Generalized theta linear series on moduli spaces of vector bundles on curves

TL;DR

This paper surveys the current understanding of pluri-theta linear series and generalized theta divisors on moduli spaces of vector bundles on curves, highlighting new techniques and recent breakthroughs like the proof of the Strange Duality conjecture.

Contribution

It provides a comprehensive overview of recent advances and techniques in the study of theta linear series on moduli spaces of vector bundles, including the use of moduli of stable maps and Fourier-Mukai transforms.

Findings

Use of moduli spaces of stable maps for Quot schemes analysis

Application of Fourier-Mukai functor to coherent sheaves on abelian varieties

Proof of the Strange Duality conjecture by Belkale and Marian-Oprea

Abstract

This article is based on lecture notes prepared for the August 2006 Cologne Summer School. The first part contains background material and references for beginners. The second (and main) part is a survey of the current status in the theory of pluri-theta linear series and generalized theta divisors on moduli spaces of vector bundles on curves. It emphasizes relatively new techniques employed in the analysis of linear series on these moduli spaces, namely the use of moduli spaces of stable maps for understanding Quot schemes, and the Fourier-Mukai functor in the study of coherent sheaves on abelian varieties. In addition, it briefly describes recent important developments, most significant of which is the proof of the Strange Duality conjecture due to Belkale and Marian-Oprea.