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Niveau: Supérieur, Doctorat, Bac+8

Heisenberg invariant quartics and SUC(2) for a curve of genus four William Oxbury and Christian Pauly The projective moduli variety SUC(2) of semistable rank 2 vector bundles with trivial determinant on a smooth projective curve C comes with a natural morphism ? to the linear series |2?| where ? is the theta divisor on the Jacobian of C. Well-known results of Narasimhan and Ramanan say that ? is an isomorphism to P3 if C has genus 2 [16], and when C is nonhyperelliptic of genus 3 it is an isomorphism to a special Heisenberg-invariant quartic QC ? P7 [18]. The present paper is an attempt to extend these results to higher genus. In the nonhyperelliptic genus 3 case the so-called Coble quartic QC ? |2?| = P7 is characterised by either of two properties: (i) QC is the unique Heisenberg-invariant quartic containing the Kummer variety, i.e. the image of Kum : JC ? |2?|, x 7? ?x + ??x, in its singular locus; and (ii) QC is precisely the set of 2?-divisors containing some translate of the curve W1 ? J1C . We shall examine, for a curve of genus 4, the analogue of each of these properties, and our first main result, analogous to (i), is the following.

Heisenberg invariant quartics and SUC(2) for a curve of genus four William Oxbury and Christian Pauly The projective moduli variety SUC(2) of semistable rank 2 vector bundles with trivial determinant on a smooth projective curve C comes with a natural morphism ? to the linear series |2?| where ? is the theta divisor on the Jacobian of C. Well-known results of Narasimhan and Ramanan say that ? is an isomorphism to P3 if C has genus 2 [16], and when C is nonhyperelliptic of genus 3 it is an isomorphism to a special Heisenberg-invariant quartic QC ? P7 [18]. The present paper is an attempt to extend these results to higher genus. In the nonhyperelliptic genus 3 case the so-called Coble quartic QC ? |2?| = P7 is characterised by either of two properties: (i) QC is the unique Heisenberg-invariant quartic containing the Kummer variety, i.e. the image of Kum : JC ? |2?|, x 7? ?x + ??x, in its singular locus; and (ii) QC is precisely the set of 2?-divisors containing some translate of the curve W1 ? J1C . We shall examine, for a curve of genus 4, the analogue of each of these properties, and our first main result, analogous to (i), is the following.

- theta characteristic
- invariant quartic
- jg?1 ?
- qc ?
- any great
- wg?d ? supp
- embedded prym- kummer variety
- without vanishing

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Publié par | profil-zyak-2012 |

Nombre de lectures | 25 |

Langue | English |

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