Quiver representations arising from degenerations of linear series, II

TL;DR

This paper characterizes all schematic limits of divisors from degenerating linear series on families of projective varieties, using quiver representations and Grassmannians, providing a detailed geometric description of degenerations.

Contribution

It introduces a framework using $ ext{Z}^n$-quivers and linked nets to describe degenerations of linear series and proves properties of associated quiver Grassmannians and their relation to Hilbert schemes.

Findings

The quiver Grassmannian $ ext{LP}(rak V)$ is a local complete intersection, reduced, and pure of dimension $r$.

There is a morphism from $ ext{LP}(rak V)$ to the Hilbert scheme $ ext{Hilb}_X$.

The image of this morphism parameterizes all schematic limits of divisors in the degenerating family.

Abstract

We describe all the schematic limits of families of divisors associated to a given family of rank-$r$ linear series on a one-dimensional family of projective varieties degenerating to a connected reduced projective scheme $X$ defined over any field, under the assumption that the total space of the family is regular along $X$. More precisely, the degenerating family gives rise to a special quiver $Q$, called a \emph{$\mathbb{Z}^n$-quiver}, a special representation $\mathfrak L$ of $Q$ in the category of line bundles over $X$, called a \emph{maximal exact linked net}, and a special subrepresentation $\mathfrak V$ of the representation $H^0(X,\mathfrak L)$ induced from $\mathfrak L$ by taking global sections, called a \emph{pure exact finitely generated linked net} of dimension $r+1$. Given $\mathfrak g=(Q,\mathfrak L,\mathfrak V)$ satisfying these properties, we prove that the quiver Grassmanian $\mathbb{LP}(\mathfrak{V})$ of subrepresentations of $\mathfrak{V}$ of pure dimension 1, called a \emph{linked projective space}, is local complete intersection, reduced and of pure dimension $r$. Furthermore, we prove that there is a morphism $\mathbb{LP}(\mathfrak{V})\to\text{Hilb}_X$, and that its image parameterizes all the schematic limits of divisors along the degenerating family of linear series if $\mathfrak g$ arises from one.