Weierstrass Points on Rational Nodal Curves of Genus 3

Abstract

We determine, except for one unsettled case, which combinations of Weierstrass weights can occur on irreducible rational nodal curves of arithmetic genus three. It is shown that the number of nonsingular Weierstrass points on such curves can be any integer between 0 and 6, except 1.

Gap sequences of 1-Weierstrass points on non-hyperelliptic curves of genus 10

“…At first, the theory of the Weierstrass points was developed only for smooth curves, and for their canonical divisors. In the last years, starting from some papers by R. Lax and C. Widland (see [10], [11], [12], [13], [14], [18]), the theory has been reformulated for Gorenstein curves, where the invertible dualizing sheaf substitutes the canonical sheaf. In this contest, the singular points of a Gorenstein curve are always Weierstrass points.…”

Distribution of Weierstrass Points on Rational Cuspidal Curves

“…Lax and Widland have also studied the Weierstrass points of the canonical sheaf on cuspidal rational curves in [4], and have noted the following general pattern. Weierstrass points on cuspidal rational curves tend to be even scarcer than on rational nodal curves, because even more of the total Weierstrass weight is accounted for by the singular points (see 3 below).…”

“…There has recently been an increase of interest in the problem of extending well-known notions from the theory of smooth algebraic curves, such as the notion of Weierstrass points for invertible sheaves, to the setting of singular curves. In the papers [2], [3], and [4], R. F. Lax and C. Widland have generalized the classical définition of the Weierstrass points of the canonical sheaf of a smooth curve via the order of vanishing of a certain Wronskian. Using this method, they have defined Weierstrass points for any invertible sheaf L with dim H°(L) > 0 on an integral, complex projective Gorenstein curve.…”

On the distribution of Weierstrass points on irreducible rational nodal curves

“…Introduction. In a recent series of papers ( [2], [3], [4]), R. F. Lax and C. Widland have defined Weierstrass points for invertible sheaves on integral, projective Gorenstein curves over C. They use a method generalizing the classical definition of the Weierstrass points of the canonical sheaf on a smooth curve via Wronskians. In particular, they show that if X is an integral, projective Gorenstein curve, and S?…”