Generalized linear systems on curves and their Weierstrass points

TL;DR

This paper extends the concept of linear systems and Weierstrass points to Gorenstein curves, analyzing their behavior under degenerations and establishing a canonical limit cycle associated with these systems.

Contribution

It introduces a generalized framework for linear systems on Gorenstein curves and studies their degenerations, defining an intrinsic Weierstrass cycle that remains canonical.

Findings

Weierstrass cycles degenerate to a subscheme containing an intrinsic component.

The intrinsic subscheme is canonically associated with the generalized linear system.

The limit of Weierstrass divisors depends on the degenerating family of line bundles.

Abstract

Let C be a projective Gorenstein curve over an algebraically closed field of characteristic 0. A generalized linear system on C is a pair (I,f) consisting of a torsion-free, rank-1 sheaf I on C and a map of vector spaces f to the space of global sections of I. If the system is nondegenerate on every irreducible component of C, we associate to it a 0-cycle W, its Weierstrass cycle. Then we show that for each one-parameter family of curves C(t) degenerating to C, and each family of linear systems (L(t),f(t)) along C(t), with L(t) invertible, degenerating to (I,f), the corresponding Weierstrass divisors degenerate to a subscheme whose associated 0-cycle is W. We show that the limit subscheme contains always an "intrinsic" subscheme, canonically associated to (I,f), but the limit itself depends on the family L(t).