Gale duality and the linearisation map for noncommutative crepant resolutions

TL;DR

This paper explores the geometric relationship between Gale duality and the linearisation map in the context of noncommutative crepant resolutions, providing new insights into Reid's recipe and related conjectures.

Contribution

It introduces a geometric interpretation of Gale duality for quiver moduli spaces and formulates Reid's recipe using integer matrices, revealing sign-coherence properties.

Findings

Gale duality map described in geometric terms for noncommutative crepant resolutions

Reid's recipe reformulated via integer-valued matrices with sign-coherence

Evidence supporting a conjecture linking relations among tautological bundles and derived equivalences

Abstract

We describe in geometric terms the map that is Gale dual to the linearisation map for quiver moduli spaces associated to noncommutative crepant resolutions in dimension three. This allows us to formulate Reid's recipe in this context in terms of a pair of integer-valued matrices, one of which appears to satisfy an attractive sign-coherence property. We provide some new evidence for a conjecture, known to hold in the toric case, which implies that a minimal generating set of relations between the determinants of the tautological bundles encodes the supports of the images of the vertex simples under the derived equivalence, and vice-versa.