTER 1. INFINITESIMAL DEFORMATIONS

be an infinitesimal deformation of X. A lifting of ξ to A˜ consists in a deformation X → X˜ ˜ ξ: ↓ ↓ ˜ Spec(k) → Spec(A) and an isomorphism of deformations X .

& φ

X

X˜ ×Spec(A) ˜ Spec(A)

−→ &

. Spec(A)

If we want to study arbitrary infinitesimal deformations, and not only first order ones, it is important to know whether, given ξ and e, a lifting of ξ to A˜ exists, and how many are there. Such an information can then be used to build an inductive procedure for the description of infinitesimal deformations. The following proposition addresses this question. Proposition 1.2.12 Given A in ob(A) and an infinitesimal deformation ξ of X over A: (i) To every small extension e of A there is associated an element oξ (e) ∈ ˜ which is 0 if and only H 2 (X, TX ) called the obstruction to lift ξ to A, ˜ if a lifting of ξ to A exists. (ii) If oξ (e) = 0 then there is a natural transitive action of H 1 (X, TX ) on ˜ the set of isomorphism classes of liftings of ξ to A. (iii) The correspondence e 7→ oξ (e) defines a k-linear map oξ : Exk (A, k) → H 2 (X, TX ) Proof. Let U = {Ui }i∈I be an affine open cover of X. We have isomorphisms θi : Ui × Spec(A) → X|Ui and consequently θij := θi−1 θj is an automorphism of the trivial deformation Uij × Spec(A). Moreover θij θjk = θik (1.13)

1.2. LOCALLY TRIVIAL DEFORMATIONS

37

on Uijk × Spec(A). To give a lifting ξ˜ of ξ to A˜ it is necessary and sufficient to give a collection of automorphisms {θ˜ij } of the trivial deformations Uij × ˜ such that Spec(A) (a) θ˜ij θ˜jk = θ˜ik (b) θ˜ij restricts to θij on Uij × Spec(A) In fact from such data we will be able to define X˜ by patching the local ˜ along the open subsets Uij × Spec(A) ˜ in the usual way. pieces Ui × Spec(A) To establish the existence of the collection {θ˜ij } let’s choose arbitrarily automorphisms {θ˜ij } satisfying the condition (b); they exist by Lemma 1.2.8(ii). Let −1 θ˜ijk = θ˜ij θ˜jk θ˜ik ˜ Since This is an automorphism of the trivial deformation Uijk × Spec(A). by (1.13) it restricts on Uijk × Spec(A) to the identity, by Lemma 1.2.6 we can identify each θ˜ijk with a d˜ijk ∈ Γ(Uijk , TX ) and it is immediate to check that {d˜ijk } ∈ Z 2 (U, TX ). If we choose different automorphisms {Φij } of the ˜ satisfying the analogous of condition (b) trivial deformations Uij × Spec(A) then Φij = θ˜ij + tdij (1.14) for some dij ∈ Γ(Uij , TX ), by Lemma 1.2.8(i). For each i, j, k the automorphism Φij Φjk Φ−1 ik corresponds to the derivation δijk = d˜ijk + (dij + djk − dik ) and therefore we see that the 2-cocycles {d˜ijk } and {δijk } are cohomologous. Their cohomology class oξ (e) ∈ H 2 (X, TX ) depends only on ξ and e and is 0 if and only if we can find a collection of automorphisms {Φij } such that δijk = 0 for all i, j, k ∈ I. In such a case {Φij } defines a lifting ξ˜ of ξ. This proves (i). Assume that oξ (e) = 0, i.e. that the lifting ξ˜ of ξ exists. Then we can choose the collection {θ˜ij } of automorphisms satisfying conditions (a) and (b) as above, in particular d˜ijk = 0, all i, j, k. Any other choice of a lifting ξ¯

38

CHAPTER 1. INFINITESIMAL DEFORMATIONS

of ξ to A˜ corresponds to a choice of automorphisms {Φij } satisfying (1.14) and the analogous of condition (b). Therefore, for all i, j, k, we have 0 = δijk = dij + djk − dik so that {dij } ∈ Z 1 (U, TX ) defines an element d¯ ∈ H 1 (X, TX ). As before ¯ it one checks that this element only depends on the isomorphism class of ξ; ˜ d) ¯ 7→ ξ¯ defines follows in a straightforward way that the correspondence (ξ, 1 a transitive action of H (X, TX ) on the set of isomorphism classes of liftings ˜ This proves (ii). of ξ to A. (iii) is left to the reader. qed Definition 1.2.13 The deformation ξ is called unobstructed if oξ is the zero map; otherwise ξ is called obstructed. X is unobstructed if every infinitesimal deformation of X is unobstructed; otherwise X is obstructed. Corollary 1.2.14 A nonsingular variety X is unobstructed if H 2 (X, TX ) = (0) The proof is obvious. Corollary 1.2.15 A nonsingular variety X is rigid if and only if H 1 (X, TX ) = (0) Proof. The hypothesis implies, by Proposition 1.2.9, that all first order deformations of X are trivial; moreover, by Proposition 1.2.12(ii), it implies that every infinitesimal deformation of X over any A in ob(A) has at most one lifting to any small extension of A. These two facts together easily give the conclusion. qed Examples 1.2.16 (i) If X is a projective nonsingular curve of genus g then from the Riemann-Roch theorem it follows that if g = 0 h (X, TX ) = 1 if g = 1  3g − 3 if g ≥ 2 1

 0

and h2 (X, TX ) = 0. In particular projective nonsingular curves are unobstructed.

1.2. LOCALLY TRIVIAL DEFORMATIONS

39

(ii) If X is a projective, irreducible and nonsingular surface X then H 2 (X, TX ) ∼ = H 0 (X, Ω1X ⊗ KX )∨ by Serre duality, and this rarely vanishes. For example a nonsingular surface of degree ≥ 5 in IP 3 satisfies H 2 (X, TX ) 6= (0), but it is nevertheless unobstructed (see example 3.2.11(i)); therefore the sufficient condition of Corollary 1.2.14 is not necessary. In general a surface such that H 2 (X, TX ) 6= (0) can be obstructed, but explicit examples are not elementary (see [104], [25], [94]). We will describe a class of such examples in Theorem 3.4.26, page 221. in §2.4 we will show how to construct examples of obstructed 3-folds (see remarks following Proposition 3.4.23, page 221). The first examples of obstructed compact complex manifolds where given in Kodaira-Spencer [120], §16: they are of the form T × IP 1 , where T is a two-dimensional complex torus. (iii) The projective space IP n is rigid for every n ≥ 1. In fact it follows immediately from the Euler sequence: 0 → OIP n → OIP n (1)n+1 → TIP n → 0 that H 1 (IP n , TIP n ) = 0. Similarly one shows that finite products IP n1 × · · · × IP nk of projective spaces are rigid. (iv) The ruled surfaces Fm are unobstructed because h2 (Fm , TFm ) = 0 (see (B.13), page 350).

40

CHAPTER 1. INFINITESIMAL DEFORMATIONS

Chapter 2 Formal deformation theory In this chapter we develop the theory of “functors of Artin rings”. The main result of this theory is a theorem of Schlessinger giving necessary and sufficient conditions for a functor of Artin rings to have a semiuniversal or a universal formal element. We then apply the functorial machinery to the construction of formal semiuniversal, or universal, deformations, which is the final goal of formal deformation theory, and we explain the relation between formal and algebraic deformations. In order to make the arguments easier to follow these applications are given only in the case of the deformation functors Def X and Def 0X of an algebraic scheme. Only in Chapter 3 we will apply the general theory to other deformation functors.

2.1

Obstructions

In this section we investigate the notion of formal smoothness in the category A∗ using the language of extensions. The results we prove are crucial for the understanding of obstructions in deformation theory. Our treatment is an expansion of [177]; for a more systematic treatment we refer to [55]. Let Λ ∈ ob(A∗ ) and µ : Λ → R be in ob(A∗Λ ). The relative obstruction space of R/Λ is o(R/Λ) := ExΛ (R, k) If Λ = k then o(R/k) is called the (absolute) obstruction space of R and simply denoted by o(R). We say that R is unobstructed (resp. obstructed) over Λ if 41

42

CHAPTER 2. FORMAL DEFORMATION THEORY

o(R/Λ) = (0) (resp. if o(R/Λ) 6= (0)); R is said to be unobstructed (resp. obstructed) if o(R) = (0) (resp. if o(R) 6= (0)). Given a homomorphism f : R → S in A∗Λ we denote by o(f /Λ) : o(S/Λ) → o(R/Λ) the linear map induced by pullback: o(f /Λ)([η]) = [f ∗ η] ∈ ExΛ (R, k) for all [η] ∈ ExΛ (S, k). Since this definition is functorial we have a contravariant functor: o(−/Λ) : A∗Λ → (vector spaces/k) When Λ = k we write o(f ) instead of o(f /k). If µ is such that o(µ) is injective one simetimes says that R is less obstructed than Λ. By applying Proposition 1.1.4 we obtain an exact sequence for each f : R → S in A∗Λ : o(f /Λ)

0 → tS/R → tS/Λ → tR/Λ → o(S/R) → o(S/Λ) −→ o(R/Λ)

(2.1)

In case Λ = k we obtain the exact sequence: o(f )

0 → tS/R → tS → tR → o(S/R) → o(S) −→ o(R)

(2.2)

which relates the absolute and the relative obstruction spaces. The next result gives a description of o(R/Λ) and an interpretation of formal smoothness of a Λ-algebra (R, m) in A∗Λ . ˆ Proposition 2.1.1 Assume that Λ is in ob(A). ˆ be the natural homomorphism (i) Let (R, m) be in ob(A∗Λ ) and let χ : R → R ˆ of R into its m-adic completion R. Then the induced map: ˆ o(χ/Λ) : o(R/Λ) → o(R/Λ) is an isomorphism. (ii) For every (R, m) in ob(A∗Λ ) let d = dimk (tR/Λ ) and let ˆ = Λ[[X1 , . . . , Xd ]]/J R

2.1. OBSTRUCTIONS

43

ˆ Then with J ⊂ (X)2 , be a presentation of the m-adic completion R. there is a natural isomorphism: ∨ o(R/Λ) ∼ = (J/(X)J)

In particular R is unobstructed over Λ if and only if it is a formally smooth Λ-algebra. Proof. (i) Let ˆ→0 η:0→k→S→R be a small extension; denote by m0 the maximal ideal of S. Claim: S is complete. ˆ is Let {fn } ⊂ S be a Cauchy sequence; then the image sequence {f¯n } in R Cauchy, hence it converges to a limit which we may assume to be zero, after possibly subtracting a constant sequence from {fn }. We have f¯n ∈ m ˆ e(n) , with limn [e(n)] = ∞. For every n we may find gn ∈ m0e(n) lying above f¯n . The sequence {gn } in S is Cauchy and converges to zero, and {fn − gn } is a Cauchy sequence in k. Since k is complete as an S-module, because it is annihilated by the maximal ideal, {fn − gn } converges to a limit f ∈ k. This is also the limit of {fn } because fn − f = (fn − gn − f ) + gn Therefore S is complete. If χ∗ (η/Λ) is trivial the section induces a homomorphism g : R → S ˆ because S is complete. Hence η is trivial. This which factors through R proves that o(χ/Λ) is injective. Given a small Λ-extension of R: (S, ϕ) :

0→k→S→R→0

ˆ = k. Therefore [S, ˆ is surjective and ker(ϕ) ˆ ϕ] the map ϕˆ : Sˆ → R ˆ =k ˆ ∈ ˆ ˆ ExΛ (R, k) and o(χ/Λ)([S, ϕ]) ˆ = [S, ϕ]: this means that o(χ/Λ) is also surjective. ˆ is a power series ring (ii) R is a formally smooth Λ-algebra if and only if R over Λ, i.e. if and only if J = (0). Therefore the last assertion follows from the fact that J/(X)J = (0) if and only if J = (0), by Nakayama’s lemma.

44

CHAPTER 2. FORMAL DEFORMATION THEORY

In order to prove the first assertion we may assume that R is in ob(AˆΛ ), ˆ since o(R/Λ) = o(R/Λ) by the first part of the proposition. Hence R = 2 Λ[[X]]/J with J ⊂ (X) . The extension of R: Φ:

0 → J/(X)J → Λ[[X]]/(X)J → R → 0

induces by pushouts a homomorphism: α : (J/(X)J)∨ d

→ 7−→

ExΛ (R, k) = o(R) [d∗ Φ]

Letting M be the maximal ideal of Λ[[X]]/(X)J we have J/(X)J ⊂ M 2 . If d ∈ (J/(X)J)∨ is such that [d∗ Φ] = 0 then we have: Φ: d∗ Φ :

0 → J/(X)J ↓d 0→ k

→ Λ[[X]]/(X)J ↓h → A

→ R →0 k → R →0

with d∗ Φ trivial. From the example 1.1.2(ii) it follows that the generator of k in A is contained in mA \m2A . Since h(J/(X)J) ⊂ m2A we deduce that d = 0. It follows that α is injective. Conversely, given a Λ-extension (A, ϕ) of R by k it is possible to find a lifting: Λ[[X]] ↓ ϕ˜ & ϕ: A → R because A is complete (see proof of 2.1.1 and C.47(ii)). From the fact that ker(ϕ) = k it follows that ker(ϕ) ˜ ⊃ (X)J and therefore we have a commutative diagram: Φ: (A, ϕ) :

0 → J/(X)J ↓d 0→ k

→ Λ[[X]]/(X)J ↓ ϕ˜ → A

→ R →0 k → R →0

in which d is the map induced by ϕ. ˜ It follows that (A, ϕ) = d∗ Φ; hence α is surjective. qed

2.1. OBSTRUCTIONS

45

Corollary 2.1.2 For every R in ob(A∗ ) the following are true: (i) (ii)

dimk [o(R)] dimk (tR ) ≥

dim(R) ≥

<∞ dimk (tR ) − dimk [o(R)]

(where dim(R) means Krull dimension of R). In (ii) the first equality holds if and only if R is formally smooth; the second equality holds if and only if ˆ = k[[X1 , . . . , Xd ]]/J, with J ⊂ (X)2 and J generated by a regular sequence. R ˆ = k[[X1 , . . . , Xd ]]/J, ˆ hence R = R Proof. We may assume that R is in ob(A); 2 ∨ ∼ with J ⊂ (X) and o(R) = (J/(X)J) . Then (i) and (ii) follow from the fact that dimk [J/(X)J] is the number of elements of a minimal set of generators of J. qed Remarks 2.1.3 The only formally smooth k-algebra in A is k itself. By 2.1.1(ii) this means that o(k) = (0) and that o(A) 6= (0) for every A 6= k in A. The following are some special cases. ˜ , a trivial extension of k by a vector space V of dimension • If A = k⊕V d, then A ∼ = k[X1 , . . . , Xd ]/(X)2 , and o(A) = [(X)2 /(X)3 ]∨ . • If A = k[X]/(X)k then o(A) = [(X)k /(X)k+1 ]∨ . In particular, if A = k[t]/(tn ), n ≥ 2, then o(A) = [(tn )/(tn+1 )]∨ is 1-dimensional; from the proof of 2.1.1(ii) it follows immediately that o(A) is generated by the class of the extension: 0 → (tn )/(tn+1 ) → k[t]/(tn+1 ) → k[t]/(tn ) → 0 We will need the following Lemma 2.1.4 (i) Let µ : Λ → R be a homomorphism in A∗ . Given a small extension η : B → A in A and a homomorphism ϕ : R → A, the condition ϕ∗ (η) ∈ ker(o(µ)) is equivalent to the existence of a commutative diagram: ϕ A ←− R (2.3) ↑η ↑µ B ←− Λ ϕ ˜

46

CHAPTER 2. FORMAL DEFORMATION THEORY Moreover ϕ∗ (η) = 0 if and only if there exists ϕ0 : R → B such that the resulting diagram ϕ A ←− R ↑η

. ϕ0

↑µ

B

←−

Λ

ϕ ˜

(2.4)

is commutative. ˆ and µ : Λ → R in ob(A∗Λ ) there exists A in ob(AΛ ) (ii) For every Λ in ob(A) and a homomorphism p : R → A such that o(p/Λ) : o(A/Λ) → o(R/Λ) is surjective. Proof. (i) is left to the reader. (ii) We will show something more precise, namely that if pn : R → R/mn+1 is the natural map, then o(pn /Λ) : o((R/mn+1 )/Λ) → o(R/Λ) is surjective for all n 0. ˆ Since o(pn /Λ) factors through o(R/Λ), which is isomorphic to o(R/Λ), we may assume that R is in AˆΛ . Let’s write: R = Λ[[X]]/J where J = (g1 , . . . , gs ) ⊂ (X)2 . Let n 0 be such that gj ∈ / (X)n+2 for all j = 1, . . . , s. Then we have: R Λ[[X]] = n+1 m (J, (X)n+1 ) and therefore o((R/mn+1 )/Λ) =

h

(J, (X)n+1 ) i∨ ((X)J, (X)n+2 )

The map o(pn /Λ) is the transpose of ın :

h (J, (X)n+1 ) i J → (X)J ((X)J, (X)n+2 )

2.1. OBSTRUCTIONS

47

induced by the inclusion J ⊂ (J, (X)n+1 ). From the hypothesis on n it follows that if γ ∈ J ∩ ((X)J, (X)n+2 ) then γ ∈ (X)J; this means that ın is injective, i.e. that o(pn /Λ) is surjective. qed The following theorem gives a characterization of formally smooth homomorphisms in A∗ . Theorem 2.1.5 Let µ : Λ → R be a homomorphism in A∗ . The following conditions are equivalent: (i) For every commutative diagram (2.3) with η a small extension in A∗ there exists ϕ0 : R → B such that diagram (2.4) is commutative. (ii) µ is formally smooth. (iii) dµ : tR → tΛ is surjective and o(µ) is injective. (iv) o(R/Λ) = (0) Proof. (i) ⇒ (ii) is trivial. (ii) ⇒ (iii) Let v ∈ tΛ be given as a k-algebra homomorphism Λ → k[]. The formal smoothness of µ implies the existence of a homomorphism w : R → k[] which makes the following diagram commutative: k ← R ↑ . ↑ k[] ← Λ and this means that dµ(w) = v. Therefore dµ is surjective. Consider a commutative diagram of k-algebra homomorphisms (2.3) with η a small extension in A. Then o(ϕ)([η]) ∈ ker(o(µ)). By the formal smoothness of µ there exists ϕ0 : R → B making (2.4) commutative: this implies that o(ϕ)([η]) = 0. Since, by Lemma 2.1.4, ϕ and η can be chosen so that o(ϕ)([η]) is an arbitrary element of ker(o(µ)), we deduce that o(µ) is injective. (iii) ⇔ (iv) follows from the exact sequence (2.2). (iii) ⇒ (i). Consider a diagram (2.3) with η a small extension in A∗ . Then o(ϕ)([η]) ∈ ker(o(µ)) By assumption o(ϕ)([η]) = 0, and therefore there exists ϕ¯ : R → B such that η ϕ¯ = ϕ. It follows that ϕ˜ − ϕµ ¯ : R → ker(η) = k is a k-derivation.

48

CHAPTER 2. FORMAL DEFORMATION THEORY

By assumption there exists a k-derivation v : R → k such that ϕ˜ − ϕµ ¯ = dµ(v) = vµ. Then ϕ0 := ϕ¯ + v : R → B is a k-homomorphism which obviously satisfies 0 ηϕ = ϕ. Moreover ϕ0 µ = (ϕ¯ + v)µ = ϕµ ¯ + vµ = ϕ˜ and therefore ϕ0 makes (2.4) commutative.

qed

In the special case Λ = k we obtain that a k-algebra R in ob(A∗ ) is unobstructed if and only if R is formally smooth, a result already proven in 2.1.1. More generally the theorem says that µ is formally smooth if and only if dµ is surjective and R is less obstructed than Λ. The following corollary is immediate. Corollary 2.1.6 (i) Let µ : Λ → R be a homomorphism in A∗ such that dµ is surjective and R is formally smooth. Then Λ and µ are formally smooth. (ii) A homomorphim µ : Λ → R in A∗ is formally etale if and only if dµ is an isomorphism and o(µ) is injective. This happens in particular if dµ is an isomorphism and R is formally smooth. ˆ is formally (iii) If R ∈ ob(A∗ ) then the natural homomorphism R → R etale. In practise it is seldom possible to compute the obstruction map o(µ) explicitly for a given µ : Λ → R in A∗ . But for the purpose of studying the formal smoothness of µ all that counts is to have informations about ker[o(µ)]. This can be achieved somehow indirectly by means of the following simple result, which turns out to be very effective in practise. Proposition 2.1.7 Let µ : Λ → R be a morphism in A∗ . Assume that there exists a k-vector space υ(R/Λ) such that for every homomorphism ϕ : R → A in A∗Λ with A in ob(AΛ ) there is a k-linear map ϕυ : ExΛ (A, k) → υ(R/Λ) satisfying ker(ϕυ ) = ker[o(ϕ/Λ)]

2.1. OBSTRUCTIONS

49

where

ϕυ

−→

ExΛ (A, k)

υ(R/Λ)

& o(ϕ/Λ) ExΛ (R, k) = o(R/Λ) Then there is a natural k-linear inclusion o(R/Λ) ⊂ υ(R/Λ) Proof. Choosing ϕ such that o(ϕ/Λ) surjects onto o(R/Λ) we obtain an inclusion o(R/Λ) ⊂ υ(R/Λ) as asserted. qed In practise this proposition will be applied as follows. Given ϕ, to give an element of ExΛ (A, k) is the same as to give a commutative diagram (2.3). Assume that to each such diagram one associates in a linear way an element of υ(R/Λ) which vanishes if and only if there is an extension ϕ0 : R → B making (2.4) commutative. Then the proposition applies. Taking Λ = k we get the following absolute version of the proposition, where for the last assertion we apply 2.1.2(ii). Corollary 2.1.8 Let R be in ob(A∗ ). Assume that there exists a k-vector space υ(R) such that for every morphism ϕ : R → A in A∗ with A in ob(A) there is a k-linear map ϕυ : o(A) → υ(R) satisfying ker(ϕυ ) = ker[o(ϕ)] Then there is a natural k-linear inclusion o(R) ⊂ υ(R) If υ(R) is finite dimensional then dimk (tR ) ≥ dim(R) ≥ dimk (tR ) − dimk [υ(R)] A k-vector space υ(R) satisfying the conditions of 2.1.8 will sometimes be called an obstruction space for R.

50

2.2

CHAPTER 2. FORMAL DEFORMATION THEORY

Functors of Artin rings

A functor of Artin rings is a covariant functor F : AΛ −→ (sets) ˆ Let A be in ob(AΛ ). An element ξ ∈ F (A) will be where Λ ∈ ob(A). called an infinitesimal deformation of ξ0 ∈ F (k) if ξ 7→ ξ0 under the map F (A) → F (k); if A = k[] then ξ is called a first order deformation of ξ0 . Examples of functors of Artin rings are obtained by fixing an R in ob(AˆΛ ) and letting: hR/Λ (A) = HomAˆΛ (R, A)

for every A in ob(AΛ )

Such a functor is clearly nothing but the restriction to AΛ of a representable functor on AˆΛ . A functor of Artin rings isomorphic to hR/Λ for some R in ob(AˆΛ ) is called prorepresentable. In case Λ = k we write hR instead of hR/k . Every representable functor hR/Λ , R in ob(AΛ ), is a (trivial) example of prorepresentable functor. Typically a prorepresentable functor of Artin rings arises as follows. One considers a scheme M and the restriction Φ : A −→ (sets) Φ(A) = Hom(Spec(A), M ) of the functor of points Hom(−, M ) : (schemes)◦ −→ (sets) Φ is a functor of Artin rings; if ϕ : Spec(A) → M is an element of Φ(A) then ϕ is an infinitesimal deformation of the composition ϕ

Spec(k) → Spec(A) −→ M where the first morphism corresponds to A → A/mA = k. For a fixed k-rational point m ∈ M , one may consider the subfunctor F : A −→ (sets) of Φ defined as follows: F (A) = Hom(Spec(A), M )m =

n

morphisms Spec(A) → M o whose image is {m}

(2.5)

2.2. FUNCTORS OF ARTIN RINGS

51

ˆM,m , so F is prorepresentable. We will We have F = hR , where R = O introduce several functors of Artin rings which are defined by deformation problems. The hope is that they are of the form (2.5) for some k-pointed algebraic scheme (M, m) which has some geometrical meaning with respect to the deformation problem under consideration (a “moduli scheme”). As we will see, this sometimes happens, but most of the times it doesn’t. We will thus have to content ourselves to prove that such functors have weaker properties. In this section and in the following one we will study prorepresentability and other properties of functors of Artin rings from an abstract point of view. Later on we will consider specific examples of such functors related with deformation problems. A prorepresentable functor F = hR/Λ satisfies the following conditions: H0 ) F (k) consists of one element (the canonical quotient R → R/mR = k) Let

A0

A00 &

. A be a diagram in AΛ and consider the natural map

(2.6)

α : F (A0 ×A A00 ) → F (A0 ) ×F (A) F (A00 ) induced by the commutative diagram: F (A0 ×A A00 ) → F (A00 ) ↓ ↓ 0 F (A ) → F (A) Then H` ) (left exactness) For every diagram (2.6) α is bijective (straightforward to check). Hf )

F (k[]) has a structure of finite dimensional k-vector space.

In fact F (k[]) = HomΛ−alg (R, k[]) = DerΛ (R, k) = tR/Λ is the relative tangent space of R over Λ (here the Λ-algebra structure of k[] is given by the composition Λ → k → k[], (see B.42(vi)). A property weaker than H` satisfied by a prorepresentable functor F is the following:

52

CHAPTER 2. FORMAL DEFORMATION THEORY

H ) α is bijective if A = k and A00 = k[]. It is interesting to observe that if F is prorepresentable the structure of k-vector space on F (k[]) can be reconstructed in a purely functorial way only using properties H0 and H , and without using the prorepresentability explicitly. Indeed it is an easy exercise to check that the homomorphism: +:

k[] ×k k[]

−→

k[]

(a + b, a + b0 ) 7−→ a + (b + b0 ) induces the operation of sum on F (k[]) as the composition F (+)

F (k[]) × F (k[]) → F (k[] ×k k[]) −→ F (k[]) where the first map is the inverse of α. Associativity is checked using H . The zero is the image of F (k) → F (k[]). The multiplication by a scalar c ∈ k is induced in F (k[]) by the morphism k[]

−→

k[]

a + b 7−→ a + (cb) We can therefore state the following Lemma 2.2.1 If F is a functor of Artin rings having properties H0 and H then the set F (k[]) has a structure of k-vector space in a functorial way. This vector space is called the tangent space of the functor F , and denoted tF . If F = hR/Λ then tF = tR/Λ . If f : F → G is a morphism of functors of Artin rings then the induced map tF → tG is called the differential of f and it is denoted df . It is straightforward to check that if F and G satisfy H0 and H then df is k-linear. Every functor of Artin rings F can be extended to a functor Fˆ : AˆΛ −→ (sets) by letting, for every R in ob(AˆΛ ): Fˆ (R) = lim F (R/mn+1 R ) ←−

2.2. FUNCTORS OF ARTIN RINGS

53

and for every ϕ : R → S: Fˆ (ϕ) : Fˆ (R) → Fˆ (S) to be the map induced by the maps F (R/mnR ) → F (S/mnS ), n ≥ 1. An element uˆ ∈ Fˆ (R) is called a formal element of F . By definition uˆ can be represented as a system of elements {un ∈ F (R/mn+1 R )}n≥0 such that for every n ≥ 1 the map n F (R/mn+1 R ) → F (R/mR )

induced by the projection R/mn+1 → R/mnR sends R un 7−→ un−1

(2.7)

uˆ is also called a formal deformation of u0 . If f : F → G is a morphism of functors of Artin rings then it can be extended in an obvious way to a ˆ morphism of functors fˆ : Fˆ → G. Lemma 2.2.2 Let R be in ob(AˆΛ ). There is a 1-1 correspondence between Fˆ (R) and the set of morphisms of functors hR/Λ −→ F

(2.8)

Proof. To a formal element uˆ ∈ Fˆ (R) there is associated a morphism of functors (2.8) in the following way. Each un ∈ F (R/mn+1 R ) defines a morphism of functors h(R/mn+1 )/Λ → F . The compatibility conditions (2.7) imply that the following diagram commutes: h(R/mn )/Λ

→ h(R/mn+1 )/Λ &

↓ F

for every n. Since for each A in ob(AΛ ) h(R/mn )/Λ (A) → h(R/mn+1 )/Λ (A) is a bijection for all n 0 we may define hR/Λ (A) → F (A)

54

CHAPTER 2. FORMAL DEFORMATION THEORY

as lim [h(R/mn+1 )/Λ (A) → F (A)]

n→∞

Conversely each morphism (2.8) defines a formal element uˆ ∈ Fˆ (R), where n+1 un ∈ F (R/mn+1 via R ) is the image of the canonical projection R → R/mR the map n+1 hR/Λ (R/mn+1 R ) → F (R/mR ) qed Definition 2.2.3 If R is in ob(AˆΛ ) and uˆ ∈ Fˆ (R), we call (R, uˆ) a formal couple for F . The differential tR/Λ → tF of the morphism hR/Λ → F defined by uˆ is called the characteristic map of uˆ (or of the formal couple (R, uˆ)) and is denoted dˆ u. If (R, uˆ) is such that the induced morphism (2.8) is an isomorphism, then F is prorepresentable, and we also say that F is prorepresented by the formal couple (R, uˆ). In this case uˆ is called a universal formal element for F , and (R, uˆ) is a universal formal couple. An ordinary couple (R, u) with R in AΛ defines a special case of formal couple, in which (Rn , un ) = (Rn+1 , un+1 ) for all n 0. A universal formal couple seldom exists; we will therefore need to introduce some weaker properties of a formal couple. They are based on the following definition. Definition 2.2.4 Let f : F → G be a morphism of functors of Artin rings. f is called smooth if for every surjection µ : B → A in AΛ the natural map: F (B) → F (A) ×G(A) G(B)

(2.9)

induced by the diagram: F (B) → G(B) ↓ ↓ F (A) → G(A) is surjective. The functor F is called smooth if the morphism from F to the constant functor G(A) = {one element} is smooth; equivalently, if F (µ) : F (B) → F (A) is surjective for every surjection µ : B → A in AΛ .

all A ∈ ob(AΛ )

2.2. FUNCTORS OF ARTIN RINGS

55

Note that, since every surjection in AΛ factors as a finite sequence of small extensions (i.e. surjections with one-dimensional kernel), for f (or F ) to be smooth it is necessary and sufficient that the defining condition is satisfied for small extensions in AΛ . Next proposition states some properties of the notion of smoothness of a morphism of functors of Artin rings. Proposition 2.2.5 (i) Let f : R → S be a homomorphism in AˆΛ . Then f is formally smooth if and only if the morphism of functors hf : hS/Λ → hR/Λ induced by f is smooth. (ii) If f : F → G is smooth then f is surjective, i.e. F (B) → G(B) is surjective for every B in ob(AΛ ). In particular the differential df : tF → tG is surjective. (iii) If f : F → G is smooth, then F is smooth if and only if G is smooth. (iv) If f : F → G is smooth then the induced morphism of functors fˆ : Fˆ → ˆ is surjective, i.e. Fˆ (R) → G(R) ˆ G is surjective for every R in ob(AˆΛ ). (v) If F → G and G → H are smooth morphisms of functors then the composition F → H is smooth. (vi) If F → G and H → G are morphisms of functors and F → G is smooth, then F ×G H → H is smooth. (vii) If F and G are smooth functors then F × G is smooth. Proof. (i)

Let B → A be a surjection in AΛ and let A ↑ B

←

S ↑f ← R

be a commutative diagram of homomorphisms of Λ-algebras. The formal smoothness of f is equivalent to the existence, for each such diagram, of a morphism S → B such that the resulting diagram A ↑ B

← S . ↑f ← R

56

CHAPTER 2. FORMAL DEFORMATION THEORY

is commutative. This is another way to express the condition that the map hS/Λ (B) −→ hS/Λ (A) ×hR/Λ (A) hR/Λ (B) is surjective, i.e. that hf is smooth. (ii) follows immediately from the definition. (iii) Let µ : B → A be a surjection in AΛ , and consider the diagram: F (B) → F (A) ×G(A) G(B) → G(B) ↓ ↓ G(µ) F (A) −→ G(A) f (A)

Suppose G is smooth and let ξ ∈ F (A). By the smoothness of G there exists η ∈ G(B) such that G(µ)(η) = f (A)(ξ). By the smoothness of f there is ζ ∈ F (B) which is mapped to (ξ, η) ∈ F (A) ×G(A) G(B). It follows that F (µ)(ζ) = ξ and F is smooth. The converse is proved similarly using the surjectivity of f (A). ˆ (iv) Let vˆ = {vn } ∈ G(R). Since f is smooth the map F (R/m2R ) → F (k) ×G(k) G(R/m2R ) = G(R/m2R ) is surjective. Therefore there exists w1 ∈ F (R/m2R ) such that f (R/m2R )(w1 ) = v1 . Let’s assume that for some n ≥ 1 there exist wi ∈ F (R/mi+1 R ), i = 1, . . . , n such that: f (R/mi+1 R )(wi ) = vi i and wi 7−→ wi−1 under F (R/mi+1 R ) −→ F (R/mR ). The surjectivity of the map: n n+1 F (R/mn+1 R ) −→ F (R/mR ) ×G(R/mn ) G(R/mR )

implies that there exists wn+1 ∈ F (R/mn+1 R ) whose image is (wn , vn+1 ). By induction we conclude that there exists wˆ ∈ Fˆ (R) whose image under fˆ is vˆ. The proofs of (v), (vi) and (vii) are straightforward. qed We now introduce the notions of “versality” and “semiuniversality”, which are slightly weaker than universality. Definition 2.2.6 Let F be a functor of Artin rings. A formal element uˆ ∈ Fˆ (R), for some R in ob(AˆΛ ), is called versal if the morphism hR/Λ → F defined by uˆ is smooth; uˆ is called semiuniversal if it is versal and moreover the differential tR/Λ → tF is bijective. We will correspondingly speak of a versal formal couple (respectively a semiuniversal formal couple).

2.2. FUNCTORS OF ARTIN RINGS

57

It is clear from the definitions that: uˆ universal ⇒ uˆ semiuniversal ⇒ uˆ versal but none of the inverse implications is true (see Examples 2.6.8). We can also describe these properties as follows. Assume given µ : (B, ξB ) → (A, ξA ), a surjection of couples for F (i.e. µ is surjective), and a homomorphism ϕ : R → A such that ϕ(ˆ u) = ξA . Then uˆ is versal if for every such data there is a lifting ψ : R → B of ϕ (i.e. µψ = ϕ) such that ψ(ˆ u) = ξB : B

ξB

ψ%

% ↓µ

R ϕ&

↓

uˆ &

A

ξA

uˆ is universal if moreover the lifting ψ is unique. uˆ is semiuniversal if it is versal and moreover the lifting ψ is unique when µ : k[] → k. Let (R, uˆ) and (S, vˆ) be two formal couples for F . A morphism of formal couples f : (R, uˆ) → (S, vˆ) is a morphism f : R → S in Aˆ such that Fˆ (f )(ˆ u) = vˆ. We will call f an isomorphism of formal couples if in addition f : R → S is an isomorphism. It is obvious that with this definition the formal couples for F and their morphisms form a category containing the category IF of couples for F as a full subcategory. Proposition 2.2.7 Let F be a functor of Artin rings. Then: (i) If (R, uˆ) and (S, vˆ) are universal formal couples for F there exists a unique isomorphism of formal couples (R, uˆ) ∼ = (S, vˆ). (ii) If (R, uˆ) and (S, vˆ) are semiuniversal formal couples for F there exists an isomorphism of formal couples (R, uˆ) ∼ = (S, vˆ), which is not necessarily unique but the induced isomorphism tS ∼ = tR is uniquely determined. (iii) If (R, uˆ) is a semiuniversal formal couple for F and (S, vˆ) is versal, then there is an isomorphism, not necessarily unique: ϕ : R[[X1 , . . . , Xr ]] → S

58

CHAPTER 2. FORMAL DEFORMATION THEORY for some r ≥ 0, such that Fˆ (ϕj)(ˆ u) = vˆ, where j : R ⊂ R[[X1 , . . . , Xr ]] is the inclusion.

Proof. (i) By the universality of (R, uˆ) for every n ≥ 1 there exists a n+1 unique fn ∈ hR/Λ (S/mn+1 S ) such that fn 7→ vn ∈ F (S/mS ) under the isomorphism associated to uˆ. In this way we obtain f = lim{fn } : (R, uˆ) → (S, vˆ) ←

which is uniquely determined. Analogously we can construct a uniquely determined g : (S, vˆ) → (R, uˆ). By universality the compositions gf : (R, uˆ) → (R, uˆ) and f g : (S, vˆ) → (S, vˆ) are the identity. (ii) Using versality, and proceeding as above we can construct morphisms of formal couples f : (R, uˆ) → (S, vˆ) and g : (S, vˆ) → (R, uˆ) We obtain a commutative diagram: tR/Λ ↑↓ tS/Λ

& −→ tF

where the vertical arrows are the differentials of f and g, and the other arrows are the characteristic maps dˆ u and dˆ v . From this diagram we deduce that −1 −1 df = (dˆ u) dˆ v and dg = (dˆ v ) dˆ u are uniquely determined. Since d(gf ) = (dg)(df ) = identity of tS/Λ and d(f g) = (df )(dg) = identity of tR/Λ it follows that f and g are bijections inverse of each other.

2.2. FUNCTORS OF ARTIN RINGS (iii)

59

By the versality of (R, uˆ) we can find a morphism of formal couples f : (R, uˆ) → (S, vˆ)

We obtain a commutative diagram: tR/Λ ↑ tS/Λ

& −→ tF

where dˆ u : tR/Λ → tF is bijective because (R, uˆ) is semiuniversal, and dˆ v : tS/Λ → tF is surjective because (S, vˆ) is versal. Hence df : tS/Λ → tR/Λ is surjective. This means that f induces an inclusion t∨R/Λ ⊂ t∨S/Λ Let x1 , . . . , xr ∈ S be elements which induce a basis of t∨S/R = mS /[m2S + f (mR )] and define: ϕ : R[[X1 , . . . , Xr ]] → S by ϕ(Xi ) = xi , i = 1, . . . , r. Let j : R ⊂ R[[X1 , . . . , Xr ]] be the inclusion. Letting wˆ = Fˆ (j)(ˆ u) ∈ Fˆ (R[[X1 , . . . , Xr ]]) we have a commutative diagram of formal couples: ϕ

(R[[X1 , . . . , Xr ]], w) ˆ

−→

↑j

%f

(S, vˆ)

(R, uˆ) such that Fˆ (ϕj)(ˆ u) = vˆ. Since ϕ1 : R[[X1 , . . . , Xr ]]/M 2 → S/m2S is an isomorphism (here M ⊂ R[[X1 , . . . , Xr ]] denotes the maximal ideal) ϕ is surjective. Let ψ1 : S/m2S → R[[X1 , . . . , Xr ]]/M 2

60

CHAPTER 2. FORMAL DEFORMATION THEORY

be the inverse of ϕ1 . We have F (ψ1 )(v1 ) = w1 hence, by the versality of (S, vˆ), it is possible to find a lifting of ψ1 : ψ : S → R[[X1 , . . . , Xr ]] such that Fˆ (ψ)(ˆ v ) = w. ˆ Since by construction d(ψϕ) = dψ dϕ = identity of tR[[X]] it follows that ψϕ is an automorphism of R[[X1 , . . . , Xr ]]. In particular we deduce that ϕ is injective. qed The following is a useful remark. Proposition 2.2.8 If a functor of Artin rings F : A → (sets) satisfying H0 has a semiuniversal formal element and tF = (0) then F = hk , the constant functor. Proof. The assumptions imply that there is a smooth morphism hk → F . Since a smooth morphism of functors satisfying H0 is surjective, the conclusion follows. qed ∗

∗

∗

∗

∗ ∗

For the rest of this section we will only consider functors of Artin rings satisfying conditions H0 and H . Definition 2.2.9 Let F be a functor of Artin rings. Suppose that υ(F ) is a k-vector space such that for every A in ob(AΛ ) and for every ξ ∈ F (A) there is a k-linear map ξυ : ExΛ (A, k) → υ(F ) with the following property: ˜ ϕ) such that ker(ξυ ) consists of the isomorphism classes of extensions (A, ˜ → F (A)] ξ ∈ Im[F (A) Then υ(F ) is called an obstruction space for the functor F . If F has (0) as an obstruction space then it is called unobstructed.

2.2. FUNCTORS OF ARTIN RINGS

61

If F is prorepresented by the formal couple (R, uˆ), then o(R/Λ) is an obstruction space for F , as it follows from the functorial characterization of o(R/Λ) given in §2.1. Conversely if υ(F ) is an obstruction space for F , and F is prorepresented by the formal couple (R, uˆ), then it follows immediately from the definition that υ(F ) is a relative obstruction space for R/Λ. One can show that under relatively mild conditions a functor F has an obstruction space. We will not discuss this matter here, and we refer the interested reader to [55]. The following are some basic properties of obstruction spaces. Proposition 2.2.10 (i) F has (0) as an obstruction space if and only if it is smooth. (ii) Let f : F → G be a smooth morphism of functors of Artin rings. If υ(G) is an obstruction space for G then it is also an obstruction space for F . Proof. (i) is obvious. (ii) Let A be in ob(AΛ ) and ξ ∈ F (A), and let ξυ := f (A)(ξ)υ : ExΛ (A, k) → υ(G) ˜ the map defined by f (A)(ξ) ∈ G(A). If A˜ → A = A/(t) defines an element ˜ of ker(ξυ ) then, since υ(G) is an obstruction space for G, there is η ∈ G(A) ˜ → G(A). From the smoothness such that η 7→ f (A)(ξ) under the map G(A) of f it follows that the map ˜ → F (A) ×G(A) G(A) ˜ F (A) ˜ which maps to the formal couple (ξ, η). is surjective, hence there is ξ˜ ∈ F (A) ˜ ˜ It follows that ξ 7→ ξ under F (A) → F (A). ˜ such that Conversely, if A˜ is such that there exists ξ˜ ∈ F (A) ˜ → F (A) F (A) ξ˜ then from the diagram:

7→

ξ

˜ → F (A) F (A) ˜ ↓ f (A)

↓ f (A)

˜ → G(A) G(A)

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CHAPTER 2. FORMAL DEFORMATION THEORY

˜ 7→ f (A)(ξ) hence f (A)(ξ) ∈ Im[G(A) ˜ ξ) ˜ → G(A)]. Therewe see that f (A)( ˜ fore the extension A → A defines an element of ker[f (A)(ξ)υ ]. This proves that ξυ satisfies the conditions of definition 2.2.9. qed

Corollary 2.2.11 Let F : A → (sets) be a functor of Artin rings, and suppose that (R, uˆ) is a versal formal couple for F . If F has a finite dimensional obstruction space υ(F ) then dimk (tR ) ≥ dim(R) ≥ dimk (tR ) − dimk [υ(F )] Proof. From the definition of versal formal couple and from Proposition 2.2.10 we deduce that υ(F ) is an obstruction space for hR , hence for R. The conclusion follows from 2.1.6. qed Corollary 2.2.11 is very important in deformation theory since it gives a basic geometrical information about a deformation problem. If F is induced by a k-pointed scheme (M, m), i.e. it is of the form (2.5), then 2.2.11 gives an information about the dimension of M at m. We refer the reader to Proposition 2.3.6 for a result about obstruction spaces which is often used in applications. Notes and Comments 1. For an interesting discussion on the geometric motivations behind the consideration of functors of Artin rings we refer the reader to [76], part C.1. 2. Corollary 2.2.11 is due to Mumford (unpublished, see [212], p. 126). A proof is outlined in [177], p. 153. The proof given here is based on such outline, and has appeared already in [183], Corollary (B2.4). ˆ ϕ : P → R a homomorphism and F : hR → hP be the 3. Let R, P be in A, corresponding morphism between the prorepresentable functors hR , hP : A → (sets) Then F (A) : hR (A) → hP (A) is injective for all A in ob(A) if and only if ϕ is surjective, i.e. if and only if R = P/J for an ideal J ⊂ P . (Proof: left to the reader).

2.3. THE THEOREM OF SCHLESSINGER

2.3

63

The theorem of Schlessinger

In this section we will prove a well known theorem of Schlessinger which gives necessary and sufficient conditions, easy to verify in practice, for the existence of a (semi)universal element for a functor of Artin rings. Before stating the theorem we want to make some introductory remarks, which can ˆ throughout this section. We be useful in what follows. Let’s fix Λ ∈ ob(A) start with a characterization of prorepresentable functors: Proposition 2.3.1 Let F : AΛ → (sets) be a functor of Artin rings satisfying condition H0 . Then F is prorepresentable if and only if it is left exact and has finite dimensional tangent space, i.e. it has properties H` and Hf . Proof. The “only if” implication is obvious (see also §2.2). So let’s assume that F satisfies H` and Hf . Then, by Proposition E.17, the category IF of couples for F is cofiltered and F = lim →

(X,ξ)

hX

Let (Ai , ξi ) ∈ ob(IF ) and consider all the subrings of Ai images of morphisms (Aj , ξj ) → (Ai , ξi ). By the descending chain condition there is (A¯i , ξ¯i ) such that \ A¯i = Im[Aj → Ai ] By construction the couples (A¯i , ξ¯i ) form a full subcategory of IF in which all maps are surjective. Moreover the corresponding category of representable functors hA¯i /Λ is clearly cofinal in (IF )◦ , and therefore F = lim hA¯i /Λ → Therefore replacing IF by this subcategory we can assume that all homomorphisms are surjective, and therefore we have: F =

[

hAi /Λ

Moreover, since Hf holds, F (k[]) = hAi (k[]) when i 0. We can discard all those Ai for which this is not true and we get again a full subcategory. Therefore F = lim hAi /Λ →

64

CHAPTER 2. FORMAL DEFORMATION THEORY

and F (k[]) = hAi /Λ (k[]) for all i. Then for each i we can find a surjection Λ[[X1 , . . . , Xr ]] → Ai with r = dim(F (k[])), and these surjections are compatible, i.e. the diagrams Λ[[X1 , . . . , Xr ]] → Ai & ↓ Aj are commutative. Define B1 = Λ[[X1 , . . . , Xr ]]/(X)2 = Ai /m2i for all i. Then fix ν ≥ 2, and set Ai,ν = Ai /mν+1 . All Ai,ν ’s are quotients of Λ[[X1 , . . . , Xr ]]/(X)ν+1 i and form a projective system: · · · → Aj,ν → Ai,ν → · · · Let i0

Bν = lim Ai,ν = Ai,ν ← Then by construction

· · · Bν+1 → Bν → · · · form a projective system and hBν /Λ F = lim → Then ˆ := lim Bν B ← is in ob(AˆΛ ) and prorepresents F .

qed

Unfortunately this characterization of prorepresentable functors is not very useful in practise because given homomorphisms A0

A00 &. A

in AΛ we have Spec(A0 ×A A00 ) = Spec(A0 )

[ Spec(A)

Spec(A00 )

2.3. THE THEOREM OF SCHLESSINGER

65

and this is not easy to visualize. That’s why left exactness of a functor of Artin rings F is hard to check. On the other hand, if at least one of the above homomorphisms is surjective then Spec(A0 ×A A00 ) is easier to describe. The theorem of Schlessinger reduces the prorepresentability to the verification of the condition of left exactness only in cases when at least one of the above maps is surjective. An analogous condition is given for the existence of a semiuniversal element. The result is the following. Theorem 2.3.2 (Schlessinger [175]) Let F : AΛ → (sets) be a functor of Artin rings satisfying condition H0 ). Let A0 → A and A00 → A be homomorphisms in AΛ and let α : F (A0 ×A A00 ) −→ F (A0 ) ×F (A) F (A00 )

(2.10)

be the natural map. Then (i) F has a semiuniversal formal element if and only if it satisfies the following conditions: ¯ if A00 → A is a small extension then the map (2.10) is surjective. H) H ) If A = k and A00 = k[] then the map (2.10) is bijective. Hf ) dimk (tF ) < ∞ (ii) F has a universal element if and only if it also satisfies the following additional condition: H) the natural map F (A0 ×A A0 ) −→ F (A0 ) ×F (A) F (A0 ) is bijective for every small extension A0 → A in AΛ . Remark 2.3.3 It will be helpful to explain the meaning of the conditions of the theorem before giving the proof. Consider a small extension in AΛ : µ

0 → (t) → A0 −→ A → 0 Assume that F ∼ = hR/Λ is prorepresentable. Then two Λ-homomorphisms 0 f, g : R → A have the same image in HomΛ−alg (R, A) if and only if there exists a Λ-derivation d : R → k (which is uniquely determined) such that g(r) = f (r) + d(r)t

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CHAPTER 2. FORMAL DEFORMATION THEORY

equivalently if and only if g and f differ by an element of DerΛ (R, k) = tR/Λ Therefore the fibres of F (µ) : HomΛ−alg (R, A0 ) → HomΛ−alg (R, A) that are non empty are principal homogeneous spaces under the above action of tR/Λ = tF . Assume now that F is just a functor of Artin rings having properties H0 and H , so that it has tangent space tF . We can define an action of tF on F (A0 ) by means of the composition α−1

F (b)

τ : tF × F (A0 ) −→ F (k[] ×k A0 ) −→ F (A0 ) where α−1 exists by property H ) and F (b) is induced by the morphism k[] ×k A0

b:

−→

A0

(x + y, a0 ) 7−→ a0 + yt The action τ maps the fibres of F (µ) into themselves. Indeed the isomorphism γ : k[] ×k A0 −→ A0 ×A A0 (x + y, a0 ) 7−→ (a0 + yt, a0 ) induces a map α−1

F (γ)

β : tF × F (A0 ) −→ F (k[] ×k A0 ) −→ F (A0 ×A A0 ) −→ F (A0 ) ×F (A) F (A0 ) which coincides with tF × F (A0 ) −→ F (A0 ) ×F (A) F (A0 ) (v, ξ)

7−→

(τ (v, ξ), ξ)

In case F is prorepresentable we have just given another description of the action of tF on the fibres of F (µ) introduced before. In general the map β is neither injective (i.e. in general the action τ is not free on the fibres of F (µ))

2.3. THE THEOREM OF SCHLESSINGER

67

nor surjective (i.e. τ is not transitive on the fibres of F (µ)). This depends on the properties of the map α0 : F (A0 ×A A0 ) −→ F (A0 ) ×F (A) F (A0 ) If F is left exact then α0 is bijective, hence β is bijective, and the action of tF on the fibres of F (µ) is free and transitive, as expected since F is prorepresentable by Prop. 2.3.1. Conversely what this analysis shows is that for τ to be free and transitive on the fibres of F (µ) we only need α0 bijective, i.e. the condition H, weaker ¯ only guarantees the transitivity of such action: the failure from than H` . H prorepresentability is therefore related to the existence of fixed points of this action. In applications this is usually due to the existence of automorphisms of geometric objects associated to an element ξ ∈ F (A) which don’t lift to automorphisms of objects associated to an element ξ 0 ∈ F (µ)−1 (ξ) (see §2.6). Since Theorem 2.3.2 guarantees that F has a semiuniversal element if it ¯ we can summarize this discussion in the following satisfies H0 , H and H, statement. Proposition 2.3.4 Let F be a functor of Artin rings satisfying conditions H0 and H and let 0 → (t) → A0 → A → 0 be a small extension in A. Then there is an action τ of tF on F (A0 ) such that (a) if F has a semiuniversal formal element then tF acts transitively on the non-empty fibres of F (A0 ) → F (A); (b) If F is prorepresentable then tF acts freely and transitively on the nonempty fibres of F (A0 ) → F (A). The following definition introduces a convenient terminology. Definition 2.3.5 Let f : F → G be a morphism of functors of Artin rings having a semiuniversal formal element and obstruction spaces υ(F ) and υ(G) respectively. An obstruction map for f is a linear map o(f ) : υ(F ) → υ(G)

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CHAPTER 2. FORMAL DEFORMATION THEORY

such that for each A in ob(A) and for each ξ ∈ F (A) the diagram: Exk (A, k) . ξυ

& ηυ o(f )

−→

υ(F )

(2.11) υ(G)

is commutative, where we have denoted η = f (A)(ξ). If f has an injective obstruction map we say that F is less obstructed than G. With this terminology we can state a result often used in concrete situations: Proposition 2.3.6 Let f : F → G be a morphism of functors of Artin rings having a semiuniversal formal element and obstruction spaces υ(F ) and υ(G) respectively. Consider the following conditions: (a) df is surjective (b) F is less obstructed than G. If they are both satisfied then f is smooth. If condition (b) is satisfied (but not necessarily (a)) and G is smooth then F is smooth. Proof. Let ϕ : A˜ → A be a small extension and assume (a) and (b) hold. Consider the map ˜ → F (A) ×G(A) G(A) ˜ F (A) (2.12) ˜ Since η˜ 7→ η = f (A)(ξ) we have ηυ (ϕ) = 0. and let (ξ, η˜) ∈ F (A) ×G(A) G(A). By (b) and the commutativity of (2.11) we also have ξυ (ϕ) = 0 and therefore ˜ ∈ G(A). ˜ such that ξ˜ 7→ ξ. Let η˜0 = f (A)( ˜ ξ) ˜ We have there exists ξ˜ ∈ F (A) η˜0

η˜ & . η

¯ there is w ∈ tG Since by Theorem 2.3.2 the functor G satisfies condition H), such that τ (w, η˜0 ) = η˜. From the surjectivity of df it follows that there is ˜ ∈ F (A) ˜ satisfies v ∈ tF such that v 7→ w. Now τ (v, ξ) ˜ 7→ (ξ, η˜) τ (v, ξ)

2.3. THE THEOREM OF SCHLESSINGER

69

because the action τ is functorial (as it easily follows from its definition). This shows that (2.12) is surjective and f is smooth. Now assume that only (b) holds and that G is smooth. We must show that ˜ → F (A) F (A) is surjective or equivalently that ξυ (ϕ) = 0. Let ξ ∈ F (A) and let η := f (A)(ξ) ∈ G(A). Since G is smooth we have ηυ (ϕ) = 0. By (b) and the commutativity of (2.11) we also have ξυ (ϕ) = 0. qed Corollary 2.3.7 Let f : F → G be a morphism of functors of Artin rings having a semiuniversal formal element. Assume that F is smooth, G has an obstruction space υ(G) and df is surjective. Then f and G are smooth. Proof. Since F is smooth it has (0) as an obstruction space and the obvious map (0) → υ(G) is an obstruction map. Therefore f is smooth by Proposition 2.3.6 and we conclude by Proposition 2.2.5(iii). qed Remark 2.3.8 When F and G are prorepresentable the first part of Proposition 2.3.6 is a reformulation of Theorem 2.1.5(iii) and Corollary 2.3.7 is a reformulation of Corollary 2.1.6(i). If in Corollary 2.3.7 we replace the condition “df surjective” by “df bijective” we cannot conclude that f is an isomorphism unless both functors are prorepresentable: if they are then f is an isomorphism by Corollary 2.1.6(ii) because a formally etale homomorphism of k-algebras in Aˆ is an isomorphism. Proof of Theorem 2.3.2 (i) Let’s assume that F has a semiuniversal formal element (R, uˆ). Consider a homomorphism f : A0 → A and a small extension π : A00 → A both in AΛ , and let (ξ 0 , ξ 00 ) ∈ F (A0 ) ×F (A) F (A00 ) with F (f )(ξ 0 ) = F (π)(ξ 00 ) =: ξ ∈ F (A) By the versality of (R, uˆ) the following maps are surjective: HomΛ−alg (R, A0 ) −→ F (A0 )

70

CHAPTER 2. FORMAL DEFORMATION THEORY HomΛ−alg (R, A00 ) −→ HomΛ−alg (R, A) ×F (A) F (A00 )

(2.13)

therefore there are g 0 ∈ HomΛ−alg (R, A0 ),

g 00 ∈ HomΛ−alg (R, A00 )

such that Fˆ (g 0 )(ˆ u) = ξ 0 and g 00 7→ (f g 0 , ξ 00 ) under the map (2.13). This last condition gives: πg 00 = f g 0 ,

Fˆ (g 00 )(ˆ u) = ξ 00

and consequently Fˆ (πg 00 )(ˆ u) = ξ. Using the morphism g 0 ×g 00 : R → A0 ×A A00 we obtain an element ζ := Fˆ (g 0 × g 00 )(ˆ u) ∈ F (A0 ×A A00 ) which, by construction, is mapped to (ξ 0 , ξ 00 ) by (2.10). This proves that ¯ (R, uˆ) satisfies condition H). If A00 = k[] and A = k the map (2.10) reduces to the bijection HomΛ−alg (R, k[]) −→ tF

(2.14)

In this case if ζ1 , ζ2 ∈ F (A0 ×A k[]) are such that α(ζ1 ) = α(ζ2 ) = (ξ 0 , ξ 00 ) choose g 0 ∈ HomΛ−alg (R, A0 ) as before. By the versality the map HomΛ−alg (R, A0 ×k k[]) −→ F (A0 ×k k[]) ×F (k[]) HomΛ−alg (R, k[]) k F (A0 ×k k[]) induced by the projection A0 ×k k[] → k[] is surjective. Hence we obtain two morphisms: g 0 × gi : R −→ A0 ×k k[] such that Fˆ (g 0 × gi )(ˆ u) = ζi i = 1, 2. But then Fˆ (gi )(ˆ u) = ξ 00 , i = 1, 2, hence, by the bijectivity of (2.14), g1 = g2 , i.e. ζ1 = ζ2 . This proves that F satisfies condition H ). Condition Hf ) is satisfied because the differential tR/Λ → tF is linear and is a bijection by definition of semiuniversal formal couple.

2.3. THE THEOREM OF SCHLESSINGER

71

¯ H ) and Hf ). We Conversely, let’s assume that F satisfies conditions H), will find a semiuniversal formal couple (R, uˆ) by constructing a projective system {Rn ; pn+1 : Rn+1 → Rn }n≥0 of Λ-algebras in AΛ and a sequence {un ∈ F (Rn )}n≥0 such that F (pn )(un ) = un−1 , n ≥ 1. We take R0 = k and u0 ∈ F (k) its unique element. Let r = dimk (tF ), {e1 , . . . , er } a basis of tF and denoting Λ[[X]] = Λ[[X1 , . . . , Xr ]] we set R1 = Λ[[X]]/((X)2 + mΛ Λ[[X]]) Since we have R1 = k[] ×k · · · ×k k[]

(r

times)

from H ) we deduce that F (R1 ) = tF × · · · × tF . Let’s take u1 = (e1 , . . . , er ) ∈ F (R1 ). Note that the map induced by u1 : kr ∼ = ((X)/(X)2 )∨ = HomΛ−alg (R1 , k[]) −→ tF is the isomorphism (λ1 , . . . , λr ) 7−→

X

λj ej

j

Let’s proceed by induction on n: assume that couples (R0 , u0 ), (R1 , u1 ), . . . , (Rn−1 , un−1 ) such that Rh = Λ[[X]]/Jh ,

uh ∈ F (Rh ),

uh 7→ uh−1

h = 1, . . . , n−1 have been already constructed. In order to construct (Rn , un ) we consider the family I of all ideals J ⊂ Λ[[X]] having the following properties: a) Jn−1 ⊃ J ⊃ (X)Jn−1 : b) there exists u ∈ F (Λ[[X]]/J) such that the map F (Λ[[X]]/J) −→ F (Rn−1 ) sends u 7→ un−1 .

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CHAPTER 2. FORMAL DEFORMATION THEORY

I 6= ∅ because Jn−1 ∈ I. Moreover I has a minimal element Jn . This will be proved if we show that I is closed with respect to finite intersections. Let I, J ∈ I and K = I ∩ J. It is obvious that K satisfies condition a). We may assume that I + J = Jn−1 (making J larger without changing K if necessary). This implies that the natural homomorphism Λ[[X]]/K −→ Λ[[X]]/I ×Rn−1 Λ[[X]]/J ¯ the map is an isomorphism. By H) α : F (Λ[[X]]/K) −→ F (Λ[[X]]/I) ×F (Rn−1 ) F (Λ[[X]]/J) is surjective, therefore there exists u ∈ F (Λ[[X]]/K) whose image in F (Rn−1 ) is un−1 ; this means that K satisfies condition b) as well, hence it is in I. We take Rn = Λ[[X]]/Jn and un ∈ F (Rn ) an element which is mapped to un−1 . By induction we have constructed a formal couple (R, uˆ). We now show that it is a semiuniversal formal couple for F . As already remarked, uˆ induces an isomorphism of tangent spaces tR/Λ ∼ = tF . Therefore we only have to prove versality, namely that the map uˆπ : HomΛ−alg (R, A0 ) −→ HomΛ−alg (R, A) ×F (A) F (A0 ) is surjective for every small extension π : A0 → A. Let (f, ξ 0 ) ∈ HomΛ−alg (R, A) ×F (A) F (A0 ), i.e. Fˆ (f )(ˆ u) = F (π)(ξ 0 ). We 0 0 0 0 must find f ∈ HomΛ−alg (R, A ) such that uˆπ (f ) = (f, ξ ), i.e. such that 1) πf 0 = f ;

2) Fˆ (f 0 )(ˆ u) = ξ 0

Let’s consider the commutative diagram: HomΛ−alg (R, k[]) × HomΛ−alg (R, A0 ) → tF × F (A0 ) ↓ β1 ↓ β2 HomΛ−alg (R, A0 ) ×Hom(R,A) HomΛ−alg (R, A0 ) → F (A0 ) ×F (A) F (A0 ) (2.15) where the horizontal arrows are induced by uˆ. The map β1 is a bijection because the action of HomΛ−alg (R, k[]) on the fibres of HomΛ−alg (R, A0 ) → HomΛ−alg (R, A) ¯ it follows that β2 is is free and transitive (see Remark 2.3.3). From H) surjective (Remark 2.3.3 again). Therefore if f 0 ∈ HomΛ−alg (R, A0 ) satisfies 1) then, letting η 0 := F (f 0 )(ˆ u) ∈ F (A0 )

2.3. THE THEOREM OF SCHLESSINGER there exist v ∈ HomΛ−alg (R, k[]) = tF , diagram (2.15) we have:

73

f 00 ∈ HomΛ−alg (R, A0 ) such that in

(v, f 0 ) − 7 → (v, η 0 ) ↓ ↓ (f 00 , f 0 ) − 7 → (ξ 0 , η 0 ) This means that uˆπ (f 00 ) = (f, ξ 0 ). It follows that it suffices to find f 0 satisfying condition 1). Let n > 0 be such that f factors as fn−1

R → Rn−1 −→ A Then f 0 exists if and only if there exists ϕ which makes the following diagram commutative: ϕ Rn −→ A0 ↓ pn ↓π Rn−1 −→ A equivalently if and only if the extension fn∗ (A0 , π) :

0 → ker(π) → Rn ×A A0 k ↓ fn0 0 → ker(π) → A0

π0

−→ Rn → 0 ↓ fn π −→ A → 0

is trivial. Suppose not. This means that π 0 induces an isomorphism of tangent spaces (see 1.1.2(ii)), i.e. that there exists an ideal I ⊂ Λ[[X]] such that Rn ×A A0 = Λ[[X]]/I By construction Jn−1 ⊃ I ⊃ (X)Jn−1 ¯ the map Moreover, since by H) F (Rn ×A A0 ) −→ F (Rn ) ×F (A) F (A0 ) is surjective, there exists u ∈ F (Rn ×A A0 ) inducing un ∈ F (Rn ), hence inducing un−1 ∈ F (Rn−1 ). It follows that I satisfies condition a) and b) and, by the minimality of Jn in I, it follows that Jn ⊂ I. But this is a contradiction because from the fact that π is a surjection with non trivial

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CHAPTER 2. FORMAL DEFORMATION THEORY

kernel it follows that I is properly contained in Jn . This proves that fn∗ (A0 , π) is trivial and concludes the proof of the fact that (R, uˆ) is semiuniversal and of part (i) of the theorem. (ii) If F is prorepresentable then it trivially satisfies conditions H), H ) and Hf ), as already remarked. ¯ H), H ) and Hf ). We Conversely, suppose that F satisfies conditions H), have just proved that F has a semiuniversal formal couple (R, uˆ). We will prove that this is a universal formal couple by showing that for every A in AΛ the map uˆ(A) : HomΛ−alg (R, A) −→ F (A) induced by uˆ is bijective. This is clearly true if A = k. We will proceed by induction on dimk (A). Let π : A0 → A be a small extension in AΛ . By the inductive hypothesis HomΛ−alg (R, A) −→ F (A) is bijective and, by the versality, the map uˆπ : HomΛ−alg (R, A0 ) → HomΛ−alg (R, A) ×F (A) F (A0 ) ∼ = F (A0 ) is surjective. The map β2 in diagram (2.15) is bijective by condition H), and this implies that uˆπ is bijective. qed Notes and Comments 1. Theorem 2.3.2 has been published in [175]. It had also appeared in [174]. See also [131]. 2. From Theorem 2.3.2 it follows that if F has a semiuniversal element then it has a tangent space which is of finite dimension, because F satisfies H0 ), H ) and Hf ). This property was not explicitly stated in the definition.

2.4 2.4.1

The local moduli functors Generalities

If X is an algebraic scheme then for every A in ob(A) we let Def X (A) = {deformations of X over A}/isomorphism

2.4. THE LOCAL MODULI FUNCTORS

75

By the functoriality properties already observed in §1.2 this defines a functor of Artin rings Def X : A → (sets) which is called the local moduli functor of X. If X = Spec(B0 ) is affine, we will often write Def B0 instead of Def X . We can define the subfunctor Def 0X : A → (sets) by Def 0X (A) = {locally trivial deformations of X over A}/isomorphism called the locally trivial moduli functor of X. Theorem 2.4.1 (i) For any algebraic scheme X the functors Def X and ¯ H of Schlessinger’s theorem. ThereDef 0X satisfy conditions H0 , H, fore, if Def X (k[]) (resp. Def 0X (k[])) is finite dimensional, then Def X (resp. Def 0X ) has a semiuniversal element. (ii) There is a canonical identification of k-vector spaces Def 0X (k[]) = H 1 (X, TX )

(2.16)

In particular if X is nonsingular then Def X (k[]) = Def 0X (k[]) = H 1 (X, TX ) (iii) If X is an arbitrary algebraic scheme then we have a natural identification Def X (k[]) = Exk (X, OX ) and an exact sequence: τ

`

ρ

0 → H 1 (X, TX ) −→ Def X (k[]) −→ H 0 (X, TX1 ) −→ H 2 (X, TX ) (2.17) In particular Def B0 (k[]) = TB1 0 if X = Spec(B0 ) is affine.

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(iv) If X is a reduced algebraic scheme then there is an isomorphism Def X (k[]) ∼ = Ext1OX (Ω1X , OX ) and the exact sequence (2.17) is isomorphic to the local-to-global exact sequence for Exts: 0 → H 1 (X, TX ) → Ext1OX (Ω1X , OX ) → → H 0 (X, Ext1OX (Ω1X , OX )) → H 2 (X, TX ) (2.18) Proof. Obviously Def X and Def 0X satisfy condition H0 . To verify the other conditions we assume first that X = Spec(B0 ) is affine. ¯ Let Let’s prove that Def B0 satisfies H. A0

A00 &. A

be homomorphisms in A, with A00 → A a small extension. Letting A¯ = A0 ×A A00 we have a commutative diagram with exact rows: 0 → () → A¯ k ↓ 0 → () → A00

→ A0 ↓ → A

→ 0 (2.19) → 0

Consider an element of Def B0 (A0 ) ×Def B0 (A) Def B0 (A00 ) which is represented by a pair of deformations f 0 : A0 → B 0 and f 00 : A00 → B 00 of B0 such that A → B 0 ⊗A0 A and A → B 00 ⊗A00 A are isomorphic deformations. Assume that the isomorphism is given by A-isomorphisms B 0 ⊗A0 A ∼ =B∼ = B 00 ⊗A00 A, where A → B is a deformation. In order to check ¯ it suffices to find a deformation f¯ : A¯ → B ¯ inducing (f 0 , f 00 ). Let H ¯ = B 0 ×B B 00 B endowed with the obvious homomorphism f¯ : ¯ A¯ A0 check that there are an A0 -isomorphism B⊗ ¯ ⊗A¯ A00 ∼ B = B 00 .

¯ It is elementary to A¯ → B. 0 ∼ = B and an A00 -isomorphism

2.4. THE LOCAL MODULI FUNCTORS

77

Therefore we only need to check that f¯ is flat. Tensoring diagram (2.19) ¯ we obtain the following diagram with exact rows: with ⊗A¯ B ¯ () ⊗A¯ B k 0 → B0

¯ B ↓ → B 00 →

→ B0 ↓ → B

→ 0 → 0

where the second row is given by Lemma A.30. This diagram shows that ¯→B ¯ () ⊗A¯ B is injective: the flatness of f¯ follows from Lemma A.30. Let’s prove that Def B0 satisfies H . Assume in the above situation that ˜ be a deformation such that A00 = k[] and A = k, and let f˜ : A¯ → B 0 00 ˜ α(f ) = (f , f ). Then the diagram ˜ B ↓ ˜ ⊗A¯ A0 ∼ B = B0

˜ ⊗A¯ A00 ∼ → B = B 00 ↓ → B0

commutes: the universal property of the fiber product implies that we have ˜ → B ¯ of deformations, hence an isomorphism by a homomorphism γ : B Lemma A.25. This proves that the fibres of α contain only one element, i.e. α is bijective. Therefore Def B0 satisfies condition H . Let’s prove (i) for X arbitrary. Let’s consider a diagram in A: A0

A00 &. A

with A00 → A a small extension and let A¯ = A0 ×A A00 . Consider an element ([X 0 ], [X 00 ]) ∈ Def X (A0 ) ×Def X (A) Def X (A00 ) Therefore we have a diagram of deformations: X0

X 00 -f

↓ Spec(A0 )

0

%f X ↓

-

↓ Spec(A00 ) %

Spec(A)

00

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CHAPTER 2. FORMAL DEFORMATION THEORY

where the morphisms f 0 and f 00 induce isomorphisms of deformations X 0 ×Spec(A0 ) Spec(A) ∼ =X ∼ = X 00 ×Spec(A00 ) Spec(A) ¯ Consider the sheaf of A-algebras OX 0 ×OX OX 00 on X. Then X¯ := (|X|, OX 0 ×OX ¯ OX 00 ) is a scheme over Spec(A) (by the proof of the affine case). Reducing ¯ Therefore X¯ is a to the affine case one shows that X¯ is flat over Spec(A). ¯ deformation of X over Spec(A) inducing the pair ([X 0 ], [X 00 ]). This shows that the map ¯ → Def X (A0 ) ×Def (A) Def X (A00 ) Def X (A) X ¯ for Def X . Moreover if the deformations X 0 and X 00 is surjective, proving H ¯ holds for the functor Def 0X are locally trivial then so is X¯ , and therefore H as well. Now assume that A00 = k[] and A = k. Then the previous diagram becomes X0 X 00 0 00 -f %f X ¯ inducing the pair In this case any X˜ → Spec(A) ([X 0 ], [X 00 ]) ∈ Def X (A0 ) ×Def X (A) Def X (A00 ) is such that the isomorphisms 0 X˜ ×Spec(A) ¯ Spec(A ) ∼ = X 0,

00 X˜ ×Spec(A) ¯ Spec(A ) ∼ = X 00

˜ fits into a induce the identity on X = X˜ ×Spec(A) ¯ Spec(k). Therefore X commutative diagram X˜ % X

-

0

X 00 -f

0

%f

00

X By the universal property of the fibered sum of schemes we then get a morphism of deformations X¯ → X˜ , which is necessarily an isomorphism. This proves property H for Def X . The proof for Def 0X is similar.

2.4. THE LOCAL MODULI FUNCTORS

79

(ii) The identification (2.16) has been proved in 1.2.9. The verification that it is k-linear is elementary and will be left to the reader. (iii) If X ⊂ X ↓ ↓ Spec(k) → Spec(k[]) is a first order deformation of X then X ⊂ X is an extension of X by OX because by the flatness of X over Spec(k[]) we have OX ∼ = OX (Lemma A.30). Conversely, given such an extension X ⊂ X we have an exact sequence j

0 → OX → OX → OX → 0 OX has a natural structure of k[]-algebra by sending, for any open U ⊂ X, 7→ j(1). It follows from Lemma A.30 that X is flat over Spec(k[]). Let’s assume first that X = Spec(B0 ) is affine. Then Exk (X, OX ) = H 0 (X, TB1 0 ) = TB1 0 and the exact sequence (2.17) reduces to the isomorphism Def B0 (k[]) ∼ = TB1 0 which holds by what we have just remarked. Let’s assume that X is general. The map τ corresponds to the inclusion Def 0X (k[]) ⊂ Def X (k[]) in view of (2.16). The map ` associates to a first order deformation ξ of X the section of TX1 defined by the restrictions {ξ|Ui } for some affine open cover {Ui }. It is clear that Im(τ ) = ker(`). We now define ρ. Let h ∈ H 0 (X, TX1 ) be represented, in a suitable affine open cover U = {Ui = Spec(Bi )} of X, by a collection of k-extensions Ei of Bi by Bi . Since h is a global section there exist isomorphisms σij : Ej|Ui ∩Uj ∼ = Ei|Ui ∩Uj . These isomorphisms patch together to give an extension E if and only if h ∈ Im(`) if and only if we can find new isomorphisms σij0 such that 0 0 σij0 σjk = σik

on Ui ∩ Uj ∩ Uk . Such isomorphisms are of the form σij0 = σij θij

(2.20)

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CHAPTER 2. FORMAL DEFORMATION THEORY

where θij is an automorphism of the extension Ej|Ui ∩Uj . The collection of automorphisms (θij ) corresponds, via Lemma 1.2.6, to a 1-cochain (tij ) ∈ C 1 (U, TX ); conversely every 1-cochain (tij ) defines a system of isomorphisms (σij0 ); and the condition (2.20) is satisfied if and only if (tij ) is a 1-cocycle. Therefore we define ρ(h) to be the class of the 2-cocycle (tij + tjk − tik ). With this definition we clearly have ker(ρ) = Im(`). We leave to the reader to verify that the definition of ρ does not depend on the choices made. (iv) Since we have a natural identification Def X (k[]) = Exk (X, OX ) we conclude by Theorem 1.1.10. The exact sequences (2.17) and (2.18) are isomorphic in view of the isomorphism TX1 ∼ = Ext1OX (Ω1X , OX ) given by Corollary 1.1.11, page 18.

qed

Corollary 2.4.2 If X is a projective scheme or an affine scheme with at most isolated singularities then Def X has a semiuniversal formal element. Proof. The condition implies that H 1 (X, TX ) and H 0 (X, TX1 ) are finite dimensional vector spaces. Therefore the conclusion follows from Theorem 2.4.1 and Schlessinger’s theorem. qed The stronger property of being prorepresentable is not satisfied in general by Def X . We will discuss this matter in §2.6. The case X affine will be taken up again and analized in detail in Section 3.1. Definition 2.4.3 If (R, uˆ) is a semiuniversal couple for Def X then the Krull dimension of R (i.e. the maximum of the dimensions of the irreducible components of Spec(R)) is called the number of moduli of X and it is denoted by µ(X). Remark 2.4.4 Let X be a reduced scheme and let ξ : X → Spec(k[]) be a first order deformation of X. Then the conormal sequence of X ⊂ X 0 → OX → Ω1X |X → Ω1X → 0

(2.21)

is exact and defines the element of Ext1OX (Ω1X , OX ) which corresponds to ξ in the identification Def X (k[]) = Ext1OX (Ω1X , OX ) of Theorem 2.4.1(iv) (see also Theorem 1.1.10).

2.4. THE LOCAL MODULI FUNCTORS

81

Given an infinitesimal deformation ξ : X → Spec(A) of X we have a Kodaira-Spencer map κξ : tA → Ext1OX (Ω1X , OX ) which associates to a tangent vector θ ∈ tA the conormal sequence 2.21 of the pullback of ξ to Spec(k[]) defined by θ. Example 2.4.5 Let X be a projective curve. Then H 2 (X, TX ) = (0) and the exact sequence (2.17) shows that if h0 (X, TX1 ) 6= 0 then Def X (k[]) 6= (0) and X is not rigid. In particular if X is reduced and has at least one nonrigid singular point then X is not rigid. By the way, it is not known whether rigid curve singularities exist at all.

