Theta divisors of stable vector bundles may be nonreduced

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 g ≥ 5, we construct stable vector bundles over X of rank r for all r ≥ 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 ≥ 3.