Moduli Spaces and Invariant Theory
MWF 9:05-9:55am, LGRT 1234.

Professor: Jenia Tevelev
Office: LGRT 1235E.
Office hours MWF 1:30-3:30
E-mail: tevelev(at)math.umass.edu

Class webpage: http://people.math.umass.edu/~tevelev/797_2017/

All lecture notes in one file

  Class Meeting    Topic     Assignments  
Jan 23 Grassmannian as a complex manifold.
Jan 25 Grassmannian as a quotient. Stiefel and Plucker coordinates.
Jan 27 Grassmannian as a projective variety.
Jan 30 Homogeneous ideal of the Grassmannian.
Feb 1 Hilbert function and degree of the Grassmannian. Topics of presentations
Feb 3 Application to a Schubert calculus problem.
Feb 6 Representable functors
Feb 8 Grassmannian as a fine moduli space
Feb 10 Moduli of algebraic curves.
Feb 13 Snow day
Feb 15 Riemann surfaces. Meromorphic forms. Genus.
Feb 17 Divisors. Canonical divisor. Riemann-Hurwitz.Homework 1 due. Notes on the Grassmannian and Homework 1
Feb 22 Linear systems and Riemann-Roch
Feb 24 Curves of genus 1
Feb 27 Complex tori
Mar 1 J-invariant
Mar 3 Elliptic fibrations Homework 2 due. Notes on algebraic curves and Homework 2
Mar 6 Cartier divisors
Mar 8 Morphisms with reduced fibers
Mar 10 Pushforwards and derived pushforwards
SPRING BREAK
Mar 20 Cohomology and base change
Mar 22 Riemann-Roch in families.
Mar 24 Invariants of finite groups
Mar 27 Properties of quotients of finite groups
Mar 29 Quotient singularities
Mar 31 Icosahedral singularity Homework 3 due. Notes on moduli of elliptic curves and Homework 3
Apr 3Linear algebraic groups
Apr 5 Reductive groups
Apr 7 Geometric and categorical quotient
Apr 10 Weighted projective space
Apr 12 Projective spectrum
Apr 14 GIT quotients and stability Homework 4 due. Notes on families of algebraic varieties, invariants of finite group and Homework 4
Apr 18Toric Varieties as Quotients of Affine Space (by Johnson, Li)
Apr 19 Stable Curves and Stable Reduction (by Stern, Torres)
Apr 21 Examples of GIT stability
Apr 24 Moduli of stable vector bundles (by T. Nakamura, Simonetti).
Apr 26 Mnev's universality and applications (by Cabrera, Fu, K. Nakamura).
Apr 28 Nagata's Counterexample (by Day, Hart).
May 1 Stability of smooth hypersurfaces Homework 5 due on May 3. Notes on quotients by reductive groups and Homework 5
Your course grade will be based on two components, the homework and the project.

Homework

There will be 5 biweekly homework sets. Problems will be worth certain number of points depending on their difficulty with a total of 30 points for each homework.

To get an A+ you have to accumulate 100 points by the end of the semester.

Homework problems can be presented in two ways. An ideal method is to come to my office and explain your solution to me. You can do it during office hours, by scheduling an appointment, or by just showing up in my office assuming I am not too busy. If you can show me a correct solution at the blackboard, you won’t have to turn in the problem in the written form. If your solution does not work, there will be no penalty and I will probably give you a hint.

Each homework will have a two-week deadline. Problems not discussed in my office to my satisfaction will have to be written down and turned in by the end of the two-week period.

Project

The second component of your grade is the project. I will announce possible topics during the second week of classes. My expectation is that you will split into teams of two, work on projects collaboratively, write a paper about your topic (about 15 pages), and present it during the last two weeks of classes. I will be happy to assist with projects (choosing literature, scope, etc.)