2012/11/05 by George H. Hitching, Hitching, George H., With an appendix by Christian Pauly +1
Mathematics · #14H60 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14H60
paper · pdf · doi:10.48550/arxiv.1211.1064
18 pages. Some statements improved. Appendix by Christian Pauly added
arxiv created 2013/06/08 · arxiv updated 2013/06/11
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 ≥ 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.