Integrable systems and moduli spaces of rank two vector bundles on a non-hyperelliptic genus 3 curve

Vanhaecke, Pol. "Integrable systems and moduli spaces of rank two vector bundles on a non-hyperelliptic genus 3 curve." Annales de l’institut Fourier 55.6 (2005): 1789-1802. <http://eudml.org/doc/116233>.

@article{Vanhaecke2005,
abstract = {We use the methods that were developed by Adler and van Moerbeke to determine explicit equations for a certain moduli space, that was studied by Narasimhan and Ramanan. Stated briefly it is, for a fixed non-hyperelliptic Riemann surface $\Gamma $ of genus $3$, the moduli space of semi-stable rank two bundles with trivial determinant on $\Gamma $. They showed that it can be realized as a projective variety, more precisely as a quartic hypersurface of $\{\mathbb \{P\}\}^7$, whose singular locus is the Kummer variety of $\Gamma $. We first construct an algebraic completely integrable system whose generic fiber of the momentum map is the Jacobian of a non-hyperelliptic Riemann surface of genus $3$. The techniques, developed by Adler and van Moerbeke then allow to compute the eight cubics that define the Kummer variety of $\Gamma $. Since the latter is the singular locus of the moduli space, we can explicitly determine an equation for the moduli space. Our final equation depends on several parameters, which account for the moduli of the Jacobians that appear in the integrable system. We thus actually find explicit equations for a whole family of moduli spaces, which is interesting from the point of view of applications to the Knizhnik-Zamolodchikov equation.},
affiliation = {Université de Poitiers, UFR Sciences SP2MI, laboratoire de mathématiques, UMR 6086 du CNRS, boulevard Marie et Pierre Curie, BP 30179, 86962 Futuroscope Chasseneuil Cedex (France)},
author = {Vanhaecke, Pol},
journal = {Annales de l’institut Fourier},
keywords = {Integrable systems; moduli spaces; Kummer variety; integrable systems; vector bundles},
language = {eng},
number = {6},
pages = {1789-1802},
publisher = {Association des Annales de l'Institut Fourier},
title = {Integrable systems and moduli spaces of rank two vector bundles on a non-hyperelliptic genus 3 curve},
url = {http://eudml.org/doc/116233},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Vanhaecke, Pol
TI - Integrable systems and moduli spaces of rank two vector bundles on a non-hyperelliptic genus 3 curve
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 6
SP - 1789
EP - 1802
AB - We use the methods that were developed by Adler and van Moerbeke to determine explicit equations for a certain moduli space, that was studied by Narasimhan and Ramanan. Stated briefly it is, for a fixed non-hyperelliptic Riemann surface $\Gamma $ of genus $3$, the moduli space of semi-stable rank two bundles with trivial determinant on $\Gamma $. They showed that it can be realized as a projective variety, more precisely as a quartic hypersurface of ${\mathbb {P}}^7$, whose singular locus is the Kummer variety of $\Gamma $. We first construct an algebraic completely integrable system whose generic fiber of the momentum map is the Jacobian of a non-hyperelliptic Riemann surface of genus $3$. The techniques, developed by Adler and van Moerbeke then allow to compute the eight cubics that define the Kummer variety of $\Gamma $. Since the latter is the singular locus of the moduli space, we can explicitly determine an equation for the moduli space. Our final equation depends on several parameters, which account for the moduli of the Jacobians that appear in the integrable system. We thus actually find explicit equations for a whole family of moduli spaces, which is interesting from the point of view of applications to the Knizhnik-Zamolodchikov equation.
LA - eng
KW - Integrable systems; moduli spaces; Kummer variety; integrable systems; vector bundles
UR - http://eudml.org/doc/116233
ER -