2.4.2

Obstruction spaces

The elementary analysis of obstructions to lift infinitesimal deformations carried out in Chapter I can be interpreted as the description of obstruction spaces for the corresponding deformation functors. More precisely we have the following Proposition 2.4.6 Let X be a nonsingular algebraic variety. Then H 2 (X, TX ) is an obstruction space for the functor Def X . If X is an arbitrary algebraic scheme then H 2 (X, TX ) is an obstruction space for the functor Def 0X . Proof. The proposition is just a rephrasing of 1.2.12.

qed

Corollary 2.4.7 Let X be a nonsingular projective algebraic variety. Then h1 (X, TX ) ≥ µ(R) ≥ h1 (X, TX ) − h2 (X, TX ) The first equality holds if and only if X is unobstructed. Proof. It is an immediate consequence of 2.4.6 and of Corollary 2.2.11. qed If X is a singular algebraic scheme the previous results give no information about obstructions of the functor Def X . The following proposition addresses the case of a reduced local complete intersection (l.c.i.) scheme. Proposition 2.4.8 Let X be a reduced l.c.i. algebraic scheme X, and assume char(k) = 0. Then Ext2OX (Ω1X , OX ) is an obstruction space for the functor Def X .

82

CHAPTER 2. FORMAL DEFORMATION THEORY Proof. Let A be in ob(A) and let ξ:

X → X ↓ ↓f Spec(k) → Spec(A)

be a deformation of X over A. We need to define a k-linear map oξ : Exk (A, k) → Ext2OX (Ω1X , OX ) having the properties of an obstruction map according to Definition 2.2.9. Consider an element of Exk (A, k) represented by an extension e:

0 → (t) → A˜ → A → 0

and the conormal sequence of e: 0 → (t) → ΩA/k ⊗A˜ A → ΩA/k → 0 ˜

(2.22)

which is exact by Lemma B.43. Since f is flat, pulling back (2.22) to X we obtain the exact sequence 0 → OX → f ∗ (Ω1Spec(A)|Spec(A) ) → f ∗ (Ω1Spec(A) ) → 0 ˜

(2.23)

Since X is a reduced l.c.i. the morphism f satisfies the hypothesis of Theorem D.8. Therefore the relative cotangent sequence of f : 0 → f ∗ (Ω1Spec(A) ) → Ω1X → Ω1X /Spec(A) → 0

(2.24)

is exact. Composing (2.23) and (2.24) we obtain the 2-term extension 0 → OX

→ f ∗ (Ω1Spec(A)|Spec(A) ) → Ω1X ˜

→ Ω1X /Spec(A)

→0

which defines an element oξ (e) ∈ Ext2OX (Ω1X /Spec(A) , OX ) = Ext2OX (Ω1X , OX ) This defines the map oξ . The linearity of oξ is a consequence of the linearity of the map Exk (A, k) → Ext1A (ΩA/k , k) associating to an extension e the conormal sequence (2.22).

2.4. THE LOCAL MODULI FUNCTORS

83

Assume that there is a lifting of ξ to A˜ i.e. that we have a diagram: X ⊂ X˜ ↓f ↓ f˜ ˜ Spec(A) ⊂ Spec(A) Then we have a commutative and exact diagram as follows: 0 ↓ 0 → OX k 0 → OX

→ f˜∗ (Ω1Spec(A) ˜ )|Spec(A) ↓ 1 → ΩX˜|X ↓ 1 (ΩX˜/Spec(A) ˜ )|X ↓ 0

→ f ∗ (Ω1Spec(A) ) → 0 ↓ → Ω1X →0 ↓ 1 = ΩX /Spec(A) ↓ 0

In this diagram the first row is the pullback of the second row and this implies that oξ (e) = 0. Conversely, assume that oξ (e) = 0. Then we have a commutative and exact diagram as follows: 0 ↓ 0 → OX k 0 → OX

→ f ∗ (Ω1Spec(A)|Spec(A) ) → f ∗ (Ω1Spec(A) ) → 0 ˜ ↓ ↓ → E → Ω1X →0 ↓ Ω1X /Spec(A) ↓ 0

for some coherent sheaf E on X . By the construction of Theorem 1.1.10 one finds a sheaf of A-algebras OX˜ and an extension of sheaves of A-algebras 0 → OX → OX˜ → OX → 0 such that E = Ω1X˜|X . It remains to be shown that OX˜ can be given a structure ˜ of sheaf of flat A-algebras. The shortest way to see this is to use Corollary

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CHAPTER 2. FORMAL DEFORMATION THEORY

3.1.13(ii), to be proved in the next chapter: since X is a l.c.i. it implies ˜ This means that the that there are no obstructions to lift X locally to A. restriction of OX˜ to every affine open subset U ⊂ X defines a lifting of X|U ˜ and therefore it is a sheaf of flat A-algebras. ˜ to A, qed Example 2.4.9 If X is a reduced l.c.i. curve then Ext1OX (Ω1X , OX ) is a torsion sheaf and Ext2OX (Ω1X , OX ) = 0; therefore H i (X, ExtjOX (Ω1X , OX )) = (0) for all i, j with i + j = 2. It follows that Ext2OX (Ω1X , OX ) = 0 and X is unobstructed by Proposition 2.4.8. For a discussion of obstructions in the affine case we refer the reader to Subsection 3.1.2.

2.4.3

Algebraic surfaces

In this subsection we will assume char(k) = 0. We will denote by S a projective nonsingular connected algebraic surface. Let (R, uˆ) be a semiuniversal formal deformation of S, and denote by µ(S) := dim(R) the number of moduli of S. Proposition 2.4.10 10χ(OS ) − 2(K 2 ) + h0 (S, TS ) ≤ µ(S) ≤ h1 (S, TS )

(2.25)

If h2 (S, TS ) = 0 then both inequalities are equalities. Proof. A direct application of the Riemann-Roch formula gives h1 (S, TS ) − h2 (S, TS ) = 10χ(OS ) − 2(K 2 ) + h0 (S, TS ) By applying Corollary 2.4.7 we obtain the conclusion. The first inequality was proved by Enriques (see [54]).

qed

2.4. THE LOCAL MODULI FUNCTORS

85

Examples 2.4.11 (i) If S is a minimal ruled surface and S → C is the ruling over a projective nonsingular curve C, then letting S(x) ∼ = IP 1 be the fibre of any x ∈ C we have h1 (S(x), TS|S(x) ) = 0 as an immediate consequence of the exact sequence 0 → TS(x)

→ TS|S(x) → NS(x)/S k OS(x)

→0

Therefore R1 p∗ TS = 0 by Corollary 4.2.6 and the Leray spectral sequence implies that h2 (S, TS ) = 0. Therefore S is unobstructed and µ(S) = h1 (S, TS ). This computation for the rational ruled surfaces Fm is also done in Example B.44(iii), page 346. (ii) Assume that S is a K3-surface. Then h2 (S, TS ) = h0 (S, Ω1S ) = h1 (S, OS ) = 0 Therefore S is unobstructed. Moreover h0 (S, TS ) = h2 (S, Ω1S ) = h1 (S, ωS ) = 0 and Def S is prorepresentable (Corollary 2.6.4). Formula (2.25 gives in this case µ(S) = h1 (S, TS ) = 20 (iii) Let π : X → S be the blow-up of S at a point s and E = π −1 (s) the exceptional curve. Then we have a standard exact sequence dπ

0 → TX −→ π ∗ TS → OE (−E) → 0 (to see it one can use the exact sequence (3.50), page 204, and note that coker(dπ) = Nπ the normal sheaf of π, which is defined on page 193). We thus see that H 2 (X, TX ) ∼ = H 2 (S, TS ) This implies for example that non-minimal rational or ruled surfaces and blow-ups of K3-surfaces are unobstructed.

86

CHAPTER 2. FORMAL DEFORMATION THEORY (iv) When the Kodaira dimension of S is ≥ 1 then in general h2 (S, TS ) = h0 (S, Ω1 ⊗ ωS ) 6= 0

and in fact such surfaces can be obstructed. Examples are given in Theorem 3.4.26, page 221. If we assume that h0 (S, TS ) = 0 the estimate for µ(S) given by Proposition 3.4.20 becomes 10χ(OS ) − 2(K 2 ) ≤ µ(S) ≤ h1 (S, TS ) = 10χ(OS ) − 2(K 2 ) + h0 (S, Ω1 ⊗ ωS ) and to give an upper bound for µ(S) amounts to giving one for h0 (S, Ω1 ⊗ωS ) in terms of pa , K 2 , q. We refer the reader to [28] for a more detailed discussion of this point. (v) If h0 (S, ωS ) > 0 and q > 0 then certainly h0 (S, Ω1 ⊗ ωS ) > 0 This is because there is a bilinear pairing H 0 (S, Ω1S ) × H 0 (S, ωS ) → H 0 (S, Ω1 ⊗ ωS ) which is non-degenerate on each factor. For example, take S = C1 ×C2 where C1 and C2 are projective nonsingular connected curves of genera g1 ≥ 2 and g2 ≥ 2 respectively. Then h0 (S, ωS ) = g1 g2 , q = g1 + g2 and TS = p∗1 ωC−11 ⊕ p∗2 ωC−12 where pi : S → Ci is the i-th projection, i = 1, 2. Therefore h0 (TS ) = 0,

h1 (TS ) = 3(g1 + g2 ) − 6,

h2 (TS ) = g2 (3g1 − 3) + g1 (3g2 − 3)

On the other hand S is unobstructed. In fact there is a natural morphism of functors f : Def C1 × Def C2 → Def S which is clearly an isomorphism on tangent spaces: in fact (Def C1 × Def C2 )(k[]) := Def C1 (k[]) × Def C2 (k[]) = = H 1 (C1 , ωC−11 ) ⊕ H 1 (C2 , ωC−12 ) = H 1 (S, TS )

2.5. FORMAL VERSUS ALGEBRAIC DEFORMATIONS

87

Therefore, since Def C1 × Def C2 is smooth because both factors are (Proposition 2.2.5(vii)), Def S is smooth by Corollary 2.3.7. Actually f is an isomorphism because both functors are prorepresentable (see Remark 2.3.8, page 69). This example can be generalized to any finite product of nonsingular projective connected curves of genera ≥ 2. If S is an abelian surface then h0 (S, Ω1 ⊗ ωS ) = 2 = h0 (S, TS ) Formula (2.25) gives 2 ≤ µ(S) ≤ h1 (S, TS ) = 4 In fact the second equality holds because abelian surfaces are unobstructed despite the fact that h2 (S, TS ) 6= 0. This is a property common to all abelian varieties (see [153]). (vi) One should keep in mind that µ(S) is defined as the number of moduli of S in a formal sense. This is because the semiuniversal formal deformation (R, uˆ) can be non-algebraizable (see §2.5 for the notion of algebraizability). For example µ(S) = 20 for a K3-surface, but every algebraic family of K3surfaces with injective Kodaira-Spencer map at every point has dimension ≤ 19 (see §3.3). Similarly an abelian surface has µ(S) = 4 but every algebraic family of abelian surfaces with injective Kodaira-Spencer map at every point has dimension ≤ 3 (see Example 3.3.13). In order to give an algebraic meaning to the number of moduli one should take the maximum dimension of a semiuniversal formal deformation of a pair (S, L) where L is an ample invertible sheaf on S. See §3.3 and the appendix by Mumford to Chapter V of [212].

2.5

Formal versus algebraic deformations

We have already mentioned (see Examples 1.2.5 and 1.2.11) that infinitesimal deformations do not explain faithfully some of the phenomena which can occur when one considers deformations parametrized by algebraic schemes or by the spectrum of an arbitrary noetherian, or even e.f.t., local ring. In this section we will make such statements precise. We start with a few definitions and some terminology. Let (S, s) be a pointed scheme. An etale neighborhood of s in S is an etale morphism of pointed schemes f : (T, t) → (S, s) such that the following

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CHAPTER 2. FORMAL DEFORMATION THEORY

diagram commutes: T t% ↓f s

Spec(k(s)) −→ S The definition implies that k(s) ∼ = k(t), i.e. f induces a trivial extension of ˆS,s ∼ ˆT,t by C.51. Affine neighborthe residue fields at s and t; therefore O =O hoods of s are particular etale neighborhoods. Given two etale neighborhoods (T, t) and (U, u) of s ∈ S a morphism (T, t) → (U, u) is given by a commutative diagram of pointed schemes: →

(T, t)

(U, u)

&

. (S, s)

Lemma 2.5.1 Let f : X → Y be an etale morphism and g : Y → X a section of f . Then g is etale. Proof. Use C.46(v).

qed

Proposition 2.5.2 Let S be a scheme. The etale neighborhoods of a given s ∈ S form a filtered system of pointed schemes. Proof. Given two etale neighborhoods (S 0 , s0 ) and (S 00 , s00 ), they are dominated by a third, namely: S 0 ×S S 00 ↓ S0

→ S 00 ↓ → S

Now let f1 , f2 : (S 00 , s00 ) → (S 0 , s0 ) be two morphisms between etale neighborhoods. Then there exists a third etale neighborhood (S 000 , s000 ) and a morphism (S 000 , s000 ) → (S 00 , s00 ) which equalizes them. In fact consider the diagram: S 00 ×S S 0 ↓ pr00 S 00

pr0

−→ S 0 ↓ → S

2.5. FORMAL VERSUS ALGEBRAIC DEFORMATIONS

89

We can shrink S 0 and S 00 so that S 0 is affine and S 00 is connected. Then the graphs Γ1 and Γ2 of f1 and f2 are closed, because S 0 is affine, and open, because images of sections of the etale morphism pr00 , which are etale. Therefore they are connected components of S 00 ×S S 0 . But (s00 , s0 ) ∈ Γ1 ∩ Γ2 and therefore Γ1 = Γ2 . It follows that f1 = f2 on S 00 = Γ1 = Γ2 . qed Definition 2.5.3 Given a scheme S and a point s ∈ S we define the local ring of S in s in the etale topology to be ˜S,s := lim O

→ (S 0 ,s0 )

OS 0 ,s0

where the limit is taken for (S 0 , s0 ) varying through all the etale neighborhoods ˜S,s is also called the henselization of OS,s . (Note that O ˜S,s of s. The ring O is a local ring, because it is a limit of a filtering system of local rings and local homomorphisms). A local ring A is called henselian if for the closed point s of S = Spec(A) one has ˜S,s = OS,s = A A˜ := O The henselization of an e.f.t. local k-algebra is called an algebraic local ring. Therefore the local ring in the etale topology of a point of an algebraic scheme is an algebraic local ring. For a given scheme S and point s ∈ S there is a canonical homomorphism ˜S,s which is flat and induces an isomorphism of the completions OS,s → O ˆS,s ∼ ˜d O =O S,s because every OS,s → OS 0 ,s0 does. Moreover ˜S,s ⊂ O ˆS,s OS,s ⊂ O ˜S,s is faithfully flat and OS,s is separated for the m-adic because OS,s → O ˜S,s if OS,s = O ˆS,s , i.e. a local topology. In particular we see that OS,s = O ˆ k-algebra in A (i.e. complete noetherian with residue field k) is henselian. Theorem 2.5.4 (Nagata) If A is a noetherian local ring then A˜ is noetherian.

90

CHAPTER 2. FORMAL DEFORMATION THEORY b˜ Proof. We have A ⊂ A˜ ⊂ Aˆ and Aˆ = A. Moreover

A˜ = lim A0 →

with A0 local algebras etale over A and inducing trivial residue field extension. To prove that A˜ is noetherian it suffices to prove that every ascending chain of finitely generated ideals of A˜ a1 ⊆ a2 ⊆ · · · ⊆ an ⊆ ˆ stabilizes because Aˆ is noetherian. Therefore stabilizes. The chain {an A} it suffices to prove that if a, b ⊂ A˜ are finitely generated ideals such that aAˆ = bAˆ then a = b. Since a and b are finitely generated one can find ˜ A0 ⊃ A as above and finitely generated ideals a0 , b0 ⊂ A0 such that a = a0 A, 0 ˜ 0 ˆ But since A0 is noetherian it follows that b = b A. It follows that a0 Aˆ = b A. 0 0 a = b and therefore a = b. qed The following proposition gives a geometrical characterization of the henselization. Proposition 2.5.5 Let A be a local ring, S = Spec(A), s ∈ S the closed point. A is henselian if and only if every morphism f : Z → S such that there is a point z ∈ Z with f (z) = s, k(s) = k(z) and f etale at z, admits a section. Proof. Assume the condition satisfied. If A → A0 is an etale homomorphism inducing an isomorphism of the residue fields then the induced morphism f : Spec(A0 ) → S admits a section, which defines an isomorphism A0 ∼ = A; therefore A is henselian. Conversely assume A henselian and let f : Z → S be a morphism satisfying the stated conditions. Then f induces an isomorphism A ∼ = OZ,z because A is henselian. The section is the composition S = Spec(A) ∼ = Spec(OZ,z ) ⊂ Z qed ∗

∗

∗

∗

∗ ∗

We will need the notion of “formal deformation of an algebraic scheme”, already introduced in §2.2 for a general functor of Artin rings. Let X be an ˆ Then a formal deformation of X algebraic scheme and let A¯ be in ob(A).

2.5. FORMAL VERSUS ALGEBRAIC DEFORMATIONS

91

d (A) ¯ for Def X : by definition it is a over A¯ is just a formal element ηˆ ∈ Def X sequence {ηn } of infinitesimal deformations of X fn

X −→ Xn ηn : ↓ ↓ πn Spec(k) → Spec(A¯n ) ¯ n+1 where A¯n = A/m ¯ , such that for all n ≥ 1 ηn induces ηn−1 by pullback A under the natural inclusion Spec(A¯n−1 ) → Spec(A¯n ), i.e. we have:

ηn−1

fn−1 X −→ Xn ⊗A¯n A¯n−1 = Xn−1 : ↓ ↓ πn−1 Spec(k) → Spec(A¯n−1 )

¯ {ηn }) or by (A, ¯ ηˆ). It can We will denote such a formal deformation by (A, be also viewed as the morphism of formal schemes ¯ π ¯:X X → Specf(A) where X X = (X, lim OXn ), ←

π ¯ = lim πn ←

(see [89] and [17] for the definition and main properties of formal schemes). Note that π ¯ is a flat morphism of formal schemes, namely for every x ∈ X the local ring OXX ,x is flat over A¯ = OSpecf(A),¯ ¯ π (x) . This is an almost immediate consequence of the general version of the local criterion of flatness ([3], Expos´e IV, Corollaire 5.8). ¯ {ηn }) will be called trivial (resp. locally trivThe formal deformation (A, ial) if each ηn is trivial (resp. locally trivial). ¯ {ηn }) is not to be confused with a deformation A formal deformation (A, ¯ ¯ is not in general of X over Spec(A), just as a formal scheme over Specf(A) ¯ (see Definition 2.5.10 below a formal completion of a scheme over Spec(A) and the discussion following it). Let X be a projective scheme and consider a flat family of deformations of X parametrized by an affine scheme S = Spec(B), with B in (k-algebras) f

X −→ X η: ↓ ↓π s Spec(k) −→ S

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namely a cartesian diagram with π projective and flat. The deformation η is called algebraic if B is a k-algebra of finite type. If B is in ob(A∗ ) then η is called a local deformation of X. If B is in ob(A) we obtain an infinitesimal deformation of X; it is simultaneously local and algebraic. We will identify the deformation η with the couple (S, η) or (B, η) and we will also denote it by (S, s, η) or (B, s, η). Given such a deformation (S, s, η) let ηn be the infinitesimal deformation induced by pulling back η under the natural closed embedding Spec(OS,s /mn+1 ) → S ˆS,s /m ˆS,s , {ηn }) is We have OS,s /mn+1 = O ˆ n+1 and therefore it follows that (O a formal deformation of X. It will be called the formal deformation defined by (or associated to) η. (S, s, η) is called formally trivial (resp. formally locally trivial) if the formal deformation defined by η is trivial (resp. locally trivial). Lemma 2.5.6 Let (S, s, η) be a deformation of an algebraic scheme X, f : ˜ s˜) → (S, s) an etale neighborhood of s in S and (S, ˜ s˜, η˜) the deformation (S, of X obtained by pulling back η by f . Then the formal deformations of X defined by η and by η˜ are isomorphic. Proof. We have a cartesian diagram of formal schemes: ˜ X X → X X ¯˜ ↓π ↓π ¯ ˆ ˆ Specf(OS,˜ ˜ s ) → Specf(OS,s ) ¯˜ and π where π ¯ are the formal deformations defined by η˜ and by η respectively. ˆS,s ∼ ˆ ˜ and the conclusion Since f is etale it induces an isomorphism O =O S,˜ s follows. qed Definition 2.5.7 A deformation (S, s, η) of X is called formally universal (resp. formally semiuniversal, formally versal) if the formal deformation ˆS,s , {ηn }n≥0 ) associated to η is universal (resp. semiuniversal, versal). An (O algebraic formally versal deformation of X is also called with general moduli. A flat family π : X → S is called formally universal (resp. formally semiuniversal, formally versal, with general moduli) at a k-rational point

2.5. FORMAL VERSUS ALGEBRAIC DEFORMATIONS

93

s ∈ S if X (s) ⊂ X η: ↓ ↓π s Spec(k) −→ S is a formally universal, (resp. formally semiuniversal, formally versal, with general moduli) deformation of X (s). The expression “general moduli” goes back to the classical geometers. Informally, it means that the family parametrizes all possible “sufficiently small” deformations of X (s); when the family π parametrizes varieties for which there is a moduli scheme, or just a moduli stack (whatever this means for the reader since we have not introduced these notions), π with general moduli means that the functorial morphism from S to the moduli scheme or stack is open at s. An expression like “consider a variety X with general moduli” is used to mean: “choose X as a general fibre in a family with general moduli”. The early literature on deformation theory of complex analytic manifolds (in the approach of Kodaira and Spencer) considered only families parametrized by complex analytic manifolds. In that context the expression “effectively parametrized” was used to mean “with injective KodairaSpencer map”, and the word “complete” was used to mean the analogous of the notion of formal versality, in the category of germs of complex analytic manifolds. Proposition 2.5.8 Let (S, s, η) be a deformation of X. Then: (i) If η is formally versal (resp. formally semiuniversal or formally universal) then the Kodaira-Spencer map κπ,s : Ts S → Def X (k[]) is surjective (resp. an isomorphism). (ii) If S is nonsingular at s and the Kodaira-Spencer map κπ,s is surjective (resp. an isomorphism) then η is formally versal (resp. formally semiuniversal) and X is unobstructed, i.e. the functor Def X is smooth. Proof. (i) is obvious in view of the definitions of versality and semiuniversality of a formal couple applied to hOˆS,s → Def X .

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(ii) follows from Proposition 2.2.5(iii) applied to f : hOˆS,s → Def X .

qed

The proposition applies in particular to an algebraic deformation, giving a criterion for it to have general moduli. A classical result (see [120]) states the completeness of a complex analytic family of compact complex manifolds if the map κπ,s is surjective. Part (ii) of Proposition 2.5.8 is the algebraic version of this result. It turns out to be very useful because it reduces the verification of formal versality and of unobstructedness to the computation of the Kodaira-Spencer map. ¯ {ηn }) of X is called algebraizDefinition 2.5.9 A formal deformation (A, able if there exists an algebraic deformation (S, s, ξ) of X and an isomorˆS,s ∼ ¯ {ηn }) is isomorphic to phism O = A¯ sending ηn to ξn for all n (i.e. (A, the formal deformation defined by ξ). The deformation (S, s, ξ) is called an ¯ {ηn }). algebraization of (A, It goes without saying that an algebraization of a formal versal (resp. semiuniversal, universal) deformation is formally versal (resp. formally semiuniversal, formally universal). The existence of algebraizations is a highly non trivial problem. It can be considered as the counterpart of the convergence step in the construction of local families of deformations in the Kodaira-Spencer theory of deformations of compact complex manifolds. But the algebraic case presents some characteristic features which make the two theories radically different in methods and in results. In particular a projective algebraic variety X defined over the field of complex numbers need not have an algebraic formally versal deformation even in the case when Def X is prorepresentable and unobstructed (see the following Example 2.5.12); but, according to the KNSK theory (see the Introduction), such a variety has a complete deformation in the complex analytic sense. ∗

∗

∗

∗

∗ ∗

For the rest of this section we will need to assume some familiarity with formal schemes: for some definitions and results not contained in [89] we will refer to [1]. The following definition introduces an important notion weaker than algebraizability. ˆ A Definition 2.5.10 Let X be an algebraic scheme and let A¯ be in ob(A). ¯ formal deformation (A, ηˆ) of X is called effective if there exists a deformation η¯ :

X → ↓ Spec(k) →

X ↓π ¯ Spec(A)

2.5. FORMAL VERSUS ALGEBRAIC DEFORMATIONS

95

of X over A¯ such that ηˆ is the formal deformation associated to η¯. Equiva¯ be the morphism of formal schemes assolently, letting π ¯ :X X → Specf(A) ¯ ηˆ), its effectivity means that there is a ciated to the formal deformation (A, deformation η¯ such that X X is the formal completion of X along X and π ¯ is the morphism of formal schemes induced by π; in symbols: ¯ ×Spec(A) X X ∼ ¯ X = Specf(A) ¯ where It is obvious that the trivial formal deformation X X → Specf(A), ¯ = (X, lim OX ⊗k A/m ¯ n) X X = X × Specf(A) ←

is effective: it is the formal completion of the trivial deformation ¯ → Spec(A) ¯ X × Spec(A) ˆ the formal projective space over A¯ In particular, for any A¯ in ob(A) PAr¯ := (IP r , lim OIPAr¯ ) ← n

¯ along the closed is effective, being the formal completion of IP r × Spec(A) r r fibre IP = IP ×Spec(k). A consequence of Grothendieck’s theorem of formal functions is that a formal deformation of a proper algebraic scheme can be effective in at most one way; more precisely we have: Theorem 2.5.11 Let X be a proper algebraic scheme. If η¯ :

X → ↓ Spec(k) →

X ↓π ¯ Spec(A)

ξ¯ :

X → ↓ Spec(k) →

Y ↓q ¯ Spec(A)

ˆ such that the associated forare two deformations of X with A¯ in ob(A) ˆ are isomorphic, then η¯ and ξ¯ are isomorphic ¯ ηˆ), (A, ¯ ξ) mal deformations (A, deformations. ¯ and Sˆ = Specf(A). ¯ For a given proper S-scheme Proof. Let S = Spec(A) ˆ f : Z → S we denote by Z the formal completion of Z along Z(s) = f −1 (s), where s ∈ S is the closed point, and by Fˆ the OZˆ -sheaf obtained as the formal completion of a quasi-coherent OZ -sheaf F. It suffices to show that the correspondence ϕ 7→ ϕˆ defines a 1-1 correspondence ˆ HomS (X , Y) → HomSˆ (Xˆ , Y)

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ˆ implies X ∼ ˆ i.e. (A, ¯ ηˆ) ∼ ¯ ξ), because it will follow that Xˆ ∼ = Y, = (A, = Y. The proof goes as follows. For any proper S-scheme f : Z → S and coherent sheaves F and G on Z the theorem of formal functions ([89], p. 277) implies that there is an isomorphism ˆ G) ˆ Hom(F, G)∧ ∼ HomOZn (Fn , Gn ) = HomZˆ (F, = lim ← ¯ Moreover, since Hom(F, G) is a finitely generated A-module, it coincides with its mA¯ -adic completion ([139], Theorem 8.7, p. 60); thus we have an isomorphism ˆ G) ˆ Hom(F, G) ∼ = HomZˆ (F, ˆ It is It maps a homomorphism u : F → G to its completion uˆ : Fˆ → G. easy to see that uˆ is injective (resp. surjective) if and only if u is injective (resp. surjective) ([1], ch. III, Cor. 5.1.3). In particular we have a one to one correspondence between sheaves of ideals of OZ and of OZˆ , equivalently ˆ between closed subschemes of Z and closed formal subschemes of Z. Now take Z = X ×S Y and view it as an S-scheme by any of the projections. Then we have a natural identification Zˆ = Xˆ ×Sˆ Yˆ ([1], ch I, Prop. 10.9.7). We deduce a one to one correspondence between closed subschemes ˆ in particular between graphs of S-morphisms of X ×S Y and of Xˆ ×Sˆ Y, ˆ X → Y and graphs of morphisms Xˆ → Yˆ of S-formal schemes. qed In general not every formal deformation of a given algebraic scheme need to be effective, just as deformations of algebraic varieties defined over the field of complex numbers need not be algebraic. Here is a typical example. Example 2.5.12 (K3-surfaces) For this example we will need a result to be proved in Chapter III. Let X be a projective K3-surface, and assume for ¯ ηˆ) be the formal simplicity that Pic(X) = ZZ[H] for some divisor H. Let (A, semiuniversal deformation of X and X X the corresponding formal scheme over ˆ ¯ S = Specf(A). By Example 2.4.11(ii) we know that A¯ ∼ = k[[X1 , . . . , X20 ]] where X1 , . . . , X20 are indeterminates. ¯ ηˆ) is not effective. Claim: (A, ¯ Otherwise there is a proper smooth morphism f : X → S = Spec(A) −1 such that Xˆ ∼ X , where Xˆ is the formal completion of X along X = f (s), =X s ∈ S the closed point. Since X is of finite type over the integral scheme

2.5. FORMAL VERSUS ALGEBRAIC DEFORMATIONS

97

S, it has a non-trivial line bundle L which we may assume to correspond to a Cartier divisor whose support does not contain X and has a non-empty intersection with X; therefore L has a non-trivial restriction to the closed fibre X, say L ⊗ OX ∼ = nH, n 6= 0. By [1], Ch. III, Cor. 5.1.6, L corresponds ˆ to a line bundle L on X X which extends L ⊗ OX ∼ = nH. But nH cannot be extended to the all of X X because it extends to a 19-dimensional space of first order deformations of X (see page 179). Therefore we have a contradiction and the claim is proved. The following is a basic result of effectivity of formal deformations. Theorem 2.5.13 (Grothendieck[75]) Let X be a projective scheme. Then: ¯ A¯ in ob(A), ˆ be a formal deformation of X. (i) Let π ¯ : X X → Specf(A), Assume that there is a closed embedding of formal schemes j : X X ⊂ PAr¯ ¯ is the projection. Then π such that π ¯ = pj where p : PAr¯ → Specf(A) ¯ is effective. (ii) Assume that h2 (X, OX ) = 0. Then every formal deformation of X is effective. Proof. (i) The structure sheaf OXX and the ideal sheaf IXX are coherent sheaves of OPAr¯ -modules. Repeating verbatim the proof of the classical result of Serre (see [89], Theorem II.5.15 pag. 121) one shows that there is an exact sequence N M i=1

g¯

OPAr¯ (−ni ) −→ OPAr¯ → OXX → 0

for some positive integers n1 , . . . , nN . We have g¯ ∈

N M i=1

H 0 (PAr¯ , OPAr¯ (ni ))

and, by the theorem on formal functions ([89], ch. III, Theorem 11.1) we have H 0 (PAr¯ , OPAr¯ (ni )) = H 0 (IPAr¯ , OIPAr¯ (ni ))∧ ¯ Since H 0 (IPAr¯ , OIPAr¯ (ni )) is an A-module of finite type we have H 0 (IPAr¯ , OIPAr¯ (ni ))∧ = H 0 (IPAr¯ , OIPAr¯ (ni ))

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([139], Theorem 8.7, p. 60) and therefore H 0 (PAr¯ , OPAr¯ (ni )) = H 0 (IPAr¯ , OIPAr¯ (ni )) This implies that the homomorphism g¯ is induced by a homomorphism g:

N M i=1

OIPAr¯ (−ni ) → OIPAr¯

whose cokernel is the structure sheaf of a closed subscheme X ⊂ IPAr¯ having ¯ follows X X as formal completion along X (o). The flatness of X over Spec(A) from Theorem A.26. ¯ {ηn }) be a formal deformation of X, where: (ii) Let (A, fn

X −→ Xn ηn : ↓ ↓ πn Spec(k) → Spec(A¯n ) For each n ≥ 1 we have an exact sequence: 0→

mnA¯ exp ρ ∗ ∗ n+1 ⊗k OX −→ OXn −→ OXn−1 → 0 mA¯

where ρ is the natural restriction, exp(σ) = 1+σ and where we have identified mnA¯ mnA¯ ⊗ O = ⊗A¯n OXn k X mn+1 mn+1 ¯ ¯ A A using the flatness of πn . From the hypothesis h2 (X, OX ) = 0 we deduce that the homomorphism of Picard groups Pic(Xn ) → Pic(Xn−1 ) is surjective. This implies that if we fix a very ample line bundle L on X such that H 1 (X, L) = 0 then for each n ≥ 1 we can find a line bundle Ln on Xn such that Ln|Xn−1 = Ln−1 . By Note 3 of §4.2 Ln is very ample and defines an embedding Xn ⊂ IPAN ¯n where N + 1 = h0 (X, L). This means that the formal scheme X X is a formal closed subscheme of PAN¯ . Now we conclude by part (i). qed After these preliminaries we can state the following special case of Artin’s algebraization theorem. ¯ ηˆ) Theorem 2.5.14 (Artin[12]) Let X be a projective scheme and let (A, ¯ be an effective formal versal deformation of X. Then (A, ηˆ) is algebraizable.

2.5. FORMAL VERSUS ALGEBRAIC DEFORMATIONS

99

¯ ηˆ) of a K3-surface is not effecThe semiuniversal formal deformation (A, tive (Example 2.5.12) and in fact it is not algebraizable: this is well known if k = C because K3-surfaces have arbitrarily small deformations which are non-algebraic complex manifolds. ∗

∗

∗

∗

∗ ∗

To give a complete proof of Theorem 2.5.14 is beyond our goals. Nevertheless we want to introduce some relevant definition and preliminary result which are needed for its general statement (see below) and proof. Consider a covariant functor F : (k-algebras) → (sets) ˆ An important question Suppose given R in ob(A∗ ) and an element u¯ ∈ F (R). is whether it is possible to approximate u¯ in some way by an element u ∈ F (R). Indeed every algebraization problem can be reduced to such a question for an appropriate functor F . In this context the following is a natural definition of approximation: ˆ and u ∈ F (R). If Definition 2.5.15 Suppose given R in ob(A∗ ), u¯ ∈ F (R) c > 0 is an integer we say that u¯ and u are conguent modulo mc , in symbols u¯ ≡ u (modulo mc ) ˆ cˆ ) = F (R/mcR ). if u¯ and u induce the same element in F (R/m R In order to make the problem tractable it turns out to be natural to impose the following finiteness condition on the functor F . Definition 2.5.16 A functor F : (k-algebras) → (sets) is said to be locally of finite presentation if for every filtering inductive system of k-algebras {Bi } the canonical map lim F (Bi ) → F (lim Bi ) → → is bijective. A functor locally of finite presentation is sometimes also called limit preserving.

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This is a natural finiteness property, first introduced in [1], ch. IV, Prop. 8.14.2, which is usually satisfied by the functors arising in algebraic geometry. An illustration of the meaning of Definition 2.5.16 is given by the following proposition and by its corollary. Proposition 2.5.17 Let R be a noetherian ring and B an R-algebra. Then B is an R-algebra of finite type if and only if for every filtering inductive system of R-algebras {Ci } the canonical map lim [HomR−alg (B, Ci )] → HomR−alg (B, lim Ci ) →

→

is bijective. Proof. Assume that the condition of the statement is satisfied, and write B = lim Bi → where the Bi ⊂ B are R-subalgebras of finite type. Then the hypothesis implies that the identity B → B factors as B → Bi → B for some i, and this implies B = Bi . Therefore B is of finite type. Conversely assume that B is of finite type, and let {t1 , . . . , tn } be a system of generators of B as an R-algebra. Let {Ci } be a filtering inductive system of R-algebras, C = lim Ci and let lim [HomR−alg (B, Ci )] → HomR−alg (B, C) →

(2.26)

be the natural map. Consider two elements of lim [HomR−alg (B, Ci )], given by two compatible systems of homomorphisms (θi ) and (θi0 ), defining the same homomorphism θ : B → C. Then, letting ϕi : Ci → C and ϕji : Ci → Cj be the canonical homomorphisms of the inductive system, for each s = 1, . . . , n there is an index is such that θ(ts ) = ϕis (θis (ts )) = ϕis (θi0s (ts )) We may assume that all the is are equal, say to i0 . In the same way we can find an index j0 ≥ i0 such that ϕj0 i0 (θi0 (ts )) = ϕj0 i0 (θi00 (ts )) for all s = 1, . . . , n; this means that θj0 (ts ) = θj0 0 (ts ), i.e. θj0 = θj0 0 . Therefore the map (2.26) is injective.

2.5. FORMAL VERSUS ALGEBRAIC DEFORMATIONS

101

Let now θ : B → C be a homomorphism, and let cs = θ(ts ), s = 1, . . . , n. Write B = R[T1 , . . . , Tn ]/J, and let {Pk }k=1,...,m be a system of generators of the ideal J. Then we have Pk (c1 , . . . , cn ) = θ(Pk (t1 , . . . , tn )) = 0,

k = 1, . . . , m

Let i0 be an index such that there exist elements x1 , . . . , xn ∈ Ci0 such that ϕi0 (xs ) = cs , i = 1, . . . , n. One has ϕi0 (Pk (x1 , . . . , xn )) = Pk (c1 , . . . , cn ) = 0

k = 1, . . . , m

Then there is an index j0 ≥ i0 such that 0 = ϕj0 i0 ((Pk (x1 , . . . , xn )) = Pk (ϕj0 i0 (x1 ), . . . , ϕj0 i0 (xn ))

k = 1, . . . , m

One deduces the existence of a k-algebra homomorphism θj0 : B → Cj0 such that θj0 (ts ) = ϕj0 i0 (xs ), s = 1, . . . , n. This implies the existence of a homomorphism θj = ϕjj0 θj0 : B → Cj for every j ≥ j0 . It follows that θ is the inductive limit of the system (θj ), and therefore the map (2.26) is also surjective. qed The following is an immediate consequence. Corollary 2.5.18 Let B be a k-algebra and F = Homk−alg (B, −) : (k-algebras) → (sets) Then F is locally of finite presentation if and only if B is a k-algebra of finite type. Therefore we see that the condition of being locally of finite presentation for a representable functor coincides with the condition of being represented by an algebra of finite type. For the statement of the algebraization theorem we need some terminology which generalizes notions we already introduced. Consider a covariant functor F : (k-algebras) → (sets) Let u0 ∈ F (k). A couple (A, u) where A is in ob(A) and u ∈ F (A) is called an infinitesimal deformation of u0 if u 7→ u0 under the map F (A) → F (A/mA ) = F (k)

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. We will denote by Fu0 : A → (sets) the functor of infinitesimal deformations of u0 , i.e.: Fu0 (A) = {u ∈ F (A) : u 7→ u0 under F (A) → F (k)} This is a functor of Artin rings and, according to the definitions given in §2.2, we have the notions of formal deformation of u0 , and of universal (resp. semiuniversal, resp. versal) formal deformation of u0 . Definition 2.5.19 A local deformation of u0 ∈ F (k) is a couple (A, u) where u ∈ F (A), A is in A∗ and u 7→ u0 under the map F (A) → F (A/mA ) = F (k) An algebraic deformation of u0 is a triple (B, x, u) where B is a k-algebra of finite type, x ∈ Spec(B) is a k-rational point and u ∈ F (B) is such that u 7→ u0 under the map F (B) → F (k) induced by the composition B → OSpec(B),x → k(x) = k To every local deformation (A, u) of u0 one associates the formal deˆ {un }), where Aˆ is the mA -adic completion of A, and un ∈ formation (A, n+1 ˆ n+1 ) is the image of u under the map F (A) → F (A/mn+1 F (A/mA ) = F (A/m A ). ˆ A ˆ We call (A, {un }) the formal deformation of u0 associated to (or defined by ) (A, u). Similarly, the formal deformation of u0 defined by an algebraic deˆSpec(B),x , {un }) where formation (B, x, u) of u0 is the formal deformation (O u 7→ un under the map ˆSpec(B),x /mn+1 ) F (B) → F (O induced by the natural homomorphism ˆSpec(B),x /mn+1 B → OSpec(B),x /mn+1 = O A local deformation is called formally universal (resp. formally semiuniversal, resp. formally versal) if the associated formal deformation has the corresponding property. A similar definition is given for algebraic deformations.

2.5. FORMAL VERSUS ALGEBRAIC DEFORMATIONS

103

¯ {un }) of u0 ∈ F (k) is called Definition 2.5.20 A formal deformation (A, algebraizable if it is associated to some algebraic deformation (B, x, u). In ¯ {un }). this case the deformation (B, x, u) is called an algebraization of (A, ¯ {un }) of u0 is called effective if there exists A formal deformation (A, ¯ ¯ u¯) an effective u¯ ∈ F (A) which induces {un }. In this case we will call (A, formal deformation as well. It is clear from the definition that an algebraic deformation (B, x, u) of ¯ u¯) if and only u0 is an algebraization of the effective formal deformation (A, ˆX,x and u ≡ u¯ mod mn for all n ≥ 0. if A¯ = O ¯ {un }) of u0 is associated to a Note that if the formal deformation (A, local deformation (A, u) then it is effective: infact it follows that {un } is also ˆ which is the image of u under the associated to the deformation u¯ ∈ F (A) ˆ Similarly, an algebraizable deformation is effective. map F (A) → F (A). The algebrization theorem states that the converse is true for versal deformations. Namely: Theorem 2.5.21 (algebraization theorem [12]) Let F : (k-algebras) → (sets) be a functor locally of finite presentation and let u0 ∈ F (k). Then every effective versal formal deformation of u0 is algebraizable. Theorem 2.5.21 is a generalization of Theorem 2.5.14 because the deformation functor Def X of a projective scheme X can be extended to a functor locally of finite presentation defined on the category (k-algebras). The main technical ingredient in the proof of the algebraization theorem is the following Theorem 2.5.22 (approximation theorem [11]) Let F : (k-algebras) → (sets) be a functor locally of finite presentation, A an e.f.t. local k-algebra and ˆ Then for every positive integer c there is an element u ∈ F (A) ˜ u¯ ∈ F (A). c such that u¯ ≡ u modulo m .

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Outline of the proof of Theorem 2.5.21 in a special case, when it follows almost directly from the Approximation Theorem. ¯ u¯) be an effective formal versal deformation of u0 . We will assume Let (A, ˆ where A is an e.f.t. local k-algebra in A∗ . Suppose that we can that A¯ = A, ˜ whose associated formal deformation is {¯ find u˜ ∈ F (A) un }. Then, since A˜ = lim OS,s →

where (S, s) runs through all the etale neighborhoods of (Spec(A), {mA }), and since F is locally of finite presentation there is an etale neighborhood (Spec(B), x) of (Spec(A), {mA }) and u ∈ F (B) such that u 7→ u˜ under ˜ Again because F is locally of finite presentation we may F (B) → F (A). assume that Spec(B) is an affine algebraic scheme: therefore (B, x, u) is an ¯ u¯). Therefore we only need to find u˜ as above. By algebraization of (A, ˜ such that u¯ ≡ u˜ mod the Approximation Theorem there exists u˜ ∈ F (A) 2 ˜ 2 sends u¯ 7→ u˜1 = u1 . m . Therefore the homomorphism ψ1 : A¯ → A/m By versality there is a compatible sequence of homomorphisms ψn : A¯ → ˜ n+1 lifting ψ1 and such that F (ψn )(¯ A/m u) = u˜n for all n. We obtain an b˜ ¯ induced local homomorphism ψ : A → A = A¯ such that F (ψ)(¯ u) ≡ u˜n mod n+1 m for all n. It suffices to prove that ψ is an isomorphism. By construction ψ is the identity mod m2 : the conclusion now follows from Lemma C.49. qed Notes and Comments 1. The terminology introduced in Definition 2.5.10 is the most commonly used today but it differs from Grothendieck’s: in [75] he calls a formal deformation “algebraizable” if it is effective. The same terminology is used in [89], p. 195. 2. Other references for Theorem 2.5.13 are [1], ch. III, Th. 5.4.5, and [3], Exp. III, Prop. 7.2.

2.6

Automorphisms and prorepresentability

... chaque fois que ... une vari´et´e de modules ... ne peut exister, malgr´e de bonnes hypoth´eses de platitude, propret´e, et non singularit´e eventuellement, la raison en est seulement l’existence d’automorphismes de la structure ... (Grothendieck [41], p. 94)

2.6. AUTOMORPHISMS AND PROREPRESENTABILITY

2.6.1

105

The automorphism functor

The following theorem gives a criterion on an algebraic scheme X to decide whether Def X , resp. Def 0X , has a universal formal element and not merely a semiuniversal one. Theorem 2.6.1 Assume that X is an algebraic scheme such that Def X has a semiuniversal element (e.g. X affine with isolated singularities or X projective). Then the following conditions are equivalent: (i) Def X is prorepresentable (ii) for each small extension A0 → A in A, and for each deformation X 0 of X over Spec(A0 ), every automorphism of the deformation X 0 ×Spec(A0 ) Spec(A) → Spec(A) is induced by an automorphism of X 0 . A similar statement holds for the functor Def 0X . Proof. (i) ⇒ (ii) Let X = X 0 ×Spec(A0 ) Spec(A) and let f : X → X 0 be the induced morphism; assume that θ is an automorphism of X . Letting A¯ = A0 ×A A0 , one can construct two deformations Z and W of X over A¯ as we did in the proof of Theorem 2.4.1 as fibered sums fitting into the two diagrams: Z % X

W -

0

% X

- fθ X

%f

0

X

-

0

X0 -f

%f X

¯ have the same image ([X 0 ], [X 0 ]) under the map Since [Z], [W] ∈ Def X (A) ¯ → Def X (A0 ) ×Def (A) Def X (A0 ) Def X (A) X and since this map is bijective by (i), we have an isomorphism of deformations ρ:Z∼ = W. The isomorphism ρ induces automorphisms θ1 and θ2 of X 0 and an automorphism ψ of X such that: θ1 f θ = f ψ, θ2 f = f ψ

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This equality implies f θ = θ1−1 θ2 f : θ−1 θ2

1 X 0 −→ ↑f θ X −→

X0 ↑f X

and therefore θ1−1 θ2 induces θ. (ii) ⇒ (i) Since Def X has a semiuniversal element, it suffices to show that it satisfies condition H of Theorem 2.3.2. Let A0 → A be a small extension in A: letting A¯ = A0 ×A A0 we must show that the map ¯ → Def X (A0 ) ×Def (A) Def X (A0 ) α : Def X (A) X is bijective. Given deformations X 0 and mation X over A, we have the “fibered fits into the diagram: X¯ % 0 X -f X

X˜ 0 of X over A0 inducing the defor¯ which sum” deformation X¯ over A,

X˜ 0 % f˜

and satisfies α([X¯ ]) = ([X 0 ], [X 0 ]). Suppose that Z is another deformation of X over A¯ such that α([Z]) = ([X 0 ], [X˜ 0 ]). We have isomorphisms of deformations induced by the two projections: 0 X0 ∼ ¯ Spec(A ) ∼ = Z ×Spec(A) = X˜ 0

There remains induced an automorphism θ of X as the composition: X ∼ ¯ Spec(A) ∼ = Z ×Spec(A) = X˜ 0 ×Spec(A0 ) Spec(A) ∼ =X = X 0 ×Spec(A0 ) Spec(A) ∼ and θ fits into the commutative diagram: Z % X

X˜ 0

0

% f˜

-f θ

X −→ X

2.6. AUTOMORPHISMS AND PROREPRESENTABILITY

107

By (ii) we can lift θ to an automorphism σ : X 0 ∼ = X 0 . Replacing the lower left map f by σf we obtain the commutative diagram Z %

X˜ 0

X0 - σf X

% f˜

By the universal property of the fibered sum we obtain an isomorphism X¯ ∼ = Z which is an isomorphism of deformations. Therefore [Z] = [X¯ ] and α is bijective. In the case of Def 0X the proof is similar. qed When X is a projective scheme condition (ii) of Theorem 2.6.1 can be stated in a different way by means of the automorphism functor, which we now introduce. Assume that X is an algebraic scheme such that Def X has a semiuniversal couple (R, uˆ). Consider the functor of Artin rings Autuˆ : AR Autuˆ (A)

→ ( sets) = the group of automorphisms of the deformation XA

where XA is the deformation induced by uˆ under the morphism R → A. Then we have the following Proposition 2.6.2 If X is projective then Autuˆ has H 0 (X, TX ) as tangent space and is prorepresented by a complete local R-algebra S. Moreover the deformation functor Def X is prorepresentable if and only if S is a formally smooth R-algebra, i.e. if it is a power series ring over R. Proof. Obviously Autuˆ satisfies condition H0 because by definition the only automorphism of the deformation Xk = X is the identity. Now consider a diagram in AR : A0 A00 &. A

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with A00 → A a small extension and let A¯ = A0 ×A A00 . There is induced a diagram of deformations: XA¯ %XA0

XA00 -% XA

and therefore a natural homomorphism: OXA¯ → OXA0 ×OXA OXA00

(2.27)

Since OXA0 = OXA¯ ⊗A¯ A0 it follows from Lemma A.25 that (2.27) is an isomorphism; in particular we obtain an induced isomorphism OX∗ A¯ ∼ = OX∗ A0 ×OX∗ OX∗ A00 A

and therefore H 0 (OX∗ A¯ ) ∼ = H 0 (OX∗ A0 ) ×H 0 (OX∗

A

)

H 0 (OX∗ A00 )

(2.28)

Now note that for every A in ob(AR ) the elements of Autuˆ (A) are identified ∗ with those elements of H 0 (OX∗ A ) which restrict to 1 ∈ OX . Hence we see that (2.28) immediately implies the bijection ¯ ∼ Autuˆ (A) = Autuˆ (A0 ) ×Autuˆ (A) Autuˆ (A00 ) Therefore the functor Autuˆ also satisfies conditions H and H . From Lemma 1.2.6 it follows that Autuˆ (k[]) ∼ = H 0 (X, TX )

(2.29)

which has finite dimension since X is projective, and also Hf is satisfied. This concludes the proof of the first part. Condition (ii) of Theorem 2.6.1 can be rephrased by saying that the functor Autuˆ is smooth over Def X . qed H 0 (X, TX ) is usually called the space of infinitesimal automorphisms of X. As a corollary we obtain:

2.6. AUTOMORPHISMS AND PROREPRESENTABILITY

109

Corollary 2.6.3 If X is a projective scheme then the following conditions are equivalent: (i) h0 (X, TX ) = 0, i.e. X has no infinitesimal automorphisms; (ii) Autuˆ ∼ = Def X ; (iii) every infinitesimal deformation of X has no non-trivial automorphisms. Proof. It is an immediate consequence of the proposition. It can be also proved directly without using Proposition 2.6.2: just observe that if H 0 (X, TX ) = 0 then using Lemma 1.2.6 one shows by induction that every infinitesimal deformation of X has no automorphisms. qed An important application of the above proposition is the following result, which can be considered as the scheme-theoretic version of a classical theorem due to Kodaira-Nirenberg-Spencer [119]: Corollary 2.6.4 If X is a projective scheme such that h0 (X, TX ) = 0 then Def X is prorepresentable. If moreover X is nonsingular and h2 (X, TX ) = 0 then Def X is prorepresented by a formal power series ring. Proof. From (2.29) it follows that S = R if h0 (X, TX ) = 0 and in particular S is a formally smooth R-algebra. Then the first part follows from Proposition 2.6.2. The last assertion is a consequence of Corollary 2.4.7. qed The condition h0 (X, TX ) = 0 (no infinitesimal automorphisms) implies that the group Aut(X) is finite. We thus see that the existence of a non-trivial automorphism group does not prevent Def X from being prorepresentable provided there are no infinitesimal automorphisms. On the other hand the existence of automorphisms whatsoever is a source of difficulties when one considers local deformations (see Subsection 2.6.2). The prorepresentability criterion for Def X given by 2.6.2 is not easy to apply when h0 (X, TX ) > 0. Note that the condition h0 (X, TX ) = 0 is not necessary for the prorepresentability of Def X . An example is given by X = IP r , r ≥ 1: in this case Def X is trivially prorepresentable because X is rigid, but h0 (X, TX ) = (r + 1)2 − 1 > 0. For another example see the following Proposition 2.6.5. Corollary 2.6.4 can be generalized in a straightforward way to conclude that any functor of Artin rings F classifying isomorphism classes of deformations of a scheme with some additional structure or of any other algebrogeometric object Ξ (a morphism, etc.) is prorepresentable provided F has

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a semiuniversal element and Ξ has no infinitesimal automorphisms. This remark will be applied in the proof of the following Proposition 2.6.5 Let X be a projective irreducible and nonsingular curve of genus 1. Then Def X is prorepresentable. Proof. Fix a closed point p ∈ X. For each A in ob(A) we define a deformation of the pointed curve (X, p) to be a pair (ξ, σ) where X → X ξ: ↓ ↓π Spec(k) → Spec(A) is an infinitesimal deformation of X over A and σ : Spec(A) → X is a section of π such that Im(σ) = {p}. We have an obvious definition of isomorphism of two deformations of (X, p) over A, and we define a functor of Artin rings Def (X,p) → (sets) by Def (X,p) (A) = {isom. classes of deformations of (X, p) over A} We have a morphism of functors: φ : Def (X,p) → Def X induced by the correspondence (ξ, σ) 7→ ξ which forgets the section σ. The proposition is a consequence of the following two facts: a) φ is an isomorphism of functors. b) Def (X,p) is prorepresentable. To prove a) let A ∈ ob(A) and consider an infinitesimal deformation ξ of X over A. The point p defines a morphism Spec(k) → X making the following diagram commutative: X p% ↓π Spec(k) → Spec(A)

2.6. AUTOMORPHISMS AND PROREPRESENTABILITY

111

By the smoothness of X over Spec(A) there is an extension of p to a section σ : Spec(A) → X of π: this proves that Def (X,p) (A) → Def X (A) is surjective. Now let (ξ, σ) and (η, τ ) be two deformations with section of X over A, where X → Y η: ↓ ↓q Spec(k) → Spec(A) and suppose that there is an isomorphism of deformations X .

& ψ

X

−→

Y

&

. Spec(A)

Then ψσ, τ : Spec(A) → Y are two sections of q. We now use that fact that Y has a structure of group scheme over Spec(A) with identity τ (in outline this can be seen as follows: X is a group scheme with identity p and the group structure is given by a multiplication morphism µ : X × X → X; the group operation on Y is defined by a morphism µA : Y ×A Y → Y which extends µ and which exists because we have a commutative diagram: X ×X ∩ Y ×A Y

µ

−→

X →

⊂

Y ↓ Spec(A)

and Y is smooth over Spec(A)). Replacing ψ by by ψ 0 = ψ(ψσ)−1 we obtain an isomorphism of deformations ψ 0 such that ψ 0 σ = τ and therefore ψ 0 defines an isomorphism of (ξ, σ) with (η, τ ): this proves that Def (X,p) (A) → Def X (A) is injective as well, and a) is proved. In particular it follows that Def (X,p) has a semiuniversal element because Def X does. Now observe that the vector space of automorphisms of the trivial deformation of (X, p) can be identified with the vector subspace of H 0 (X, TX ) = H 0 (X, OX ) consisting of the derivations D : OX → OX vanishing at p, and this is equal to H 0 (X, OX (−p)) = (0). Now the remark following Corollary 2.6.4 applies to conclude that Def (X,p) is prorepresentable, i.e. b) holds. qed The following corollary computes in particular the number of moduli of projective nonsingular curves.

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Corollary 2.6.6 If X is a projective nonsingular connected curve of genus g then Def X is prorepresentable. More precisely, Def X = hR where   k

if g = 0 if g = 1 R =  k[[X]]  k[[X , . . . , X 1 3g−3 ]] if g ≥ 2 . Proof. X is unobstructed by Proposition 2.4.6 and if g = 0 h (X, TX ) = 1 if g = 1  3g − 3 if g ≥ 2 1

 0

So it remains to be shown that Def X is prorepresentable. In case g = 0 this is because IP 1 is rigid; in case g ≥ 2 since deg(TX ) = 2 − 2g < 0 we have H 0 (X, TX ) = 0 and therefore 2.6.4 applies. If g = 1 we use 2.6.5. qed Recalling Theorems 2.5.13 and 2.5.14 we deduce the following: Theorem 2.6.7 Let X be a projective nonsingular connected curve. Then X has an algebraic formally universal deformation. Examples 2.6.8 (i) ([174]) Let C = Spec(B), where B = k[x, y]/(xy), be a reducible affine plane conic. Then Def C is not prorepresentable although C has a semiuniversal deformation by Corollary 2.4.2. In fact consider the deformation of C over k[] given by xy + = 0 and its automorphism: x → 7 x + x y → 7 y This automorphism does not extend to an automorphism of xy + t¯ = 0 over k[t]/(t3 ): if it did it would be of the form x → 7 x + xt¯ + at¯2 y → 7 y + bt¯2 for some a, b ∈ k[x, y]. But this implies that bx + ay = −1 in k[x, y], which is impossible. From Theorem 2.6.1 we deduce that Def C is not prorepresentable. This holds more generally for the union of the coordinate axes in An , n ≥ 2 (see [174]).

2.6. AUTOMORPHISMS AND PROREPRESENTABILITY

113

(ii) The condition of Corollary 2.6.4 is not satisfied by the rational ruled surfaces Fm , m ≥ 0 (see Example B.44(iii). Since h1 (TF0 ) = 0 = h1 (TF1 ) we find that F0 and F1 are rigid; in particular Def F0 and Def F1 are prorepresentable. On the other hand Def Fm is unobstructed when m ≥ 2 (since h2 (TFm ) = 0) and has a semiuniversal element but it is not prorepresentable. To see it we can argue as follows. We can identify Fm with the hypersurface Σm of IP 2 × IP 1 of equation xm v − y m u = 0 where (u, v, w; x, y) are bihomogeneous coordinates in IP 2 × IP 1 (for the elementary proof of this fact see [5], p. 55). For simplicity let’s consider the case m = 2. The linear pencil V ⊂ IP 2 × IP 1 × A1 of equation: x2 v − y 2 u − txyw = 0 defines a flat family V → A1 such that V(0) = Σ2 and V(t) ∼ = Σ0 for all t 6= 0. In fact we have an isomorphism V\V(0) → V(1) × A1 \{0} over A1 \{0} sending (x, y; u, v, w; t) 7→ (x, y; u, v, tw; t); on the other hand Σ0 ∼ = V(1) by the isomorphism (x, y; u, v, w) 7→ (x, y; −x2 uw − xyw2 , xyu2 + y 2 vw, x2 u2 + 2xyuw + y 2 w2 ) (V is essentially the family considered in Example 1.2.11(iii) for m = 2). The pullback V = V (x2 v − y 2 u − xyw) ⊂ IP 2 × IP 1 × Spec(k[]) has the automorphism defined by sending w 7→ w + u and leaving all the other coordinates unchanged. We leave to the reader to check that this automorphism does not extend to the pullback of V over Spec(k[t]/(t3 )). From Theorem 2.6.1 we deduce that Def F2 is not prorepresentable.

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2.6.2

CHAPTER 2. FORMAL DEFORMATION THEORY

Isotriviality

The notions of triviality and of formal triviality of an algebraic deformation are related in a quite subtle way, as shown by Example 1.2.11(ii). Such example is a special case of an important phenomenon, called isotriviality, considered for the first time in the literature in [186]. Definition 2.6.9 Let π : X → S be a flat family of schemes. Then (i) π is called isotrivial if there is an etale cover (i.e. a finite surjective etale morphism) f : S 0 → S such that the family πS 0 : S 0 ×S X → S 0 is trivial. If S 0 ×S X ∼ = X × S 0 we say that π is isotrivial with fibre X. (ii) If s ∈ S is a k-rational point then π is called locally isotrivial at s if there is an etale neighborhood f : (S 0 , s0 ) → (S, s) such that the pullback πS 0 : S 0 ×S X → S 0 of π is trivial. π is called locally isotrivial if it is locally isotrivial at every k-rational point of S. Every trivial family is isotrivial. A rational ruled surface π : Fm → IP 1 with m ≥ 2 is locally isotrivial because locally around each point of IP 1 it is the product family with fibre IP 1 ; on the other hand π is not isotrivial because it is not trivial and the identity IP 1 → IP 1 is the only connected etale cover of IP 1 : in particular the trivial family IP 1 × IP 1 → IP 1 cannot be obtained by pulling back π by an etale cover of IP 1 . Therefore the notions of isotriviality and of local isotriviality are different. If π : X → S is isotrivial then all the fibres over the k-rational points are isomorphic. The next proposition considers the opposite implication. Proposition 2.6.10 Let π : X → S be a flat family of algebraic schemes, and let s ∈ S be a closed point. If π is locally isotrivial at s then the formal deformation of X (s) associated to π is trivial. If the morphism π is projective then the converse is also true. Proof. The first part follows immediately from Lemma 2.5.6. Conversely, let X = X (s) and A = OS,s , and assume that π is projective and that the ˆ ηˆ) of X associated to the family π is trivial. Let (A, ˆ η¯) formal deformation (A, ˆ induced by π under the morphism be the deformation of X over Spec(A) ˆ → S. By Theorem 2.5.11 this deformation is uniquely determined Spec(A) ˆ ηˆ) and therefore it is trivial. by the formal deformation (A,

2.6. AUTOMORPHISMS AND PROREPRESENTABILITY

115

Extend Def X to a locally of finite presentation functor F defined on (kalgebras). We may assume that S = Spec(B) is affine. We have a commutative diagram of functors defined on the category (k-algebras): hAˆ .

&

hB

hB &

. F

The lower arrows are induced by the two algebraic deformations π : X → S and p : X × S → S of X (by Yoneda’s lemma), while the upper arrows ˆ There are the same and correspond to the natural homomorphism B → A. remains induced a morphism of functors hAˆ → hB ×F hB which corresponds to an effective formal element for the functor hB ×F hB . Since F is locally of finite presentation and B is of finite type the functor hB ×F hB is locally of finite presentation as well (a fibered product of functors locally of finite presentation is again locally of finite presentation: see [12], p. 33) so that we can apply the approximation theorem 2.5.22. Therefore we can find an etale neighborhood Spec(B 0 ) → Spec(B) such that both deformations π and p pullback to the same deformation over Spec(B 0 ): in particular π pulls back to a trivial deformation over Spec(B 0 ). qed The inverse implication of Proposition 2.6.10 is false in general: families of nonsingular affine schemes are formally trivial but not isotrivial in general, as shown by Example 1.2.5, page 28. The following is immediate. Corollary 2.6.11 If π : X → S is a smooth projective family of algebraic schemes which is locally isotrivial at a k-rational point s ∈ S then the Kodaira-Spencer map κπ,s : Ts S → H 1 (X (s), TX (s) ) is 0. Example 2.6.12 The converse of Corollary 2.6.11 is false. For example consider a smooth projective family π : X → Spec(k[t]) such that the induced

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deformations over k[t]/(tn ) are non-trivial for n 0; in particular π is not locally isotrivial at the point t = 0. Let q ] : Spec(k[u]) → Spec(k[t]) be the morphism defined by q : k[t] → k[u] sending t 7→ u2 . Then q ] has zero differential at 0 and therefore the pulled back family X ×Spec(k[t]) Spec(k[u]) → Spec(k[u]) has a vanishing Kodaira-Spencer map at 0. But the restriction of this family to k[u]/(un ) is non-trivial for n 0 and therefore the family is not locally isotrivial at s by Proposition 2.6.10. The existence of non-trivial isotrivial deformations of a scheme X is closely related with the existence of non-trivial automorphisms of X. Before investigating this fact we give some examples. Examples 2.6.13 (i) Let X be a quasi-projective scheme such that there is a finite non trivial subgroup G ⊂ Aut(X). Let S 0 be a quasi-projective scheme on which G acts freely and let S := S 0 /G be the quotient. Then G acts on X × S 0 componentwise and the action is easily seen to be free. The quotient X := (X × S 0 )/G exists and is an algebraic scheme (see [187] or [3], exp. V). Since the projection X × S 0 → S 0 is G-equivariant it induces a morphism π:X →S and we have a commutative diagram: X × S0 ↓ S0

→ X ↓π → S

(2.30)

where the horizontal arrows are etale morphisms and all the fibres of π over the closed points are isomorphic to X. Moreover (2.30) is a cartesian diagram (there is an S 0 -morphism X × S 0 → S 0 ×S X which is easily seen to be an isomorphism) and π is flat (use A.26). Claim : The family π is not trivial. In fact, since (2.30) is cartesian, if π were trivial the action of G on X × S 0 would be trivial on the first factor, a contradiction. It follows that π is an isotrivial non-trivial family. (ii) Let G be a non-trivial finite group scheme and Z a quasi-projective scheme on which G acts freely. Then the quotient scheme Z/G exists (see

2.6. AUTOMORPHISMS AND PROREPRESENTABILITY

117

[187] or [3], exp. V). Let π : Z → Z/G be the canonical morphism. Then π is an etale cover, in particular it is flat. Moreover we have a commutative diagram Z ×Z/G Z → Z 0 ↓π ↓π Z → Z/G and since the action is free we have an isomorphism G × Z → Z ×Z/G Z induced by the map (g, z) 7→ (z, gz). Therefore π 0 is the trivial family and π is isotrivial. If Z is integral then π has no sections and it follows that π is not trivial. The morphism π is called a principal G-bundle. (iii) Perhaps the simplest examples of isotrivial non-trivial families are those described in Example 1.2.11(ii), page 34. We leave to the reader to verify that they are isotrivial. We need the following well known lemma ([186], n. 1.5). Lemma 2.6.14 Let f : Z → Y be an etale cover of algebraic schemes. Then there is an etale cover ϕ : P → Z such that the composition fϕ : P → Z → Y is a principal G-bundle with respect to a finite group G. In particular f is isotrivial and, if Z is integral and deg(f ) > 1, it is non-trivial. Proof. Let n = deg(f ). In the n-fold fiber product Z ×Y Z ×Y · · · ×Y Z consider the set P of points (z1 , . . . , zn ) such that zi 6= zj for all i 6= j. Then P is a union of connected components of Z ×Y Z ×Y · · · ×Y Z which is stable under the natural action of the symmetric group Sn . The natural morphism φ : P → Y is an etale cover of degree n! and therefore it induces an isomorphism P/Sn ∼ = Y . Therefore φ is a principal Sn -bundle. Moreover the first projection ϕ : P → Z is etale of degree (n − 1)! and satisfies f ϕ = φ. In order to prove the last assertion recall that by Example 2.6.13(ii) φ is isotrivial. More precisely we have a commutative diagram: P ×Y P ∼ = Sn × P ↓ P

→ P ↓φ → Y

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The left vertical morphism is the projection and, because of the factorization f ϕ = φ, it factors as Sn × P → P ×Y Z → P It follows that P ×Y Z → P is trivial as well and this imples that f is isotrivial. If Z is integral and deg(f ) > 1 then f cannot be trivial because it has no sections. qed Theorem 2.6.15 The following conditions are equivalent on a quasi-projective scheme X: (a) There exists a non-trivial isotrivial algebraic family with fibre X. (b) Aut(X) contains a non-trivial finite subgroup. Proof. (b) ⇒ (a) has been proved in Example 2.6.13(i). (a) ⇒ (b). Let π : X → S be a family as in (a). By hypothesis there is an etale cover φ : P → S such that πP : P ×S X → P is trivial with fibre X; let’s identify X × P = P ×S X and let ψ : X × P → X be the projection: X ×P ↓ P

ψ

−→ X ↓π φ −→ S

For each p ∈ P denote by ψp : X → X the morphism defined by ψp (x) = ψ(x, p) By Lemma 2.6.14 we may assume that φ is a principal G-bundle with respect to some finite group G. We define an action of G on X × P by the following rule: −1 g(x, p) = (ψgp ψ(x, p), gp) (X × P )/G exists and, since the action is clearly free and transitive on the fibres of ψ, there is an induced morphism (X × P )/G → X which is an etale cover of degree one, i.e. it is an isomorphism. Fix p ∈ P and define G×X →X −1 by gx = ψgp ψ(x, p). Then one checks immediately that this is an action of G on X. Since π is non-trivial it is quite clear that this action cannot be

2.6. AUTOMORPHISMS AND PROREPRESENTABILITY

119

trivial for all p ∈ P . Therefore for some p ∈ P the above action defines a homomorphism G → Aut(X) whose image is 6= {1}. qed If a scheme X has an isotrivial local deformation η which is non-trivial then the local moduli functor Def X : A∗ → (sets) considered in the Introduction cannot be representable, i.e. the local deformation υ considered there cannot exist. Assume by contradiction that υ exists; then, since η is non-trivial it must be pulled back from υ by a nonconstant morphism g : Spec(A) → Spec(O). On the other hand, since η ˜ is trivial and is therefore obtained by is isotrivial its pullback to Spec(A) pulling back υ in two different ways: by the constant morphism and by the composition ϕ g ˜ −→ Spec(A) Spec(A) −→ O which is non-constant because ϕ is faithfully flat hence surjective; this contradicts the universality of υ. These remarks explain why we cannot expect to be able to construct families representing functors defined on A∗ or on (k-algebras) or on (schemes), which classify isomorphism classes of schemes having non-trivial isotrivial deformations; and the existence of such deformations is closely related to the existence of non-trivial automorphisms of such schemes, as we saw in Theorem 2.6.15. This discussion suggests that while the consideration of isomorphism classes of deformations is not a drawback when one is studying infinitesimal deformations, it becomes inadequate for the classification of algebraic deformations and for global moduli problems. In other words, because of the presence of non-trivial automorphisms we cannot in general expect to find a scheme structure on the set M of isomorphism classes of objects we want to classify in such a way that it reflects faithfully the functorial properties of families. For example, in the case of projective nonsingular curves of genus 0 one should have M = Spec(k) and the only candidate for being the universal family is IP 1 → Spec(k) because there is only one isomorphism class of such curves; but the families Fm → IP 1 (Example B.44(iii)) cannot be pulled back from it, thus a universal family cannot exist in this case. That’s why it would be more natural, instead of taking isomorphism classes of deformations, to consider all families together and analize them

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and their isomorphisms. This will result in a more general structure, called a stack , which contains all the informations about families and deformations of the objects of the set M we want to classify. We refer to [44], [206] and [128] for foundational material about stacks, and to [19], [68] for expository treatments.

Chapter 3 Examples of deformation functors This chapter is devoted to the study of the most important functors of Artin rings which arise when one considers deformations of various algebrogeometric objects. We will verify Schlessinger’s conditions and we will describe first order deformations, i.e. the tangent spaces of these functors, and obstruction spaces. We will focus on several examples and applications. It will emerge from the treatment a pattern common to almost all examples: the tangent space and the obstruction space of a given functor will be respectively isomorphic to H i and to H i+1 of a certain sheaf which depends on the functor. It will be i = 0 if the deformation problem has no automorphisms, while i = 1 if there are automorphisms; in this case the H 0 will classify the infinitesimal automorphisms.

3.1

Affine schemes

In this section we study the deformation functor of an affine scheme. We ¯ already know that such a functor verifies Schlessinger’s conditions H0 , H, H and we computed its tangent space (Proposition 2.4.1). In particular we proved that it has a semiuniversal element if the scheme has isolated singularities (Corollary 2.4.2). Here we will analyze this case in more detail. We will start by recalling the description of the tangent space. 121

122

3.1.1

CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

First order deformations

Let B0 be a k-algebra, and let X0 = Spec(B0 ). We continue the study of infinitesimal deformations of X0 , equivalently of B0 , started in Section 1.2 in the nonsingular case. Let’s recall the following fact (see 2.4.1(iii)). Proposition 3.1.1 There is a natural isomorphism Def B0 (k[]) ∼ = TB1 0 where the class of trivial deformations corresponds to 0 ∈ TB1 0 . If B0 is reduced then Def B0 (k[]) ∼ = Ext1k (ΩB0 /k , B0 ) Proof. A first order deformation of B0 consists of a flat k[]-algebra B, plus a k-isomorphism B ⊗k[] k ∼ = B0 . This set of data determines a k-extension: 0 →

B0 k B

j

−→ B

→ B0

→ 0 (3.1)

obtained after tensoring the exact sequence 0 → () → k[] → k → 0 by ⊗k[] B. Isomorphic deformations give rise to isomorphic extensions. Conversely, given a k-extension (3.1), B has a structure of k[]-algebra given by 7→ j(1) B is k[]-flat by Lemma A.30. If B0 is reduced then TB1 0 = Ext1k (ΩB0 /k , B0 ) by Corollary 1.1.11.

qed

The following is an immediate consequence of 3.1.1, 2.2.8 and of Schlessinger’s Theorem 2.3.2. Corollary 3.1.2 If dimk (TB1 0 ) < ∞ then Def B0 has a semiuniversal formal element. This happens in particular if X0 = Spec(B0 ) has isolated singularities. If TB1 0 = 0 then B0 is rigid.

3.1. AFFINE SCHEMES

123

We will give some indications for the practical computation of TB1 0 when B0 is e.f.t.. Let B0 = P/J, where P is a smooth k-algebra of the form P = ∆−1 k[X1 , . . . , Xd ] for some multiplicative system ∆ ⊂ k[X1 , . . . , Xd ], and J ⊂ P is an ideal. Consider the exact sequence δ∨

0 → Hom(ΩB0 /k , B0 ) → Hom(ΩP/k ⊗B0 , B0 ) −→ Hom(J/J 2 , B0 ) −→ TB1 0 → 0 The module Hom(ΩP/k ⊗ B0 , B0 ) = Derk (P, B0 ) consists of all derivations D of the form D(p) =

d X j=1

bj

∂p ∂Xj

for given bj ∈ B0 , and δ ∨ (D)(f¯) = D(f ) =

d X j=1

bj

∂f ∂Xj

f ∈J

Assume that J = (f1 , . . . , fn ) and let ι

j

0 → R −→ P n −→ J → 0 be the corresponding presentation. We have the exact sequence: j∨

ι∨

0 → Hom(J/J 2 , B0 ) −→ Hom(B0n , B0 ) −→ Hom(R, B0 ) where j ∨ identifies Hom(J/J 2 , B0 ) with the submodule of Hom(B0n , B0 ) consisting of those homomorphisms which are 0 on R. Identifying Hom(B0n , B0 ) = B0n , thereby viewing its elements as column vectors, we see that the condition for   q1  ..  q =  .  ∈ B0n qn

124

CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

to be in Hom(J/J 2 , B0 ) is that t q ·r = 0 for each r ∈ R (where we are viewing R as consisting of column vectors as well). j ∨ associates to a homomorphism J/J 2 bj f¯j

ϕ:

P

the column vector

→ B0 P 7 → bj ϕ(f¯j )

ϕ(f¯1 )  ..   .  ϕ(f¯n ) 



Therefore Im(δ ∨ ) ⊂ B0n is generated by the column vectors corresponding to ∂ ∂ ), . . ., δ ∨ ( ∂X ), i.e. by the classes mod J of: δ ∨ ( ∂X 1 d  ∂f1 ∂Xd    , · · · ,  ..   .

 ∂f1 1  ∂X  ..  .



∂fn ∂Xd

∂fn ∂X1

   

If J = (f1 , . . . , fn ) is generated by a regular sequence then ι∨ = 0, equivalently j ∨ is an isomorphism, and it follows that TB1 0 ∼ =    

Pn  ∂f1 ∂Xd    , · · · ,  ..   .

∂f1 ∂X1



.. .

∂fn ∂Xd

∂fn ∂X1



⊗P B0

  

In particular, if B0 = P/(f ) then TB1 0 ∼ =

P (f,

∂f ∂f ) , · · · , ∂X ∂X1 d

It follows from this description that the hypersurface V (f ) is rigid if and only if it is nonsingular. A similar remark holds for complete intersections. In particular, recalling the definition of Subsection D.1, we can state the following, for future reference: Proposition 3.1.3 An e.f.t. complete intersection ring B0 such that Spec(B0 ) is singular is not rigid. Example 3.1.4 Let P be the local ring of a nonsingular algebraic surface X at a k-rational point p, m = (x, y) its maximal ideal, and B0 = P/(f ) the local ring of a curve C ⊂ X at p. Let’s compute TB1 0 in some cases.

