Projectivity and effective global generation of determinantal line bundles on quiver moduli

TL;DR

This paper develops a moduli-theoretic approach to quiver representation spaces, establishing their projectivity and global generation properties of associated line bundles without relying on geometric invariant theory.

Contribution

It introduces a new moduli-theoretic framework for quiver representations, proving properness, projectivity, and effective bounds for line bundle global generation.

Findings

Moduli spaces of semistable quiver representations are proper over semisimple moduli.

Constructs a natural determinantal line bundle that is semiample and, for acyclic quivers, ample.

Provides new effective bounds for the global generation of these line bundles.

Abstract

We give a moduli-theoretic treatment of the existence and properties of moduli spaces of semistable quiver representations, avoiding methods from geometric invariant theory. Using the existence criteria of Alper--Halpern-Leistner--Heinloth, we show that for many stability functions, the stack of semistable representations admits an adequate moduli space, and prove that this moduli space is proper over the moduli space of semisimple representations. We construct a natural determinantal line bundle that descends to a semiample line bundle on the moduli space and provide new effective bounds for global generation. For an acyclic quiver, we show that this line bundle is ample, thus giving a modern proof of the fact that the moduli space is projective.