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
Full-text PDF

Abstract | References | Additional Information

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 $\mathcal{C}$.


References [Enhancements On Off] (What's this?)


Additional Information

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

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is sponsored by the Department of Mathematical Sciences
of Tsinghua University
and is distributed by the American Mathematical Society
for University Press, Inc.
Online ISSN 1534-7486; Print ISSN 1056-3911
© 2017 University Press, Inc.
Comments: jag-query@ams.org
AMS Website