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A conic bundle degenerating on the Kummer surface

A conic bundle degenerating on the Kummer surface 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

A conic bundle degenerating on the Kummer surface

Mathematische Zeitschrift , Volume 261 (1) – Feb 8, 2008

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References (10)

Publisher
Springer Journals
Copyright
Copyright © 2008 by Springer-Verlag
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/s00209-008-0319-4
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Mathematische ZeitschriftSpringer Journals

Published: Feb 8, 2008

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