## Introduction to Algebraic Geometry (Math 8620), Fall 2014
Instructor: Andrew Obus
email: obus [at] virginia.edu
office: Kerchof Hall 208 |

Problem session

There will be an (almost) weekly one-hour problem session where students will present homework solutions on the board, and where we can discuss the solutions. The problem session will meet Fridays from 9-10 AM in Kerchof 326.

Textbook

*The Red Book of Varieties and Schemes*, by David
Mumford. There should also be a copy on reserve in the
library. This book has great exposition and a lot of geomteric
intuition, while still taking a very algebraic approach. It also
has some nonstandard terminology/notation and unfortunately more than
its share of typos. I'll try to point both of these out as much
as possible!

- Algebraic Geometry,
by Robin Hartshorne (The (abridged) Bible. Great source of
exercises. More difficult than the Red Book and contains more
material. A must for those seriously interested in algebraic geometry.)

- Éléments de Géométrie Algébrique,
by Alexander Grothendieck (The unabridged Bible. 1800 pages over 4
volumes, but very logical and proves everything in complete
detail. Also includes the necessary commutative algebra.)

- Foundations of Algebraic Geometry, by Ravi Vakil (lecture notes, more abstract from the beginning. Does not start with varieties.)
- The Geometry of Schemes, by David Eisenbud and Joe Harris (very geometric and example-based)

- Algebraic Geometry and Arithmetic Curves, by Qing Liu (lots of material related to number theory and my research!)

In particular, I hope to cover chapters I and II of the Red Book (some of which you already know from Algebra IV). If there is time left over, I might do a small unit on divisors or on differentials. One semester is not a lot of time for algebraic geometry, and I've had to make some decisions about what to exclude. Unfortunately, there will not be a great "capstone theorem" for this course --- the goal is more to get you conversant and comfortable with the language of schemes and to give you intuition. If there is demand, I will run a reading course in the Spring to cover further topics.

The prerequisite for this course is knowledge of commutative algebra at the level of Atiyah-MacDonald (including many of the exercises there). In particular, you should be comfortable with the idea of a ring homomorphism A --> B giving rise to a continuous map Spec B --> Spec A of topological spaces. If you need to brush up, Chapters 1, 2, 3, 5, 7, 9, and 10 are the most important for this course.

Mondays 1-2, Fridays 12-1. Kerchof Hall 208 (my office). If these times do not work for you, please make an appointment with me.

There will be roughly 6-7 homework assignments over the course of the semester. Some of the problems will be difficult! The homework assignments will be posted on Collab.

If you have (anonymous) comments for me about teaching style or anything related to the course, you can make them on the Collab page for the course.