Algebraic Geometry

Spring 2005

- 1/11/05: The room has been moved to Math 320.

This is a continuation of Math 536A, with the theme "geometry for arithmetic". More precisely, the goal is to introduce students to some advanced topics in algebraic geometry which are useful in number theory, i.e., some arithmetical algebraic geometry. By the end, students should have a fighting chance at following a seminar that starts out "Let **X** be a regular proper model over **Z** of a curve X over **Q**" or "Consider a finite flat group scheme G over Spec **F**_{p}" or "We view the l-adic cohomology group H^{1}(X,**Q**_{l}) as a representation of the absolute Galois group of k."

The main topics will be:

- Abstract varieties with their structure sheaf.
- Basic notions and examples of scheme theory.
- Coherent sheaves and their cohomology. Projective embeddings.
- Etale and other Grothendieck topologies and their cohomology theories.

The emphasis will be on examples and techniques that help to study low-dimensional schemes of arithmetic interest: curves over fields, rings of algebraic integers, and curves and group schemes over fields, number rings, and curves.

**Class meetings:**The course will be partly lecture, partly seminar style, with reports from students. Attendance and active participation are essential. Meetings are currently scheduled for Wednesdays and Fridays, 3-4:15 in Math East 241. This can be adjusted if necessary.**Texts:**The main text for technical issues will be selected with input from the students. Some of the options are Shafarevich volume II, unpublished notes of Mumford-Lang, Eisenbud-Harris, Ueno, Hartshorne, and EGA. The book "An Invitation to Arithmetic Geometry" by Lorenzini is recommended for examples and motivation.**Prerequisites:**Math 536A and the math core, or equivalent knowledge, are required. Math 531 would be helpful but is not required.**Homework:**There will be regular substantial homework assignments.**Project:**There will be a required final project in which students will master an additional topic and write a short expository paper on it, suitable for use by fellow students.**Grades:**Grades will be based on homework, class participation, and the project, all weighted equally.**Other policies:**Please read my standard policies about other matters.**Two money-back guarantees:**This course will be hard work and students who take it seriously will learn a lot of beautiful and useful mathematics.

**Instructor:**Douglas Ulmer, Professor of Mathematics**Office:**Math 204**Phone:**621-6861**E-mail:**ulmer@math.arizona.edu**Office hours:**Check my home page for the latest office hours.