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On the computation of Weierstrass gap sequences. (English) Zbl 1007.14003

Let \(C\) be a smooth algebraic curve of genus \(g\) defined over the field of complex numbers, and let \(P\) be a point of \(C\). If one asks for which positive integers \(n\) there is no function having a pole only at \(P\), and of order exactly \(n\), the answer is that, except for a finite number of points, called Weierstrass points, these integers are \(1, 2, \dots, g\). At a Weierstrass point, the sequence of such integers is different, although it also contains \(g\) numbers, and it is called the Weierstrass gap sequence (WGS) at the point.
More generally, given a linear system \(V\) on \(C\), of dimension \(r\), one may use it to attach to each point of \(C\) a sequence of \(r\) different positive integers, such that the sequence is the same except for a finite set of points, also called Weierstrass points (moreover, when \(V\) is the canonical linear system, one recovers the former Weierstrass points and their sequences). These WGS sequences have been the object of many investigations, and, among their applications is an important stratification of the moduli space of curves of genus \(g\). In a series of works, R. Lax and C. Widland have extended this theory to a special type of singular curves, the so-called Gorenstein curves, and in Can. Math. Bull. 30, 286-294 (1987; Zbl 0594.14014) they also defined a WGS for Gorenstein singularities that reduces to the usual one if the point is nonsingular. In Geom. Dedicata 54, No. 3, 267-300 (1995; Zbl 0847.14017), L. Gatto defined WGS for Gorenstein points in a way completely different from that by Lax and Widland; his WGS also reduces to the expected sequence in the non-singular case and moreover, relates nicely with (the computation of) the Weierstrass weights at the singular points.
In the present paper, the author presents an algorithm to calculate the WGS, as defined by Gatto, at a point \(P\) of a (possibly singular) plane (and hence Gorenstein) curve \(C\), with respect to a linear system \(V \subset H^{0}(C, {\mathcal O}_{C}(n))\). The algorithm may be implemented in any computer algebra system that does the factorization of multivariate polynomials and finds the solution of algebraic equations of arbitrary degrees. The whole procedure boils down to the calculation, for all the branches of \(C\) through the point \(P\), of the intersection multiplicity at \(P\) of a branch and curves related to linear subsystems of \(V\). This can be done by using certain osculating curves, which the author shows how to construct.
The paper ends with a series of examples that illustrate the method.

MSC:

14H55 Riemann surfaces; Weierstrass points; gap sequences
14Q05 Computational aspects of algebraic curves

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