Deformation Theory and Moduli in Algebraic Geometry

July 23, 2007 to August 03, 2007
Organized By: Max Lieblich (Princeton), Martin Olsson (Berkeley), Brian Osserman (Berkeley), Ravi Vakil (Stanford)
Background material
Background exercises
Background lecture topics
Background material
We will expect students to have a background in modern algebraic geometry prior to the start of the workshop. For convenience, we use Hartshorne's book Algebraic Geometry as a common starting point in describing the background material. The more of Hartshorne you are familiar with, the easier it will be to understand important examples and motivation, and the more you will get out of the workshop. However, for the sake of expediency we have isolated several key topics which play a crucial role in deformation theory, and you should be sure to familiarize yourself with them prior to the workshop. The background lectures, which will be assigned in advance and given by the participants in small groups during the first week, will also have their topics drawn roughly from this material.

The most important background material for the workshop is the following three areas:
  • Non-reduced schemes
  • Flatness
  • Cohomology
The topic of non-reduced schemes is the vaguest, but we wish to underline that we will be working extensively with them, and it will not be enough to be familiar only with the naive, point-oriented point of view of affine or projective varieties over algebraically closed fields.

A basic introduction to flatness is contained in the section in Hartshorne, which is Section 9 of Chapter III (excepting Proposition 9.3 and Corollary 9.4, which assume familiarity with higher derived pushforwards).

Finally, for cohomology, the crucial basics are covered in Sections 1-4 of Chapter III of Hartshorne, while we recommend that students familiarize themselves as well with Sections 5-7, covering through Serre duality.

Additional material that will be helpful, but at a lower priority, is the remainder of Chapter III of Hartshorne, including smoothness, the theorem on formal functions, and the semicontinuity theorem and theory of cohomology and base change. If you still find yourself with extra time, you can read up on the background lecture topics listed below.
Background exercises
The following exercises are provided to help explore the required background topics listed above, and to supplement their treatment in Hartshorne. They will not be collected or graded. We will periodically update this page with additional exercises.

Non-reduced schemes:
  • Classify all non-reduced subschemes of the affine plane having length 3. You may find it helpful to first consider the case of schemes supported at a single point, and then those supported at at least two points.
  • Classify (up to isomorphism) all Artin local schemes (i.e., obtained as Spec of an Artin local ring) of length 2 and residue field Fp.
  • Exercise 9.3 of Hartshorne, Chapter III.
  • Suppose that X and Y are schemes flat over Spec A, for A a local Artin ring, and f:X -> Y is a morphism. Let k be the residue field of A. Prove that if f is an isomorphism after restriction to Spec k, then f is an isomorphism.
Related to deformation theory:
  • Exercise 2.8 of Hartshorne, Chapter II.
  • Exercise 8.6 of Hartshorne, Chapter II.
  • Exercise 8.7 of Hartshorne, Chapter II.
Background lecture topics
The background lectures will each be a half hour, with two each afternoon from Monday through Thursday in the first week. Following are the background lecture topics. Most of them are not covered in Hartshorne, and we will provide additional references as necessary, as well as additional details on what we expect you to cover. Click on each topic to see a more detailed description.

Please rank your top three choices in order of preference, and email your choices to no later than 5:00 PM west coast time on Thursday, July 5. We will attempt to assign topics by Friday, July 6.
  1. Flatness, and the local criterion for flatness. Wednesday, July 25
  2. Smoothness, and the formal criterion for smoothness. Monday, July 23
  3. Etaleness, and the formal criterion for etaleness. Wednesday, July 25
  4. Categories and functors; abelian categories. Monday, July 23
  5. Derived functors. Monday, July 23
  6. Sheaf cohomology: basic properties and computation of examples. Tuesday, July 24
  7. Cohomology on curves: Riemann-Roch and computation of examples. Tuesday, July 24
  8. Grothendieck's existence theorem for coherent sheaves. Thursday, July 26