169 Pages
English

History-Social Science

Gain access to the library to view online
Learn more

Description

  • cours - matière potentielle : number principles
  • cours - matière potentielle : number
  • cours - matière : economics
  • cours - matière potentielle : number history
  • cours - matière potentielle : contract
  • cours - matière potentielle : number world history courses
  • cours - matière potentielle : listing
  • cours - matière : geography
History-Social Science Course Listing and Textbook Matrix Course Number History-Social Science Courses and Related Textbook Options Publisher or Author Credit Value Level N/A Primary History-Social Science A/B Reflections: Our World, Now and Long Ago Harcourt N/A K Reflections: A Child's View Harcourt N/A 1 Reflections: People We Know Harcourt N/A 2 Reflections: Our Communities Harcourt N/A 3 Living in Our Communities, Level C Steck-Vaughn N/A 3 Homes and Families, Level A Steck-Vaughn N/A K-1 People and Places Nearby, Level B Steck-Vaughn N/
  • world mcdougal-littell
  • -12 pacemaker u.s. history globe fearon
  • -12 world history
  • action glencoe mcgraw
  • 3 mcgraw hill
  • mcgraw hill
  • social science
  • world of history
  • world history
  • history
  • hill

Subjects

Informations

Published by
Reads 7
Language English

Math 274
Lectures on
Deformation Theory
Robin Hartshorne
c2004Preface
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
iiiCONTENTS iii
Contents
Preface .................................................. i
Chapter 1. Getting Started ....................................... 1
1. Introduction ......................................... 1
2. Structures over the dual numbers .................... 4
i3. The T functors ..................................... 11
4. The in nitesimal 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 atness ....................... 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 .................................... 117CONTENTS 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 ...................................................... 157CHAPTER 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 in nitesimal study of a family in the
neighborhood of a given element. A typical situation would be a at
morphism of schemes f : X→ T. For varying t∈ T we regard the
bres X asafamilyofschemes. Deformationtheoryisthein nitesimalt
study of the family in the neighborhood of a special bre X .0
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 X . Even if the variety of moduli does not exist,0
deformation theory can be useful for the classi cation 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 eld k. Then
(a) ThereexistsaschemeH, calledtheHilbertscheme, parametrizing
closed subschemes of X with the same Hilbert polynomial P as
Y, and there exists a universal subscheme WXH, at over
H, such that the bres of W over points h∈ H are all closed
subschemes of X with the same Hilbert polynomial P and which
0is universal in the sense that if T is any other scheme, if W CHAPTER 1: GETTING STARTED 2
XT is a closed subscheme, at over T, all of whose bres are
subschemes of X with the same Hilbert polynomial P, then there
0exists a unique morphism ϕ :T→H, such that W =W T.H
(b) The Zariski tangent space to H at the point y∈H corresponding
0to Y is given by H (Y,N) whereN is the normal bundle of Y in
X.
1(c) If H (Y,N) = 0, then H is nonsingular at the point y, of dimen-
0 0sion equal to h (Y,N) = dim H (Y,N).k
0(d) In any case, the dimension of H at y is at least h (Y,N)
1h (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 rst 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
lecturesisratherthelocaltheorywhichisrelevanttoparts(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 bene t derived from Grothendieck’s
insistence on the systematic use of nilpotent elements. For let D =
2k[t]/t betheringofdualnumbers. TakingDasourparameterscheme,
0we see that the at families Y XD with closed bre Y are in one-
to-one correspondence with the morphisms of schemes SpecD→ H
that send the unique point to y. This set Hom (D,H) in turn can bey
interpreted as the Zariski tangent space to H at y. Thus to prove (b)
0of the theorem, we have only to classify schemes Y XD, at over
D, whose closed bre 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
in nitesimal deformation de ned over an Artin ring A, to extend the
deformation further there is usually some obstruction, whose vanishing
isnecessaryandsu cientfortheexistenceofanextendeddeformation.CHAPTER 1: GETTING STARTED 3
1InthiscasetheobstructionslieinH (Y,N). Ifthatgroupiszero,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 xed scheme X. The problem in this case is to
deform the subscheme while keeping the ambient scheme xed.
This leads to the Hilbert scheme mentioned above.
B. Line bundles on a xed 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. Vectorbundlesona xedscheme X. Hereone ndsthevarietyof
moduli of stable vector bundles. This suggests another question
toinvestigateintheselectures. Wewillseethatthedeformations
ofagivenvectorbundleE overthedualnumbersareclassi edby
1H (X,EndE), whereEndE =Hom(E,E) is the sheaf of endo-
2morphisms of E, and that the obstructions lie in H (X,EndE).
Thus we can conclude, if the functor of stable vector bundles is
1a representable functor, that H 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-
formationcanweobtainifE isunstableandthevarietyofmoduli
does not exist at all?
E. Deformations of singularities. In this case we consider deforma-
tions of an a ne scheme to see what happens to its singularities.
We will show that deformations of an a ne 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 localCHAPTER 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. Ifa nemodulivarietydoesnotexist,wewilltrytounderstand
why.
Along the way, as examples and applications of the theory, I hope
to include the following.
31. Mumford’s example [56] of a curve in P with obstructed defor-
mations, i.e., whose Hilbert scheme is nonreduced.
2. Examples of rigid singularities and questions of smoothing singu-
larities [77].
3. Mori’s lower bound on the dimension of the Hilbert scheme, used
in his proof that a variety with an ample tangent bundle must be
projective space [54].
4. Tannenbaum’s proof of the existence of irreducible plane curves
1of degree d and r nodes, for any 0r (d 1)(d 2).
2
5. Examples of obstructed surface deformations (Kas [36], Burns
and Wahl [9]).
6. Applicationstothemoduliofvectorbundlesonprojectivespaces.
7. Theproblemofliftingschemesfromcharacteristicptocharacter-
istic 0. A typical question here is the following: IfW is a discrete
valuation ring of characteristic 0, whose residue eld k is a eld
of characteristic p > 0, and if X is a scheme over k, does there0
exist a scheme X at over W, whose closed