3.1. AFFINE SCHEMES

125

ˆ0 ∼ (a) Node (ordinary double point) - By definition B = k[[X, Y ]]/(X 2 + Y 2 ). Then f = x2 + y 2 + higher order terms, and B0 ∼ TB1 0 ∼ = =k (x, y) if char(k) 6= 2. ˆ0 ∼ (b) Ordinary cusp - In this case B = k[[X, Y ]]/(X 2 + Y 3 ). Then f = x2 + y 3 + higher order terms, and TB1 0 ∼ =

B0 ∼ 2 =k (y 2 , x)

if char(k) 6= 2, 3. ˆ0 ∼ (c) Tacnode - We have in this case B = k[[X, Y ]]/(Y (Y + X 2 )) and TB1 0 ∼ =

(x2

B0 ∼ = k3 + 2y, xy)

if char(k) 6= 2. Conversely we have the following result: Proposition 3.1.5 Assume char(k) = 0. Let P be the local ring of a nonsingular algebraic surface X at a k-rational point p, m = (x, y) its maximal ideal, and B0 = P/(f ) the local ring of a curve C ⊂ X at p; let t = dimk TB1 0 . Then (a) t = 0 if and only if B0 is regular (a DVR). (b) t = 1 if and only if B0 is the local ring of a node. (c) t = 2 if and only if B0 is the local ring of an ordinary cusp. (d) t = 3 if and only if B0 is the local ring of a tacnode. Proof. The “if” implication follows from the above computations. We have t = dimk P/(f, fx , fy ) f ∈ m3P immediately implies t ≥ 4; then f ∈ m2P and, after suitable choice of generators of mP we may suppose f = y 2 + xn + higher order terms, n ≥ 2 or f = y(y + xn )+ higher order terms, n ≥ 2. Now the conclusion follows easily. qed

126

CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

Example 3.1.6 The affine cone over IP 1 × IP 2 - Let P = k[X0 , X1 , X2 , Y0 , Y1 , Y2 ] J = (X1 Y2 − X2 Y1 , X2 Y0 − X0 Y2 , X0 Y1 − X1 Y0 ) Then B0 = P/J is the coordinate ring of the affine cone over the Segre embedding IP 1 × IP 2 ⊂ IP 5 . We have the following presentation: A

0 → P 2 −→ P 3 → J → 0 where 

X0  A =  X1 X2



Y0 Y1   Y2

A direct computation shows that Hom(J/J 2 , B0 ) is generated by the following column vectors: Y1 −Y0 0

Y2 0 −Y0

0 Y2 −Y1

X1 −X0 0

X2 0 −X0

0 X2 −X1

Since these vectors are, up to permutation, δ∨(

∂ ∂ ∂ ∂ ∂ ∂ ), δ ∨ ( ), δ ∨ ( ), δ ∨ ( ), δ ∨ ( ), δ ∨ ( ) ∂X0 ∂X1 ∂X2 ∂Y0 ∂Y1 ∂Y2

we see that TB1 0 = 0. This implies that B0 is rigid (see Corollary 3.1.2). More generally one can prove that the coordinate ring of the affine cone over the Segre embedding IP n × IP m ⊂ IP (n+1)(m+1)−1 is rigid whenever n + m ≥ 3. This has been computed for the first time in [71] in the case n = m ≥ 2; the general case is in [177] (see Corollary 3.1.20 below). Example 3.1.7 Let P = k[X1 , X2 , X3 ](X) , J = (X2 X3 , X1 X3 , X1 X2 ). Then B0 = P/J is the local ring at the origin of the union of the coordinate axes in A3 . We have the presentation A

P 3 −→ P 3 → J → 0

3.1. AFFINE SCHEMES where

127





X1 X1 0   A =  −X2 0 X2  0 −X3 −X3 and the columns of A generate R. Hom(J/J 2 , B0 ) is generated by the following column vectors mod J: X2 0 0

X3 0 0

0 X1 0

0 X3 0

0 0 X1

0 0 X2

and Im(δ ∨ ) is generated by the column vectors mod J: 0 X3 X2

X3 0 X1

X2 X1 0

It follows at once that TB1 0 = Hom(J/J 2 , B0 )/Im(δ ∨ ) ∼ = k3 because there are 3 generators of Hom(J/J 2 , B0 ) which are linearly independent modulo the generators of Im(δ ∨ ), and all other elements of Hom(J/J 2 , B0 ) are in Im(δ ∨ ). In a similar vein one can consider, for any d ≥ 3 B0 = k[X1 , . . . , Xd ](X) /J where ˆ i · Xd , . . .)i=1,...,d J = (. . . , X1 X2 · X Then B0 is the local ring at the origin of the union of the coordinate axes in Ad . One computes easily, along the same lines of the case d = 3, that T1 ∼ = kd(d−2) B0

Example 3.1.8 Let P be the local ring of a nonsingular algebraic variety V of dimension n ≥ 2 at a k-rational point p and let B0 = P/(f ) be the local ring of a hypersurface X ⊂ V at p. We call p a node for X if ˆ0 ∼ B = k[[X1 , . . . , Xn ]]/(X 2 + · · · + X 2 ) 1

n

equivalently if we can choose generators x1 , . . . , xn of the maximal ideal mP P so that f = x2i + higher order terms. It is immediate to prove that if p is 1 ∼ a node for X and char(k) 6= 2 then TB1 0 ∼ = k. In particular Tk[] = k.

128

3.1.2

CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

The second cotangent module and obstructions

We will now describe an obstruction space for the functor Def B0 . Assume that B0 = P/J for a smooth k-algebra P , and an ideal J ⊂ P . Consider a presentation: η:

ι

j

0 → R −→ F −→ J → 0

where F is a finitely generated free P -module. Let λ : by: λ(x ∧ y) = (jx)y − (jy)x

(3.2) V2

F → F be defined

and Rtr = Im(λ). Obviously Rtr ⊂ R and Rtr ⊂ JF . If J = (f1 , . . . , fn ) then F = P n and R is the module of relations among f1 , . . . , fn . Rtr is called the module of trivial (or Koszul) relations; it is generated by the relations of the form (0, . . . , −fj , . . . , fi , . . . , 0) i

j

It follows that R/Rtr = H1 (K• (f1 , . . . , fn )) the first homology module of the Koszul complex associated to f1 , . . . , fn . Lemma 3.1.9 The P -module R/Rtr is annihilated by J and therefore it is a B0 -module in a natural way. Proof. Let x ∈ R, a ∈ J. Let y ∈ F be such that j(y) = a. Then ax = j(y)x = j(y)x − j(x)y = λ(y ∧ x) ∈ Rtr qed Since Rtr ⊂ JF the presentation (3.2) induces an exact sequence of B0 modules: ¯ j ¯ ι R/Rtr −→ F ⊗P B0 −→ J/J 2 → 0 (3.3) Definition 3.1.10 The second cotangent module of B0 is the B0 -module TB2 0 defined by the exact sequence: HomB0 (J/J 2 , B0 ) → HomB0 (F ⊗P B0 , B0 ) → HomB0 (R/Rtr , B0 ) → TB2 0 → 0 induced by (3.3). Obviously TB2 0 is a B0 -module of finite type.

3.1. AFFINE SCHEMES

129

Lemma 3.1.11 For every e.f.t. k-algebra B0 the B0 -module TB2 0 is independent of the presentation (3.2). Proof. Assume that F ∼ = P n and that j : P n → J is defined by the system of generators f1 , . . . , fn of J. Let 0 → S → Pm → J → 0 be another presentation of J, defined by the system of generators g1 , . . . , gm . We may assume that m ≥ n and that gk = fk , k = 1, . . . , n. Let gk =

X

bks fs

k = n + 1, . . . , m

s

for some bks ∈ P . Denote by α : P n → P m the map α(a1 , . . . , an ) = (a1 , . . . , an , 0, . . . , 0) and by β : P m → P n the map β(a1 , . . . , am ) = (a1 +

m X s=n+1

b1s as , . . . , an +

m X

bns as )

s=n+1

Evidently α(R) ⊂ S and α(Rtr ) ⊂ Str . It is easy to verify that β(S) ⊂ R and β(Str ) ⊂ Rtr . It follows that α and β induce homomorphisms β ? : Hom(R/Rtr , B0 ) → Hom(S/Str , B0 ) α? : Hom(S/Str , B0 ) → Hom(R/Rtr , B0 ) whence homomorphisms β˜ : Hom(R/Rtr , B0 )/Hom(P n , B0 ) → Hom(S/Str , B0 )/Hom(P m , B0 ) α ˜ : Hom(S/Str , B0 )/Hom(P m , B0 ) → Hom(R/Rtr , B0 )/Hom(P n , B0 ) Since α? β ? = identity of Hom(R/Rtr , B0 ) it follows that α ˜ β˜ = identity of Hom(R/Rtr , B0 )/Hom(P n , B0 ) We now prove that β˜α ˜ = identity of Hom(S/Str , B0 )/Hom(P m , B0 )

(3.4)

130

CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

Let g ∈ Hom(S/Str , B0 ) be induced by the homomorphism G : S → B0 . Then, if (a1 , . . . , am ) ∈ S and (¯ a1 , . . . , a ¯m ) ∈ S/Str is its class, we have: (β ? α? )(g)(¯ a1 , . . . , a ¯m ) = G(α(β(a1 , . . . , am ))) = G(a1 +

Pm

s=n+1 b1s as , . . . , an

Pm

G(a1 , . . . , am ) + G(

s=n+1 b1s as , . . . ,

+

Pm

s=n+1 bns as , 0, . . . , 0)

=

Pm

s=n+1 bns as , −an+1 , . . . , −am )

= (∗)

Now note that (

m X

m X

b1s ps , . . . ,

s=n+1

bns ps , −pn+1 , . . . , −pm ) ∈ S

s=n+1

for every (p1 , . . . , pm ) ∈ P m . Therefore letting τ (p1 , . . . , pm ) = G(

m X

s=n+1

b1s ps , . . . ,

m X

bns ps , −pn+1 , . . . , −pm )

s=n+1

we define a homomorphism τ : P m → B0 . It follows that (∗) = g(¯ a1 , . . . , a ¯m ) + τ (a1 , . . . , am ) Hence (β ? α? )(g) − g ∈ Im[Hom(P m , B0 ) → Hom(S/Str , B0 )] or equivalently (3.4) holds.

qed

From the definition it easily follows that TB2 0 localizes. Namely, for every multiplicative subset ∆ ⊂ P we have: ∆−1 TB2 0 = T∆2 −1 B0 It follows that for any scheme X we can define in an obvious way the second cotangent sheaf which we will denote by TX2 . It satisfies 2 = TO2 X,x TX,x

Proposition 3.1.12 Assume that B0 = P/J for a smooth k-algebra P . Then TB2 0 is an obstruction space for the functor Def B0 .

3.1. AFFINE SCHEMES

131

Proof. Let A be an object of A and let ξ:

→ B0 ↑ → k

B ↑ A

be an infinitesimal deformation of B0 over A. We must associate to ξ a k-linear map ξv : Exk (A, k) → TB2 0 satisfying the conditions of Definition 2.2.9. Let B0 = P/J for a smooth k-algebra P and an ideal J = (f1 , . . . , fn ) ⊂ P . We have an exact sequence: f

0 → R → P n −→ J → 0 Then, by the smoothness, in particular flatness, of P we have B = (P ⊗k A)/(F1 , . . . , Fn ) where fj = Fj (mod mA ), j = 1, . . . , n, by Theorem A.31. The flatness of B over A implies that for every r = (r1 , . . . , rn ) ∈ R there exist R1 , . . . , Rn ∈ P P ⊗k A such that rj = Rj (mod. mA ), j = 1, . . . , n, and j Rj Fj = 0, again by Theorem A.31. Let [γ] ∈ Exk (A, k) be represented by an extension γ:

φ 0 → (t) → A˜ −→ A → 0

˜1, . . . , R ˜ n ∈ P ⊗k A˜ liftings of F1 , . . . , Fn , R1 , . . . , Rn Choose F˜1 , . . . , F˜n , R respectively; then X

˜ j F˜j ∈ ker[P ⊗k A˜ → P ⊗k A] ∼ R =P

j

˜1, . . . , R ˜ n or of F˜1 , . . . , F˜n It is easy to check that a different choice of R P ˜ ˜ P modifies j Rj Fj by an element of J or by one of the form j qj rj , where qj ∈ P , respectively. Therefore sending r 7→

X j

˜ j F˜j R

(3.5)

132

CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

defines an element of coker[Hom(B0n , B0 ) → Hom(R, B0 )] Moreover, since if r = rij = (0, . . . , fj , . . . , −fi , . . . , 0) we can take ˜1, . . . , R ˜ n ) = (0, . . . , F˜j , . . . , −F˜i , . . . , 0) (R P ˜ ˜ tr and we get j R j Fj = 0, it follows that (3.5) is zero on Hom(R , B0 ). Therefore the n-tuple of liftings (F˜ ) = (F˜1 , . . . , F˜n ) defines an element ξv (γ) of coker[Hom(B0n , B0 ) → Hom(R/Rtr , B0 )] = TB2 0

Let’s prove that the map γ 7→ ξv (γ) is k-linear. Let [ζ] ∈ Exk (A, k) be another element defined by the extension: 0 → (t) → A0 → A → 0

ζ:

and let (F 0 ) = (F10 , . . . , Fn0 ), Fj0 ∈ P ⊗k A0 be the corresponding lifting, which defines ξv (ζ). Then ξv (γ) + ξv (ζ) is defined by r 7→

X

˜ j F˜j + R

j

X

Rj0 Fj0

j

where R10 , . . . , Rn0 ∈ P ⊗k A0 are liftings of R1 , . . . , Rn . Consider the diagram: 0 → k ⊕ k → A˜ ×A A0 ↓δ ↓σ γ+ζ : 0→ k → C

→ A →0 k → A →0

Then ξv (γ + ζ) is defined by r 7→

X

Ψj Φj

j

where Ψ1 , . . . , Ψn , Φ1 , . . . , Φn ∈ P ⊗k C are liftings of R1 , . . . , Rn , F1 , . . . , Fn . Since ˜ ×A (P ⊗k A0 ) P ⊗k (A˜ ×A A0 ) ∼ = (P ⊗k A) letting ρ : P ⊗k (A˜ ×A A0 ) → P ⊗k C be the homomorphism induced by σ, we may assume that Φj = ρ(F˜j , Fj0 );

˜ j , Rj0 ) Ψj = ρ(R

3.1. AFFINE SCHEMES

133

Then: X

Ψj Φj =

j

=

X

˜ j , Rj0 )ρ(F˜j , Fj0 ) = ρ[ ρ(R

X

˜ j , Rj0 )(F˜j , Fj0 )] = (R

j

j

X

= ρ(

˜ j F˜j , R

j

X

Rj0 , Fj0 ) = δ(

X

X

j

j

=

˜ j F˜j , R

X

˜ j F˜j + R

j

X

Rj0 , Fj0 ) =

j

Rj0 , Fj0

j

This proves that ξv (γ + ζ) = ξv (γ) + ξv (ζ). A similar argument shows that ξv (λγ) = λξv (γ), λ ∈ k. Now assume that [γ] ∈ Exk (A, k) is such that there exists an infinitesimal deformation ˜ → B0 B ˜ ξ: ↑ ↑ ˜ A → k such that

˜ → Def B0 (A) Def B0 (A) ξ˜

7→

ξ

It follows that there exist liftings F˜j ∈ P ⊗k A˜ of the Fj ’s such that every ˜ ∈ (P ⊗k A) ˜ N such that Pj R ˜ j F˜j = 0. This means that r ∈ R has a lifting R ξv (γ) = 0. Conversely, assume that ξv (γ) = 0, and let F˜1 , . . . , F˜n ∈ P ⊗k A˜ be arbitrary liftings of F1 , . . . , Fn . Then there exists (h1 , . . . , hn ) ∈ P n such ˜ ∈ (P ⊗k A) ˜ n of a relation r ∈ R we have: that for every choice of a lifting R X j

˜ j F˜j = −t R

X

rj hj = −

j

X

˜ j hj R

j

This means that the ideal (F˜1 + th1 , . . . , F˜n + thn ) ⊂ P ⊗k A˜ defines a flat deformation of B0 over A˜ lifting the deformation B = (P ⊗k A)/(F1 , . . . , Fn ). Any other choice of a lifting of the deformation ξ over A˜ is of the form (F˜1 + t(h1 + k1 ), . . . , F˜n + t(hn + kn )) where k = (k1 , . . . , kn ) ∈ B0n satisfy j rj kj = 0 for every relation r ∈ R. Therefore k ∈ Hom(J/J 2 , B0 ). It is straightforward to verify that if k ∈ P

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˜ Im(δ ∨ ) then F˜ + th and F˜ + t(h + k) define isomorphic liftings of ξ over A. 1 ˜ This means that we have an action of TB0 on the set of liftings of ξ over A. By construction it follows that this action is transitive. qed Corollary 3.1.13 Let X0 = Spec(B0 ) be an affine algebraic scheme such that dimk (TB1 0 ) < ∞, and let (R, {ηn }) be a semiuniversal formal deformation of X0 . Then (i) dimk (TB1 0 ) ≥ dim(R) ≥ dimk (TB1 0 ) − dimk (TB2 0 ) The first equality holds if and only if X0 is unobstructed. (ii) If B0 is an e.f.t. local complete intersection k-algebra then it is unobstructed. In particular hypersurface singularities are unobstructed. Proof. (i) follows from Corollary 2.2.11. (ii) If J is generated by a regular sequence then R = Rtr and therefore 2 TB0 = (0). qed Proposition 3.1.14 Assume that B0 = P/J for a smooth k-algebra P . Then: (i) If Spec(B0 ) is reduced then TB2 0 ∼ = Ext1B0 (J/J 2 , B0 ) (ii) If Spec(B0 ) is reduced and has depth at least 2 along the locus where it is not a l.c.i. (e.g. Spec(B0 ) is normal of dimension ≥ 2) there is an isomorphism TB2 0 ∼ = Ext2 (ΩB0 /k , B0 ) Proof. We keep the notations introduced on page 128. (i) From the exact sequence (3.3) we deduce the following commutative diagram with exact rows and columns: Hom(ker(¯ι), B0 ) ↑ ¯ j∨

Hom(J/J 2 , B0 ) → Hom(F ⊗ B0 , B0 ) −→ k k 2 Hom(J/J , B0 ) → Hom(F ⊗ B0 , B0 ) →

Hom(R/Rtr , B0 ) → T 2 → 0 ∪ ∪ Hom(Im(¯ι), B0 ) → E 1 → 0

3.1. AFFINE SCHEMES

135

where E 1 = Ext1 (J/J 2 , B0 ) and T 2 = TB2 0 . Since the exact sequence η:

j

ι

0 → R −→ F −→ J → 0

from which (3.3) is obtained localizes, we see that ker(¯ι) is supported on the locus where B0 is not a l.c.i.; in particular ker(¯ι) is torsion. Therefore we have Hom(ker(¯ι), B0 ) = (0) and the conclusion follows. (ii) Consider the conormal sequence δ

J/J 2 −→ ΩP/k ⊗ B0 → ΩB0 /k → 0 Since Spec(B0 ) is reduced ker(δ) is supported in the locus where Spec(B0 ) ⊂ Spec(P ) is not a regular embedding (Proposition D.4), and this locus coincides with the locus where Spec(B0 ) is not a l.c.i. (Proposition D.5). From the assumption about the depth of Spec(B0 ) it follows that Hom(ker(δ), B0 ) = Ext1 (ker(δ), B0 ) = (0) Using this fact and recalling that Exti (ΩP/k ⊗B0 , B0 ) = (0), i > 0, we obtain: Ext2 (ΩB0 /k , B0 ) ∼ = Ext1 (Im(δ), B0 ) ∼ = Ext1 (J/J 2 , B0 ) qed Example 3.1.15 (Schlessinger[174]) An obstructed affine curve - Let s be an indeterminate and let B0 = k[s7 , s8 , s9 , s10 ] ⊂ k[s] be the coordinate ring of the affine rational curve C ⊂ A4 = Spec(k[x, y, z, w]) having parametric equations: x = s7 , y = s8 , z = s9 , w = s10 Write P = k[x, y, z, w] and B0 = P/I for an ideal I ⊂ P . One can check, using for example a computer algebra package, that I is generated by the six 2 × 2 minors of the following matrix:

x y y z

z w

w2 x3

i.e. by the following polynomials: f1 = y 2 − xz;

f2 = xw − yz;

f3 = z 2 − yw;

f4 = x4 − w2 y; f5 = x3 y − zw2 ; f6 = w3 − x3 z

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This ideal is prime of height 3. Consider a presentation: 0→R→F →I→0 where F ∼ = P 6 with generators say e1 , . . . , e6 , so that ei 7→ fi . To describe R one can use the beginning of the free resolution of I given by the EagonNorthcott complex (see [49]). One obtains a set of generators for R given by the rows of the following matrix: R1 R2 R3 R4 R5 R6 R7 R8

: z : w2 : 0 : 0 : w : x3 : 0 : 0

y 0 0 −w2 z 0 0 −x3

x 0 w2 0 y 0 x3 0

0 y 0 z 0 z 0 w

0 0 −x 0 z y 0 x 0 0 −y 0 w z 0 y

Here each row gives the coefficients ai of the linear combination We then have an exact sequence

P

i

ai ei ∈ R.

¯ → F¯ → I/I 2 → 0 0→R ¯ = R/(IF ∩R). Reducing mod I the above relations where F¯ = F/IF and R ¯ as elements of F¯ : one gets the following set of generators of R r1 r2 r3 r4 r5 r6 r7 r8

: s9 : s20 : 0 : 0 : s10 : s21 : 0 : 0

s8 0 0 −s20 s9 0 0 −s21

s7 0 s20 0 s8 0 s21 0

0 s8 0 s9 0 s9 0 s10

0 −s7 s9 0 0 −s8 s10 0

0 0 s8 s7 0 0 s9 s8

(3.6)

Since B0 is reduced we have ¯ B0 )/Hom(F¯ , B0 ) TB2 0 = Ext1B0 (I/I 2 , B0 ) = Hom(R, ¯ B0 ) as the 8-tuple (h(r1 ), . . . , h(r8 )) ∈ Representing an element h ∈ Hom(R, 8 B0 we see that the submodule Hom(F¯ , B0 ) is generated by the columns of

3.1. AFFINE SCHEMES

137

(3.6). Therefore to prove that B0 is obstructed it will suffice to produce a first order deformation ξ ∈ Def B0 (k[]) whose obstruction to lift to k[t]/(t3 ) ¯ → B0 not in the submodule generated by the is represented by an h : R columns of (3.6). We define ξ by the ideal (f1 + ∆f1 , . . . , f6 + ∆f6 ) ⊂ k[, x, y, z, w] where ∆f := (∆f1 , . . . , ∆f6 ) = (0, 0, 0, zw, w2 , −x3 ) this defines a deformation because Rj · ∆f ∈ I for all j = 1, . . . , 8. More precisely: (R1 · ∆f, . . . , R8 · ∆f ) = (0, −wf2 , −f5 , f4 − wf3 , 0, wf3 , f6 , −f5 ) Therefore we see that the obstruction to lift ξ to second order is defined by ¯ → B0 represented by the homomorphism h : R (0, 0, −t20 , t19 , 0, 0, −t21 , −t20 ) Now it is immediate to check that this vector is not in Hom(F¯ , B0 ) and therefore the deformation ξ cannot be lifted: thus B0 is obstructed.

3.1.3

Comparison with deformations of the nonsingular locus

Under certain conditions it is possible to compare the deformations of an affine scheme with the deformations of the open subscheme of its nonsingular points. In this and the following subsections we will describe the analysis made in [176] and [177], with applications to the study of certain quotient singularities. We will need a preliminary lemma. Lemma 3.1.16 Let X be an affine scheme, Z ⊂ X a closed subscheme and G a coherent sheaf on X. Let G∨ = Hom(G, OX ). If depthZ (OX ) ≥ 2 then depthZ (G∨ ) ≥ 2 and therefore H 0 (X, G∨ ) ∼ = H 0 (X\Z, G∨ ) Proof. Consider a presentation 0→R→F →G→0

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where F is a free OX -module. Then we obtain an exact sequence 0 → G∨ → F ∨ → Q → 0

(3.7)

where Q ⊂ R∨ . Since F ∨ is free we have depthZ (F ∨ ) = depthZ (OX ) ≥ 2 and therefore HZ0 (F ∨ ) = 0 = HZ1 (F ∨ ) ([79], Theorem 3.8, p. 44); it follows that HZ0 (G∨ ) = 0. Similarly one proves that HZ0 (R∨ ) = 0, and therefore HZ0 (Q) = 0. From the sequence of local cohomology associated to (3.7) we obtain HZ1 (G∨ ) = 0 and therefore depthZ (G∨ ) ≥ 2 by [79], Theorem 3.8, p. 44. The last assertion follows from the exact sequence 1 0 → H 0 (X, G∨ ) → H 0 (X\Z, G∨ ) → HX (G∨ )

(see [89], p. 212).

qed

Consider an affine scheme X = Spec(B) where B = P/J for a smooth k-algebra P . Let Z = Sing(X) be the singular locus of X and U = X\Z. Let Y = Spec(P ) and consider the exact sequence 0 → TX → TY |X → NX → TX1 → 0

(3.8)

where NX = NX/Y . Since TX1 is supported on Z, by restricting to U we get the exact sequence: 0 → TU → TY |U → NX|U → 0

(3.9)

Proposition 3.1.17 (i) If depthZ (OX ) ≥ 2 (e.g. X is normal of dimension ≥ 2) we have an exact sequence 0 → TB1 → H 1 (U, TU ) → H 1 (U, TY |U ) (ii) If depthZ (OX ) ≥ 3 then TB1 ∼ = H 1 (U, TU ) Proof. (i) We have the local cohomology exact sequences (see [89], p. 212): 0 → H 0 (X, NX ) → H 0 (U, NX|U ) → HZ1 (NX ) 0 → H 0 (X, TY |X ) → H 0 (U, TY |U ) → HZ1 (TY |X )

3.1. AFFINE SCHEMES

139

If depthZ (OX ) ≥ 2 then from Lemma 3.1.16 we deduce that depthZ (NX ) ≥ 2 and depthZ (TY |X ) ≥ 2. Therefore we have HZ1 (NX ) = 0 = HZ1 (TY |X ) ([79], Theorem 3.8, p. 44) and H 0 (X, NX ) ∼ = H 0 (U, NX|U ),

H 0 (X, TY |X ) ∼ = H 0 (U, TY |U )

Comparing the exact cohomology sequences of (3.8) and (3.9) we get an exact and commutative diagram: H 0 (X, TY |X ) → H 0 (X, NX ) → TB1 → 0 k k ∩ H 0 (U, TY |U ) → H 0 (U, NX|U ) → H 1 (U, TU ) → H 1 (U, TY |U ) which proves (i). If depthZ (OX ) ≥ 3 then X is normal and HZ1 (TY |X ) = 0 = HZ2 (TY |X ) because TY |X is locally free; from the local cohomology exact sequence we get H 1 (U, TY |U ) ∼ = H 1 (X, TY |X ) = 0 because X is affine. Using (i) we deduce (ii).

qed

The above proposition can be applied to prove the rigidity of a large class of cones over projective varieties. We will need the following well known lemmas, which we include for the reader’s convenience. Lemma 3.1.18 Let W ⊂ IP r be a projective nonsingular variety, CW the affine cone over W , v ∈ CW the vertex, U = CW \{v} and p : U → W the projection. If G is a coherent sheaf on CW such that G|U = p∗ F for some coherent F 6= (0) on W , then the following conditions are equivalent: (i) depthv (G) ≥ d for some d ≥ 2 (ii) H 0 (CW, G) = ⊕ν∈ZZ H 0 (W, F (ν)) and H k (W, F (ν)) = 0 for all 1 ≤ k ≤ d − 2 and ν ∈ ZZ Proof. We will use the equivalence depthv (G) ≥ d ⇔ Hvk (G) = 0,

k
([79], Theorem 3.8, p. 44). We have an exact local cohomology sequence: 0 → Hv0 (G) → H 0 (CW, G) → H 0 (U, G|U ) → Hv1 (G) → 0

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and isomorphisms: H k−1 (U, G|U ) ∼ = Hvk (G)

k≥2

Since G|U = p∗ F with F 6= (0) we have depthv (G) ≥ 1, thus Hv0 (G) = 0. On the other hand, since p∗ G|U = p∗ p∗ F = ⊕ν∈ZZ F (ν), we have H 0 (U, G|U ) = ⊕ν∈ZZ H 0 (W, F (ν)). Now the conclusion follows. qed Lemma 3.1.19 Let 0 = [0, . . . , 0, 1] ∈ IP r+1 , V = IP r+1 \{0} and let π : V → IP r be the projection. Then TV /IP r = π ∗ O(1) Proof. It is an immediate consequence of the commutative exact diagram 0 ↑ TV 0 → TV /IP r → k ↑ 0 → OV (1) → OV (1)r+2 ↑ O ↑ 0

0 ↑ → π ∗ TIP r ↑ → OV (1)r+1 ↑ O ↑ 0

→0 →0

where the vertical sequences are restrictions of Euler sequences.

qed

Corollary 3.1.20 Let W ⊂ IP r be a projective nonsingular variety of dimension ≥ 2. Assume that (i) H 0 (IP r , O(ν)) → H 0 (W, OW (ν)) surjective for all ν ∈ ZZ (W is projectively normal) (ii) H 1 (W, OW (ν)) = 0 for all ν ∈ ZZ. (iii) H 1 (W, TW (ν)) = 0 for all ν ∈ ZZ. Then the affine cone CW over W is rigid.

3.1. AFFINE SCHEMES

141

Proof. CW has dimension ≥ 3 and hypothesis (i) implies that it is normal ([89], p. 126). Hypothesis (i) and (ii) imply that depthv (OCW ) ≥ 3, by Lemma 3.1.18. Therefore by 3.1.17(ii) it suffices to show that H 1 (U, TU ) = 0 where U = CW \{v}. Let p : U → W be the projection. We have H 1 (U, OU ) = ⊕ν∈ZZ H 1 (W, OW (ν)) = 0 (3.10) 1

∗

1

H (U, p TW ) = ⊕ν∈ZZ H (W, TW (ν)) = 0 by conditions (ii) and (iii). The relative tangent sequence of p takes the following form: 0 → OU → TU → p∗ TW → 0

(3.11)

In fact it follows from Lemma 3.1.19 and from Proposition B.34(i) that we have Tπ−1 (W )/W = π ∗ OW (1) and therefore, since U = π −1 (W )\W , we have TU/W = p∗ OW (1) and this is clearly equal to OU . The conclusion follows from (3.10) and from the cohomology sequence of (3.11). qed

Corollary 3.1.21 (i) The affine cone over IP n ×IP m in its Segre embedding is rigid for every positive n, m such that n + m ≥ 3. (ii) The affine cone over any Veronese embedding of IP n , n ≥ 2, is rigid. (iii) If W ⊂ IP r is a projective nonsingular variety of dimension ≥ 2, such that h1 (W, OW ) = 0 = h1 (W, TW ) then the affine cone over the m-th Veronese embedding W (m) of W is rigid for every m 0. Proof. Using 3.1.20 (i) and (ii) are easy computations and (iii) follows from Serre’s vanishing Theorem ([89], Th. III.5.2). qed The affine cone over the quadric IP 1 × IP 1 ⊂ IP 3 is not rigid because it is a hypersurface (see Proposition 3.1.3, page 124); it does not satisfy both conditions (ii) and (iii) of Corollary 3.1.20. The affine cone over a rational normal curve Γr ⊂ IP r , r ≥ 2, is not rigid either (for r = 2 it is a singular quadric surface in A3 ; for r ≥ 3 see [146], [159], [160]). For a generalization of Corollary 3.1.20 and applications see [198].

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3.1.4

CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

Quotient singularities

The analysis of the previous subsection can be applied to the study of deformations of a class of affine singular schemes obtained as quotients of nonsingular ones by the action of a finite group. Let Y = Spec(P ) be an affine nonsingular algebraic variety on which a finite group G acts. Let X = Y /G be the quotient variety, q : Y → X the projection. Assume that the action is free outside a G-invariant closed subscheme W ⊂ Y . Set V = Y \W . Proposition 3.1.22 Assume that depthW (OY ) ≥ 2. Then TX ∼ = (q∗ TY )G where G acts on TY = Derk (P, P )∼ by D 7→ Dg := gDg −1 for all D ∈ Derk (P, P ) and g ∈ G. Proof. Consider the exact sequence of coherent sheaves on Y : 0 → TY /X → TY → q ∗ TX → TY1/X Ω1Y /X is supported on W since q is etale outside W ; then we have TY /X = 0. Similarly TY1/X is supported on W so that from the above exact sequence restricted to V we deduce an isomorphism H 0 (V, TY ) ∼ = H 0 (V, q ∗ TX ) Then by Lemma 3.1.16 we deduce that H 0 (Y, TY ) ∼ = H 0 (Y, q ∗ TX ) Note that, letting A = P G the ring of invariant elements, we have X = Spec(A) and the above isomorphism is equivalent to an isomorphism Derk (P, P ) ∼ = Derk (A, P ) Therefore it will suffice to show that Derk (A, A) ∼ = Derk (A, P )G

3.1. AFFINE SCHEMES

143

So let D ∈ Derk (A, P ) be such that D = gDg −1 for all g ∈ G. Then for every a ∈ A we have D(a) = g(D(g −1 a)) = g(D(a)) so D(a) ∈ A and therefore Derk (A, P )G ⊂ Derk (A, A). Conversely if D ∈ Derk (A, A) then it defines a k-derivation of A in P which is clearly Ginvariant and we also have Derk (A, A) ⊂ Derk (A, P )G . qed Corollary 3.1.23 Let n be the order of G. Under the same assumptions of 3.1.22 if char(k) does not divide n then TX is a direct summand of q∗ TY . Proof. Define a homomorphism q∗ TY → TX = (q∗ TY )G by D 7→

1 X g D n g∈G

This defines the splitting.

qed

Theorem 3.1.24 In the above situation, if the action is free outside a Ginvariant closed subscheme W of codimension ≥ 3, and char(k) does not divide the order of G then X = Y /G is rigid. Proof. Let Z = q(W ) where q : Y → X is the projection, V = Y \W , U = X\Z = V /G. We have depthZ OX ≥ 2 because X is normal, being the quotient of a nonsingular variety by a finite group (for this elementary fact see e.g. [187], p. 58). Therefore T 1 ⊂ H 1 (U, TU ) ∼ (3.12) = H 1 (U, (q∗ TY )G ) X

where the inclusion follows from 3.1.17(i) and the isomorphism is Proposition 3.1.22. We also have an exact sequence H 1 (Y, TY ) →

2 H 1 (V, TY ) → HW (TY ) k 1 H (U, q∗ TY )

where the left vector space is 0 because Y is affine and the right one is 0 because of the depth assumption on W . It follows that H 1 (U, q∗ TY ) = 0 and therefore H 1 (U, (q∗ TY )G ) = 0 as well because it is a direct summand of it by Corollary 3.1.23. The conclusion now follows from (3.12). qed In the theorem the hypothesis on the codimension of W cannot be removed. In fact all rational two-dimensional double points are quotient singularities and they are hypersurfaces, therefore they are not rigid (see [16]).

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Example 3.1.25 (Kollar[123]) Let n ≥ 3. Consider the weighted projective space X = IP (1, 1, 1, a3 , . . . , an ) = Proj(k[X0 , . . . , Xn ]) where the Xi ’s are indeterminates with weights ai = 1 if i = 0, 1, 2 and ai ≥ 1 if i = 3, . . . , n (see [47] for details on weighted projective spaces). Then X is locally the quotient of an affine space by a finite cyclic group and by the choice of the weights all its singularities have codimension ≥ 3. It follows from Theorem 3.1.24 that TX1 = 0. Therefore the first order deformations of X are classified by H 1 (X, TX ) by the exact sequence (2.17). But the exact sequence on X: M O(ai ) → TX → 0 0 → OX → i 1

implies H (X, TX ) = 0. Therefore X is rigid. Notes and Comments 1. The main references for this section are [174], [132], [15]. 2. ([195]) Let B be an e.f.t. local k-algebra, I ⊂ B an ideal generated by a regular sequence. Prove that if B/I is rigid then B is rigid as well.

3.2 3.2.1

Closed subschemes The local Hilbert functor

Let X ⊂ Y be a closed embedding of algebraic schemes. A cartesian diagram of morphisms of schemes η:

X → X ⊂ Y ×S ↓ ↓π Spec(k) → S

where π is flat, and it is induced by the projection from Y × S, is called a (flat) family of deformations of X in Y parametrized by S, or over S. We call S and X respectively the parameter scheme and the total scheme of the family. When S = Spec(A) with A in ob(A∗ ) (resp. in ob(A), resp. A = k[]) we say that η is a local, resp. infinitesimal, resp. first order family

3.2. CLOSED SUBSCHEMES

145

of deformations of X in Y . We will also say that η is a deformation of X in Y over A. The family is called trivial if X = X × S. X is rigid in Y if every infinitesimal deformation of X in Y is trivial. When considering a family η we speak generally of a family of closed subschemes of Y . Let X ⊂ Y be a closed embedding of algebraic schemes. For each A in ob(A) we let Y HX (A) = {deformations of X in Y over A}

It is immediate to verify that this defines a functor of Artin rings Y HX : A → (sets)

called the local Hilbert functor of X in Y . Proposition 3.2.1 Given a closed embedding of algebraic schemes X ⊂ Y then: Y ¯ H of Theorem (i) the local Hilbert functor HX satisfies conditions H0 , H , H, 2.3.2.

(ii) There is a natural identification Y HX (k[]) = H 0 (X, NX/Y )

where NX/Y is the normal sheaf of X in Y . Y Proof. (i) Obviously HX satisfies condition H0 . Let

A0

A00 &. A

be homomorphisms in A, with A00 → A a small extension. Letting A¯ = A0 ×A A00 we have a commutative diagram with exact rows: 0 → () → A¯ k ↓ 0 → () → A00

→ A0 ↓ → A

Take an element of Y Y HX (A0 ) ×HXY (A) HX (A00 )

→ 0 → 0

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which is represented by a pair of deformations X 0 ⊂ Y × Spec(A0 ) and X 00 ⊂ Y × Spec(A00 ) such that X 0 ×Spec(A0 ) Spec(A) = X 00 ×Spec(A00 ) Spec(A) ⊂ Y × Spec(A) ¯ Consider the sheaf of A-algebras OX 0 ×OX OX 00 on X. Then X¯ := (|X|, OX 0 ×OX ¯ flat over Spec(A) ¯ (see the proof of 2.4.1). OX 00 ) is a scheme over Spec(A), ¯ inducing X 0 and X 00 . We Therefore X¯ is a deformation of X over Spec(A) have a commutative diagram: X ↓ X 00

X0 ↓ ¯ → Y × Spec(A) →

and the universal property of the fibered sum implies that there is a morphism ¯ Φ : X¯ → Y × Spec(A) Pulling back Φ over Spec(A00 ) (resp. Spec(A0 )) we obtain the closed embedding X 00 ⊂ Y × Spec(A00 ) (resp. X 0 ⊂ Y × Spec(A0 )). Since Spec(A00 ) ⊂ ¯ is a closed embedding defined by a square zero ideal, it follows that Spec(A) Φ is a closed embedding as well (details are left to the reader). Therefore Y ¯ ¯ defines an element of HX Φ : X¯ ⊂ Y × Spec(A) (A) which is mapped to 0 00 (X , X ) by the map: Y Y α : HX (A0 ×A A00 ) → hYX (A0 ) ×HXY (A) HX (A00 )

¯ be another element It follows that α is surjective. Now let X˜ ⊂ Y × Spec(A) Y ¯ 0 00 of HX (A) which is mapped to (X , X ). Then by the universal property of the fibered sum there is a morphism τ : X¯ → X˜ which, as in the proof of 2.4.1, is easily seen to be an isomorphism. Moreover the following diagram commutes: ¯ Y × Spec(A) % τ ¯ X −→ X˜ Since the diagonal arrows are closed embeddings it follows that X¯ = X˜ as ¯ This proves that α is actually a bijection closed subschemes of Y × Spec(A). and (i) follows.

3.2. CLOSED SUBSCHEMES

147

(ii) Let I ⊂ OY be the ideal sheaf of X. A first order deformation of X in Y , i.e. a flat family: X → X ⊂ Y × Spec(k[]) ↓ ↓ Spec(k) → Spec(k[])

(3.13)

is defined by a sheaf OX of flat k[]-algebras, with an isomorphism OX ⊗k[] k∼ = OX . The closed embedding X ⊂ Y × Spec(k[]) is determined by a sheaf of ideals I ⊂ OY [] := OY ⊗k k[] such that OX = OY []/I . The above data are obtained by gluing together their restrictions to an affine open cover. On an affine open set U = Spec(P ) ⊂ Y , let X ∩ U = Spec(B) where B = P/J for an ideal J = (f1 , . . . , fN ) ⊂ P . Consider the exact sequence v

f

0 → R −→ P N −→ J → 0 where R is the module of relations among f1 , . . . , fN . Taking HomP (−, B) we obtain the exact sequence: v∨

0 → HomB (J/J 2 , B) → HomP (P N , B) −→ HomP (R, B) which identifies HomB (J/J 2 , B) with ker(v ∨ ). An element of ker(v ∨ ) can be represented as an N -tuple h = (h1 , . . . , hN ) of elements of P which, interpreted as an element of HomP (P N , B) by scalar product (i.e. h(p1 , . . . , pN ) = P j hj pj mod J) must be zero on R. Hence X

hj rj ∈ J for every (r1 , . . . , rN ) ∈ R

j

This means that there exist ∆r1 , . . . , ∆rN ∈ P such that X j

hj rj = −

X

fj ∆rj

j

or, equivalently, such that (f + h) t (r + ∆r) = 0 in P ⊗k k[]. Therefore from Corollary A.32 it follows that f + h generates an ideal in P ⊗k k[] which defines a first order deformation of Spec(B) in

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Spec(P ) because every relation among f1 , . . . , fN extends to a relation among f1 + h1 , . . . , fN + hN . Using the same argument backwards one sees that every first order deformation of Spec(B) in Spec(P ) defines an element of HomB (J/J 2 , B). It follows that at the global level we have a canonical 1-1 correspondence between first order deformations of X in Y and H 0 (X, NX/Y ). qed Corollary 3.2.2 Let X ⊂ Y be a closed embedding of algebraic schemes. Y If h0 (X, NX/Y ) < ∞, for example if X is projective, then HX is prorepresentable. Proof. Follows from Proposition 3.2.1 and from Theorem 2.3.2.

qed

If X ⊂ Y is a closed embedding of projective schemes then the prorepY resentability of HX follows directly from the existence of the Hilbert scheme Y Y ˆ Y Hilb because HX is prorepresented by the complete local ring O Hilb ,[X] (see §4.3). ˆ A formal deformation of X in Y is a sequence Let A¯ be in ob(A). Xn ⊂ Y × Spec(A¯n ) ξn : ↓ Spec(A¯n ) ¯ n+1 of infinitesimal deformations of X in Y over A¯n = A/m such that ξn ¯ A induces ξn−1 by pullback under the natural inclusion Spec(A¯n−1 ) → Spec(A¯n ) ¯ {ξn }) of X in Y for all n ≥ 1. We can describe the formal deformation (A, as a diagram of formal schemes ¯ X X ⊂ Y × Specf(A) ↓ ¯ Specf(A) ¯ {ξn }) of X in As in the case of the functor Def X , a formal deformation (A, d Y Y ¯ ¯ ˆ Y defines an element ξˆ ∈ H X (A), i.e. a formal couple (A, ξ) for HX , and conversely every such element is defined by a formal deformation of X in Y . Considering not necessarily infinitesimal deformations of a closed subscheme one has the notion of characteristic map, as follows.

3.2. CLOSED SUBSCHEMES

149

Definition 3.2.3 Let X ⊂ Y be a closed embedding of algebraic schemes, (S, s) a pointed scheme and let X ⊂ X ⊂ Y ×S ξ: ↓ ↓π s Spec(k) −→ S be a family of deformations of X in Y . By pulling back this family by morphisms Spec(k[]) → S with image s and applying Proposition 3.2.1 we obtain a linear map χξ : Ts S → H 0 (X, NX/Y ) called the characteristic map of the family ξ at s. The characteristic map is the analogous for embedded deformations of the Kodaira-Spencer map for abstract ones. Examples 3.2.4 (i) If X ⊂ IP r , r ≥ 1, is the complete intersection of r − n hypersurfaces f1 , . . . , fr−n of degrees d1 ≤ d2 ≤ . . . ≤ dr−n respectively, we have a presentation 2 ^

f

[⊕j O(−dj )] −→ ⊕j O(−dj ) → IX → 0

(3.14)

Taking Hom(−, OX ) we obtain an exact sequence: ˇ f

0 → NX → ⊕j OX (dj ) −→

2 ^

[⊕j OX (dj )]

where NX = NX/IP r . Since each nonzero entry of the matrix defining f is one of the fj ’s, the map ˇf is zero. Therefore NX ∼ = ⊕j OX (dj ) Equivalently one can remark that (3.14) induces a surjective homomorphism 2 ⊕j OX (−dj ) → IX /IX

of locally free sheaves of the same rank, which must therefore be an isomorphism. In particular if X is a hypersurface of degree d we have NX ∼ = OX (d) and therefore ! d+r 0 h (X, NX ) = −1 r

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confirming the fact that X can be deformed in IP r only inside the linear system of hypersurfaces of degree d, which is a projective space of dimension d+r − 1. r If X is a linear subspace, i.e. d1 = · · · = dr−n = 1, then NX ∼ = OX (1)⊕r−n and therefore h0 (X, NX ) = (r − n)(n + 1) as expected, since such linear subspaces X are parametrized by the grassmannian G(n + 1, r + 1), which is nonsingular of dimension (r − n)(n + 1). (ii) Let X be a Cartier divisor on a connected projective scheme Y . Consider the exact sequence 0 → OY → OY (X) → NX/Y → 0 and the cohomology sequence: χ

0 → H 0 (Y, OY ) → H 0 (Y, OY (X)) −→ H 0 (X, NX/Y ) → H 1 (Y, OY ) (3.15) Classically the map χ was called the characteristic map of the linear system |X|. By definition Im(χ) ∼ = H 0 (Y, OY (X))/H 0 (Y, OY ) is naturally identified with the tangent space to the linear system |X|. We can verify that this is so by identifying Im(χ) as a subvector space of first order deformations of X in Y , as follows. Assume that X is defined by a system of local equations {fi }, fi ∈ Γ(Ui , OY ) not 0-divisor, with respect to an affine cover {Ui } of Y . We have fij := fi fj−1 ∈ Γ(Ui ∩ Uj , OY∗ ) for all i, j, and {fij } is a Cech 1-cocycle which defines the line bundle OY (X). A first order deformation of X in Y is a Cartier divisor X ⊂ Y × Spec(k[]) which is determined by a system {Fi = fi + gi }, gi ∈ Γ(Ui , OY ), such that there exist Fij = fij + gij ∈ Γ(Ui ∩ Uj , OY∗ ×Spec(k[]) ) (hence gij ∈ Γ(Ui ∩ Uj , OY )) satisfying Fi = Fij Fj for all i, j. Therefore on Ui ∩ Uj we have: fi + gi = (fij + gij )(fj + gj ) which is equivalent to the identity: gi = fij gj + gij fj This identity shows that the system {¯ gi = gi mod IX } defines a section of NX/Y , as expected. We also see that {¯ gi } ∈ Im(χ) if and only if gi = fij gj ,

3.2. CLOSED SUBSCHEMES

151

in which case Fij = fij , i.e. X deforms inside the linear system |X|, as asserted. From (3.15) we see that if H 1 (Y, OY ) 6= (0) then there are first order deformations of X in Y which are not contained in the linear system |X|. (iii) In IP 2 × IP 1 with bihomogeneous coordinates (u, v, w; x, y) consider the hypersurface Σm , m ≥ 0, defined by the equation: xm v − y m u = 0 One can show that Σm ∼ = Fm , and that the structure morphism π : Fm → IP 1 is induced by the projection IP 2 × IP 1 → IP 1 . Therefore the surface Σm is a realization of the rational ruled surface Fm as the total scheme of a family of lines of IP 2 parametrized by IP 1 . When m > 0 this family is non-trivial because Fm ∼ 6= IP 1 × IP 1 (for details see [5]). Examples 3.2.5 The following set of examples deals with properties of projective curves. Proofs are straightforward and are left to the reader. (i) If Y is a projective scheme and C ⊂ Y is a projective integral l.c.i. curve, then the normal sheaf NC/Y is torsion free. If C is nonsingular then NC/Y is locally free. (ii) Consider a nonsingular curve C ⊂ IP 3 and a (possibly singular) surface S ⊂ IP 3 of degree n containing C. Prove that there is an exact sequence of locally free sheaves on C: ψ

0 → a−1 ⊗ KC (−n + 4) → NC/IP 3 −→ OC (n) → [OC /a](n) → 0

(3.16)

where a ⊂ OC is the ideal sheaf generated by the restriction to C of the partial derivatives ∂F ∂F ,..., ∂X0 ∂X3 where F = 0 is an equation of S. (Hint: Im(ψ) = OC ⊗ Im(TIP 3 |S → NS/IP 3 )). In case S is nonsingular we obtain the sequence: 0 → KC (−n + 4) → NC → OC (n) → 0

(3.17)

Deduce from (3.16) that if a 6= OC (i.e. if C ∩ Sing(S) 6= ∅) then C is not regularly embedded in S (yet the normal sheaf NC/S is locally free).

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(iii) Consider a nonsingular curve C ⊂ IP 3 and a point p ∈ IP 3 \C. Then there is an exact sequence 0 → OC (1) → NC/IP 3 → ωC (3) → 0 which is obtained as a special case of (3.16) by taking as S the cone projecting C from p. Deduce that h1 (C, NC/IP 3 ) ≤ h1 (C, OC (1)) (iv) Let C ⊂ IP r , r ≥ 4, be a nonsingular irreducible projective curve. Let p1 , . . . , pk ∈ IP r , 1 ≤ k ≤ r − 3, be general points and let π : C → IP r−k be the projection with center the linear span hp1 , . . . , pk i. Prove that there is an exact sequence 0 → OC (1)k → NC/IP r → π ∗ Nπ(C)/IP r−k → 0

(3.18)

which splits if and only if C is contained in a hyperplane. Deduce that h1 (C, NC/IP r ) ≤ (r − 2)h1 (C, OC (1))

(3.19)

In particular H 1 (C, NC/IP r ) = (0) if h1 (C, OC (1)) = 0. (Solution: see [183], Prop. 11.2). Inequality (3.19) was already known classically as a bound on the dimension of the Hilbert scheme ([190], s. 8). For a sharpening of (3.19) see [48]. (v) Let C ⊂ IP r be a nonsingular irreducible projective curve and let L = OC (1). Show that if the Petri map µ0 (L) (see Example 3.3.9(ii)) is injective then H 1 (C, NC/IP r ) = (0). (vi) Let C ⊂ IP r be a projective irreducible nonsingular curve of degree d and genus g. Prove that χ(NC/IP r ) := h0 (C, NC/IP r ) − h1 (C, NC/IP r ) = (r + 1)d + (r − 3)(1 − g) In particular for a canonical curve of genus r + 1 ≥ 3 and degree 2r in IP r we have χ(NC/IP r ) = h0 (C, NC/IP r ) = r(r + 5) because h1 (C, NC/IP r ) = 0 since the Petri map µ0 (ωC ) is injective (see (v) above).

3.2. CLOSED SUBSCHEMES

3.2.2

153

Obstructions

Let X ⊂ Y be a closed embedding, A in ob(A) and let ξ:

X → X ⊂ Y × Spec(A) ↓ ↓f Spec(k) → Spec(A)

be a deformation of X in Y over A. Let e : 0 → k → A˜ → A → 0 ˜ be a small extension. A lifting of ξ to A˜ is a deformation of X in Y over A: ξ˜ :

˜ X → X˜ ⊂ Y × Spec(A) ↓ ↓f ˜ Spec(k) → Spec(A)

whose pullback to Spec(A) is ξ. Proposition 3.2.6 Let X ⊂ Y be a regular closed embedding (Definition D.1) of algebraic schemes with X projective. Then H 1 (X, NX/Y ) is an obY struction space for the local Hilbert functor HX . Proof. Given ξ:

X → X ⊂ Y × Spec(A) ↓ ↓f Spec(k) → Spec(A)

where A is in ob(A), an infinitesimal deformation of X in Y , we will show that there is a natural linear map oξ/Y : Exk (A, k) → H 1 (X, NX/Y ) such that, for every extension e : 0 → k → A˜ → A → 0 ˜ this will prove we have oξ/Y (e) = 0 if and only if ξ has a lifting to A; the proposition. Since X is regularly embedded in Y we can find an affine

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open cover U = {Ui }i∈I of Y such that Xi := X ∩ Ui is a complete intersection in Ui for each i. Let Ui = Spec(Pi ), Xi = Spec(Pi /Ii ) where Ii = (fi1 , . . . , fiN ) with {fi1 , . . . , fiN } a regular sequence in Pi . We then have X|Ui = Spec(PiA /IiA ) where IiA = (Fi1 , . . . , FiN ) ⊂ PiA := Pi ⊗ A and fiα = Fiα mod mA , α = 1, . . . , N . Choose arbitrarily F˜i1 , . . . , F˜iN ∈ PiA˜ such that Fiα = F˜iα mod . By Example A.33 {Fi1 , . . . , FiN } and {F˜i1 , . . . , F˜iN } are regular sequences in PiA and in PiA˜ respectively; in particular, letting IiA˜ = (F˜i1 , . . . , F˜iN ), ˜ X˜i := Spec(PiA˜ /IiA˜ ) ⊂ Ui × Spec(A) ˜ In order to find a lifting of X to A˜ is a lifting of X|Ui ⊂ Ui × Spec(A) to A. we must be able to choose the F˜iα ’s in such a way that ˜ X˜i|Uij = X˜j|Uij ⊂ Uij × Spec(A)

(3.20)

for each i, j ∈ I. Letting Uij = Spec(Pij ) and viewing the fiα ’s and fjα ’s as elements of Pij via the natural maps Pi

Pj &

. Pij

we have F˜jα − F˜iα =: hijα and hij := (hij1 , . . . , hijN ) ∈ Γ(Uij , NX/Y ), because NX/Y is locally free of rank N and is trivial on each Ui . By construction {hij } ∈ Z 1 (U, NX/Y ). The condition (3.20) means that we can choose the F˜iα ’s so that hij = 0 all i, j. A different choice of the F˜iα ’s is of the form F˜iα +hiα and hi := (hi1 , . . . , hiN ) ∈ Γ(Ui , NX/Y ). Since we have (F˜jα + hjα ) − (F˜iα + hiα ) = (hijα + hjα − hiα )

(3.21)

we see that {hij } defines an element oξ/Y (e) ∈ H 1 (X, NX/Y ) which is zero if ˜ and only if the X˜i ’s satisfy condition (3.20) and define a lifting X˜ of X to A. It is easy to show that the map oξ is k-linear. qed

3.2. CLOSED SUBSCHEMES

155

The element oξ/Y (e) ∈ H 1 (X, NX/Y ) is called the obstruction to lift ξ to ˜ we call ξ obstructed if oξ/Y (e) 6= 0 for some e ∈ Exk (A, k); otherwise it A; is unobstructed. X is said to be unobstructed in Y if all its infinitesimal deformations in Y are unobstructed; otherwise X is said to be obstructed in Y . Examples of obstructed closed subschemes are usually quite subtle, especially if one is interested in nonsingular obstructed subvarieties. In order to be able to describe them in a natural way it is necessary to know the existence of the Hilbert scheme of a projective scheme. We will give examples in §4.6. Corollary 3.2.7 Let j : X ⊂ Y be a regular closed embedding of algebraic schemes with X projective and let (R, {ξn }) be the formal universal deformation of X in Y . Then (i) h0 (X, NX/Y ) ≥ dim(R) ≥ h0 (X, NX/Y ) − h1 (X, NX/Y ) The first equality holds if and only if X is unobstructed in Y . (ii) X is rigid in Y if and only if H 0 (X, NX/Y ) = 0. (iii) If H 1 (X, NX/Y ) = (0) then X is unobstructed in Y . The proof is left to the reader. In the case of a closed embedding X ⊂ Y which is not regular Proposition 3.2.6 says nothing about the obstruction Y space of HX . We refer the reader to §4.4 and §4.6 for some information about the general case. Examples 3.2.8 (i) Let C be a projective nonsingular curve contained in a nonsingular surface S, and assume that C is negatively embedded in S, i.e. deg(OC (C)) < 0. Then H 0 (C, OC (C)) = 0 and therefore C is rigid in S. This happens in particular when C ∼ = IP 1 is an exceptional curve of the first kind. Another example is when C has genus g ≥ 2, S = C × C and C is identified to the diagonal ∆ ⊂ S. In this case NC/S = TC which has degree 2 − 2g < 0. Note that H 1 (C, TC ) 6= (0) but C is unobstructed in S, being rigid in S. This example shows that the sufficient condition of Corollary 3.2.7 is not necessary. (ii) Hypersurfaces of IP r are unobstructed. In fact, if X ⊂ IP r has degree d then h1 (X, NX/IP r ) = h1 (X, OX (d)) = 0

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More generally complete intersections in IP r are unobstructed (see Subsection 4.6.1). (iii) Let Q ⊂ IP 3 be a quadric cone with vertex v, and L ⊂ Q a line. Then we have an inclusion NL/Q ⊂ NL/IP 3 = OL (1) ⊕ OL (1) whose cokernel is OL (2)(−v) (see Example 3.2.5(ii)). It follows that NL/Q = OL (1), in particular it is locally trivial and H 1 (L, NL/Q ) = 0, despite the fact that L ⊂ Q is not a regular embedding (Example 3.2.5(ii)) and L is obstructed in Q (see Note 1 of §4.6). (iv) Let C ⊂ IP r be a nonsingular irreducible projective curve and let L = OC (1). If the Petri map µ0 (L) (see Example 3.3.9(ii)) is injective then C is unobstructed in IP r because h1 (C, NC/IP r ) = 0 (Example 3.2.5(v)). In particular canonical curves of genus g ≥ 3 are unobstructed in IP g−1 (Example 3.2.5(vi)).

3.2.3

The forgetful morphism

Let X ⊂ Y be a closed embedding of algebraic schemes. The forgetful morphism Y Φ : HX → Def X is the morphism which associates to an infinitesimal deformation of X in Y : X → X ⊂ Y × Spec(A) ξ: ↓ ↓ Spec(k) → Spec(A) the isomorphism class of the deformation of X: X → X ↓ ↓ Spec(k) → Spec(A) Proposition 3.2.9 Assume that X and Y are nonsingular and that X is projective. Consider the exact sequence 0 → TX → TY |X → NX/Y → 0 Then

(3.22)

3.2. CLOSED SUBSCHEMES

157

(i) dΦ = δ : H 0 (X, NX/Y ) → H 1 (X, TX ) is the coboundary map coming from (3.22). (ii) The coboundary map δ1 : H 1 (X, NX/Y ) → H 2 (X, TX ) arising from the exact sequence (3.22) is an obstruction map for Φ (see Definition 2.3.5). (iii) If H 1 (X, TY |X ) = 0 then Φ is smooth. (iv) If X is unobstructed in Y and δ is surjective then Φ is smooth. In particular X is unobstructed as an abstract variety and has rk(δ) number of moduli. Proof. (i) Let ξ:

X ⊂ X ⊂ Y × Spec(k[]) ↓ ↓π s Spec(k) −→ Spec(k[])

be a first order deformation of X in Y . We must show that δχ(ξ) = κ(ξ). Let χ(ξ) = h ∈ H 0 (X, NX/Y ). Consider an affine open cover U = {Ui = Spec(Pi )} of Y , and let Xi = X ∩ Ui = Spec(Pi /(fi1 , . . . , fiN )). We have Xi := X|Ui = Spec(P []/(fi1 + hi1 , . . . , fiN + hiN )) Then (hi1 , . . . , hiN ) =: hi = h|Ui ∈ Γ(Ui , NX/Y ). Since Xi is affine and nonsingular the abstract deformation Xi of Xi is trivial: thus there exist isomorphisms θi : Xi × Spec(k[]) → Xi and κ(ξ) ∈ H 1 (X, TX ) is defined by the 1-cocycle {dij } ∈ Z 1 (U, TX ) corresponding to the system of automorphisms θij = θi−1 θj : Xij × Spec(k[]) → Xij × Spec(k[]) where Xij = X ∩ Uij . Let’s compute δ(h). The isomorphism θi is given by an isomorphism of k[]-algebras ti : Pi []/(fi1 + hi1 , . . . , fiN + hiN ) → Pi []/(fi1 , . . . , fiN ) which, by the smoothness of Pi , is induced by a k[]-automorphism Ti : Pi [] → Pi []

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

of the form Ti (p + q) = p + (q + di (p)) where di ∈ Derk (Pi , Pi ) = Γ(Ui , TY ) is such that di (hiα ) = −fiα . We have {di } ∈ C 0 (U, TY ) and δ(h) is defined by {(dj − di )|Xi }. Since (dj − di )|Xi = dij we conclude that δ(h) = κ(ξ). (ii) One must show that, given an infinitesimal deformation ξ:

X ⊂ X ⊂ Y × Spec(A) ↓ ↓π s Spec(k) −→ Spec(A)

where A is in A, we have a commutative diagram Exk (A, k) . oξ/Y H 1 (X, NX/Y )

& oξ δ1

H 2 (X, TX )

−→

The proof of this fact is similar to the proof of part (i) and will be omitted. (iii) The exact sequence (3.22) shows that the hypothesis implies dΦ : H 0 (X, NX/Y ) → H 1 (X, TX )

surjective and

δ1 : H 1 (X, NX/Y ) → H 2 (X, TX )

injective

Therefore the assertion follows from Proposition 2.3.6. (iv) is left to the reader.

qed

Remark 3.2.10 Let X ⊂ Y be a regular embedding of algebraic schemes with X reduced and Y nonsingular, I ⊂ OY the ideal sheaf of X, and let Y Φ : HX → Def X be the forgetful morphism. The differential dΦ : H 0 (X, NX/Y ) = HomOX (I/I 2 , OX ) → Ext1OX (Ω1X , OX ) is the k-linear map which associates to σ : I/I 2 → OX the pushout σ∗ (S) where S : 0 → I/I 2 → Ω1Y |X → Ω1X → 0 is the conormal sequence of X ⊂ Y . This generalizes Proposition 3.2.9. The proof consists in considering, for a first order deformation of X in Y X ⊂ X ⊂ Y × Spec(k[])

3.2. CLOSED SUBSCHEMES

159

the induced diagram of conormal sequences 0 → I/I 2 ⊕ OX ↓ 0→ OX

→ Ω1Y |X ⊕ OX ↓ 1 → ΩX |X

→ Ω1X k → Ω1X

→0 →0

and in recognizing the second row as the pushout of the first. Part (iii) of the proposition often gives a very effective way of proving that a given X ⊂ Y is unobstructed as an abstract variety. If X is a curve in IP r the vanishing of H 1 (X, TIP r |X ) is related with the Petri map (see Example 3.3.9(ii)). The following examples are further applications of this principle. Examples 3.2.11 (i) ([120]) Let X ⊂ IP r , r ≥ 3, be a nonsingular hypersurface of degree d ≥ 2. Then h1 (NX/IP r ) = h1 (OX (d)) = 0 and therefore X is unobstructed in IP r . On the other hand from the exact sequence: 0 → TIP r (−d) → TIP r → TIP r |X → 0 and the Euler sequence: 0 → OIP r → OIP r (1)r+1 → TIP r → 0 we deduce that: h1 (TIP r |X ) = h2 (TIP r (−d)) = 0 if r ≥ 4 while for r = 3 we have the exact sequence: 0 ← H 2 (TIP 3 (−d))∨ k H 1 (TIP 3 |X )∨

← H 0 (OIP 3 (d − 4)) ← H 0 (OIP 3 (d − 5))4

Therefore we see that h1 (TIP r |X ) =

1 if r = 3 and d = 4; 0 otherwise r

IP → Def X is smooth and X is unobFrom 3.2.9 we therefore deduce that HX structed as an abstract variety unless r = 3 and d = 4 (this is precisely the

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case when X is a K3 surface). An analogous result holds more generally for complete intersections ([181]). Using (3.22) one computes easily that H 2 (TX ) 6= 0 if X is a nonsingular surface of degree d ≥ 5 in IP 3 : therefore the unobstructedness of X could not have been deduced from 2.4.6 in this case. One can generalize to singular hypersurfaces as follows. Consider a reduced hypersurface X ⊂ IP r , r ≥ 3, of degree d ≥ 2. Then the conormal sequence is 0 → OX (−d) → Ω1IP r |X → Ω1X → 0 so that H 1 (X, NX/IP r ) = 0 and we have the exact sequence dΦ

H 0 (X, NX/IP r ) −→ Ext1OX (Ω1X , OX ) → Ext1 (Ω1IP r |X , OX ) k k 0 1 H (X, OX (d)) H (X, TIP r |X ) where the equality on the right is because Ω1IP r |X is locally free. Therefore Y as before we see that Φ : HX → Def X is smooth and X is unobstructed if (r, d) 6= (3, 4). (ii) The previous example can be easily generalized to nonsingular hypersurfaces of IP n × IP m , 1 ≤ n ≤ m, n + m ≥ 3. Let IP n × IP m ↓p IP n

q

−→ IP m

be the projections. Consider a nonsingular hypersurface X ⊂ IP n × IP m of bidegree (a, b), i.e. defined by an equation σ = 0 for some σ ∈ H 0 (O(a, b)), where O(a, b) := p∗ O(a) ⊗ q ∗ O(b) From the exact sequence 0 → O → O(a, b) → NX/IP n ×IP m → 0 one deduces that H 1 (NX/IP n ×IP m ) = (0) and therefore X is unobstructed in IP n × IP m . For any coherent sheaf F on IP n × IP m we use the notation F(α, β) = F ⊗ O(α, β)

3.2. CLOSED SUBSCHEMES

161

Using the fact that TIP n ×IP m = p∗ TIP n ⊕ q ∗ TIP m and the Leray spectral sequence with respect to any one of the projections, one easily computes that hi (TIP n ×IP m (α, β)) = 0 when n+m ≥ 4, i = 1, 2 and (α, β) arbitrary. Moreover, when (n, m) = (1, 2) one finds: hi (TIP 1 ×IP 2 ) =0 h2 (TIP 1 ×IP 2 (−a, −b)) = 0

all i ≥ 1 unless (a, b) = (2, 3)

Putting all these informations together and using the exact sequence 0 → TIP n ×IP m (−a, −b) → TIP n ×IP m → TIP n ×IP m |X → 0 one deduces that h1 (TIP n ×IP m |X ) = 0 unless (n, m) = (1, 2) and (a, b) = (2, 3) (this is precisely the case when X is a K3 surface). Now as before we conclude that the forgetful morphism IP n ×IP m HX → Def X is smooth and X is unobstructed as an abstract variety.

3.2.4

The local relative Hilbert functor

Given a projective morphism p : X → S of schemes and a k-rational point s ∈ S, consider the fibre X (s) and a closed subscheme Z ⊂ X (s). For each A in ob(A) an infinitesimal deformation of Z in X relative to p parametrized by A is a commutative diagram: Z

⊂ &

XA → X ↓ ↓p s Spec(A) −→ S

where the right square is cartesian, the left diagonal morphism is flat and its closed fibre is Z; this means in particular that the morphism s has image {s} and therefore that A is an OS,s -algebra. Then, letting Λ = OS,s , we can define the local relative Hilbert functor X /S

HZ

: AΛ → (sets)

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by X /S

HZ

(A) =

infinitesimal deformations of Z in X relative to p parametrized by A

We have the following generalization of Corollaries 3.2.2 and 2.4.7: Theorem 3.2.12 Let p : X → S be a projective morphism of schemes, s ∈ S a k-rational point, and Z ⊂ X (s) a closed subscheme of the fibre X (s). Denote by Λ = OS,s . Then X /S

(i) the local relative Hilbert functor HZ : AΛ → (sets) is prorepresentable and has tangent space H 0 (Z, NZ/X (s) ). (ii) If Z is regularly embedded in X (s) and p is flat then H 1 (Z, NZ/X (s) ) is X /S an obstruction space for HZ , and we have an exact sequence: 0 → H 0 (Z, NZ/X (s) ) → tR → Ts S → H 1 (Z, NZ/X (s) ) X /S

where R is the local Λ-algebra prorepresenting HZ

(3.23)

.

Proof. (i) The proof of 3.2.1 can be followed almost verbatim showing X /S ¯ H and that H 0 (Z, NZ/X (s) ) is its that HZ satisfies conditions H0 , H , H, tangent space. (ii) The proof of 2.4.6 can be easily adapted to this case. The exact sequence (3.23) follows from the above and from (2.2). qed Another generalization of the local Hilbert functors can be obtained with no extra effort. Consider a projective scheme X and a formal deformation of X π ¯:X X → Specf(R) ˆ and π where R is in ob(A) ¯ is a flat projective morphism of formal schemes; let Z ⊂ X be a closed subscheme. For each A in ob(A) define an infinitesimal deformation of Z in X X relative to π ¯ parametrized by A as a commutative diagram: Z ⊂ X XA → X X & ↓ ↓π ¯ s Spec(A) −→ Specf(R) where the right square is cartesian, the left diagonal morphism is flat and its closed fibre is Z. Note that Spec(A) = Specf(A) and the morphism s

3.3. INVERTIBLE SHEAVES

163

is defined by a surjective homomorphism R → A, so that X XA is just an ordinary scheme projective and flat over Spec(A). We can define the local relative Hilbert functor X X /Specf(R)

HZ

: AR → (sets)

as above. A result analogous to 3.2.12 can be proved in this case as well with a similar proof. Details of this straightforward generalization are left to the reader.

3.3 3.3.1

Invertible sheaves The local Picard functors

The local Picard functors are related to the Picard schemes in the same way as the local Hilbert functors are to the Hilbert schemes, in the sense that if the Picard scheme of a given scheme exists then the local Picard functors describe its infinitesimal and local properties. In this section we will prove the basic properties of the local Picard functors. The Picard schemes will not be treated here. For a full treatment the reader is referred to Kleiman [111]. Let X be a scheme. Recall that the Picard group of X is defined to be the group Pic(X) of isomorphism classes of invertible sheaves on X; we have the ∗ identification Pic(X) = H 1 (X, OX ) (see [89]). We denote by [L] ∈ Pic(X) the class of an invertible sheaf L. For every A in ob(A) we will write for brevity XA instead of X × Spec(A). For every morphism A → B in A, and for every invertible sheaf L on XA we denote by L ⊗A B the invertible sheaf it induces on XB by pullback. Similarly, given λ ∈ Pic(XA ) we denote by λ ⊗A B ∈ Pic(XB ) its image under the pullback operation. This defines a homomorphism of group ∗ ∗ Pic(XA ) = H 1 (XA , OX ) → H 1 (XB , OX ) = Pic(XB ) A B

which makes A 7→ Pic(XA ) a covariant functor on A with values in the category of abelian groups. We fix an element λ0 ∈ Pic(X) once and for all and, for each A in ob(A), we let Pλ0 (A) := {λ ∈ Pic(XA ) : λ ⊗A k = λ0 }

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Then Pλ0 (A) is a subset of Pic(XA ) whose definition is functorial in A so that we have a functor of Artin rings: Pλ0 : A → (sets) which will be called the local Picard functor of (X, λ0 ). Let L be an invertible sheaf on X such that λ0 = [L]. For a given A ∈ ob(A) the elements of Pλ0 (A) are the isomorphism classes of infinitesimal deformations of L over A: an infinitesimal deformation of L over A is an invertible sheaf L on X ×Spec(A) such that L = L|X×Spec(k) . In case A = k[] we speak of a first order deformation of L. The main result about this functor is the following. Theorem 3.3.1 Let X be an algebraic scheme, λ0 ∈ Pic(X). Assume that the following conditions are satisfied: (a) H 0 (X, OX ) ∼ =k (b) dimk H 1 (X, OX ) < ∞ Then Pλ0 is prorepresentable and Pλ0 (k[]) = H 1 (X, OX ). Moreover H 2 (X, OX ) is an obstruction space for Pλ0 . Proof. Let’s check the conditions of Theorem 2.3.2. It is clear that P = Pλ0 satisfies condition H0 . Let A0

A00 &

. A

be a diagram in A with A00 → A surjective. Consider an element (λ0 , λ00 ) ∈ P (A0 ) ×P (A) P (A00 ) and let λ = λ0 ⊗A0 A = λ00 ⊗A00 A. Let L0 , L00 , L be invertible sheaves on XA0 , XA00 , XA respectively such that [L0 ] = λ0 , [L00 ] = λ00 , [L] = λ. Then we have homomorphisms L0 → L and L00 → L of sheaves on |X| inducing isomorphisms L0 ⊗A0 A ∼ = L, L00 ⊗A00 A ∼ = L. Let B = A0 ×A A00 . Claim: OX ∼ = OX 0 ×O OX 00 B

A

XA

A

3.3. INVERTIBLE SHEAVES

165

For every open set U ⊂ |X| we have by definition: [OXA0 ×OXA OXA00 ](U ) = OXA0 (U ) ×OXA (U ) OXA00 (U ) and by the universal property of the fibered sum we have a homomorphism φ : OXB → OXA0 ×OXA OXA00 which is induced by the homomorphisms OXB → OXA0 and OXB → OXA00 coming from B → A0 and B → A00 respectively. Since A00 → A is surjective φ ⊗B A0 : OXB ⊗B A0 → OXA0 is an isomorphism. From Lemma A.25 it follows that φ is an isomorphism. From the claim we deduce that N := L0 ×L L00 is an invertible sheaf on XB and the projections induce isomorphisms N ⊗B A0 ∼ = L0 , N ⊗B A00 ∼ = L00 . Therefore P (B) 3 [N ] 7→ (λ0 , λ00 ) ∈ P (A0 ) ×P (A) P (A00 ) This shows that the map α : P (B) → P (A0 ) ×P (A) P (A00 ) is surjective. Assume that M is an invertible sheaf such that α([M ]) = α([N ]). This means that there are homomorphisms M → L0 and M → L00 inducing isomorphisms M ⊗B A0 ∼ = L0 , M ⊗B A00 ∼ = L00 . It follows that we have an automorphism θ : L → L given by the composition: L∼ = L0 ⊗A0 A ∼ = M ⊗B A ∼ = L00 ⊗A00 A ∼ =L which makes the following diagram commutative: M .q

0

& q 00

L0

L00 & u0

. u00 θ

L −→ L By hypothesis (a) the isomorphism θ is the multiplication by a unit a ∈ A. Since A00 → A is surjective we can lift a to a00 ∈ A00 and we can change q 00 to

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a00 q 00 thus assuming that u0 q 0 = u00 q 00 . It follows that we have a commutative diagram M . & L0 L00 & . L ∼ and this implies that M = N . This shows that α is bijective. Therefore P satisfies also conditions H and H. We have an exact sequence exp

∗ ∗ 0 → OX −→ OX → OX →0 k[]

where exp(f ) = 1 + f . It follows that ∗ ∗ P (k[]) = ker[H 1 (Xk[] , OX ) → H 1 (X, OX )] = H 1 (X, OX ) k[]

Finally, given A in ob(A) and [e] ∈ Exk (A, k) represented by an extension 0 → (t) → A˜ → A → 0 we have an exact sequence: ∗ ∗ 0 → tOX → OX → OX →0 ˜ A A

which induces an exact sequence δ

∗ ∗ H 1 (XA˜ , OX ) → H 1 (XA , OX ) −→ H 2 (X, OX ) ˜ A A

˜ ∈ P (A) ˜ if and only if δ(λ) = 0 ∈ Given λ ∈ P (A) it can be lifted to a λ 2 2 H (X, OX ). This shows that H (X, OX ) is an obstruction space for P . qed Corollary 3.3.2 Let X be a projective integral scheme. Then Pλ0 is prorepresentable for every λ0 ∈ Pic(X). Proof. A projective integral scheme satisfies both conditions (a) and (b) of the theorem. qed Remark 3.3.3 Let X be an algebraic scheme and λ0 ∈ Pic(X). Then the tangent and obstruction spaces of the functor Pλ0 , as described by Theorem 3.3.1, depend only on X and not on λ0 . This is because, given any λ0 , µ0 ∈ Pic(X), there is a canonical isomorphism of functors Pλ0 ∼ = Pµ0 . We leave to the reader the easy proof of this fact.

