Algebraic Geometry Notes

This is a course in algebraic geometry. Actually, it is two courses, consisting of notes from a yearlong course that didn't get far enough (2002-03), and a brisk treatment from a course (in progress, Spring 2010) in which I am *determined* to get to the cohomology of coherent sheaves, as well as some interesting properties of curves, surfaces and toric varieties. Of course, this means risking the wrath of the Bourbaki police. It is pre-Hartshorne, a cross between Mumford's Red Book (minus schemes) and Shafarevich's Basic Algebraic Geometry. My feeling is that by focusing on complex varieties, I am able to more clearly point out the relationship between commutative algebra and the geometry of varieties, and to get to properties of curves and surfaces without drowning the students in abstraction. Obviously, it would make better "Bourbaki" sense to start with Grothendieck's theory of schemes from the beginning, but it has been my experience that this approach (even in Mumford's Red Book) overwhelms all but the very best graduate students and results in an unsatisfying first-year course. It remains to be seen, of course, whether this approach fares any better. But hope springs eternal.

The Slow Notes (from 2002).

I. Homage to Hilbert

Hilbert's Basis Theorem and Nullstellensatz

Hilbert's Polynomial Growth Theorem

II. Introducing Varieties

Affine Varieties

Projective Varieties

Examples of Varieties


III. First Properties of Varieties

Regular Maps


Normal Varieties

Nonsingular Varieties

Codimension 1 Phenomena

The Brisk Treatment (in progress, Spring 2010)

An Overview

Complex Varieties: Affine, Abstract, Projective and Toric

Affine Varieties

Abstract Varieties

Projective Varieties

Smooth Curves

Basics of Smooth Curves




First Steps Towards a Classification

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Last modified on Tuesday, 16-Feb-2010 20:59:56 MST