Gale duality and the linearisation map for noncommutative crepant resolutions

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.C containing the generic stability parameter θ, so we write it aswhere Θ is a corank-one lattice in K(A) and NS(X) is the Néron-Severi group of X. Note that L C (θ) is the polarising ample line bundle on M θ (A, v) given by the GIT quotient construction.On the other hand, the vertex simple modules S i for i ∈ Q 0 determine objects Ψ(S i ) in the bounded derived category of coherent sheaves with compact support on X; here, Ψ is the tilting equivalence determined by the (dual to the) tautological bundle on M θ (A, v). The numerical Grothendieck group K num c (X) for objects with compact support on X admits a dimension filtration, and we demonstrate that the map Gale dual (see Section 4.1) to the linearisation map L C is a map of the formthat is defined in terms of the support (with multiplicities) of the objects Ψ(S i ) for i ∈ Q 0 .This paper describes the relationship between the line bundles det(T i ) and the objects Ψ(S i ) for i ∈ Q 0 via Gale duality between L C and G C . For a specific GIT chamber C + , these two collections of geometric objects have been encoded before, at least in the toric case, via rival interpretations of the combinatorial cookery known as Reid's recipe [31,13]. Here we reconcile these rival interpretations by demonstrating they are equivalent. Our approach via L C + or G C + allows us to formulate Reid's recipe in the non-toric case, and it suggests that a matrix representing G C + is 'sign-coherent'; this surprising phenomenon would follow from a proof of the Trichotomy Conjecture 1.6. 1.2. The main result and an illuminating example. We now present the results in more detail. We assume for now that A is an NCCR 1 satisfying a technical condition (see Assumption 3.2) which implies in particular that the vertex simple modules exist. In addition, assume that the dimension vector v has a component v 0 = 1, and write 0 ∈ Q 0 for the corresponding vertex.