Math 203B - Algebraic Geometry

Welcome to Math 203B!

Course description:

This course provides an introduction to algebraic geometry. Algebraic geometry is a central subject in modern mathematics, and an active area of research. It has connections with number theory, differential geometry, symplectic geometry, mathematical physics, string theory, representation theory, combinatorics and others.

Math 203 is a two quarter sequence. Math 203b will cover vector bundles, sheaf cohomology, algebraic curves, Riemann-Roch, Serre duality. Other possible topics: Chow groups, Chern classes, intersection theory.

Instructor: Dragos Oprea, doprea "at" math.you-know-where.edu, AP&M 6-101.

Lectures: MW, 2pm-3:20pm, AP&M 7-421 (will most likely change).

Office hours:

Monday 3:30-4:30, Wednesday 1-2.

I am available for questions after lecture or by appointment. Also, feel free to drop in if you see me in my office.

Textbook: There is no required textbook. I will follow Andreas Gathamnn's notes available online .

Other useful texts are

• Igor Shafarevich, Basic Algebraic Geometry I, Varieties in Projective Space
• Joe Harris, Algebraic Geometry: a first course.
• David Mumford, Algebraic Geometry I, Complex Projective Varieties

More advanced but useful references are:

• Robin Hartshorne, Algebraic geometry
• David Mumford, The red book of varieties and schemes.
There will be no exams or homeworks for this class.

Lecture Summaries

• Lecture 1: Sheaves of O_X-modules. Locally free sheaves and vector bundles. Functorial properties of sheaves. Kernels. Cokernels. Sheafification. Short exact sequences.
• Lecture 2: More on sheafification and short exact sequences. Direct sums, tensor products, determinants, wedge products and other natural operations with vector bundles and sheaves.
• Lecture 3: Quasi-coherent sheaves. Functorial constructions. Ideal sheaves. Pullbacks.
• Lecture 4: The sheaf of differentials.
• Lecture 5: Examples: the canonical sheaf of P^n and hypersurfaces in P^n.
• Lecture 6: Divisors and rational equivalence. Line bundles associated to divisors. Statemenet of Riemann-Roch.
• Lecture 7: Riemann-Hurwitz (algebraic and topological versions). Ramification divisors. The degree of the canonical divisor.
• Lecture 8: Proof of Riemann-Roch via the residue theorem.
• Lecture 9: Applications of Riemann-Roch. Morphisms to projective space; globally generated and very ample line bundles. Examples.
• Lecture 10: More applications of Riemann-Roch: genus 0 curves are isomorphic to P^1. Genus 1 curves are cubics. Addition law on genus 1 curves. Genus 2 curves and hyperelliptic curves. Canonical embedding of nonhyperelliptic curves.
• Lecture 11: Cohomology of sheaves. Cech cohomology. First examples.
• Lecture 12: Independence of affine cover. Long exact sequence in cohomology.
• Lecture 13: Kodaira vanishing. More on Riemann-Roch. Serre duality.
• Lecture 14: Cohomology of projective space. Serre's theorems.
• Lecture 15: Intersection theory. Chow groups. The excision sequence. Chow group of P^n. Weil and Cartier divisors.
• Lecture 16: Line bundles and Cartier divisors. Intersections with Cartier divisors. Intersection pairing on surfaces. Curves on surfaces and their genus.
• Lecture 17: Riemann-Roch for surfaces. Hirzebruch-Riemann-Roch stated. Axiomatics of Chern classes. Todd genus. Chern character.
• Lecture 18: Construction of Chern classes. Applications.