Geometric helices on del Pezzo surfaces from tilting

TL;DR

This paper demonstrates that all geometric helices on del Pezzo surfaces are interconnected through elementary operations, linking non-commutative resolutions via mutations and interpreting tilting as cluster transformations.

Contribution

It establishes a comprehensive framework connecting geometric helices, tilting operations, and cluster transformations on del Pezzo surfaces.

Findings

All geometric helices are related by elementary operations.

Non-commutative crepant resolutions are connected by mutations.

Tilting operations correspond to cluster transformations.

Abstract

We prove that all geometric helices in the derived category of coherent sheaves on a del Pezzo surface are related by a sequence of elementary operations: rotation, shifting, orthogonal reordering, tensoring by a line bundle, and tilting. As a consequence, any two non-commutative crepant resolutions of the affine cone over a del Pezzo surface are related by mutations. The proof relies on a geometric interpretation of tilting operations as cluster transformations acting on toric models of a log Calabi--Yau surface mirror to the del Pezzo surface.