Limit canonical series

TL;DR

This paper studies the behavior of canonical series limits on degenerating families of curves, introducing a fan structure on the dual graph's edge lengths to parametrize these limits across various topologies.

Contribution

It extends existing theories by defining a fan structure and constructing a projective variety that parametrizes limits of canonical series for all topologies of degenerating curves.

Findings

Developed a fan structure on the dual graph's edge lengths.

Constructed a projective variety parametrizing limit canonical series.

Extended previous work to all topologies of degenerating curves.

Abstract

We describe the limits of canonical series along families of curves degenerating to a nodal curve which is general for its topology, in the weak sense that the branches over nodes on each of its components are in general position. We define a fan structure on the space of edge lengths on the dual graph of the limit curve, and construct a projective variety parametrizing the limits, organized in strata associated to the cones of this fan. This extends to all topologies the works by Eisenbud-Harris (Invent. Math. 87: 496-515, 1987) on curves of compact type and Esteves-Medeiros (Invent. Math. 149: 267-338, 2002) on two-component curves.