Smoothing of limit linear series on curves and metrized complexes of pseudocompact type

TL;DR

This paper explores the relationship between different notions of limit linear series on curves and metrized complexes, establishing conditions for smoothability and applying these results to divisor lifting and generalizations.

Contribution

It introduces a notion of pre-limit linear series, connects Osserman and Amini-Baker limit series, and proves smoothability is equivalent to a weak glueing condition under certain circumstances.

Findings

Smoothability is equivalent to a weak glueing condition for certain limit series.

The weak glueing condition is necessary for smoothability in general.

Confirmed lifting property of specific divisors and generalized previous results for divisors of rank one.

Abstract

We investigate the connection between Osserman limit series (on curves of pseudocompact type) and Amini-Baker limit linear series (on metrized complexes with corresponding underlying curve) via a notion of pre-limit linear series on curves of the same type. Then, applying the smoothing theorems of Osserman limit linear series, we deduce that, fixing certain metrized complexes, or for certain types of Amini-Baker limit linear series, the smoothability is equivalent to a certain "weak glueing condition". Also for arbitrary metrized complexes of pseudocompact type the weak glueing condition (when it applies) is necessary for smoothability. As an application we confirm the lifting property of specific divisors on the metric graph associated to a certain regular smoothing family, and give a new proof of the main theorem in the paper of Cartwright, Jensen and Payne for vertex avoiding divisors, and generalize loc.cit. for divisors of rank one in the sense that, for the metric graph, there could be at most three edges (instead of two) between any pair of adjacent vertices.