A conic bundle degenerating on the Kummer surface
Abstract
Let $C$ be a genus 2 curve and $\su$ the moduli space of semistable 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 $\omega$ of $C$ with $\omega_C^{1}$. 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 February 2007
 arXiv:
 arXiv:math/0702525
 Bibcode:
 2007math......2525B
 Keywords:

 Mathematics  Algebraic Geometry;
 14H60;
 14J70
 EPrint:
 29 pages