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.

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.
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.