Smoothing of Limit Linear Series on Curves and Metrized Complexes of Pseudocompact Type

Abstract

AbstractWe 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 with a certain regular smoothing family, and give a new proof of a result of Cartright, Jensen, and Payne for vertex-avoiding divisors, and generalize it 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.