University of Michigan
Department of Mathematics
 Fall 2009 Topics in Algebraic Geometry Seminar
Mondays 2:00PM-3:00PM, 3866 East Hall


This semester we will be learning about vector bundles on curves. Our goal is to cover the necessary background to understand the recent work of Belkale and Marian-Oprea on Strange Duality.

The co-organizers are Jesse Kass and Yusuf Mustopa. Please contact either one of us if you would like to get on the e-mail list or volunteer to give a talk.

DateSpeakerTopicAbstract
September 14th Organizational meeting
September 21st Yusuf Mustopa Introductory Talk I will discuss the classification of vector bundles on P^1 as well as some basic concepts necessary for the study of the positive genus case.
September 28th Jesse Kass Vector Bundles on Elliptic Curves I will discuss Atiyah's paper.
October 12th Eugene Eisenstein Stability of M-bundles
October 19th Mihai Fulger Crash Course in GIT
October 26th Yusuf Mustopa The Hilbert-Mumford Criterion and Slope-Semistability
November 9th Amanda Knecht Characterizing Semistable Sheaves Among Coherent Sheaves

List of References:

Generalized theta linear series on moduli spaces of vector bundles on curves, M. Popa. This contains a nice survey of the progress on Strange Duality.

Lectures on Vector Bundles, J. Le Potier, Cambridge UP, 1996. The first half of this book treats the moduli of semistable vector bundles on curves in detail, and the second half treats semistable sheaves on P^2.

An Introduction to Invariants and Moduli, S. Mukai, Cambridge UP, 2003. Chapters 10 through 12 deal almost exclusively with vector bundles on curves. A partial list of errata may be found here.

Vector bundles over an elliptic curve, M.F. Atiyah, Proc. London Math. Soc, 1957

Vector Bundles on Curves, M.Teixidor i Bigas. This is a nice set of elementary notes.

Positivity in Algebraic Geometry II: Positivity for Vector Bundles, and Multiplier Ideals, R. Lazarsfeld, Springer-Verlag, 2003. The first three chapters are a survey of the concept of amplitude for vector bundles, and section 6.4 in particular treats ample vector bundles on curves.

Brill-Noether Theory for stable vector bundles, I. Grzegorczyk and M. Teixidor i Bigas, a survey article on the titular subject.

Current Topics in Complex Algebraic Geometry, H. Clemens, J. Kollár, ed., MSRI Publications Vol. 28. Of particular interest to us are the articles by Beauville and Mukai.

Spectral curves and the generalised theta divisor, A. Beauville, M.S. Narasimhan, and S. Ramanan.