Stable rank three vector bundles without theta divisors over bielliptic curves

TL;DR

This paper constructs stable rank three vector bundles without theta divisors over bielliptic curves of genus at least 5, providing a counterexample to a general expectation and addressing a question by Beauville.

Contribution

It demonstrates the existence of stable rank three bundles without theta divisors on special bielliptic curves, contrasting with known results for general curves.

Findings

Stable rank three bundles without theta divisors exist on bielliptic curves of genus ≥ 5.

Counterexamples show the failure of a general property for special curves.

Addresses a question posed by Beauville regarding theta divisors.

Abstract

Raynaud has shown that over a general curve of genus $g \ge 2$, every semistable bundle of rank three and integral slope admits a theta divisor. We show that this can fail for special curves: Over any bielliptic curve of genus $g \ge 5$, we construct a stable rank three bundle of trivial determinant with no theta divisor. This gives a partial answer to a question of Beauville.