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

Abstract

We investigate the connection between Osserman limit series [Oss] (on curves of pseudocompact type) and Amini-Baker limit linear series [AB15] (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 [CJP15, Theorem 1.1] 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.

Limit linear series and the Amini–Baker construction

“…We thus see that we cannot have a limit g 1 2 with such a multidegree, regardless of any generality hypotheses. Now, since the above argument never used gluing conditions, it follows from Theorem 3.9 of [He17] that we also have that CpX 0 , nq is non-hyperelliptic from the Amini-Baker perspective. More precisely, it follows that there is no Amini-Baker limit g 1 2 on CpX 0 , nq which can come from a divisor supported at integral points of p Γ.…”

“…We show in Proposition 5.9 that for these curves, the construction of our forgetful map will yield an Amini-Baker limit linear series even if our gluing condition is not satisfied. In a companion paper, Xiang He [He17] shows that conversely, an Amini-Baker limit linear series on a curve of pseudocompact type satisfies our generalized vanishing condition, so that on such curves, Amini-Baker limit linear series are equivalent to tuples of linear series which satisfy our generalized vanishing condition. He also examines cases in which the forgetful map is and is not surjective, proving new results on smoothability (and non-smoothability) of Amini-Baker limit linear series, with some applications also to smoothability of tropical linear series.…”