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Deformation Theory Math 274 Lectures on Deformation Theory Robin Hartshorne c 2004 Preface My goal in these notes is to give an introduction to deformation theory by doing some basic constructions in careful detail in their sim- plest cases, by explaining why people do things the way they do, with examples, and then giving some typical interesting applications. The early sections of these notes are based on a course I gave in the Fall of 1979. Warning: The present state of these notes is rough. The notation and numbering systems are not consistent (though I hope they are consis- tent within each separate section). The cross-references and references to the literature are largely missing. Assumptions may vary from one section to another. The safest way to read these notes would be as a loosely connected series of short essays on deformation theory. The order of the sections is somewhat arbitrary, because the material does not naturally fall into any linear order. I will appreciate comments, suggestions, with particular reference to where I may have fallen into error, or where the text is confusing or misleading. Berkeley, September 6, 2004 i ii CONTENTS iii Contents Preface . i Chapter 1. Getting Started . 1 1. Introduction . 1 2. Structures over the dual numbers . 4 3. The T i functors . 11 4. The infinitesimal lifting property . 18 5. Deformation of rings . 25 Chapter 2. Higher Order Deformations . 33 6. Higher order deformations and obstruction theory . 33 7. Obstruction theory for a local ring . 39 8. Cohen–Macaulay in codimension two . 43 9. Complete intersections and Gorenstein in codimension three . 54 10. Obstructions to deformations of schemes . 58 11. Dimensions of families of space curves . 64 12. A non-reduced component of the Hilbert scheme . 68 Chapter 3. Formal Moduli . 75 13. Plane curve singularities . 75 14. Functors of Artin rings . 82 15. Schlessinger’s criterion . 86 16. Fibred products and flatness . 91 17. Hilb and Pic are pro-representable . 94 18. Miniversal and universal deformations of schemes . 96 19. Deformations of sheaves and the Quot functor . 104 20. Versal families of sheaves . 108 21. Comparison of embedded and abstract deformations . 111 Chapter 4. Globe Questions . 117 CONTENTS iv 22. Introduction to moduli questions . 117 23. Curves of genus zero . 122 24. Deformations of a morphism . 126 25. Lifting from characteristic p to characteristic 0 . 131 26. Moduli of elliptic curves . 138 27. Moduli of curves . 150 References . 157 CHAPTER 1: GETTING STARTED 1 CHAPTER 1 Getting Started 1 Introduction Deformation theory is the local study of deformations. Or, seen from another point of view, it is the infinitesimal study of a family in the neighborhood of a given element. A typical situation would be a flat morphism of schemes f : X → T . For varying t ∈ T we regard the fibres Xt as a family of schemes. Deformation theory is the infinitesimal study of the family in the neighborhood of a special fibre X0. Closely connected with deformation theory is the question of ex- istence of varieties of moduli. Suppose we try to classify some set of objects, such as curves of genus g. Not only do we want to describe the set of isomorphism classes of curves as a set, but also we wish to describe families of curves. So we seek a universal family of curves, parametrized by a variety of moduli M, such that each isomorphism class of curves occurs exactly once in the family. Deformation theory would then help us infer properties of the variety of moduli M in the neighborhood of a point 0 ∈ M by studying deformations of the cor- responding curve X0. Even if the variety of moduli does not exist, deformation theory can be useful for the classification problem. The purpose of these lectures is to establish the basic techniques of deformation theory, to see how they work in various standard situa- tions, and to give some interesting examples and applications from the literature. Here is a typical theorem which I hope to elucidate in the course of these lectures. Theorem 1.1. Let Y be a nonsingular closed subvariety of a nonsin- gular projective variety X over a field k. Then (a) There exists a scheme H, called the Hilbert scheme, parametrizing closed subschemes of X with the same Hilbert polynomial P as Y , and there exists a universal subscheme W ⊆ X × H, flat over H, such that the fibres of W over points h ∈ H are all closed subschemes of X with the same Hilbert polynomial P and which is universal in the sense that if T is any other scheme, if W 0 ⊆ CHAPTER 