Maps between moduli spaces of vector bundles and the base locus of the theta divisor

TL;DR

This paper explores the relationships between moduli spaces of vector bundles on curves and the base locus of the theta divisor, introducing new maps derived from vector bundle constructions.

Contribution

It investigates the properties of maps between moduli spaces of vector bundles and the theta divisor's base locus, providing new insights into their structure.

Findings

Constructs new maps between moduli spaces and the theta divisor

Analyzes conditions for global validity of vector bundle operations

Provides foundational results on the geometry of these maps

Abstract

Given a vector bundle $E$ of rank $r$ and degree $d$ on a curve $C$ of genus $g$, one can associate to $E$ in a natural way several other vector bundles. For example, one can take wedge powers of $E$. If $E$ is generated by global sections, the kernel of the evaluation map of sections is again a vector bundle. Also, new vector bundles can be produced by taking elementary transformations centered at a fixed point. Under suitable conditions on degree and rank, these constructions can be carried out globally. While all this processes seem quite elementary, very little is known about the resulting maps. The purpose of this paper is to fill in this gap.