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

Pol Vanhaecke^{[1]}

- [1] 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)

Annales de l’institut Fourier (2005)

- Volume: 55, Issue: 6, page 1789-1802
- ISSN: 0373-0956

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topVanhaecke, 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 -

## References

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- M. Adler, P. van Moerbeke, P. Vanhaecke, Algebraic integrability, Painlevé geometry and Lie algebras, 47 (2004), Springer-Verlag, Berlin Zbl1083.37001
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- U.V. Desale, S. Ramanan, Classification of vector bundles of rank 2 on hyperelliptic curves, Invent. Math. 38 (1976/77), 161-185 Zbl0323.14012MR429897
- R. Friedman, Algebraic surfaces and holomorphic vector bundles, (1998), Springer-Verlag, Berlin Zbl0902.14029MR1600388
- J. Huebschmann, Poisson structures on certain moduli spaces for bundles on a surface, Ann. Inst. Fourier 45 (1995), 65-91 Zbl0819.58010MR1324125
- M.S. Narasimhan, S. Ramanan, Moduli of vector bundles on a compact Riemann surface, Ann. of Math. 89 (1969), 14-51 Zbl0186.54902MR242185
- M.S. Narasimhan, S. Ramanan, $2\theta $-linear systems on Abelian varieties, Vector bundles on algebraic varieties, (Bombay, 1984) 11 (1987), 415-427 Zbl0685.14023
- P. Vanhaecke, Integrable systems in the realm of algebraic geometry, 2nd ed., 1638 (2001), Springer-Verlag, Berlin Zbl0997.37032MR1850713

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