Level-$\delta$ limit linear series

TL;DR

This paper introduces level-$oldsymbol{\delta}$ limit linear series, a new framework for understanding degenerations of linear series on curves, constructing moduli spaces, and describing divisor limits on singular degenerations.

Contribution

It defines level-$oldsymbol{\delta}$ limit linear series, constructs their moduli spaces as new compactifications, and generalizes divisor limit descriptions for degenerations to singular curves.

Findings

Constructed projective moduli spaces $G^r_{d,oldsymbol{\delta}}(X)$ for level-$oldsymbol{\delta}$ limit linear series.

Showed that these spaces compactify Osserman's limit linear series spaces.

Described limits of divisors on degenerating families of smooth curves to singular curves.

Abstract

We introduce the notion of level-$\delta$ limit linear series, which describe limits of linear series along families of smooth curves degenerating to a singular curve $X$. We treat here only the simplest case where $X$ is the union of two smooth components meeting transversely at a point $P$. The integer $\delta$ stands for the singularity degree of the total space of the degeneration at $P$. If the total space is regular, we get level-1 limit linear series, which are precisely those introduced by Osserman in 2006. We construct a projective moduli space $G^r_{d,\delta}(X)$ parameterizing level-$\delta$ limit linear series of rank $r$ and degree $d$ on $X$, and show that it is a new compactification, for each $\delta$, of the moduli space of Osserman exact limit linear series, an open subscheme $G^{r,*}_{d,1}(X)$ of the space $G^r_{d,1}(X)$ already constructed by Osserman. Finally, we generalize work by Esteves and Osserman by associating to each exact level-$\delta$ limit linear series $\mathfrak g$ on $X$ a closed subscheme $\mathbb P(\mathfrak g)\subseteq X^{(d)}$ of the $d$th symmetric product of $X$, and showing that $\mathbb P(\mathfrak g)$ is the limit of the spaces of divisors associated to linear series on smooth curves degenerating to $\mathfrak g$ on $X$, if such degenerations exist. In particular, we describe completely limits of divisors along degenerations to such a curve $X$.