Quiver representations arising from degenerations of linear series, I

TL;DR

This paper characterizes when certain quiver representations, especially those from degenerations of linear series, are semisimple, using local criteria within semisimple Abelian categories, to understand schematic limits of divisor families.

Contribution

It provides a local criterion for semisimplicity of quiver representations arising from degenerations of linear series, linking geometric degenerations to quiver Grassmannians.

Findings

Characterization of semisimple quiver representations in Abelian categories.

Application to degenerations of linear series on projective varieties.

Framework for describing schematic limits via quiver Grassmannians.

Abstract

We give a local characterization for when certain quiver representations in semisimple Abelian categories are semisimple, among them those arising from degenerations of linear series. This paper is the first of two, aimed to describe all the schematic limits of families of divisors associated to a given family of 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 $X$, by means of certain quiver Grassmannians.