Weierstrass points on Gorenstein curves

Widland, Carl; Lax, R. F.

doi:10.2140/pjm.1990.142.197

Cited by 21 publications

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“…The next theorem treats the weight of a unibranch singularity on an arbitrary (in particular, not necessarily rational) Gorenstein curve. This theorem generalizes a result of C. Widland [21] in the case of a simple cusp.…”

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confidence: 85%

“…In the case of an ordinary node, we have I = 1, δ 1 = δ 2 = 0 and Theorem (4.17) reduces to the following result of Widland [21].…”

mentioning

confidence: 92%

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confidence: 99%

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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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confidence: 85%

“…In the case of an ordinary node, we have I = 1, δ 1 = δ 2 = 0 and Theorem (4.17) reduces to the following result of Widland [21].…”

mentioning

confidence: 92%

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Weierstrass weight of Gorenstein singularities with one or two branches

Garcia¹,

Lax

1993

Manuscripta Math

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“…The classical theory of Weierstrass points on smooth curves was extended to Gorenstein curves in [14,15,7]. (For a further extension to integral curves, see [2].)…”

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confidence: 99%

“…Widland [14] showed that the rational curve with three nodes obtained from P 1 C by identifying 0 with ∞, 1 with −1, and i with −i has no smooth Weierstrass points. This curve may be realized as the projective plane curve x 2 y 2 + y 2 z 2 = x 2 z 2 , which has "biflecnodes" at the points (1, 0, 0), (0, 1, 0), and (0, 0, 1), with each biflecnode being a Weierstrass point of weight 8.…”

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Rational nodal curves with no smooth Weierstrass points

Garcia

Lax

1996

Proc. Amer. Math. Soc.

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Abstract. Let X denote the rational curve with n+1 nodes obtained from the Riemann sphere by identifying 0 with ∞ and ζ j with −ζ j for j = 0, 1, . . . , n−1, where ζ is a primitive (2n)th root of unity. We show that if n is even, then X has no smooth Weierstrass points, while if n is odd, then X has 2n smooth Weierstrass points.C. Widland [14] showed that the rational curve with three nodes obtained from P 1 C by identifying 0 with ∞, 1 with −1, and i with −i has no smooth Weierstrass points. This curve may be realized as the projective plane curve x 2 y 2 + y 2 z 2 = x 2 z 2 , which has "biflecnodes" at the points (1, 0, 0), (0, 1, 0), and (0, 0, 1), with each biflecnode being a Weierstrass point of weight 8. In this note, we show that this curve is one member of an infinite family of rational nodal curves with no smooth Weierstrass points. We remark that a general rational nodal curve with n nodes has n(n − 1) smooth Weierstrass points [9].Let n be a positive integer. Let ζ denote a primitive (2n)th root of unity. Let X denote the rational curve with n + 1 nodes obtained from P 1 C by identifying 0 with ∞, and ζ j with −ζ j for j = 0, 1, . . . , n − 1. Our main result is:(0.1) Theorem.(1) If n is even, then X has no smooth Weierstrass points.(2) If n is odd, then X has 2n smooth Weierstrass points at the (2n)th roots of −1.In the first section, we prove a theorem that can be used to show that fixed points of an automorphism are frequently Weierstrass points, which extends known results for smooth curves. This is then used in the second section to prove the main result.We thank Pierre Conner for very helpful discussions. 1In this section, X and Y will denote integral projective curves over the complex numbers. If C is a curve, we let p a (C) denote the arithmetic genus of C and p g (C) denote the geometric genus of C. If P ∈ C, then we letÕ P denote the normalization

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On the poles of regular differentials of singular curves

Stöhr

1993

Bol. Soc. Bras. Mat

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Weierstrass points on Gorenstein curves

Cited by 21 publications

References 5 publications

Weierstrass weight of Gorenstein singularities with one or two branches

Weierstrass weight of Gorenstein singularities with one or two branches

Rational nodal curves with no smooth Weierstrass points

On the poles of regular differentials of singular curves

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