Weierstrass points and gap sequences for families of curves

Laksov, Dan; Thorup, Anders

doi:10.1007/bf02559578

Cited by 19 publications

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“…For the first part, see Hubbard [9] or Laksov-Thorup [11]; the second part is classic (see [8,Theorem C.1.4]). …”

Section: Specializationmentioning

confidence: 99%

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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“…For the first part, see Hubbard [9] or Laksov-Thorup [11]; the second part is classic (see [8,Theorem C.1.4]). …”

Section: Specializationmentioning

confidence: 99%

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

Girard

2006

Math. Comp.

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“…In [LT1] and [LT2] such a relative wronskian is defined regardless of the characteristic of the ground field the curves are defined over. What we do, in section 2, is to consider not only the relative wronskian but also its derivatives in the following sense.…”

Section: 2mentioning

confidence: 99%

“…The purpose of this section is to define a suitable notion of "derivative" of the relative wronskian in the sense, e.g., of [LT1], [LT2]. We recall briefly, for the reader's convenience, the basic framework which stays behind the extended notion of wronskians for families of curves.…”

Section: Derivatives Of Relative Wronskiansmentioning

confidence: 99%

“…The purpose now is to define the higher order derivatives with respect to the generator σ α (compare, however, with [LT1] and [LT2]). We set u…”

Section: 4mentioning

confidence: 99%

“…constructed above, will be said to be the n-th derivative of the wronskian relative to the family π : X −→ S. D 0 W π is the usual wronskian as defined in [LT1] and [LT2]. The n-th derivative of the π-relative wronskian can be alternatively seen as …”

Section: End Of the Constructionmentioning

confidence: 99%

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Derivatives of Wronskians with applications to families of special Weierstrass points

Gatto

Ponza

1999

Trans. Amer. Math. Soc.

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Abstract. Let π : X −→ S be a flat proper family of smooth connected projective curves parametrized by some smooth scheme of finite type over C. On every such a family, suitable derivatives "along the fibers" (in the sense of Lax) of the relative wronskian, as defined by Laksov and Thorup, are constructed. They are sections of suitable jets extensions of the g(g + 1)/2-th tensor power of the relative canonical bundle of the family itself.The geometrical meaning of such sections is discussed: the zero schemes of the (k − 1)-th derivative (k ≥ 1) of a relative wronskian correspond to families of Weierstrass Points (WP's) having weight at least k.The locus in Mg, the coarse moduli space of smooth projective curves of genus g, of curves possessing a WP of weight at least k, is denoted by wt(k). The fact that wt(2) has the expected dimension for all g ≥ 2 was implicitly known in the literature. The main result of this paper hence consists in showing that wt(3) has the expected dimension for all g ≥ 4. As an application we compute the codimension 2 Chow (Q-)class of wt(3) for all g ≥ 4, the main ingredient being the definition of the k-th derivative of a relative wronskian, which is the crucial tool which the paper is built on. In the concluding remarks we show how this result may be used to get relations among some codimension 2 Chow (Q-)classes in M 4 (g ≥ 4), corresponding to varieties of curves having a point P with a suitable prescribed Weierstrass Gap Sequence, relating to previous work of Lax.0. Introduction 0.1. The main goal that motivated the investigations which led to this paper was to deal with loci of the (coarse) moduli space M g of the smooth projective curves of genus g (defined over an algebraically closed field of characteristic zero), whose points correspond to curves possessing some special Weierstrass points.We are, in fact, especially interested in the computation of the classes in the Chow ring of M g with rational coefficients (as defined by Mumford in [Mu2], see section 1.5) of some of such loci. To achieve this goal we have been in some sense forced to introduce and to study a locus in M g which does not seem to have been considered in the earlier literature. In the moduli space M g define: i.e. the locus of (isomorphism classes of) curves in M g possessing a Weierstrass point with weight at least k. In order to study such a locus, above defined only as a set, one needs to put on it a reasonable structure of scheme and to study how its dimension varies, by varying k. For instance it is quite clear that wt(1) must be all of M g , a fancy way of saying that every curve of genus g ≥ 2 has at least 1 Weierstrass point. 0.2.In order to put on wt(k) a scheme structure, one has to deal with some natural but very useful tools, developed in section 2, whose construction, indeed, forms the conceptual core of the paper. As is well known, the Weierstrass Points (WP's in the following) on a smooth curve of genus g, can be detected as the zero-locus of the wronskian section of the bundle K, K being the ca...

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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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Weierstrass points and gap sequences for families of curves

Abstract: The theory of Weierstrass points and gap sequences for linear series on smooth curves is generalized to smooth families of curves with geometrically irreducible fibers, and over an arbitrary base scheme.

Cited by 19 publications

References 26 publications

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

Derivatives of Wronskians with applications to families of special Weierstrass points

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

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