Math 216: Foundations of algebraic geometry 2009-10

Spring quarter: Wednesday Friday 9:00-10:15 in 380-F (with many exceptions)

An updated version of the notes will be gradually posted here, starting roughly September 1, 2010. See this mathoverflow page for interesting related discussion.

The table of contents (and sketchy introduction) for the notes as of June 15, 2010 (much of the course) is here. Topics near the end are quite rough (see the bottom of this page for further discussion). I'll drop off copies in the mailboxes of many of the people in the class late in the week of June 15. A paper copy of the current version will be available outside my office then too.

There are several types of courses that can go under the name of "introduction to algebraic geometry": complex geometry; the theory of varieties; a non-rigorous examples-based course; algebraic geometry for number theorists (perhaps focusing on elliptic curves); and more. There is a place for each of these courses. This course will deal with schemes, and will attempt to be faster and more complete and rigorous than most, but with enough examples and calculations to help develop intuition for the machinery. Such a course is normally a "second course" in algebraic geometry, and in an ideal world, people would learn this material over many years. We do not live in an ideal world. To make things worse, I am experimenting with the material, and trying to see if a non-traditional presentation will make it possible to help people learn this material better, so this year's course is only an approximation. (See here for an earlier version.)

This course is for mathematicians intending to get near the boundary of current research, in algebraic geometry or a related part of mathematics. It is not intended for undergraduates or people in other fields; for that, people should take Maryam Mirzakhani's class, or else wait for a later incarnation of Math 216 (which will vary in style over the years).

In short, this not a course to take casually. But if you have the interest and time and energy, I will do my best to make this rewarding.

Office hours: Because of the nature of this class, I'd like to be as open as possible about office hours, and not have them restricted to a few hours per week. So if you would like to chat, please let me know, and I'll be most likely happy to meet on a couple of days' notice. I am almost always available to meet immediately after class.

References: I hope to periodically release notes, perhaps once per week. (The most recent version should be outside my office door. I will also give each new version out in class as they become available.) You should take notes yourself, and not count on these. The notes from the class two years ago are available here. It may be useful having Hartshorne's Algebraic Geometry, and possibly Mumford's Red Book of Varieties and Schemes (the original edition is better, as Springer introduced errors into the second edition by retyping it). Mumford (second edition) is availble online (with a Stanford account) from Springer. For background on commutative algebra, I'd suggest consulting Eisenbud's Commutative Algebra with a View toward Algebraic Geometry or Atiyah and MacDonald's Commutative Algebra. For background on abstract nonsense, Weibel's Introduction to Homological Algebra is good to have handy. Freyd's Abelian Categories is available online (free and legally) here.

Homework: Unlike most advanced graduate courses, there will be homework. It is important --- this material is very dense, and the only way to understand it is to grapple with it at close range. There will be a problem set most weeks. Your grade will depend on the problem sets.

Fall quarter

Winter quarter Spring quarter There are a few topics we didn't get to. In the notes, I've written some up, including a proof of cohomology and base change theorem and related facts, a full proof of Serre duality (including that the determinant of Omega is dualizing), Chow's Lemma (and the fact that coherent sheaves push forward to coherent sheaves under proper morphisms, with civilized hypotheses), and blowing up. There are some other topics I intend to type up in the medium term (some by later this summer), notably ampleness and possibly Cohen-Macaulayness. I would like to write up some notes on things related to Stein factorization (and other things related to when the pushforward of the structure sheaf is the structure sheaf) and Zariski's Main Theorem. But I'm not sure when I'll get to that. I may try to do some examples (surfaces, K3 surfaces and Calabi-Yau varieties, abelian varieties, toric varieties). If there are other things that you think I should do (even if you didn't take the class), please let me know!
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