Geometric helices on del Pezzo surfaces from tilting

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.