Genus 2 curves and generalized theta divisors

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.