3.3. INVERTIBLE SHEAVES

3.3.2

167

Deformations of sections, I

Let X be a projective integral scheme, and let L be an invertible sheaf on X. One can define a homomorphism of sheaves: m : OX → H 0 (X, L)∨ ⊗ L

(3.24)

as follows. For every open set U ⊂ X m(U ) : O(U ) → H 0 (X, L)∨ ⊗ Γ(U, L) f

7→

[s 7→ f s|U ]

for every s ∈ H 0 (X, L). The induced maps on global sections are just given by cup product: mi : H i (X, OX ) → Hom(H 0 (X, L), H i (X, L)) a 7→ [s 7→ a ∪ s] If L is base point free and ϕL : X → IP := IP (H 0 (X, L)∨ ) is the morphism defined by the sections of L, then it is easy to check that the homomorphism (3.24) is the same as the one appearing in the pulled back Euler sequence of IP : 0 → OX → H 0 (X, L)∨ ⊗ L → ϕ∗L TIP → 0 We leave this to the reader. The linear maps between cohomology groups induced by these sheaf homomorphisms have deformation theoretic interpretations which we will now explain. Consider a deformation L of L over A ∈ ob(A): we have an induced restriction map ρL : H 0 (XA , L) → H 0 (X, L) We say that a section σ ∈ H 0 (X, L) extends to L if σ ∈ Im(ρL ). Proposition 3.3.4 Let La be a first order deformation of L, corresponding to an element a ∈ H 1 (X, OX ). A section s ∈ H 0 (X, L) extends to La if and only if a ∪ s = 0 ∈ H 1 (X, L)

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

equivalently if and only if s ∈ ker[m1 (a)] where m1 : H 1 (X, OX ) → Hom(H 0 (X, L), H 1 (X, L)) is the map induced by (3.24). Proof. Let U = {Uα } be an affine open covering of X such that L is ∗ represented by a system of transition functions {fαβ }, fαβ ∈ Γ(Uαβ , OX ). Then the first order deformation La of L can be represented, in the same covering {Uα } of X × Spec(k[]), by transition functions: ∗ f˜αβ ∈ Γ(Uαβ , OX×Spec(k[]) )

such that f˜αβ f˜βγ = f˜αγ

(3.25)

and which restrict to the fαβ ’s modulo . ∗ ∗ Since OX×Spec(k[]) = OX + OX we can write f˜αβ = fαβ + gαβ

(3.26)

for suitable gαβ ∈ Γ(Uαβ , OX ). Identity (3.25) gives gαβ gβγ gαγ + = fαβ fβγ fαγ g ˇ and the system ( fαβ ) is a Cech 1-cocycle which defines the element a ∈ αβ H 1 (X, OX ). Let’s assume that s ∈ H 0 (X, L) is represented by the cocycle (sα ), sα ∈ Γ(Uα , OX ), such that sα = fαβ sβ on Uαβ . For s to extend to a section

s˜ ∈ H 0 (X × Spec(k[]), La ) it is necessary and sufficient that there exist tα ∈ Γ(Uα , OX ) such that sα + tα = f˜αβ (sβ + tβ ) on Uαβ . After replacing the f˜αβ ’s by the expressions (3.26) we obtain the identities sα + tα = (fαβ + gαβ )(sβ + tβ )

3.3. INVERTIBLE SHEAVES

169

which are equivalent to: gαβ sβ = tα − fαβ tβ These identities can be also written as: gαβ sα = tα − fαβ tβ fαβ and they mean exactly that the 1-cocycle (

gαβ sα ) ∈ Z 1 (U, L) fαβ

is a coboundary, i.e. that a ∪ s = 0.

qed

Corollary 3.3.5 In the above situation (i) If a ∈ ker(m1 ) then all the sections of L extend to La . (ii) For every a ∈ H 1 (X, OX ), at least max{0, h0 (X, L) − h1 (X, L)} linearly independent sections of L extend to La . Proof. Immediate.

qed

Let’s fix a section s ∈ H 0 (X, L) and let D = div(s) ⊂ X be the divisor of X s. Consider the local Hilbert functor HD . We have a morphism of functors: X aD : HD → P[L]

associating to a deformation of D in X over A in ob(A), given by an effective Cartier divisor D ⊂ XA , the element [OXA (D)] ∈ P[L] (A). aD is called the Abel-Jacobi morphism of D ⊂ X. Consider the exact sequence s

0 → OX −→ L → LD → 0 It induces linear maps: δ0 : H 0 (D, LD ) → H 1 (X, OX ) δ1 : H 1 (D, LD ) → H 2 (X, OX ) Noting that LD = ND/X we have:

(3.27)

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

Proposition 3.3.6 In the above situation δ0 is the differential of aD and δ1 is an obstruction map for aD . Proof. We keep the notations of the proof of Proposition 3.3.4 and therefore we assume D defined by local equations sα = 0 where (sα ) is a 0-cocycle defining s with respect to the covering U; in particular sα = fαβ sβ . A first order deformation D ⊂ X × Spec(k[]) of D is defined by local equations sα + tα = 0 which satisfy the cocycle conditions: sα + tα = (fαβ + gαβ )(sβ + tβ ) These conditions can be also written as: tα = fαβ tβ +

gαβ sα fαβ

(3.28)

They mean that (t¯α = tα mod sα ) define a section t¯ ∈ H 0 (D, LD ) which is the one corresponding to the first order deformation D of D. From the identities (3.28) we also see that g ). Since this cocycle represents δ0 (t¯) is represented by the 1-cocycle ( fαβ αβ 1 ¯ aD (D) ∈ H (X, OX ) this proves that δ0 (t) = daD (t¯). The proof that δ1 is an obstruction map is similar and will be left to the reader. qed Corollary 3.3.7 Under the assumptions of Proposition 3.3.6: (i) If H 1 (X, L) = 0 then aD is smooth. (ii) If the natural map H 1 (X, L) → H 1 (D, LD ) X is zero then HD is less obstructed than P[L] . If moreover P[L] is smooth X then HD is smooth.

Proof. (i) If H 1 (X, L) = 0 then (3.27) implies that δ0 is surjective and δ1 is injective. Therefore aD is smooth by Proposition 2.3.6. The proof of (ii) is similar using Proposition 2.3.6 again. qed

3.3. INVERTIBLE SHEAVES

171

Remarks 3.3.8 (i) Assume X to be projective and nonsingular. A Cartier divisor D on X is called semiregular if the natural map H 1 (X, OX (D)) → H 1 (D, OD (D)) is zero, i.e. if condition (ii) of the corollary is satisfied. Part (ii) of the corollary can thus be rephrased by saying that D is unobstructed in X if it is semiregular and P[L] is smooth. The smoothness of P[L] is known to be true if char(k) = 0, by a general theorem of Cartier: we thus recover a celebrated theorem of Severi-Kodaira-Spencer claiming the unobstructedness of semiregular divisors on a projective nonsingular complex variety X. These matters are covered in full detail in [144]. The original sources are [193] and [121]. For further developments and applications of the notion of semiregularity see [22], [166] and [24]. (ii) In view of Proposition 3.3.6 the cohomology sequence of (3.27) can be interpreted as a sequence of tangent spaces and differentials as follows: 0→

H 0 (X,L) hσi

δ

0 → H 0 (D, LD ) −→ H 1 (X, OX )

k 0→

TD |D|

k →

X HD (k[])

k da

D −→

P[L] (k[])

0

(X,L) with the tangent space to the where we used the identification of H hσi linear system |D| at D proved in Example 3.2.4(ii).

Example 3.3.9 Let X be a Gorenstein curve (e.g. a local complete intersection curve). Then the map m1 is dual to the map: µ0 (L) : H 0 (X, L) ⊗ H 0 (X, ωX L−1 ) → H 0 (X, ωX )

(3.29)

given by multiplication of global sections (ωX denotes the dualizing sheaf of X). µ0 (L) is called the Petri map of L. More generally, if V ⊂ H 0 (X, L) is a vector space of sections of L, one can consider the multiplication map: µ0 (V ) : V ⊗ H 0 (X, ωX L−1 ) → H 0 (X, ωX ) which is called the Petri map of V . From Proposition 3.3.4 it follows that coker(µ0 (V ))⊥ ⊂ H 1 (X, OX ) is the space of first order deformations of L

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

to which all sections of V extend. In particular, if h0 (X, L) = r + 1 and deg(L) = d then coker(µ0 (L))⊥ is the tangent space at L to the scheme Wdr (X) of linear systems of degree d and dimension ≥ r on X (see [9] for details).

3.3.3

Deformations of pairs (X, L)

Let X be a nonsingular projective algebraic variety and let d : OX → Ω1X be the canonical derivation. We can define a homomorphism of sheaves of abelian groups ∗ OX → Ω1X by the rule du u ∗ for all open sets U ⊂ X and u ∈ Γ(U, OX ). We have an induced group homomorphism: ∗ c : H 1 (X, OX ) → H 1 (X, Ω1X ) u 7→

To simplify the notation we write c(L) instead of c([L]) for a given invertible sheaf L on X. If k = C then c(L) is, up to a multiplicative constant, the Chern class of L (see [142], p. 127). Since Ω1X is locally free we have an identification H 1 (X, Ω1X ) = Ext1OX (TX , OX ) so that we can associate to c(L) an extension 0 → OX → EL → TX → 0

(3.30)

defined up to isomorphism, called the Atiyah extension of L. The sheaf EL is locally free of rank dim(X) + 1 and PL := EL∨ ⊗OX L is called the sheaf of (first order) principal parts of L. Let U = {Uα } be an affine open covering of X such that L is represented ∗ by a system of transition functions {fαβ }, fαβ ∈ Γ(Uαβ , OX ). Then c(L) is ˇ represented by the Cech 1-cocycle df

αβ

fαβ

∈ Z 1 (U, Ω1X )

3.3. INVERTIBLE SHEAVES

173

The sheaf EL|Uα is isomorphic to OUα ⊕ TX|Uα . A section (aα , dα ) of OUα ⊕ TX|Uα and a section (aβ , dβ ) of OUβ ⊕ TX|Uβ are identified on Uαβ if and only d (f ) if dα = dβ and aβ − aα = αfαβαβ . Remark 3.3.10 We have c(L ⊗ M ) = c(L) + c(M ), in particular c(Ln ) = nc(L) for any n ∈ ZZ. Therefore the Atiyah extension of Ln is a constant multiple of the extension (3.30). This means in particular that if n 6= 0 then ELn ∼ = EL . Consider for example X = IP := IP (V ) for some finite dimensional kvector space V . Then one can easily compute that the Euler sequence 0 → OIP → V ⊗ OIP (1) → TIP → 0 is the Atiyah extension of OIP (1). Therefore EL ∼ = EO(1) = V ⊗ OIP (1) for every non-trivial line bundle L on IP . Let A be in ob(A). An infinitesimal deformation of the pair (X, L) over A consists of a pair (ξ, L) (also denoted (X , L)), where X → X ξ: ↓ ↓ Spec(k) → Spec(A) is an infinitesimal deformation of X over A and L is an invertible sheaf on X such that L = L|X . One can also say that L is a deformation of L along ξ. In case A = k[] we speak of a first order deformation of (X, L). Two deformations (X , L) and (X 0 , L0 ) of (X, L) over A will be called isomorphic if there is an isomorphism of deformations f : X → X 0 and an isomorphism L → f ∗ L0 . By letting n

o

Def (X,L) (A) = deformations of (X, L) over A /isomorphism we define a functor of Artin rings Def (X,L) : A → (sets) called the functor of infinitesimal deformations of the pair (X, L). We will denote by [X , L] ∈ Def (X,L) (A) the isomorphism class of a deformation (X , L) of (X, L) over A ∈ ob(A).

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

Theorem 3.3.11 Let (X, L) be a pair consisting of a nonsingular projective algebraic variety X and an invertible sheaf L on X. Then: (i) The functor Def (X,L) has a semiuniversal formal element. (ii) there is a canonical isomorphism Def (X,L) (k[]) =

{1-st order deformations of (X, L)} ∼ 1 = H (X, EL ) isomorphism

and H 2 (X, EL ) is an obstruction space for Def (X,L) . (iii) Given a first order deformation ξ of X, there is a first order deformation of L along ξ if and only if κ(ξ) · c(L) = 0 where “·” denotes the composition: ∪

H 1 (X, TX ) × H 1 (X, Ω1X ) −→ H 2 (X, TX ⊗ Ω1X ) → H 2 (X, OX ) of the cup product of cohomology classes ∪ with the map induced by the duality pairing TX ⊗ Ω1X → OX (therefore the left hand side is an element of H 2 (X, OX )). Proof. Let’s check the conditions of Theorem 2.3.2. It is clear that Def (X,L) satisfies condition H0 . Let A0

A00 &

. A

be a diagram in A with A00 → A surjective. Consider an element ([X 0 , L0 ], [X 00 , L00 ]) ∈ Def (X,L) (A0 ) ×Def (X,L) (A) Def (X,L) (A00 ) Then there is an [X , L] ∈ Def (X,L) (A) and a diagram of deformations: X0

X 00 -f

↓ Spec(A0 )

0

%f X ↓

-

↓ Spec(A00 ) %

Spec(A)

00

3.3. INVERTIBLE SHEAVES

175

where the morphisms f 0 and f 00 induce isomorphisms of deformations: X 0 ×Spec(A0 ) Spec(A) ∼ =X ∼ = X 00 ×Spec(A00 ) Spec(A) Moreover we have isomorphisms f 0∗ L0 ∼ =L∼ = f 00∗ L00 Let B = A0 ×A A00 . Then, as in the proof of Theorem 2.4.1, one sees that X¯ := (|X|, OX 0 ×OX OX 00 ) is a deformation of X over B inducing the pair ([X 0 ], [X 00 ]). Define L¯ := L0 ×L L00 Then L¯ is an invertible sheaf over X¯ which restricts on X 0 to L0 and on X 00 ¯ defines an element of Def (X,L) (B) such that to L00 . Therefore the pair (X¯ , L) ¯ 7→ ([X 0 , L0 ], [X 00 , L00 ]) [X¯ , L]

(3.31)

under the map Def (X,L) (B) → Def(X,L) (A0 ) ×Def (X,L) (A) Def (X,L) (A00 ) ¯ of Theorem 2.3.2. This proves that Def (X,L) satisfies condition H In order to show that Def (X,L) satisfies condition H we must prove that ¯ constructed above is the unique one satisfying (3.31) if we the element [X¯ , L] assume that A00 = k[]. From the proof of Theorem 2.4.1 we know that [X¯ ] ∈ Def X (B) is the unique element inducing ([X 0 ], [X 00 ]) ∈ Def X (A0 ) ×Def X (A) Def X (A00 ). Using this fact now the proof can be completed along the lines of the proof of Theorem 3.3.1. We still need to verify that Def (X,L) satisfies condition Hf . This will result as a consequence of part (ii) to be proved next, because X is projective. (ii) Let (ξ, L) be a first order deformation of (X, L), where X → X ξ: ↓ ↓ Spec(k) → Spec(k[]) Let U = {Uα } be an affine open covering such that L is given by a system ∗ of transition functions fαβ ∈ Z 1 (U, OX ) and κ(ξ) ∈ H 1 (X, TX ) is given by a

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

ˇ Cech 1-cocycle (dαβ ) ∈ Z 1 (U, TX ). Let θαβ = 1 + dαβ be the automorphism of Uαβ × Spec(k[]) corresponding to dαβ . The invertible sheaf L can be given by a system of transition functions (Fαβ ) ∈ Z 1 (U, OX∗ ) which reduces to {fαβ } mod . Therefore it can be represented on Uαβ × Spec(k[]) as gαβ ∈ Γ(Uαβ , OX )

Fαβ = fαβ + gαβ ,

and the cocycle condition translates into Fαβ θαβ (Fβγ ) = Fαγ equivalently: Fαβ (Fβγ + dαβ Fβγ ) = Fαγ which means: (fαβ + gαβ )[fβγ + gβγ + dαβ (fβγ + gβγ ))] = fαγ + gαγ After dividing by fαγ and equating the coefficients of we obtain: gαγ dαβ fβγ gαβ gβγ + − + =0 fαβ fβγ fαγ fβγ

(3.32)

g

This identity means that the data {( fαβ , dαβ )} define an element of Z 1 (U, EL ), αβ and that conversely such an element defines a first order deformation of (X, L). This proves that Def (X,L) (k[]) ∼ = H 1 (X, EL ) modulo verifying that this correspondence is independent from the choice of the covering U and of the cocycles representing (X, L); we leave this to the reader. Consider a small extension e : 0 → (t) → A˜ → A → 0 in A and let (ξ, L) be an infinitesimal deformation of (X, L) over A, where ξ:

X → X ↓ ↓ Spec(k) → Spec(A)

3.3. INVERTIBLE SHEAVES

177

Let U = {Uα } be an affine open cover of X; let θα : Uα × Spec(A) → X|Uα be isomorphisms so that θαβ := θα−1 θβ is an automorphism of the trivial deformation Uαβ × Spec(A). We may assume that L is given by a system of transition functions (Fαβ ), where each Fαβ is a nowhere zero function on Uαβ × Spec(A), such that Fαβ θαβ (Fβγ ) = Fαγ ˜ L) ˜ of (ξ, L) to Spec(A) ˜ exists we choose In order to see if a lifting (ξ, arbitrarily a collection {θ˜αβ , F˜αβ } where, for each α, β, γ: ˜ which (a) θ˜αβ is an automorphism of the product family Uαβ × Spec(A) restricts to θαβ on Uαβ × Spec(A). ˜ which restricts to Fαβ (b) F˜αβ is a nowhere zero function on Uαβ × Spec(A) on Uαβ × Spec(A). Such collection exists by Lemma 1.2.8. Because of (a) we have −1 θ˜αβ θ˜βγ θ˜αγ = id + tdαβγ

where “id” here means the identity of Γ(Uαβγ , OX ), and (dαβγ ) ∈ Z 2 (U, TX ) ˜ (see is a 2-cocycle which represents the obstruction to lift ξ over Spec(A) the proof of Proposition 1.2.12). Because of (b), for each α, β, γ there is gαβγ ∈ Γ(Uαβγ , OX ) such that −1 F˜αβ θ˜αβ (F˜βγ )F˜αγ = 1 + tgαβγ

Therefore we have:

−1 θ˜αβ [F˜βγ θ˜βγ (F˜γδ )F˜βδ ] −1 −1 ˜ ˜ −1 ˜ ˜ −1 −1 ][Fαβ θαβ (F˜βγ )F˜αγ ] = [F˜αγ θ˜αγ (F˜γδ )F˜αδ ] [Fαβ θαβ (F˜βδ )F˜αδ

= 1 + t(gβγδ − gαγδ + gαβδ − gαβγ )

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

On the other hand the left side can be written as dαβγ (fγδ ) θ˜αβ θ˜βγ (F˜γδ )[θ˜αγ (F˜γδ )]−1 = t fγδ Therefore we see that (gαβγ , dαβγ ) ∈ Z 2 (U, EL )

(3.33)

We are free to modify our choice by replacing the θ˜αβ ’s and the F˜αβ ’s by Φαβ = θ˜αβ + tdαβ ,

Gαβ = F˜αβ + tgαβ

(3.34)

for some dαβ ∈ Γ(Uαβ , TX ) and gαβ ∈ Γ(Uαβ , OX ). One checks easily that the cocycle (3.33) is replaced by the cohomologous one: (gαβγ + gαβ − gαγ + gβγ , dαβγ + dαβ − dαγ + dβγ )

(3.35)

˜ L) ˜ exists if and only if the data (3.34) can and it is clear that the lifting (ξ, be determined so that (3.35) is zero. Therefore associating to the extension e the cohomology class o(ξ,L) (e) ∈ H 2 (X, EL ) defined by the cocycle (3.33) we have defined an obstruction map Exk (A, k) → H 2 (X, EL ) which we leave to the reader to verify to be linear. This makes H 2 (X, EL ) an obstruction space for the functor Def (X,L) . (iii) Observing that d

αβ fβγ

fβγ

∈ Z 2 (U, OX )

represents κ(ξ) · c(L), the identity (3.32) expresses the condition that this 2-cocycle is a coboundary, and proves (iii). qed If X is a connected nonsingular projective curve then H 2 (X, OX ) = 0 so that every line bundle can be deformed along any first order deformation of X. Moreover H 2 (X, EL ) = 0 for any L, thus Def (X,L) is smooth and from the exact sequence (3.30) it follows that h1 (X, EL ) = 4g − 3

3.3. INVERTIBLE SHEAVES

179

if X has genus g ≥ 2. For higher dimensional varieties the situation is more complicated in general. For example if X is a K3-surface then the cup product H 1 (X, TX ) × H 1 (X, Ω1X ) → H 2 (X, OX ) ∼ =k coincides with Serre duality. Therefore ·c(L)

H 1 (X, TX ) −→ H 2 (X, OX ) is surjective for every nontrivial line bundle L. This means that L deforms along a 19-dimensional subspace of H 1 (X, TX ), because h1 (X, TX ) = 20 (see Example 2.4.11(ii), page 85). We have a natural forgetful morphism Φ : Def (X,L) → Def X defined in the obvious way. The differential and the obstruction map of this morphism are described as follows. Proposition 3.3.12 In the situation of Theorem 3.3.11: (i) The differential dΦ : Def (X,L) (k[]) → Def X (k[]) coincides with the linear map H 1 (X, EL ) → H 1 (X, TX ) coming from the exact sequence (3.30). (ii) The map H 2 (X, EL ) → H 2 (X, TX ) coming from the exact sequence (3.30) is an obstruction map for Φ. (iii) If H 2 (X, OX ) = 0 the morphism Φ is smooth. Proof. (i) and (ii) are left to the reader. For (iii) use Proposition 2.3.6. qed

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

Example 3.3.13 Assume k = C. Let A = V /Λ be an abelian variety of dimension g, represented as the quotient of a g-dimensional complex vector space V by a lattice Λ ⊂ V . Then, letting Ω = V ∨ we have Ω1A = Ω ⊗ OA

TA = V ⊗ OA , Moreover

¯ := HomC¯ (V, C) H 1 (A, OA ) = Ω is the space of C-antilinear forms on V and H 2 (A, OA ) =

2 ^

¯ Ω

(see [21], Theorem 1.4.1). Therefore ¯ H 1 (A, TA ) = V ⊗ H 1 (A, OA ) = V ⊗ Ω In particular h1 (A, TA ) = g 2 . We also have: ¯ H 1 (A, Ω1A ) = Ω ⊗ H 1 (A, OA ) = Ω ⊗ Ω which can be identified with the space of hermitian forms on V . Let L be an ample invertible sheaf on X. Then ¯ c(L) ∈ H 1 (A, Ω1A ) = Ω ⊗ Ω is identified with a positive definite hermitian form hL : V × V → C ([21], §4.1). It follows that the map ·c(L)

H 1 (A, TA ) −→ H 2 (A, OA ) of Theorem 3.3.11 is just the composition of the map ¯ → ¯ ⊗Ω ¯ V ⊗Ω Ω v ⊗ ` 7→ hL (v, −) ⊗ `

(3.36)

(3.37)

with the canonical surjection ¯ ⊗Ω ¯→ Ω

2 ^

¯ Ω

Since hL is positive definite the map (3.37) is an isomorphism and it follows that the map (3.36) is surjective. The conclusion is that L deforms along a subspace of H 1 (A, TA ) of dimension g g(g + 1) = g2 − 2 2

!

For the related case of jacobians see Example 3.4.24(iii), page 217.

3.3. INVERTIBLE SHEAVES

3.3.4

181

Deformations of sections, II

Consider again a projective nonsingular variety X and an invertible sheaf L on X. Assume as above that L is given, in an affine open cover U = {Uα }, ∗ by transition functions (fαβ ) ∈ Z 1 (U, OX ). We can define a homomorphism of sheaves M : EL → H 0 (X, L)∨ ⊗ L in the following way. Consider a section η ∈ Γ(U, EL ), where U ⊂ X is an open set; it is given by a system (aα , dα ) where aα ∈ Γ(U ∩ Uα , OX ), dα ∈ Γ(U ∩ Uα , TX ), subject to the conditions that dβ = dα and aβ − aα = dα (fαβ )/fαβ on U ∩ Uα ∩ Uβ . Then, for every s = (sα ) ∈ H 0 (X, L) we let M (η)(sα ) = aα sα + dα (sα ) On U ∩ Uα ∩ Uβ we find: fαβ M (η)(sβ ) = fαβ (aβ sβ + dβ (sβ ) = aβ sα + fαβ dβ (sβ ) = = sα (aα + dα (fαβ )/fαβ ) + fαβ dβ (sβ ) = sα aα + dα (fαβ )sβ + fαβ dβ (sβ ) = = sα aα + dα (fαβ sβ ) = sα aα + dα (sα ) = M (η)(sα ) Therefore the functions M (η)(sα ) ∈ Γ(U ∩Uα , OX ) patch together to define a section M (η)(s) ∈ Γ(U, L). This defines M . It is obvious from the definition that we have a commutative diagram: m : OX → H 0 (X, L)∨ ⊗ L ↓ k 0 M : EL → H (X, L)∨ ⊗ L

(3.38)

namely M extends m (see 3.24, page 167). The linear map M1 : H 1 (X, EL ) → Hom(H 0 (X, L), H 1 (X, L)) induced by M can be explicitely described as follows. Let η1 ∈ H 1 (X, EL ) be ˇ represented by the Cech cocycle (aαβ , dαβ ) ∈ Z 1 (U, EL ). Then M1 (η1 ) : H 0 (X, L) → (sα )

H 1 (X, L)

7→ (aαβ sα + dαβ sα )

The map M1 has the following deformation theoretic interpretation. Let A ∈ ob(A) and let (X , L) be an infinitesimal deformation of (X, L) over Spec(A). Then we say that a section s ∈ H 0 (X, L) extends to L if s ∈ Im[H 0 (X , L) → H 0 (X, L)]

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

Proposition 3.3.14 Let X be a projective nonsingular variety, L a line bundle on X, and (X , L) a first order deformation of the pair (X, L) defined by a cohomology class η1 ∈ H 1 (X, EL ) according to Theorem 3.3.11(ii). Then a section s ∈ H 0 (X, L) extends to L if and only if s ∈ ker(M1 (η1 )). Proof. With respect to an affine cover U = {Uα } of X we assume L rep∗ resented by (fαβ ) ∈ Z 1 (U, OX ) and η1 by (aαβ , dαβ ) ∈ Z 1 (U, EL ). Then, according to the description given in the proof of Theorem 3.3.11, X is determined by glueing the products Uα × Spec(k[]) along the open subsets Uαβ × Spec(k[]) by means of the automorphisms θαβ = 1 + dαβ , and L is determined by transition functions of the form Fαβ = fαβ (1+aαβ ) satisfying the patching identities on Uαβγ × Spec(k[]): Fαβ θαβ (Fβγ ) = Fαγ The condition that s = (sα ) ∈ H 0 (X, L) extends to L is equivalent to the existence of functions tα ∈ Γ(Uα , OX ) such that θαβ [Fαβ (sβ + tβ )] = sα + tα for all α, β. After replacing the expressions for Fαβ and θαβ we obtain the identities: fαβ (1 + aαβ )(sβ + tβ ) + dαβ (fαβ sβ ) = sα + tα By equating the coefficients of on both sides we obtain: aαβ sα + dαβ (sα ) = tα − fαβ tβ and this means exactly that M1 (η1 )(s) = 0.

qed

Corollary 3.3.15 In the above situation (i) If η1 ∈ ker(M1 ) then all the sections of L extend to the first order deformation of (X, L) corresponding to η1 . (ii) For any η1 ∈ H 1 (X, EL ) at least max{0, h0 (X, L) − h1 (X, L)} linearly independent sections of L extend to the first order deformation of (X, L) corresponding to η1 .

3.3. INVERTIBLE SHEAVES Proof. Immediate.

183 qed

This statement, as well as Proposition 3.3.14, show the deformation theoretic interest of the map M1 . It can be useful to have a closer picture of the relation between the maps m and M . Assuming that L is base point free and denoting by ϕL : X → IP := IP (H 0 (X, L)∨ ) the morphism defined by the sections of L, we have a commutative and exact diagram, which extends (3.38):

0 ↓ 0 → OX ↓ 0 → EL ↓ TX ↓ 0

0 ↓ TX ↓ m −→ H 0 (X, L)∨ ⊗ L → ϕ∗L TIP → 0 k ↓ M 0 ∨ −→ H (X, L) ⊗ L → NϕL → 0 ↓ 0

(3.39)

The first column is the Atiyah extension. The sheaf NϕL is the normal sheaf of the morphism ϕL . It will be considered more sistematically in §3.4, page 193. In case ϕL is a closed embedding NϕL = NϕL (X)/IP and this diagram shows that coker(m1 ) ⊂ H 1 (X, ϕ∗ TIP ) where we recall that m1 : H 1 (X, OX ) → Hom(H 0 (X, L), H 1 (X, L)) is the map induced by m (see Proposition 3.3.4, page 167). Therefore, according to Proposition 3.2.9(iii), h2 (X, OX ) = 0 and the surjectivity of m1 are sufficient conditions for the smoothness of the forgetful morphism Φ : HϕIPL (X) → Def X . In particular if X is a nonsingular curve and m1 is surjective then Φ is smooth (see also Example 3.2.5(v)). Example 3.3.16 Let X be a connected projective nonsingular curve of genus g. Then the map M1 is dual to PL : H 0 (X, L) ⊗ H 0 (X, ωX L−1 ) → H 0 (X, ωX ⊗ EL∨ )

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

More generally for any vector subspace V ⊂ H 0 (X, L) we can consider the map, obtained by restricting PL : PV : V ⊗ H 0 (X, ωX L−1 ) → H 0 (X, ωX ⊗ EL∨ ) It follows directly from Proposition 3.3.14 that coker(PV )⊥ ⊂ H 1 (X, EL ) is the space of first order deformations of (X, L) to which all sections of V extend. PL (resp. PV ) is called the extended Petri map of L (resp. of V ). After dualizing the cohomology diagram of (3.39) we deduce the following commutative diagram which relates the Petri map and the extended Petri map in case L is globally generated:

0→

0 ↓ H 1 (NϕL )∨ ↓

0 ↓ 0 2 H (ωX ) ↓ PL → H 0 (L) ⊗ H 0 (ωX L−1 ) −→ H 0 (ωX ⊗ EL∨ ) k ↓ µ0 (L)

0 → H 1 (ϕ∗L TIP )∨ → H 0 (L) ⊗ H 0 (ωX L−1 ) −→

H 0 (ωX )

(3.40) If ϕL : X ⊂ IP is an embedding then this diagram contains some significant IP informations on the relations between HX and Def X . From what has been remarked just before this example it follows that the injectivity of µ0 (L), being equivalent to the surjectivity of its dual m1 , is a sufficient condition for IP the smoothness of the forgetful morphism Φ : HX → Def X . Diagram (3.40) also shows that we have an exact sequence µ1 (L)

2 0 → H 1 (NX/IP )∨ → H 1 (TIP |X )∨ −→ H 0 (ωX ) k k ker(PL ) ker(µ0 (L))

thus the injectivity of PL is implied by the injectivity of µ0 (L) or more generally by the injectivity of µ1 (L). For more about this topic we refer the reader to [8]. Notes and Comments 1. The coboundary maps δk in the cohomology sequence of the Atiyah extension (3.30) are induced by cup product with c(L), since the extension is defined

3.4. MORPHISMS

185

by c(L). In particular 3.3.11(iii) just says that L deforms along ξ if and only if κ(ξ) ∈ ker[δ1 : H 1 (X, TX ) → H 2 (X, OX )], which is obvious in view of (i) and (ii). 2. The content of Theorem 3.3.11(iii) is outlined in the Appendix by Mumford to Chapter V of [212]. See also [96]. This result is related with the notion of deformation of a polarization (see [147]). 3. For a discussion of Example 3.3.13 for abelian varieties defined over a field of positive characteristic see Oort [153], remark on page 226. 4. The Petri map has been considered classically in [157] and it has reappeared in the modern literature for the first time in [10]. It is of great importance in the study of the Brill-Noether loci on a curve and of their relation with moduli (see [9] for more details).

3.4

Morphisms

In this section we study deformations of a morphism between algebraic schemes. In the analytic case the corresponding theory has been developed by Horikawa in [92], [93], [95] and [96]. For a treatment in the analytic case we refer to [150]. Related work in the algebraic case is in [127] and [165]. Definition 3.4.1 Let f : X → Y be a morphism of algebraic schemes and let A be in ob(A) (resp. A = k[], resp. A in ob(A∗ )). An infinitesimal (resp. first order, resp. local) family of deformations of f parametrized by A or by S (shortly a deformation of f over A or over S ) is a cartesian diagram X

→

↓f Y

X ↓F

→

↓

Y

(3.41)

↓ψ s

Spec(k) −→ S where S = Spec(A), and ψ and ψF are flat (“cartesian diagram” in this case means that the horizontal morphisms induce an isomorphism of the left column with the pullback of the right column by s). If we replace S by a pointed scheme (S, s) we will call (3.41) a family of deformations of f .

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

Essentially a deformation of f consists of a morphism F between deformations of X and of Y and an assigned identification with f of the restriction of F to the closed fibre. Note that if Y = Spec(k) then a family of deformations of f : X → Spec(k) is just a family of deformations of X in the sense of the definition given at the beginning of §1.2. The notion of trivial deformation of f can be given in a obvious way, as well as the notion of rigid morphism. Given an infinitesimal deformation (3.41) and a small extension

e:

0 → (t) → A˜ → A → 0

a lifting of (3.41) to A˜ is a cartesian diagram

X

→

↓f Y

X

→

↓ F˜

↓F →

↓

Y ↓ψ

X˜

→

Y˜ ↓ ψ˜

s

˜ Spec(k) −→ Spec(A) → Spec(A)

(i.e. the horizontal morphisms induce an isomorphisms of each column with the pullback of the column on the right by the lowest horizontal morphisms) where F˜ and ψ˜F˜ are flat. Of course such a lifting defines in particular a ˜ deformation of f over A. Definition 3.4.1 gives the most general notion of deformation of a morphism. It can be modified in several ways so to obtain more restricted notions each having independent interest and applications. We will study some of them starting from the simplest ones. In each situation we will consider a corresponding functor of Artin rings which classifies deformations modulo an appropriate equivalence relation.

3.4. MORPHISMS

3.4.1

187

Deformations of a morphism leaving domain and target fixed

If in Definition 3.4.1 we impose that both X and Y are the product families, i.e. if we consider a cartesian diagram of the form: X

→ X ×S

↓f

↓F

Y

→ Y ×S

↓

↓ψ

Spec(k) →

S

where S = Spec(A) and ψ is the projection, then we obtain the notion of deformation of f with fixed domain and target (shortly wfdat). Deformations of f wfdat can be interpreted as deformations of the graph of f in X × Y so that the methods introduced in §3.2 apply. Precisely let’s define a functor of Artin rings by setting n

o

Def X/f /Y (A) = deformations of f over A wfdat for all A in ob(A). Then we have the following:

Proposition 3.4.2 Let f : X → Y be a morphism of algebraic schemes, with X projective and reduced and Y nonsingular. Then (i) there is a natural isomorphism of functors Def X/f /Y ∼ = HΓX×Y f where Γf ⊂ X × Y is the graph of f . In particular Def X/f /Y is prorepresentable. (ii) We have a natural isomorphism of vector spaces: Def X/f /Y (k[]) → H 0 (X, f ∗ TY ) (iii) H 1 (X, f ∗ TY ) is an obstruction space for the functor Def X/f /Y .

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

Proof. (i) Let A be in ob(A) and let F : X × Spec(A) → Y × Spec(A) be a deformation of f . Its graph ΓF ⊂ X × Y × Spec(A) defines a deformation of Γf ⊂ X × Y over A because ΓF ∼ = X × Spec(A) and the projection ΓF → Spec(A) equals the composition ΓF ∼ = X × Spec(A) → Spec(A) in particular it is flat. Therefore we can define Def X/f /Y (A) → HΓX×Y (A) f by F 7→ (ΓF ⊂ X × Y × Spec(A)) This is an isomomorphism of functors. In fact given a deformation ΓA ⊂ X × Y × Spec(A) of ΓF in X × Y the projection X × Y × Spec(A) → X × Spec(A) induces a morphism ΓA → X × Spec(A) which is an isomorphism of deformations. Therefore the composition X × Spec(A) ∼ = ΓA ⊂ X × Y × Spec(A) → Y × Spec(A) can be identified with a deformation of f . This defines the inverse of the morphism of functors of the statement. (ii) We have natural isomorphisms of vector spaces: Def X/f /Y (k[]) ∼ (k[]) ∼ = HΓX×Y = H 0 (Γ, NΓ/X×Y ) f Since the projection p : X × Y → X is smooth and the composition pj : Γ → X is an isomorphism, from Proposition D.5 it follows that j is a regular embedding. Therefore applying Proposition D.4 we obtain the exact sequence: 0 → IΓ /IΓ2 → j ∗ Ω1X×Y → Ω1Γ → 0 On the other hand we have the exact sequence: 0 → f ∗ Ω1Y → j ∗ Ω1X×Y → (pj)∗ Ω1X → 0 obtained by restricting to Γ the sequence 0 → q ∗ Ω1Y → Ω1X×Y → p∗ Ω1X → 0

3.4. MORPHISMS

189

(where q : X × Y → Y is the second projection). Since (pj)∗ Ω1X ∼ = Ω1Γ , comparing the two sequences we deduce that f ∗ Ω1Y ∼ = IΓ /IΓ2 . Therefore H 0 (Γ, NΓ/X×Y ) = Hom(IΓ /IΓ2 , OΓ ) ∼ = Hom(f ∗ Ω1Y , OX ) = H 0 (X, f ∗ TY ) and (ii) follows. Similarly H 1 (Γ, NΓ/X×Y ) = H 1 (X, f ∗ TY ) and (iii) follows as well.

qed

The notions of obstructed (resp. unobstructed) deformation, and of obstructed (resp. unobstructed) morphism can be given in the usual way. One can give the notion of rigid morphism wfdat in an obvious way. We leave to the reader the task of proving that, under the hypothesis of Proposition 3.4.2, H 0 (X, f ∗ TY ) = 0 implies that f is rigid wfdat. Let f : X → Y be a morphism of algebraic schemes with X reduced and Y nonsingular; let (3.41) be a family of deformations of f wfdat parametrized by a pointed scheme (S, s). To every tangent vector t ∈ TS,s , viewed as a morphism Spec(k[]) → S with image {s} we can associate the pullback of the family F over Spec(k[]) by t and, using the correspondence of 3.4.2(i), we obtain an element of H 0 (X, f ∗ TY ) associated to s. This defines a natural linear map TS,s → H 0 (X, f ∗ TY ) which will be called the characteristic map of the family (3.41) wfdat. Next corollary follows immediately from the proposition. It can also be deduced as a consequence of Lemma 1.2.6. Corollary 3.4.3 If X is a nonsingular projective scheme, then the space of first order deformations of the identity X → X is H 0 (X, TX ). Examples 3.4.4 (i) Let f : X → Y be a nonconstant morphism of projective nonsingular connected curves, with g(Y ) ≥ 2. Then deg(TY ) < 0 and therefore h0 (X, f ∗ TY ) = 0. Thus f is rigid as a morphism wfdat. (ii) A morphism f from a scheme X to a projective space is given by a a linear system on X: deformations of f wfdat can thus be interpreted as “deformations of linear systems” in an appropriate way. We briefly discuss the case of curves. Let X be a projective irreducible and nonsingular curve, f : X → IP r a morphism and let L = f ∗ OIP r (1), deg(L) = n. Then f is defined by a vector subspace V ⊂ H 0 (X, L) of dimension r +1 plus the choice of a basis of V . From the Euler sequence pulled back to X we have: χ(f ∗ TIP r ) = (r + 1)χ(L) − χ(OX ) = ρ(g, r, n) + r(r + 2)

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

where ρ(g, r, n) := g − (r + 1)(g − n + r) is the Brill-Noether number and r(r + 2) = h0 (TIP r ) = dim[P GL(r + 1)] Assume that r + 1 = h0 (L), i.e. that f is defined by the complete linear series |L|, and consider the exact sequence H 1 (OX ) → H 1 (L)r+1 → H 1 (f ∗ TIP r ) → 0 obtained from the Euler sequence. It dualizes as: µ0 (L)

0 → H 1 (f ∗ TIP r )∨ → H 0 (L) ⊗ H 0 (ωX L−1 ) −→ H 0 (ωX ) where µ0 (L) is the Petri map (see Example 3.3.9, page 171). Therefore we see that f is unobstructed if µ0 (L) is injective. A necessary condition for this to be true is that (r + 1)(g − n + r) = dim[H 0 (L) ⊗ H 0 (ωX L−1 )] ≤ h0 (ωX ) = g i.e. that ρ(g, r, n) ≥ 0. This necessary condition is not sufficient. The simplest example is given by a nonsingular complete intersection X = Q∩S ⊂ IP 3 of a quadric cone Q and of a cubic surface S. Then X is a canonical curve of genus 4 and the projection from the vertex of the cone defines a complete g31 |L| such that ωX L−1 ∼ = L. In this case ρ(4, 1, 3) = 0 but dim[ker(µ0 (L))] = 1. The morphism f : X → IP 1 is obstructed because h0 (f ∗ TIP 1 ) = 4 but L is the unique g31 on X. Therefore the unobstructed first order deformations of f are only those in the 3-dimensional space coming from the automorphisms of IP 1 . (iii) If IP 1 ∼ = E ⊂ S is a nonsingular projective rational curve negatively embedded in a projective nonsingular surface S with E 2 = −n < 0, n ≥ 1, then we have h0 (E, NE/S ) = 0 and h0 (E, TS|E ) = 3. More precisely the exact sequence 0 → TE → TS|E → NE/S → 0 splits because Ext1OE (NE/S , TE ) = H 1 (E, OE (n + 2)) = 0: therefore TS|E ∼ = OE (2) ⊕ OE (−n)

3.4. MORPHISMS

191

This means that, despite the fact that E is rigid in S, the morphism f : E → S has a 3-dimensional family of deformations obtained by composing it with the automorphisms of E. More generally, whenever we have an embedding f : IP 1 → Y with Y nonsingular algebraic variety, we have an inclusion H 0 (IP 1 , TIP 1 ) ⊂ H 0 (IP 1 , f ∗ TY ) which implies h0 (IP 1 , f ∗ TY ) ≥ 3. In general, for any non-costant morphism f : IP 1 → Y the sheaf f ∗ TY is locally free and splits as a direct sum of dim(Y )−1 invertible sheaves, by the structure theorem (see [152] p. 22). The study of such morphisms is closely related with the notions of uniruledness and rational connectedness. We refer to Debarre[43] and to Kollar[122] for a detailed treatment of these matters. (iv) Similarly, if E ⊂ Y is an embedding of a projective nonsingular curve of genus 1 into a nonsingular algebraic variety Y , from the inclusion OE = TE ⊂ TY |E we deduce H 0 (E, OE ) ⊂ H 0 (E, TY |E ) which implies h0 (E, TY |E ) ≥ 1.

3.4.2

Deformations of a morphism leaving the target fixed

In this subsection we will follow quite faithfully the treatment given in [93]. Given a morphism f : X → Y of algebraic schemes, a notion slightly more general than the previous one is that of a deformation of f with target Y , obtained by specializing Definition 3.4.1 to the case when Y is the product family, i.e. by considering a cartesian diagram of the form X

→

↓f

X ↓F

Y

→ Y ×S

↓

↓ψ

Spec(k) →

S

192

CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

where S = Spec(A) with A in ob(A) (resp. in ob(A∗ )) and ψ is the projection. Such a deformation will be shortly denoted by the diagram F

X

−→ &

Y ×S (3.42)

. S

The deformation (3.42) will be called locally trivial if its domain X defines a locally trivial deformation of X. Given a deformation (3.42) and another deformation of f with target Y parametrized by S: F0

X0

−→ &

Y ×S .

S an isomorphism between them is an isomorphism of deformations of X: X .

& Φ

X

X0

−→ &

. S

which makes the following diagram commutative: X & ↓Φ

Y ×S %

X0

3.4. MORPHISMS

193

Definition 3.4.5 To a morphism f : X → Y of algebraic schemes there is associated an exact sequence of coherent sheaves on X J

df

P

0 → TX/Y −→ TX −→ Hom(f ∗ Ω1Y , OX ) −→ Nf → 0

(3.43)

which defines the sheaf Nf called the normal sheaf of f . The morphism f is called non-degenerate when TX/Y = 0. The sequence (3.43) is of course related with the exact sequence (1.5) 1 on page 16. Comparing (3.43) with (1.5) we see that Nf = TX/Y if X is nonsingular. If f is smooth then Nf = 0. The condition that f is nondegeneratee is equivalent to f being unramified on a dense open subset of X. 1 We now introduce vector spaces DX/Y and DX/Y which will be used to describe infinitesimal deformations of the morphism f . Definition 3.4.6 Let f : X → Y be a morphism between algebraic schemes, with X projective. Let U = {Ui }i∈I be an affine open cover of X and define DX/Y =

{(v, t) ∈ C 0 (U, f ∗ TY ) × Z 1 (U, TX ) : δv = df (t)} {(df (w), δw) : w ∈ C 0 (U, TX )}

and

{(ζ, s) ∈ C 1 (U, f ∗ TY ) × Z 2 (U, TX ) : δζ = df (s)} = {(df (u), δu) : u ∈ C 1 (U, TX )} ˇ where δ is the coboundary map in Cech cohomology. 1 DX/Y

Lemma 3.4.7 In the situation of Definition 3.4.6: 1 (i) DX/Y and DX/Y don’t depend on the choice of the affine cover U of X.

(ii) We have the following exact sequences: (a) H 0 (X, TX ) → H 0 (X, f ∗ TY ) → DX/Y → H 1 (X, TX ) → H 1 (X, f ∗ TY ) (b)

0 → H 1 (X, TX/Y ) → DX/Y → H 0 (X, Nf ) → H 2 (X, TX/Y )

(c)

1 H 1 (X, TX ) → H 1 (X, f ∗ TY ) → DX/Y → H 2 (X, TX ) → H 2 (X, f ∗ TY )

(d)

1 0 → H 2 (X, TX/Y ) → DX/Y → H 1 (X, Nf ) → H 3 (X, TX/Y )

(3.44)

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

(iii) If f is non-degenerate then DX/Y ∼ = H 0 (X, Nf ) 1 ∼ DX/Y = H 1 (X, Nf )

(iv) If f is smooth then

DX/Y ∼ = H 1 (X, TX/Y ) 1 ∼ DX/Y = H 2 (X, TX/Y )

Proof. It is clear that (ii) ⇒ (i). Moreover since f non-degenerate implies TX/Y = 0, (iii) follows from the exact sequences (3.44)(b) and (3.44)(d). Similarly, f smooth implies Nf = 0 and (iv) follows again from (3.44)(b) and (3.44)(d). Therefore it suffices to prove (ii). In the first sequence the homomorphism DX/Y → H 1 (X, TX ) is given by sending (v, t) 7→ t; similarly we have H 0 (X, f ∗ TY ) → DX/Y which sends u 7→ (u, 0). The proof of exactness is left to the reader. In the second exact sequence the map H 1 (X, TX/Y ) → DX/Y is given by Z 1 (U, TX/Y ) 3 ψ 7→ (0, Jψ) ∈ DX/Y The map DX/Y → H 0 (X, Nf ) sends (v, t) 7→ P v. Every element of H 0 (X, Nf ) is represented by some v ∈ C 0 (U, f ∗ TY ) such that δv = df (t) for some t ∈ C 1 (U, TX ). We have df (δt) = δδt = 0 so that δt can be viewed as an element of Z 2 (U, TX/Y ). This defines the map H 0 (X, Nf ) → H 2 (X, TX/Y )

3.4. MORPHISMS

195

The proof of exactness is left to the reader. The other two sequences are defined and their exactness is checked similarly. qed Let f : X → Y be a morphism between algebraic schemes. Define functors of Artin rings: Def f /Y , Def 0f /Y : A → (sets) by

isomorphism classes of deformations of f over A with fixed target

isomorphism classes of locally trivial deformations of f over A with fixed target

Def f /Y (A) = Def 0f /Y (A)

=

for all A in ob(A). Obviously Def f /Y = Def 0f /Y when X is nonsingular. We will consider the locally trivial case. The main general result about Def 0f /Y is the following: Theorem 3.4.8 Let f : X → Y be a morphism of algebraic schemes with X projective. Then Def 0f /Y has a formal semiuniversal deformation. Its tangent 1 space is DX/Y and DX/Y is an obstruction space (see Definition 3.4.6). Proof. Let’s check the conditions of Theorem 2.3.2. Def 0f /Y trivially satisfies condition H0 . Consider a diagram in A: A0

A00 &. A

with A00 → A a small extension and let A¯ = A0 ×A A00 . Let (fA0 , fA00 ) ∈ Def 0f /Y (A0 ) ×Def 0f /Y (A) Def 0f /Y (A00 ) where fA0 : X 0 → Y × Spec(A0 ),

fA00 : X 00 → Y × Spec(A00 )

¯ there is a deformation X¯ of X over A¯ such that Since Def 0X satisfies H 0 X0 ∼ ¯ Spec(A ), = X¯ ×Spec(A)

00 X 00 ∼ ¯ Spec(A ) = X¯ ×Spec(A)

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

and we have a commutative diagram: X 00 ↓ → X¯

X0 ↓ fA0

f

00

A −→

Y × Spec(A00 ) .

¯ → Y × Spec(A)

Y × Spec(A0 )

By the universal property of the fibered sum we obtain a morphism ¯ fA¯ : X¯ → Y × Spec(A) which pulls back over Spec(A0 ) and Spec(A00 ) to fA0 and fA00 respectively. Therefore fA¯ 7→ (fA0 , fA00 ) under the map ¯ → Def 0f /Y (A0 ) ×Def 0 (A) Def 0f /Y (A00 ) α : Def 0f /Y (A) f /Y ¯ and this proves H. ¯ and Let A = k and A00 = k[] and suppose that fA¯ : X¯ → Y × Spec(A) 0 ¯ are elements of Def f /Y (A) ¯ mapped to (fA0 , fA00 ) by α. f˜ : X˜ → Y × Spec(A) By the universal property of X¯ we have a morphism X¯ → X˜ which, being an isomorphism mod , is an isomorphism. Moreover we have a commutative diagram X¯ → X˜ & ↓ ¯ Y × Spec(A) ¯ this implies and therefore fA¯ and f˜ define the same element of Def 0f /Y (A); that α is bijective in this case and H holds. In order to describe the tangent space to Def 0f , consider a first order locally trivial deformation X

→

↓f Y

X ↓F

→ Y × Spec(k[])

↓ Spec(k) →

↓ψ Spec(k[])

3.4. MORPHISMS

197

Then the deformation X

→

↓

X ↓ ψF

(3.45)

Spec(k) → Spec(k[]) is locally trivial. Choose an affine open cover U = {Ui }i∈I of X and, for each index i, let θi : Ui × Spec(k[]) → X|Ui be an isomorphism of deformations. The composition Fi := F θi : Ui × Spec(k[]) → Y × Spec(k[]) is a deformation wfdat of fi := f|Ui and therefore it corresponds to an element vi ∈ Γ(Ui , fi∗ TY ) = Γ(Ui , f ∗ TY ) by Proposition 3.4.2. Therefore we get an element v = {vi } ∈ C 0 (U, f ∗ TY ). Restricting to Uij we have: Fi|Uij θi−1 θj = Fj|Uij

(3.46)

and, by Lemma 1.2.6, θi−1 θj corresponds to a section tij ∈ Γ(Uij , TX ); the collection {tij } is an element t ∈ Z 1 (U, TX ) which defines the Kodaira-Spencer class of the deformation (3.45). The identity (3.46) means that vj −vi = d(tij ) or, equivalently, that the pair (v, t) satisfies δv = df (t). The pair (v, t) is defined up a choice of the trivializations θi or, equivalently, up to an element of the form (df (w), δw), w ∈ C 0 (U, TX ). Similarly if we replace F by an isomorphic deformation of the form F 0 = F σ, where σ : X × Spec(k[]) → X × Spec(k[]) is an automorphism of the trivial deformation of X. Conversely, suppose given an element of DX/Y , represented by a pair (v, t) = ({vi }, {tij }) ∈ C 0 (U, f ∗ TY ) × Z 1 (U, TX ) such that δv = df (t). The class t¯ ∈ H 1 (X, TX ) defines a first order deformation of X X ↓

→

X ↓

Spec(k) → Spec(k[])

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

which is locally trivial. The 1-cocycle t = {tij } defines local trivializations θi : Ui × Spec(k[]) → X|Ui and each vi defines a deformation wfdat Fi of fi and by composition we get a morphism: θ−1

F

i i X|Ui −→ Ui × Spec(k[]) −→ Y × Spec(k[])

By construction Fi|Uij θi−1 θj = Fj|Uij and therefore Fi|Uij θi−1 = Fj|Uij θj−1 This means that the morphisms Fi θi−1 patch together and define a morphism F : X → Y ×Spec(k[]). This obviously gives rise to a first order deformation of f . It is a straightforward task to verify that the correspondences F 7→ v¯ and v¯ 7→ F are inverse of each other. Therefore Def 0f /Y (k[]) ∼ = DX/Y and in particular Hf holds. Now we want to prove the assertion about obstructions to lifting deformations. We have to show that for every locally trivial infinitesimal deformation F

X

−→ &

Y ×S .

S of f with target Y over S = Spec(A) there is a natural map 1 oF/Y : Exk (A, k) → DX/Y

such that, for a given extension e : 0 → k → A˜ → A → 0 we have oF/Y (e) = 0 if and only if F has a lifting to A˜ which is a locally trivial deformation of f with target Y . Choose U = {Ui }i∈I an affine open cover of X and trivializations θi : Ui × Spec(A) → X|Ui

3.4. MORPHISMS

199

Since H 1 (Ui , f ∗ TY ) = 0 for each i ∈ I the morphism Fi := F θi : Ui × Spec(A) → Y × Spec(A) has a lifting as a deformation of fi wfdat: F

i Ui × Spec(A) −→ Y × Spec(A)

T

T F˜

i ˜ −→ ˜ Ui × Spec(A) Y × Spec(A)

If we restrict to Uij we have the identity: Fi|Uij θij = Fj|Uij where we have written θij := θi−1 θj : Uij × Spec(A) → Uij × Spec(A) Let ˜ → Uij × Spec(A) ˜ θ˜ij : Uij × Spec(A) be an automorphism which restricts to θij on Uij ×Spec(A). Then F˜i|Uij θ˜ij and F˜j|Uij are both liftings of Fj|Uij . Therefore, since Γ(Uij , f ∗ TY ) acts faithfully and transitively on the set of such liftings (Proposition 3.4.2 (iii)), there is a ζij ∈ Γ(Uij , f ∗ TY ) which carries F˜j|Uij into F˜i|Uij θ˜ij ; set ζ = {ζij } ∈ C 1 (U, f ∗ TY ). Let now −1 ˜ → Uijk × Spec(A) ˜ θ˜ijk = θ˜ij θ˜jk θ˜ik : Uijk × Spec(A)

Since θ˜ijk restricts to the identity on Uijk × Spec(A), by Lemma 1.2.6 it corresponds to an sijk ∈ Γ(Uijk , TX ); by construction s := {sijk } is a 2cocycle, i.e. it is an element of Z 2 (U, TX ), and the pair (ζ, s) satisfies δ(ζ) = df (s) 1 Define oF/Y (e) = (ζ, s) ∈ DX/Y . This definition is well posed because a different choice of the θ˜ij ’s will replace (ζ, s) by (ζ + df (u), s + δu) for some u ∈ C 1 (U, TX ).

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

If oF/Y (e) = 0 then (ζ, s) = (df (u), δu) for some u ∈ C 1 (U, TX ). The condition s = δu means that for the deformation X → X ↓ ↓ ξ: Spec(k) → Spec(A) ˜ Such we have oξ (e) = s = 0 ∈ H 2 (X, TX ) so that ξ has a lifting to Spec(A). a lifting X → X → X˜ ˜ ↓ ↓ ↓ ξ: ˜ Spec(k) → Spec(A) → Spec(A) is defined by an appropriate choice of the automorphisms θ˜ij such that θ˜ij θ˜jk = θ˜ik . The condition ζ = df (u) means that the liftings F˜i and F˜j can be chosen so that F˜i|U θ˜ij = F˜j|U ij

ij

and this means that they patch together to define a lifting F˜ of F . Conversely, if such a lifting exists then one shows in the same way that oF/Y (e) = 0. qed oF/Y (e) is called the obstruction to lift F to A˜ as a deformation with target Y . The notions of obstructed/unobstructed deformation, obstructed/unobstructed morphism with fixed target can be given as usual. One can give the notion of rigid morphism with fixed target in an obvious way. We have the following Corollary 3.4.9 Under the hypotheses of Theorem 3.4.8 we have: 1 (i) If DX/Y = 0 then f is rigid as a morphism with fixed target. If DX/Y =0 then f is unobstructed as a morphism with fixed target.

(ii) If f is non-degenerate then in the statement of 3.4.8 we can replace 1 DX/Y and DX/Y by H 0 (X, Nf ) and H 1 (X, Nf ) respectively. (iii) If f is smooth then in the statement of 3.4.8 we can replace DX/Y and 1 DX/Y by H 1 (X, TX/Y ) and H 2 (X, TX/Y ) respectively. Proof. The proof is immediate in view of Lemma 3.4.7.

qed

3.4. MORPHISMS

201

Remarks 3.4.10 (i) If f is a closed embedding of projective nonsingular algebraic schemes then Nf = NX/Y and Theorem 3.4.8 asserts that first order deformations, resp. obstructions, of f with target Y coincide with first order deformations, resp. obstructions, of X in Y . This is because, more generally, every infinitesimal deformation of f with target Y is an infinitesimal deformation of X in Y : a proof is given in Note 3. (ii) If f : X → Y is a morphism between algebraic varieties with X projective, and if (3.42) is a family of locally trivial deformations of f with target Y , then, using Theorem 3.4.8, we can define a linear map TS,s → DX/Y which associates to a tangent vector t : Spec(k[]) → S at s the element of DX/Y corresponding to the first order deformation obtained by pulling back F by t. This map is called the characteristic map of the family F . Given a morphism f : X → Y between algebraic varieties with X projective we have a natural morphism of functors Φf : Def 0f /Y → Def 0X called the forgetful morphism which associates to a locally trivial deformation (3.42) over S = Spec(A), A in ob(A), the family of deformations of X obtained by forgetting the morphism F . This morphism of functors generalizes the analogous forgetful morphism defined for the local Hilbert functor in Subsection 3.2.3. We have the following generalization of Proposition 3.2.9. Proposition 3.4.11 Let f : X → Y be a morphism between algebraic varieties with X projective, and let Φf : Def 0f /Y → Def 0X be the forgetful morphism. Then (i) dΦf : DX/Y → H 1 (X, TX ) is the map occurring in the exact sequence (3.44)(a). (ii) The map 1 DX/Y → H 2 (X, TX )

occurring in the exact sequence (3.44)(c) is an obstruction map for Φf .

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

(iii) Assume that H 1 (X, f ∗ TY ) = 0 Then Φf is smooth. Proof. The proofs of (i) and (ii) are straightforward. (iii) The hypothesis and the exact sequences (3.44)(a) and (3.44)(c) imply that DX/Y → H 1 (X, TX ) is surjective and 1 DX/Y → H 2 (X, TX )

is injective

Now, using parts (i) and (ii), the conclusion follows from Proposition 2.3.6. qed Corollary 3.4.12 (i) Let f : X → Y be a non-degenerate morphism of algebraic schemes with X projective. Then dΦf : H 0 (X, Nf ) → H 1 (X, TX ) and o(Φf ) : H 1 (X, Nf ) → H 2 (X, TX ) are the coboundary maps coming from the exact sequence 0 → TX → f ∗ TY → Nf → 0 (ii) Let f : X → Y be a smooth morphism of projective nonsingular algebraic schemes. Then dΦf : H 1 (X, TX/Y ) → H 1 (X, TX ) and o(Φf ) : H 2 (X, TX/Y ) → H 2 (X, TX ) are the maps induced by the natural inclusion of sheaves TX/Y ⊂ TX . Proof. The corollary is just a special case of the proposition.

qed

3.4. MORPHISMS

203

Examples 3.4.13 (i) Let m ≥ 0 be an integer, Fm = IP (Fm ) where F is the locally free rank two sheaf on IP 1 : Fm = OIP 1 (m) ⊕ OIP 1 and let π : Fm → IP 1 be the projection. Then we have H 1 (Fm , π ∗ TIP 1 ) = 0 by an easy calculation using the Leray spectral sequence. Therefore, since π is smooth, we can apply Proposition 3.4.11 to conclude that Φπ is smooth. Moreover, since Fm is unobstructed as an abstract variety because h2 (Fm , TFm ) = 0 (see (B.13)), it follows from Proposition 2.2.5(iii) that π is unobstructed. We can actually be more precise because we have an exact sequence of locally free sheaves on IP 1 : ∨ 0 → OIP 1 → Fm ⊗ Fm → π∗ TFm /IP 1 → 0

which can be deduced easily from the exact sequence (4.28). Since ∨ ∼ Fm ⊗ Fm = O⊕2 ⊕ O(m) ⊕ O(−m)

using the Leray spectral sequence we deduce that h1 (Fm , TFm /IP 1 ) = m − 1 for m ≥ 1 Therefore, recalling (B.12), we see that the map H 1 (Fm , TFm /IP 1 ) → H 1 (Fm , TFm ) is not only surjective but it is actually an isomorphism. (ii) Let f : X → Y be a smooth family of projective curves of genus ≥ 2 with X a projective nonsingular surface and Y a projective nonsingular connected curve. Assume that f is non-isotrivial (see Definition 2.6.9). Then H 1 (X, TX/Y ) = 0 and therefore, by (i) and (iii) of Corollary 3.4.9, f is rigid as a morphism with fixed target. This theorem is due to Parshin [155] in char 0. For an exposition we refer the reader to Szpiro [199], where the theorem, and its generalization due to Arakelov [6], is proved without the restriction char(k) = 0.

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

(iii) Let f : X → Y be an etale morphism of projective nonsingular schemes of dimension n. Then f is non-degenerate and df : TX → f ∗ TY is an isomorphism; therefore Nf = 0 and f is rigid as a morphism with fixed target. If we only assume f to be non-degenerate, but not necessarily etale, then df degenerates on a divisor R ⊂ X (which is the divisor Dn−1 (df ) of Example 4.2.8), called the ramification divisor of f . Nf is supported on R, and in general f is not rigid as a morphism with fixed target. For the case when X and Y are curves see Subsection 3.4.3. (iv) Let Y be a projective nonsingular variety, γ ⊂ Y a nonsingular closed subvariety of pure codimension r ≥ 2 and π : X → Y the blow-up of Y with center γ. Let E = π −1 (γ) ⊂ X be the exceptional divisor. Then E∼ = IP (Nγ/Y ) is a projective bundle over γ: let q : E → γ be the structure morphism. Then NE/X = OE (E) and it is well known that the restriction of NE/X to each fibre IP of q is OIP (−1). Therefore by the Leray spectral sequence of q we immediately deduce that hi (E, NE/X ) = 0

(3.47)

for all i. We have TX/Y = 0 because TX/Y is a subsheaf of the locally free TX and is supported on E; therefore π is non-degenerate. Since π∗ OX = OY and Ri π∗ OX = 0 for i ≥ 1, from the Leray spectral sequence we deduce that H i (X, π ∗ TY ) = H i (Y, (π∗ π ∗ OX ) ⊗ TY ) = H i (Y, TY ),

i≥0

(3.48)

We have an exact and commutative diagram of locally free sheaves on E:

0 → TE/γ k 0 → TE/γ 0

0 ↓ → TE ↓ → TX|E ↓ → NE/X ↓ 0

0 ↓ → q ∗ Tγ ↓ → q ∗ TY ↓ ∗ → q Nγ/Y ↓ 0

→

0

→ Nπ k → Nπ

→0

(3.49)

→0

In particular we see that we have an exact sequence of locally free sheaves on E: 0 → NE/X → q ∗ Nγ/Y → Nπ → 0 (3.50)

3.4. MORPHISMS

205

The verification of these facts is straightforward and it is left to the reader. For example, let π : X = Bl[1,0,0] IP 2 → IP 2 be the blow-up of IP 2 with center the point [1, 0, 0]. From the exact sequence (3.50) we deduce that Nπ = OE (1). Therefore h0 (X, Nπ ) = 2,

hi (X, Nπ ) = 0, i ≥ 1

In particular π is unobstructed. Moreover h0 (X, TX ) = h0 (IP 2 , TIP 2 ⊗ I[1,0,0] ) = 6 as can be easily checked using the Euler sequence. Therefore from the exact sequence (3.43) we see that h1 (X, TX ) = 0, i.e. X is rigid.

3.4.3

Morphisms from a nonsingular curve with fixed target

Theorem 3.4.8 applies in particular to a morphism ϕ:C→Y where C and Y are projective and nonsingular, C is a curve, and ϕ is not constant on each component of C. Consider the exact sequence dϕ

0 → TC −→ ϕ∗ TY → Nϕ → 0 The vanishing divisor (see Definition 4.2.8 page 239) Z := D0 (dϕ) of dϕ is called the ramification divisor of ϕ; the index of ramification of ϕ at p ∈ C is the coefficient of p in Z. ϕ is unramified if and only if Z = 0. The homomorphism dϕ extends to a homomorphism TC (Z) → ϕ∗ TY ¯ϕ ; it is locally free. We have whose cokernel we denote by N ¯ϕ = Nϕ /Hϕ N

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

where Hϕ ⊂ Nϕ is the torsion subsheaf; it is supported on Z. The following commutative and exact diagram summarizes the situation:

dϕ

TC −→ ϕ∗ TY ↓ k ∗ 0 → TC (Z) → ϕ TY 0→

0 ↓ Hϕ ↓ → Nϕ ↓ ¯ → Nϕ ↓ 0

→0

(3.51)

→0

We obtain: χ(Nϕ ) = χ(ϕ∗ TY ) + 3g − 3

(3.52)

Example 3.4.14 Assume that C is connected of genus g, that Y is a projective connected nonsingular curve of genus γ, and that ϕ has degree d; then ¯ϕ = 0, Nϕ = Hϕ = OZ N where O(Z) = ϕ∗ (TY ) ⊗ KC , so that χ(Nϕ ) = h0 (Nϕ ) = deg(Z) = 2[g − 1 + (1 − γ)d] and ϕ is unobstructed because h1 (Nϕ ) = 0. This corresponds to the fact that the deformations of ϕ leaving Y fixed are obtained by varying the branch points of ϕ. Note that ϕ is rigid as a morphism with fixed target if g ≥ 2 and Z = 0, i.e. if it is unramified. ∗

∗

∗

∗

∗ ∗

Assume now that ϕ:C→S is a non-constant morphism from an irreducible projective nonsingular curve C of genus g to a projective nonsingular surface S and that ϕ is birational

3.4. MORPHISMS

207

onto its image; let Γ = ϕ(C) ⊂ S. Then we have a commutative and exact diagram 0 ↓ dϕ 0 → TC −→ ϕ∗ TS → Nϕ →0 ↓ k ↓j 0 → ϕ∗ TΓ → ϕ∗ TS → ϕ∗ NΓ/S ¯ϕ and the Since ϕ∗ NΓ/S is invertible the homomorphism j factors through N above diagram gives rise to the following: 0 ↓ 0 → TC (Z) −→ ϕ∗ TS ↓ k ∗ ∗ 0 → ϕ TΓ → ϕ TS

¯ϕ N ↓ ∗ 0 → ϕ NΓ/S

→

→0 →0

0 where Z is the ramification divisor of ϕ and NΓ/S = ker[NΓ/S → TΓ1 ] is the equisingular normal sheaf (see also §4.7). This diagram implies the following isomorphisms:

TC (Z) ∼ = ϕ∗ TΓ ,

0 ¯ϕ ∼ N , = ϕ∗ NΓ/S

Hϕ ∼ = coker[TC → ϕ∗ (TΓ )] =: Nϕ¯

where we have denoted by ϕ¯ : C → Γ the morphism induced by ϕ. In 0 particular ϕ∗ TΓ and ϕ∗ NΓ/S are invertible and ϕ∗ [TC (Z)] ∼ = TΓ ⊗ ϕ∗ OC ,

0 ¯ϕ ∼ ϕ∗ N ⊗ ϕ ∗ OC = NΓ/S

On Γ we have a natural exact sequence: 0 → OΓ → ϕ∗ OC → t → 0 where t is a torsion sheaf supported on the singular locus of Γ. Since NΓ/S is invertible the homomorphism NΓ/S → NΓ/S ⊗ ϕ∗ OC is injective and it follows that we have an exact sequence 0 0 → NΓ/S

→

¯ϕ ϕ∗ N k 0 NΓ/S ⊗ ϕ∗ OC

0 → NΓ/S ⊗t →0

(3.53)

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

This sequence implies in particular: 0 ¯ϕ ) ≤ h0 (Nϕ ) h0 (NΓ/S ) ≤ h0 (N 0 ¯ϕ ) = h1 (Nϕ ) h1 (NΓ/S ) ≥ h1 (N

Lemma 3.4.15 If the singularities of Γ are nodes and ordinary cusps then 0 NΓ/S ⊗ t = 0, equivalently 0 ∼ ¯ϕ NΓ/S = ϕ∗ N In particular if Γ has only nodes as singularities then 0 ∼ NΓ/S = ϕ ∗ Nϕ

Proof. The exact sequence (3.53) can be embedded in the following exact and commutative diagram: 0 ↓ 0 0 → NΓ/S ↓ 0 → NΓ/S ↓ 0 → TΓ1 ↓ 0

0 ↓ ¯ϕ → ϕ∗ N ↓a → NΓ/S ⊗ ϕ∗ OC ↓ b 1 −→ TΓ ⊗ ϕ∗ OC ↓ 0

0 → NΓ/S ⊗t →0 ↓d → NΓ/S ⊗ t → 0 ↓c 1 → TΓ ⊗ t → 0 ↓ 0

The arrow a is injective because it is a nonzero homomorphism of torsion free rank one sheaves. Because of the assumptions made on the singularities, at each singular point p ∈ Γ we have tp = ϕ∗ OC,p /OΓ,p ∼ = k. Therefore the 1 ∼ ∼ arrow c is an isomorphism because NΓ/S ⊗ t = t = TΓ ⊗ t. Thus d = 0. The arrow b is injective because at each singular point p ∈ Γ we have (TΓ1

⊗ ϕ∗ OC )p = ϕ∗ ϕ

∗

1 (TΓ,p )

∼ =

2 k 3

k

if p is a node if p is a cusp

(proved by easy local computation) while 1 ∼ TΓ,p =

k k2

if p is a node if p is a cusp

3.4. MORPHISMS

209

(recall Example 3.1.4). The conclusion now follows from the “Snake Lemma”. ¯ϕ = Nϕ and we deduce that N 0 ∼ If Γ has only nodes then N qed Γ/S = ϕ∗ Nϕ . 0 It is possible to show that conversely if NΓ/S ⊗t = 0 then the singularities of Γ are nodes and ordinary cusps (see [72]). If S = IP 2 then, letting L = ϕ∗ O(1), d = deg(L), from the Euler sequence restricted to C: 0 → OC → L⊕3 → ϕ∗ TIP 2 → 0

we deduce χ(ϕ∗ TIP 2 ) = 3d + 2 − 2g and from (3.52) χ(Nϕ ) = 3d + g − 1

(3.54)

By Corollary 3.4.9 the unobstructedness of ϕ is related to the vanishing of H 1 (C, Nϕ ). From (3.51) we see that deg(Nϕ ) = c1 (ϕ∗ TIP 2 ) − deg(TC ) = 3d + 2g − 2 and that ¯ϕ ) h1 (Nϕ ) = h1 (N But ¯ϕ ) = c1 (Nϕ ) − deg(Z) deg(N ¯ϕ is a non-special line bundle whenever deg(Z) < 3d. We can and therefore N therefore state the following result: Proposition 3.4.16 Let ϕ : C → IP 2 be a morphism from an irreducible projective nonsingular curve C of genus g, birational onto its image. Let d = deg(ϕ∗ O(1)), and let Z be the ramification divisor of ϕ. Then h0 (C, Nϕ ) ≥ 3d + g − 1 If deg(Z) < 3d then ϕ is unobstructed and the above inequality is an equality. In particular if ϕ(C) is a plane curve having nodes and cusps as its only singularities and the the number κ of cusps satisfies κ < 3d then h0 (C, Nϕ ) = 3d + g − 1,

h1 (C, Nϕ ) = 0

210

3.4.4

CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

Deformations of a closed embedding

The deformation theory of morphisms is more subtle if we want to allow both the domain and the target to deform non-trivially. In this subsection we will address this case, considering only the simplest situation of a closed embedding. Let j : X ⊂ Y be a closed embedding of algebraic schemes. If J

X

−→ &

Y .

S where S = Spec(A), A in ob(A), is an infinitesimal deformation of j then J is a closed embedding (see Note 3 at the end of of this section). It is obvious that an infinitesimal deformation of j with fixed target is nothing but a deformation of X in Y . Given another infinitesimal deformation of j: J0

X0

Y0

−→ &

. S

over the same S = Spec(A), an isomorphism between them is a pair of isomorphisms of deformations: α : X → X 0,

β : Y → Y0

which make the diagram X

J

−→ Y

↓α

↓β J0

X 0 −→ Y 0 commutative. We define

Def j (A) =

isomorphism classes of deformations of j over A

3.4. MORPHISMS

211

Def 0j (A)

=

isomorphism classes of locally trivial deformations of j over A

for each A in ob(A). These are the functor of infinitesimal deformations of j and of locally trivial infinitesimal deformations of j respectively. The locally trivial infinitesimal deformations of a closed embedding are studied by means of a sheaf which we now introduce. Let’s now assume that Y is nonsingular and let IX ⊂ OY be the ideal sheaf of X. Let TY hXi ⊂ TY be the inverse image of TX ⊂ TY |X under the natural restriction homomorphism TY → TY |X . Then TY hXi is called the sheaf of germs of tangent vectors to Y which are tangent to X. We clearly have an inclusion IX TY ⊂ TY hXi such that TX = TY hXi/IX TY and an exact sequence 0 0 → TY hXi → TY → NX/Y →0

(3.55)

0 where NX/Y ⊂ NX/Y is the equisingular normal sheaf of X in Y (introduced in Proposition 1.1.9, page 16). From the definition it follows that, for every open set U ⊂ Y , Γ(U, TY hXi) consists of those k-derivations D ∈ Γ(U, TY ) such that D(g) ∈ Γ(U, IX ) for every g ∈ Γ(U, IX ). We also have the following exact commutative diagram:

0 ↓ IX TY

0 ↓ = IX TY

↓

↓

0 → TY hXi → ↓ 0→

TX ↓ 0

TY ↓

→

TY |X ↓ 0

0 → NX/Y

→0

k 0 → NX/Y

→0

(3.56)

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0 Of course NX/Y is replaced by NX/Y in case X is nonsingular. We will describe locally trivial infinitesimal deformations of a closed embedding by means of the sheaf TY hXi.

Proposition 3.4.17 Let j : X ⊂ Y be a closed embedding of projective algebraic schemes with Y nonsingular. Then Def 0j has a formal semiuniversal deformation. Its tangent space is H 1 (Y, TY hXi) and H 2 (Y, TY hXi) is an obstruction space. ¯ and Proof. The proof that Def 0j satisfies Schlessinger’s conditions H0 , H H is similar to the proof given in Theorem 3.4.8 and will be left to the reader. Since Y is projective the existence of a semiuniversal formal deformation will follow if we will prove the assertion about the tangent space of Def 0j , because H 1 (Y, TY hXi) is finite dimensional. Let U = {Ui }i∈I be an affine open cover of Y and V = {Vi = X ∩ Ui }i∈I the induced affine open cover of X. Every locally trivial first order deformation of j is obtained by glueing the trivial deformations Vi ⊂ Vi × Spec(k[]) T

T

Ui ⊂ Ui × Spec(k[]) along Vij ×Spec(k[]) and Uij ×Spec(k[]). It is therefore necessary to describe the automorphisms of the trivial deformations Vij ⊂ Vij × Spec(k[]) T

T

Uij ⊂ Uij × Spec(k[]) Every such automorphism Aij consists of a pair (θij , Θij ) where θij : Vij × Spec(k[]) → Vij × Spec(k[]) and Θij : Uij × Spec(k[]) → Uij × Spec(k[])

3.4. MORPHISMS

213

are automorphisms of deformations such that the following diagram commutes: θij Vij × Spec(k[]) −→ Vij × Spec(k[]) T

T Θij

Uij × Spec(k[]) −→ Uij × Spec(k[]) equivalently such that θij = Θij|Vij . According to Lemma 1.2.6 Θij and θij correspond to sections Dij ∈ Γ(Uij , TY ) and dij ∈ Γ(Vij , TX ) respectively such that Dij 7→ dij when restricted to X. It follows that Dij ∈ Γ(Uij , TY hXi) and that to give Aij is the same as to give Dij . This said, the proof of the statement about tangent and obstruction spaces of Def 0j now proceeds in a straightforward way along the lines of the analogous proofs of 1.2.9 and of 1.2.12. We omit the details. qed Also in this case we have the notions of obstructed (resp. unobstructed) deformation, obstructed (resp. unobstructed) embedding, and of rigid embedding. It follows from Proposition 3.4.17 that a closed embedding j : X ⊂ Y of projective nonsingular varieties is rigid if and only if H 1 (Y, TY hXi) = 0 Let

J

X

−→ &

Y .

