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

International audienceLet $X$ 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 $X$ (see \cite{Se}). 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 $\SU_X(r,L) \subset U_X(r,d)$ 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 $\SU_X(3,\mathcal O _X)$, where $X$ is a genus 2 curve. We call this variety the generalized Kummer variety of $X$ and denote it by $\Kum_3(X)$. Recall that the classical Kummer variety of $X$ is defined as the quotient of the Jacobian variety $\Jac(X) = U_X(1,0)$ by the involution $L\mapsto L^{-1}$. It turns out that our $\Kum_3(X)$ has a similar description as a quotient of $\Jac(X) \times \Jac(X)$ which justifies the name. We will see that the first definition allows one to define a natural embedding of $\Kum_3(X)$ in a projective space (see section \ref{deg}). The second approach is useful in order to give local description of $\Kum_3(X)$ by following the theory developed in \cite{Be} (section \ref{sing})