Theta divisors of stable vector bundles may be nonreduced

TL;DR

This paper constructs stable vector bundles of various ranks over algebraic curves with reducible and nonreduced theta divisors, expanding known examples and exploring their properties in different geometric contexts.

Contribution

It provides new explicit constructions of stable bundles with nonreduced theta divisors for genus at least 5 and extends Raynaud's examples to higher genus bi-elliptic curves.

Findings

Constructed stable bundles with reducible, nonreduced theta divisors for genus ≥ 5.

Extended Raynaud's example to bi-elliptic curves of genus ≥ 3.

Demonstrated the existence of such bundles across all ranks r ≥ 5.

Abstract

A generic strictly semistable bundle of degree zero over a curve X has a reducible theta divisor, given by the sum of the theta divisors of the stable summands of the associated graded bundle. The converse is not true: Beauville and Raynaud have each constructed stable bundles with reducible theta divisors. For X of genus at least 5, we construct stable vector bundles over X of rank $r$ for all $r \geq 5$, with reducible and nonreduced theta divisors. We also adapt the construction to symplectic bundles. In the appendix, Raynaud's original example of a stable rank 2 vector bundle with reducible theta divisor over a bi-elliptic curve of genus 3 is generalized to bi-elliptic curves of genus $g \geq 3$.