S be a locally trivial deformation of j : X ⊂ Y parametrized by a pointed scheme (S, s). Then we can define a characteristic map χJ : TS,s → H 1 (Y, TY hXi) by associating to a tangent vector t : Spec(k[]) → S at s the element of H 1 (Y, TY hXi) corresponding to the first order deformation of f obtained by pulling back J by t. Remark 3.4.18 Let j : X ⊂ Y be a closed embedding of projective schemes with Y nonsingular. If H 0 (Y, TY hXi) = 0

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

then Def 0j is prorepresentable. In fact, letting (R, uˆ) be the formal semiuniversal deformation of Def 0j , one can generalize Theorem 2.6.1 and its corollaries by introducing an automorphism functor Autuˆ : AˆR → (sets) in an obvious way and proving that it is prorepresentable with tangent space H 0 (Y, TY hXi). This is called the space of infinitesimal automorphisms of j. The details of this straightforward generalization are left to the reader. Example 3.4.19 Let Y be a projective scheme, p ∈ Y a closed point and j : {p} ⊂ Y . Then TY (hpi) = Ip TY where Ip ⊂ OY is the ideal sheaf of p. In this case Def 0j is the functor of locally trivial deformations of the pointed scheme (Y, p). If for example Y is a projective nonsingular connected curve of genus g then TY (hpi) = TY (−p) and we get:  if g = 0 0 1 h (Y, TY (−p)) = 1 if g = 1  3g − 2 if g ≥ 2 while h2 (Y, TY (−p)) = 0. Of course we can generalize by considering a set of m distinct closed points {p1 , . . . , pm } of Y , and the inclusion j : {p1 , . . . , pm } → Y . Then Def 0j is the functor of locally trivial deformations of the m-pointed scheme (Y ; p1 , . . . , pm ).

3.4.5

Stability and costability

Whenever we have a locally trivial infinitesimal deformation X

→

Y

&

. Spec(A)

of a closed embedding j : X ⊂ Y of projective schemes we also have a deformation of X and a deformation of Y (both locally trivial): X → X ↓ ↓ ξ: Spec(k) → Spec(A)

Y → Y ↓ ↓ η: Spec(k) → Spec(A)

3.4. MORPHISMS

215

This means that we have two forgetful morphisms of functors: Φ

Y Def 0j −→ Def 0Y ↓ ΦX Def 0X

The differentials and obstruction maps of these morphisms are described as follows. Proposition 3.4.20 If j : X → Y is a closed embedding of projective schemes with Y nonsingular then (i) dΦY : H 1 (Y, TY hXi) → H 1 (Y, TY ) and o(ΦY ) : H 2 (Y, TY hXi) → H 2 (Y, TY ) are the maps induced in cohomology by the inclusion TY hXi ⊂ TY . (ii) dΦX : H 1 (Y, TY hXi) → H 1 (X, TX ) and o(ΦX ) : H 2 (Y, TY hXi) → H 2 (X, TX ) are the maps induced in cohomology by the restriction TY hXi → TX . Proof. It is a straightforward consequence of the above analysis. Details are left to the reader. qed Remark 3.4.21 Let j : X ⊂ Y be a closed embedding of projective nonsingular schemes. Then there is a natural morphism of functors Y HX → Def j

whose differential is easily seen to be the coboundary map δ : H 0 (X, NX/Y ) → H 1 (Y, TY hXi) determined by the exact sequence (3.55). It follows from the proposition that ker(dΦY ) = Im(δ)

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

as expected, because deformations of X in Y are precisely those deformations of the embedding X ⊂ Y which induce the trivial deformation of Y . The proposition implies also that β

ker(dΦX ) = Im[H 1 (Y, IX TY ) −→ H 1 (Y, TY hXi)] where β is the map induced by the inclusion IX TY ⊂ TY hXi. This kernel consists of the first order deformations of j which induce the trivial deformation of X. If in particular H 0 (X, TX ) = 0 then ker(dΦX ) = H 1 (Y, IX TY ) Definition 3.4.22 If j : X ⊂ Y is a closed embedding of projective schemes then X is called stable in Y if the morphism of functors ΦY : Def 0j → Def 0Y is smooth. X is called costable in Y if the morphism of functors ΦX : Def 0j → Def 0X is smooth. The notion of stability was introduced and studied in [116] for a compact complex submanifold of a complex manifold. As stated in [116], stability means that “no local deformation of Y makes X disappear”. Our definition of stability implies that every infinitesimal locally trivial deformation of Y is induced by a locally trivial deformation of j. Costability implies that every infinitesimal locally trivial deformation of X is induced by a locally trivial deformation of j. The notion of costability has been introduced in [95]. Proposition 3.4.23 Let j : X ⊂ Y be a closed embedding of projective schemes, Y nonsingular. 0 (i ) If H 1 (X, NX/Y ) = (0) then X is stable in Y .

(ii) If H 2 (Y, IX TY ) = 0 then X is costable in Y . (iii) If X is nonsingular and is both stable and costable in Y then X is obstructed if and only if Y is obstructed (as abstract varieties).

3.4. MORPHISMS

217

Proof. (i) From the exact sequence (3.55) it follows that dΦY is surjective and that H 2 (Y, TY hXi) → H 2 (Y, TY ) is injective; but by Proposition 3.4.20(i) this last condition means that Def 0f is less obstructed than Def Y and the conclusion follows from Proposition 2.3.6. (ii) The proof is similar using the exact cohomology sequence of the first column of diagram (3.56), Proposition 3.4.20 and Proposition 2.3.6. (iii) Since the morphisms of functors Φ

Φ

X Y Def X ←− Def j −→ Def Y

are both smooth we deduce that any one of the functors Def X , Def j , Def Y is smooth if and only if the others are. qed Examples 3.4.24 (i) (Kodaira[116], Th. 5) Let Y be a projective nonsingular variety, γ ⊂ Y a nonsingular closed subvariety and π : X → Y the blow-up of Y with center γ. Let E = π −1 (γ) ⊂ X be the exceptional divisor; then hi (E, NE/X ) = 0 for all i (see (3.47)). From Proposition 3.4.23 we obtain that E is a stable subvariety of X. This is remarkable because γ has not been required to be stable in Y . (ii) Let X be a projective nonsingular algebraic surface and Z ⊂ X an irreducible nonsingular rational curve with self intersection ν = Z 2 . Then Z is stable in X if ν ≥ −1 because H 1 (Z, NZ/X ) = 0 in this case. On the other hand if ν ≤ −2 then in general Z is not stable in X. An example is provided by the negative section E in the rational ruled surface Fm , for m ≥ 2. In fact E 2 = −m and we have seen in Example 1.2.2(ii) that there is a family f : W → A1 of deformations of Fm for which [E] does not extend to the other fibres W(t), t 6= 0, since they are isomorphic to Fn for some 0 ≤ n < m. This implies that E is not stable. (iii) Assume k = C. Let C be a projective irreducible nonsingular curve of genus g ≥ 3, and let α : C → JC be the Abel-Jacobi embedding of C into its jacobian variety. Then • H 1 (JC, OJC ) ∼ = H 1 (C, OC ) ([21], Lemma 11.3.1)

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

• TJC = H 1 (C, OC ) ⊗ OJC . Thus the restriction TJC → TJC|C induces isomorphisms H 0 (JC, TJC ) ∼ = = H 0 (C, TJC|C ) ∼

H 1 (C, OC )

H (JC, TJC ) ∼ = H 1 (C, TJC|C ) ∼ = H 1 (C, OC ) ⊗ H 1 (C, OC )

(3.57)

1

In particular, in view of the second column of diagram (3.56), we have H 0 (JC, IC TJC ) = 0 = H 1 (JC, IC TJC ) and therefore, taking cohomology of (3.56), we obtain the commutative and exact diagram: H 0 (NC/JC ) → H 1 (JC, TJC hCi) k ∩ H 0 (NC/JC ) → H 1 (C, TC )

dΦJC

−→ σ

−→

H 1 (JC, TJC ) → H 1 (NC/JC ) → 0 k k H 1 (C, TJC|C ) → H 1 (NC/JC ) → 0 (3.58)

which implies that H 1 (JC, TJC hCi) ∼ = H 1 (C, TC )

(3.59)

H 2 (JC, TJC hCi) ⊂ H 2 (JC, TJC )

(3.60)

and Since (3.60) is the obstruction map of ΦJC : Def α → Def JC from Proposition 2.3.6 we deduce that Def α is smooth, being less obstructed than Def JC which is smooth (see Example 2.4.11(v), page 87). On the other hand, since (3.59) is the differential of the forgetful morphism ΦC : Def α → Def C and Def α is smooth we deduce by Corollary 2.3.7 that ΦC is smooth , i.e. C is costable in JC. Note that H 0 (JC, TJC hCi) = 0 by the first column of diagram (3.56): therefore α has no infinitesimal automorphisms and Def α is prorepresentable (Remark 3.4.18). It follows that ΦC is actually an isomorphism of functors (Remark 2.3.8).

3.4. MORPHISMS

219

We can identify Def α with Def C and the differential dΦJC : H 1 (JC, TJC hCi) → H 1 (JC, TJC ) with the map σ in diagram (3.58). Therefore ΦJC is a closed embedding if and only if σ is injective. In view of the isomorphisms (3.57) σ is Serre-dual to the natural multiplication map: σ ∨ : H 0 (C, KC ) ⊗ H 0 (C, KC ) → H 0 (C, 2KC ) This map is surjective if and only if C is non-hyperelliptic: therefore in this case Def C is a closed smooth subfunctor of Def JC . In the natural decomposition 1

1

2

1

H (C, OC ) ⊗ H (C, OC ) = S H (OC ) ⊕

2 ^

H 1 (OC )

we have Im(σ) ⊂ S 2 H 1 (OC ) because 2 H 1 (OC ) ⊂ ker(σ ∨ ). Therefore Im(dΦJC ) is contained in S 2 H 1 (OC ) which is the space of first order deformations of JC preserving the principal polarization (compare with Example 3.3.13, page 180, and observe that in ¯ = H 1 (OC ) = V by the Hodge decomposition of H 1 (C, C)). this case Ω The validity of the condition “σ injective” is called the infinitesimal Torelli theorem: thus it holds if and only if C is non-hyperelliptic. It is not difficult to show that α(C) is unobstructed in JC if and only if C is non-hyperelliptic (see [74], [126]). V

The following result, due to Kodaira [116], gives the possibility of relating a local Hilbert functor with the deformation functor of an abstract variety. Proposition 3.4.25 Let Y be a projective nonsingular variety, γ ⊂ Y a closed nonsingular subvariety of pure codimension r ≥ 2, and let π : X → Y be the blow-up of Y with center γ. Then: (i) There is a natural isomorphism of functors B : HγY → Def π/Y In particular Def π/Y is prorepresentable.

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

(ii) Assume that H 1 (Y, TY ) = 0, i.e. that Y is rigid. Then the forgetful morphism Φπ : Def π/Y → Def X is smooth. Therefore we have a smooth morphism of functors BΦπ : HγY → Def X . In particular, if γ is obstructed in Y then X is obstructed as an abstract variety. Proof. (i) We define B : HγY → Def π/Y by associating to a family of deformations γA ⊂ Y × Spec(A) ↓ Spec(A) of γ in Y over A the blow-up πA : XA := BlγA (Y × Spec(A)) → Y × Spec(A) of Y × Spec(A) along γA . Note that, since OγA is A-flat and we have the exact sequence on Y × Spec(A): 0 → IγA → OY ⊗ A → OγA → 0 the sheaf IγA is A-flat as well (Proposition A.23(VI)); moreover IγkA /Iγk+1 is A locally free over OY ⊗ A for all k ≥ 1 because γA is regularly embedded in Y × Spec(A) by Lemma D.3. From this it is easy to deduce that IγkA is A-flat for all k ≥ 1 and therefore XA = Proj

M

IγkA

k

is A-flat by Proposition A.23(V). We leave to the reader to check the functoriality of B. The differential of B is the composition dB : H 0 (γ, Nγ/Y ) ∼ = H 0 (E, q ∗ Nγ/Y ) → H 0 (X, Nπ ) where the first map is the obvious isomorphism and the second one comes from the exact sequence (3.50); in a similar way one describes the obstruction map of B as the one induced by the composition H 1 (γ, Nγ/Y ) ∼ = H 1 (E, q ∗ Nγ/Y ) → H 1 (X, Nπ )

3.4. MORPHISMS

221

deduced from the exact sequence (3.50). These facts can be easily verified by chasing diagram (3.49). Since H i (E, NE/X ) = 0 all i we see that these maps are both bijective, and the conclusion follows. (ii) From (3.48) it follows that H 1 (X, π ∗ TY ) = 0. The exact sequence 0 → TX → π ∗ TY → Nπ → 0 and Proposition 3.4.20(i) imply that dΦπ is surjective and o(Φπ ) is injective. The conclusion is a consequence of Proposition 2.3.6. The last assertion is an obvious consequence of the fact that the composition Φπ B : HγY → Def X is smooth. qed By applying Proposition 3.4.25 to any obstructed nonsingular curve γ ⊂ IP 3 (e.g. the curve of degree 14 and genus 24 described in §4.6) we obtain an example of obstructed projective variety of dimension 3. As an application we obtain the following result, which gives examples of obstructed surfaces. Theorem 3.4.26 (Horikawa [95]) Let γ ⊂ IP 3 be an obstructed nonsingular curve, and X the blow-up of IP 3 with center γ. If S ⊂ X is a sufficiently ample nonsingular surface then S is obstructed as an abstract variety. Proof. By Proposition 3.4.23(iii) it is sufficient to show that S is both stable and costable in X. We have h2 (X, OX ) = 0 and, by the ampleness of S, h1 (X, OX (S)) = 0 by Serre’s vanishing theorem. From the exact sequence 0 → OX → OX (S) → NS/X → 0 we deduce that h1 (S, NS/X ) = 0 and therefore S is stable in X. On the other hand we have H 2 (X, TX (−S)) = 0 by Serre’s vanishing theorem again. Therefore S is costable in X as well. qed It is immediate to verify, using the adjunction formula, that the surfaces S constructed in the theorem are regular and with ample canonical class. Notes and Comments 1. The analysis of morphisms from a nonsingular curve is taken from [8]. For Lemma 3.4.15 see also [201].

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CHAPTER 3. EXAMPLES OF DEFORMATION FUNCTORS

2. For the study of the functors Def f /Y and Def f under more general assumptions than those of Theorem 3.4.8 a more careful analysis of first order deformations and obstructions is needed. We refer the reader to [165] and [167] for more information about this. The deformation theory of closed embeddings is studied in [150] in the analytic category. 3. Let X

Φ

−→ &. S

Y

be a commutative diagram of morphisms of algebraic schemes, with X and Y S-flat and Φ projective. Assume that Φo : X (o) → Y(o) is a closed embedding, for some k-rational point o ∈ S. Then there is an open neighborhood U ⊂ S of o such that the restriction Φ(U ) : X (U ) → Y(U ) is a closed embedding. Proof. Let K = coker[OY → Φ∗ (OX )]. Since Φ is projective Φ∗ (OX ) is a coherent sheaf and so is K. Moreover K(o) = (0) because Φo is a closed embedding. It follows that there is an open subset U ⊂ S containing o such that K|Y(U ) = (0). Let Z = Spec(Φ∗ (OX )), h : Z → Y the induced S-morphism and X

Φ

−→ g &% h Z

Y

the Stein factorization of Φ. Then it follows that h(U ) : Z(U ) → Y(U ) is a closed embedding. Moreover, since g has connected fibres and is bijective over Z(o), it follows that, modulo shrinking U if necessary, g(U ) : X (U ) → Z(U ) is an isomorphism. The conclusion follows. qed

Chapter 4 The Hilbert schemes and the Quot schemes Even though this book is centered around the theme of infinitesimal and local deformations, in this chapter we turn our attention to global deformations. We will introduce the Hilbert schemes and other related objects, which are important examples of parameter schemes for global families of deformations of algebro-geometric objects. They are used to describe and classify “extrinsic” deformations, i.e. deformations of objects within a given ambient space (e.g. closed subschemes of a given scheme). Their study is preliminary to the construction of “moduli schemes”. Moreover they provide some of the most typical examples of constructions in algebraic geometry by the functorial approach. We will study some of their properties and consider a few applications of the local theory developed so far.

4.1

Castelnuovo-Mumford regularity

In this section we introduce the notion of m-regularity, also called CastelnuovoMumford regularity, and we prove its main properties. They will be needed for the construction of the Hilbert schemes and of the Quot schemes. Let m ∈ ZZ. A coherent sheaf F on IP r is m-regular if H i (F(m − i)) = (0) for all i ≥ 1. 223

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CHAPTER 4. HILBERT AND QUOT SCHEMES

Because of Serre’s vanishing theorem, every coherent sheaf F on IP r is m-regular for some m ∈ ZZ. The definition of m-regularity makes sense for a coherent sheaf on any projective scheme X endowed with a very ample line bundle O(1). For semplicity we will consider the case X = IP r only, leaving to the reader the obvious modifications of the statements and of the proofs in the general case. Proposition 4.1.1 If F is m-regular then (i) the natural map H 0 (F(k)) ⊗k H 0 (O(1)) → H 0 (F(k + 1)) is surjective for all k ≥ m. (ii) H i (F(k)) = (0) for all i ≥ 1 and k ≥ m − i; in particular F is n-regular for all n ≥ m. (iii) F(m), and therefore also F(k) for all k ≥ m, is generated by its global sections. Proof. We prove (i) and (ii) by induction on r. If r = 0 there is nothing to prove. Assume r ≥ 1 and let H be a hyperplane not containing any point of Ass(F); it exists because Ass(F) is a finite set. Tensoring by F(k) the exact sequence: 0 → O(−H) → O → OH → 0 we get an exact sequence: 0 → F(k − 1) → F(k) → FH (k) → 0 where FH = F ⊗ OH . For each i > 0 we obtain an exact sequence H i (F(m − i)) → H i (FH (m − i)) → H i+1 (F(m − i − 1)) which implies that FH is m-regular on H. It follows by induction that (i) and (ii) are true for FH . Let’s consider the exact sequence H i+1 (F(m − i − 1)) → H i+1 (F(m − i)) → H i+1 (FH (m − i))

4.1. CASTELNUOVO-MUMFORD REGULARITY

225

If i ≥ 0 the two extremes are zero (the right one by (ii) for FH , the left one by the m regularity of F), therefore F is (m + 1)-regular. By iteration this proves (ii). To prove (i) we consider the commutative diagram: H 0 (F(k)) ⊗k H 0 (O(1)) −→ H 0 (FH (k)) ⊗k H 0 (OH (1)) u ↓w ↓t 0 0 0 H (F(k)) → H (F(k + 1)) −→ H (FH (k + 1)) v

The map u is surjective for k ≥ m because H 1 (F(k − 1)) = (0); moreover t is surjective for k ≥ m by (i) for FH . Therefore vw is surjective. It follows that H 0 (F(k + 1)) is generated by Im(w) and by H 0 (F(k)) for all k ≥ m. But H 0 (F(k)) ⊂ Im(w) because the inclusion H 0 (F(k)) ⊂ H 0 (F(k + 1)) is multiplication by H. Therefore w is surjective. Let’s prove (iii). Let h 0 be such that F(m + h) is generated by its global sections. Then the composition H 0 (F(m)) ⊗k H 0 (O(h)) ⊗k O → H 0 (F(m + h)) ⊗k O → F(m + h) is surjective because from (i) it follows that the first map is; we deduce that the composition H 0 (F(m)) ⊗k H 0 (O(h)) ⊗k O(−h) → H 0 (F(m)) ⊗k O → F(m) is also surjective, hence the second map is surjective too.

qed

Note that if F is m-regular then the graded k[X1 , . . . , Xr ]-module Γ∗ (F) :=

M

H 0 (F(k))

k∈ZZ

can be generated by elements of degree ≤ m. In fact this is equivalent to the surjectivity of the multiplication maps H 0 (F(m)) ⊗k H 0 (O(h)) → H 0 (F(m + h)) for h ≥ 1, and follows from part (i) of the proposition. In particular, if an ideal sheaf I ⊂ OIP r is m-regular then the homogeneous ideal I = Γ∗ (I) ⊂ k[X0 , . . . , Xr ] is generated by elements of degree ≤ m. Note also that in the way of proving 4.1.1 we have proved the following:

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Proposition 4.1.2 If F is m-regular and 0 → F(−1) → F → G → 0 is an exact sequence, then G is m-regular. Conversely, we have the following: Proposition 4.1.3 Let 0 → F(−1) → F → G → 0 be an exact sequence of coherent sheaves on IP r , and assume that G is mregular. Then: (i) H i (F(k)) = 0 for i ≥ 2 and k ≥ m − i (ii) h1 (F(k − 1)) ≥ h1 (F(k)) for k ≥ m − 1 (iii) H 1 (F(k)) = 0 for k ≥ (m − 1) + h1 (F(m − 1)) In particular F is m + h1 (F(m − 1))-regular. Proof. (i) In the exact sequence H i−1 (G(k)) → H i (F(k − 1)) → H i (F(k)) → H i (G(k)) the first and the last group are zero for i ≥ 2 and k ≥ m − (i − 1). Therefore H i (F(m − i)) ∼ = H i (F(m − i + 1)) ∼ = H i (F(m − i + 2)) ∼ = ··· From Serre’s vanishing theorem we get H i (F(m − i + h)) = 0 for all h 0 and (i) follows. (ii) For k ≥ m − 1 we have the exact sequence v

k H 0 (G(k)) → H 1 (F(k−1)) → H 1 (F(k)) → 0 0 → H 0 (F(k−1)) → H 0 (F(k)) −→

which implies (ii). (iii) Assume vk surjective, and consider the commutative diagram: H 0 (F(k)) ⊗ H 0 (O(1)) ↓ 0 H (F(k + 1))

vk ⊗id

−→ vk+1

−→

H 0 (G(k)) ⊗ H 0 (O(1)) ↓ wk 0 H (G(k + 1))

4.1. CASTELNUOVO-MUMFORD REGULARITY

227

Since wk is surjective for k ≥ m, we have that vk+1 is surjective too. Therefore H 1 (F(k − 1)) ∼ = H 1 (F(k)) ∼ = H 1 (F(k + 1)) ∼ = ··· ∼ =0 If vk is not surjective then h1 (F(k − 1)) > h1 (F(k)). Therefore the function k 7→ h1 (F(k)) is strictly decreasing for k ≥ m − 1, and this implies (iii). qed The following is a useful characterization of m-regularity. Theorem 4.1.4 A coherent sheaf F on IP r is m-regular if and only if it has a resolution of the form: 0 → O(−m − r − 1)br+1 → · · · → O(−m − 1)b1 → O(−m)b0 → F → 0 (4.1) for some b0 , . . . , br+1 ≥ 1. Proof. Assume that F has a resolution (4.1) and let R1 = ker[O(−m)b0 → F] Rj = ker[O(−m − j + 1)bj−1 → O(−m − j + 2)bj−2 ]

j = 2, . . . , r

Rr+1 = O(−m − r)br+1 Using the short exact sequences: 0 → R1 (m − i) → O(−i)b0 → F(m − i) → 0 0 → Rj (m − i) → O(−i − j + 1)bj−1 → Rj−1 (m − i) → 0 0 → O(−i − r − 1)br+1 → O(−i − r)br → Rr (m − i) → 0 we see that for all 1 ≤ i ≤ r we have: H i (F(m − i)) ∼ = H i+1 (R1 (m − i)) ∼ = ··· ··· ∼ = H r (Rr−i (m − i)) ∼ = H r+1 (Rr−i+1 (m − i)) = (0) and F is m-regular. Assume conversely that F is m-regular. By 4.1.1(iii) we have an exact sequence: 0 → R1 → O(−m)b0 → F → 0 with b0 = h0 (F(m)), which defines R1 .

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If R1 = 0 we are done; if R1 6= 0 from the sequences: 0 → R1 (m − i + 1) → O(−i + 1)b0 → F(m − i + 1) → 0 we deduce that H i (R1 (m − i + 1)) ∼ = H i−1 (F(m − i + 1))

1≤i≤r

hence R1 is (m + 1)-regular. Applying the same argument to R1 we find an exact sequence: 0 → R2 → O(−m − 1)b1 → O(−m)b0 → F → 0 with R2 (m+2)-regular. This process can be repeated for at most r +1 steps, by the Hilbert syzygy theorem, and gives a resolution as required. qed We will now turn to the problem of finding numerical criteria of mregularity for a coherent sheaf F on IP r . Consider a sequence σ1 , . . . , σN of N sections of OIP r (1). We will call it F-regular if the sequences of sheaf homomorphisms induced by multiplication by σ1 , . . . , σN : σ1 0 → F(−1) −→ F → F1 → 0 σ

2 0 → F1 (−1) −→ F1 → F2 → 0

etc., are exact. By choosing σi+1 not containing any point of Ass(Fi ) one shows that F-regular sequences of any lenght exist. Therefore any general N -tuple (σ1 , . . . , σN ) ∈ H 0 (OIP r (1))N is an F-sequence. Definition 4.1.5 Let F be a coherent sheaf on IP r , and (b) = (b0 , b1 , . . . , bN ) a sequence of nonnegative integers such that N ≥ dim[Supp(F)]. We will call F a (b)-sheaf if there exists an F-regular sequence σ1 , . . . , σN of sections of OIP r (1) such that h0 (Fi (−1)) ≤ bi , i = 0, . . . , N where F0 = F, and Fi = F/(σ1 , . . . , σi )F(−1), i ≥ 1. Note that from the definition it follows immediately that if F is a (b)sheaf then F1 is a (b1 , . . . , bN )-sheaf. It is likewise clear that for every coherent sheaf F on IP r there is a sequence (b) such that F is a (b)-sheaf. Moreover a subsheaf of a (b)-sheaf is easily seen to be a (b)-sheaf. For example, every ideal sheaf I ⊂ OIP r is a (0)-sheaf, because OIP r is clearly a (0)-sheaf.

4.1. CASTELNUOVO-MUMFORD REGULARITY

229

Lemma 4.1.6 Let 0 → F(−1) → F → G → 0 be an exact sequence of coherent sheaves on IP r . If r X

k+i ai χ(F(k)) = i i=0 then χ(G(k)) =

r−1 X

ai+1

i=0

!

k+i i

!

The proof is left to the reader. Proposition 4.1.7 Let F be a (b)-sheaf, let s = dim[Supp(F)] and s X

k+i χ(F(k)) = ai i i=0

!

Then (i) For each k ≥ −1 we have h0 (F(k)) ≤

Ps

i=0 bi

k+i i

.

(ii) as ≤ bs and F is also a (b0 , . . . , bs−1 , as )-sheaf. Proof. (i) By induction on s. If s = 0 then a0 = h0 (F) = h0 (F(−1)) ≤ b0 and the conclusion is obvious. Assume s ≥ 1. We have an exact sequence 0 → F(−1) → F → F1 → 0 with F1 a (b1 , . . . , bN )-sheaf and dim[Supp(F1 )] = s − 1. Then: h0 (F(k)) − h0 (F(k − 1)) ≤ h0 (F1 (k)) and 0

h (F1 (k)) ≤

s−1 X i=0

bi+1

k+i i

!

by the inductive hypothesis. Since h0 (F(−1)) ≤ b0 by induction on k ≥ −1 we get the conclusion. (ii) By Lemma 4.1.6 and by induction on s we get as ≤ bs and F1 is a (b1 , . . . , bs−1 , as )-sheaf. The conclusion follows. qed

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Definition 4.1.8 The following polynomials, defined by induction for each integer r ≥ −1: P−1 = 0 P r )−1+i Pr (X0 , . . . , Xr ) = Pr−1 (X1 , . . . , Xr ) + ri=0 Xi Pr−1 (X1 ,...,X i are called (b)-polynomials. One immediately sees that Pr (X0 , . . . , Xt , 0, . . . , 0) = Pt (X0 , . . . , Xt )

(4.2)

for each t < r. The following theorem gives a numerical criterion of m-regularity. Theorem 4.1.9 Let F be a (b)-sheaf on IP r , with (b) = (b0 , b1 , . . . , bN ), and let ! r X k+i χ(F(k)) = ai i i=0 be its Hilbert polynomial. Let (c0 , . . . , cr ) be a sequence of integers such that ci ≥ bi − ai , for i = 0, . . . , r, and m = Pr (c0 , . . . , cr ). Then m ≥ 0 and F is m-regular. In particular F is Ps−1 (c0 , . . . , cs−1 )-regular, if s = dim[Supp(F)]. Proof. By induction on r. If r = 0 then m = 0 and F is n-regular for every n ∈ ZZ, so the theorem is true in this case. Assume r ≥ 1. We have an exact sequence: 0 → F(−1) → F → F1 → 0 with F1 a (b1 , . . . , bN )-sheaf supported on IP r−1 . From Lemma 4.1.6 and from the inductive hypothesis we deduce that n ≥ 0 and F1 is n-regular, where n = Pr−1 (c1 , . . . , cr ). From 4.1.3 we deduce that F is [n+h1 (F(n−1)]-regular and hi (F(n − 1)) = 0 for i ≥ 2. Therefore: r X

n−1+i h (F(n − 1)) = h (F(n − 1)) − χ(F(n − 1)) ≤ (bi − ai ) i i=0 1

0

!

by 4.1.7(i). It follows that F is n + ri=0 ci n−1+i -regular, by 4.1.1(ii). This i proves the first assertion. The last assertion follows from 4.1.7(ii) and from (4.2). qed P

4.1. CASTELNUOVO-MUMFORD REGULARITY

231

Note that the integer m in the statement of the theorem depends on the coefficients of the Hilbert polynomial of F as well as on the integers bi . In the special case when F is a sheaf of ideals we can determine an m for which F is m-regular which depends only on the Hilbert polynomial of F, as stated in the next corollary. Corollary 4.1.10 For each r ≥ 0 there exists a polynomial Fr (X0 , . . . , Xr ) such that every sheaf of ideals I ⊂ OP r having Hilbert polynomial r X

k+r χ(I(k)) = ai i i=0

!

is m-regular, where m = Fr (a0 , . . . , ar ), and m ≥ 0. Proof. It suffices to observe that I is a (0)-sheaf. Therefore the corollary follows from Theorem 4.1.9 taking Fr (X0 , . . . , Xr ) = Pr (−X0 , . . . , −Xr ). qed Notes and Comments 1. Corollary 4.1.10 is in general false for coherent sheaves which are not sheaves of ideals. An example from [144] is F = OIP 1 (k) ⊕ OIP 1 (−k) In fact χ(F) = 2 is independent of k but the least m such F is m-regular is |k|. 2. If I is the sheaf of ideals of the closed subscheme X ⊂ IP r and I is m-regular with m ≥ 0, then OX is (m − 1)-regular. Conversely, if OX is (m − 1)-regular and the restriction map H 0 (IP r , OIP r (m − 1)) → H 0 (X, OX (m − 1)) is surjective, then I is m-regular. This follows from the exact sequences 0 → I(k) → OIP r (k) → OX (k) → 0 k ≥ m − 1. 3. The notion of m-regularity is related with that of bounded collection of sheaves, important in moduli theory. A collection of coherent sheaves {Fj }j∈J on a projective scheme X is said to be bounded if there is an algebraic scheme S and a coherent sheaf F on X × S such that for each j ∈ J there is a closed point s ∈ S such that Fj is isomorphic to the

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sheaf F(s) = F|X×{s} . One also says that the collection {Fj }j∈J is bounded by the sheaf F on X × S. For details we refer to [108]. 4. The notion of Castelnuovo-Mumford regularity has been introduced in [144]. Castelnuovo studied the properties of m-regularity of projective curves in [27], where his upper bound for the genus of projective curves was proved. The treatment of (b)-sheaves has been taken from [108].

4.2

Flatness in the projective case

This section is devoted to some properties of flat families of projective schemes which will be needed in this chapter. In particular we will prove a powerful technical result due to Grothendieck and Mumford, the existence of flattening stratifications, which is a key ingredient in the construction of the Hilbert schemes, of the Quot schemes, and of related schemes like the Severi varieties. The treatment of stratifications closely follows Lecture 8 of [144].

4.2.1

Flatness and Hilbert polynomials

The following result gives the name to the “Hilbert scheme”. Proposition 4.2.1 (i) Let S be a scheme, F a coherent sheaf on IP r × S and p : IP r × S → S the projection. Then F is flat over S if and only if p∗ F(h) is locally free on S for all h 0. (ii) Assume that S is connected. For each s ∈ S let Ps (t) = χ(F(s)(t)) =

X

(−1)i hi (IP r (s), F(s)(t))

i

be the Hilbert polynomial of F(s). If F is flat over S then Ps (t) is independent of s ∈ S. Conversely, if S is integral and Ps (t) is independent of s for all s ∈ S, then F is flat over S. If S is integral and algebraic and Ps (t) is independent of s for all closed s ∈ S, then F is flat over S. For the proof of this proposition we refer the reader to [89], Theorem III.9.9.

4.2. FLATNESS IN THE PROJECTIVE CASE

233

Corollary 4.2.2 If X ⊂ IP r × S ↓ S is a flat family of closed subschemes of IP r with S connected, then all fibres X (s) have the same Hilbert polynomial; in particular they have the same degree. Proof. It follows from 4.2.1 applied to F = OX .

qed

Examples 4.2.3 (i) Let Ui = {(z0 , z1 ) ∈ IP 1 : zi 6= 0}, U = U0 U1 and f : U → IP 1 the natural morphism. Then f is flat surjective and quasifinite. The fibres of f are 0-dimensional, hence projective, but their degree is not constant. This is not a contradiction with Corollary 4.2.2 because the morphism f is not projective, since U is an affine variety. `

(ii) In IP 3 with homogeneous coordinates X = (X0 , X1 , X2 , X3 ) consider the curve Cu = Proj(k[X]/(X2 , X3 )) ∪ Proj(k[X]/(X1 , X3 − uX0 )) for every u ∈ A1 . If u 6= 0 then Cu consists of two disjoint lines, while C0 = Proj(k[X]/(X1 X2 , X3 )) is a reducible conic in the plane X3 = 0. The Hilbert polynomials are Pu (t) = 2t + 2 u 6= 0 P0 (t) = 2t + 1 From Corollary 4.2.2 it follows that {Cu } cannot be the set of fibres of a flat family of closed subschemes of IP 3 . We may try to construct a morphism whose fibres are the Cu ’s by considering the closed subscheme X ⊂ IP 3 × A1 defined by the ideal J = (X2 , X3 )∩(X1 , X3 −uX0 ) = (X1 X2 , X1 X3 , X2 (X3 −uX0 ), X3 (X3 −uX0 )) of k[u, X0 , . . . , X3 ]. From [89], Prop. III.9.7, it follows that X is flat over A1 . We have: X (u) = Cu ,

u 6= 0

X (0) = Proj(k[X]/(X1 X2 , X1 X3 , X2 X3 , X32 ))

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and X (0) 6= C0 : indeed X (0) = C0 ∪ Proj(k[X]/(X1 , X2 , X32 )) is a non-reduced scheme obtained from C0 by adjoining an embedded point in (1, 0, 0, 0). In particular we see that X (0) and C0 have the same support. Prop. III.9.8 of [89] implies that X (0) is uniquely determined by the other fibres, i.e. by X ∩ [IP 3 × (A1 \{0})]. Fix a scheme S and a coherent sheaf F on IP r × S. Consider a morphism g : T → S and the diagram IP r × T ↓q T

h

−→ IP r × S ↓p g −→ S

where h = id × g. For every open set U ⊂ S we have homomorphisms H j (IP r × U, F) → H j (IP r × g −1 (U ), h∗ F) → H 0 (g −1 (U ), Rj q∗ (h∗ F)) and therefore a homomorphism Rj p∗ F → g∗ [Rj q∗ (h∗ F)] which corresponds to a homomorphism g ∗ (Rj p∗ F) → Rj q∗ (h∗ F) In case j = 0 we have the following asymptotic result which will be applied later in this section: Proposition 4.2.4 For all m 0 the homomorphism g ∗ (p∗ F(m)) → q∗ (h∗ F(m)) is an isomorphism and, if T is noetherian, Rj q∗ (h∗ F(m)) = 0 all j ≥ 1. Proof. We have h∗ F = Γ∗ (h∗ F)e := [⊕m q∗ (h∗ F(m))]e

4.2. FLATNESS IN THE PROJECTIVE CASE

235

Since F = Γ∗ (F)e we also have h∗ F = h∗ [Γ∗ (F)e ] = [⊕m g ∗ (p∗ F)(m)]e and therefore for all m 0 g ∗ (p∗ F(m)) ∼ = q∗ (h∗ F(m)) For the last assertion cover T by finitely many affine open sets and apply Theorem III.5.2 of [89]. qed The homomorphism of Proposition 4.2.4 is particularly important when g : Spec(k(s)) → S is the inclusion in S of a point s ∈ S; it is denoted tj (s) : Rj p∗ (F)s ⊗ k(s) → H j (IP r (s), F(s)) The study of these homomorphisms is carried out in [1], Ch. III2 (see also Chapter III, Section 12, of [89]). Their main properties are summarized in the following theorem and in its corollary. Theorem 4.2.5 Let S be a scheme, F a coherent sheaf on IP r × S, flat over S, s ∈ S and j ≥ 0 an integer. Then: (i) If tj (s) is surjective then it is an isomorphism. (ii) If tj+1 (s) is an isomorphism then Rj+1 p∗ (F) is free at s if and only if tj (s) is an isomorphism. (iii) If Rj p∗ (F) is free at s for all j ≥ j0 + 1 then tj (s) is an isomorphism for all j ≥ j0 . Proof. (i) and (ii) are Theorem III.12.11 of [89]. (iii) follows from (i) and (ii) by descending induction on j0 . qed Corollary 4.2.6 Let X → S be a projective morphism, and let F be a coherent sheaf on X , flat over S. Then: (i) If H j+1 (X (s), F(s)) = 0 for some s ∈ S and j ≥ 0 then Rj+1 p∗ (F)s = (0), and tj (s) : Rj p∗ (F)s ⊗ k(s) → H j (X (s), F(s)) is an isomorphism.

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(ii) Let j0 be an integer such that H j (X (s), F(s)) = 0 for all j ≥ j0 + 1 and s ∈ S (e.g. j0 = maxs∈S {dim[Supp(F(s))]}). Then tj0 (s) is an isomorphism for all s ∈ S. (iii) Let j0 ≥ 0 be an integer. Then there is a non empty open set U ⊂ S such that tj0 (s) is an isomorphism for all s ∈ U . Proof. (i) follows immediately from 4.2.5. (ii) is a special case of (i). T (iii) It is the open set U = j≥j0 Uj , where Uj = {s ∈ S : Rj p∗ (F)s is free} (apply 4.2.5(iii)). qed

4.2.2

Stratifications

Let S be a scheme. A stratification of S consists of a set of finitely many locally closed subschemes {S1 , . . . , Sn } of S, called strata, pairwise disjoint and such that S = S1 ∪ . . . ∪ Sn i.e. such that we have a surjective morphism a

Si → S

i

Let F be a coherent sheaf on S and for each s ∈ S let e(s) := dimk(s) [Fs ⊗ k(s)] Fix a point s ∈ S, let e = e(s) and let a1 , . . . , ae ∈ Fs be such that their images in Fs ⊗k(s) form a basis. From Nakayama’s lemma it follows that the e homomorphism fs : OS,s → Fs defined by a1 , . . . , ae is surjective; therefore there is an open neighborhood U of s to which f extends defining a surjective homomorphism f : OUe → F|U . With a similar argument applied to ker(fs ) we may find an affine open neighborhood U (s) of s contained in U and an exact sequence g f OUd (s) −→ OUe (s) −→ F|U (s) → 0 (4.3) It follows that:

4.2. FLATNESS IN THE PROJECTIVE CASE

237

(i) e(s0 ) ≤ e(s) for all s0 ∈ U (s): therefore s 7→ e(s) is an upper semicontinuous function from S to ZZ. (ii) Let (gij ) be the e × d matrix with entries in H 0 (U (s), OS ) which defines g. The ideal generated by the gij ’s in H 0 (U (s), OS ) defines a closed subscheme Zs of U (s) with support equal to Ye ∩ U (s), where for each e ≥ 0 we have set Ye = {s ∈ S : e(s) = e}. In particular Ye is a locally closed subset of S. Moreover (iii) If q : T → U (s) is a morphism; q ∗ (F) is locally free of rank e if and only if q factors through the subscheme Zs . Proof. q factors through Zs if and only if all the functions q ∗ (gij ) are zero on T . Since the sequence q ∗ (g)

q ∗ (f )

OTd −→ OTe −→ q ∗ (F) → 0 is exact on T , this is equivalent to q ∗ (f ) being an isomorphism and this condition implies that q ∗ (F) is locally free of rank e. Conversely if q ∗ (F) is locally free of rank e, let G = ker[q ∗ (f )]. At every point t ∈ T we have an exact sequence: 0 → G ⊗ k(t) → k(t)e → q ∗ (F) ⊗ k(t) → 0 Since q ∗ (F) ⊗ k(t) is a vector space of dimension e we have G ⊗ k(t) = (0). By Nakayama’s lemma G = (0) in a neighborhood of t and therefore G = (0) everywhere. (iv) Since property (iii) characterizes the scheme Zs and does not depend on the presentation (4.3), for any s, s0 ∈ S the schemes Zs and Zs0 coincide on U (s)∩U (s0 ); therefore the collection of schemes {Zs : s ∈ S} defines a locally closed subscheme Ze of S supported on Ye . Evidently {Ze : e ≥ 0} is a stratification of S. S

(v) Because of (i), for each e the closure of Ze is contained in e0 ≥e Ze0 . In particular, if E is the highest integer such that ZE 6= ∅, then ZE is closed.

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(vi) By the right exactness of tensor product, the construction of the schemes Ze commutes with base change (the proof is similar to that of (iii)). In other words, if q : T → S is a morphism then Z˜e = f −1 (Ze ) for all e, where Z˜e ⊂ T is the locally closed stratum associated to the sheaf q ∗ F. We have proved the following Theorem 4.2.7 Let S be a scheme and F a coherent sheaf on S. There is a unique stratification {Ze }e≥0 of S such that if q : T → S is a morphism the sheaf q ∗ (F) is locally free if and only if q factors through the disjoint union ` of the Ze ’s: T → e Ze → S. Moreover the strata Z0 , Z1 , . . . are indexed so that for each e = 0, 1, . . . the restriction of F to Ze is locally free of rank e. S For a given e, Z¯e ⊂ e0 ≥e Ze0 . In particular, if E is the highest integer such that ZE 6= ∅, then ZE is closed. The stratification {Ze }e≥0 commutes with base change. Theorem 4.2.7 describes a natural way to construct stratifications on a scheme. {Ze }e≥0 is called the stratification defined by the sheaf F. An alternative approach to the construction of this stratification is by the “Fitting ideals” of the sheaf F. Let k ≥ 0; the k-th Fitting ideal of F is the ideal sheaf F ittF k ⊂ OS locally defined by the minors of order (e − k) of the matrix (gij ) in the presentation (4.3). The proof of Theorem 4.2.7 essentially shows that the Fitting ideals are independent of the choice of the presentations (4.3). The closed subscheme of S defined by F ittF k−1 is denoted by Nk (F). It follows directly from the definition that Supp(Nk (F)) = {s ∈ S : dimk(s) (Fs ⊗ k(s)) ≥ k} and that Nk (F) commutes with base change. Therefore the stratification defined in Theorem 4.2.7 can be also described as follows: Ze = Ne (F)\Ne+1 (F) For details about the properties of the Fitting ideals see [49].

(4.4)

4.2. FLATNESS IN THE PROJECTIVE CASE

239

Example 4.2.8 Let ϕ : A → B be a homomorphism of locally free sheaves on the scheme S, of ranks a and b respectively. Applying Theorem 4.2.7 to coker(ϕ) we obtain a stratification of S with the property that Zb−e is supported on the locus {s ∈ S : rk[ϕ(s) : A(s) → B(s)] = e} The scheme Zb−e of this stratification will be denoted De (ϕ). Note that in particular the subscheme D0 (ϕ), called the vanishing scheme of ϕ, is closed in S because of (v) above. It has the property that a morphism f : T → S satisfies f ∗ (ϕ) = 0 is and only if f factors through D0 (ϕ). The ideal sheaf of D0 (ϕ) is locally generated by the entries of a matrix representing ϕ. More intrinsecally it can be obtained as follows. Since ϕ ∈ Hom(A, B), it induces by adjunction a homomorphism: ϕ∨

Hom(B, A) −→ OS whose image is just the ideal sheaf of D0 (ϕ). Example 4.2.9 Let f : X → S be a finite morphism of algebraic schemes. Then f∗ OX is a coherent sheaf on S. The scheme Nk (f∗ OX ) ⊂ S is supported on the set of points of S having ≥ k preimages (counting multiplicities). It is usually denoted by Nk (f ) and it is called the k-th multiple point scheme of f . The corresponding stratification is the multiple point stratification of S relative to f . There is a vast literature on this stratification. For more about it we refer the reader to [64] and to the literature quoted there.

4.2.3

Flattening stratifications

Definition 4.2.10 Let S be a scheme and F a coherent sheaf on IP r × S. A flattening stratification for F is a stratification {S1 , . . . , Sn } of S such that for every morphism g : T → S the sheaf Fg := (1 × g)∗ (F) on IP r × T is flat over T if and only if g factors through

`

Si .

Note that if such a stratification exists it is clearly unique. In the special case r = 0 we obtain again the notion of stratification defined by the sheaf F. The following is a basic technical result.

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Theorem 4.2.11 For every coherent sheaf F on IP r ×S the flattening stratification exists. Proof. The theorem has already been proved in the case r = 0 (Theorem 4.2.7). Therefore we may assume r ≥ 1. We will proceed in several steps. Step 1) There are finitely many locally closed subsets Y 1 , . . . , Y k of S such that for each i = 1, . . . , k if we consider on Y i the reduced scheme structure then F ⊗ OY i ×IP r is flat over Y i . It follows immediately from a repeated use of the fact that there is a nonempty open subset U ⊂ S such that F|IP r ×Ured is flat over Ured (see Note 7). Step 2) Only finitely many polynomials P 1 , . . . , P h occur as Hilbert polynomials of the sheaves F(s), s ∈ S. In fact from Corollary 4.2.2 it follows that at most as many Hilbert polynomials occur as the number of connected components of the sets Y 1 , . . . , Y k . Step 3) There is an integer N such that for every m ≥ 0 and for every s ∈ S we have: H j (IP r (s), F(s)(N + m)) = (0) for j ≥ 1 and the natural map: [p∗ F(N + m)]s ⊗ k(s) → H 0 (IP r (s), F(s)(N + m)) is an isomorphism, where p : IP r × S → S is the projection. For each i = 1, . . . , k consider the diagram hi : IP r × Y i ↓ pi Yi

→ IP r × S ↓p → S

(4.5)

and let ni 0 be so that Rj pi∗ [hi∗ F(ni + m)] = (0) for all m ≥ 0 and all j ≥ 1 (apply Proposition 4.2.4). Letting N max{n1 , . . . , nk }

4.2. FLATNESS IN THE PROJECTIVE CASE

241

we may apply Proposition 4.2.4 to the diagrams (4.5) and to the sheaf F and we obtain isomorphisms [p∗ F(N + m)] ⊗ OY i ∼ = pi∗ [hi∗ F(N + m)] for all s ∈ Y i and for all i = 1, . . . , k. In particular we have isomorphisms [p∗ F(N + m)] ⊗ k(s) ∼ = pi∗ [hi∗ F(N + m)]s ⊗ k(s)

(4.6)

for all s ∈ Y i and for all i = 1, . . . , k. We may also apply Corollary 4.2.6 to the sheaves hi∗ F and to the projections pi for j0 = 0 to deduce that H j (IP r (s), F(s)(N + m)) = (0) for all s ∈ S, j ≥ 1 and m ≥ 0, and that pi∗ [hi∗ F(N + m)]s ⊗ k(s) ∼ = H 0 (IP r (s), F(s)(N + m))

(4.7)

for all s ∈ Y i and for all i = 1, . . . , k and all m ≥ 0. Comparing (4.6) and (4.7) we obtain the conclusion. Step 4) Let N be as in Step 3, and let g : T → S be a morphism. Then Fg is flat over T if and only if g ∗ [p∗ F(N + m)] is locally free for all m ≥ 0. Suppose that Fg is flat over T and let q : IP r × T → T be the projection. Since H j (IP r (t), Fg (t)(N + m)) = H j (IP r (g(t)), F(g(t))(N + m)) = (0) for all t ∈ T , m ≥ 0 and j ≥ 1, from Corollary 4.2.6(ii) we deduce that q∗ Fg (N + m)t ⊗ k(t) → H 0 (IP r (g(t)), F(g(t))(N + m))

(4.8)

is an isomorphism for all t ∈ T . Theorem 4.2.5(ii) applied for j = −1 implies that q∗ Fg (N + m) is locally free for all m ≥ 0. For all t ∈ T the natural homomorphism ϕ : g ∗ [p∗ F(N + m)] → q∗ Fg (N + m) induces an isomorphism: g ∗ [p∗ F(N + m)]t ⊗ k(t) ∼ = q∗ Fg (N + m)t ⊗ k(t)

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because both sides are isomorphic to H 0 (IP r (g(t)), F(g(t))(N + m)) (the first because of Step 3, the second because of (4.8)). From the fact that q∗ Fg (N + m) is locally free and from Nakayama’s lemma it follows that ϕ is an isomorphism. Therefore g ∗ [p∗ F(N + m)] is locally free for every m ≥ 0. Conversely suppose that g ∗ [p∗ F(N + m)] is locally free for all m ≥ 0. Since for all m 0 the natural map ϕ is an isomorphism (Prop. 4.2.4) it follows that q∗ Fg (N + m) is locally free for all m 0: Proposition 4.2.1 implies that Fg is flat. Step 5) For every m ≥ 0 apply Theorem 4.2.7 to the sheaf p∗ F(N + m) and let Ym,j be the component of the corresponding stratification of S where p∗ F(N + m) becomes locally free of rank j. Then for each j = 1, . . . , h we have the following equality of subsets of S: \

Supp(Ym,P i (N +m) ) =

\

Supp(Ym,P i (N +m) )

m=0,...,r

m≥0

The inclusion ⊂ is obvious. For s ∈ S let Ps (t) be the Hilbert polynomial of F(s). Then s ∈ ∩m≥0 Supp(Ym,P i (N +m) ) if and only if Ps (N +m) = h0 (IP r (s), F(s)(N +m)) = dim[p∗ F(N +m)s ⊗k(s)] = P i (N +m) for all m ≥ 0, and this happens if and only if Ps (t) = P i (t) as polynomials. On the other hand s ∈ ∩m=0,...,r Supp(Ym,P i (N +m) ) if and only if Ps (N + m) = P i (N + m) for m = 0, . . . , r. Since both Ps (t) and P i (t) have degree ≤ r, it follows that Ps (t) = P i (t) and therefore s ∈ ∩m≥0 Supp(Ym,P i (N +m) ). Step 6) Fix i between 1 and h. For each integer c ≥ 0 the finite intersection \

Ym,P i (N +m)

m=0,...,c

is a well defined locally closed subscheme of S. Because of Step 5 the subschemes \ Ym,P i (N +m) c = r, r + 1, . . . m=0,...,c

for a descending chain with fixed support; in particular they form a descending chain of closed subschemes of a fixed open set V ⊂ S, and therefore they stabilize. In other words the intersection Zi =

\ m≥0

Supp(Ym,P i (N +m) )

4.2. FLATNESS IN THE PROJECTIVE CASE

243

is a well defined locally closed subscheme of S. By Step 5 we have: Supp(Z i ) = {s ∈ S : Ps (t) = Pi (t)} Step 7) The subschemes Z 1 , . . . , Z h form a stratification of S. It follows immediately from Step 4 that this is the flattening stratification for F. This concludes the proof of Theorem 4.2.11. qed

Notes and Comments 1. From the proof of Theorem 4.2.11 it follows that the strata Z 1 , . . . , Z h of the flattening stratification for F are indexed by the Hilbert polynomials of the sheaves F(s), s ∈ S. 2. ( The see-saw theorem) Let X be a projective scheme such that H 0 (X, OX ) = k, S an algebraic scheme and L an invertible sheaf on X × S. Then (i) the locus S0 = {s ∈ S : L|X×{s} ∼ = OX } is closed in S. (ii) Letting p0 : X × S0 → S0 be the projection, there is an invertible sheaf M on S0 such that L|X×S0 ∼ = p∗0 M Proof. A line bundle L on X is trivial if and only if it satisfies h0 (X, L) > 0 and h0 (X, L−1 ) > 0. In fact non-zero sections σ ∈ H 0 (X, L) and τ ∈ H 0 (X, L−1 ) correspond to homomorphisms σ

τ∨

OX −→ L −→ OX whose composition is multiplication by a costant (by the assumption H 0 (X, OX ) = k), so that they are isomorphisms. Therefore S0 = S + ∩ S − where S + = {s ∈ S : h0 (X × {s}, L|X×{s} ) > 0} S − = {s ∈ S : h0 (X × {s}, L−1 |X×{s} ) > 0} It follows from the semicontinuity theorem that both S + and S − are closed in S, and (i) follows.

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(ii) Since h0 (X × {s}, L|X×{s} ) = 1 is constant in s ∈ S0 , from Corollary 12.9 of [89] we deduce that p0∗ L ⊗ k(s) ∼ = H 0 (X × {s}, L|X×{s} ) for all s ∈ S0 and in particular p0∗ L =: M is an invertible sheaf. We have a homomorphism of invertible sheaves on X × S0 : p∗0 M = p∗0 p0∗ L → L|X×S0 which is surjective because L|X×{s} is globally generated for all s ∈ S0 . Then it is an isomorphism. qed 3. Let X → S be a flat projective morphism of algebraic schemes, and L an invertible sheaf on X . Assume that for some k-rational point o ∈ S the sheaf L(o) is very ample on X (o) and satisfies H 1 (X (o), L(o)) = 0. Then there is an open neighborhood V ⊂ S of o such that LV := L|X (V ) is very ample relative to V . In particular L(s) is very ample on X (s) for every s ∈ V . Proof. By Corollary 4.2.6 there is an open neighborhood U ⊂ S of o such that 1 (R f∗ L)|U = (0) and t0 (u) : (f∗ L)u ⊗ k(u) → H 0 (X (u), L(u)) is an isomorphism for all u ∈ U . We may even assume that f∗ L is locally free of rank h0 (X (o), L(o)) on U . From the surjectivity of the map t0 (o) and from the fact that L(o) is globally generated we deduce that the canonical homomorphism : f ∗ (f∗ L) → L is surjective on X (o). Since f is projective it follows that there is an open W ⊂ U containing o such that [f ∗ (f∗ L)]|X (W ) → LW (4.9) is surjective and moreover [f ∗ (f∗ L)]|X (W ) is locally free. The homomorphism (4.9) defines a W -morphism X (W )

IP (f ∗ (f∗ L))|X (W )

→ &

.

(4.10)

W whose restriction to X (o) is the embedding defined by the global sections of L(o). From Note 3 of §3.4 above it follows that there is an open subset V ⊂ W containing o such that the restriction of (4.10) to X (V ) is an embedding. This implies the conclusion. (see [129], Theorem 1.2.17 for the proof of a more general statement without flatness assumption). 4. Let E be a locally free sheaf over IP 1 ×S, with S an algebraic integral scheme. Let o ∈ S be a k-rational point, and E(o) ∼ = ⊕i O(ni0 ) the fibre over o. Then

4.2. FLATNESS IN THE PROJECTIVE CASE

245

(i) there is an open set U ⊂ S such that for each s ∈ U we have E(s) ∼ = ⊕i O(nis ) with maxi {nis } ≤ maxi {nio } and

mini {nis } ≥ mini {nio }

Moreover if E(o) is balanced (i.e. nio = njo for all i, j) then E(s) ∼ = E(o) for all s ∈ U . (ii) For each s ∈ S we have X i

nis =

X

ni0

i

Proof. ([26]) (i) By the structure theorem for locally free sheaves on IP 1 (see [152]) we know that for each s ∈ S we have an isomorphism E(s) ∼ = ⊕i O(nis ) i i for some integers ns . Let M0 = maxi {n0 } and consider the sheaf E¯ := E ⊗ ¯ p∗ O(−M0 − 1), where p : IP 1 × S → IP 1 is the projection. Since h0 (E(0)) = 0, from the semicontinuity theorem it follows that there is an open neighborhood U ¯ of 0 such that h0 (E(s)) = 0 for all s ∈ U ; but this means that maxi {nis } ≤ M0 for all s ∈ U , which is the first statement of the proposition. The statement about the minimum is proved similarly after replacing E by its dual. The last assertion is obvious. (ii) Applying (i) to det(E) we find that every point t ∈ S has an open neighP P borhood Ut where i nis = i nit for all s ∈ Ut . Since S is connected we deduce P that i nis is constant. 5. Let X → S be a flat projective morphism with S an algebraic scheme, and let o ∈ S be a k-rational point. Prove that: (i) If X (o) is connected and X (s) is disconnected for all s 6= o in an open neighborhood of o then X (o) is non-reduced. In particular: (ii) If X (o) is connected and reduced then X (s) is connected for all s in an open neighborhood of o. (iii) If X (o) is disconnected then X (s) is disconnected for all s in an open neighborhood of o. 6. Let f : X → Y be a proper morphism of algebraic schemes with finite fibres. Let g : Y 0 → Y be an arbitrary morphism, X 0 = X ×Y Y 0 , f 0 : X 0 → Y 0 and

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g 0 : X 0 → X the projections. Then for every quasi-coherent OX -module F we have a canonical isomorphism g ∗ (f∗ F) ∼ = f∗0 (g 0∗ F) Proof. Since it is proper and quasi-finite, f is finite, in particular it is affine. The conclusion follows from [1] Ch. II, 1.5.2. qed 7. (Generic flatness) Let f : X → S be a morphism of finite type with S integral, and let F be a coherent sheaf on X . There is a dense open subset U ⊂ S such that the restriction of F to f −1 (U ) is flat over U . Proof. See [1], ch. IV, Th. 6.9.1 or [49], Theorem 14.4, p. 308. qed Note that if f is not dominant then U = S\f (X ) and f −1 (U ) = ∅.

4.3

Hilbert schemes

4.3.1

Generalities

Consider a projective scheme Y , and a closed embedding Y ⊂ IP r . Let’s fix a numerical polynomial of degree ≤ r, i.e. a polynomial P (t) ∈ Q[t] of the form: ! r X t+r P (t) = ai i i=0 with ai ∈ ZZ for all i. For every scheme S we let: ( flat

HilbYP (t) (S) =

families X ⊂ Y × S of closed subschemes ) of Y parametrized by S with fibres having Hilbert polynomial P (t)

Since flatness is preserved under base change, this defines a contravariant functor HilbYP (t) : (schemes)◦ → (sets) called the Hilbert functor of Y relative to P (t). In case Y = IP r we will denote the Hilbert functor with the symbol HilbrP (t) . If the functor HilbYP (t) is representable, the scheme representing it will be called the Hilbert scheme of Y relative to P (t), and will be denoted by

4.3. HILBERT SCHEMES

247

HilbYP (t) (or HilbrP (t) in case Y = IP r ). If P (t) = n a constant polynomial then HilbYP (t) is usually denoted by Y [n] . If the Hilbert scheme HilbYP (t) exists then there is a universal element, i.e. there is a flat family of closed subschemes of Y having Hilbert polynomial equal to P (t): W ⊂ Y × HilbYP (t) (4.11) parametrized by HilbYP (t) and possessing the following Universal property: for each scheme S and for each flat family X ⊂ Y ×S of closed subschemes of Y having Hilbert polynomial P (t) there is a unique morphism S → HilbYP (t) , called the classifying morphism, such that X = S ×HilbY W ⊂ Y × S P (t)

The family (4.11) is called the universal family, and the pair (HilbYP (t) , W) represents the functor HilbYP (t) . The family W is the universal element of HilbYP (t) (HilbYP (t) ), namely the element corresponding to the identity under the identification Hom(HilbYP (t) , HilbYP (t) ) = HilbYP (t) (HilbYP (t) ) Example 4.3.1 Consider the constant polynomial P (t) = 1. Then we have a canonical identification Y [1] = Y and the universal family is the diagonal ∆⊂Y ×Y. To prove it consider an element of Y [1] (S) for some scheme S: Γ ⊂ S×Y ↓f S Then f is an isomorphism: in fact it is a one-to-one morphism and OS → f∗ OΓ is an isomorphism since f∗ OΓ is an OS -algebra which is locally free of rank one over OS . We therefore have the well defined morphism g = qf −1 : S → Y where q : S × Y → Y is the projection. The morphism (gf, q) : Γ → Y × Y

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CHAPTER 4. HILBERT AND QUOT SCHEMES

factors through ∆ and induces a commutative diagram Γ ↓ S

→

∆ ↓ −→ Y g

such that Γ ∼ = g ∆. Therefore the family Γ is induced by ∆ via the morphism g. ∗

Before proving the existence of the Hilbert schemes in general we will consider two important special cases.

4.3.2

Linear systems

If X ⊂ IP r is a hypersurface of degree d it has Hilbert polynomial !

!

t+r t+r−d d h(t) = − = tr−1 + · · · (r − 1)! r r Conversely, if a closed subscheme Y of IP r has Hilbert polynomial h(t) then it is a hypersurface of degree d. In fact, since h(t) has degree r − 1, Y has dimension r − 1, so Y = Y1 ∪ Z, with Y1 a hypersurface and dim(Z) < r − 1. We have the exact sequence: 0 → IY1 /IY → OY → OY1 → 0 where IY , IY1 ⊂ OIP r are the ideal sheaves of Y and Y1 . We deduce that h(t) = h1 (t) + k(t) where h1 (t) is the Hilbert polynomial of Y1 and k(t) is the Hilbert polynomial of IY1 /IY . Since this sheaf is supported on Z, we have deg(k(t)) < r − 1; therefore we see that h1 (t) =

d tr−1 + · · · (r − 1)!

so Y1 is a hypersurface of degree d, and therefore h1 (t) = h(t). It follows that k(t) = 0, i.e. IY1 = IY , equivalently Y = Y1 .

4.3. HILBERT SCHEMES

249

Therefore Hilbrh(t) , if it exists, parametrizes a universal family of hypersurfaces of degree d in IP r . To prove its existence let V := H 0 (IP r , O(d)) and in IP (V ) take homogeneous coordinates (. . . , ci(0),...,i(r) , . . .)i(0)+···+i(r)=d The hypersurface H ⊂ IP r × IP (V ) defined by the equation X

i(0)

ci(0),...,i(r) X0

· · · Xri(r) = 0

projects onto IP (V ) with fibres hypersurfaces of degree d. It follows from Proposition 4.2.1 that H is flat over IP (V ). Let’s denote by p : IP r ×IP (V ) → IP (V ) the projection, and let IH ⊂ OIP r ×IP (V ) be the ideal sheaf of H. For all x ∈ IP (V ) we have 1 = h0 (IP r (x), IH(x) (d)) = h0 (IP r (x), IH (d)(x)) and

0 = hi (IP r (x), IH(x) (d)) = hi (IP r (x), IH (d)(x)) 0 = hi (H(x), OH(x) (d))

for all i ≥ 1. Applying 4.2.5 and 4.2.2 we deduce that a) R1 p∗ IH (d) = 0 b) p∗ IH (d) is an invertible subsheaf of p∗ OIP r ×IP (V ) (d) = V ⊗k OIP (V ) c) p∗ OIP r ×IP (V ) (d)/p∗ IH (d) = p∗ OH (d) is locally free. It follows that p∗ IH (d) = OIP (V ) (−1) the tautological invertible sheaf on IP (V ), and the natural map p∗ p∗ IH (d) → IH (d) is an isomorphism. Therefore IH = [p∗ OIP (V ) (−1)](−d) Let’s prove that H ⊂ IP r × IP (V ) is a universal family.

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Suppose that X ⊂ IP r × S ↓f S is a flat family of closed subschemes of IP r with Hilbert polynomial h(t), i.e. hypersurfaces of degree d, and let IX ⊂ OIP r ×S be the ideal sheaf of X . Arguing as above we deduce that f∗ IX (d) is an invertible subsheaf of V ⊗k OS with locally free cokernel f∗ OX (d), and that IX = [f ∗ f∗ IX (d)](−d) We have an induced morphism g : S → IP (V ) such that g ∗ [OIP (V ) (−1)] = f∗ IX (d) The subscheme S ×IP (V ) H ⊂ IP r × S is defined by the ideal sheaf (1 × g)∗ IH = (1 × g)∗ [OIP (V ) (−1)(−d)] = = f ∗ [g ∗ OIP (V ) (−1)](−d) = [f ∗ f∗ IX (d)](−d) = IX Hence S ×IP (V ) H = X . The proof of the uniqueness of g having this property is left to the reader. Therefore we see that H ⊂ IP r × IP (V ) is a universal family, and Hilbrh(t) = IP (V ).

4.3.3

Grassmannians

The classical grassmannians are special cases of Hilbert schemes, since they parametrize linear spaces, which are the closed subschemes with Hilbert poly t+n−1 nomials of the form n−1 , n − 1 being their dimension. Let’s fix a k-vector space V of dimension N , and let 1 ≤ n ≤ N . Letting GV,n (S) = {loc. free rk n quotients of the free sheaf V ∨ ⊗k OS on S} we define a contravariant functor: GV,n : (schemes)→(sets) called the Grassmann functor; we will denote it simply by G when no confusion is possible.

4.3. HILBERT SCHEMES

251

Theorem 4.3.2 The Grassmann functor G is represented by a scheme Gn (V ) together with a locally free quotient of rank n V ∨ ⊗k OGn (V ) → Q called universal quotient bundle. Proof. Given a scheme S and an open cover {Ui } of S, to give a locally free rank n quotient of V ∨ ⊗k OS is equivalent to give one such quotient over each open set Ui so that they patch together on the intersections Ui ∩ Uj . Therefore G is a sheaf. Let’s fix a basis {ek } of V ∨ and choose a set J of n distinct indices in {1, . . . , N }. We have an induced decomposition V ∨ = E 0 ⊕E 00 , with E 0 (resp. E 00 ) a vector subspace of rank n (resp. N − n). We can define a subfunctor GJ of G letting:

GJ (S) =

locally free rk n quotients V ∨ ⊗k OS → F inducing E 0 ⊗k OS → F surjective

Let S be any scheme and f : Hom(−, S) → G a morphism of functors corresponding to a locally free rank n quotient V ∨ ⊗k OS → F The fibered product SJ := Hom(−, S) ×G GJ is clearly represented by the open subscheme of S supported on the points where the map E 0 ⊗k OS → F is surjective; this proves that GJ is an open subfunctor of G. Since clearly the SJ ’s cover S, we also see that the family of subfunctors {GJ } is an open covering of G. To prove that GJ is representable note that if q : V ∨ ⊗k OS → F is an element of G(S) then the induced map η : E 0 ⊗k OS → F is surjective if and only if it is an isomorphism; in this case the composition η −1 ◦ q : V ∨ ⊗k OS → E 0 ⊗k OS

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CHAPTER 4. HILBERT AND QUOT SCHEMES

restricts to the identity on E 0 ⊗k OS , hence it is determined by the composition E 00 ⊗k OS → V ∨ ⊗k OS → E 0 ⊗k OS It follows that we can identify GJ (S) = Hom(E 00 ⊗k OS , E 0 ⊗k OS ) = Hom(E 00 , E 0 ) ⊗k OS This proves that GJ is isomorphic to Hom(−, An(N −n) ), hence it is representable. Now the theorem follows from Proposition E.18. qed Gn (V ) is called the grassmannian of n-dimensional subspaces of V ; it is also called the grassmannian of (n − 1)-dimensional projective subspaces of IP (V ). When V = kN the grassmannian Gn (kN ) is denoted G(n, N ). When n = 1 the functor GV,1 is represented by G1 (V ) = IP (V ) = Proj(Sym(V ∨ )), the (N − 1)-dimensional projective space associated to V . In this case Q = OIP (V ) (1). From Theorem 4.3.2 it follows that for all schemes S the morphisms f : S → Gn (V ) are in 1-1 correspondence with the locally free rank n quotients V ∨ ⊗k OS → F via f ↔ f ∗ Q. This is the universal property of Gn (V ). The universal quotient bundle defines an exact sequence of locally free sheaves on Gn (V ): 0 → K → V ∨ ⊗k OGn (V ) → Q → 0 called the tautological exact sequence; K is called the universal subbundle. Let S be a scheme. Associating to every locally free quotient of rank n V ∨ ⊗k OS → F the quotient (∧n V ∨ ) ⊗k OS → ∧n F we define a morphism of functors GV,n → G∧n V,1 , which is induced by a morphism π : Gn (V ) → IP (∧n V ) π is called the Pl¨ ucker morphism.

4.3. HILBERT SCHEMES

253

Proposition 4.3.3 The Pl¨ ucker morphism is a closed embedding. In particular Gn (V ) is a projective variety. Proof. As in the proof of 4.3.2 we fix a basis of V ∨ and we choose a set J of n distinct indices in {1, . . . , N }. We obtain a decomposition V ∨ = E 0 ⊕ E 00 with dim(E 0 ) = n, dim(E 00 ) = N − n, and an induced one: ∧n V ∨ = ⊕ni=0 (∧n−i E 0 ) ⊗k ∧i E 00 = ∧n E 0 ⊕ F where F = ⊕ni=1 (∧n−i E 0 ) ⊗k ∧i E 00 . For every scheme S let

IPJ (S) =

locally free rk 1 quotients ∧n V ∨ → L s.t. the induced ∧n E 0 → L is surjective

We obtain a subfunctor IPJ of G∧n V,1 . As in the proof of E.18 we see that the IPJ ’s form an open cover of G∧n V,1 by functors representable by affine spaces. Note that for every locally free rank n quotient V ∨ ⊗k OS → F the induced homomorphism: E 0 ⊗k OS → F is surjective if and only if ∧n E 0 → ∧n F is. Therefore GJ = π −1 (IPJ ) and it suffices to prove that π : GJ → IPJ is a closed embedding. We may identify GJ with (the affine space associated to) Homk (E 0 , E 00 ) and IPJ with Homk (F, ∧n E 0 ). Considering that Homk (∧n−i E 0 , ∧n E 0 ) ∼ = ∧i E 0 canonically via the perfect pairing: ∧i E 0 × ∧n−i E 0 → ∧n E 0 we have: Homk (F, ∧n E 0 ) = ⊕ni=1 Homk ((∧n−i E 0 ) ⊗k ∧i E 00 , ∧n E 0 ) = = ⊕ni=1 Homk (∧i E 00 , Homk (∧n−i E 0 , ∧n E 0 )) = ⊕ni=1 Homk (∧i E 00 , ∧i E 0 ) and the map π : Homk (E 00 , E 0 ) → Homk (F, ∧n E 0 ) is λ 7→ (λ, ∧2 λ, . . . , ∧n λ)

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CHAPTER 4. HILBERT AND QUOT SCHEMES

This is the graph of a morphism of affine schemes, hence it is a closed embedding. qed For some 1 ≤ n ≤ r, let G = G(n + 1, r + 1) be the grassmannian of n-dimensional projective subspaces of IP r . Consider the projective bundle over G: I := IP (Q∨ ) = Proj(Sym(Q)) where Q is the universal quotient bundle on G. Because of the surjection r+1 OG → Q we have a closed embedding: I ⊂ IP r × G ↓p G r+1 note that Q∨ = p∗ II (1) ⊂ p∗ OIP r ×G (1) = OG . For every closed point v ∈ G the fibre I(v) is the projective subspace IP (v) ⊂ IP r ; for this reason I is called the incidence relation. Since all fibres of p have Hilbert polynomial t+n , n from Proposition 4.2.1 it follows that p is a flat family. Suppose now that

Λ ⊂ IP r × S ↓q S is another flat family whose fibres have Hilbert polynomial an inclusion of sheaves on S

t+n n

. We have

q∗ IΛ (1) ⊂ q∗ OIP r ×S (1) = OSr+1 which has locally free cokernel q∗ OΛ (1). By the universal property of G the above inclusion induces a unique morphism g:S→G such that g ∗ (Q∨ ) = q∗ IΛ (1). Since Λ = IP (q∗ IΛ (1)) it follows that Λ = S ×G I namely the family q is obtained by base change from the incidence relation via the morphism g. This proves that G(n + 1, r + 1) = Hilbr(t+n) n

4.3. HILBERT SCHEMES

4.3.4

255

Existence

Theorem 4.3.4 For every projective scheme Y ⊂ IP r and every numerical polynomial P (t), the Hilbert scheme HilbYP (t) exists and is a projective scheme. Proof. We will first prove the theorem in the case Y = IP r . From Corollary 4.1.10 it follows that there is an integer m0 such that for every closed subscheme X ⊂ IP r with Hilbert polynomial P (t) the sheaf of ideals IX is m0 -regular. It suffices to take m0 = Fr (−a0 , . . . , −ar−1 , 1 − ar ) It follows that for every k ≥ m0 hi (IP r , IX (k)) = 0

(4.12)

for i ≥ 1 and !

k+r h (IP , IX (k)) = − P (k) r 0

r

depends only on P (k). Moreover by Note 2 of Section 4.1 we have hi (X, OX (k)) = 0

(4.13)

all k ≥ m0 and all i ≥ 1. Let !

m0 + r N= − P (m0 ) r V = H 0 (IP r , OIP r (m0 )) Consider the grassmannian G = GN (V ) of N -dimensional vector subspaces of V . Let V ∨ ⊗k OG → Q be the universal quotient bundle, which is locally free sheaf of rank N on G, and p : IP r × G → G the projection. We may identify V ⊗k OG = p∗ [OIP r ×G (m0 )]. The image of the composition p∗ Q∨ (−m0 ) −→

V ⊗k OIP r ×G (−m0 ) −→ OIP r ×G k ∗ p p∗ [OIP r ×G (m0 )] ⊗ OIP r ×G (−m0 )

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CHAPTER 4. HILBERT AND QUOT SCHEMES

is a sheaf of ideals: we will denote it by J. Let Z ⊂ IP r × G be the closed subscheme defined by J and denote by q : Z → G the restriction of p to Z. Consider the flattening stratification G1

a

G2

a

... ⊂ G

for OZ and let H be the stratum relative to the polynomial P (t). We will prove that H = HilbrP (t) and that the universal family is the pullback of q to H: W := H ×G Z → Z ↓π ↓q H → G By the choice of H, W defines a flat family of closed subschemes of IP r with Hilbert polynomial equal to P (t). Let’s prove that W has the universal property. Consider a flat family of closed subschemes of IP r with Hilbert polynomial P (t): X ⊂ IP r × S ↓f S From (4.12) and (4.13) and from Theorem 4.2.5 it follows that R1 f∗ IX (m0 ) = (0) = R1 f∗ OX (m0 ) In particular we have an exact sequence on S: 0 → f∗ IX (m0 ) → f∗ OIP r ×S (m0 ) → f∗ OX (m0 ) → 0 k V ⊗k OS If we apply Theorem 4.2.5 for j = −1 we deduce that f∗ IX (m0 ) and f∗ OX (m0 ) are locally free and f∗ IX (m0 ) has rank N . From the universal property of G it follows that there exists a unique morphism g : S → G such that f∗ IX (m0 ) = g ∗ Q∨ Claim: For all m m0 we have f∗ OX (m) = g ∗ p∗ OZ (m).

4.3. HILBERT SCHEMES

257

Proof of the Claim: For all m m0 we have exact sequences: 0 → p∗ J(m) → q∗ OIP r ×G (m) → p∗ OZ (m) → 0 on G and 0 → f∗ IX (m) → f∗ OIP r ×S (m) → f∗ OX (m) → 0 on S; since g ∗ p∗ OIP r ×G (m) = f∗ OIP r ×S (m) it suffices to show that: f∗ IX (m) ∼ = g ∗ p∗ J(m) for all m m0 . For all such m we have the equality on G: p∗ J(m) = Im[Q∨ ⊗ p∗ O(m − m0 ) → p∗ OIP r ×G (m)] induced by the surjections p∗ Q∨ (m − m0 ) → J(m) of sheaves on IP r × G. Hence for all m m0 we have: g ∗ p∗ J(m) = g ∗ Im[Q∨ ⊗ p∗ OIP r ×G (m − m0 ) → p∗ OIP r ×G (m)] = = Im[g ∗ Q∨ ⊗ f∗ OIP r ×S (m − m0 ) → f∗ OIP r ×S (m)] = = Im[f∗ IX (m0 ) ⊗ f∗ OIP r ×S (m − m0 ) → f∗ OIP r ×S (m)] = f∗ IX (m) and this proves the Claim. From the Claim it follows that (i) g factors through H. Indeed from 4.2.4 it follows that for all m m0 : g ∗ q∗ OZ (m) = f∗ (1 × g)∗ OZ (m) Since g ∗ p∗ OZ (m) = f∗ OX (m) is locally free of rank P (m) for all such m Proposition 4.2.1 implies that (1 × g)∗ OZ is flat over S with Hilbert polynomial P (t). Hence g factors by the definition of H. (ii) X = S ×H W.

