Welcome to Math 206!
Instructor: Dragos Oprea, doprea "at" math.youknowwhere.edu
Lectures: MW 10:0011:20, APM 5402
Course description:

We will construct and study different parameter spaces of sheaves over curves and surfaces. Time permitting, the topics could include the Quot scheme, the moduli of vector bundles over curves, Hilbert scheme of points over surfaces and others.
Prerequisites:
 I will make an attempt to be as selfcontained 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.
 All instruction is required to be fully remote during weeks 1 and 2. Lectures will be given via zoom. The link can be found in Canvas.
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.
 Lecture 3: Schur polynomials and some examples. Pieri, JacobiTrudi and Cauchy's formulas. Schubert cycles and examples. "Two" additive bases for the cohomology of Grassmannians.
 Lecture 4: Connections between the various classes on the Grassmannian. Examples of enumerative calculations via Pieri/Giambelli. Transversality of general translates.