Linear systems and ramification points on reducible nodal curves

Abstract

In the 80's D. Eisenbud and J. Harris developed the general theory of limit linear series, Invent. math. 85 (1986), in order to understand what happens to linear systems and their ramification points on families of non-singular curves degenerating to curves of compact type. They applied their theory to the study of limits of Weierstrass points, among other endeavours. In one of their articles, Invent. math. 87 (1987), they asked: What are the limits of Weierstrass points in families of curves degenerating to stable curves not of compact type? In this eprint we address this question within a more general framework. More precisely, given a family of linear systems on a family of non-singular curves degenerating to a nodal curve we give a formula for the limit of the associated ramification divisors in terms of certain limits of the family of linear systems. In contrast with the theory of limit linear series of Eisenbud's and Harris', we do not need to blow up the family to swerve the degenerating ramification points away from the nodes of the limit curve. Indeed, we can assign the adequate weight to the limit ramification divisor at any point of the limit curve. In a forthcoming submission we shall deal with the specific question of limits of Weierstrass points, assuming certain generic conditions.