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Indeed X = Proj[ = Proj[

L

m0

L

m0

f∗ OX (m)] = Proj[

g ∗ π∗ OW (m)] = S ×H Proj[

L

L

m0

m0

g ∗ p∗ OZ (m)] =

π∗ OW (m)] = S ×H W

Properties (i) and (ii) imply that H = HilbrP (t) and that π is the universal family. By construction HilbrP (t) is a quasi-projective scheme. To prove that it is projective it suffices to show that it is proper over k. We will use the valuative criterion of properness. Let A be a discrete valuation k-algebra with quotient field L and residue field K, and let ϕ : Spec(L) → HilbrP (t) be any morphism. We must show that ϕ extends to a morphism ϕ˜ : Spec(A) → HilbrP (t) Pulling back the universal family by ϕ we obtain a flat family X ⊂ IP r × Spec(L) of closed subschemes of IP r with Hilbert polynomial P (t). Since Spec(A) is nonsingular of dimension one and Spec(L) = Spec(A)\{closed point} Proposition III.9.8 of [89] implies the existence of a flat family X 0 ⊂ IP r × Spec(A) which extends X . By the universal property of HilbrP (t) this family corresponds to a morphism ϕ˜ : Spec(A) → HilbrP (t) which extends ϕ. This concludes the proof of the theorem in the case Y = IP r . Let’s now assume that Y is an arbitrary closed subscheme of IP r . It will suffice to show that the functor HilbYP (t) is represented by a closed subscheme of HilbrP (t) . Applying Corollary 4.1.10 twice we can find an integer µ such that IY ⊂ OIP r is µ-regular and such that for every closed subscheme X ⊂ IP r with Hilbert polynomial P (t) the ideal sheaf IX ⊂ OIP r is µ-regular. Let V = H 0 (IP r , OIP r (µ)),

U = H 0 (IP r , IY (µ))

4.3. HILBERT SCHEMES

259

It follows from 4.2.5 and 4.2.6 that π∗ IW (µ) is a locally free subsheaf of V ⊗k OHilb with locally free cokernel. On HilbrP (t) consider the composition Ψ : U ⊗k OHilb → V ⊗k OHilb → V ⊗k OHilb /π∗ IW (µ) Let Z ⊂ HilbrP (t) be the closed subscheme defined by the condition Ψ = 0, or equivalently by the condition U ⊗k OZ ⊂ π∗ IW (µ) ⊗ OZ

(4.14)

Letting j : Z → HilbrP (t) be the inclusion, one easily sees that condition (4.14) implies that IY ×Z ⊂ (1 × j)∗ IW ⊂ OIP r ×Z hence that Z ×Hilb W ⊂ Y × Z ⊂ IP r × Z

(4.15)

It is straightforward to check that Z = HilbYP (t) and that (4.15) is the universal family. This concludes the proof of Theorem 4.3.4. qed For any projective scheme Y ⊂ IP r it is often convenient to consider the functor: HilbY : (schemes) → (sets) defined as: HilbY (S) =

a

HilbYP (t) (S)

P (t)

This functor is represented by the disjoint union HilbY =

a

HilbYP (t)

P (t)

which is a scheme locally of finite type (but not of finite type because it has infinitely many connected components unless dim(Y ) = 0). It is the Hilbert scheme of Y . One convenient feature of HilbY is that it is independent on the projective embedding of Y , even though the indexing of its components HilbYP (t) by Hilbert polynomials does depend on the embedding. For this reason, when considering HilbY we will not need to specify a projective embedding of Y . Let’s fix a projective scheme Y , and in the Hilbert scheme HilbY let’s consider a k-rational point [X] which parametrizes a closed subscheme X ⊂

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Y . Denote by I ⊂ OY the ideal sheaf of X in Y . The local Hilbert functor Y HX is a subfunctor of the restriction to A of the Hilbert functor; since HilbY represents the Hilbert functor we have, with the notation introduced in §2.2: Y HX (A) = Hom(Spec(A), HilbY )[X] Y for every A in ob(A). In particular HX is prorepresented by the local ring ˆ OHilb,[X] . We can therefore apply the results proved in §2.4 to obtain information about the local properties of HilbY at [X]. In particular we have the following:

Theorem 4.3.5 (i) There is a canonical isomorphism of k-vector spaces: T[X] HilbY ∼ = H 0 (X, NX/Y ) where NX/Y = HomOX (I/I 2 , OX ) is the normal sheaf of X in Y . (ii) If X ⊂ Y is a regular embedding then the obstruction space of OHilbY ,[X] is a subspace of H 1 (X, NX/Y ). The simplest illustration of Theorem 4.3.5 is for Y [1] = Y . In this case 4.3.5(i) simply says that Hom(mp /m2p , k) is the Zariski tangent space of Y at a k-rational point p ∈ Y . The obstruction space is o(OY [1] ,[p] ) = o(OY,p ). Of course if p is a singular point then it is not regularly embedded in Y , and H 1 (p, Np/Y ) = 0 is not an obstruction space for the local Hilbert functor. Consider a flat family of closed subschemes of Y : X ⊂ Y ×S ↓f S It induces a functorial morphism χ : S → HilbY (the classifying morphism of the family) whose differential at a k-rational point s ∈ S is a linear map dχs : Ts S → H 0 (X (s), NX (s)/Y )

(4.16)

called the characteristic map of the family f . Obviously the surjectivity of dχs is a necessary condition for the smoothness of χ at s. We have the following more precise result.

4.3. HILBERT SCHEMES

261

Proposition 4.3.6 Let Y be a projective scheme, X ⊂

Y ×S ↓f S

a flat family of closed subschemes of Y with S algebraic, and χ : S → HilbY the classifying map of the family. Then (i) If s is a nonsingular point of S and if the characteristic map dχs : Ts S → H 0 (X (s), NX (s)/Y ) is surjective then χ is smooth at s and X (s) is unobstructed in Y . (ii) If moreover h1 (Y, TY |X (s) ) = 0 then f has general moduli at s and X (s) is unobstructed as an abstract variety. Proof. (i) The smoothness of χ is a consequence of Theorem 2.1.5 and of the nonsingularity of S at s. The unobstructedness of X (s) in Y , i.e. the nonsingularity of HilbY at χ(s) follows from the smoothness of χ and the nonsingularity of S at s. (ii) The condition h1 (Y, TY |X (s) ) = 0 implies that the forgetful morphism Y HX (s) → Def X (s) is smooth (Proposition 3.2.9) and therefore the KodairaSpencer map of f at s is surjective being the composition of two surjective maps. We obtain the conclusion from Proposition 2.5.8. qed The following criterion follows at once from Proposition 3.2.9: Proposition 4.3.7 Let X ⊂ Y be a closed embedding of projective schemes such that h1 (X, TY |X ) = 0. Then the universal family X ↓ HilbY

⊂ Y × HilbY

has general moduli at the point [X].

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CHAPTER 4. HILBERT AND QUOT SCHEMES

Notes and Comments 1. The construction of the grassmannian given here is taken from [107]. 2. It is a classical result of Hartshorne that HilbrP (t) is connected for all r and P (t) (see [88] and [156]). For general Y this is no longer true: for example, if Y ⊂ IP 3 is a nonsingular quadric then HilbYt+1 has two connected components. 3. Let X ⊂ Y be a closed embedding of projective schemes. It can be easily verified that for any closed subscheme Z ⊂ X, the induced injective linear map H 0 (Z, NZ/X ) → H 0 (Z, NZ/Y ) coincides with the differential at [Z] of the closed embedding HilbX ⊂ HilbY 4. If Z ⊂ Y is a closed embedding of projective schemes and P (t) a numerical polynomial, then one can define a functor Hilb(−Z)YP (t) : (schemes/S)◦ → (sets) by Hilb(−Z)YP (t) (S) =

n flat families X ⊂ Y × S with Hilbert polynomial o

P (t) and such that X (s) ∩ Z 6= ∅, ∀s ∈ S

Then HilbY (−Z)YP (t) is representable by a closed subscheme Hilb(−Z)YP (t) ⊂ HilbYP (t) Let

Y \Z

HilbP (t) := HilbYP (t) \Hilb(−Z)YP (t) ⊂ HilbYP (t) be the corresponding open subscheme and define HilbY \Z :=

a

Y \Z

HilbP (t)

P (t)

Then this is a scheme locally of finite type which depends only on W := Y \Z and not on the embedding W ⊂ Y . In case P (t) = n a constant polynomial we denote [n] HilbW P (t) by W . If W is affine then HilbW =

a

W [n]

n

The proofs of these facts are left to the reader. See also [85].

4.4. QUOT SCHEMES

4.4 4.4.1

263

Quot schemes Existence

We will now introduce an important class of schemes, the so called Quot schemes, which generalize the Hilbert schemes. As special cases we will obtain the relative Hilbert schemes. Let p : X → S be a projective morphism of algebraic schemes, and let OX (1) be a line bundle on X very ample with respect to p. Fix a coherent sheaf H on X and a numerical polynomial P (t) ∈ Q[t]. We define a functor X/S

QuotH,P (t) : (schemes/S)◦ → (sets) called the Quot functor of X/S relative to H and P (t), in the following way: X/S

QuotH,P (t) (Z → S) =

quotients HZ → F , flat over Z, with o Hilbert polyn. P (t) on the fibres of XZ → Z

n coherent

where we have denoted XZ = Z ×S X and HZ the pullback of H on XZ , as X/Spec(k) usual. When S = Spec(k) we write QuotX . H,P (t) instead of QuotH,P (t) This definition generalizes the Hilbert functors which are obtained in the case S = Spec(k) and H = OX . X/S

Theorem 4.4.1 The functor Quot = QuotH,P (t) is represented by a projective S-scheme X/S QuotH,P (t) → S Proof. We first consider the case S = Spec(k) and X = IP r . From Theorem 4.1.9 it follows that there is an integer m such that for each scheme Z and for each (ϕ : HZ → F ) ∈ Quot(Z), letting N = ker(ϕ), all the sheaves N (z), H(z) = H, F (z), z ∈ Z, are m-regular. Therefore, letting pZ : IP r × Z → Z be the projection, we obtain an exact sequence of locally free sheaves on Z: 0 → pZ∗ N (m) → H 0 (IP r , H(m)) ⊗k OZ → pZ∗ F (m) → 0 Moreover, for each m0 ≥ m there is an exact sequence H 0 (IP r , O(m0 − m)) ⊗k pZ∗ N (m) → H 0 (IP r , H(m0 )) ⊗k OZ → pZ∗ F (m0 ) → 0

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CHAPTER 4. HILBERT AND QUOT SCHEMES

where the first map is given by multiplication of sections. This shows that pZ∗ F (m) uniquely determines the sheaf of graded OZ [X0 , . . . , Xr ]-modules L 0 r k≥m pZ∗ F (k), which in turn determines F . Therefore, letting H (IP , H(m)) = V , we have an injective morphism of functors: Quot → GVˇ ,P (m) given by: Quot(Z) → GVˇ ,P (m) (Z) (HZ → F ) 7→ [V ⊗k OZ → pZ∗ F (m)] On G = GP (m) (Vˇ ) consider the tautological exact sequence 0 → K → V ⊗k OG → Q → 0 Let moreover p2 : IP r ×G → G and p1 : IP r ×G → IP r be the projections. On G we have Γ∗ (H) ⊗k OG , which is a sheaf of graded OG [X0 , . . . , Xr ]-modules, and determines p∗1 (H). Consider the subsheaf KOG [X0 , . . . , Xr ] and the sheaf F on G × IP r corresponding to the quotient Γ∗ (H) ⊗k OG /KOG [X0 , . . . , Xr ], and let GP ⊂ G be the stratum corresponding to P of the flattening stratification of F. Then we claim that a morphism of schemes f : Z → G defines an element of Quot(Z) if and only if f factors through GP , and therefore Quot is represented by GP . The proof of this fact is similar to the one given for the proof of Theorem 4.3.4 and will be left to the reader. Since GP is quasi-projective, to prove that it is projective amounts to prove that it is proper over k, and this can be done using the valuative criterion of properness. Let A be a discrete valuation k-algebra with quotient field L and residue field K, and let ϕ : Spec(L) → GP be any morphism. We must show that ϕ extends to a morphism ϕ˜ : Spec(A) → GP The datum of ϕ corresponds to an element (ϕL : HL → FL ) of Quot(Spec(L)). The existence of ϕ˜ will be proved if there is a quotient ϕA : HA → FA on IP r × Spec(A) which is flat over Spec(A) and which restricts to FL over IP r × Spec(L). Let i : IP r × Spec(L) → IP r × Spec(A) be the inclusion, and take FA = i∗ (FL ). Obviously FA restricts to FL . Moreover, if KL = ker(ϕL ), we have R1 i∗ (KL ) = 0 and therefore a surjection HA = i∗ (HL ) → FA . We need the following

4.4. QUOT SCHEMES

265

Lemma 4.4.2 Let X be a scheme, U an open subset of X and i : U → X the inclusion. Then for every coherent sheaf F on U we have Ass(i∗ (F )) = Ass(F ) Proof. Since i∗ (F )|U = F we have Ass(i∗ (F )) ∩ U = Ass(F ). Therefore we only need to prove that Ass(i∗ (F )) ⊂ U . We may assume that X = Spec(A) and U = Spec(B) are affine. The inclusion i corresponds to an injective homomorphism A → B and F = M ∼ for a f.g. B-module M . Let x ∈ Ass(i∗ (F )) and assume that x ∈ X\U . Then the ideal px ⊂ A annihilates an element mx ∈ i∗ (F )x which corresponds to a section m ∈ Γ(V, i∗ (F )) for some open neighborhood V of x. Up to shrinking X we may assume V = X, so that m ∈ Γ(X, i∗ (F )) = Γ(U, F ) = M is annihilated by the ideal px B. But px B = B because x ∈ / U and therefore m = 0: this is a contradiction. The lemma is proved. From the lemma it follows that Ass(FA ) = Ass(FL ) : therefore, using the fact that FL is flat over Spec(L) and [89], Prop. III.9.7), we deduce that FA is flat over Spec(A). This concludes the proof of the theorem in the case S = Spec(k) and X = IP r . Assume now that S and X are arbitrary. Consider the closed embedding j : X → IP r × S determined by OX (1). Replacing H by j∗ H we can assume that X = IP r × S. Let h, h0 0 be such that we have an exact sequence: 0

OIP r ×S (−h0 )M → OIP r ×S (−h)M → H → 0 for some M, M 0 . Then for each S-scheme Z → S and for each IP r ×S/S

(HZ → F ) ∈ QuotH,P (t) (Z → S) we obtain that the composition OIP r ×Z (−h)M → HZ → F → 0 IP r ×S/S

is a surjection, i.e. is an element of QuotO(−h)M ,P (t) (Z → S). This proves IP r ×S/S

IP r ×S/S

that the functor QuotH,P (t) is a subfunctor of the functor QuotO(−h)M ,P (t) , Pr and this functor is evidently represented by QuotIO(−h) M ,P (t) × S. Conversely, a quotient IP r ×S/S

(OIP r ×Z (−h)M → F ) ∈ QuotOX (−h)M ,P (t) (Z → S)

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CHAPTER 4. HILBERT AND QUOT SCHEMES IP r ×S/S

is in QuotH,P (t) (Z → S) if and only if the composition 0

OIP r ×Z (−h0 )M → OIP r ×Z (−h)M → F is zero. This means that the condition for an S-morphism IP r /k

IP r ×S/S

Z → QuotO(−h)M ,P (t) = QuotO(−h)M ,P (t) × S IP r ×S/S

to define an element of QuotH,P (t) (Z → S) is that it factors through the closed subscheme defined by the entries of the matrix of the homomorphism: 0

OIP r ×Quot (−h0 )M → OIP r ×Quot (−h)M IP r ×S/S

and this is a closed condition. This proves that QuotH,P (t) by a closed subscheme of

IP r /k QuotO(−h)M ,P (t)

is represented

× S.

qed X/S

X/S

From the fact that QuotH,P (t) represents the functor QuotH,P (t) it follows that there is a universal quotient X/S

X/S

(HQuot → F) ∈ QuotH,P (t) (QuotH,P (t) ) corresponding to the identity morphism under the identification Hom(Quot, Quot) = Quot(Quot) X/S

X/S

In case H = OX the scheme QuotOX ,P (t) is denoted HilbP (t) and called the relative Hilbert scheme of X/S with respect to the polynomial P (t). It will be sometimes convenient to consider the functor X/S

QuotH

: (schemes/S)◦ → (sets)

defined as: X/S

QuotH (Z → S) =

a

X/S

QuotH,P (t) (Z → S)

P (t)

This functor is represented by the disjoint union X/S

QuotH

=

a

X/S

QuotH,P (t)

P (t)

which is a scheme locally of finite type, called the Quot scheme of X over S relative to H; it carries a universal quotient HQuot → F.

4.4. QUOT SCHEMES

267

Similarly we will consider the relative Hilbert scheme of X over S: HilbX/S =

a

X/S

HilbP (t)

P (t)

The construction of the Quot scheme commutes with base change; this is a result which follows quite directly from the definition, but it is worth pointing it out: Proposition 4.4.3 (base change property) Given a projective morphism X → S, a coherent sheaf H on X, and a morphism T → S, there is a natural identification: X/S X /T QuotHTT = T ×S QuotH Proof. Consider the product diagram X/S

T × QuotH ↓ T

X/S

→ QuotH ↓ → S

X/S

The universal quotient HX/S → F on QuotH pullsback to a quotient X/S X/S HXT /T → FT on T ×QuotH . It is immediate that the T -scheme T ×QuotH X /T qed endowed with this quotient represents the functor QuotHTT .

4.4.2

Local properties

Proposition 4.4.4 Let X → S be a projective morphism of algebraic schemes, X/S H a coherent sheaf on X, flat over S, and π : Q = QuotH → S the associated Quot scheme over S. Let s ∈ S be a k-rational point and q ∈ π −1 (s) = Q(s) corresponding to a coherent quotient f : H → F with kernel K. Let fs : H(s) → F(s) be the restriction of f to the fibre X(s), whose kernel is K(s) = K ⊗ OX(s) (by the flatness of F). Then there is an exact sequence dπq

0 → Hom(K(s), F(s)) → tq Q −→ ts S → Ext1OX(s) (K(s), F(s)) and an inclusion: ker[o(πq] )] ⊂ Ext1OX(s) (K(s), F(s))

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CHAPTER 4. HILBERT AND QUOT SCHEMES

where o(πq] ) : o(OQ,q ) → o(OS,s ) is the obstruction map of the local homomorphism πq] : OS,s → OQ,q . In particular π is smooth at q if Ext1OX(s) (K(s), F(s)) = 0. Proof. A vector in ker(dπq ) corresponds to a commutative diagram: Spec(k[]) → Q ↓ ↓π Spec(k) → S such that the upper horizontal arrow has image {q}. The above diagram corresponds to an exact and commutative diagram of sheaves on X(s): 0 0 ↓ ↓ K(s) −→ K(s) → 0 ↓ ↓ 0 → H(s) → H(s)[] −→ H(s) → 0 ↓ ↓ ↓ 0 → F(s) → F(s) −→ F(s) → 0 ↓ ↓ ↓ 0 0 0 where the middle row is exact by the flatness of H. Replacing the middle row by its pushout under H(s) → F(s) we see that this diagram is equivalent to the following one: 0 ↓ K(s) ↓ 0 → F(s) → P k ↓ 0 → F(s) → F(s) ↓ ↓ 0 0

0 ↓ = K(s) → 0 ↓ −→ H(s) → 0 ↓ −→ F(s) → 0 ↓ 0

and therefore we deduce that ker(dπq ) = Hom(K(s), F(s)). Now consider A in A and a commutative diagram ϕ

A ←− OQ,q ↑η ↑ πq] B ←− OS,s ϕ ˜

(4.17)

4.4. QUOT SCHEMES

269

where η is a small extension in A. This diagram corresponds to an exact diagram of sheaves on X:

γ : 0 → H(s) → H ⊗k B

0 ↓ KA ↓ −→ H ⊗k A → 0 ↓ FA ↓ 0

(4.18)

where the row is exact by the flatness of H over S. By pushing out by the quotient fs : H(s) → F(s) and then pulling back by α : KA → H ⊗k A we obtain an element [α∗ fs∗ (γ)] ∈ Ext1OX ⊗A (KA , F(s)) = Ext1OX(s) (K(s), F(s)) By construction this element vanishes if and only if the previous diagram can be embedded in a commutative diagram with exact rows and columns 0 0 ↓ ↓ 0 → K(s) → KB ↓ ↓ 0 → H(s) → H ⊗k B ↓ ↓ 0 → F(s) → FB ↓ ↓ 0 0

0 ↓ → KA →0 ↓ −→ H ⊗k A → 0 ↓ → FA ↓ 0 X/S

The middle column of this diagram is an element of QuotH (Spec(B)), which corresponds to a homomorphism ϕ0 : OQ,q → B making the diagram ϕ

A

←−

↑η

. ϕ0

B

←− ϕ ˜

OQ,q ↑ πq] OS,s

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CHAPTER 4. HILBERT AND QUOT SCHEMES

commutative. Therefore we have associated an element of Ext1OX(s) (K(s), F(s)) to each diagram (4.18). It is straightforward to check that this correspondence is linear. Taking η : k[] → k we get an inclusion coker(dπq ) ⊂ Ext1OX(s) (K(s), F(s)) Taking any small extension η in A we can apply Proposition 2.1.7 to yield the conclusion. qed Corollary 4.4.5 Under the same assumptions of 4.4.4, if Ext1OX(s) (K(s), F(s)) = 0 then π : Q → S is smooth at q of relative dimension dim[Hom(K(s), F(s))]. When S = Spec(k) we obtain the following “absolute” version of Proposition 2.1.7. Corollary 4.4.6 If X is a projective scheme, H a coherent sheaf on X and f : H → F a coherent quotient of H with ker(f ) = K then, letting Q = QuotX H , we have: T[f ] Q = Hom(K, F) and the obstruction space of OQ,[f ] is a subspace of Ext1 (K, F). In particular, if Ext1 (K, F) = 0 then Q is nonsingular of dimension dim(Hom(K, F)) at [f ]. A special case of Proposition 4.4.4 is the following: Proposition 4.4.7 Let p : X → S be a projective flat morphism of algebraic schemes, and π : HilbX /S → S the relative Hilbert scheme. For a closed point s ∈ S let X = X (s) be the fibre over s and let Z ⊂ X be a closed subscheme with ideal sheaf I ⊂ OX . Then there is an exact sequence: dπ[Z]

0 → H 0 (Z, NZ/X ) → T[Z] HilbX /S −→ Ts S → Ext1OX (I, OZ ) If moreover Z ⊂ X is a regular embedding then the above exact sequence becomes: dπ[Z]

0 → H 0 (Z, NZ/X ) → T[Z] HilbX /S −→ Ts S → H 1 (Z, NZ/X )

(4.19)

If Ext1OX (I, OZ ) = (0) (resp. H 1 (Z, NZ/X ) = (0) in case Z ⊂ X is a regular embedding) then π is smooth at [Z] of relative dimension h0 (Z, NZ/X ).

4.5. FLAG HILBERT SCHEMES

271

Notes and Comments 1. One should compare the statement of 4.4.7 with 3.2.12, since the local relative Hilbert functor of Z in X relative to X → S is prorepresented by the local ˆ X/S . ring O Hilb [Z]

2. Our presentation of the Quot schemes is an adaptation of the one given in [98]. For a description of the sheaf of differentials of the Quot schemes see [130]. 3. When p : X → S is a projective flat morphism of integral schemes the relative Hilbert scheme HilbX /S has a component isomorphic to S, parametrizing the fibres of p, and a component isomorphic to X , identified with X [1] .

4.5

Flag Hilbert schemes

4.5.1

Existence

Fix an integer r ≥ 1 and an m-tuple of numerical polynomials P(t) = (P1 (t), . . . , Pm (t)),

m≥1

For every scheme S we let: ( r F HP(t) (S) =

) X1 ⊂ · · · ⊂ Xm ⊂ IP r × S (X1 , . . . , Xm ) : S-flat closed subschemes with Hilbert polynomials P(t)

This clearly defines a contravariant functor: r F HP(t) : (schemes)◦ → (sets)

called the flag Hilbert functor of IP r relative to P(t). When m = 1 the flag Hilbert functors are just ordinary Hilbert functors. Theorem 4.5.1 For every r ≥ 1 and P(t) as above the flag Hilbert functor r F HP(t) is represented by a projective scheme FHrP(t) , called the flag Hilbert scheme of IP r relative to P(t), and by a universal family: W1 ⊂ · · · ⊂ Wm ⊂ IP r × FHrP(t) ↓ FHrP(t)

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CHAPTER 4. HILBERT AND QUOT SCHEMES

Proof. We will only prove the theorem in the case m = 2, leaving to the reader the task of extending the proof to the general case. By applying Corollary 4.1.10 twice we can find an integer µ such that simultaneously for i = 1, 2 we have that for every closed subscheme Xi ⊂ IP r with Hilbert polynomial Pi (t) the sheaf of ideals IXi is µ-regular. For every such X1 and X2 we thus have: !

µ+r h (IP , IXi (µ)) = − Pi (t) =: Ni r 0

r

i = 1, 2

Let V = H 0 (IP r , OIP r (µ)). Consider the Hilbert scheme Hi = HilbrP1 (t) with universal family Vi ⊂ IP r × Hi , i = 1, 2. On the product H1 × H2 consider the pullback of the two universal families with respect to the projections: Vi ×Hi (H1 × H2 ) ⊂ IP r × H1 × H2 ,

i = 1, 2

and denote by q : IP r × H1 × H2 → H1 × H2 the projection. Because of the choice of µ and by Theorem 4.2.5 we have that q∗ Ii (µ) is a locally free subsheaf of V ⊗k OH1 ×H2 of rank Ni , with locally free cokernel, i = 1, 2. Consider the composition ϕ : q∗ I2 (µ) ⊂ V ⊗k OH1 ×H2 → V ⊗k OH1 ×H2 /q∗ I1 (µ) and let F ⊂ H1 × H2 be the vanishing scheme of ϕ (Example 4.2.8). Note that we have q∗ I2 (µ) ⊗ OF ⊂ q∗ I1 (µ) ⊗ OF ⊂ V ⊗ OF (4.20) We now pullback to F the two universal families, i = 1, 2: IP r × F IP r × Hi ∪ ∪ Wi := Vi ×Hi F → Vi ↓k F

↓ →

Hi

Claim: q∗ Ii (µ) ⊗ OF = k∗ IWi (µ), i = 1, 2. We have natural homomorphisms: βi : q∗ Ii (µ) ⊗ OF → k∗ IWi (µ)

4.5. FLAG HILBERT SCHEMES

273

Because of the µ-regularity and of Theorem 4.2.5 for every x ∈ F we have isomorphisms q∗ Ii (µ)x ⊗ k(x) ∼ = H 0 (IP r , IWi (x) (µ)) ∼ = k∗ IWi (µ)x ⊗ k(x) Since the sheaf k∗ IWi (µ) is locally free from Nakayama’s lemma it follows that β1 and β2 are isomorphisms, and this proves the claim. From the claim and from (4.20) we deduce that k∗ IW2 (µ) ⊂ k∗ IW1 (µ)

(4.21)

The µ-regularity and Proposition 4.1.1(iii) imply that the natural homomorphisms k ∗ k∗ IWi (µ) → IWi (µ), i = 1, 2 are surjective; because of (4.21) we deduce that IW2 ⊂ IW1 , hence W1 ⊂ W2 . r Therefore (W1 , W2 ) ∈ F HP(t) (F). r Claim: The pair (F, (W1 , W2 )) represents the functor F HP(t) . r Let S be a scheme and let (X1 , X2 ) ∈ F HP(t) (S). By definition X1 ⊂ r X2 ⊂ IP × S are flat over S with Hilbert polynomials P1 (t) and P2 (t) respectively. Let f : IP r × S → S be the projection. We have induced classifying morphisms g1 : S → H1 , g2 : S → H2

which together define a morphism g : S → H 1 × H2 Arguing as before we see that g ∗ q∗ Ii (µ) ∼ = f∗ IXi (µ),

i = 1, 2

(4.22)

The fact that X1 ⊂ X2 implies that IX2 (µ) ⊂ IX1 (µ), hence that f∗ IX2 (µ) ⊂ f∗ IX1 (µ) This, together with (4.22), in turn implies that g factors through F. Since clearly we have X1 = S ×F W1 , X2 = S ×F W2 the claim follows, hence F = FHrP(t) . Moreover FHrP(t) is projective because it is a closed subscheme of H1 × H2 . qed

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From the definition it follows that the closed points of FHrP(t) are in 1-1 correspondence with the m-tuples (X1 , . . . , Xm ) of closed subschemes of IP r such that Xi has Hilbert polynomial Pi (t) and X1 ⊂ X2 ⊂ · · · ⊂ Xm Such an m-tuple is called a flag of closed subschemes of IP r . We will denote by [X1 , . . . , Xm ] the point of FHrP(t) parametrizing such a flag. From the proof of Theorem 4.5.1 it follows that FHrP(t) is a closed subscheme of m Y

HilbrPi (t)

i=1

We will denote the projections by pri : FHrP(t) → HilbrPi (t) ,

i = 1, . . . , m

For every subset I ⊂ {1, . . . , m} with cardinality µ we can consider the µ-tuple of polynomials PI (t) = (Pi1 , . . . , Piµ ) and the flag Hilbert scheme FHrPI (t) . We have natural projection morphisms prI : FHrP(t) → FHrPI (t) of which the pri ’s are special cases. The flag Hilbert schemes are generalizations of the flag varieties (fibr´es en drapeaux in [81], ch. 1, §9.9), which parametrize flags of linear subspaces of IP r . If Z ⊂ IP r is a closed subscheme having Hilbert polynomial Q(t), and P(t) is an m-tuple of numerical polynomials as above, one can define the flag Hilbert scheme of Z relative to P(t) Z F HP(t) : (schemes)◦ → (sets) r by an obvious modification of the definition of F HP(t) . It is straightforward Z to prove that F HP(t) is represented by a closed subscheme FHZP(t) of the scheme FHrP(t) ˜ , where

˜ P(t) = (P1 (t), . . . , Pm (t), Q(t)) Precisely, letting r prQ : F HP(t) → HilbrQ(t) ˜

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275

be the projection, one has an identification of FHZP(t) with the scheme−1 theoretic fibre prQ ([Z]) of prQ over the point [Z] ∈ HilbrQ(t) . It is convenient to consider the disjoint union FHZ =

a

FHZP(t)

P(t)

which is a scheme locally of finite type, and call it the flag Hilbert scheme of Z. Another variation on the same theme is the following. Given closed subschemes X ⊂ Z ⊂ IP r , and an m-tuple of numerical polynomials P(t), one can consider flags of closed subschemes of Z containing X, namely m-tuples (Y1 , . . . , Ym ) of closed subschemes of Z having Hilbert polynomials P(t) and such that X ⊂ Y1 ⊂ Y2 · · · ⊂ Ym ⊂ Z Again there is an obvious generalization of the definition of the corresponding flag Hilbert functor and of the proof of its representability by a projective scheme. These generalized flag Hilbert schemes can be fruitfully used in concrete geometrical situations, like for example in the study of families of pointed closed subschemes, or of reducible closed subschemes, of a given projective scheme.

4.5.2

Local properties

Consider a projective scheme Z and let (X1 , . . . , Xm ) be a flag of closed subschemes of Z; let Ii ⊂ OZ be the ideal sheaf of Xi . We have inclusions Im ⊂ Im−1 ⊂ · · · ⊂ I1 and surjections OXm → OXm−1 → · · · → OX1 → 0 Definition 4.5.2 The normal sheaf of (X1 , . . . , Xm ) in Z is the sheaf of germs of commutative diagrams of homomorphisms of OZ -modules of the following form: Im ⊂ Im−1 ⊂ · · · ⊂ I1 ↓ σm ↓ σm−1 ↓ σ1 OXm → OXm−1 → · · · → OX1 It is denoted N(X1 ,...,Xm )/Z .

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Note that we have an obvious homomorphism of “projection”: N(X1 ,...,Xm )/Z → N(Xi1 ,...,Xiµ )/Z for each choice of a subset I = {i1 , . . . , iµ } ⊂ {1, . . . , m}. In particular we have homomorphisms: N(X1 ,...,Xm )/Z → NXi /Z ,

i = 1, . . . , m

Proposition 4.5.3 Let Z be a projective scheme and let (X1 , . . . , Xm ) be a flag of closed subschemes of Z. Then (i) there is a natural identification: T[X1 ,...,Xm ] FHZ = H 0 (Z, N(X1 ,...,Xm )/Z )

(4.23)

(ii) If Xi ⊂ Xi+1 is a regular embedding for all i = 1, . . . , m − 1 and Xm is regularly embedded in Z then the obstruction space of the local ring OFHZ ,[X1 ,...,Xm ] is contained in H 1 (Z, N(X1 ,...,Xm )/Z ). Proof. For simplicity we give the proof in the case m = 2, i.e. in the case of a flag (X, Y ) of closed subschemes of Z. The general case can be treated similarly. (i) Let IX ⊂ OZ and IY ⊂ OZ be the ideal sheaves of X and of Y . We can represent a first order deformation Y ⊂ Z × Spec(k[]) of Y in Z by an ideal sheaf IY ⊂ OZ [] fitting into the commutative and exact diagram:

0 → OZ ↓ 0 → OY

0 0 ↓ ↓ IY → IY → 0 ↓ ↓ → OZ [] → OZ → 0 ↓ ↓ → OY → OY → 0

The flatness of Y over Spec(k[]) follows from the exactness of the last row (Lemma A.30). This diagram is equivalent to the following one, deduced

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277

after pushing out the second row by the homomorphism OZ → OY :

0 → OY k 0 → OY

0 0 ↓ ↓ IY = IY → 0 ↓ ↓ → OY ⊕ OZ → OZ → 0 ↓ ↓ → OY → OY → 0

Therefore we have a convenient representation of the given first order deformation as the middle vertical exact sequence of the previous diagram, which we rewrite: 0 → IY → OY ⊕ OZ → OY → 0 Composing the first inclusion with the first projection OY ⊕ OZ → OY we obtain the section σY ∈ H 0 (Y, NY /Z ) = HomOZ (IY , OY ) corresponding to Y. Let’s assume now that we have an element (X , Y) ∈ FHZ (k[]). By putting together the above constructions for X and for Y we obtain the following commutative diagram: 0 → IY → OY ⊕ OZ → OY → 0 ∩ ↓ ↓ 0 → IX → OX ⊕ OZ → OX → 0 where the condition X ⊂ Y corresponds to the condition OY → OX , and this is in turn equivalent to the condition that composing the inclusions on the left with the first projections in the middle terms we obtain a commutative diagram: IY ⊂ IX ↓ σY ↓ σX OY → OX This is the global section of N(X,Y )/Z corresponding to (X , Y) in (4.23). Conversely, given a global section IY ⊂ IX ↓σ ↓τ OY → OX

∈ H 0 (Z, N(X,Y )/Z )

one finds a first order deformation of (X, Y ) by repeating backwards the above construction.

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(ii) Let A be in ob(A) and let X ⊂ Y ⊂ Z × Spec(A)

ξ:

be an infinitesimal deformation of the flag (X, Y ). Let 0 → k → A˜ → A → 0

η:

define an element [e] ∈ Exk (A, k), where mA˜ = 0. Since the embeddings X ⊂ Y and Y ⊂ Z are regular we can find an affine open cover U = {Ui = Spec(Pi )}i∈I of Z such that Yi := Y ∩ Ui is a complete intersection in Ui and Xi := X ∩Ui is a complete intersections in Yi . Therefore, letting Xi = X ∩Ui , Yi = Y ∩ Ui , there are liftings ˜ Y˜i ⊂ Ui × Spec(A) ∪ ∪ Yi ⊂ Ui × Spec(A) and X˜i ⊂ Y˜i ∪ ∪ Xi ⊂ Y i which together give local liftings of the flags (Xi , Yi ): ˜ X˜i ⊂ Y˜i ⊂ Ui × Spec(A) ∪ ∪ ∪ Xi ⊂ Yi ⊂ Ui × Spec(A) ˜ of (X , Y) we must be able to choose the local In order to find a lifting (X˜ , Y) liftings (X˜i , Y˜i ) so that on every Uij := Ui ∩ Uj we have (X˜i|Uij , Y˜i|Uij ) = (X˜j|Uij , Y˜j|Uij ) At the level of ideals we have: IYi ⊂ IXi ⊂ Pi IYi ⊂ IXi ⊂ PiA := Pi ⊗k A IY˜i ⊂ IX˜i ⊂ PiA˜ := Pi ⊗k A˜

4.5. FLAG HILBERT SCHEMES

279

We have a commutative and exact diagram:

0→

Pi ↓ 0 → Pi /IXi

IX˜i → IXi →0 ↓ ↓ → PiA˜ → PiA →0 ↓ ↓ → PiA˜ /IX˜i → PiA /IXi → 0

which, after pushing out the middle row by Pi → Pi /IXi , gives the following one: IXi = IXi ↓ ↓ 0 → Pi /IXi → QiA → PiA →0 k ↓ ↓ 0 → Pi /IXi → PiA˜ /IX˜i → PiA /IXi → 0 where we have denoted QiA = (Pi /IXi )

a

PiA

Pi

therefore the datum of X˜i corresponds to the middle vertical sequence of this diagram: (4.24) 0 → IXi → QiA → PiA˜ /IX˜i → 0 Repeating the analogous construction for Y˜i we obtain that Y˜i is determined by an exact sequence: 0 → IYi → QiA → PiA˜ /IY˜i → 0

(4.25)

Since X˜i ⊂ Y˜i the exact sequences (4.24) and (4.25) fit together: 0 → IYi → QiA → PiA˜ /IY˜i → 0 ∩ k ↓ 0 → IXi → QiA → PiA˜ /IX˜i → 0 A different choice of the lifting X˜i corresponds to a different homomorphism IXi → QiA , and they differ by an element σX˜i ∈ HomPiA˜ (IXi , Pi /IXi ) = HomPi (IXi , Pi /IXi )

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CHAPTER 4. HILBERT AND QUOT SCHEMES

Similarly, a different choice of the lifting Y˜i corresponds to an element σY˜i ∈ HomPi (IYi , Pi /IYi ) The condition that the pair (σX˜i , σY˜i ) defines another lifting of the flag (Xi , Yi ) is that the following diagram IYi ⊂ IXi ↓ σX˜i ↓ σY˜i Pi /IYi → Pi /IXi commutes. This is precisely the condition that (σX˜i , σY˜i ) ∈ Γ(Ui , N(X,Y )/Z ). ˜ is in 1-1 correspondence Therefore the set of liftings of (Xi , Yi ) over Spec(A) with Γ(Ui , N(X,Y )/Z ). It follows from this analysis that for all i, j ∈ I we have a section (σij , τij ) ∈ Γ(Uij , N(X,Y )/Z ) such that (X˜j|Uij , Y˜j|Uij ) is obtained from (X˜i|Uij , Y˜i|Uij ) after modifying it by the section (σij , τij ). The collection of these sections is a 1-cocycle {(σij , τij )} ∈ Z 1 (U, N(X,Y )/Z ) which defines a cohomology class o(X ,Y) ([e]). It is straightforward to verify that this class is independent on the choices made and that o(X ,Y) ([e]) = 0 if ˜ exists. and only if a lifting (X˜ , Y) qed If a flag (X1 , . . . , Xm ) of closed subschemes of a projective scheme Z satisfies the conditions of Proposition 4.5.3(ii), i.e; if Xi ⊂ Xi+1 is a regular embedding for all i = 1, . . . , m − 1 and Xm is regularly embedded in Z, we say that the flag (X1 , . . . , Xm ) is regularly embedded in Z. Remarks 4.5.4 (i) One can adapt the above proof to the case m = 1 to obtain another, more intrinsic, proof of Propositions 3.2.1(ii) and 3.2.6. (ii) In the case m = 2 considered in the proof of Proposition 4.5.3, to the flag X ⊂ Y of closed subschemes of Z there is associated a diagram of normal sheaves:

0 → NX/Y → NX/Z

NY /Z ↓ → NY /Z ⊗OZ OX

and it is immediate to check that there is a natural identification H 0 (Z, N(X,Y )/Z ) = H 0 (X, NX/Z ) ×H 0 (X,NY /Z ⊗OZ OX ) H 0 (Y, NY /Z )

4.6. EXAMPLES AND APPLICATIONS

281

(iii) It is an immediate consequence of the definition that, given a flag (X1 , . . . , Xm ) of closed subschemes of a projective scheme Z, the normal sheaf N(X1 ,...,Xm )/Z is an OXm -module. This implies that in the case when we have a regularly embedded flag consisting of 0-dimensional subschemes of Z, we have H 1 (Z, N(X1 ,...,Xm )/Z ) = H 1 (Xm , N(X1 ,...,Xm )/Z ) = 0 and from Proposition 4.5.3(ii) we deduce that FHZ is nonsingular at [X1 , . . . , Xm ]. For example if we consider a projective nonsingular curve Z then it follows that FHZ(n1 ,...,nm ) , which parametrizes flags of effective divisors (D1 , . . . , Dm ) of degrees n1 < n2 < · · · < nm , is nonsingular. (iv) It is possibile to give a notion of flag Quot scheme generalizing the flag Hilbert schemes. This seems not to have been considered in the literature yet.

Notes and Comments 1. The flag Hilbert schemes have been considered in [114]. The proof of Theorem 4.5.1 given here has appeared in [183]. More recent references about flag Hilbert schemes are [32] and [65] (where they are called “nested Hilbert schemes”) and [168].

4.6 4.6.1

Examples and applications Complete intersections

We have already discussed some properties of the local Hilbert functor of a complete intersection X ⊂ IP r , which of course correspond to local properties of the Hilbert scheme Hilbr at [X]. It is easy to check that, despite the fact that H 1 (X, NX ) 6= (0) in general, every complete intersection X is unobstructed in IP r . We may assume dim(X) > 0. Let’s suppose that X ⊂ IP r , r ≥ 2, is the complete intersection of r − n hypersurfaces f1 , . . . , fr−n of degrees d1 ≤ d2 ≤ . . . ≤ dr−n respectively, n < r. Consider a basis Φ(1) , . . . , Φ(m) of ⊕j H 0 (IP r , O(dj )) where (h)

(h)

Φ(h) = (φ1 , . . . , φr−n )

282

CHAPTER 4. HILBERT AND QUOT SCHEMES (h)

h = 1, . . . , m, and the φj ∈ k[X0 , . . . , Xr ]. Consider indeterminates u1 , . . . , um and the (r − n)-tuple f+

m X

(1)

(m)

(1)

(m)

uh Φ(h) = (f1 + u1 φ1 + · · · + um φ1 , . . . , fr−n + u1 φr−n + · · · + um φr−n )

h=1

(4.26) of elements of the polynomial ring k[u, x] = k[u1 , . . . , um , X0 , . . . , Xr ]. P Let K• (f + h uh Φ(h) ) be the Koszul complex relative to (4.26) and ∆ := Supp[H1 (K• (f +

X

uh Φ(h) ))] ⊂ Am+r+1 = Spec(k[u, x])

h

Denoting by p : Am+r+1 → Am the projection, U := Am \p(∆) is the set P of points u ∈ Am such that K• (f + h th Φ(h) ) is exact; U is an open set containing the origin. In IP r × Am consider the closed subscheme

(1)

(m)

(1)

(m)

X = Proj k[u, x]/(f1 +u1 φ1 +· · ·+um φ1 , . . . , fr−n +u1 φr−n +· · ·+um φr−n )

the projection π : X → Am and its restriction πU : XU → U , where XU := π −1 (U ). All the fibres of πU are complete intersections of multidegree (d1 , . . . , dr−n ) and X (0) = X. The Hilbert polynomial of a complete intersection depends only on its multidegree because it can be computed using the Koszul complex: it follows that all the fibres of πU have the same Hilbert polynomial P (t) and therefore πU is a flat family of deformations of X in IP r . In an obvious way the tangent space of U at 0 can be identified with ⊕j H 0 (IP r , O(dj )), and the characteristic map with the restriction ϕ : ⊕j H 0 (IP r , O(dj )) → ⊕j H 0 (X, OX (dj )) Since dim(X) > 0 the map ϕ is surjective, as one easily verifies using the Koszul complex; since moreover U is nonsingular at 0 from Proposition 4.3.6 it follows that HilbrP (t) is smooth at [X] and the classifying map is smooth. From this it also follows that complete intersections are parametrized by an open subset of HilbrP (t) . It is interesting to observe that the closure of this open set may contain points parametrizing nonsingular subschemes of IP r which are not complete intersections. An example of such a subscheme is given by a trigonal canonical curve C ⊂ IP 4 : the quadrics containing C intersect in a rational cubic surface S, so it is not a complete intersection since it has degree 8; but [C]

4.6. EXAMPLES AND APPLICATIONS

283

is in the closure of the family of complete intersections of three quadrics. It is apparently unknown whether a similar phenomenon may occur in IP 3 , namely whether there are nonsingular curves in IP 3 which are flat limits of complete intersections without being complete intersections. See [52] for more about this. The Kodaira-Spencer map of the families πU has been studied in [181] in the case of complete intersections of dimension ≥ 2: πU has general moduli except for surfaces of multidegrees (4), (2, 3), (2, 2, 2) (respectively in IP 3 , in IP 4 and in IP 5 ), i.e. for complete intersection K3-surfaces. The special case of hypersurfaces had already been considered in [120] (see Example 3.2.11(i), page 159).

4.6.2

An obstructed nonsingular curve in IP 3 3

We will prove that the Hilbert scheme HilbIP has an everywhere non-reduced component Σ whose general point parametrizes a nonsingular curve of degree 14 and genus 24. It will follow that every curve parametrized by a general point of Σ is obstructed in IP 3 . This example is due to Mumford ([143]). A general element of Σ is constructed as follows. Let F ⊂ IP 3 be a nonsingular cubic surface, E, H ⊂ F respectively a line and a plane section in F . Let C ⊂ F be a general member of the linear system |4H + 2E|. Using Bertini’s theorem one easily checks that C is irreducible and nonsingular; its degree and genus are (C · H) = 14 and 21 (C − H · C) + 1 = 24. From the exact sequence: 0 → KC (H) → NC → OC (3H) → 0 we see that h1 (C, NC ) = h1 (C, OC (3H)) = h0 (C, KC (−3H)) = h0 (C, OC (2E)) = 1 where the last equality follows easily from the exact sequence 0 → OF (−4H) → OF (2E) → OC (2E) → 0 and from h0 (OF (−4H)) = 0 = h1 (OF (−4H)) and h0 (OF (2E)) = h0 (OF (E)) = 1. Moreover the linear system |C| = |4H + 2E| has dimension dim(|C|) = 1 + dim(|C|C ) = h0 (C, KC (H)) = 37

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CHAPTER 4. HILBERT AND QUOT SCHEMES

and therefore, since every curve C is contained in a unique cubic surface (because 9 < 14), the dimension of the family W of all curves C we are considering is 19 + 37 = 56 = 4 · 14 but they satisfy h0 (C, NC ) = 56 + h1 (C, NC ) = 57. We will prove that W is an open set of a component Σ of 3 HilbIP and this will imply that Σ is everywhere nonreduced. Our assertion will be proved if we show that our curves C are not contained in a family whose general member is a curve D not contained in a cubic surface. But on every such D the line bundle OD (4) is nonspecial and therefore, by RiemannRoch, h0 (D, OD (4)) = 33, hence D is contained in a pencil of quartic surfaces. Let G1 , G2 be two linearly independent quartics containing D: they are both irreducible because otherwise D would be either contained in a plane or in a quadric, which is not the case because there are no nonsingular curves of degree 14 and genus 24 on such surfaces. We have G1 ∩ G2 = D ∪ q where q is a conic; since q has at most double points D has at most triple points and therefore G1 and G2 cannot be simultaneously singular at any point of D, thus the general quartic surface G containing D is nonsingular along D. By applying Riemann-Roch on G we obtain dim(|D|G ) = 24. Therefore, since G is not a general quartic surface (because D is not a complete intersection), we see that the family of pairs (D, G) has dimension ≤ 33 + 24 = 57 so that the family Z of curves D has dimension ≤ 56. This shows that the family W , which has dimension 56, cannot be in the closure of Z and this proves the assertion. It is instructive to observe that we can write the linear system |C| on a nonsingular cubic surface F as |4H + 2E| = |6H − 2(H − E)| and this means that we can find a sextic surface F6 such that F ∩ F6 = C ∪ q1 ∪ q2 where q1 and q2 are disjoint conics; if [C] ∈ Σ is general then one can show that q1 , q2 and F6 can be chosen to be nonsingular. 3 There is another component R of HilbIP whose general point parametrizes a nonsingular curve C 0 of degree 14 and genus 24 such that C 0 ∪ E ∪ Γ = F3 ∩ F6 where E is a line and Γ is a rational normal cubic which are disjoint. We have in this case |C 0 | = |6H − E − Γ| and h1 (C 0 , NC 0 ) = h1 (C 0 , OC 0 (3H)) = h0 (C 0 , KC 0 (−3H)) = = h0 (C 0 , OC 0 (2H − E − Γ)) = h0 (F3 , OF3 (2H − E − Γ)) = 0 Thus C 0 is unobstructed.

4.6. EXAMPLES AND APPLICATIONS

285

We refer the reader to [42] for another point of view about this example. This has been the first published example of an obstructed space curve. Many others have appeared in the literature since thereafter (see [82], [182], [83], [51], [115], [210], [23], [137], [84]). A final word to the search for pathologies of this kind is contained in [204], where it is shown that virtually every 3 singularity can appear as a point of HilbIP parametrizing a nonsingular curve.

4.6.3

An obstructed (non-reduced) scheme

In IP 3 consider the scheme

X = Proj k[X0 , . . . , X3 ]/J

where J = (X1 X2 , X1 X3 , X2 X3 , X32 ) X is supported on the reducible conic defined by the equations X1 X2 = 0,

X3 = 0

has an embedded point at (1, 0, 0, 0) and has Hilbert polynomial 2(t + 1) (see Example 4.2.3(ii)). As in 4.2.3(ii) we consider the flat family parametrized by A1 :

X = Proj k[u, X]/(X1 X2 , X1 X3 , X2 (X3 − uX0 ), X3 (X2 − uX0 )) ⊂ IP 3 × A1 where k[u, X] = k[u, X0 , . . . , X3 ]. We have X = X (0). If u 6= 0 then X (u) is a pair of disjoint lines. Let g : A1 → Hilb32(t+1) be the classifying map. If u 6= 0 we have h1 (X (u), NX (u) ) = 0;

h0 (X (u), NX (u) ) = 8

Therefore g(u) is a smooth point and the tangent space has dimension 8. In order to show that X is obstructed it suffices to show that h0 (X, NX ) > 8

(4.27)

because g(0) and g(u) belong to the same irreducible component of Hilb32(t+1) .

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Consider the surjection f : OIP 3 (−2)⊕4 → IX → 0 determined by the four equations of degree two which define X. Elementary computations, based on the fact that the generators of the ideal are monomials, lead to the following resolution of OX which extends f : B

A

f

0 → OIP 3 (−4) −→ OIP 3 (−3)⊕4 −→ OIP 3 (−2)⊕4 −→ OIP 3 → OX → 0 (4.28) A and B being given by the following matrices: X3  −X  3 B=    X2  −X1 



X3  −X 2 A=   0 0 

X3 0 −X1 0

0 X3 0 −X1

0 0    X3  −X2 

By taking Hom(−, OX ) we obtain the following exact sequence: tA

0 → NX → OX (2)⊕4 −→ OX (3)⊕4 from which we deduce that tA

H 0 (X, NX ) = ker[H 0 (OX (2))⊕4 −→ H 0 (X, OX (3))⊕4 ]

(4.29)

Using resolution (4.28) it is easy to show that the restriction maps ϕn : H 0 (IP 3 , O(n)) → H 0 (X, OX (n)) are surjective if n ≥ 2. This allows us to identify H 0 (X, OX (2)) and H 0 (X, OX (3)) with the homogeneous parts of degree 2 and 3 respectively of k[X0 , . . . , X3 ]/J. Hence using (4.29) we can represent H 0 (X, NX ) by 4-tuples of polynomials. Precisely H 0 (X, NX ) is, modulo J, the vector space of 4-tuples q = (q1 , q2 , q3 , q4 ) of homogeneous polynomials of degree 2 such that A t q ∈ (J3 )4 . It is easy to find all of them because J is generated by monomials. Computing one finds that a basis of H 0 (X, NX ) is defined by the following column vectors: X12 0 0 0

X1 X0 0 0 0

X22 0 0 0

X2 X0 0 0 0

X3 X0 0 0 0

0 X12 0 0

0 X1 X0 0 0

0 X3 X0 0 0

0 0 X22 0

0 0 X2 X0 0

0 0 X3 X0 0

0 0 0 X3 X0

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287

In particular we see that h0 (X, NX ) = 12, and this proves (4.27). A little extra work shows that [X] = g(0) belongs to two irreducible components of Hilb32(t+1) . We already know one of them of dimension 8: it contains g(u), u 6= 0, and a general point of it parametrizes a pair of disjoint lines. The other component has dimension 11 and a general point of it parametrizes the disjoint union Y = Q ∪ {p} of a conic Q and a point p. Note that h0 (Y, NY ) = h0 (Q, NQ ) + h0 (p, Np ) = 8 + 3 = 11 and h1 (Y, NY ) = 0. Hence Y is a smooth point of a component of dimension 11 of Hilb32(t+1) . Therefore it suffices to produce a flat family parametrized by an irreducible curve, e.g. A1 , Y ⊂ IP 3 × A1 such that Y(0) = X, Y(1) = Y . Here it is:

Y = Proj k[v, X0 , X1 , X2 , X3 ]/I where

I = (X1 X2 , X1 X3 + vX1 X0 , X2 X3 + vX2 X0 , X32 − v 2 X02 ) Clearly Y(0) = X; since I = (X1 , X2 , X3 − vX0 ) ∩ (X3 + vX0 , X1 X2 ) it follows that for all v 6= 0 Y(v) is the disjoint union of a conic and a point. The flatness of Y follows from [89], Prop. III.9.8. This example shows that in general the Hilbert schemes are reducible and not equidimensional. For a description of Hilb33t+1 , which presents several analogies with the one given here of Hilb32(t+1) , we refer to [158].

4.6.4

Relative grassmannians and projective bundles

Consider a coherent sheaf E on an algebraic scheme S, and let P (t) = n, S/S where n is a positive integer, be a constant polynomial. Then QuotE,n is a projective S-scheme which will be denoted Quotn (E) in what follows. We will denote by ρ : Quotn (E) → S

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the structural projective morphism and by ρ∗ E → Q the universal quotient bundle; Q is locally free of rank n. If n = 1 then Q is an invertible sheaf: it will be denoted by OQuot1 (E) (1), or simply by O(1) if no confusion arises. The pair (Quotn (E)/S, Q) represents the functor Quotn (E) : (schemes/S)◦ → (sets) defined by: Quotn (E)(f : T → S) = {locally free rk n quotients f ∗ E → F on T } On Quotn (E) we have a tautological exact sequence 0 → K → ρ∗ (E) → Q → 0 If E is locally free then K is locally free as well and it is called the universal subbundle. If E is locally free we define Gn (E) := Quotn (E ∨ ) and call it the grassmannian bundle of subbundles of rank n of E; for n = 1 we have G1 (E) = IP (E) the projective bundle associated to E, according to our definition (which differs from the one adopted in [89], p. 162). The tautological exact sequence on IP (E) is: 0 → K → ρ∗ (E ∨ ) → OIP (E) (1) → 0 In particular for a finite dimensional k-vector space V we have IP (V ⊗k OS ) = IP (V ) × S and more generally Gn (V ⊗k OS ) = Gn (V ) × S Therefore, if E is locally free on S then Quotn (E) is locally the product of S by a grassmannian; in particular the projection ρ : Quotn (E) → S is a smooth morphism.

4.6. EXAMPLES AND APPLICATIONS

289

Proposition 4.6.1 Let E be a locally free sheaf on the algebraic scheme S, and let 0 → K → ρ∗ (E) → Q → 0 (4.30) be the tautological exact sequence on Quotn (E) for some 1 ≤ n ≤ rk(E). Then there is a natural isomorphism Ω1Quotn (E)/S ∼ = Hom(Q, K) and therefore

TQuotn (E)/S ∼ = Hom(K, Q)

Proof. Letting B = Quotn (E) consider the product B ×S B with projections pri : B ×S B → B, i = 1, 2, and let EB×S B be the pullback of E on B ×S B. Denote by I∆ ⊂ OB×S B the ideal sheaf of the diagonal ∆ ⊂ B ×S B. The tautological exact sequence (4.30) pulls back to two exact sequences: ∨ 0 → pr∗i K → EB× → pr∗i Q → 0 SB

on B ×S B whose restrictions to ∆ coincide, and ∆ is characterized by this property. This can be also expressed by saying that ∆ is the vanishing scheme of the composition ∨ pr∗1 K → EB× → pr∗2 Q SB Therefore we have a surjective homomorphism: Hom(pr∗2 Q, pr∗1 K) → I∆ (see 4.2.8) which, restricted to ∆, gives a surjective homomorphism: 2 Hom(Q, K) → I∆ /I∆ = Ω1B/S

which has to be an isomorphism since both sheaves are locally free and have the same rank. qed Proposition 4.6.2 Let α

β

0 → E −→ F −→ G → 0 be an exact sequence of locally free sheaves on the algebraic scheme S, and n ≥ 1 an integer. Then there is a closed regular embedding Quotn (G) ⊂ Quotn (F)

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and a natural identification: NQuotn (G)/Quotn (F ) = ρ∗ E ∨ ⊗ Q ⊗ OQuotn (G) where ρ : Quotn (F) → S is the structure morphism and ρ∗ F → Q is the universal quotient. In particular, if n = 1 we have NQuot1 (G)/Quot1 (F ) = ρ∗ E ∨ ⊗ O(1) ⊗ OQuot1 (G) Proof. Let f : T → S be a morphism. For every locally free rank n quotient (f ∗ G → H) ∈ Quotn (G)(T ) there is associated, by composition with the surjective homomorphism f ∗ (β) : f ∗ F → f ∗ G, an element (f ∗ F → H) ∈ Quotn (F)(T ) Therefore Quotn (G) is a subfunctor of Quotn (F). Consider the diagram of homomorphisms on Quotn (F): ρ∗ (E) ↓ ρ∗ (α) γ

ρ∗ (F) −→ Q → 0 ↓ ρ∗ (β) ρ∗ (G) ↓ 0 Given a morphism f : T → S, an element of Quotn (F)(T ) = HomS (T, Quotn (F)) belongs to Quotn (G)(T ) if and only if it factors through the closed subscheme D0 (γρ∗ (α)) of Quotn (F). This proves that Quotn (G) is a closed subfunctor of Quotn (F), and therefore the embedding Quotn (G) ⊂ Quotn (F)

4.6. EXAMPLES AND APPLICATIONS

291

is closed. More precisely, this analysis shows that Quotn (G) = D0 (γρ∗ (α)). Since Quotn (G) has codimension rk(E)n in Quotn (F) it follows that it is regularly embedded. According to Example 4.2.8 we have a surjective homomorphism: Hom(Q, ρ∗ (E)) → I where I ⊂ OQuotn (F ) is the ideal sheaf of Quotn (G). Quotn (G) we obtain a surjective homomorphism:

By restricting to

Hom(Q, ρ∗ E) ⊗ OQuotn (G) → I/I 2 → 0 which is an isomorphism because both are locally free and of the same rank. qed Corollary 4.6.3 Let 0→E →F →G→0

(4.31)

be an exact sequence of locally free sheaves on the algebraic scheme S. Then there is a closed immersion IP (E) ⊂ IP (F) and a natural identification: NIP (E)/IP (F ) = ρ∗ G ⊗ OIP (E) (1) In particular to every exact sequence (4.31) with E an invertible sheaf there corresponds a section σ : S → IP (E) ⊂ IP (F) of the projective bundle IP (F) → S whose normal bundle is G ⊗ E ∨ . Proof. Only the last assertion requires a proof. It follows by observing that OIP (F ) (1) ⊗ OIP (E) = OIP (E) (1) = E ∨ and therefore the formula for the normal bundle is a direct consequence of Proposition 4.6.2. qed

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Remark 4.6.4 Let E be a locally free sheaf on an algebraic scheme S, and let 0 → Q∨ → ρ∗ (E) → K∨ → 0 (4.32) be the dual of the tautological exact sequence (4.30) on Gn (E) = Quotn (E ∨ ). Tensoring with Q we obtain the exact sequence: 0 → Q∨ ⊗ Q → ρ∗ (E) ⊗ Q →

K∨ ⊗ Q k TGn (E)/S

→0 (4.33)

In the case n = 1 and S = Spec(k) we have E = V a vector space and G1 (V ) = IP (V ) = IP ; the dual of the tautological sequence is 0 → OIP (−1) → V ⊗ OIP → TIP (−1) → 0

(4.34)

and the sequence (4.33) is the Euler sequence 0 → OIP → V ⊗ OIP (1) → TIP → 0 Therefore (4.33) is a generalization of the Euler sequence. Dualizing (4.34) we get an inclusion I = IP (Ω1IP ) ⊂ IP × IP ∨ where I := {(x, H) : x ∈ H} is the incidence relation (see Note 2 of Appendix B). From Corollary 4.6.3 we obtain NI/IP ×IP ∨ = p∗1 OIP (1) ⊗ p∗2 OIP ∨ (1) ⊗ OI where

p

p

1 2 IP ←− IP × IP ∨ −→ IP ∨

are the projections. Example 4.6.5 Let X be a projective nonsingular variety and L an invertible sheaf on X. Consider the Atiyah extension 0 → OX → EL → TX → 0 (see page 172). Then ρ : IP (EL ) → X is a IP r -bundle, r = dim(X), and IP (OX ) ⊂ IP (EL ) is a section of ρ with normal bundle TX . In the case X = IP := IP (V ), where V is a finite dimensional vector space, and L = O(1)

4.6. EXAMPLES AND APPLICATIONS

293

the Atiyah extension coincides with the Euler sequence (Remark 3.3.10) so that we have IP (EL ) = IP × IP and IP = IP (OIP ) ⊂ IP (EL ) = IP × IP is the diagonal embedding. Example 4.6.6 Let V be a finite dimensional k-vector space, X ⊂ IP := IP (V ) a projective irreducible nonsingular variety and I ⊂ OIP its ideal sheaf. Then we have inclusions of locally free sheaves on X: ∨ 2 1 ∨ NX/I P = I/I ⊂ ΩIP |X ⊂ V ⊗ OX

which induce closed embeddings of projective bundles: ∨ IP (NX/I P) ⊂

IP (Ω1IP |X ) ⊂ X × IP ∨ k {(x, H) : x ∈ H}

Recalling the exact sequence ∨ 1 1 0 → NX/I P → ΩIP |X → ΩX → 0

(4.35)

we see that we have an identification: ∨ ∨ IP (NX/I P ) = {(x, H) : Tx X ⊂ H} ⊂ X × IP

(4.36)

Letting ∨ ρ : IP (NX/I P) → X

be the structural morphism, n = dim(X), and r = dim(IP ), from the description (4.36) we deduce that for every x ∈ X, the fibre of ρ is identified with the (r − n − 1)-dimensional linear system of hyperplanes which are tangent to X at x. Let’s consider the special case when X is a rational normal curve in IP = IP r , r ≥ 2. Denote by λ the invertible sheaf of degree 1 on X, so that OX (1) = λr . We have a morphism r−2 ∼ ∨ q : IP (NX/I | = IP r−2 P ) → |λ

which associates to (x, H) the divisor q(H) of degree r − 2 such that H · X = 2x + q(H)

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∨ r−2 ∼ The existence of q implies that IP (NX/I and therefore P ) = X × IP ∨ s NX/I ⊕ ·{z · · ⊕ λ}s P = λ | r−1

for some s ∈ ZZ. From the exact sequence (4.35) one immediately computes that (r − 1)s = (r + 2)(1 − r) i.e. s = −(r + 2). The conclusion is the following formula for the normal bundle of a rational normal curve X ⊂ IP r : r+2 NX/IP r ∼ ⊕ ·{z · · ⊕ λr+2} =λ |

(4.37)

r−1

For another approach to the same computation see [146]. The normal bundles of rational curves of any degree in IP 3 (resp. in IP r ) have been computed in [66] and [50] (resp. in [173]). Example 4.6.7 (Zappa[211]) Let C be a projective nonsingular connected curve of genus 1 and let 0 → OC → E → OC → 0

(4.38)

be an exact sequence of locally free sheaves corresponding to a non-zero element of Ext1OC (OC , OC ) = H 1 (C, OC ); in particular (4.38) does not split. Consider the ruled surface ρ : IP (E) → C and let C 0 := IP (OC ) ⊂ IP (E) be the section of ρ corresponding to the subsheaf OC ⊂ E in (4.38). By Corollary 4.6.3 we have N := NC 0 /IP (E) = OC 0 and therefore h0 (C 0 , N ) = 1. If C 00 ⊂ IP (E) is a curve such that [C 0 ] and [C 00 ] belong to the same component of HilbIP (E) then, since N is the trivial sheaf, either C 0 = C 00 or C 00 ∩ C 0 = ∅. But both C 0 and C 00 are sections of ρ and the second possibility implies that the exact sequence (4.38) splits. This is a contradiction, and we conclude that [C 0 ] is isolated in HilbIP (E) , with a 1-dimensional Zariski tangent space. Therefore C 0 is obstructed in IP (E). ∗

∗

∗

∗

∗ ∗

Let H be a coherent sheaf on the projective scheme X, and let P (t) = n be a constant polynomial, where n is a positive integer. Then we have two different Quot schemes associated to these data.

4.6. EXAMPLES AND APPLICATIONS

295

X/X

The first one is QuotH,n , whose k-rational points are quotients H → F which are locally free of rank n: it has just been considered. The other one is QuotX H,n . A k-rational point of this scheme is nothing but a quotient H → F such that F is a torsion sheaf with finite support and h0 (F) = n. When n = 1 then F ∼ = k(x) for some closed point x ∈ X: therefore we have a natural morphism q:

QuotX H,1

→

X

(H → F) 7→ Supp(F) and QuotX H,1 is a scheme over X. Let H → F be a k-rational point of X QuotH,1 ; then ker[H → F] is called an elementary transform of H. The process of passing from H to ker[H → F] is called an elementary transformation centered at x. This construction is classical when X is a projective nonsingular curve and H is locally free. For generalizations of it see [138]. QuotX H,1 is the scheme of elementary transformations of H. Proposition 4.6.8 Assume that Supp(H) is connected. Then QuotX H,1 is connected. Proof. The natural morphism q:

QuotX H,1

→

X

(H → F) 7→ Supp(F) has image Supp(H). Every H → F factors as H → H ⊗ Ox ↓ F ↓ 0

∼ = Ox

and therefore the fibre q −1 (x) is identified with IP (H 0 (H ⊗ Ox ))∨ which is connected. The conclusion follows. qed

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4.6.5

Hilbert schemes of points

The Hilbert schemes parametrizing 0-dimensional subschemes of a given scheme are inexpectedly complicated and have a variety of applications. Since it is beyond our scopes to give an exhaustive overview of the subject, we will only explain some of the basic facts. The reader is referred to [99], [100], [149] for more details. See also [85] for another approach to Hilbert schemes of points. Consider a projective scheme Y and, for a positive integer n, the Hilbert scheme Y [n] . We have already seen in §4.3 that Y [1] ∼ = Y : we will therefore assume n ≥ 2. Let Z ⊂ Y be a closed subscheme of length n. Then H 0 (Z, NZ ) =

M

Hom(IZy , OZy )

y∈Supp(Z)

and H 1 (Z, NZ ) = (0) Moreover from the spectral sequence of Ext’s (see [67], p. 265) we see that Ext1 (IZ , OZ ) = H 0 (Y, Ext1 (IZ , OZ )) =

M

Ext1 (IZy , OZy )

y∈Supp(Z)

It follows that the local properties of Y [n] at [Z] are determined by the independent contributions from each of the points of Supp([Z]). The following properties follow at once: (a) If Z is reduced and supported at n distinct points of Y then [Z] is a nonsingular point of Y [n] if and only if it is supported at nonsingular points of Y . (b) If Y is reduced then the set of [Z]’s with Z supported at n nonsingular points of Y is an open set of dimension n dim(Y ) contained in the nonsingular locus of Y [n] . Another important property is the following: (c) [61] If Y is connected then Y [n] is also connected.

4.6. EXAMPLES AND APPLICATIONS

297

Proof. Let n ≥ 1 and let I ⊂ OY ×Y [n] be the ideal sheaf of the universal family in Y × Y [n] . Then we have a diagram of morphisms: QuotYI,1×Y

[n]

.p

&q

Y [n+1]

Y × Y [n]

where q is the natural morphism, which is surjective because Supp(I) = Y × Y [n] . The morphism p is defined as follows. Let (y, [Z]) ∈ Y × Y [n] be a k-rational point and let γ : I → k(y, [Z]) be a [n] quotient, which is a k-rational point of QuotYI,1×Y . Then ker(γ) ⊂ OY ×Y [n] is an ideal sheaf such that ker(γ)OY ×[Z] has colength 1 in I ⊗OY ×[Z] . Therefore ker(γ)OY ×[Z] defines a subscheme W ⊂ Y of length n+1 containing Z and x; we define p(γ) = [W ]. The morphism p is clearly surjective. Since Y × Y [n] is connected by induction, we conclude that Y [n+1] is connected by Proposition 4.6.8. qed In general Y [n] is singular and reducible even if Y is nonsingular connected. Notable exceptions are the cases dim(Y ) = 1, 2. If C is a projective nonsingular curve and n ≥ 1 an integer, every closed subscheme D ⊂ C of length n is a Cartier divisor, therefore regularly embedded in C. It follows that C [n] is nonsingular of dimension h0 (D, ND ) = h0 (D, OD ) = n because H 1 (D, ND ) = (0) is an obstruction space for OC [n] ,[D] (Theorem 4.3.5(ii)). Actually C [n] is naturally isomorphic to C (n) , the n-th symmetric product of C, the isomorphism being given by the cycle map C [n] → C (n) which maps a closed D ⊂ C to the associated Weil divisor. The case of surfaces is more subtle. Theorem 4.6.9 (Fogarty[61]) If Y is a projective nonsingular connected surface then Y [n] is nonsingular connected of dimension 2n.

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Proof. Let [Z] ∈ Y [n] . We then have: Exti (IZ , OZ ) = (0)

i≥3

Moreover from the exact sequence: 0 → IZ → OY → OZ → 0 we obtain the sequence: 0 → Hom(OZ , OZ ) → Hom(OY , OZ ) → Hom(IZ , OZ ) → → Ext1 (OZ , OZ ) → Ext1 (OY , OZ ) → Ext1 (IZ , OZ ) → → Ext2 (OZ , OZ ) → Ext2 (OY , OZ ) → Ext2 (IZ , OZ ) → 0 Since Exti (OY , OZ ) = H i (Y, OZ ) = (0) for i ≥ 1 we see that Ext2 (IZ , OZ ) = (0) and Ext1 (IZ , OZ ) ∼ = Ext2 (OZ , OZ )

= Hom(OZ , OZ ⊗ ωY )∨ =

= Hom(OZ , OZ )∨ = H 0 (Z, OZ )∨ Therefore 2 X

(−1)i dim[Exti (IZ , OZ )] = h0 (Z, NZ ) − h0 (Z, OZ ) = h0 (Z, NZ ) − n

i=0

Since the left hand side is independent of Z, it follows that h0 (Z, NZ ) is also independent of Z. But Y [n] is connected and has an open set which is nonsingular and of dimension 2n: the conclusion follows. qed To see that Y [n] is singular if dim(Y ) = 3 consider IP 3 with homogeneous coordinates X0 , . . . , X3 and the subscheme Z = V (X12 , X22 , X32 , X1 X2 , X1 X3 , X2 X3 ) Then [Z] ∈ (IP 3 )[4] . A computation similar to that of the example of Subsection 4.6.3 shows that the Zariski tangent space of (IP 3 )[4] at [Z] has dimension 18. But (IP 3 )[4] is connected and has a component which is nonsingular of dimension 12 at its general point: it follows that (IP 3 )[4] is singular. For more examples and for a useful detailed introduction to punctual Hilbert schemes we refer the reader to [99].

4.6. EXAMPLES AND APPLICATIONS

4.6.6

299

Schemes of morphisms

Let X and Y be schemes, with X projective and Y quasi-projective. For every scheme S let: F (S) = HomS (X × S, Y × S) This defines a contravariant functor: F : (schemes)◦ → (sets) called the functor of morphisms from X to Y . For every Φ ∈ F (S) let ΓΦ ⊂ X ×Y ×S be its graph. Then ΓΦ ∼ = X ×S is flat over S and therefore defines a flat family of closed subschemes of X × Y parametrized by S. This means that F is a subfunctor of HilbX×Y . If G ⊂ X × Y × S is a flat family of closed subschemes of X × Y , proper over S, then the projection π : G → Y × S is a family of morphisms into Y and the locus of points s ∈ S such that π(s) is an isomorphism is open (Note 2 of §4.2). This means that F is an open subfunctor of HilbX×Y , represented by an open subscheme of HilbX×Y , which we denote Hom(X, Y ). It is called the scheme of morphisms from X to Y . The scheme Hom(X, Y ) contains an open and closed subscheme isomorphic to Y and consisting of the constant morphisms. In particular if the only morphisms f : X → Y are the constant ones then Hom(X, Y ) ∼ =Y Let X and Y be as above, and consider the contravariant functor G : (schemes)◦ → (sets) defined as follows: G(S) = {S-isomorphisms X × S → Y × S} Clearly G is a subfunctor of F . It is easy to prove that G is represented by an open subscheme Isom(X, Y ) of Hom(X, Y ), called the scheme of isomorphisms from X to Y . When X = Y it is denoted Aut(X) and called the scheme of automorphisms of X. It is a group scheme. The following result follows immediately from Proposition 3.4.2 and Corollary 3.4.3:

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Proposition 4.6.10 Let f : X → Y be a morphism of algebraic schemes, with X reduced and projective and Y nonsingular and quasiprojective. Then T[f ] (Hom(X, Y )) ∼ = H 0 (X, f ∗ TY ) and the obstruction space of Hom(X, Y ) at [f ] is contained in H 1 (X, f ∗ TY ). If X is nonsingular then the tangent space to Aut(X) at 1X is H 0 (X, TX ). Let j : X ⊂ Y be a closed embedding of projective nonsingular schemes. Then j induces an inclusion J : Aut(X) ⊂ Hom(X, Y ) such that J(1X ) = j and which is induced by the closed embedding 1X × j : X × X ⊂ X × Y . It follows that J is a closed embedding. Its differential at 1X is the injective linear map H 0 (TX ) → H 0 (TY |X ) coming from the natural inclusion TX ⊂ TY |X . In fact from the diagram of inclusions: X ×X ⊂ X ×Y ∪ ∪ ∼ ∆ Γj = we deduce the commutative diagram: N∆/X×X k TX

⊂ NΓj /X×Y k ⊂ TY |X

and we conclude according to §4.3, Note 3.

qed

Example 4.6.11 Consider X = IP 1 , Y = IP r and j : IP 1 → IP r the r-th Veronese embedding. Locally around [j] we have a well defined morphism M : Hom(IP 1 , IP r ) → HilbIP

r

sending [j] 7→ [j(IP 1 )] with fibre M −1 ([j(IP 1 )]) an open neighborhood of the identity in Aut(IP 1 ). Consider the following diagram consisting of two exact

4.6. EXAMPLES AND APPLICATIONS

301

sequences:

0 → TIP 1

0 ↑ ∗ → j TIP r ↑ OIP 1 (r)r+1 ↑ OIP 1 ↑ 0

→ Nj

→0

From the vertical sequence (the Euler sequence restricted to IP 1 ) we get h1 (j ∗ TIP r ) = 0,

h0 (j ∗ TIP r ) = r(r + 2)

Since h1 (TIP 1 ) = 0 from the other sequence we obtain h1 (Nj ) = 0 and the exact sequence 0→

q

H 0 (TIP 1 ) → H 0 (j ∗ TIP r ) −→ H 0 (Nj ) k k k r 1 1 r T1IP 1 Aut(IP ) T[j] Hom(IP , IP ) Tj(IP 1 )] HilbIP

→0

Since the map q can be identified with dM[j] we see that M and Hom(IP 1 , IP r ) r are smooth at [j] and HilbIP is smooth at [j(IP 1 )]; moreover r

dim[j] (Hom(IP 1 , IP r ) = r(r + 2) = dimj(IP 1 ) (HilbIP ) + 3 For more on the schemes Hom(IP 1 , X) and applications to uniruledness see [43].