1: GETTING STARTED 2 X × T is a closed subscheme, flat over T , all of whose fibres are subschemes of X with the same Hilbert polynomial P , then there 0 exists a unique morphism ϕ : T → H, such that W = W ×H T . (b) The Zariski tangent space to H at the point y ∈ H corresponding to Y is given by H0(Y, N ) where N is the normal bundle of Y in X. (c) If H1(Y, N ) = 0, then H is nonsingular at the point y, of dimen- 0 0 sion equal to h (Y, N ) = dimk H (Y, N ). (d) In any case, the dimension of H at y is at least h0(Y, N ) − h1(Y, N ). Parts (a), (b), (c) of this theorem are due to Grothendieck [22]. For part (d) there are recent proofs due to Laudal [46] and Mori [54]. I do not know if there is an earlier reference. Let me make a few remarks about this theorem. The first part (a) deals with a global existence question of a parameter variety. In these lectures I will probably not prove any global existence theorems, but I will state what is known and give references. The purpose of these lectures is rather the local theory which is relevant to parts (b), (c), (d) of the theorem. It is worthwhile noting, however, that for this particu- lar moduli question, a parameter scheme exists, which has a universal family. In other words, the corresponding functor is representable. In this case we see clearly the benefit derived from Grothendieck’s insistence on the systematic use of nilpotent elements. For let D = k[t]/t2 be the ring of dual numbers. Taking D as our parameter scheme, we see that the flat families Y 0 ⊆ X × D with closed fibre Y are in one- to-one correspondence with the morphisms of schemes Spec D → H that send the unique point to y. This set Homy(D, H) in turn can be interpreted as the Zariski tangent space to H at y. Thus to prove (b) of the theorem, we have only to classify schemes Y 0 ⊆ X × D, flat over D, whose closed fibre is Y . In §2 of these lectures we will therefore make a systematic study of structures over the dual numbers. Part (c) of the theorem is related to obstruction theory. Given an infinitesimal deformation defined over an Artin ring A, to extend the deformation further there is usually some obstruction, whose vanishing is necessary and sufficient for the existence of an extended deformation. CHAPTER 1: GETTING STARTED 3 In this case the obstructions lie in H1(Y, N ). If that group is zero, there are no obstructions, and one can show that the corresponding moduli space is nonsingular. Now I will describe the program of these lectures. There are several standard situations which we will keep in mind as examples of the general theory. A. Subschemes of a fixed scheme X. The problem in this case is to deform the subscheme while keeping the ambient scheme fixed. This leads to the Hilbert scheme mentioned above. B. Line bundles on a fixed scheme X. This leads to the Picard variety of X. C. Deformations of nonsingular projective varieties X, in particular curves. This leads to the variety of moduli of curves. D. Vector bundles on a fixed scheme X. Here one finds the variety of moduli of stable vector bundles. This suggests another question to investigate in these lectures. We will see that the deformations of a given vector bundle E over the dual numbers are classified by H1(X, End E), where End E = Hom(E, E) is the sheaf of endo- morphisms of E, and that the obstructions lie in H2(X, End E). Thus we can conclude, if the functor of stable vector bundles is a representable functor, that H1 gives the Zariski tangent space to the moduli. But what can we conclude if the functor is not representable, but only has a coarse moduli space? And what in- formation can we obtain if E is unstable and the variety of moduli does not exist at all? E. Deformations of singularities. In this case we consider deforma- tions of an affine scheme to see what happens to its singularities. We will show that deformations of an affine nonsingular scheme are all trivial. For each of these situations we will study a range of questions. The most local question is to study extensions of these structures over the dual numbers. Next we study the obstruction theory and structures over Artin rings. In the limit these give structures over complete local CHAPTER 1: GETTING STARTED 4 rings, and we will study Schlessinger’s theory of prorepresentability. Then we will at least report on the question of existence of global moduli. If a fine moduli variety does not exist, we will try to understand why. Along the way, as examples and applications of the theory, I hope to include the following.
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