Math 273: Introduction to Algebraic Geometry

Instructor: Chad Schoen

Prerequisites:

Math 251

Texts:

The first roughly 5 lectures will follow Chapter 8 of Ideals, Varieties and Algorithms by Cox, Little and O'Shea.
The main text is Basic Algebraic Geometry I by Shafarevich.

Homework:

There will be weekly homework assignments.

About this course:

This is a one semester course at the first year graduate level. It will begin where the brief introduction to affine varieties in Math 251 left off. The main topic will be subvarieties of projective space. These will be studied using algebraic techniques (hence the name ALGEBRAIC geometry). Such techniques generally work over any base field, although the connection between the algebra and the geometry is simplist when the base field is algebraically closed.

Course goals:

0. Introduce the participants to what algebraic geometry is and how it fits in with the rest of mathematics.
1. Become familiar with general geometric properties of projective and quasi-projective varieties: Dimension, decomposition into irreducible components, singular locus.
2. Understand the nature of maps between the objects discussed in 1: Rational maps and regular maps. The equivalence relations birational and biregular. The image of a projective variety under a morphism, the image of a quasi-projective variety, how the dimensions of fibers behave and why this is significant, families of varieties, incidence correspondences and their uses, parametrizability and non-parametrizability.
3. Become familiar with specific geometric objects which are important in many areas of pure mathematics: Grassmannians, the simplest flag varieties, Segre, and Veronese varieties, quadric hypersurfaces.
4. Introduce the local and infinitesimal study of algebraic varieties: Tangent space, singular locus, closedness of singular locus, tangent variety. Local ring of a variety at a point, local parameters. Tangent cone and associated graded ring. Completion of the local ring, power series rings, Weierstrass preparation theorem. Proof that geometric regular local rings are UFDs.
5. Introduce blowing up.
6. Introduce the notions of Weil divisor and Cartier divisor. Develope the bare rudiments of intersection theory for divisors on non-singular varieties. Prove Bezout's theorem for plane curves.
7. Become acquainted with basic techniques in commutative algebra in the course of developing 1-6: Localization, completion, integral extension, integral closure, Nakayama's Lemma, the significance of unique factorization, transcendence degree of field extensions, Lueroth's theorem.
8. Provide a foundation for further study in algebraic geometry and related areas.

This course is intended for students interested in Algebraic Geometry, Commutative Algebra, Complex Differential Geometry, Mathematical Physics, Number Theory, and related areas.