On the distribution of Weierstrass points on singular curves

Lax, R. F.

doi:10.1007/bf02769464

Israel J. Math.

1987

DOI: 10.1007/bf02769464

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On the distribution of Weierstrass points on singular curves

R. F. Lax

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Cited by 16 publications

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“…Let X denote an integral, projective Gorenstein curve of arithmetic genus g > 1 defined over an algebraically closed field k. In previous articles ( [6,11]), we have defined Weierstrass points on such a curve if k = C. We stated that the notions of Weierstrass gaps and nongaps did not seem to apply at a singular point P, since one is now interested in all 0-dimensional subschemes supported at P and not the Weil divisors nP. Here, we present a generalization of gaps and nongaps to singular points by considering certain chains of ideals in the local ring at P.…”

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confidence: 99%

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Gap sequences at a singularity

Lax¹,

Widland²

1991

Pacific J. Math.

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“…17 τ. In the notation of [6], we then have F\ 9 \ = 1, F\ >2 = t 3 , F\ 9 3 = t 4 in a neighborhood of P. To compute the "wronskian" in [6], we must differentiate "with respect to τ." For example, we have dF\^ = 3t 2 dt = 3ί 8 τ, and so i%2 = 3ί 8 .…”

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Gap sequences at a singularity

Lax¹,

Widland²

1991

Pacific J. Math.

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“…where 0 denotes the local ring of Pon Xs-Let 0 denote the normalization of 0 and put 6 -dim 0/0. The following result was shown in [5]. PROPOSITION 1.…”

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“…Widland [8] extended the classical notion of Weierstrass point to integral, projective Gorenstein curves. We considered Weierstrass points of invertible sheaves on such curves in [5] and showed that a singular point is always a Weierstrass point of high weight of any invertible sheaf with at least two linearly independent global sections. We remarked that this may be interpreted to mean that as a family of smooth curves degenerates to an irreducible Gorenstein curve, then many of the Weierstrass points tend towards the singularities.…”

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Weierstrass Weight and Degenerations

Lax¹

1987

Proceedings of the American Mathematical Society

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Weierstrass weight of Gorenstein singularities with one or two branches

Garcia¹,

Lax

1993

Manuscripta Math

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Let X denote an integral, projective Gorenstein curve over an algebraically closed field k. In the case when k is of characteristic zero, C. Widland and the second author ([22], [21], [13]) have defined Weierstrass points of a line bundle on X. In the first section, we extend this by defining Weierstrass points of linear systems in arbitrary characteristic. This definition may be viewed as a generalization of the definitions of Laksov [10] and to the Gorenstein case. Recently Laksov and Thorup [11,12] have given a more general definition of Weierstrass points of "Wronski systems," and our definition may be viewed as a concrete realization in our setting of their rather abstract definition.In the second section, we give an example illustrating our definition. This example is a plane curve of arithmetic genus 3 in characteristic 2 such that the gap sequence at every smooth point (with respect to the dualizing bundle) is 1, 2, 5 and there are no smooth Weierstrass points. Since every smooth curve of genus 3 in characteristic 2 is classical, this gives us an example of a singular nonclassical curve that is the limit of nonsingular classical curves. We also compute the Weierstrass weights of points on a rational curve with a single unibranch singularity whose local ring has an especially simple form.In the third section, we compute the Weierstrass weight of a unibranch singularity (on a not necessarily rational curve) in terms of its semigroup of values. In order to arrive at a nice formula, we make the assumption that the characteristic is zero. We also compute the number of smooth Weierstrass points on a general rational curve with only unibranch singularities.In the final section, we compute the Weierstrass weight of a singularity with precisely two branches (again assuming that the characteristic is zero). This depends heavily on the structure of the semigroup of values of the singularity. These semigroups have been studied by the first author [4] and by F. Delgado [1, 2], among others. In the course of this argument, we construct a basis for the dualizing differentials that is analogous to a (Hermitian) basis of regular differentials adapted to a point in the smooth case.1. Let k denote an algebraically closed field of arbitrary characteristic. Let X be an integral, projective Gorenstein curve over k of arithmetic genus g > 0.(Gorenstein curves include any curve that is locally a complete intersection; so any curve that lies on a smooth surface is Gorenstein.) Let K denote the field of rational functions on X. Let π : Y → X denote the normalization of X. Let ω = ω X denote the sheaf of dualizing differentials on X and let O X,P denote the local ring of the structure sheaf O X at the point P ∈ X. We recall (cf. [18]) that if P ∈ X, then ω P consists of all rational differentials τ on X such that Q→P Res Q (f τ ) = 0 for all f ∈ O X,P ,

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On the distribution of Weierstrass points on singular curves

Cited by 16 publications

References 8 publications

Gap sequences at a singularity

Gap sequences at a singularity

Weierstrass Weight and Degenerations

Weierstrass weight of Gorenstein singularities with one or two branches

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