Limit linear series for curves not of compact type

TL;DR

This paper extends the theory of limit linear series to certain non-compact type curves, providing a new moduli space construction, a specialization result, and a smoothing theorem that improves existing results.

Contribution

It introduces a new notion of limit linear series for non-compact type curves, along with a moduli space construction and a smoothing theorem that generalizes and improves prior results.

Findings

Constructed a moduli space of limit linear series for non-compact type curves.

Proved a specialization result for these limit linear series.

Established a smoothing theorem that surpasses previous results even for compact type curves.

Abstract

We introduce a notion of limit linear series for nodal curves which are not of compact type. We give a construction of a moduli space of limit linear series, which works also in smoothing families, and we prove a corresponding specialization result. For a more restricted class of curves which simultaneously generalizes two-component curves and curves of compact type, we give an equivalent definition of limit linear series, which is visibly a generalization of the Eisenbud-Harris definition. Finally, for the same class of curves, we prove a smoothing theorem which constitutes an improvement over known results even in the compact-type case.