4.6.7

Focal loci

We assume char(k) = 0. Consider a flat family of closed subschemes of a projective scheme Y : q

1 Ξ ⊂ Y × B −→ Y ↓ q2 B

parametrized by a scheme B. Let f :Ξ⊂Y ×B →Y

(4.39)

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CHAPTER 4. HILBERT AND QUOT SCHEMES

be the composition of the inclusion (4.39) with the projection q1 . Denote by N := NΞ/Y ×B the normal sheaf of Ξ in Y × B, and let (q2∗ TB )|Ξ be the restriction to Ξ of the tangent sheaf along the fibres of q1 . The global characteristic map of the family (4.39) is the homomorphism: χ : (q2∗ TB )|Ξ → N defined by the following exact and commutative diagram: 0 ↓ (q2∗ TB )|Ξ ↓ 0 → TΞ → TY ×B|Ξ ↓ df ↓ f ∗ TY = f ∗ TY

χ

−→ N k → N

(4.40) → 0

For each b ∈ B the homomorphism χ induces a homomorphism χb : TB,b ⊗ OΞ(b) → NΞ(b)/Y called the local characteristic map of the family (4.39) at the point b. Let ϕ : B → HilbY be the classifying morphism induced by the flat family (4.39). Then if Ξ(b) is connected, the linear map: H 0 (χb ) : TB,b → H 0 (NΞ(b)/Y ) is dϕb , the characteristic map of (4.39) at the point b (see page 260). Assume that Y, B and the family (4.39) are smooth. In this case all the sheaves in (4.40) are locally free. From a diagram-chasing it follows that ker(χ) = ker(df ) and therefore

4.6. EXAMPLES AND APPLICATIONS

303

Proposition 4.6.12 dim[f (Ξ)] = dim(Ξ) − rk[ker(χ)] Let’s denote by V (χ) the closed subscheme of Ξ defined by the condition: rk(χ) < min{rk[(q2∗ TB )|Ξ ], rk(N )} = min{dim(B), codimY ×B (Ξ)} We will call the points of V (χ) first order foci of the family (4.39). V (χ) is the scheme of first order foci, and the fiber of V (χ) over a point b ∈ B: V (χ)b = V (χb ) ⊂ Ξ(b) is the scheme of first order foci at b. If χ has maximal rank, i.e. if χ is injective or has torsion cokernel, then V (χ) is a proper closed subscheme of Ξ. If χ does not have maximal rank then V (χ) = Ξ. The definition of first order foci at a point b depends only on the geometry of the family (4.39) in a neighborhood of b. A focus y ∈ V (χ)b is a point where there is an intersection between the fiber Ξ(b) and the infinitesimally near ones. One defines higher order foci inductively: second order foci are the first order foci of the family of first order foci, and so on. Focal loci have been studied classically in the case of families of linear spaces (see e.g. [178]). Recently they have been applied to the geometry of the theta divisor of an algebraic curve in [38] and [39]. Related work is [33], [35], [40]. Notes and Comments 1. Let Q ⊂ IP 3 be a quadric cone with vertex v, and L ⊂ Q a line. Then NL/Q = OL (1) ⊂ NL/IP 3 = OL (1) ⊕ OL (1) (see Example 3.2.8(iii), page 156); in particular H 0 (L, NL/Q ) = 2 and H 1 (L, NL/Q ) = 0. On the other hand the Hilbert scheme HilbQ is 1-dimensional at the point [L] since L moves in a 1-dimensional family. It follows that L is obstructed in Q (see [46] for generalizations of this example). 2. Another example of reducible Hilbert scheme is the following, which appears in [191]. Let Y = C × C 0 where C and C 0 are projective nonsingular connected curves of genera g and g 0 respectively and let p0

Y −→ ↓p C

C0

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be the projections; assume g, g 0 ≥ 2. Consider an effective divisor D = x1 +· · ·+xg of degree g on C, and an effective divisor D0 = x01 + · · · + x0g0 of degree g 0 on C 0 , both consisting of distinct points, and let Γ = p−1 (D) + p0−1 (D0 ) = Cx0 1 + · · · + Cx0 g + Cx01 + · · · + Cx0 0 g

where Cx0 = p−1 (x) and Cx0 = p0−1 (x0 ). Γ is a reduced divisor, has gg 0 nodes and no other singularity. If either D or D0 is non-special the curve Γ belongs to an irreducible component H1 of HilbY of dimension g + g 0 generically consisting of curves of the same form, obtained by moving D and D0 . When both D and D0 are special divisors the curve Γ belongs to a linear system of dimension ≥ 3 whose general member is a nonsingular curve and therefore belongs to another irreducible component H2 of HilbY which has dimension g + g 0 − 1. The intersection H1 ∩ H2 is irreducible of dimension g + g 0 − 2.

4.7

Plane curves

An important refinement of the Hilbert functors derives from the consideration of flat families of closed subschemes of a projective scheme Y having prescribed singularities, i.e. of families all of whose members have the same type of singularity in some specified sense. This leads to the notion of equisingularity and to a related vast area of research. In this section we will only concentrate on the specific case of families of plane curves with assigned number of nodes and cusps: we will show how to construct universal families of such curves, whose parameter schemes are called Severi varieties for historical reasons. This is a subject with a long history and a wealth of important results, both classical and modern. Here we will limit ourselves to prove a few basic results and to indicate some of their generalizations and the main references in the literature. We will assume char(k) = 0 in this section.

4.7.1

Equisingular infinitesimal deformations

Let Y be a projective nonsingular variety and X ⊂ Y a closed subscheme whith ideal sheaf IX ⊂ OY . Recall ((1.3)) that on X we have an exact sequence of coherent sheaves: 0 → TX → TY |X → NX/Y → TX1 → 0

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Recall that the sheaf 0 NX/Y := ker[NX/Y → TX1 ]

is called the equisingular normal sheaf of X in Y (see Proposition 1.1.9, page 0 16). Clearly NX/Y = NX/Y if X is nonsingular. By definition sections of the equisingular normal sheaf parametrize first order deformations of X in Y which are locally trivial, because they induce trivial deformations around every point of X. We recall that an alternative description of the equisingular normal sheaf can be given by means of the sheaf of germs of tangent vectors to Y which are tangent to X TY hXi ⊂ TY introduced in §3.4. In fact there is an exact sequence (see (3.55)) 0 0 → TY hXi → TY → NX/Y →0

(4.41)

From the definition it follows that, for every open set U ⊂ Y , Γ(U, TY hXi) consists of those k-derivations D ∈ Γ(U, TY ) such that D(g) ∈ Γ(U, I) for every g ∈ Γ(U, I). Examples 4.7.1 (i) Assume that X is a hypersurface in Y ; then NX/Y ∼ = OX (X). Locally on an affine open set U ⊂ Y we have (NX/Y )|U ∩X ∼ = OU ∩X . Assume that X can be represented by an equation f (x1 , . . . , xn ) = 0 in local 0 coordinates on U ; then from the definition of TX1 it follows that (NX/Y )|U ∩X ⊂ OU ∩X is the image of the ideal sheaf (∂f /∂x1 , . . . , ∂f /∂xn ) ⊂ OU . We deduce that an equisingular first order deformation of X in Y corresponding to a 0 local section g¯ of NX/Y can be written locally as f (x) + g(x) = 0 where g(x) = a1 (x)

∂f ∂f + · · · + an (x) ∂x1 ∂xn

restricts to g¯. Therefore if Y = IP n and X is a hypersurface of degree d we have an exact sequence 0 0 → OIP n → I(d) → NX/I Pn → 0

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where I ⊂ OIP n is the ideal sheaf locally generated by the partial derivatives of a local equation of X. In the special case of a curve X in a surface Y assume that p ∈ X is a singular point and let f (x, y) = 0 be a local equation of X around p. If p is a node then (∂f /∂x, ∂f /∂y) = mp is just the maximal ideal of p; if p is an ordinary cusp with principal tangent say y = 0 then (∂f /∂x, ∂f /∂y) = (x, y 2 ) is an ideal of colength 2. (ii) Let X ⊂ IP 2 be a (possibly reducible) plane curve of degree d of equation F (X0 , X1 , X2 ) = 0, having δ nodes p1 , . . . , pδ and no other singularity. This case is important because every nonsingular projective curve is birationally equivalent to a nodal plane curve. Denote by ∆ = {p1 , . . . , pδ } ⊂ IP 2 the 0-dimensional reduced scheme of the nodes of X and by ν : C → X the normalization map. The above analysis shows that sections of H 0 (I∆ (d)), 0 i.e. curves of degree d which are adjoint to X, cut on X sections of NX/I P2. This means that 0 ∗ ν ∗ (NX/I P 2 ) = ν [OX (d) ⊗ I∆ ] =

= ν ∗ OX (d)(−p01 − p001 − · · · − p0δ − p00δ ) = ωC ⊗ ν ∗ O(3) where ν −1 (pi ) = {p0i , p00i }, i = 1, . . . , δ, and therefore we have 0 h0 (C, ν ∗ (NX/I P 2 )) = 3d + g − 1,

0 h1 (C, ν ∗ (NX/I P 2 )) = 0

where g is the geometric genus of X. Moreover, since 0 ∗ ν∗ [ν ∗ (NX/I P 2 )] = ν∗ [ν (OX (d) ⊗ I∆ ] = 0 OX (d) ⊗ ν∗ OC (−p01 − p001 − · · · − p0δ − p00δ ) = OX (d) ⊗ I∆ = NX/I P2

we have 0 0 ∗ 0 h0 (X, NX/I P 2 ) = h (C, ν (NX/IP 2 )) = 3d + g − 1 =

d+2 2

−δ−1 (4.42)

0 1 ∗ 0 h1 (X, NX/I P 2 ) = h (C, ν (NX/IP 2 )) = 0

Finally, since !

d+2 h0 (I∆ (d)) ≥ − δ = 3d + g 2 0 and H 0 (I∆ (d))/(F ) ⊂ H 0 (NX/I P 2 ), comparing with (4.42) we see that ∆ imposes independent conditions to curves of degree d and that 0 0 H 0 (NX/I P 2 ) = H (I∆ (d))/(F )

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307

or, equivalently, the restriction map 0 H 0 (I∆ (d)) → H 0 (NX/I P2) 0 is surjective. Note that NX/I P 2 is a non-invertible subsheaf of NX/IP 2 .

(iii) Another interesting case is obtained by taking an irreducible curve X ⊂ IP 2 of degree d having δ nodes p1 , . . . , pδ and κ ordinary cusps q1 , . . . , qκ as its only singularities. This case is important because branch curves of generic projection on IP 2 of projective nonsingular surfaces are curves of this type. Let ν : C → X be the normalization map. Denoting q¯j = ν −1 (qj ), j = 1, . . . , κ, we have in this case, according to the above description 0 0 00 0 00 ν ∗ (NX/P q1 − · · · − 3¯ qκ ) = 2 ) = OC (d)(−p1 − p1 − · · · − pδ − pδ − 3¯

= ωC ⊗ ν ∗ OX (3)(−¯ q1 − · · · − q¯κ ) 0 0 As before one shows that ν∗ [ν ∗ (NX/P 2 )] = NX/P 2 and therefore

!

d+2 h = h (C, ωC ⊗ ν OX (3)(−¯ q1 − · · · − q¯κ )) ≥ − δ − 2κ − 1 2 (4.43) 0 and in general we may have strict inequality and h1 (X, NX/I ) = 6 0 because 2 P the invertible sheaf ωC ⊗ ν ∗ OX (3)(−¯ q1 − · · · − q¯κ ) can be special. But if κ < 3d then it is certainly non special and therefore in such a case we have 0

0 (X, NX/I P2)

∗

0

0 h0 (X, NX/I P2) =

d+2 2

− δ − 2κ − 1 = 3d + g − 1 − κ

0 h1 (X, NX/I P2) = 0

4.7.2

The Severi varieties

Given an integer d > 0 consider the complete linear system |O(d)| of plane curves of degree d. It is a flat family of closed subschemes of IP 2 parametrized by the projective space Σd = IP [H 0 (IP 2 , O(d))]: H |O(d)| : ↓ Σd

⊂ IP 2 × Σd

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The linear system |O(d)| has a universal property with respect to families of plane curves of degree d because the pair (Σd , |O(d)|) represents the Hilbert functor: Λd : (algschemes)◦ → (sets) given by:

Λd (S) =

flat families C ⊂ IP 2 × S of plane curves of degree d parametrized by S

(see §4.3). In this subsection we want to consider the problem of constructing a universal family of reduced curves in IP 2 having degree d, an assigned number δ of nodes and κ of ordinary cusps and no other singularity. If such a universal family exists it is parametrized by a scheme which we denote by Vdδ,κ . These schemes have been studied classically: the foundations of their theory were given in [192] and they are therefore called Severi schemes or Severi varieties. If the Severi scheme Vdδ,κ exists then, by the universal property, there is a functorially defined morphism Vdδ,κ → Σd

(4.44)

We start from the definition of the functor we want to represent. Definition 4.7.2 Let d, δ, κ as above. Then Vdδ,κ : (algschemes)◦ → (sets) is defined as follows. For each algebraic scheme S families C ⊂ IP 2 × S of plane curves of deg. d formally ) locally trivial at each k-rational s ∈ S whose geometric fibres are curves with δ nodes and κ cusps as singularities

( flat

Vdδ,κ (S)

=

(see §2.5 for the definition of formal local triviality). Obviously Vdδ,κ is a subfunctor of Λd . The main result about Vdδ,κ is the following Theorem 4.7.3 For each d, δ, κ as above the functor Vdδ,κ is represented by an algebraic scheme Vdδ,κ which is a (possibly empty) locally closed subscheme of Σd .

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309

In case κ = 0 we write Vdδ instead of Vdδ,0 . The first published proof of this result is in [209]. We will not reproduce it in full generality here, but we will only consider the case κ = 0, i.e. the case of nodal curves. This assumption allows a technically simpler argument without changing the structure of the original proof. We need some lemmas. Lemma 4.7.4 Let p ∈ IP 2 , O the local ring of IP 2 at p, B0 = O/(f0 ) the local ring of a plane curve having a node at p. Assume that for some A in 1 ob(A) we have a deformation A → B of B0 over A such that TB/A is A-flat. 1 ∼ Then B is trivial and TB/A = A. Proof. By induction on dimk (A). The case A = k is trivial because TB1 0 = O/(f0 , f0X , f0Y ) = O/(f0X , f0Y ) ∼ =k (see 3.1.4). In the general case consider a small extension 0 → () → A → A0 → 0 and the induced deformation A0 → B 0 . We have B = (A ⊗k O)/(f ) for 1 some f which reduces to f0 modulo mA , and TB/A = B/(fX , fY ). Therefore 0 0 0 0 B = (A ⊗k O)/(f ) where f is obtained from f by reducing the coefficients to A0 , and 1 TB1 0 /A0 = B 0 /(fX0 , fY0 ) = B/(fX , fY ) ⊗A A0 = TB/A ⊗A A0

It follows that TB1 0 /A0 is A0 -flat and, by induction, we have B 0 = (A0 ⊗k O)/(f0 ) and TB1 0 /A0 = A0 ⊗k [O/(f0 , f0X , f0Y )] = A0 Thus f = f0 + g where g ∈ k. We have: 1 = (A ⊗k O)/(f0 + g, f0X , f0Y ) = A/(g) TB/A

where the last equality follows from the fact that f0 ∈ (f0X , f0Y ). Since A/(g) is A-flat if and only if g = 0 it follows that B = (A ⊗k O)/(f0 ) = 1 A ⊗k (O/(f0 )) is the trivial deformation and TB/A = A. qed

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Lemma 4.7.5 Let f : X → S be a flat morphism of algebraic schemes which factors as j X −→ Y & ↓q S where j is a regular embedding of codimension 1 and q is smooth. Then for every morphism of algebraic schemes ϕ : S 0 → S we have 1 TS10 ×S X/S 0 ∼ = Φ∗ TX/S 1 where Φ : S 0 ×S X → X is the projection (i.e. TX/S commutes with base change).

Proof. Since the question is local we may reduce to a diagram of k-algebras of the form: B = P/(f ) → B 0 = PA0 /(f 0 ) ↑ ↑ A → A0 where P is a smooth A-algebra, f ∈ P is a regular element, and f 0 is the image of f in PA0 = P ⊗A A0 . Then we have (f )/(f 2 ) ∼ =B f¯ and

(f 0 )/(f 02 ) ∼ = B0 f¯0

δ

−→ ΩP/A ⊗P B 7→ df ⊗ 1 δ0

−→ ΩP 0 /A0 ⊗P 0 B 0 7→ df 0 ⊗ 1

But since ΩP 0 /A0 ⊗P 0 B 0 = (ΩP/A ⊗P B) ⊗B B 0 we have δ 0 = δ ⊗B B 0 and 1 TB/A ⊗B B 0 = coker(δ ∨ ) ⊗B B 0 = coker(δ 0∨ ) = TB1 0 /A0

qed Lemma 4.7.6 Let S be an algebraic scheme and C ⊂ IP 2 × S a flat family of plane curves of degree d. Let s ∈ S be a k-rational point such that the fibre 1 C(s) is a curve having at most nodes as singularities. Then TC/S is S-flat at a point p ∈ C(s) if and only if the family is formally locally trivial at s around p.

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311

Proof. If p is a nonsingular point of C(s) then C → S is smooth at p, hence T 1 (C/S, OC )p = 0 and the assertion is obvious. Let’s assume that p is a node of C(s). Let A = OS,s , Aα := A/mα , Sα = Spec(Aα ) and Cα → Sα the induced infinitesimal deformation of C1 = C(s); denote by B = OC,p and by Bα = OCα ,p . By Lemma 4.7.5 we have: 1 TC1α /Sα ,p = TB1 α /Aα ∼ ⊗B Bα = T 1 (C/S, OC )p ⊗OC,p Bα = TB/A

(4.45)

1 Assume that T 1 (C/S, OC )p = TB/A is A-flat. Then TB1 α /Aα is Aα -flat by (4.45) and therefore Bα is the trivial deformation of B0 = B/mA B = OC(s),p , by Lemma 4.7.4. Since this is true for every α we conclude that the family C → S is formally locally trivial at p. Conversely, assume that C → S is formally locally trivial at p. Then Bα ∼ = B0 ⊗k Aα for all α and TB1 α /Aα is Aα -flat by Lemma 4.7.4. But by (4.45) we have 1 1 TB1 α /Aα = TB/A ⊗B Bα = TB/A ⊗A Aα 1 and from the local criterion of flatness we deduce that TB/A = T 1 (C/S, OC )p is A-flat. qed

Proof of Theorem 4.7.3 Consider the universal family H ⊂ IP 2 × Σd of plane curves of degree d 1 and let {Wi } be the flattening stratification of TH/Σ . Let W = Wi be a d stratum containing a k-rational point s ∈ Σd parametrizing a reduced curve H(s) having δ nodes and no other singularity, and let H0 ⊂ IP 2 × W be the induced family of degree d curves. By Lemma 4.7.5 we have 1 TH1 0 /W ∼ ⊗ OH 0 = TH/Σ d

and therefore, by construction, TH1 0 /W is flat over W . Moreover, since H0 ⊂ IP 2 × W is a regular embedding of codimension 1, TH1 0 /W is of the form OV for some closed subscheme V ⊂ IP 2 × W . By applying Lemma 4.7.5 again we deduce 1 OV (s) = TH1 0 /W ⊗ k ∼ = TH(s) which is a reduced scheme of length δ supported at Sing(H(s)). This implies that V → W is etale at the δ points of V (s). Therefore there is an open neighborhood U of s ∈ W such that V (U ) → U is etale of degree δ. If u ∈ U 1 ∼ is a k-rational point then H(u) is a curve such that TH(u) = V (u), hence H(u) has δ singular points p1 , . . . , pδ and no other singularity, such that TO1 pj ∼ = k.

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From Proposition 3.1.5 it follows that H(u) is a δ-nodal curve. Therefore by applying Lemma 4.7.6 we see that the family H0 (U ) → U is an element of Vdδ,0 (U ). Putting together all these open sets we obtain a locally closed subset Ui ⊂ Wi such that the induced family Hi0 ⊂ IP 2 × Ui defines an element of Vdδ,0 (Ui ). Now let [ Vdδ = Ui i 2

Vdδ

and let H ⊂ IP × be the induced family. If C ⊂ IP 2 × S is an element of Vdδ,0 (S) for an algebraic scheme S then by the universal property of |O(d)| we obtain a unique morphism S → Σd inducing the given family by pullback. By Lemma 4.7.6 and the defining property of the flattening stratification such morphism factors through Vdδ . Thus (Vdδ , H) represents the functor Vdδ,0 . qed We now consider the local properties of the Severi varieties. Proposition 4.7.7 Let C ⊂ IP 2 be a reduced curve having degree d, δ nodes and κ ordinary cusps and no other singularity. Let [C] ∈ V = Vdδ,κ be the point parametrizing C. Then there is a natural identification: 0 T[C] V = H 0 (C, NC/I P2) 0 and H 1 (C, NC/I P 2 ) is an obstruction space for OV,[C] .

Proof. T[C] V is the subspace of T[C] Σd = H 0 (C, OC (d)) corresponding to locally trivial first order deformations, and these are the elements of 0 0 H 0 (C, NC/I P 2 ) by the very definition of NC/IP 2 . From the proof of Proposition 3.2.6 it is obvious that obstructions to deforming locally trivial deformations 0 lie in the space H 1 (C, NC/I qed P 2 ). According to the classical terminology, we call Vdδ,κ regular at a point δ,κ 0 [C] if H 1 (C, NC/I is called superabundant at [C]. An P 2 ) = 0; otherwise Vd δ,κ irreducible component W of Vd is called regular (resp. superabundant) if it is regular (resp. superabundant) on a non-empty open subset. Vdδ,κ is called regular if all its components are regular; otherwise it is called superabundant. From 4.7.7 and from Example 4.7.1(iii) it follows that if a component W of Vdδ,κ is regular then it is generically nonsingular of dimension 3d + g − 1 − κ, where g is the geometric genus of C, i.e. of pure codimension δ + 2κ in Σd .

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313

Corollary 4.7.8 If κ < 3d then Vdδ,κ is regular at every point. In particular Vdδ is regular at every point, thus it is nonsingular of pure dimension !

d+2 3d + g − 1 = −1−δ 2 if it is non-empty. Proof. It follows from Proposition 4.7.7 and from Example 4.7.1(iii). qed Remark 4.7.9 Note that the corollary does not claim that Vdδ,κ 6= ∅. The non-emptiness of the Severi varieties will be discussed in the next subsection. Corollary 4.7.8 follows also from Proposition 3.4.16 recalling that, by Lemma 3.4.15, we have 0 ∼ i ˜ H i (C, NC/I P 2 ) = H (C, Nϕ ),

i = 0, 1

where ϕ : C˜ → C is the normalization map. The description of the deformations of C given by Proposition 3.4.16 can be considered as the “parametric” counterpart of the “cartesian” point of view of Corollary 4.7.8. The Severi varieties Vdδ,κ may have a complicated structure. If there are 0 too many cusps then in general a [C] ∈ Vdδ,κ satisfies H 1 (C, NC/I P 2 ) 6= (0) δ,κ (see Example 4.7.10 below) and in fact Vd can be singular at such a [C]. To decide whether this effectively happens has been a long standing classical problem (see [212], ch. VIII, where this topic is discussed). The first example of a singular point of a Severi variety was given by Wahl [209]: it is a plane irreducible curve of degree 104 with 3636 nodes and 900 cusps. For other examples see [134], [84] and [204]. Example 4.7.10 If κ > 3d then Vdδ,κ can be superabundant. The following classical example is due to B. Segre (see [179] and [212], p. 220). Consider plane curves of the following type: C : [f2m (x, y)]3 + [f3m (x, y)]2 = 0 where f2m (x, y) and f3m (x, y) are general polynomials of the indicated degrees, and m > 2. Then d = deg(C) = 6m, δ = 0 and κ = 6m2 because the only singularities of C are the points of intersection of the curves f2m = 0 and

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f3m = 0 and they are easily seen to be cusps. C is irreducible of geometric genus ! 6m − 1 g= − 6m2 2 The dimension of the family of curves C is !

!

2m + 2 3m + 2 1 R := + − 1 = (13m + 2)(m + 1) 2 2 2 which is larger than r=

6m(6m + 3) − 2κ = 6m2 + 9m 2 2

0,6m In fact R − r = m−1 . Therefore V6m is superabundant at all points [C]. 2 1 0 Let’s compute h (C, NC/IP 2 ). By the analysis of Example 4.7.1(iii) we 0 know that h1 (C, NC/I P 2 ) equals the index of speciality ι of the linear system cut on the normalization C˜ of C by the curves of degree 6m passing through the cusps and tangent there to the cuspidal tangents. It is an easy computation (see [212] p. 220 for details) that

ι=R−r The conclusion is that each [C] is a nonsingular point of a superbundant 0,6m2 of dimension R. component of V6m For a modern treatment of this example see [201].

4.7.3

Non-emptiness of Severi varieties

Even though we have proved precise results about the structure of the Severi varieties, it is not clear that nodal curves of given degree and number of nodes exist at all: the task of writing down explicitly the equation of such a curve is too concrete and precise to be within the reach of known techniques. Nevertheless Severi himself outlined a method to prove the existence of nodal curves. His approach is based on the notion of “analytic branch” δ and consists in analyzing the local structure of V d along Vdδ+1 . While his proof makes perfectly good sense over the field of complex numbers, it is not straightforward to translate it into an algebraic proof. In this subsection we

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315

will show that the Severi varieties Vdδ of nodal curves are non-empty in the expected range of δ using a different method which is entirely algebraic and elementary. It is based on the theory of multiple point schemes, which we will now recall. Consider a finite unramified morphism f : X → Y of algebraic schemes and let Nk (f ) ⊂ Y be the k-th multiple point scheme of f (for the definition see Example 4.2.9, page 239). Define Mk (f ) = f −1 (Nk (f )) ⊂ X (scheme theoretic inverse image). Note that in particular N1 (f ) is supported on the image of f and M1 (f ) = X. Let X ×Y X = ∆

a

X2

(where ∆ is the diagonal and the union is disjoint because f is unramified) and let f1 : X2 → X be the morphism induced by the first projection. Then f1 is called the first iteration morphism of f . Lemma 4.7.11 Let f : X → Y be a finite unramified morphism of algebraic schemes. Then (i) The first iteration morphism f1 is finite and unramified. (ii) Nk−1 (f1 ) = Mk (f ) for all k ≥ 2. Proof. (i) f1 is finite and unramified because both properties are invariant under base change and composition with a closed embedding. (ii) By the base change property of the Fitting ideals we have Mk (f ) = f −1 (Nk (f )) = Nk (π1 ) = Nk−1 (f1 ) where π1 : X ×Y X → X is the first projection.

qed

Lemma 4.7.12 Let f : X → Y be a finite unramified morphism of purely dimensional algebraic schemes, such that dim(X) = dim(Y ) − 1 and Y is ˜ 2 of X2 satisfies nonsingular. Then every irreducible component X ˜ 2 ) ≥ dim(Y ) − 2 dim(X

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Proof. We have X ×Y X = (f × f )−1 (∆Y ) where ∆Y ⊂ Y × Y is the diagonal. Since Y is nonsingular, ∆Y is regularly embedded of codimension dim(Y ) in Y × Y . It follows that every component of X ×Y X has codimension ≤ dim(Y ) in X × X (Lemma D.2). We deduce ˜ 2 of X2 satisfies that every component X ˜ 2 ) ≥ 2 dim(X) − dim(Y ) = dim(Y ) − 2 dim(X qed If the morphism f satisfies some further assumptions then one can describe its behaviour quite precisely. Definition 4.7.13 Let f : X → Y be a finite unramified morphism of algebraic schemes. Then f is called self-transverse of codimension one if X and Y are nonsingular and purely dimensional, dim(X) = dim(Y ) − 1, and for any closed point y ∈ Y and for any r distinct points x1 , . . . , xr ∈ f −1 (y) the tangent spaces Tx1 X, . . . , Txr X, viewed as subspaces of Ty Y , are in general position (i.e. their intersection has codimension r). Lemma 4.7.14 If f : X → Y is a self-transverse codimension one morphism then f1 : X2 → X is a self-transverse codimension one morphism. Proof. Let (x1 , x2 ) ∈ X2 and let y = f1 (x1 , x2 ) = f (x1 ) = f (x2 ). Then T(x1 ,x2 ) X2 = Tx1 X ×Ty Y Tx2 X Since f is self-transverse we have dim(T(x1 ,x2 ) X2 ) = dim(Y ) − 2. On the other hand dim(X2 ) ≥ dim(Y ) − 2, by Lemma 4.7.12. It follows that X2 is nonsingular of pure dimension dim(Y ) − 2. For any x ∈ X and (x, x0 ) ∈ f1−1 (x) the differential df1(x,x0 ) : T(x,x0 ) X2 → Tx X is an injection of codimension one. For any (x, x2 ), . . . , (x, xs ) ∈ f1−1 (x) we have df1(x,x2 ) (T(x,x2 ) X2 )

\

···

\

df1(x,xs ) (T(x,xs ) X2 ) = Tx X

\

Tx2 X

\

···

\

Txs X

viewed as subspaces of Ty Y . The self-transversality of f1 now follows from the analogous property of f . qed With this terminology we can state the following useful result.

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Proposition 4.7.15 Assume that f : X → Y is a self-transverse codimension one morphism and that Nr (f ) 6= ∅ for some r ≥ 2. Then Nr (f ) has pure codimension r in Y and Ns (f ) 6= ∅ for all 1 ≤ s ≤ r − 1. In particular Ns (f ) has pure codimension s for all 1 ≤ s ≤ r − 1. Proof. By induction on r. If r = 2 then M2 (f ) = N1 (f1 ), by Lemma 4.7.11(ii), and has pure dimension equal to dim(X2 ) = dim(Y )−2 by Lemma 4.7.14: therefore N2 (f ) is of pure codimension 2. Moreover N1 (f ) = f (X) 6= ∅ and has pure dimension dim(X) = dim(Y ) − 1. Now assume r ≥ 3. Nr (f ) has pure codimension r in Y if and only if Mr (f ) has pure codimension r − 1 in X. By Lemma 4.7.11(ii) we have Mr (f ) = Nr−1 (f1 ) and by the inductive hypothesis Nr−1 (f1 ) has pure codimension r −1 because f1 is self-transverse of codimension one. Again by the inductive hypothesis Ms (f ) = Ns−1 (f1 ) 6= ∅ for all 1 ≤ s − 1 ≤ r − 2. This implies that Ns (f ) 6= ∅ for all 2 ≤ s ≤ r − 1. The case s = 1 is a consequence of the first part of the proof. qed Next we will see how Proposition 4.7.15 can be applied to the study of the Severi varieties of nodal curves. ∗

∗

∗

∗

∗ ∗ 2

Consider the universal family H ⊂ IP × Σd of plane curves of degree 1 d ≥ 2 and the cotangent sheaf TH/Σ . Since H ⊂ IP 2 × Σd is a regular d 1 embedding of codimension one, TH/Σ is a quotient of OIP 2 ×Σd . Therefore we d 1 can identify TH/Σd = OZ where Z ⊂ H is a closed subscheme; evidently the set of closed points of Z is {(p, s) : p is a singular point of H(s)} ⊂ IP 2 × Σd Z can be described more precisely as follows. We let F (X0 , X1 , X2 ) =

X

An0 ,n1 ,n2 X0n0 X1n1 X2n2 = 0

n0 +n1 +n2 =d

be the equation of the universal curve H inside IP 2 × Σd . Then Z is defined by the three equations: ∂F ∂F ∂F = 0, = 0, =0 ∂X0 ∂X1 ∂X2

318

CHAPTER 4. HILBERT AND QUOT SCHEMES

Denote by π

π

1 2 IP 2 ←− Z −→ Σd

the projections. Lemma 4.7.16 (i) Z is irreducible, nonsingular, rational of codimension three in IP 2 × Σd . (ii) π2 maps Z birationally onto its image W ⊂ Σd , which is an irreducible rational divisor parametrizing all singular curves of degree d. Proof. π1 is surjective with fibres linear systems of dimension dim(Σd ) − 3 because for each p ∈ IP 2 the fibre π1−1 (p) is the linear system Σd (−2p) of all curves of degree d which are singular at p. This proves (i). Moreover by an easy application of Bertini’s theorem one gets that a general element of Σd (−2p) is a curve having a node at p as its only singular point. This takes care of (ii). qed Z contains an open subset Z1 whose set of closed points is {(p, s) : H(s) has only nodes as singularities and p is a node of H(s)} From the proof of Lemma 4.7.16 it follows immediately that Z1 ⊂ Z is a non-empty (dense) open subset. Let W1 := π2 (Z1 ) W1 is a dense open subset of W . It parametrizes all singular curves of degree d having only nodes as singularities; a general point of W1 parametrizes a curve of degree d having one node and no other singularities. Consider the closed subset Bd := π2 (W \W1 ) ⊂ Σd , which parametrizes all singular non-nodal curves of degree d, and the morphism π : Z1 → Σd \Bd obtained by restricting π2 . Proposition 4.7.17 π is birational onto its image W1 , finite, unramified and self-transverse of codimension one.

4.7.

PLANE CURVES

319

Proof. If s ∈ W1 = π(Z1 ) is a closed point then π −1 (s) is the scheme of nodes of H(s) which is finite and reduced. Therefore π is unramified. Moreover π is the restriction over an open subset of Σd of a projective morphism. Therefore π is finite. The birationality onto its image follows from that of π2 (Lemma 4.7.16). In order to prove that π is self-transverse of codimension one, let X ⊂ IP 2 be a curve of degree d having δ nodes {p1 , . . . , pδ } and no other singularity. Then (pi , [X]) ∈ Z1 , i = 1, . . . , δ. The local analysis of Example 4.7.1(i) and the fact that π is unramified show that we have Im[dπ(pi ,[X]) ] = H 0 (X, NX/IP 2 (−pi )) where 1 NX/IP 2 (−pi ) = ker[NX/IP 2 → TX,p ] i

Therefore

δ \

0 δ Im[dπ(pi ,[X]) ] = H 0 (X, NX/I P 2 ) = T[X] Vd

i=1

Since T[X] Vdδ has codimension δ by Corollary 4.7.8, this equality means that the subspaces Im[dπ(pi ,[X]) ] are hyperplanes of T[X] Σd = H 0 (X, NX/IP 2 ) in general position. qed We can now prove the main result of this subsection. Theorem 4.7.18 (Severi[194]) Let X ⊂ IP 2 be a curve of degree d having δ nodes and no other singularity, i.e. such that [X] ∈ Vdδ . Then δ−1

∅= 6 Vdδ ⊂ V d s

1

0

⊂ · · · ⊂ V d ⊂ V d = Σd s−1

and V d has pure codimension 1 in V d , for every s = 1, . . . , δ. Proof. From the previous analysis it follows that for each s Vds = Ns (π)\Ns+1 (π) The existence of X implies that Vdδ 6= ∅. The theorem is now an immediate consequence of Propositions 4.7.17 and 4.7.15. qed Corollary 4.7.19 (i) For every d ≥ 2 and !

d 0≤δ≤ 2 the Severi variety Vdδ is non-empty.

320

CHAPTER 4. HILBERT AND QUOT SCHEMES

(ii) For every d ≥ 2 and !

d−1 0≤δ≤ 2

the Severi variety Vdδ contains irreducible curves. Proof. (i) A curve X consisting of d distinct lines no three of which pass δo through the same point defines a point of Vd with δo = d2 and the nonemptiness of Vdδ for δ ≤ δo follows from the theorem. d (ii) A general projection in IP 2 of a rational normal curve Γd ⊂ IP is an irreducible curve Y of degree d having d−1 nodes. The theorem implies 2

that Y is in the closure of Vdδ for every δ ≤ d−1 . The following lemma 2 guarantees that for each such δ we can find irreducible curves in Vdδ . qed Lemma 4.7.20 Let C ⊂ IP 2 × S be a flat family of plane curves of degree d ≥ 2, with S an algebraic scheme, and let o ∈ S be a k-rational point. Assume that the fibre C(o) is reduced and irreducible. Then C(s) is reduced and irreducible for all s ∈ S in an open neighborhood of o. Proof. Denote by f : C → S the projection. By contradiction and after possibly shrinking S we may assume that C(s) is reducible and/or nonreduced for all s 6= o in S. Since H i (C(s), OC(s) (m)) = 0 for all m ≥ d − 2, i ≥ 1, and for all s ∈ S, by Theorem 4.2.5 t0m (s) : f∗ OC (m)s ⊗ k(s) → H 0 (C(s), OC(s) (m)) is bijective and f∗ OC (m)s is free for all m ≥ d − 2 and s ∈ S. Since C(o) is reduced and irreducible the multiplication map µo : H 0 (OC(o) (d − 1)) ⊗ H 0 (OC(o) (d − 1)) → H 0 (OC(o) (2d − 2)) is injective. Then the kernel of the multiplication map µ : f∗ OC (d − 1) ⊗ f∗ OC (d − 1) → f∗ OC (2d − 2) is zero in a neighborhood of o. By the bijectivity of the maps t0m (s) and t0m (o) for m = d − 1, 2d − 2, it follows that the multiplication map µs : H 0 (OC(s) (d − 1)) ⊗ H 0 (OC(s) (d − 1)) → H 0 (OC(s) (2d − 2)) is injective for all s in a neighborhood of o. This is impossible because C(s) is reducible and/or non-reduced for all s 6= o. qed

4.7.

PLANE CURVES

321

Remarks 4.7.21 (i) It is easy to see that Vdδ is reducible in general. For example V43 has two irreducible components and V69 has five irreducible components. An important classical problem, known as the “Severi problem”, has been to decide about the irreducibility of the open set of Vdδ parametrizing irreducible nodal curves. This problem has been solved affermatively in [87] and, independently, in [164] (see also [203]). A report of Harris’ proof is given in [133]. It is known that the open set of Vdδ,κ parametrizing irreducible curves is reducible in general if κ > 0. For examples see [180] (such examples are also reported in [212]). (ii) Theorem 4.7.18 and some generalizations of it have been reconsidered in [200], but the proof given there is based on infinitesimal considerations which seem to need a further insight. For an interesting discussion see [63]. The analogous problem for the varieties Vdδ,κ is still open, i.e. we don’t know a characterization of the values of d, δ, κ for which Vdδ,κ 6= ∅. For partial results see [103]; for a classical discussion see [179]. The results on multiple point schemes used here are special cases of a general theory for which we refer the reader to [110], [113] and to the references quoted there. (iii) The proof of Theorem 4.7.3 can be easily modified to prove the existence of universal families of curves with nodes and cusps (generalized Severi varieties) on a projective nonsingular surface Y . In such a proof one replaces Σd by HilbY and uses the existence and the universal property of HilbY . Such generalized Severi varieties behave in a way relatively similar to the Vdδ,κ ’s as long as Y has Kodaira dimension ≤ 0 (see [200], [125]). On surfaces of general type the situation changes radically. On such a surface Y the generalized Severi varieties can be superabundant even when κ = 0 and it is not known in which range of δ they are not empty. A systematic study of them has started relatively recently. We refer the reader to [36], [73], [34], [58], [59] for details.

322

CHAPTER 4. HILBERT AND QUOT SCHEMES

Appendix A Flatness The algebraic notion of flatness, introduced for the first time in [185], is the basic technical tool for the study of families of algebraic varieties and schemes. In this appendix we will overview the main algebraic results needed. For the properties of flat morphisms between schemes we refer to [89]. See also §4.2. A module M over a ring A is A-flat (or flat over A, or simply flat) if the functor N 7→ M ⊗A N from the category of A-modules into itself is exact. Since this functor is always right exact, the flatness means that it takes monomorphisms into monomorphisms. An A-algebra B is flat over A if B is flat as an A-module. The A-module M is said to be faithfully flat if for every sequence of A-modules N 0 → N → N 00 the sequence M ⊗A N 0 → M ⊗A N → M ⊗A N 00 is exact if and only if the original sequence is exact. Obviously, if M is faithfully flat then it is flat. In a similar way we give the notion of faithfully flat A-algebra. It is straightforward to check that if A → B is a local homomorphism of local rings, then a B-module of finite type is faithfully A-flat if and only if it is A-flat and nonzero. Recall that the flatness of an A-module M is equivalent to any of the following conditions: (1) TorA i (M, N ) = (0) for all i > 0 and for every A-module N . (2) TorA 1 (M, N ) = (0) for every A-module N . (3) TorA 1 (M, N ) = (0) for every finitely generated A-module N . 323

324

APPENDIX A. FLATNESS

(4) TorA 1 (M, A/I) = (0) for every ideal I ⊂ A. (5) I ⊗A M → M is injective for every ideal I ⊂ A. (6) I ⊗A M → IM is an isomorphism for every ideal I ⊂ A. Example A.22 Let k be a ring, u, v indeterminates and f : k[u, uv] → k[u, v] the inclusion. Then k[u, uv] k[u, uv] u = k[u] −→ k[u] = (uv) (uv) is injective. Tensoring by ⊗k[u,uv] k[u, v] we obtain: k[u, v] u k[u, v] −→ (uv) (uv) which is not injective. Therefore f is not flat. We list without proof a few basic properties of flat modules: Proposition A.23 (I) M is A-flat if and only if Mp is Ap -flat for every prime ideal p. (II) Every projective module is flat. (III) Assume M is finitely generated. Then M is flat if and only if it is projective; if A is local then M is flat if and only if it is free. (IV) If S ⊂ A is a multiplicative subset then AS is A-flat. (V) A direct sum M = ⊕i∈I Mi is flat if and only if all Mi ’s are flat. (VI) Let 0 → M 0 → M → M 00 → 0 be an exact sequence of A-modules with M 00 flat. Then M is flat if and only if M 0 is flat. (VII) Base change: if M is A-flat and f : A → B is a ring homomorphism, then M ⊗A B is B-flat.

325 (VIII) Transitivity: if B is a flat A-algebra and N is a flat B-module, then N is A-flat. (IX) If A is a noetherian ring and I is an ideal, the I-adic completion Aˆ is a flat A-algebra. If I is contained in the Jacobson radical of A then Aˆ is a faithfully flat A-algebra. (X) If B is an A-algebra and if there exists a B-module M which is faithfully flat, then the morphism Spec(B) → Spec(A) is surjective. (XI) If X1 , . . . , Xr are indeterminates, then A[X1 , . . . , Xr ] and A[[X1 , . . . , Xr ]] are A-flat. The following result is frequently used: Proposition A.24 If A is an artinian local ring with residue field k the following are equivalent for an A-module M : (i) M is free (ii) M is flat (iii) TorA 1 (M, k) = (0) Proof. (i) ⇒ (ii) ⇒ (iii) are clear. (iii) ⇒ (ii). Let N be a finitely generated A-module and let N = N0 ⊃ · · · ⊃ Nn = (0) be a composition series for N such that Ni /Ni+1 ∼ =k for i = 0, . . . , n − 1. Using the Tor exact sequences from the hypothesis (iii) we deduce that Tor1 (M, N ) = (0) and the flatness of M follows from (3). Let’s now prove (ii) ⇒ (i). Let {ej }j∈J be a system of elements of M which induces a basis of M ⊗A k over k. The system {ej } defines a homomorphism f : AJ → M which induces an isomorphism k J → M ⊗A k. From the following lemma it follows that f is an isomorphism, and therefore M is free. qed

326

APPENDIX A. FLATNESS

Lemma A.25 Let R be a ring, I an ideal and f : F → G a homomorphism of R-modules with G flat. Assume that one of the following conditions is satisfied: (a) I is nilpotent. (b) R is noetherian, I is contained in the Jacobson radical of R and F and G are finitely generated. If the induced homomorphism F/IF → G/IG is an isomorphism, then f is an isomorphism. Proof. Let K = coker(f ). Tensoring the exact sequence F →G→K→0 with R/I we get K/IK = 0: from Nakayama’s lemma (which holds in either of the hypothesis (a) and (b)) it follows that K = 0, and therefore F is surjective. Letting H = ker(f ) we deduce an exact sequence 0 → H/IH → F/IF → G/IG → 0 using the flatness of G. By Nakayama again we deduce H = 0 and the conclusion follows. qed The following is a basic criterion of flatness. Theorem A.26 (Local criterion of flatness) Suppose that ϕ : A → B is a local homomorphism of local noetherian rings, and let k = A/mA be the residue field of A. If M is a finitely generated B-module, then the following conditions are equivalent: (i) M is A-flat (ii) TorA 1 (M, k) = 0. (iii) M ⊗A (A/mnA ) is flat over A/mnA for every integer n ≥ 1. (iv) M ⊗A (A/mnA ) is free over A/mnA for every integer n ≥ 1.

327 Proof. (i) ⇒ (ii) is obvious. (ii) ⇒ (i) see [49], Th. 6.8, p. 167. (i) ⇒ (iii) is obvious. (iii) ⇒ (i) It suffices to show that for every inclusion N 0 → N of Amodules of finite type we have an inclusion M ⊗A N 0 → M ⊗A N . For this purpose it suffices to show that the kernel of this last map is contained in Kn := ker[M ⊗A N 0 → M ⊗A (N 0 /N 0 ∩ mnA N )] for all n, because rows: 0 → Kn

T

n

Kn = (0). We have a commutative diagram with exact

→ M ⊗A N 0 ↓ M ⊗A N

→ M ⊗A (N 0 /N 0 ∩ mnA N ) → 0 ↓ → M ⊗A (N/mnA N ) →0

The last vertical arrow coincides with the map obtained from the injection N 0 /N 0 ∩ mnA N → N/mnA N after tensoring over A/mnA with the A/mnA -flat module M ⊗A (A/mnA ), and therefore it is injective. The conclusion follows from the above diagram. (iii) ⇔ (iv) follows from Proposition A.24 because A/mnA is artinian.qed For a more general version of the local criterion we refer to [3], exp. IV, Th´eor`eme 5.6. Note that A.24 is a special case of A.26. Corollary A.27 Suppose that ϕ : A → B is a local homomorphism of local noetherian rings, let k = A/mA be the residue field of A, M, N two finitely generated B-modules, and suppose that N is A-flat. Let u : M → N be a B-homomorphism. Then the following are equivalent: (i) u is injective and coker(u) is A-flat. (ii) u ⊗ 1 : M ⊗ k → N ⊗ k is injective. Proof. (i) ⇒ (ii). Let G = Coker(u). Tensoring by k the exact sequence u

0 → M −→ N → G → 0 by k we obtain the exact sequence: u⊗1

TorA 1 (G, k) → M ⊗A k −→ N ⊗A k → G ⊗A k → 0

328

APPENDIX A. FLATNESS

Since G is A-flat we have TorA 1 (G, k) = 0, and it follows that u⊗1 is injective. (ii) ⇒ (i). Factor u ⊗ 1 as β

α

M ⊗A k −→ Im(u) ⊗A k −→ N ⊗A k Then α is an isomorphism and β is injective. Tensoring by k the exact sequence 0 → Im(u) → N → G → 0 (A.1) we obtain the exact sequence: β

A TorA 1 (N, k) → Tor1 (G, k) → Im(u) ⊗A k −→ N ⊗A k → G ⊗A k → 0

Since N is A-flat we have TorA 1 (N, k) = 0; from the injectivity of β we deduce A Tor1 (G, k) = 0 and from A.26 it follows that G is A-flat. Applying (VI) to the exact sequence (A.1) we deduce that Im(u) is A-flat as well. Consider the exact sequence: 0 → ker(u) → M → Im(u) → 0 and tensor by k. We obtain the exact sequence: α

0 → ker(u) ⊗A k → M ⊗A k −→ Im(u) ⊗A k → 0 Since α is an isomorphism we deduce that ker(u) ⊗A k = 0, and therefore ker(u) = 0 by Nakayama’s lemma. qed A related result is the following: Lemma A.28 Let B be a local ring with residue field K, and let d : G → F be a homomorphism of finitely generated B-modules, with F free. Then d is split injective if and only if d ⊗B K : G ⊗B K → F ⊗B K is injective. In such a case also G is free. Proof. d is split injective if and only if coker(d) is free and d is injective. If this last condition is satisfied then clearly d ⊗B K is injective. Conversely, assume that d ⊗B K is injective, and factor d as G → Im(d) → F We see that G ⊗B K Im(d) ⊗B K

→ Im(d) ⊗B K → F ⊗B K

is bijective is injective

329 From the exact sequence 0 → Im(d) → F → coker(d) → 0 we get 0 → Tor1 (coker(d), K) → Im(d) ⊗B K → F ⊗B K so Tor1 (coker(d), K) = (0) and this implies that coker(d) is free. From the above exact sequence we deduce that Im(d) is free as well, so that 0 → ker(d) → G → Im(d) → 0 is split exact. Recalling that G⊗B K ∼ = Im(d)⊗B K we deduce that ker(d)⊗B K = (0), hence ker(d) = (0) by Nakayama. qed For the reader’s convenience we include the proof of the following well known lemma: Lemma A.29 Let (B, m) be a noetherian local integral domain, with residue field K and quotient field L. If M is a finitely generated B-module and if dimK (M ⊗B K) = dimL (M ⊗B L) = r then M is free of rank r. Proof. Let m1 , . . . , mr ∈ M be such that their images in M ⊗B K = M/mM form a basis. Then they define a homomorphism ϕ : B r → M and we have an exact sequence: ϕ

0 → N → B r −→ M → Q → 0 where N and Q are kernel and cokernel of ϕ. Since tensoring with K we get ϕ ¯

K r −→ M/mM → Q/mQ → 0 and ϕ¯ is surjective, we get Q/mQ = (0) and from Nakayama’s lemma it follows that Q = (0): hence ϕ is surjective. Now we tensor the above exact sequence with L, which is flat over B (by (IV)), and we obtain the exact sequence: ϕ ˜ 0 → N ⊗B L → Lr −→ M ⊗B L → 0 Since M ⊗B L ∼ ˜ = (0). = Lr and ϕ˜ is surjective, it follows that N ⊗B L = ker(ϕ) Therefore N is a torsion module. But N ⊂ B r and therefore N = (0). qed We have the following useful criterion:

330

APPENDIX A. FLATNESS

Lemma A.30 Let A → A0 be a small extension in A, and let g : A → R be a homomorphism of k-algebras. Let R0 = R ⊗A k. Then g is flat if and only if ker(R → R ⊗A A0 ) ∼ = R0 and the homomorphism g 0 : A0 → R ⊗A A0 induced by g is flat. Proof. Assume that g is flat. Then since R ⊗A () ∼ = R ⊗A k = R0 and A 0 Tor1 (R, A ) = 0, from the exact sequence 0 0 0 → TorA 1 (R, A ) → R ⊗A () → R → R ⊗A A → 0

(A.2)

we deduce that the first condition is satisfied. The flatness of g 0 is obvious. Assume conversely that the conditions of the statement are satisfied. 0 0 Then the sequence (A.2) implies that TorA 1 (R, A ) = 0. If A = k the conclusion follows from A.24. If not, from the exact sequence 0 → mA0 → A0 → k → 0 one gets the exact sequence: ∂

A 0 TorA R ⊗A mA0 → A0 → R ⊗A k → 0 1 (R, A ) → Tor1 (R, k) −→ k k k 0 0 0 R ⊗A0 mA0 R ⊗A0 k

From the flatness of R0 over A0 we deduce that ∂ = 0, hence TorA 1 (R, k) = 0, and we conclude by A.24. qed ∗

∗

∗

∗

∗ ∗

Flatness in terms of generators and relations. Let P be a noetherian k-algebra, J ⊂ P an ideal. Let A be in ob(A), PA = P ⊗k A, and J ⊂ PA an ideal such that (PA /J) ⊗A k ∼ = P/J. We want to find the conditions J has to satisfy so that PA /J is A-flat. We have the following Theorem A.31 Let Π0 : P n → P N → P → P/J → 0 be a presentation of P/J as a P -module. Then the following conditions are equivalent for an ideal J ⊂ PA :

331 (i) PA /J is A-flat and (PA /J) ⊗A k ∼ = P/J. (ii) There is an exact sequence Π : PAn → PAN → PA → PA /J → 0 such that Π0 = Π ⊗A k (= Π/mA Π). (iii) There is a complex ϕ

Π : PAn −→ PAN → PA → PA /J → 0 which is exact except possibly at PAN , such that

Π0 = Π ⊗A k.

Proof. (ii) ⇒ (i). We have: TorA 1 (PA /J, k) = H1 (Π ⊗ k) = H1 (Π0 ) = (0) From A.24 it follows that PA /J is A-flat. Moreover (ii) implies that (PA /J)⊗A k∼ = P/J. (i) ⇒ (ii). Choose a PA -homomorphism p : PAN → J which makes the following diagram commute: PAN ↓ p0 : P N p:

→

J ↓ → J

where p0 is the surjective homomorphism defined by the presentation Π0 . From the flatness of PA /J it follows that TorA 1 (PA /J, k) = (0); hence the exact sequence 0 → TorA 1 (PA /J, k) → J ⊗ k → PA ⊗ k → (PA /J) ⊗A k → 0 k k P P/J implies that J ⊗ k = J. It follows that p ⊗A k = p0 and therefore coker(p) ⊗A k = coker(p0 ) = (0) so that coker(p) = (0) by Nakayama’s lemma. Hence p is surjective.

332

APPENDIX A. FLATNESS

Now consider the exact sequence 0 → ker(p) → PAN → J → 0 and the associated Tor sequence: N TorA →J →0 1 (J, k) → ker(p)/mA ker(p) → P

(A.3)

From the flatness of PA /J and from the exact sequence 0 → J → PA → PA /J → 0 A we have TorA 1 (J, k) = Tor2 (PA /J, k) = (0). Therefore from (A.3) we see that ker(p)/mA ker(p) ∼ = ker(p0 )

Arguing as before we can find a surjective homomorphism q : PAn → ker(p) which makes the following diagram commutative: PAn ↓ Pn

q

−→ →

ker(p) ↓ ker(p0 )

(ii) ⇒ (iii) is obvious. (iii) ⇒ (i) If Π is not exact at PAN then we can add finitely many generators of the kernel of PAN → PA to obtain an exact sequence 0

Π0 :

ϕ0

PAn −→ PAN → PA → PA /J → 0

Then Π0 ⊗A k has the form: 0

ϕ0 ⊗k

P n −→ P N → P → P/J → 0 Since Im(ϕ ⊗ k) ⊂ Im(ϕ0 ⊗ k) ⊂ ker[P N → P ] we see that Im(ϕ0 ⊗ k) = ker[P N → P ] and therefore Π0 ⊗A k is exact. Now (i) follows from A.24. qed

333 Corollary A.32 Assume that J = (f1 , . . . , fN ) ⊂ P and that J = (F1 , . . . , FN ) ⊂ PA with fj = Fj (mod mA PA ), j = 1, . . . , N . Then every relation among f1 , . . . , fN lifts to a relation among F1 , . . . , FN if and only if PA /J is A-flat and (PA /J) ⊗A k ∼ = P/J. Proof. The condition that the Fj ’s reduce to the fj ’s modulo mA PA implies that the exact sequence F

PAN −→ PA → PA /J → 0 reduces to f

P N −→ P → P/J → 0

(A.4)

when tensored by ⊗A k. Complete (A.4) to a presentation Π0 of P/J. The condition that every relation among f1 , . . . , fN lifts to a relation among F1 , . . . , FN is a restatement of condition (iii) of A.31. Therefore the conclusion follows from Theorem A.31. qed Example A.33 Let A be in ob(A). Suppose that f1 , . . . , fN ∈ P form a regular sequence, and let F1 , . . . , FN ∈ PA be any liftings of f1 , . . . , fN , i.e. such that fj = Fj (mod mPA ), j = 1, . . . , N . Then J = (F1 , . . . , FN ) ⊂ PA defines a flat family of deformations of X = Spec(P/J), where J = (f1 , . . . , fN ). In fact every relation among f1 , . . . , fN is a linear combination of the trivial ones rij = (0, . . . , fj , . . . , −fi , . . . , 0)

1≤i<j≤N

and these can be lifted to the corresponding trivial relations Rij = (0, . . . , Fj , . . . , −Fi , . . . , 0) among F1 , . . . , FN . Applying Corollary A.32 it is easy to show that F1 , . . . , FN form a regular sequence.

334

APPENDIX A. FLATNESS

Notes and Comments 1. In the proof of Theorem A.31 the condition that A is artinian has only been used in the proof of (i) ⇒ (ii) in order to apply Nakayama’s Lemma. In particular the implications (ii) ⇒ (i), (iii) ⇒ (i) and (ii) ⇒ (iii) hold for any A ∈ ob(A∗ ). Using the local criterion of flatness it is easy to verify that the implication (i) ⇒ (ii) (and therefore the equivalence of the three conditions) holds ˆ as well if A is in ob(A).

Appendix B Differentials Let A → B be a ring homomorphism. As usual, we will denote by ΩB/A the module of differentials of B over A, and by dB/A : B → ΩB/A the canonical A-derivation. Recall that ΩB/A := I/I 2 µ

where I = ker(B ⊗A B −→ B) is the natural map, and for each b ∈ B dB/A (b) = b ⊗ 1 − 1 ⊗ b is called the differential of b. We have a natural isomorphism of B-modules DerA (B, M ) ∼ = HomB (ΩB/A , M ) Note that the exact sequence µ0

0 → ΩB/A → (B ⊗A B)/I 2 −→ B → 0

(B.1)

where µ0 is induced by µ, is an A-extension of B. The ring PB/A := (B ⊗A B)/I 2 is called the algebra of principal parts of B over A. The A-extension (B.1) is trivial because we have splittings: λ1 , λ2 : B → PB/A defined by λ1 (b) = b ⊗ 1, λ2 (b) = 1 ⊗ b; note that dB/A = λ1 − λ2 . We will consider PB/A as a B-algebra via λ1 . The following are some fundamental properties of the modules of differentials: 335

336

APPENDIX B. DIFFERENTIALS

Proposition B.34 (i) If B ↑ A

A0

−→

are ring homomorphisms, then: ΩB/A ⊗A A0 ∼ = ΩB⊗A A0 /A0 (ii) If A → B is a ring homomorphism and ∆ ⊂ B is a multiplicative system, then: Ω∆−1 B/A ∼ = ∆−1 ΩB/A (iii) Let K → L be a finitely generated extension of fields. Then dimL (ΩL/K ) ≥ trdeg(L/K) and equality holds if and only if L is separably generated over K. In particular ΩL/K = (0) if and only if K ⊂ L is a finite algebraic separable extension. Proof. See [49].

qed

We now introduce two standard exact sequences. Theorem B.35 (Relative cotangent sequence) Given ring homomorphisms f

g

A −→ B −→ C there is an exact sequence of C-modules: α

β

ΩB/A ⊗B C −→ ΩC/A −→ ΩC/B → 0

(B.2)

where the maps are given by: α(dB/A (b) ⊗ c) = cdC/A (g(b)); β(dC/A (r)) = dC/B (r) Proof. See [49], prop. 16.2.

b ∈ B, c ∈ C qed

When B → C is surjective we have ΩC/B = (0) and the next theorem describes ker(α).

337 Theorem B.36 (Conormal sequence) Let f

g

A −→ B −→ C be ring homomorphisms with g surjective, and let J = ker(g), so that C = B/J. Then: (i) We have an exact sequence δ

α

J/J 2 −→ ΩB/A ⊗B C −→ ΩC/A → 0

(B.3)

where δ is the C-linear map defined by δ(¯ x) = dB/A (x) ⊗ 1. (ii) There is an isomorphism Ω(B/J 2 )/A ⊗(B/J 2 ) C ∼ = ΩB/A ⊗B C In other words the conormal sequence (B.3) depends only on the first infinitesimal neighborhood of Spec(C) in Spec(B). (iii) The map δ is a split injection if and only if there is a map of A-algebras C → B/J 2 splitting the projection B/J 2 → C. Proof. (i) see e.g. [49], prop. 16.3. (ii) Comparing the exact sequence (B.3) with the analogous sequence associated to A → B/J 2 → C we get a commutative diagram: J/J 2 k J/J 2

→

ΩB/A ⊗B C ↓ → Ω(B/J 2 )/A ⊗(B/J 2 ) C

→ ΩC/A k → ΩC/A

→0 →0

and the vertical arrow, which is induced by B → B/J 2 , must be an isomorphism. (iii) By (ii) we may assume that J 2 = 0, i.e. that 0 → J → B → C → 0 is an A-extension. Assume that δ : J → ΩB/A ⊗B C is a split injection, and let σ : ΩB/A ⊗B C → J be a splitting. Then the composition d¯

σ

B −→ ΩB/A ⊗B C −→ J is an A-derivation. It follows that 1 − σ d¯ : B → B is an A-homomorphism ¯ such that (1 − σ d)(J) = 0 and therefore it induces an A-homomorphism C → B which splits g.

338

APPENDIX B. DIFFERENTIALS

Conversely assume that g : B → C has a section τ : C → B. Then we have a derivation D : B → J ⊕ ΩC/A given by D(b) = (b − (τ g)(b), dC/A (g(b))). One easily checks that D induces an isomorphism ΩB/A ⊗B C ∼ qed = J ⊕ ΩC/A , thus proving the assertion. As an application we have the following: Proposition B.37 Let K be a field and (B, m) a local K-algebra with residue field B/m = K 0 . Then the map δ : m/m2 → ΩB/K ⊗B K 0 in the exact sequence (B.3) relative to K → B → K 0 is injective if and only if K ⊂ K 0 is a separable field extension. In particular, if B/m = K then δ : m/m2 → ΩB/K ⊗B K is an isomorphism. Therefore dim(B) ≤ dimK (ΩB/K ⊗B K) Proof. See [49], cor. 16.13. The last assertion follows from the conormal sequence relative to K → B → K. qed The following theorem describes the module of differentials for regular local rings. Theorem B.38 Assume that K is a field and B is a local noetherian Kalgebra with residue field B/m = K. If ΩB/K is a free B-module of rank equal to dim(B) then B is a regular local ring. If K is perfect (e.g. algebraically closed) and B is e.f.t. over K then the converse is also true. Proof. Assume first that ΩB/K is free of rank equal to dim(B). Then dimK (m/m2 ) = dim(B) by B.37, so B is a regular local ring. Assume conversely that K is perfect and that B is a regular local ring, e.f.t. over K. Then we have dimK (ΩB/K ⊗B K) = dimK (m/m2 ) = dim(B)

339 Let L be the quotient field of B. Then, by B.34(3), we have ΩB/K ⊗B L = ΩL/K and dimL (ΩL/K ) = trdeg(L/K) = dim(B) because L is separably algebraic over K, since K is perfect. Therefore we have dimK (ΩB/K ⊗B K) = dim(B) = dimL ΩB/K ⊗B L Since B is e.f.t. over K, ΩB/K is a finitely generated B-module, and from Lemma A.29 it follows that it is free of rank equal to dim(B). qed In particular we have the following: Corollary B.39 Let k be an algebraically closed field, and let B be an integral k-algebra of finite type. Then B is a regular ring if and only if ΩB/k is a projective B-module of rank equal to dim(B). Proof. Both conditions are satisfied if and only if they are satisfied after localizing at the maximal ideals of B. For every maximal ideal m ⊂ B the local ring Bm is a k-algebra e.f.t. with residue field k. By B.38 Bm is a regular local ring if and only if ΩBm /k = (ΩB/k )m is free of rank equal to dim(B). The conclusion follows. qed Proposition B.40 If the ring homomorphism A → B is e.f.t. then ΩB/A is a B-module of finite type. If in particular B = S −1 A[X1 , . . . , Xn ] for some multiplicative system S, then ΩB/A is a free B-module of rank n with basis {dB/A (X1 ), . . . , dB/A (Xn )}. Proof. The last assertion is elementary (see [49]). To prove the first, let B = (S −1 P )/J, where P = A[X1 , . . . , Xn ] and S ⊂ P is a multiplicative system. Then ΩB/A is a quotient of ΩS −1 P/A ⊗S −1 P B, by the conormal sequence. qed Remark B.41 If A and B are only assumed to be noetherian then ΩB/A is not necessarily a B-module of finite type even if A is a field. An example is given by ΩQ[[X]]/Q (see [1] ch. 0IV , n. 20.7.16).

340

APPENDIX B. DIFFERENTIALS

Examples B.42 (i) Assume that B = S −1 A[X1 , . . . , Xn ] for some multiplicative system S. Then DerA (B, B) = HomB (ΩB/A , B) is a free module of rank n with basis n ∂ ∂ o ,..., ∂X1 ∂Xn which is the dual of the basis {dB/A (X1 ), . . . , dB/A (Xn )} ∂ of ΩB/A , and where ∂X : B → B is the partial A-derivation with respect to j Xj . Let Y1 , . . . , Yn ∈ B be such that the jacobian determinant

det

∂Yi ∂Xj

is a unit in B. Then {dB/A (Y1 ), . . . , dB/A (Yn )} is another basis of ΩB/A and we have: dB/A (Xj ) =

∂Xj ∂Xj dB/A (Y1 ) + · · · + dB/A (Yn ) ∂Y1 ∂Yn

Dually: ∂Y1 ∂ ∂Yn ∂ ∂ = + ··· + ∂Xj ∂Xj ∂Y1 ∂Xj ∂Yn

(B.4)

The proof of these statements is straightforward. (ii) Let k be a field and let B = k[X, Y ]/(XY ), where X, Y are indeterminates. Then, since Ωk[X,Y ]/K ⊗ B ∼ = BdX ⊕ BdY , using the conormal sequence we deduce that BdX ⊕ BdY ΩB/k ∼ = (Y dX ⊕ XdY ) It follows that the element Y dX = −XdY is killed by the maximal ideal (X, Y ) and therefore it generates a torsion submodule T := (Y dX) ∼ = k ⊂ ΩB/k

341 The quotient is ΩB/k BdX ⊕ BdY ∼ = = k[X]dX ⊕ k[Y ]dY ∼ = (X, Y ) ⊂ B T (Y dX, XdY ) where the last isomorphism is given by f (X)dX ⊕ g(Y )dY 7→ f (X)X + g(Y )Y Therefore we have an exact sequence: 0 → T → ΩB/k → B → k → 0 (iii) Let k be a field and let B = k[t, X, Y ]/(f ) where t, X, Y are indeterminates and f = XY + t. Then arguing as before we see that BdX ⊕ BdY ΩB/k[t] ∼ = (Y dX ⊕ XdY ) The element Y dX = −XdY is not killed by any b ∈ B; therefore ΩB/k[t] is torsion free of rank one. The homomorphism ΩB/k[t] → B sending f (t, X)dX ⊕ g(t, Y )dY 7→ f (t, X)X + g(t, Y )Y is bijective onto the maximal ideal (t, X, Y ) so that we have an exact sequence: 0 → ΩB/k[t] → B → k → 0 (iv) Let k be a field and let k[] := k[t]/(t2 ), where we have denoted by the class of t mod (t2 ). Then the conormal sequence of k → k[t] → k[] is δ

(t2 )/(t4 ) −→ Ωk[t]/k ⊗k[t] k[] → Ωk[]/k → 0 and the middle term is isomorphic to k[]. The map δ acts as t¯2 t¯3

7→ 2 7 → 0

In particular we see that δ is not injective. Therefore

Ωk[]/k =

kd if char(k) 6= 2; k[]d if char(k) = 2

342

APPENDIX B. DIFFERENTIALS

and d : k[] → Ωk[]/k acts as d(α + β) = βd. (v) An obvious generalization of the above computation shows that if A = k[t]/(tn ), n ≥ 2 and char(k) = 0 or char(k) > n then ΩA/k = A/(t¯n−1 ) (vi) If B ∈ ob(A∗ ) then t∨B := mB /m2B and tB := (mB /m2B )∨ are the (Zariski) cotangent space respectively tangent space of B. We have mB /m2B ∼ = ΩB/k ⊗B k by Prop. B.37, and therefore Derk (B, k) = HomB (ΩB/k , k) = Homk (ΩB/k ⊗B k, k) = (mB /m2B )∨ Moreover there is a natural identification Derk (B, k) = Homk−alg (B, k[]) which we leave to the reader to verify. If µ : Λ → B is a homomorphism in A∗ , the induced homomorphism dµ∨ : mΛ /m2Λ → mB /m2B is the codifferential of µ, while its transpose dµ : tB → tΛ is the differential of µ. We define the relative cotangent space of B over Λ to be t∨B/Λ := coker(dµ∨ ) = mB /(m2B + mΛ B) and the relative tangent space of B over Λ as its dual: tB/Λ = ker(dµ) = [mB /(m2B + mΛ B)]∨ From the exact sequence ΩΛ/k ⊗Λ B → ΩB/k → ΩB/Λ → 0 tensored by k we deduce an identification t∨B/Λ = ΩB/Λ ⊗B k and therefore tB/Λ = HomB (ΩB/Λ , k) = DerΛ (B, k) = HomΛ−alg (B, k[]) where the Λ-algebra structure on k[] is defined by the composition Λ → k → k[] (the last equality is straightforward to verify).

343 The following Lemma describes a situation where the conormal sequence is exact. Lemma B.43 Assume char(k) = 0. Let e:

0 → (t) → R0 → R → 0

be a small extension in A. Then the conormal sequence δ

η : 0 → (t) −→ ΩR0 /k ⊗R0 R → ΩR/k → 0 is exact also on the left. ˜ Then the codifProof. Assume first that e is trivial, so that R0 = R⊕k. ferential ΩR0 /k ⊗R0 k → ΩR/k ⊗R k k k 2 mR0 /mR0 mR /m2R has a non-trivial kernel (Example 1.1.2) so that a fortiori ΩR0 /k ⊗R0 R → ΩR/k has a non-trivial kernel. Assume now that e is not trivial. Then, letting m = dim(tR0 ) = dim(tR ), we can write R0 = P/J 0 , R = P/J where P = k[X1 , . . . , Xm ], a polynomial algebra, and J 0 , J ⊂ (X)2 ⊂ P ideals such that J 0 ⊂ J and J/J 0 ∼ = (t). Let T ∈ J be such that t = T + J0 / (X)J. Since e is small we have (X)J ⊂ J 0 and therefore T ∈ Claim: We can choose T so that ∂T ∈ /J ∂Xi If

∂T ∂Xi

for some i

∂T ∈ J then Xi ∂X ∈ J 0 so that we can replace T by i

T1 := T − Xi

∂T ∂Xi

344

APPENDIX B. DIFFERENTIALS

If

∂T1 ∂2T ∈ /J = 2 ∂Xi ∂Xi we are done, otherwise we replace T1 by −Xi

T2 := T1 −

Xi ∂T1 ∂T X 2 ∂2T = T − Xi + i 2 ∂Xi ∂Xi 2 ∂Xi2

and we apply the same argument. After ν steps of this process we obtain Tν := Tν−1 −

ν X X s ∂sT Xi ∂Tν−1 = T + (−1)s i ν ∂Xi s! ∂Xis s≥1

∂Tν ∂Tν Since ∂X = 0 for ν 0 we see that either ∂X ∈ / J for some ν and we replace i i T by the first Tν with this property, or we can replace T by a Tν which is constant with respect to Xi . Repeating this process for every index i we will end up by replacing T by a T¯ having the required property or otherwise constant with respect to every variable, which is clearly a contradiction. The Claim is proved.

