Weierstrass weight of Gorenstein singularities with one or two branches

Abstract

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 ,

Rational nodal curves with no smooth Weierstrass points

“…[14]. (For a generalization of this result to arbitrary Gorenstein singularities see [6,8].) (2.2) Corollary.…”