NCCRs of cones over del Pezzo surfaces

TL;DR

This paper classifies non-commutative crepant resolutions of anticanonical cones over del Pezzo surfaces and proves their interconnectedness via mutations, extending known results beyond terminal singularities.

Contribution

It provides a classification of NCCRs for these cones and demonstrates their connectivity through mutations, using geometric helices and polygons associated with exceptional collections.

Findings

Classified NCCRs of anticanonical del Pezzo cones.

Proved all NCCRs are connected by mutations.

Linked polygons to exceptional collections on del Pezzo surfaces.

Abstract

Non-commutative crepant resolutions (NCCRs) are non-commutative versions of classical crepant resolutions in algebraic geometry. For 3-dimensional terminal Gorenstein singularities Iyama and Wemyss proved that all NCCRs are connected by mutations, which may be viewed as a non-commutative analogue of Kawamata's result that all crepant resolutions are connected by flops. In this paper we prove the corresponding result for a class of canonical Gorenstein singularities which are not terminal, namely anticanonical cones over del Pezzo surfaces. More precisely, we first obtain a classification of NCCRs of anticanonical del Pezzo cones, showing that every NCCR arises from a geometric helix on the corresponding del Pezzo surface. We then prove that all such geometric helices are connected to each other by mutations, up to simple operations which include tensoring by line bundles and shifts. A crucial ingredient in our proofs is the polygons that can be associated to exceptional collections on del Pezzo surfaces following the works of Hille and Perling. We obtain some interesting observations about these polygons which may be of independent interest.