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 lecture topics|
|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:
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.
|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
|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 email@example.com no later than 5:00 PM west coast time on Thursday, July 5. We will attempt to assign topics by Friday, July 6.