Weight sequences versus gap sequences at singular points of Gorenstein curves
The main purpose of this paper is to complete the conceptual background of the theory of Weierstraß points on Gorenstein curves developed by \textit{C. Widland} and \textit{R. F. Lax}. There are several equivalent ways of saying that a point \(P\) on a smooth projective curve \(C\) is a Weierstraß point (WP in the following). The most common is to say that \(P\) is a WP if there exists a rational function regular on \(C\backslash\{P\}\) having at \(P\) a pole of order smaller or equal than \(g\). This last proposition is equivalent to the existence of an integer \(n \in \{1, \dots, g\}\) such that \(h^0 (C, (n - 1) P) < h^0 (C,nP)\), by a simple application of Riemann-Roch theorem. Such an integer is said to be a Weierstraß gap. However, Lax and Widland's papers show that, really, the most intrinsic way for defining a WP is using the so-called Wronskian determinant. In fact one can define the Weierstraß points on a singular Gorenstein curve as the zero locus of a Wronskian constructed by means of the global sections of the (invertible) dualizing sheaf. Following this theory, it turns out that all singular points have to be considered as Weierstraß points, because the Wronskian vanishes at them. However, Lax and Widland, although their theory specializes in the classical one for smooth curves, are not really able to recover the concept of (a unique) WGS (= Weierstraß gap sequence) at a singular point \(P\), because the definition given above for smooth curves does not appear so easily generalizable. Their attempt to do it [cf. \textit{R. F. Lax} and \textit{C. Widland}, Pac. J. Math. 150, No. 1, 111-122 (1991; Zbl 0686.14033)], has not been very successful because in principle they attach, to each singular point \(P\), more than one WGS without stating any relation with the computation of the Weierstraß weight of the point itself, which is, indeed, one of the most important open problems raised in that paper. In a sense, among other several results, this paper fills this gap. And the solution offered in this paper is based on concentrating once more the attention, in the same spirit of Lax and Widland, on the Wronskian as the main tool for studying Weierstraß points, which indeed contains much more information than it seems to do.
scientific article; zbMATH DE number 806212
scientific article