Rank loci in representation spaces of quivers

TL;DR

This paper studies rank functions on quiver representations, showing that rank loci are constructible subvarieties in various representation spaces despite the lack of semicontinuity.

Contribution

It establishes the geometric property that rank loci in quiver representation spaces are constructible subvarieties, extending understanding of rank functions beyond classical semicontinuity.

Findings

Rank loci are constructible subvarieties in representation spaces.

This property holds for loci in Schofield's subrepresentation bundles and quiver Grassmannians.

Rank functions in quivers, though not semicontinuous, still define well-behaved geometric loci.

Abstract

Rank functors on a quiver $Q$ are certain additive functors from the category of representations of $Q$ to the category of finite-dimensional vector spaces. Composing with the dimension function on vector spaces gives a rank function on $Q$. These induce functions on $\rep(Q, \alpha)$, the variety of representations of $Q$ of dimension vector $\alpha$, and thus can be used to define "rank loci" in $\rep(Q, \alpha)$ as collections of points satisfying finite lists of linear inequalities of rank functions. Although quiver rank functions are not generally semicontinuous like the rank of a linear map, we show here that they do have the geometric property that these rank loci are constructible subvarieties. The same is true for loci defined by rank functions in Schofield's subrepresentation bundles on $\rep(Q, \alpha)$, and in quiver Grassmannians.