A conic bundle degenerating on the Kummer surface

Let C be a genus 2 curve and Su(C)(2) the moduli space of semi-stable rank 2 vector bundles on C with trivial determinant. In Bolognesi (Adv Geom 7(1): 113-144, 2007) we described the parameter space of non stable extension classes of the canonical sheaf omega of C by omega(-1). In this paper, we study the classifying rational map phi : PExt(1)(omega, omega(-1)) congruent to P-4 -> Su(C)(2) congruent to P-3 that sends an extension class to the corresponding rank two vector bundle. Moreover, we prove that, if we blow up P-4 along a certain cubic surface S and Su(C)(2) at the point p corresponding to the bundle O circle plus O, then the induced morphism (phi) over tilde : Bl(S) -> Bl(p)Su(C)(2) defines a conic bundle that degenerates on the blow up (at p) of the Kummer surface naturally contained in Su(C)(2). Furthermore we construct the P-2-bundle that contains the conic bundle and we discuss the stability and deformations of one of its components.