Genus 2 curves and generalized theta divisors

Abstract

In this paper we investigate generalized theta divisors ${\Theta }_r$ in the moduli spaces ${\mathcal{U}}_C(r,r)$ of semistable vector bundles on a curve $C$ of genus $2$. We provide a desingularization $\Phi$ of ${\Theta }_r$ in terms of a projective bundle $\pi :\mathbb{P}(\mathcal{V})\to {\mathcal{U}}_C(r-1,r)$ which parametrizes extensions of stable vector bundles on the base by ${\mathcal{O}}_C$. Then, we study the composition of $\Phi$ with the well known theta map $\theta$. We prove that, when it is restricted to the general fiber of $\pi$, we obtain a linear embedding.

19 pages. Last version has been accepted for publication on "Bulletin des Sciences Mathématiques" on May 2019

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