Geometric helices on del Pezzo surfaces from tiltingSeminar Algebraic Geometry (SAG)
by
MPIM, Vivatsgasse, 7 - Seminar Room
Max Planck Institute for Mathematics
Geometric helices on a surface S are sequences of objects in the derived category of coherent sheaves on S that provide a way to describe the derived category of coherent sheaves on the local surface K_S in terms of a quiver with potential. 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.