Deformations of Algebraic Schemes

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

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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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CHAPTER 2. FORMAL DEFORMATION THEORY

(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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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