Generalized Linear Systems on Curves and Their Weierstrass Points

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, ε) consisting of a torsion-free, rank-1 sheaf ℐ on C, and a map of vector spaces ε: V → Γ(C, 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 , ε t ) along C t , with ℒ t invertible, degenerating to (ℐ, ε), 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, ε), but the limit itself depends on the family ℒ t .