Non-commutative desingularizations and partial resolutions

Lieven Lebruyn

We start from the (partial) resolution of singularities

G − Hilb ⟼ ℂ^{n}/G

of the quotient singularity by means of the Hilbert scheme of G-clusters and generalize this to the following setting. Let (Q, R) be a quiver Q with relations R and let α be a dimension vector of a simple representation of (Q, R). We assume that there is a stability structure θ such that the set rep_{α}^{θ}(Q, R) of all θ-semistable representations of (Q, R) is a smooth variety and consider the projective morphism

moduli_{α}^{θ}(Q, R) ⟼ iss_{α}(Q, R)

from the moduli space of θ-semistable representations to the quotient variety of semi-simple representations of (Q, R). In the above case, (Q, α) is the McKay quiver-setting, R the set of commuting matrix-relations and θ a suitable stability structure. We show that there is a sheaf of smooth noncommutative algebras A over moduli_{α}^{θ}(Q, R) such that its non-commutative variety

spec A ⟼ moduli_{α}^{θ}(Q, R) ⟼ iss_{α}(Q, R)

can be viewed as a non-commutative desingularization of iss_{α}(Q, R). However, the central scheme moduli_{α}^{θ}(Q, R) may still have singularities. We classify the types of singularities with can occur for a given dimension n of iss_{α}(Q, R). If n = 2 there are none, if n = 3 there is just one type (the conifold), for n = 4 there are three, for n = 5 ten etc.