Quivers of sections on toric orbifolds

TL;DR

This paper extends classical linear series to toric orbifolds using quivers and moduli stacks, enabling new geometric insights and describing GIT wall crossings for certain orbifold resolutions.

Contribution

It introduces a stacky analogue of linear series on toric orbifolds via refined quiver representations and constructs associated moduli stacks, extending classical concepts.

Findings

Constructed moduli stacks of refined quiver representations for line bundles on toric orbifolds.

Defined a notion of stacky linear series and associated maps to ambient stacks.

Described GIT wall crossings between quotient stacks and G-Hilb for abelian groups in low dimensions.

Abstract

Starting from a collection of line bundles on a projective toric orbifold X, we introduce a stacky analogue of the classical linear series. Our first main result extends work of King by building moduli stacks of refined representations of labelled quivers. We associate one such stack to any collection of line bundles on X to obtain our notion of a stacky linear series; as in the classical case, X maps to the ambient stack by evaluating sections of line bundles in the collection. As a further application, we describe a finite sequence of GIT wall crossings between [A^n/G] and G-Hilb(A^n) for a finite abelian subgroup G of SL(n,k) where n is less than or equal to 3.