Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

A structure theorem for $ \mathcal{SU}_C(2)$ and the moduli of pointed rational curves


Authors: Alberto Alzati and Michele Bolognesi
Journal: J. Algebraic Geom. 24 (2015), 283-310
DOI: https://doi.org/10.1090/S1056-3911-2014-00659-7
Published electronically: June 11, 2014
MathSciNet review: 3311585
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Abstract: Let $ \mathcal {SU}_C(2)$ be the moduli space of rank 2 semistable vector bundles with trivial determinant on a smooth complex algebraic curve $ C$ of genus $ g>1$. We assume $ C$ nonhyperellptic if $ g>2$. In this paper we construct large families of pointed rational normal curves over certain linear sections of $ \mathcal {SU}_C(2)$. This allows us to give an interpretation of these subvarieties of $ \mathcal {SU}_C(2)$ in terms of the moduli space of curves $ \mathcal {M}_{0,2g}$. In fact, there exists a natural linear map $ \mathcal {SU}_C(2) \to \mathbb{P}^g$ with modular meaning, whose fibers are birational to $ \mathcal {M}_{0,2g}$, the moduli space of $ 2g$-pointed genus zero curves. If $ g<4$, these modular fibers are even isomorphic to the GIT compactification $ \mathcal {M}_{0,2g}^{GIT}$. The families of pointed rational normal curves are recovered as the fibers of the maps that classify extensions of line bundles associated to some effective divisors.


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Alberto Alzati
Affiliation: Dipartimento di Matematica “F. Enriques”, Via Saldini 50, 20133 Milano, Italy
Email: alberto.alzati@unimi.it

Michele Bolognesi
Affiliation: IRMAR, Université de Rennes 1, 263 Av. du Général Leclerc, 35042 Rennes Cedex, France
Email: michele.bolognesi@univ-rennes1.fr

DOI: https://doi.org/10.1090/S1056-3911-2014-00659-7
Received by editor(s): October 17, 2011
Received by editor(s) in revised form: March 21, 2013, and October 21, 2013
Published electronically: June 11, 2014
Article copyright: © Copyright 2014

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