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

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.

33 pages; made a few minor corrections and improvements, and added some new results in the last section; written for the Handbook of Moduli

References in corpus (2)

• The Verlinde bundles and the semihomogeneous Wirtinger duality
• Effective divisors on \bar{M}_g, curves on K3 surfaces and the Slope Conjecture

Cited by in corpus (9)

• The geometry of Ulrich bundles on del Pezzo surfaces
• Coordinate rings for the moduli of $SL_2(\C)$ quasi-parabolic principal bundles on a curve and toric fiber products
• On kernel bundles over reducible curves with a node
• Strange duality and the Hitchin/WZW connection
• Ulrich bundles on intersections of two 4-dimensional quadrics
• Vector bundles on genus 2 curves and trivectors
• Linear series on curves: stability and Clifford index
• New examples of reducible theta divisors for some Syzygy bundles
• Strange Duality of Verlinde spaces for G_2 and F_4