Genus 2 curves and generalized theta divisors

In this paper we investigate generalized theta divisors ${\mathrm{\Theta }}_r$ in the moduli spaces ${\mathcal{U}}_C(r,r)$ of semistable vector bundles on a curve $C$ of genus $2$. We provide a desingularization $\mathrm{\Phi }$ of ${\mathrm{\Theta }}_r$ 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 $\mathrm{\Phi }$ 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.