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Principles of Algebraic Geometry
Let C be a genus 2 curve and $${\mathcal{SU}}_C(2)$$ 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 ω of C by ω−1. In this paper, we study the classifying rational map $$\varphi: {\mathbb{P}} Ext^1(\omega,\omega^{-1})\cong {\mathbb{P}}^4 \dashrightarrow {\mathcal{SU}}_C(2)\cong {\mathbb{P}}^3$$ that sends an extension class to the corresponding rank two vector bundle. Moreover, we prove that, if we blow up $${\mathbb{P}}^4$$ along a certain cubic surface S and $${\mathcal{SU}}_C(2)$$ at the point p corresponding to the bundle $$\mathcal{O} \oplus \mathcal{O}$$ , then the induced morphism $$\tilde\varphi: Bl_S \rightarrow Bl_p \mathcal{SU}_C(2)$$ defines a conic bundle that degenerates on the blow up (at p) of the Kummer surface naturally contained in $${\mathcal{SU}}_C(2)$$ . Furthermore we construct the $${\mathbb{P}}^2$$ -bundle that contains the conic bundle and we discuss the stability and deformations of one of its components.
Mathematische Zeitschrift – Springer Journals
Published: Feb 8, 2008
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