On generalized Kummer of rank-3 vector bundles over a genus 2 curve

Let $X$ be a smooth projective complex curve and let $U_X(r,d)$ be the moduli space of semi-stable vector bundles of rank $r$ and degree $d$ on $X$. It contains an open Zariski subset $U_X(r,d)^s$ which is the coarse moduli space of stable bundles, i.e. vector bundles satisfying inequality $\frac{d_F}{r_F} < \frac{d_E}{r_E}.$ The complement $U_X(r,d)\setminus U_X(r,d)^s$ parametrizes certain equivalence classes of strictly semi-stable vector bundles which satisfy the equality $\frac{d_F}{r_F} = \frac{d_E}{r_E}$. Each equivalence class contains a unique representative isomorphic to the direct sum of stable bundles. Furthermore one considers subvarieties $\mathrm{SU}_X(r,L) \subset U_X(r,d)$ of vector bundle of rank $r$ with determinant isomorphic to a fixed line bundle $L$ of degree $d$. In this work we study the variety of strictly semi-stable bundles in $\mathrm{SU}_X(3,{\mathcal O}_X)$, where $X$ is a genus 2 curve. We call this variety the generalized Kummer variety of $X$ and denote it by $\mathrm{Kum}_3(X)$. Recall that the classical Kummer variety of $X$ is defined as the quotient of the Jacobian variety $\mathrm{Jac}(X) = U_X(1,0)$ by the involution $L\mapsto L^{-1}$. It turns out that our $\mathrm{Kum}_3(X)$ has a similar description as a quotient of $\mathrm{Jac}(X) \times \mathrm{Jac}(X)$ which justifies the name. We will see that the first definition allows one to define a natural embedding of $\mathrm{Kum}_3(X)$ in a projective space. The second approach is useful in order to give local description of $\mathrm{Kum}_3(X)$.