Groupe des points de Weierstrass sur une famille de quartiques lisses

Girard, Martine

doi:10.4064/aa105-4-1

Cited by 4 publications

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“…From these results, using the specialization theorem, Theorem 2.1, we can deduce that the rank is at least 5 and the torsion part is at most (Z/4Z) 4 or (Z/4Z) 5 for all strata containing one of the above curves. For curves with four hyperflexes or less, we obtain sharper results.…”

Section: Moduli Space Of Curvesmentioning

confidence: 87%

“…The curves Ω i of Theorem 3.1 are in Y 5 . The specialization theorem, Theorem 2.1, implies that the torsion part is at most (Z/4Z) 4 or (Z/4Z) 5 respectively, and gives bounds on the rank.…”

Section: Elliptic Factors Of the Jacobianmentioning

confidence: 99%

“…In particular, in [3], we showed that for a generic quartic, it is a free abelian group of rank at least 11. In in Section 6, we pursue the study carried out in [4], [3] and improve some of the bounds previously obtained.…”

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confidence: 98%

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The group of Weierstrass points of a plane quartic with at least eight hyperflexes

Girard

2006

Math. Comp.

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Abstract. The group generated by the Weierstrass points of a smooth curve in its Jacobian is an intrinsic invariant of the curve. We determine this group for all smooth quartics with eight hyperflexes or more. Since Weierstrass points are closely related to moduli spaces of curves, as an application, we get bounds on both the rank and the torsion part of this group for a generic quartic having a fixed number of hyperflexes in the moduli space M 3 of curves of genus 3.The Weierstrass points of an algebraic curve, which can be defined over some extension of the base field, form a distinguished set of points of the curve having the property of being geometrically intrinsic. Curves of genus 0 or 1 have no Weierstrass points, and for hyperelliptic curves, the Weierstrass points can easily be characterized as the ramification points under the hyperelliptic involution, which generate the 2-torsion subgroup of the Jacobian. For nonhyperelliptic curves of genus 3, which admit a plane quartic model, the structure of the Weierstrass subgroup, i.e., the subgroup of the Jacobian generated by the images of the Weierstrass points under the Abel-Jacobi map, cannot be characterized so easily. This structure is known for curves with many automorphisms ([10], [12], [13]), in which cases the groups are finite. As with Weierstrass points, the Weierstrass subgroup is a geometric invariant of the curve, which we study in this paper. More precisely, we determine the structure of this group for all curves having either eight (Theorems 4.1 and 5.1) or nine hyperflexes (Theorem 3.1), i.e., points at which the tangent line to the curve meets the curve with multiplicity four. These are the first nontrivial cases of interest since there are no curves with either ten or eleven hyperflexes, and the two possible cases of curves with twelve hyperflexes have already been treated ([7], [13]). We show:

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Section: Moduli Space Of Curvesmentioning

confidence: 87%

Section: Elliptic Factors Of the Jacobianmentioning

confidence: 99%

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The group of Weierstrass points of a plane quartic with at least eight hyperflexes

Girard

2006

Math. Comp.

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“…Nous donnons une démonstration de ce résultat en appendice. Remarquons que les arguments de spécialisation décrits au paragraphe 2 permettent de montrer relativement facilement que le groupe associé à la courbe générique noté W g est de la forme Z r × (Z/4Z) 2 , avec r \ 5 (en effet, la partie de torsion de W s'injecte dans (Z/4Z) 5 En utilisant la stratification de Vermeulen, nous pouvons poursuivre cette étude, ce qui fera l'objet d'un article ultérieur.…”

Section: Nous Verrons Que Lorsque T¨{−1 2 1/2 (1 ± I`3 )/2}unclassified

Géométrie du groupe des points de Weierstrass d'une quartique lisse

Girard¹

2002

Journal of Number Theory

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Nous déterminons, pour une famille de courbes lisses de genre 3, la structure du groupe engendré par les points de Weierstrass dans la jacobienne. Cela fournit un exemple où ce groupe est infini. Plus précisément, nous exhibons une famille où ce groupe est isomorphe àPuis, nous déterminons un encadrement du rang et de la partie de torsion pour une quartique générique en fonction de son nombre de points d'hyper-inflexion.We describe the group generated by the Weierstrass points in the Jacobian, for a family of smooth curves of genus 3. This gives an example where this group is not finite. More precisely, we construct a family with a group isomorphic to. Thus we can conclude that there exists a family whose group is Z r with 11 [ r [ 23. Then we determine bounds for the rank and the finite part in the case of a generic quartic depending on the number of hyperflexes. © 2002 Elsevier Science (USA)

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“…Finally, in [1], a family of smooth curves of genus 3 is determined for which W = Z 4 × (Z/3Z) 5 . In this note, we consider the smooth curve C of genus 3 given by the following equation:…”

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Group generated by the Weierstrass points of a plane quartic

Girard

Tzermias

2001

Proc. Amer. Math. Soc.

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Abstract. We describe the group generated by the Weierstrass points in the Jacobian of the curve X 4 +Y 4 +Z 4 +3 (X 2 Y 2 +X 2 Z 2 +Y 2 Z 2 ) = 0. This curve is the only curve of genus 3, apart from the fourth Fermat curve, possessing exactly twelve Weierstrass points.Let C be a smooth projective curve of genus g. For D a divisor, let [D] be its class in Pic 0 (C). Let J be the Jacobian of C which we identify with Pic 0 (C). We choose a Weierstrass point ∞ as base point for the Abel-Jacobi mapwhich can be extended linearly to divisors Div(C).We study the structure of the group W generated by the images under j of Weierstrass points in the Jacobian of C. This structure does not depend on the Weierstrass point chosen as base-point for the map j and thus provides an interesting geometric invariant.The simplest case is the case of a hyperelliptic curve, where it is easy to see that). For non-hyperelliptic curves of genus 3 -that is, plane quartics -some particular cases have already been considered. Each of these curves has a large automorphism group. More precisely, for the Klein curve (given by. For the Fermat quartic (given by, a family of smooth curves of genus 3 is determined for which W = Z 4 × (Z/3Z) 5 . In this note, we consider the smooth curve C of genus 3 given by the following equation:This curve has 24 automorphisms [4]. We prove the following result:Theorem 0.1. Let W be the group generated by the images of the Weierstrass points in the Jacobian J of C. Then W ∼ = (Z/4Z) 5 .

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Groupe des points de Weierstrass sur une famille de quartiques lisses

Cited by 4 publications

References 1 publication

The group of Weierstrass points of a plane quartic with at least eight hyperflexes

The group of Weierstrass points of a plane quartic with at least eight hyperflexes

Géométrie du groupe des points de Weierstrass d'une quartique lisse

Group generated by the Weierstrass points of a plane quartic

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