Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

On the moduli space of rank $3$ vector bundles on a genus $2$ curve and the Coble cubic


Author: Angela Ortega
Journal: J. Algebraic Geom. 14 (2005), 327-356
DOI: https://doi.org/10.1090/S1056-3911-04-00387-X
Published electronically: November 18, 2004
MathSciNet review: 2123233
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Abstract: We prove a conjecture about the moduli space $\mathcal{SU}_C(3)$ of semi-stable rank 3 vector bundles with trivial determinant on a genus 2 curve $C$, due to I. Dolgachev. Given $C$ a smooth projective curve of genus 2, and the embedding of the Jacobian $JC$ into $\vert 3\Theta\vert$, A. Coble proved, at the beginning of the 20th century, that there exists a unique cubic hypersurface $\mathcal{C}$ in $\vert 3\Theta\vert^* \simeq \mathbb{P} ^8$, $JC[3]$-invariant and singular along $JC$. On the other hand, we have a map of degree 2 from $\mathcal{SU}_C(3)$ over $\vert 3\Theta\vert \simeq \mathbb{P} ^{8*}$, ramified along a sextic hypersurface $\mathcal{B}$. Dolgachev's conjecture affirms that the sextic $\mathcal{B}$ is the dual variety of Coble's cubic $\mathcal{C}$.


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Angela Ortega
Affiliation: Laboratoire J.-A. Dieudonné, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France
Address at time of publication: Instituto de Matemáticas, UNAM Unidad Morelia, Apartado Postal 61-3 Xangari, CP 58089 Morelia, Mich., Mexico
Email: ortega@math.unice.fr

DOI: https://doi.org/10.1090/S1056-3911-04-00387-X
Received by editor(s): November 19, 2003
Received by editor(s) in revised form: January 20, 2004
Published electronically: November 18, 2004

American Mathematical Society