Abstract
Let
C
be a genus 2 curve and $\su$ the moduli space of semi-stable rank 2 vector bundles on
C
with trivial determinant. In \cite{bol:wed} we described the parameter space of non stable extension classes (invariant with respect to the hyperelliptic involution) of the canonical sheaf
ω
of
C
with
ω
−1
C
. In this paper we study the classifying rational map $\phi: \pr Ext^1(\omega,\omega^{-1})\cong \pr^4 \dashrightarrow \su\cong \pr^3$ that sends an extension class on the corresponding rank two vector bundle. Moreover we prove that, if we blow up $\pr^4$ along a certain cubic surface
S
and $\su$ at the point
p
corresponding to the bundle $\OO \oplus \OO$, then the induced morphism $\tilde{\phi}: Bl_S \ra Bl_p\su$ defines a conic bundle that degenerates on the blow up (at
p
) of the Kummer surface naturally contained in $\su$. Furthermore we construct the $\pr^2$-bundle that contains the conic bundle and we discuss the stability and deformations of one of its components.