MATH 7670 - Algebraic Geometry

Mike Stillman, spring 2015.

This course is a first course in algebraic geometry that should be accessible if you have had a graduate algebra course. Our plan is to introduce the main concepts of algebraic geometry, and toric varieties will be our key examples.

Toric varieties are wonderful examples of affine, projective, and even more general varieties. They provide an extremely useful link of algebraic geometry with combinatorics. Toric varieties appear in many places, including in physics.

We will cover a selection of topics from chapters 1-4 and 6, in the book "Toric Varieties", by Cox, Little and Schenck. (Hal Schenck was my Ph.D. student in the 1990s!)

In each chapter, "section 0" introduces a number of important algebraic geometry concepts, and then these are described in the context of toric varieties in the rest of the chapter.

Algebraic geometry is a subject where you must "learn by doing". I will assign regular homework, and we will discuss the solutions during class time. The book has a number of excellent exercises.

A potential list of topic areas (warning, the exact list is subject to change!):

1. Affine varieties, affine toric varieties. Includes regular maps, smoothness, normality
2. Projective varieties, projective toric varieties
3. Abstract varieties (done via toric varieties), toric varieties defined by fans
4. Divisors: Weil and Cartier divisors, sheaves
5. Line bundles, including ample and very ampleness, and perhaps the nef and Mori cones. These have a pleasing explicit computable form for toric varieties.