Math 390C: Theory of Schemes (58105)

Prof. David Helm
RLM 9.118
dhelm@math.utexas.edu
Office Hours: Mon 1-3

Course time and location: TuTh 12:30-2 in RLM 12.166

Assignments

Problem Set 1 (Due 9-8-09)
Problem Set 2 (Due 9-15-09)
Problem Set 3 (Due 9-24-09)
Problem Set 4 (Due 10-1-09)
Problem Set 5 (Due 10-8-09)
Problem Set 6 (Due 10-20-09)
Problem Set 7 (Due 10-27-09)
Problem Set 8 (Due 11-10-09)
Problem Set 9 (Due 11-17-09)

Lecture Notes

Mohammad Haque has been kind enough to make his notes from the lectures available; I've posted them below, but have not edited them at all.
Lecture 2
Lecture 3
Lecture 4
Lecture 5
Lecture 6
Lecture 7
Lecture 8
Lecture 9
Lecture 10
Lecture 11
Lecture 12
Lecture 13
Lecture 14
Lecture 15
Lecture 16
Lecture 17
Lecture 18
Lecture 19
Lecture 20
Lecture 21
Lecture 22
Lecture 23
Lecture 24
Lecture 25

Course content, format and syllabus

The theory of schemes provides an incredibly flexible framework for dealing with a wide variety of geometric, algebraic, and even number-theoretic phenomena. Concepts as diverse as the reduction mod p of a system of equations, the "limit" of a family of algebraic varieties, and the intersection of cycles on a variety are all best understood from a scheme-theoretic point of view.

Unfortunately, this perspective relies heavily on some fairly abstract and technical formal machinery. As a result the subject has developed a somewhat daunting reputation! The goal of this course is to develop this theory from a rigorous point of view without losing sight of the intuition that lies behind it.

As far as prerequisites go, some previous familiarity with varieties will be helpful, primarily as a source of examples. Algebraic geometry also relies very heavily on commutative algebra; I don't want to spend too much time getting bogged down in this, so I'll probably end up quoting key results from commutative algebra to use as "black boxes". It therefore shouldn't be necessary to have a strong commutative algebra background to get something out of the course (although of course it wouldn't hurt!).

Grades in the course will be based on weekly problem sets. Unfortunately it's nearly impossible to get comfortable with the concepts and language of this course without working through a lot of problems.

Here's a rough overview of the topics I hope to cover.

  1. Review of Varieties over an Algebraically Closed Field

  2. Foundational Definitions

  3. Properties of Schemes

  4. Sheaves of Modules

  5. Line Bundles

  6. Differentials

  7. Families of schemes

  8. Moduli spaces and deformation theory (time permitting)

  9. Cohomology of sheaves (time permitting)

Textbook and Problem Sets

The official textbook for the course is Hartshorne, Algebraic Geometry. Many of the homework problems I assign will be from Hartshorne (some I'll write myself).

Useful Resources

J. S. Milne's algebraic geometry notes- These cover only the theory of varieties, but from a point of view similar to ours.
Ravi Vakil's lecture notes- These will cover nearly all the material we'll be discussing, and more besides! One of the best sources for learning about schemes today.
Atiyah-Macdonald, Introduction to Commutative Algebra- This is one of the best introductory sources for the commutative algebra that is relevant to algebraic geometry. It's almost maddeningly concise, but it manages to cover almost all of the important results in a very small space.
Eisenbud, Commutative Algebra With a View Towards Algebraic Geometry- By contrast to Atiyah-Macdonald, Eisenbud goes to the other extreme. Pretty much everything you'd want to know about commutative algebra can be found- somewhere- in Eisenbud.
Eisenbud and Harris, The geometry of schemes- This is an alternative introduction to the theory of schemes, written with an eye towards building up geometric intuition for them.