Séminaires : Séminaire d'Algèbre

Non-commutative crepant resolutions (NCCRs) are a natural analogue of classical crepant resolutions in algebraic geometry. As in the commutative setting, NCCRs are far from unique. However, by a result of Kawamata, all commutative crepant resolutions are connected by flops. On the non-commutative side, Iyama and Wemyss proved that for three-dimensional terminal Gorenstein singularities all NCCRs are connected by mutations, which may be viewed as non-commutative counterparts of flops.

We prove an analogous result for a class of canonical Gorenstein singularities which are not terminal, namely the anticanonical affine cones over del Pezzo surfaces. We also obtain a classification of such NCCRs, establishing a correspondence between NCCRs and certain full exceptional collections on del Pezzo surfaces. In this setting, mutations of NCCRs have several incarnations: as mutations of quivers with potential, as sequences of mutations of exceptional collections, and as mutations of convex polygons in a two-dimensional lattice, following Hille and Perling. All of these interpretations are used in the proof, which I will try to sketch.

The talk is based on joint work with Michel Van den Bergh. It will take place in hybrid mode at the Institut Henri Poincaré.