On the base locus of the linear system of generalized theta functions

TL;DR

This paper investigates the base locus of the linear system of the determinant line bundle over moduli spaces of semi-stable vector bundles, constructing explicit base points for certain ranks and curve types.

Contribution

It constructs explicit base points in the base locus for specific ranks and types of curves, advancing understanding of theta divisors in moduli spaces.

Findings

Constructed base points in rac{g+2}{r} over any curve C.

Identified base points in rac{4}{r} over hyperelliptic curves.

Analyzed the structure of the base locus of the determinant line bundle.

Abstract

Let $\cM_r$ denote the moduli space of semi-stable rank-$r$ vector bundles with trivial determinant over a smooth projective curve $C$ of genus $g$. In this paper we study the base locus $\cB_r \subset \cM_r$ of the linear system of the determinant line bundle $\cL$ over $\cM_r$, i.e., the set of semi-stable rank-$r$ vector bundles without theta divisor. We construct base points in $\cB_{g+2}$ over any curve $C$, and base points in $\cB_4$ over any hyperelliptic curve.