Math 206 - Topics in Algebraic Geometry

Welcome to Math 206!

Instructor: Dragos Oprea, doprea "at" math.you-know-where.edu

Lectures: MW 10:00-11:20, APM 5402

Course description:

We will construct and study different parameter spaces of sheaves. Time permitting, the topics could include the Quot scheme, GIT, the moduli of vector bundles over curves and others.

Office hours: I am available for questions after lecture or by appointment.

Prerequisites:

I will make an attempt to be as self-contained as the topic permits. However, I will assume working knowledge of algebraic geometry at least at the level of Math 203.

Grading:

The course is intended for graduate students in algebraic geometry, though everybody with the right prerequisites is welcome.

Remote instruction:

All instruction is required to be fully remote during weeks 1-4. Lectures will be given via zoom. The link can be found in Canvas.

We will return to in-person lectures on February 2.

Textbook: None.

Important dates:

First class: Wednesday, January 3
University Holiday: Monday, January 17
University Holiday: Monday, February 21
Last class: Wednesday, March 9

Lecture Summaries

Lecture 1: Introduction to enumerative geometry and moduli problems. Moduli functors including the Grassmannian and the Hilbert/Quot functors. Fine and coarse moduli spaces.

Lecture 2: Representability of the Grassmannian functor. Presentation of the cohomology ring of the Grassmannian in terms of generators and relations.

Lecture 3: Schur polynomials and some examples. Pieri, Jacobi-Trudi and Cauchy's formulas for Schur polynomials. Schubert cycles and examples. "Two" additive bases for the cohomology of Grassmannians.

Lecture 4: Connections between the various classes on the Grassmannian. Pieri/Giambelli for Schubert cycles and examples of enumerative calculations. Transversality of general translates.

Lecture 5: Construction of the Quot scheme. General strategy. Castelnuovo-Mumford regularity. Examples and some properties.

Lecture 6: Global generation and regularity. Uniform bounds on regularity of subsheaves. Review of cohomology and base change.

Lecture 7: Embedding of the Quot functor into the Grassmannian. Construction of the Quot scheme via the flattening stratification. Generic flatness.

Lecture 8: Construction of the flattening stratification in the non-relative setting. More on cohomology and base-change, especially in the absence of flatness. Flattening stratification in general. Dimensional reduction via pushforwards.

Lecture 9: The construction of the Quot scheme for all targets. Projectivity. Some examples.

Lecture 10: Tangent space to the Quot scheme and smoothness. A more complicated example: Quot scheme of the trivial bundle over P^1. Description using quiver representations. Torus actions and the Bialynicki-Birula decomposition.

Lecture 11: An introduction to quantum cohomology with emphasis on the Grassmannian.

Lecture 12: Quantum Schubert calculus. The formula of Vafa-Intriligator and its proof.

Lecture 13: GIT. Types of quotients: categorical, good, geometric. Good quotients are categorical. Affine quotients are good. If the action is closed, we obtain a geometric quotient.

Lecture 14: Construction of the projective and quasiprojective GIT quotient. Stable and semistable points.

Lecture 15: One parameter subgroups. Hilbert-Mumford criterion. Examples.

Lecture 16: Slope stability and semistability. S-equivalence. The moduli space of stable and semistable bundles. GIT construction. Analysis of stability. Le Potier's theorem.

Lecture 17: Theta line bundles, theta divisors and theta functions over the Jacobian. Generalized theta line bundles/divisors over the moduli of higher rank bundles. Descent from the Quot scheme.

Lecture 18: The Picard group of the moduli space. Generalized theta functions and their use. Popa's conjectures on basepointfreeness and very ampleness of the theta bundles. The Verlinde formula. Witten's volume formula.