Gap sequences at a singularity

Lax, R. F.; Widland, Carl

doi:10.2140/pjm.1991.150.111

Cited by 9 publications

(4 citation statements)

References 9 publications

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“…In this direction we present in the last section for a Gorenstein curve the Wronski systems that form the natural generalization of the systems of principal parts for a smooth curve. Our results generalize those of [27] to arbitrary characteristics and globalize the wronskian determinant used in their work.…”

Section: O Introductionsupporting

confidence: 75%

Weierstrass points and gap sequences for families of curves

Laksov¹,

Thorup²

1994

Ark. Mat.

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Section: O Introductionsupporting

confidence: 75%

Weierstrass points and gap sequences for families of curves

Laksov¹,

Thorup²

1994

Ark. Mat.

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“…This follows from the fact that Hom O P (C, O P ) =Õ P (cf. the proof of Proposition (2.2) of [14]).…”

mentioning

confidence: 89%

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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“…(3) AD m (1) (f). As we explain in § 3.3.2, our work on automatic degeneracy in the weight-1 case is related to the seminal work of C. Widland and R. Lax, who developed a general method for determining the number of weight-1 inflection points limiting to an ICIS in a family of Gorenstein curves over the course of their series of papers [Wid84, Lax87, LW87, LW88, LW90a, LW91,LW90b].…”

Section: Introductionmentioning

confidence: 99%

Inflectionary Invariants for Isolated Complete Intersection Curve Singularities

Patel¹,

Swaminathan²

2017

Preprint

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We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let N ≥ 2, and consider an isolated complete intersection curve singularity germ f : (C N , 0) → (C N−1 , 0). We introduce a numerical function m → AD m(2) (f) that arises as an error term when counting m th -order weight-2 inflection points with ramification sequence (0, . . . , 0, 2) in a 1parameter family of curves acquiring the singularity f = 0, and we compute AD m(2) (f) for various (f, m). Particularly, for a node defined by f : (x, y) → xy, we prove that AD m(2) (xy) = m+1 4, and we deduce as a corollary that AD mfor any f, where mult 0 ∆ f is the multiplicity of the discriminant ∆ f at the origin in the deformation space. Furthermore, we show that the function m → AD m(2is an analytic invariant measuring how much the singularity "counts as" an inflection point. We obtain similar results for weight-2 inflection points with ramification sequence (0, . . . , 0, 1, 1) and for weight-1 inflection points, and we apply our results to solve various related enumerative problems. CONTENTSANAND P. PATEL AND ASHVIN A. SWAMINATHAN 6.3. Computation of AD 2 (1,1) (f) 50 6.4. Computation of AD 3 (2) (y t − x s ) 51 6.5. Computation of AD 2 (1) (y t − x s ) 55 6.6. Computation of AD 4 (2) (y 2 − x s ) 59 6.7. Computation of AD 3 (1) (y 2 − x s ) 62 6.8.

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Gap sequences at a singularity

Cited by 9 publications

References 9 publications

Weierstrass points and gap sequences for families of curves

Weierstrass points and gap sequences for families of curves

Weierstrass weight of Gorenstein singularities with one or two branches

Inflectionary Invariants for Isolated Complete Intersection Curve Singularities

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