From the claim we deduce that dT =

X i

∂T dXi ∈ / JΩP/k ∂Xi

(B.5)

where d = dP/k : P → ΩP/k is the universal derivation. But we have: ΩR0 /k = ΩP/k /(J 0 ΩP/k + (dg 0 )g0 ∈J 0 ) so that ΩR0 /k ⊗R0 R = ΩP/k /(JΩP/k + (dg 0 )g0 ∈J 0 ) But since clearly dt 6= dg 0 for all g 0 ∈ J 0 , (B.5) implies that dt 6= 0 ∈ ΩR0 /k ⊗R0 R qed ∗

∗

∗

∗

∗ ∗

If f : X → Y is a morphism of schemes, we denote by Ω1X/Y the sheaf of relative differentials, or the relative cotangent sheaf, on X. It satisfies Ω1X/Y,x = ΩOX,x /OY,f (x)

345 for all x ∈ X. If f : Spec(B) → Spec(A) is a morphism of affine schemes then Ω1Spec(B)/Spec(A) = (ΩB/A )∼ We denote by TX/Y := Hom(Ω1X/Y , OX ) the sheaf of relative derivations, or the relative tangent sheaf of f . We will write Ω1X and TX instead of Ω1X/Spec(k) and TX/Spec(k) respectively; they are the cotangent sheaf and the tangent sheaf of X, respectively (cotangent and tangent bundles if locally free). If X is algebraic and x ∈ X is closed then, by B.37: Ω1X,x ⊗ k(x) =

mX,x m2X,x

is the cotangent space of X at x, and Tx X := TX,x ⊗ k(x) =

m

X,x ∨ m2X,x

∼ = Derk (OX,x , k)

is the Zariski tangent space of X at x. Let S be a scheme and g X −→ Y a morphism of S-schemes. The induced homomorphism of sheaves on X: g ∗ Ω1Y /S → Ω1X/S is called the relative codifferential of g. The dual homomorphism: TX/S → Hom(g ∗ Ω1Y /S , OX ) is the relative differential of g. When S = Spec(k) we have g ∗ Ω1Y → Ω1X , which is the codifferential of g, while its dual dg : TX → Hom(g ∗ Ω1Y , OX ) is the differential of g. Note that if Ω1Y /S is locally free then Hom(g ∗ Ω1Y /S , OX ) = g ∗ Hom(Ω1Y /S , OY ) = g ∗ TY /S

346

APPENDIX B. DIFFERENTIALS

but in general the first and the second sheaves are different. The relative cotangent sequence is g ∗ Ω1Y /S → Ω1X/S → Ω1X/Y → 0

(B.6)

Conditions for the injectivity of the first map in this sequence are given in Theorem C.59, page 363 and Theorem D.8, page 371. If X ⊂ Y is an embedding of schemes and I = IX/Y ⊂ OY is the ideal sheaf of X in Y , then I/I 2 is a sheaf of OX -modules in a natural way, called the conormal sheaf of X in Y . Its dual NX/Y := HomOX (I/I 2 , OX ) = HomOY (I, OX ) is called the normal sheaf of X in Y . NX/Y (resp. I/I 2 ) is called the normal bundle (resp. the conormal bundle) of X in Y if it is locally free. Given a closed embedding of S-schemes i : X ⊂ Y , we have an exact sequence of sheaves on X: I/I 2 → i∗ Ω1Y /S → Ω1X/S → 0 (B.7) where I ⊂ OY is the ideal sheaf of X in Y . (B.7) is called the relative conormal sequence. When S = Spec(k) we obtain the conormal sequence I/I 2 → i∗ Ω1Y → Ω1X → 0 Conditions for the injectivity of the first map in these sequences are given in Proposition D.4, page 367, and Theorem D.7, page 371. Examples B.44 In the following examples we will describe the global vector fields on the given schemes by exhibiting their restrictions to an affine open set. All will be done by explicit computation. (i) H 0 (TIP 1 ) can be described explicitly as follows. Consider IP 1 = U0 ∪U1 where U0 = Spec(k[ξ]) and U1 = Spec(k[η]) with η = ξ −1 on U0 ∩ U1 . We have ∂ ∂ξ ∂ 1 ∂ ∂ = =− 2 = −ξ 2 ∂η ∂η ∂ξ η ∂ξ ∂ξ on U0 ∩ U1 . Let θ ∈ H 0 (TIP 1 ); then θ|U0 = g(ξ)

∂ ∂ξ

g(ξ) ∈ k[ξ]

347 and θ|U1 = h(η)

∂ , ∂η

h(η) ∈ k[η]

On U0 ∩ U1 we have g(ξ)

∂ ∂ ∂ = h(η) = −h(ξ −1 )ξ 2 ∂ξ ∂η ∂ξ

and therefore g(ξ) = −h(ξ −1 )ξ 2 . It follows that g(ξ) = a0 + a1 ξ + a2 ξ 2 and h(η) = −(a0 η 2 + a1 η + a2 ), with a0 , a1 , a2 ∈ k. In particular H 0 (TIP 1 ) ∼ = k3 . Moreover H i (TIP 1 ) = 0 if i ≥ 1. For i ≥ 2 it is obvious. Let θ ∈ H 1 (TIP 1 ) ˇ be represented by a Cech 1-cocycle defined by θ01 ∈ Γ(U0 ∩ U1 , TIP 1 ). It can be written as n X

θ01 =

ai ξ i

i=−m

Letting θ1 =

P−1

i=−m

ai η −i and θ0 = −

Pn

i=0

ai ξ i we obtain:

θ01 = θ1 − θ0 so (θ01 ) is a coboundary. (ii) We want to describe H 0 (TA1 ×IP 1 ). Let A1 × IP 1 = V0 ∪ V1 where V0 = A1 × U0 = Spec[z, ξ]) V1 = A1 × U1 = Spec[z, η]) and η = ξ −1 on V0 ∩ V1 = Spec(k[z, ξ, ξ −1 ]). We have ∂ ∂ξ ∂ 1 ∂ ∂ = =− 2 = −ξ 2 ∂η ∂η ∂ξ η ∂ξ ∂ξ on V0 ∩ V1 . Let θ ∈ H 0 (TA1 ×IP 1 ); then ∂ ∂ + h(z, ξ) ∂z ∂ξ

g(z, ξ), h(z, ξ) ∈ k[z, ξ]

∂ ∂ + χ(z, η) ∂z ∂η

γ(z, η), χ(z, η) ∈ k[z, η]

θ|V0 = g(z, ξ) θ|V1 = γ(z, η) On V0 ∩ V1 we have:

g(z, ξ) = γ(z, ξ −1 )

348

APPENDIX B. DIFFERENTIALS

and therefore g(z, ξ) = g(z) is constant with respect to ξ. Moreover h(z, ξ)

∂ ∂ ∂ = χ(z, η) = −χ(z, ξ −1 )ξ 2 ∂ξ ∂η ∂ξ

and therefore h(z, ξ) = −χ(z, ξ −1 )ξ 2 It follows that h(z, ξ) = a(z) + b(z)ξ + c(z)ξ 2 , with a(z), b(z), c(z) ∈ k[z]. In conclusion every θ ∈ H 0 (TA1 ×IP 1 ) restricts to V0 as a vector field of the form θ|V0 = g(z)

∂ ∂ + (a(z) + b(z)ξ + c(z)ξ 2 ) ∂z ∂ξ

(B.8)

with g(z), a(z), b(z), c(z) ∈ k[z], and conversely every such vector field is the restriction of a global section of TA1 ×IP 1 . As in example (i) we also deduce that H i (TA1 ×IP 1 ) = 0 if i ≥ 1. In a similar way one describes H 0 (T(A1 \{0})×IP 1 ) by showing that the image of the restriction H 0 (T(A1 \{0})×IP 1 ) → H 0 (T(A1 \{0})×U0 ) consists of the vector fields of the form (B.8) with g(z), a(z), b(z), c(z) ∈ k[z, z −1 ]. (iii) We now consider, for a given integer m ≥ 0, the rational ruled surface Fm = IP (OIP 1 (m) ⊕ OIP 1 ) Let π : Fm → IP 1 be the projection. Then Fm can be represented as Fm = π −1 (U ) ∪ π −1 (U 0 ) = (U × IP 1 ) ∪ (U 0 × IP 1 ) where U = Spec(k[z]), U 0 = Spec(k[z 0 ]) and z 0 = z −1 on U ∩ U 0 . We consider the affine open sets V0 = Spec(k[z, ξ]) ⊂ U × IP 1 V00 = Spec(k[z 0 , ξ 0 ]) ⊂ U 0 × IP 1 where on V0 ∩ V00 = Spec(k[z, z −1 , ξ]) = Spec(k[z 0 , z 0−1 , ξ 0 ]) we have: z 0 = z −1 , ξ 0 = z m ξ

349 Therefore we have:

∂ ∂z 0 ∂ ∂ξ 0

∂ ∂ + mzξ ∂ξ = −z 2 ∂z ∂ = z −m ∂ξ

(B.9)

We will describe a typical element θ ∈ H 0 (TFm ) by describing its restriction to the open sets V0 and V00 . We have, by example (ii) above: θ|V0 = g(z)

∂ ∂ + (a(z) + b(z)ξ + c(z)ξ 2 ) ∂z ∂ξ

with g(z), a(z), b(z), c(z) ∈ k[z] and similarly θ|V00 = ρ(z 0 )

∂ 0 0 0 0 02 ∂ + (α(z ) + β(z )ξ + γ(z )ξ ) 0 ∂z 0 ∂ξ

with ρ(z 0 ), α(z 0 ), β(z 0 ), γ(z 0 ) ∈ k[z 0 ]. Imposing their equality on V0 ∩ V00 and using (B.9) we obtain the following conditions: g(z) = a(z) = b(z) = c(z) =

−ρ(z −1 )z 2 α(z −1 )z −m β(z −1 ) + ρ(z −1 )mz γ(z −1 )z m

(B.10)

We distinguish the cases m = 0 and m > 0. If m = 0 (B.10) give: g0 + g1 z + g2 z 2 a b c

g0 , g1 , g2 , a, b, c ∈ k

g0 + g1 z + g2 z 2 0 b − mz(g1 + g2 z) c0 + c1 z + · · · + cm z m

g0 , g1 , g2 , b, c0 , . . . , cm ∈ k

g(z) = a(z) = b(z) = c(z) = In case m > 0 we have: g(z) = a(z) = b(z) = c(z) =

Since the restriction H 0 (TFm ) → H 0 (TV0 ) is injective and we have described its image, we can conclude: H 0 (TF0 ) H 0 (TFm )

∼ = k6 ∼ = km+5

350

APPENDIX B. DIFFERENTIALS

In particular Fm and Fn are not isomorphic if m 6= n. (note that F0 ∼ = IP 1 × IP 1 is not isomorphic to F1 ∼ = Bl(1,0,0) IP 2 ). Since, by the calculations of the previous example (ii) hi (TU ×IP 1 ) = hi (TU 0 ×IP 1 ) = hi (T(U ∩U 0 )×IP 1 ) = 0, i ≥ 1

(B.11)

we deduce that: H 1 (TFm ) = H 0 (T(U ∩U 0 )×IP 1 )/H 0 (TU ×IP 1 ) + H 0 (TU 0 ×IP 1 ) An easy computation based on (B.10) shows that, for m ≥ 1, H 1 (TFm ) consists of the classes, modulo H 0 (TU ×IP 1 ) + H 0 (TU 0 ×IP 1 ), of the vector fields (b1 z + · · · + bm−1 z m−1 ) In particular

∂ ∂ξ

H 1 (TFm ) ∼ = km−1

(B.12)

It also follows from (B.11) that H 2 (TFm ) = (0)

(B.13)

Notes and Comments 1. Let X → Y be a morphism of algebraic schemes. Prove that there is an exact sequence 1 0 → Ω1X/Y → PX/Y → OX → 0 (B.14) 1 which globalizes (B.1). PX/Y is called the sheaf of principal parts of X over Y , 1 denoted by PX if Y = Spec(k). Let X = IP (V ) for a finite dimensional k-vector space V . Then the exact sequence (B.14) is the dual of the Euler sequence; in particular ∼ P1 = OIP (V ) (−1) ⊗ V ∨ IP (V )

Therefore (B.14) is a generalization of the Euler sequence to any X → Y . 2. Consider IP = IP (V ) for a finite dimensional k-vector space V and the incidence relation: I = {(x, H) : x ∈ H} ⊂ IP × IP ∨ (B.15) Consider the twisted and dualized Euler sequence: 0 → Ω1IP (V ) (1) → OIP (V ) ⊗ V ∨ → OIP (V ) (1) → 0 From its definition it follows that I = IP (Ω1IP (V ) (1)) and IP ×IP ∨ = IP (OIP (V ) ⊗V ∨ ) and the inclusion in (B.15) is induced by the first homomorphism in the above sequence.

Appendix C Smoothness The notion of “formal smoothness”, introduced in [1], Ch. IV §17, is of crucial importance in deformation theory, and therefore plays a special role in this book. In this appendix we introduce this concept from scratch, and we show how it is related to the notion of “smooth morphism” as introduced in [3] Expos´e II, and [89]. We will not give a systematic treatment of the properties of smooth morphisms in algebraic geometry: the reader is referred to the above quoted references for them. For more details on the approach taken in this section the reader can also consult [14] and [102]. Definition C.45 A ring homomorphism f : R → B is called formally smooth, and B is called a formally smooth R-algebra, if for every exact sequence: η 0 → I → A −→ A0 → 0 (C.1) where A and A0 are local artinian R-algebras, each R-algebra homomorphism B → A0 has a lifting B → A; equivalently if the map: HomR−alg (B, A) → HomR−alg (B, A0 )

(C.2)

is surjective. f is called smooth if it is formally smooth and essentially of finite type (shortly e.f.t.). If the map (C.2) is bijective (instead of only being surjective) for all exact sequences (C.1), then f is formally etale; f is etale if it is formally etale and e.f.t.. 351

352

APPENDIX C. SMOOTHNESS

Recall that e.f.t. means that B is a localization of an R-algebra of finite type (see e.g. [139]). It is easy to prove by induction that it suffices to check the above conditions only for the exact sequences (C.1) such that I 2 = (0), i.e. for extensions of local artinian R-algebras. Proposition C.46 (i) If B is a ring and ∆ ⊂ B is a multiplicative system, then B → ∆−1 B is formally etale. In particular B is a formally etale B-algebra. (ii) The composition of formally smooth (resp. formally etale) homomorphisms is formally smooth (resp. formally etale). (iii) If f : R → B is formally smooth (resp. formally etale) and C is an R-algebra, then C → C ⊗R B is formally smooth (resp. formally etale). (iv) A finitely generated field extension K ⊂ L is smooth if and only if L is separable over K. f

g

(v) Let R −→ B −→ C be ring homomorphisms, and assume that f is formally etale. Then gf is formally smooth (resp. formally etale) if and only if g is formally smooth (resp. formally etale). Proof. (i)

Given an exact sequence (C.1) and a commutative diagram B → ∆−1 B ↓ ϕ0 ↓ϕ A −→ A0 p

we must find ϕ˜ : ∆−1 B → A which makes it commutative. For every s ∈ ∆ choose as ∈ A such that ϕ(s)−1 = p(as ). Since p(ϕ0 (s)as ) = ϕ(s)ϕ(s)−1 = 1A0 we have ϕ0 (s)as = 1A + is for every s ∈ ∆. Therefore ϕ0 (s)as (1A − is ) = 1A Hence ϕ0 (s) ∈ A is invertible. Now define ϕ(r/s) ˜ = ϕ0 (r)ϕ0 (s)−1 .

is ∈ I

353 Noting that ϕ˜ is uniquely determined by ϕ0 we get the assertion. (ii) and (iii) are straightforward. (iv) Assume first that K ⊂ L is separable. By (ii) it suffices to consider the cases L = K(X) and L = K[X]/(f (X)) where f is irreducible and f 0 (x) 6= 0. The first case is left to the reader (see remark C.47(i)). In the second case consider an extension A¯ = A/I of local artinian Kalgebras, where I ⊂ A is an ideal with I 2 = (0). Let ϕ : K[X]/(f (X)) → A¯ ¯ 7→ α be a homomorphism, sending X ¯ . Choose arbitrarily α ∈ A such that α ¯ = α mod I. It will suffice to find e ∈ I such that f (α + e) = 0 We have f (α + e) = f (α) + f 0 (α)e. Since f 0 (α) is a unit mod I it is also a unit in A, and therefore we can take e = −f (α)/f 0 (α). Assume conversely that K ⊂ L is smooth. Then L = F [X]/J where F is a purely transcendental extension of K and J is a principal ideal. We have an exact sequence of finite dimensional L-vector spaces: J/J 2 → ΩF [X]/K ⊗ L → ΩL/K → 0 where J/J 2 is 1-dimensional. By the first part of the proof F is smooth over K and by B.36(ii) the left map is injective because, by the smoothness of L over K, the surjection F [X]/J 2 → L splits. It follows that dim(ΩL/K )

= dim(ΩF [X]/K ⊗ L) − 1 = trdegK (F [X]) − 1 =

= trdegK (F ) = trdegK (L) From B.34(iii) it follows that K ⊂ L is separable. (v) “if” follows immediately from (ii); “only if” is left to the reader. qed

Remarks C.47 (i) Any polynomial algebra R[X1 , X2 . . .] is trivially a formally smooth R-algebra. From C.46(i) it follows that a localization of a polynomial R-algebra is also a formally smooth R-algebra. More precisely, a localization P = S −1 R[X1 , X2 , . . .] of a polynomial algebra over a ring R satisfies the following condition, stronger than formal smoothness:

354

APPENDIX C. SMOOTHNESS For every extension of R-algebras: 0 → I → A → A0 → 0 where A and A0 are R-algebras and I 2 = 0 the map HomR−alg (P, A) → HomR−alg (P, A0 ) is surjective.

Every R-algebra B is a quotient of a formally smooth R-algebra, because it is a quotient of a polynomial R-algebra. From C.46(i) it follows that every e.f.t. R-algebra is a quotient of a smooth R-algebra. This is trivial for polynomial rings, and in the general case it can be proved adapting the proof of C.46(i) in an obvious way. ˆ then every formal power series ring R[[X1 , X2 , . . .]] (ii) if R is in ob(A) is a formally smooth R-algebra, because local artinian R-algebras are complete. More precisely a formal power series ring R[[X1 , X2 , . . .]] satisfies the following condition, stronger than formal smoothness over R: For every extension: 0 → I → A → A0 → 0 of complete local R-algebras the map HomR−alg (P, A) → HomR−alg (P, A0 ) is surjective. The proof is straightforward and is left to the reader. The following result characterizes an important class of formally smooth algebras. Theorem C.48 Let k be a field and let (B, m) be a noetherian local k-algebra with residue field K. Suppose that K is finitely generated and separable over k. Then the following are equivalent: (i) B is regular. ˆ∼ (ii) B = K[[X1 , . . . , Xd ]], where d = dim(B).

355 (iii) B is a formally smooth k-algebra. Proof. (i) ⇔ (ii) is standard (see [49], prop. 10.16 and exercise 19.1). (ii) ⇒ (iii). It follows directly from the definition that B is formally smooth ˆ is. Since B ˆ is formally smooth over K (remark over k if and only if B C.47(ii)), and since K is smooth over k by C.46(iv), the conclusion follows by transitivity. (iii) ⇒ (i). Let {x1 , . . . , xd } be a system of generators of m. Then, since B/m2 is complete and K is separable over k, B/m2 contains a coefficient field ([49], Theorem 7.8). Therefore there exists an isomorphism v1 : B/m2 ∼ = K[X1 , . . . , Xd ]/M 2

M = (X1 , . . . , Xd )

v

1 Let v : B → B/m2 −→ K[X1 , . . . , Xd ]/M 2 . By the formal smoothness of B and by induction we can find a lifting of v:

vn : B → K[X1 , . . . , Xd ]/M n+1 for every n ≥ 2. Consider the elements vn (x1 ), . . . , vn (xd ) ∈ M/M n+1 Their classes generate M/M 2 , hence they generate M/M n+1 , by Nakayama. Then we have: K[X1 , . . . , Xd ]/M n+1 = vn (B) +

P

i

= vn (B) + (M/M n+1 ) =

vn (xi )[vn (B) + (M/M n+1 )] = vn (B) + (M/M n+1 )2 = · · ·

· · · = vn (B) + (M/M n+1 )n+1

= vn (B)

hence vn is surjective. Since mn+1 ⊂ ker(vn ) we have: `(B/m

n+1

) ≥ `(K[X1 , . . . , Xd ]/M

n+1

d+n )= d

!

and this implies that dim(B) ≥ d. Since m is generated by d elements it follows that B is regular. qed For the reader’s convenience we include the proof of the following well known

356

APPENDIX C. SMOOTHNESS

Lemma C.49 (i) A surjective endomorphism f : A → A of a noetherian ring is an automorphism. (ii) Let A be a complete noetherian local ring and ψ : A → A an endomorphism inducing an isomorphism ψ1 : A/m2A → A/m2A . Then ψ is an automorphism. Proof. (i) We have an ascending chain of ideals ker(f ) ⊆ ker(f 2 ) ⊆ ker(f 3 ) ⊆ · · · Since A is noetherian we have ker(f n ) = ker(f n+1 ) = ker(f n+2 ) = · · · for some n, and it suffices to prove that ker(f n ) = (0). After replacing f by f n we may assume ker(f ) = ker(f 2 ). Let a ∈ ker(f ); by assumption there exists b ∈ A such that a = f (b). Then 0 = f (a) = f 2 (b) and therefore b ∈ ker(f 2 ) = ker(f ), i.e. a = f (b) = 0. (ii) Let gr(A) = A/m ⊕ m/m2 ⊕ · · · be the associated graded ring. Since gr(A) is generated by m/m2 over A/m the endomorphism gr(ψ) : gr(A) → gr(A) induced by ψ is surjective. It follows that also ψ is surjective. In fact given a ∈ A the surjectivity of gr(ψ) implies that there are a1 , a2 , a3 , . . . , b1 , b2 , b3 , . . . ∈ A such that ai ∈ mi−1 , bi ∈ mi , and a = f (a1 ) + b1 , b1 = f (a2 ) + b2 , b2 = f (a3 ) + b3 , . . . We obtain a convergent power series a ¯ = a1 + a2 + a3 + · · · such that a − ψ(a1 + a2 + · · · + an ) = bn ∈ mn+1 On the limit we therefore get a = ψ(¯ a). The conclusion is now a consequence of (i). qed Proposition C.50 Let f : R → B be a local homomorphism of noetherian local rings containing a field k isomorphic to their residue fields. Then the following conditions are equivalent: (i) f is formally smooth. ˆ is isomorphic to a formal power series ring over R. ˆ (ii) B ˆ→B ˆ induced by f is formally smooth. (iii) The homomorphism fˆ : R

357 Proof. (i) ⇒ (ii). Let m ⊂ B and n ⊂ R be the maximal ideals. Choose ˆ inducing a k-basis of B/( ˆ m elements x1 , . . . , xd ∈ B ˆ 2 + fˆ(ˆ n)), and let F = ˆ R[[X1 , . . . , Xd ]], where X1 , . . . , Xd are indeterminates. Denote by M ⊂ F the maximal ideal. The homomorphism ˆ u: F → B Xi 7→ xi induces an isomorphism ˆ m u1 : F/(M 2 + n ˆ F ) → B/( ˆ 2 + fˆ(ˆ n)) By the formal smoothness of f the composition −1

u1 ˆ → B/( ˆ m v1 : B → B ˆ 2 + fˆ(ˆ n)) −→ F/(M 2 + n ˆF )

can be lifted to an R-homomorphism vk : B → F/M k ˆ for each k ≥ 2. Therefore the sequence {vk } defines an R-homomorphism ˆ→F v:B ˆ→B ˆ induce isomorphisms (vu)1 : F/M 2 → such that vu : F → F and uv : B 2 2 2 ˆ m ˆ m F/M and (uv)1 : B/ ˆ → B/ ˆ respectively. From Lemma C.49 it follows that u and v are isomorphisms. (ii) ⇒ (iii) is obvious. (iii) ⇒ (i) is left to the reader. qed Corollary C.51 Let f : R → B be a local homomorphism of noetherian local rings containing a field k isomorphic to their residue fields. Then the following conditions are equivalent: (i) f is formally etale. ˆ→B ˆ induced by f is an isomorphism (ii) The homomorphism fˆ : R Proof. left to the reader. ˆ is formally Corollary C.52 Let R be in ob(A∗ ). The inclusion f : R → R etale.

358

APPENDIX C. SMOOTHNESS

The proof is obvious. We now restrict our attention to smooth homomorphisms, i.e. we add the condition that the homomorphism is e.f.t.. In this case the module of differentials comes into play; moreover the defining condition of Definition C.45 can be replaced by the more general condition (i) in the following statement. Theorem C.53 Let f : R → B be an e.f.t. ring homomorphism. Then the following conditions are equivalent: (i) For every extension of R-algebras: 0 → I → A → A0 → 0

(C.3)

the map HomR−alg (B, A) → HomR−alg (B, A0 ) is surjective. (ii) If B = P/J, where P = S −1 R[X1 , . . . , Xd ], S ⊂ R[X1 , . . . , Xd ] is a multiplicative system and J ⊂ P is an ideal, the conormal sequence δ

0 → J/J 2 −→ ΩP/R ⊗P B → ΩB/R → 0 is split exact. In particular J/J 2 and ΩB/R are finitely generated projective B-modules. (iii) B is a smooth R-algebra. (iv) (Jacobian criterion of smoothness) If P and J are as in (ii) the map δ⊗B K(p)

(J/J 2 ) ⊗B K(p) −→ ΩP/R ⊗P K(p)

where K(p) = Bp /mBp

is injective for every prime ideal p ⊂ B. Proof. (i) ⇒ (ii). The hypothesis implies that the extension: 0 → J/J 2 → P/J 2 → B → 0 splits. Therefore the conormal sequence is split exact by B.36(iii) and it follows that J/J 2 and ΩB/R are finitely generated projective because the module ΩP/R ⊗P B is free of finite rank.

359 (ii) ⇒ (i). Consider an exact sequence (C.3) and a homomorphism of Ralgebras f 0 : B → A0 . By Remark C.47(i) there exists an R-homomorphism g : P → A making the following diagram commute: P → B ↓g ↓ f0 A → A0 Since g(J) ⊂ I, we see that g factors through P/J 2 , so that we have a commutative diagram: P/J 2 → B ↓ g¯ ↓ f0 A → A0 The hypothesis implies, via B.36(iii), that there exists h : B → P/J 2 a splitting of P/J 2 → B. The composition f = g¯h : B → A gives a lifting of f 0. (i) ⇒ (iii) is obvious. (iii) ⇒ (iv). We may assume B and P local with residue field K. To prove that δ ⊗B K is injective, it suffices to show that for every K-vector space V the map induced by δ: HomK (ΩP/R ⊗P K, V ) → HomK ((J/J 2 ) ⊗B K, V ) k k DerR (P, V ) HomB (J/J 2 , V ) is surjective. Consider a homomorphism g : J/J 2 → V , and the associated pushout diagram (see §1.1 for the definition): 0 → J/J 2 → P/J 2 → B → 0 ↓g ↓ k g∗ (Λ) : 0 → V → Q → B → 0 We can write mQ = V ⊕ m0 , where m0 ⊂ Q is an ideal, because V is annihilated by mQ . Therefore the previous diagram can be embedded in the following: P ↓ & Λ: 0 → J/J 2 → P/J 2 → B → 0 ↓g ↓ k g∗ (Λ) : 0 → V → Q → B → 0 k ↓ ↓ v¯ 0 η: 0 → V → Q/m → K → 0 Λ:

360

APPENDIX C. SMOOTHNESS

where η is an extension of local artinian R-algebras. From the smoothness of B we deduce the existence of v : B → Q/m0 lifting the projection v¯ : B → K. Denoting by r : P → B the natural map, and by w : P → P/J 2 → Q → Q/m0 the composition, consider the homomorphism: d = w − vr : P → V It is easy to show that this is an R-derivation, which induces g. (iv) ⇒ (ii). From Nakayama’s lemma it follows that ker(δ) ⊗ Bp = (0) for all prime ideals p ⊂ B and therefore ker(δ) = (0). Moreover, since ΩP/R ⊗B Bp B is free and finitely generated it follows that Tor1 p (ΩB/R ⊗B Bp , K(p)) = 0: it follows that ΩB/R ⊗B Bp is flat, and therefore, being finitely generated, it is free. Thus ΩB/R is projective, δ has a splitting and J/J 2 is also projective and finitely generated. qed The following result follows easily from what we have seen so far. Theorem C.54 Let B be an integral k-algebra of finite type and of dimension d. Then the following are equivalent: (i) Bp is smooth over k for each prime ideal p ∈ Spec(B). (ii) B is a regular ring. (iii) ΩB/k is projective of rank d. (iv) B is smooth over k. Proof. (ii) ⇔ (iii) is Corollary B.39. (i) ⇔ (ii) follows from Theorem C.48. (iv) ⇒ (i). for each p ∈ Spec(B), Bp is smooth over B by Proposition C.46(i); from Proposition C.46(ii) it follows that Bp is smooth over k. (i) ⇒ (iv). (i) implies that condition (iv) of Theorem C.53 is satisfied for all p ∈ Spec(B), so that B is smooth by Theorem C.53. qed From now on we will freely replace the defining property for smooth homomorphisms given in Definition C.45 by condition (i) of Theorem C.53. Here is a first example.

361 Proposition C.55 Let R be a ring, P an R-algebra and B = P/J for an ideal J ⊂ P . If B is a smooth R-algebra the conormal sequence 0 → J/J 2 → ΩP/R ⊗P B → ΩB/R → 0 is split exact and ΩB/R is projective and finitely generated. If moreover P is a smooth R-algebra then J/J 2 is finitely generated and projective as well. Proof. Since B is smooth the R-algebra extension 0 → J/J 2 → P/J 2 → B → 0 splits. Therefore the conormal sequence splits by Theorem B.36(iii) and ΩB/R is finitely generated and projective by Theorem C.53. If P is smooth then ΩP/R is finitely generated and projective as well and so is J/J 2 . qed Corollary C.56 Let P be an e.f.t. k-algebra and B = P/J for an ideal J ⊂ P . Assume that B is reduced. Then in the conormal sequence δ

J/J 2 −→ ΩP/k ⊗P B → ΩB/k → 0

(C.4)

ker(δ) is a torsion B-module whose support is contained in the singular locus of Spec(B). If J/J 2 is torsion free then δ is injective. Proof. Since B is reduced there is a dense open subset U ⊂ Spec(B) such that Bp is a regular local ring for all p ∈ U . From Theorem C.48 it follows that Bp is a smooth k-algebra for all such p and, by Propositions C.55 and B.34(ii) , the conormal sequence (C.4) localized at p is split exact. It follows that ker(δ)p = (0) for all p ∈ U and the conclusion follows. The last assertion is an obvious consequence of the first part. qed The next result explains the relation between smoothness and the relative cotangent sequence. f

g

Theorem C.57 Let K −→ R −→ B be ring homomorphisms, with g smooth. Then the relative cotangent sequence: α

0 → ΩR/K ⊗R B −→ ΩB/K → ΩB/R → 0 is split exact.

362

APPENDIX C. SMOOTHNESS

Proof. By Theorem B.35 it suffices to prove that α is a split injection; this is equivalent to showing that, for any B-module M , the induced map: α∨

HomB (ΩB/K , M ) −→ HomB (ΩR/K ⊗R B, M ) k k DerK (B, M ) DerK (R, M ) D0

7→

D0 g

is split surjective. Let D : R → M be a K-derivation and consider the commutative diagram: 1B B −→ B ↑g ↑ γ ˜ R −→ B ⊕M where γ(r) = (g(r), D(r)), r ∈ R. By the smoothness of g we can find a ˜ homomorphism of R-algebras ψ : B → B ⊕M making the diagram 1

B −→ B &ψ ↑ ˜ −→ B ⊕M γ

B ↑g R

commutative. The homomorphism ψ is necessarily of the form: ψ(b) = (b, D0 (b)) and D0 : B → M is a K-derivation such that D = D0 g. This proves the surjectivity of α∨ . Now take M = ΩR/K ⊗R B and D = dR/K ⊗ g : R → ΩR/K ⊗R B and let α0 : ΩB/K → ΩR/K ⊗R B be the B-linear map corresponding to D0 : B → ΩR/K ⊗R B. Then α0 α = 1M and this proves that α is split injective. qed f

g

Corollary C.58 Let K −→ R −→ B be ring homomorphisms, with g etale. Then ΩR/K ⊗R B ∼ = ΩB/K and ΩB/R = (0)

363 Proof. By the relative cotangent sequence the two assertions are equivalent. We will prove the first. Keeping the notations of the proof of C.57, the hypothesis that g is etale implies that the derivation D0 is unique and consequently α is an isomorphism. qed ∗

∗

∗

∗

∗ ∗

A morphism ϕ : X → Y of algebraic schemes is smooth at a point x ∈ X if OX,x is a smooth OY,ϕ(x) -algebra; ϕ is smooth if it is smooth at every point. The definition of etale morphism is given similarly. This definition is equivalent to the definition of smooth (resp. etale) morphism as given in [3] and in [89]. The equivalence can be seen by means of the jacobian criterion of smoothness, proved in Theorem C.53, and using [3], Expos´e II, Corollaire 5.9. By translating into geometrical language the algebraic results proved above we deduce in particular the following. Theorem C.59 Let S be an algebraic scheme, and ϕ : X → Y a morphism of algebraic S-schemes. Then: (i) If ϕ is smooth at x ∈ X then the relative cotangent sequence 0 → ϕ∗ Ω1Y /S → Ω1X/S → Ω1X/Y → 0 is split exact at x and Ω1X/Y is locally free at x. The rank of the free module Ω1X/Y,x is called the relative dimension of ϕ at x. (ii) ϕ is etale at x ∈ X if and only if it is smooth of relative dimension zero at x. In particular Ω1X/Y,x = 0 (i.e. ϕ is unramified at x) and therefore we have an isomorphism ϕ∗ Ω1Y /S,x ∼ = Ω1X/S,x (iii) If X is smooth over S at x and ϕ is a closed embedding with ideal sheaf I ⊂ OY then the relative conormal sequence O → I/I 2 → ϕ∗ Ω1Y /S → Ω1X/S → 0 is exact at x and Ω1X/S is free at x; if moreover Y is also smooth over S at ϕ(x) then I/I 2 is free at x as well. The exactness of the relative conormal sequence in part (iii) holds under more general assumptions as well (see Theorem D.7).

364

APPENDIX C. SMOOTHNESS

Appendix D Complete intersections D.1

Regular embeddings

Definition D.1 An embedding of schemes j : X ⊂ Y is a regular embedding of codimension n at the point x ∈ X if j(x) has an affine open neighborhood Spec(R) in Y such that the ideal of j(X) ∩ Spec(R) in R can be generated by a regular sequence of length n. If this happens at every point of X we say that j is a regular embedding of codimension n. An open embedding is a regular embedding of codimension 0. If X and Y are both nonsingular then X ⊂ Y is a regular embedding. The set of points of X where an embedding j : X ⊂ Y is regular is open. If X ⊂ Y is a regular embedding of codimension n then I/I 2 and NX/Y are both locally free of rank n ([49], Exercise 17.12, p. 440). It follows from standard facts in commutative algebra (see [49], Exercise 17.16, p. 441) that I k /I k+1 is locally free as well for every k ≥ 2. A ring B is called a complete intersection if Spec(B) can be regularly embedded in Spec(R) where R is a regular ring. A scheme X is a local complete intersection (l.c.i.) if every local ring OX,x is a complete intersection ring. A nonsingular scheme X, i.e. a scheme all of whose local rings are regular, is an example of a l.c.i. scheme. If X ⊂ Y is a regular embedding and Y is a l.c.i. scheme, then X is a l.c.i. scheme. Lemma D.2 Let f : X → Y be a morphism of schemes and let Z ⊂ Y be a regular embedding of codimension n. Then the induced embedding j : 365

366

APPENDIX D. COMPLETE INTERSECTIONS

X ×Y Z ⊂ X has codimension ≤ n at every point and if equality holds at a point x ∈ X ×Y Z then j is regular at x. Proof. If IZ ⊂ OY is the ideal sheaf of Z in Y then the ideal sheaf f −1 IZ of X ×Y Z ⊂ X is locally generated at a point x by the n images of the local generators of IZ,f (x) . The conclusion follows easily from this fact. qed If we have a flag of embeddings of schemes X ⊂ Y ⊂ Z and IY ⊂ IX ⊂ OZ are the ideal sheaves of X and Y , we have the exact sequence 0 → IY → IX → IX/Y → 0

(D.1)

where IX/Y ⊂ OY is the ideal sheaf of X in Y . After tensoring by ⊗OZ OX we obtain an exact sequence of coherent OX -modules: IX/Y IY α IX ⊗ OX −→ 2 → 2 →0 2 IY IX IX/Y

(D.2)

Its dual is the sequence: 0 → NX/Y → NX/Z → NY /Z ⊗ OX

(D.3)

Lemma D.3 (i) If f : X ⊂ Y and g : Y ⊂ Z are regular embeddings of codimensions m and n respectively, then gf : X → Z is a regular embedding of codimension m + n. (ii) If the embeddings f and g are both regular then we have exact sequences of locally free sheaves on X: 0→

IX/Y IY α IX ⊗ OX −→ 2 → 2 →0 2 IY IX IX/Y

0 → NX/Y → NX/Z → NY /Z ⊗ OX → 0

(D.4) (D.5)

Proof. (i) left to the reader. (ii) All sheaves in (D.4) are locally free because they are conormal bundles of regular embeddings. Since Im(α) is a torsion free sheaf of the same rank of (IY /IY2 ) ⊗ OX , it follows that α must be injective. The sequence (D.5) is 2 exact because Ext1OX (IX/Y /IX/Y , OX ) = 0. qed

D.1. REGULAR EMBEDDINGS

367

Proposition D.4 Let j : X ⊂ Y be an embedding of algebraic schemes, with X reduced and Y nonsingular. Consider the conormal sequence δ

I/I 2 −→ Ω1Y |X → Ω1X → 0

(D.6)

(where I ⊂ OY is the ideal sheaf of X) Then: (i) The homomorphism δ is injective on the open set where j is a regular embedding. (ii) If X and Y are nonsingular then the dual sequence 0 → TX → TY |X → NX/Y → 0

(D.7)

is exact. Proof. (i) It suffices to show that δ is injective under the assumption that j is a regular embedding. In this case the conormal sheaf I/I 2 is locally free of rank equal to the codimension of X. The sequence (D.6) is exact at every nonsingular point x ∈ X by Theorem C.59(iii). Since X is reduced this happens on a dense open subset so that ker(δ) is supported on a nowhere dense subset. But X has no embedded points because it is regularly embedded in Y : it follows that ker(δ) = 0. (ii) Under the stated hypothesis j is a regular embedding and Ω1X is locally free, so we have Ext1 (Ω1X , OX ) = 0 and the exactness of (D.7) follows. qed Remark D.5 If we don’t assume X reduced part (i) of the proposition is false in general. An example is given by the closed regular embedding of codimension 1: Spec(k[]) ⊂ Spec(k[t]) = A1 (see Example B.42(iv)). A morphism f : X → Y of schemes will be called a cover (or a covering) if it is finite and surjective. Recall that a morphism of schemes f : X → Y is called unramified at a point x ∈ X if Ω1X/Y,x = 0; f is unramified if it is unramified at every x ∈ X. After identifying X with the diagonal ∆ ⊂ X ×Y X, we see that Ω1X/Y gets identified with the conormal sheaf of this embedding. It follows that f is unramified at x if and only if ∆ ⊂ X ×Y X is an open embedding at x, and that the locus of x ∈ X such that f is unramified at x is open. Moreover f is unramified if and only if ∆ is both open and closed in X ×Y X.

368

D.2

APPENDIX D. COMPLETE INTERSECTIONS

Relative complete intersection morphisms

We now introduce a natural class of morphisms which generalize smooth morphisms and behave well with respect to differentials and base change. Definition D.1 A flat morphism of finite type f : X → S is called a relative complete intersection (r.c.i.) morphism at the point x ∈ X if there is an open neighborhood U of x such that the restriction of f to U can be obtained as a composition j g U −→ V −→ S where j is a regular embedding and g is smooth. If f is a r.c.i. morphism at every point we call it a r.c.i. morphism, and we call X a complete intersection over S. This definition is equivalent to Def. 19.3.6 of Ch. IV of [1]; the equivalence is proved in [20], Prop. 1.4. Note that in case S = Spec(k) the morphism f is a r.c.i. if and only if X is a l.c.i. of finite type. If X → S is a flat morphism of finite type of nonsingular varieties then f is a r.c.i. because it factors as X →X ×S →S where the first morphism is the graph of f . Before discussing the main properties of this notion we need two lemmas. Lemma D.2 Let A → B be a ring homomorphism, M a B-module and f1 , . . . , fn an M -regular sequence of elements of B. Assume that for each P i = 1, . . . , n the module M/( i−1 j=1 fj M ) is A-flat. Then, for every ring ho0 momorphism A → A , letting B 0 = B ⊗A A0 , M 0 = M ⊗A A0 , and fi0 = fi ⊗ 1 (1 ≤ i ≤ n), the sequence f10 , . . . , fn0 of elements of B 0 is M 0 -regular and the P 0 0 0 modules M 0 /( i−1 j=1 fj M ) are A -flat. Proof. Consider the exact sequence: f

1 0 → M −→ M → M/f1 M → 0

Since M/f1 M is A-flat, the sequence: f ⊗1

1 0 → M ⊗A A0 −→ M ⊗A A0 → (M/f1 M ) ⊗A A0 → 0

D.2. RELATIVE COMPLETE INTERSECTION MORPHISMS

369

is exact, and therefore f10 is not a zero-divisor for M 0 . Let Mi = M/( ij=1 fj M ), P Mi0 = M 0 /( ij=1 fj0 M 0 ); then we have Mi0 = Mi ⊗A A0 , Mi+1 = Mi /fi+1 Mi , 0 0 Mi+1 = Mi0 /fi+1 Mi0 . Replacing M and f1 by Mi and fi+1 in the above argu0 ment, one deduces that fi+1 is not a zero-divisor for Mi0 , thereby proving the first assertion by induction. The last assertion follows from A(VII). qed P

Lemma D.3 Let A → B be a local homomorphism of noetherian local rings, M a B-module of finite type, flat over A, and f1 , . . . , fn ∈ mB . For 1 ≤ i ≤ n let gi be the image of fi in B ⊗A k, where k = A/mA is the residue field of A. Then the following conditions are equivalent: Pi

(i) f1 , . . . , fn is an M -regular sequence, and Mi = M/( for all 1 ≤ i ≤ n.

j=1

fj M ) is A-flat

(ii) g1 , . . . , gn is an (M ⊗A k)-regular sequence. Proof. (i) ⇒ (ii) follows from D.2 applied to A0 = k. (ii) ⇒ (i) Applying Corollary A.27, from the injectivity of g1 : M ⊗A k → M ⊗A k we deduce that f1 : M → M is injective and that M1 = M/f1 M is Aflat. Proceeding by induction on i, assume Mi flat over A. Since gi+1 : Mi ⊗A k → Mi ⊗A k is injective from A.27 again we deduce that fi+1 : Mi → Mi is injective and that Mi+1 is A-flat. qed In the next proposition some general properties of r.c.i. morphisms are proved. Proposition D.4 (i) An open embedding is a r.c.i. morphism. A smooth morphism of finite type is a r.c.i. morphism. (ii) If f : X → S is a r.c.i. morphism and h : S 0 → S is a morphism, then the morphism f 0 : X ×S S 0 → S 0 induced by f after base change is a r.c.i. morphism. Proof. (i) is an immediate consequence of the definition and (ii) follows easily from Lemma D.2. qed From D.4(ii) it follows in particular that if f : X → S is a r.c.i. morphism then Xs is a l.c.i. for every k-rational point s ∈ S. So for example, a non l.c.i. algebraic scheme cannot be the fibre of a flat morphism of algebraic nonsingular varieties. The next result gives a useful characterization of r.c.i. morphisms.

370

APPENDIX D. COMPLETE INTERSECTIONS

Proposition D.5 Let j

−→

X &f

Y .g

(D.8)

S be a commutative diagram of morphisms of algebraic schemes, where f is flat, g is smooth and j is an embedding. Then the following conditions are equivalent for a k-rational point x ∈ X: (i) f is a r.c.i. morphism at x. (ii) Letting s = f (x), the fibre Xs is a l.c.i. at x. (iii) j is a regular embedding at x. Proof. (i) ⇒ (ii) follows from D.4(ii) and (iii) ⇒ (i) is obvious. (ii) ⇒ (iii) From (ii) it follows that the embedding js : Xs ⊂ Ys is regular at x. Let I ⊂ OY be the ideal sheaf of X. Tensoring the exact sequence 0 → I → OY → OX → 0 by − ⊗OS k we obtain the sequence 0 → I ⊗OS k → OYs → OXs → 0 which is exact because f is flat. Therefore I ⊗OS k is the ideal sheaf of j(Xs ) in Ys . Consider a sequence f1 , . . . , fn of sections of I in an open 2 neighborhood of j(x) which induce a basis of Ij(x) /(ms Ij(x) + Ij(x) ) as a OY,j(x) /(ms OY,j(x) + Ij(x) )-module. Then the images f1 ⊗ 1 = g1 , . . . , fn ⊗ 1 = gn are generating sections of I ⊗OS k in an open neighborhood of j(x) in Ys which form a regular sequence in j(x). From Nakayama’s lemma it follows that f1 , . . . , fn generate I in an open neighborhood of j(x) in Y . From Lemma D.3 it follows that f1 , . . . , fn form a regular sequence in j(x) and therefore (iii) holds. (iii) ⇒ (i) is true by definition. qed Corollary D.6 Under the hypothesis of Proposition D.5, the locus of points x ∈ X such that f is a r.c.i. at x is open. If f is proper then the locus of points s ∈ S such that Xs is a l.c.i. is open.

D.2. RELATIVE COMPLETE INTERSECTION MORPHISMS

371

Proof. The last assertion follows from the first because a proper map is closed. The first assertion can be proved using characterization D.5(iii) of r.c.i. morphism and the fact that the locus where an embedding is regular is open. qed We conclude this section with two results about the relative conormal sequence and cotangent sequence for r.c.i. morphisms. Theorem D.7 Let X

j

−→

Y

&f

↓g S

be a commutative diagram of morphisms of algebraic schemes, with f a r.c.i., j an immersion and g smooth. Let J ⊂ OY be the ideal sheaf of j(X). If f is smooth on a dense open subset intersecting every fibre then the relative conormal sequence δ

0 → J /J 2 −→ j ∗ Ω1Y /S → Ω1X/S → 0 is exact and J /J 2 is locally free. Proof. From the equivalence (i) ⇔ (iii) in Proposition D.5 it follows that j is a regular embedding and therefore J /J 2 is locally free. Moreover the support of ker(δ) does not contain any generic point of X nor any fibre of f because it is contained in the locus where f is not smooth. Since f is generically smooth and j is a regular embedding X has no embedded components except possibly for some union of fibres. It follows that δ is injective. qed Theorem D.8 Let f : X → S be a .r.c.i morphism of algebraic schemes, and assume f smooth on a dense open subset intersecting every fibre. Then the relative cotangent sequence 0 → f ∗ Ω1S → Ω1X → Ω1X/S → 0 is exact.

(D.9)

372

APPENDIX D. COMPLETE INTERSECTIONS

Proof. We only have to prove the injectivity of the left homomorphism and the question is local on X. Since all schemes are algebraic, locally on X we can construct the following commutative diagram: X

j

i

−→ V

−→ U

&f ↓ψ

↓ϕ h

S −→ W where W, U, ψ, ϕ are smooth, i, h are closed embeddings and j is a regular closed embedding. From the smooth morphism ϕ we deduce the exact sequence of locally free sheaves on U : 0 → ϕ∗ Ω1W → Ω1U → Ω1U/W → 0 which restricts on X to the exact sequence: 0 → (hf )∗ Ω1W → (ij)∗ Ω1U → j ∗ Ω1V /S → 0 Let J ⊂ OV and I ⊂ OU be the ideal sheaves of the embeddings j and ij respectively, and H ⊂ OW the ideal sheaf of the embedding h. Then we have an exact and commutative diagram of coherent sheaves on X:

f ∗ (H/H2 ) ↓ 0 → (hf )∗ Ω1W ↓ f ∗ Ω1S ↓ 0

→ → df ∨

−→

I/I 2 ↓ (ij)∗ Ω1U ↓ Ω1X ↓ 0

0 ↓ → J /J 2 → 0 ↓ δj → j ∗ Ω1V /S → 0 ↓ →

Ω1X/S ↓ 0

→0

where the second and the third columns are the relative conormal sequences of ij and of j respectively; the first column is the pullback to X of the conormal sequence of h; the first row is exact because ψ ∗ (H/H2 ) is the conormal sheaf of i; the map δj is injective by Theorem D.7 and by the assumptions made on X and f . A diagram chasing shows that the codifferential df ∨ is injective and proves the theorem. qed

D.2. RELATIVE COMPLETE INTERSECTION MORPHISMS

373

Notes and Comments 1. An algebraic scheme can have different embeddings in IP r , i.e. by means of non-isomorphic invertible sheaves, but with same normal sheaf. An example is given by a projective nonsingular curve C of genus 1, and by the embeddings in IP 3 given by two non isomorphic invertible sheaves L1 and L2 of degree 4 such that L21 = L22 . Then C is embedded as a nonsingular complete intersection of two quadrics by both sheaves, and the normal bundles are L21 ⊕ L21 = L22 ⊕ L22 . 2. Let S be a scheme, and X, Y smooth over S. Prove that every closed Sembedding X ⊂ Y is regular. In particular every section of a smooth morphism f : Y → S is a regular embedding of codimension equal to the relative dimension of f . 3. Let f : X → S be a morphism of finite type and s ∈ S a k-rational point. Let ms ⊂ OS,s be the maximal ideal and I = IX (s) the ideal sheaf of the fibre X (s) of f over s. Prove that we have a surjective homomorphism ms ⊗k OX (s) → I/I 2 m2s and an injection: NX (s)/X ⊂ TS,s ⊗k OX (s) If f is flat then they are isomorphisms; in particular, if f is flat then NX (s)/X is free.

374

APPENDIX D. COMPLETE INTERSECTIONS

Appendix E Functorial language Let C be a category. A covariant (resp. contravariant) functor F from C to (sets) is said to be representable if there is an object X in C such that F is isomorphic to the functor Y 7→ Hom(X, Y )

(E.1)

(resp. Y 7→ Hom(Y, X)). We will denote by hX a functor of the form E.1. The representable functors are a full subcategory, isomorphic to C ◦ (resp. to C in the contravariant case), of the category F unct(C, (sets)) of covariant functors (resp. F unct(C ◦ , (sets)) of contravariant functors) from C to (sets). To fix ideas let’s consider covariant functors. In order to investigate conditions for the representability of a given functor F it is convenient to study functorial morphisms hX → F . Such morphisms turn out to be easy to describe, thanks to the elementary: Lemma E.9 (Yoneda) Let F : C → (sets) be a covariant functor. For each object X in C there is a canonical bijection: Hom(hX , F ) ↔ Φ 7→

F (X) Φ(X)(1X )

Let’s mention, on passing, that functorial morphisms F → hX are much harder to control. They are related to the notion of “coarse moduli scheme”. We may consider couples of the form (X, ξ), where X is an object of C and ξ ∈ F (X). Yoneda’s lemma implies that to give such a couple is equivalent to giving a morphism of functors hX → F ; if this morphism is an isomorphism 375

376

APPENDIX E. FUNCTORIAL LANGUAGE

then (X, ξ) is called a universal couple, and ξ a universal element, for F . The existence of a universal couple is equivalent to the representability of F . The couples for F are the objects of a category in which a morphism (X, ξ) → (Y, η) between two couples is by definition a morphism f : X → Y in C such that F (f )(ξ) = η. We denote this category by IF . A morphism f : (X, ξ) → (Y, η) in IF corresponds to a commutative diagram of morphisms of functors: ξ hX −→ F ↑f %η hY We have an obvious “forgetful functor” IF → C The fibres of this functor are precisely the sets F (X), which are embedded as subcategories of IF by ξ 7→ (X, ξ). (recall that, given a functor G : C → D, the fibre G−1 (D) of G over an object D of D is a subcategory of C, consisting of all objects C such that G(C) = D and of all morphisms f such that G(f ) = 1D . A set can be viewed as a category whose objects are its elements and the only morphisms are the identity morphisms). Lemma E.10 The functor F is representable if and only if the category IF has an initial object (X, ξ). If this is the case, (X, ξ) is a universal couple for F . The proof is immediate. Note that, since an initial object is unique up to isomorphism, it follows that a representable functor has a unique universal couple, up to isomorphism. ∗

∗

∗

∗

∗ ∗

Let I and D be two categories. Given an object A of D, the constant functor cA : I → D is defined as cA (i) = A for each object i of I and cA (f ) = 1A for each morphism f in I. Note that cA is both covariant and contravariant. Every morphism α : A → B in D induces an obvious morphism of functors cα : cA → cB . Consider a covariant functor Φ : I → D. An inductive limit of Φ is an object A of D and a functorial morphism λ : Φ → cA such that for

377 every other morphism µ : Φ → cB there is a morphism α : A → B such that µ = cα λ. λ Φ −→ cA & µ ↓ cα cB From the definition it follows that an inductive limit of Φ, if it exists, is unique up to unique isomorphism, and is denoted lim Φ →

In practice an inductive limit is an object A of D such that there is a morphism Φ(i) → A for each i ∈ Ob(I) with the condition that the diagram Φ(i) → A ↓ Φ(f ) % Φ(j) is commutative for each morphism f : i → j in I; moreover these data must satisfy a universal property. Dually one has the notion of projective limit of a covariant functor Φ : I → D: it is an object A of D and a morphism π : cA → Φ such that for every other morphism ρ : cB → Φ there is a morphism β : B → A such that ρ = πcβ . The projective limit of Φ, if it exists, is denoted lim Φ ← The above notions can be defined without changes replacing the covariant functor Φ by a contravariant one. We will write Φi for Φ(i), for each object i of I, and sometimes lim Φi →

(resp. lim Φi ) instead of ←

lim Φ (resp. lim Φ) → ←

Example E.11 Let J be a partially ordered set. We define a category Ord(J) as follows. The objects of Ord(J) are the elements of J; for any i, j ∈ J the set HomOrd(J) (i, j) consists of one element if i ≤ j and is ∅ otherwise. A covariant (resp. contravariant) functor Φ : Ord(J) → D is called an inductive system (resp. a projective system) in D indexed by J; in case D =(sets), we obtain the usual notions of inductive (projective) system and of inductive (projective) limit.

378

APPENDIX E. FUNCTORIAL LANGUAGE

If I is a set and Φ : I → D is a functor, where D is a category with arbitrary coproducts, then a Φi lim Φ = →

i

Similarly, if D has products then lim Φ =

Y

←

Φi

i

Proposition E.12 The inductive limit and projective limit exist for every functor Φ : I → (sets) from any category I. Proof. We take lim Φ = →

a

Φi /R

i

where R is the equivalence relation generated by pairs (x, y), x ∈ Φi and y ∈ Φj , such that there exists ϕ : i → j with Φ(x) = y. Similarly for the projective limit. qed Example E.13 Let F : C → (sets) be a covariant functor, and let IF be the category of couples for F . Then we have a contravariant functor Φ : IF → F unct(C, (sets)) which sends a couple (X, ξ) to the functor hX : C → (sets), and a morphism f : (X, ξ) → (Y, η) to the functorial morphism hf : hY → hX induced by f . By construction there is a morphism Φ → cF . This morphism makes F the inductive limit of the functor Φ (the proof is an easy exercise). We will write: F = lim hX → (X,ξ)

Definition E.14 A category I is filtered if (a) for every pair of objects i, j in I there exists an object k in I and morphisms: i ↓ j → k

379 (b) each pair of morphisms i

→ → j has a coequalizer i j → k. → →

The category I is cofiltered if the dual category I ◦ is filtered. Assume from now on that C is a category with products and fibered products. Definition E.15 A covariant functor F : C → (sets) is called left exact if F (B × C) = F (B) × F (C) and F (B ×A C) = F (B) ×F (A) F (C) for each diagram C ↓ B → A in C (i.e. F commutes with finite products and finite fibered products). Every representable functor is left exact by definition of product and fibered product. Lemma E.16 Let I be a filtered category and Φ : I → F unct(C, (sets)) a covariant functor. Then, for each diagram in C:

B

→

C ↓ A

there is a bijection: lim Φi (B) ×lim Φi (A) lim Φi (C) ∼ [Φi (B) ×Φi (A) φi (C)] = lim → → → The proof of this lemma is straightforward and we omit it. The following result is a useful characterization of left exact functors. Proposition E.17 A covariant functor F : C → (sets) is left exact if and only if the category IF is cofiltered. Proof. Assume that IF is cofiltered. Applying Lemma E.16 to the functor Φ of Example E.13, we see that the inductive limit F = lim(X,ξ) hX is left exact because each functor hX is left exact.

380

APPENDIX E. FUNCTORIAL LANGUAGE

Conversely assume that F is left exact. Let (X, ξ), (Y, η) ∈ Ob(IF ); we must find (Z, ζ) → (X, ξ) ↓ (Y, η) Take (Z, ζ) = (X × Y, (ξ, η)). Now consider (X, ξ)

→ (Y, η) coming from →

φ, ψ : X → Y . We have F (φ)(ξ) = F (ψ)(ξ) = η Consider the diagram: Γφ

−→ X × Y ↑ Γψ → X

X ↑ K

where Γφ = (1X , φ) and Γψ = (1X , ψ) and K = X ×X×Y X. Since F is left exact F (K) = F (X) ×F (X×Y ) F (X) and there is χ ∈ F (K) corresponding to (ξ, ξ): F (Γφ )

7−→

ξ

(ξ, η)

↑

↑ F (Γψ ) 7−→

χ

ξ

Then (K, χ) is the equalizer of φ and ψ. Therefore IF is cofiltered.

qed

Let I be a category. A full subcategory J of I is cofinal if for each i ∈ Ob(I) there is a morphism f : i → j for some j ∈ Ob(J). It follows immediately from the definitions that if Φ : I → D is a covariant functor and ΦJ : J → D is its restriction, then lim Φ = lim ΦJ → → ∗

∗

∗

∗

∗ ∗

381 Let Z be a scheme. In this subsection we will consider contravariant functors defined on (schemes/Z). All we will say holds, with obvious modifications, for functors defined on (algschemes/Z), the full subcategory of algebraic Z-schemes. A contravariant functor F : (schemes/Z)◦ → (sets) defines on every Z-scheme S a presheaf of sets: U 7→ F (U ) for all open sets U ⊂ S. For this reason a functor as above is also called a presheaf. F is called a sheaf (more precisely a sheaf in the Zariski topology) if it defines a sheaf on every scheme; namely if for all Z-schemes S and for all open coverings {Ui } of S the following is an exact sequence of sets: F (S) →

Y i

F (Ui )

→Y F (Ui ∩ Uj ) → i,j

The most important sheaves are the representable functors, i.e. functors isomorphic to one of the form: S 7→ HomZ (S, X) for some Z-scheme X. Such a functor is called the functor of points of X/Z. It is very important to have conditions, easy to verify in practice, for a contravariant functor F : (schemes/Z) → (sets) to be representable. Certainly a necessary condition is that F is a sheaf. Another necessary condition is the following. Recall that a subfunctor G of F is said to be an open (resp. closed) subfunctor if for every scheme S and for every morphism of functors Hom(−, S) → F the fibered product Hom(−, S) ×F G, which is a subfunctor of Hom(−, S), is represented by an open (resp. closed) subscheme of S. A family of open subfunctors {Gi } of F is a covering of F if for every Z-scheme S and for every morphism of functors Hom(−, S) → F the family {Hom(−, S) ×F Gi } of subschemes of S is an open covering of S. An obvious example is obtained by considering an open (resp. closed) subscheme X 0 of a Z-scheme X: correspondingly we obtain an open (resp.

382

APPENDIX E. FUNCTORIAL LANGUAGE

closed) subfunctor Hom(−, X 0 ) of Hom(−, X). An open cover {Xi } of X defines a cover of Hom(−, X) by open subfunctors. Therefore a second obvious necessary condition for a functor F to be representable is that it can be covered by representable open subfunctors. We will now show that these two necessary conditions are also sufficient. Proposition E.18 Let F : (schemes/Z)◦ → (sets) be a contravariant functor. Suppose that: (a) F is a sheaf; (b) F admits a covering by representable open subfunctors Fi . Then F is representable. Proof. Letting Fij = Fi ×F Fj , by (b) the projections Fij → Fi correspond to open embedding of schemes Xij → Xi . Therefore the Fi ’s patch together to form a representable functor Hom(−, X), where X is the scheme obtained by patching the Xi ’s together along the Xij ’s. By (a), F and Hom(−, X) are isomorphic. qed The following is an easy but important remark. Lemma E.19 If F is a sheaf then F is determined by its restriction to the category of affine schemes. Proof. In fact, if S is any Z-scheme we can consider an affine open cover {Ui }. For any i, j we take an affine open cover {Vi,j,α } of Ui ∩ Uj ; composing the map →Y F (Ui ) F (Ui ∩ Uj ) → i,j with the inclusions F (Ui ∩ Uj ) → F (S) →

Y i

Q

α

F (Vi,j,α ) we obtain the exact sequence:

F (Ui )

→ Y F (Vi,j,α ) → i,j,α

which shows that F (S) is determined by its values on affine schemes.

qed

383 This lemma implies that the functor of points of a scheme X F = HomZ (−, X) : (schemes)◦ → (sets) is determined by its restriction to the category of affine schemes, or equivalently, by its covariant version: F : (k-algebras) → (sets) Since the category of schemes is isomorphic to the category of functors of points, this means that we can define schemes as certain types of functors on (k-algebras). Thanks to Proposition E.18, we can say that these functors are precisely the sheaves admitting an open cover by affine schemes, i.e. by representable functors. This point of view is very fruitful because it gives the possibility of generalizing the notion of scheme by considering more general functors. The notion of algebraic space is such a generalization (see Artin [14]).

384

APPENDIX E. FUNCTORIAL LANGUAGE

Bibliography [1] El´ements de G´eom´etrie Alg´ebrique, par A. Grothendieck, r´edig´es avec la collaboration de J. Dieudonn´e, Publ. Math. IHES, n. 4, 8, 11, 17, 20, 24, 32. [2] Fondements de la G´eom´etrie Alg´ebrique, by A. Grothendieck, expos´es at S´eminaire Bourbaki 1957-1962, Secretariat Math. Paris 1962 (some of them are listed separately below). [3] Revetements Etales et Groupe Fondamentale, S´eminaire de G´eom´etrie Alg´ebrique du Bois Marie 1960/61, Springer Verlag Lecture Notes in Mathematics vol. 224. [4] Andr´e M.: Homologie des alg´ebres commutative, Springer Verlag Grundlehren b. 206 (1974). [5] Andreotti A.: On the complex structures on a class of simply-connected manifolds, Algebraic Geometry and Topology - A symposium in honor of S. Lefschetz, Princeton U.P. (1957). [6] Arakelov S.: Families of algebraic curves with fixed degeneracies, Math. USSR Izv. 5 (1971), 1277-1302. [7] Arbarello E., Cornalba M.: Su una propriet`a notevole dei morfismi di una curva a moduli generali in uno spazio proiettivo. Rend. Sem. Mat. Univers. Politecn. Torino 38 (1980), 87-99. [8] Arbarello E., Cornalba M.: Su una congettura di Petri, Comm. Math. Helvetici 56 (1981), 1-38. [9] Arbarello E., Cornalba M., Griffiths Ph., Harris J.: Geometry of Algebraic Curves, vol. I, Springer Verlag Grundlehren b. 267 (1985). 385

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List of Symbols k, ix k(s), ix ob(C), ix C ◦ , ix A, ix ˆ ix A, A∗ , ix AΛ , x A∗Λ , x AˆΛ , x E ∨, x IP (V ), x IP (E), x (R0 , ϕ), 7 ˜ 8 R⊕I, A[], 9 ExA (R, I), 11 1 , 14 TR/A 1 TR , 14 Ex(X/S, I), 15 ˜ 15 OX ⊕I, 1 TX/S , 15 TX1 , 16 0 , 16 NX/Y Fm , 23 κ, 31 κ(ξ), 33 κξ , 33 403

404 κf,s , 33 κX /S,s , 33 oξ (e), 36 o(R/Λ), 41 o(R), 41 o(f /Λ), 42 o(f ), 42 tF , 52 df , 52 Fˆ , 52 (R, uˆ), 54 Def X , 75 Def 0X , 75 µ(X), 80 ˜S,s , 89 O ˜ 89 A, d Def X , 91 ¯ {ηn }), 91 (A, ¯ ηˆ), 91 (A, X X , 91 (S, s, η), 92 (B, s, η), 92 PAr¯ , 95 Autuˆ , 107 Def (X,p) , 110 TB2 0 , 128 TX2 , 130 Y HX , 145 oξ/Y , 153 X /S HZ , 161 X X /Specf(R) HZ , 163 XA , 163 L ⊗A B, 163 Pλ0 , 164 m1 , 167 aD , 169 µ0 (L), 171

BIBLIOGRAPHY

BIBLIOGRAPHY µ0 (V ), 171 c(L), 172 EL , 172 PL , 172 Def (X,L) , 173 M1 , 181 PL , 183 PV , 184 wfdat, 187 Def X/f /Y , 187 ρ(g, r, n), 190 Nf , 193 DX/Y , 193 1 DX/Y , 193 Def f /Y , 195 Def 0f /Y , 195 oF/Y , 198 Φf , 201 ¯ϕ , 205 N Hϕ , 205 Def j , 210 Def 0j , 210 TY hXi, 211 F ittF k , 238 Nk (F), 238 De (ϕ), 239 Nk (f ), 239 HilbYP (t) , 246 HilbrP (t) , 246 HilbYP (t) , 247 HilbrP (t) , 247 Y [n] , 247 GV,n , 250 G, 250 Gn (V ), 251 G(n, N ), 252 I, 254

405

406

BIBLIOGRAPHY

HilbY , 259 HilbY , 259 Hilb(−Z)YP (t) , 262 Hilb(−Z)YP (t) , 262 Y \Z

HilbP (t) , 262 HilbY \Z , 262 X/S QuotH,P (t) , 263 QuotX H,P (t) , 263 X/S

QuotH,P (t) , 263 X/S

HilbP (t) , 266 X/S

QuotH , 266 X/S QuotH , 266 X/S Hilb , 267 P(t), 271 r F HP(t) , 271 r FHP(t) , 271 [X1 , . . . , Xm ], 274 pri , 274 PI (t), 274 prI , 274 Z F HP(t) , 274 Z FHP(t) , 274 ˜ P(t), 274 Z FH , 275 N(X1 ,...,Xm )/Z , 275 Quotn (E), 287 Quotn (E), 288 Gn (E), 288 Hom(X, Y ), 299 Isom(X, Y ), 299 Aut(X), 299 Σd , 307 |O(d)|, 307 Λd , 308 Vdδ,κ , 308 Vdδ,κ , 308

BIBLIOGRAPHY

407

Mk (f ), 315 ΩB/A , 335 dB/A , 335 PB/A , 335 k[], 341 tB/Λ , 342 Ω1X/Y , 344 TX/Y , 345 Ω1X , 345 TX , 345 Tx X, 345 dg, 345 NX/Y , 346 1 PX/Y , 350 1 PX , 350 hX , 375 F unct(C, (sets)), 375 (X, ξ), 375 IF , 376 lim Φ, →

377

lim Φ 377 ←

Index Abel-Jacobi embedding, 217 morphism, 169 abelian variety, 180 action, 66 algebra etale, 351 formally etale, 351 formally smooth, 351 less obstructed, 42 obstructed/unobstructed, 41 of dual numbers, 9 of principal parts, 335 rigid, 26, 28, 122, 144 smooth, 351 algebraic local ring, 89 algebraic space, 383 algebraization, 94 algebraization theorem, 98, 103 ample canonical class, 221 surface, 221 approximation theorem, 103 Arakelov, 203 Artin, 98, 383 Artin’s algebraization theorem, 98 Atiyah extension, 172, 183, 292 automorphism functor, 107 Bertini’s theorem, 283 blow-up, 85, 204, 217, 219, 221

bounded collection of sheaves, 231 Brill-Noether number, 190 carpet, 15 Cartier, 171 Castelnuovo, 232 Castelnuovo-Mumford regularity, 232 numerical criterion, 230 Cauchy sequence, 43 characteristic map, 54, 149, 189, 201, 213, 260, 261, 302 global/local, 302 of a linear system, 150 Chern class, 172 codifferential, 345 relative, 345 cofinal subcategory, 380 complete intersection, 149, 160, 190, 281, 365, 368 K3 surface, 283 non-rigid, 124 complex torus, 39 cone, 156 over IP 1 × IP 2 , 126 over IP n × IP m , 126, 141 over Veronese embedding of IP n , 141 rigid, 126, 140, 141 congruence, 99 conic, 112, 233, 287 conormal 408

INDEX bundle, 346 sequence, 159, 337, 346, 367 sheaf, 346 coordinate axes, 112, 126 costable subscheme, 216 cotangent module first, 14 second, 128 upper/lower, 20 sequence (relative), 336, 346, 363, 371 sheaf first, 16 second, 130 sheaf/bundle, 344 space, 342 couple, 375 formal, 54, 148 universal, 376 cover/covering, 367 covering of a functor, 381 curve, 183 affine obstructed, 135 canonical, 152, 156 canonical of genus 4, 190 cubic, 28 elliptic, 28 Gorenstein, 171 negatively embedded, 155 non-hyperelliptic, 219 nonsingular, 112, 178 not rigid, 81 obstructed, 221 obstructed in IP 3 , 283 of genus ≥ 3, 217 of genus 0, 119 of genus 1, 110

409 on a surface in IP 3 , 151 pointed, 110 rational, 190, 217 rational normal, 141 unobstructed, 38 cusp, 125, 208 cycle map, 297 Debarre, 191 deformation algebraic, 92, 102 complete, 93 effectively parametrized, 93 first order/infinitesimal, see first order/infinitesimal deformation formal, 53, 90, 91, 94, 98, 102 formally locally trivial, 92 formally trivial, 92 formally versal/universal/semiuniversal, 92, 102 local, 21, 92, 102, 109, 144, 185 locally trivial, 22 of a morphism, 185 of a morphism wfdat, 187 of a morphism with fixed target, 191 of a polarization, 185 of a scheme, 21 of a subscheme, 144 product, 22 trivial, 22, 145, 186 with general moduli, 92 depth, 137 diagonal, 155, 247, 315, 367 differential of aD , 170 of a morphism of functors, 52 of a morphism of schemes, 345

410

INDEX

of the forgetful morphism, 157, extension of schemes, 15 158, 179, 201, 202, 215 trivial, 15 divisor Cartier, 150 faithfully flat module, 323 ramification, 204, 205 family semiregular, 171 isotrivial, 114 duality pairing, 174 locally isotrivial, 114 dualizing sheaf, 171 non-isotrivial, 203 of closed subschemes, 145 Eagon-Northcott complex, 136 product, 22 element universal, 247, 271 universal, 247 with general moduli, 92, 261 elementary transformation, 295 family of deformations, see deformaembedded point, 234 tion embedding filtered category, 378 regular, 365 first iteration morphism, see morphism, rigid, 213 first iteration Segre, 126, 141 first order foci, 303 Veronese, 141, 300 first order/infinitesimal deformation Enriques, 84 of ξ0 ∈ F (k), 50 equisingularity, 304 of a morphism, 185 etale of a pair (X, L), 173 cover, 114 of a scheme, 21 homomorphism, 351 of a subscheme, 144 morphism, 204, 363 of an algebra, 26 neighborhood, 87 of an invertible sheaf, 164 topology, 89 Euler sequence, 39, 140, 159, 173, 189, Fitting ideal, 238, 315 flag 205, 209, 293, 301, 350 Hilbert functor, 271 generalization of, 292 Hilbert scheme, 271, 274 extended Petri map, 184 of closed subschemes, 274 extension of algebras, 7 Quot scheme, 281 pullback of, 11 regularly embedded, 280 pushout of, 11 variety, 274 small, 9 flat module, 323 splits/splitting, 8 flatness criterion trivial, 8 versal, 9 by Hilbert polynomial, 232

INDEX in terms of generators and relations, 330 local, 326 foci, 303 Fogarty’s theorem, 297 forgetful morphism, 156, 158, 179, 183, 201, 215, 218, 220 formal couple, 148 scheme, 148 formal couple, 54 universal, 54 versal/semiuniversal, 56 formal deformation, 53, 90, 102 algebraizable, 94, 98, 104 effective, 94, 98, 103 locally trivial, 91 of a subscheme, 148 semiuniversal, 87, 102, 134, 195, 212 trivial, 91 universal, 102, 112 versal, 98, 102 formal element, 53 semiuniversal, 60, 65, 67, 68, 80, 122, 174 universal, 54 versal/semiuniversal, 56 formal projective space, 95 formal scheme, 91 formally etale homomorphism, 351 formally smooth homomorphism, 351 functor automorphism, 107 criterion of representability, 382 flag Hilbert, 271 Grassmann, 250 left exact, 51, 379

411 less obstructed, 68, 170, 218 local Hilbert, 145 local moduli, 75 local Picard, 164 local relative Hilbert, 161 locally of finite presentation, 99, 115 of deformations of a pair (X, L), 173 of Artin rings, 50 of deformations of a closed embedding, 211 of infinitesimal deformations of u0 , 102 of morphisms, 299 of points, 381 prorepresentable, 50, 54, 69, 148, 162, 164, 187, 214, 219 Quot, 263 representable, 50, 375, 381 smooth, 54, 178 unobstructed, 60 general moduli, 92, 261 generic flatness, 246 graph, 96, 187, 254, 299, 368 Grassmann functor, 250 grassmannian, 150, 252 grassmannian bundle, 288 Grothendieck, 97, 104, 232 Harris, 321 Hartshorne, 262 henselian local ring, 89 henselization, 89 hermitian form, 180 Hilbert functor, 246 local, 145

412 local relative, 161, 163 Hilbert scheme, 246, 259 bound on the dimension, 152 connectedness, 262 existence theorem, 255 flag, 274 nested, 281 non-reduced, 283 reducible, 303 relative, 266, 267 Hilbert syzygy theorem, 228 Hodge decomposition, 219 homogeneous space, 29 formal principal, 30 principal, 29, 66 homomorphism essentially of finite type, ix, 351 etale, 351 formally etale, 351 formally smooth, 351 obstructed/unobstructed, 41 of extensions, 8 smooth, 351 Horikawa, 185, 221 hypersurface of IP n × IP m , 160 of IP r , 159, 248 unobstructed, 134, 155

INDEX of extensions, 7 jacobian criterion of smoothness, 358 jacobian variety, 217 Kleiman, 163 Kodaira, 217, 219 dimension, 86, 321 Kodaira-Nirenberg-Spencer, 109 Kodaira-Spencer, 94 class, 33, 197 correspondence, 31 map, 33, 81, 87, 93, 115, 149, 283 map (vanishing), 116 Kollar, 144, 191 Koszul complex, 128, 282 relations, 128 Krull dimension, 45, 80

lifting, 153 limit inductive, 376 projective, 377 line in a quadric cone, 156 linear system, 150, 171, 248 deformation of, 189 local complete intersection, 365 incidence relation, 254, 292, 350 criterion of flatness, 326, 334 index of ramification, 205 deformation, 109 infinitesimal automorphisms, 108 deformation of a scheme, 21 infinitesimal deformation, see asofirst Hilbert functor, 145 order/infinitesimal deformationx Picard functor, 164 of u0 ∈ F (k), 101 relative Hilbert functor, 161, 163 infinitesimal Torelli theorem, 219 ring isomorphism algebraic, 89 henselian, 89 of deformations, 21

INDEX

413

in the etale topology, 89 local-to-global exact sequence, 76

number of moduli, 80, 84, 87, 111, 157

module of differentials, 335 moduli scheme, 51, 93, 223 coarse, 375 moduli stack, 93 morphism Abel-Jacobi, 169 classifying, 247 etale, 204, 363 first iteration, 315 forgetful, see forgetful morphism non-degenerate, 193, 204 obstructed/unobstructed, 189 of nonsingular curves, 189 Pl¨ ucker, 252 quasi-finite, 233 relative complete intersection, 368 rigid, 186, 189, 200, 203 self transverse codimension 1, 316 smooth, 363 unramified, 193, 363, 367 multiple point scheme, 239, 315 stratification, 239 Mumford, 62, 87, 185, 232 example of, 283

obstructed non-reduced scheme, 285 affine curve, 135 curve, 221 curve in IP 3 , 283 surface, 221 variety of dimension 3, 221 obstructed/unobstructed embedding, 213 morphism, 189 scheme, 38 subscheme, 155 surface, 39 obstructed/unobstructed deformation of a morphism, 189 of a morphism with fixed target, 200 of a scheme, 38 of a subscheme, 155 of an embedding, 213 obstruction, 36, 155, 200 map, 67, 268 for aD , 170 for the forgetful morphism, 157, 179, 201, 202, 215 space for Def B0 , 130 for Def 0j , 212 for Def 0X , 81 for Def 0f /Y , 195 for Def X , 81 for Def (X,L) , 174 for Def X/f /Y , 187 for Hom(X, Y ), 300 Y for HX , 153

Nagata, 89 node, 125, 127, 208 non-degenerate morphism, 193 normal bundle, 291, 346 of a rational normal curve, 294 normal sheaf, 145, 151, 346 equisingular, 16, 207, 211, 305 of (X1 , . . . , Xm ), 275 of a morphism, 183, 193

414

INDEX X /S

for HZ , 162 for Pλ0 , 164 for a functor, 60, 67 for an algebra, 49 for Quot, 270 of OHilbY ,[X] , 260 of OFHZ ,[X1 ,...,Xm ] , 276 relative/absolute, 41 Oort, 185

of an extension of algebras, 11 quadric, 23, 141 cone, 156, 190 Quot functor, 263 scheme, 266 flag, 281

ramification divisor, 204, 205 rational connectedness, 191 parameter scheme, 21, 144 regular Parshin, 203 embedding, 365 Petri map, 152, 156, 159, 171, 190 sequence, 45 extended, 184 Severi variety, 312 Picard functor (local), 164 relative Picard group, 163 codifferential, 345 Pl¨ ucker morphism, 252 complete intersection morphism, pointed scheme, 22, 185, 189, 213, 368 214 conormal sequence, 346, 363, 371 polarization, 185 cotangent sequence, 336, 346, 363, presheaf, 381 371 principal G-bundle, 117 cotangent sheaf, 344 principal parts derivations, 345 algebra of, 335 differential, 345 sheaf of, 172, 350 differentials (sheaf of), 344 product of curves, 86 dimension of a smooth morphism, projection 363 of a curve, 152 Hilbert scheme, 266 projective bundle, x, 288 tangent sheaf, 345 prorepresentability ribbon, 15 of Def X , 105, 109, 112 rigid of Def 0X , 105 Y algebra, 26, 28, 122, 144 of HX , 148 cone, 126, 140, 141 of Pλ0 , 166 curve singularity, 81 pseudo-torsor, 30 embedding, 213 pullback morphism, 186, 189, 200, 203, 204 of an extension of algebras, 11 pushout, 159 nonsingular variety, 38

INDEX product of projective spaces, 39 projective space, 39 scheme, 22, 143 subscheme, 145, 155, 220 weighted projective space, 144

415

sheaf, 251, 381 (b), 228 m-regular, 223 Castelnuovo-Mumford regular, 223 conormal, 346 dualizing, 171 scheme normal, 151, 346 flag Hilbert, 271 of germs of vectors tangent along formal, 91, 148 a subvariety, 211 Hilbert, 246 of principal parts, 172, 350 multiple point, 239, 315 smooth obstructed/unobstructed, 38 functor, 54, 178 of elementary transformations, 295 homomorphism, 351 of isomorphisms/automorphisms, morphism, 363 299 morphism of functors, 54 of morphisms, 299 smoothness of forgetful morphism, 184 pointed, 22, 185, 189, 213, 214 stable subscheme, 216 Quot, 266 stack, 120 rigid, 22, 143 stratification, 236 Severi, 308 defined by a sheaf, 238 unobstructed, 93 flattening, 239 vanishing, 239 multiple point, 239 Schlessinger, 135 subfunctor, 253, 290 theorem of, 65, 75, 122 open/closed, 381 see-saw theorem, 243 subscheme Segre costable, 216 embedding, 126, 141 obstructed/unobstructed, 155 example of, 313 rigid, 145, 155, 220 semiregular, 171 stable, 216 Serre, 97 unobstructed, 155 duality, 179 superabundant Severi variety, 312 vanishing theorem, 221, 224, 226 surface, 125, 217 Severi, 314 abelian, 87 problem, 321 K3, 85, 87, 96, 160, 161, 179, 283 scheme/variety, 308 minimal ruled, 85 variety obstructed, 221 regular/superabundant, 312 obstructed/unobstructed, 39 of general type, 321 Severi-Kodaira-Spencer, 171

416

INDEX

rational, 85 property, 247, 252 quotient, 266 rational ruled, 23, 34, 39, 85, 113, quotient bundle, 251, 288 114, 151, 348 subbundle, 252, 288 ruled, 85 unobstructed symmetric product, 297 canonical curve, 156 system curve, 38 inductive/projective, 377 functor, 60 Szpiro, 203 hypersurface, 134, 155 tacnode, 125 local complete intersection, 134 tangent morphism, 203 space, 342 nonsingular projective variety, 81 tangent space rational ruled surface, 39, 203 of Def 0j , 212 scheme, 93 0 of Def X , 75 subscheme, 155 of Def X , 75 unramified covering of curves, 34 X /S of HZ , 162 unramified morphism, 193, 363, 367 0 of Def f /Y , 195 upper semicontinuous function, 237 of Def (X,L) , 174 vanishing scheme, 239 of Def X/f /Y , 187 Y Veronese embedding, 141, 300 of Hilb , 260 of Hom(X, Y ), 300 Wahl, 313 Y of HX , 145 weighted projective space, 144 of Pλ0 , 164 of FHZ , 276 Yoneda’s lemma, 115, 375 of a functor, 52 Zappa’s example, 294 of Quot, 270 tautological exact sequence, 252, 288, 292 invertible sheaf, 249 torsor, 29 total scheme, 21, 144 truncated cotangent complex, 19 uniruledness, 191, 301 universal family, deformation, element, see family, deformation, element