On exceptional collections on some log Del Pezzo surfaces with one singular point

TL;DR

This paper constructs full exceptional collections of sheaves on specific log Del Pezzo surfaces viewed as smooth stacks, expanding understanding of their derived categories beyond toric cases.

Contribution

It introduces a method to build full exceptional collections on non-toric log Del Pezzo surfaces embedded as hypersurfaces in weighted projective spaces.

Findings

Constructed full exceptional collections on certain log Del Pezzo surfaces.

Surfaces are embedded as quasi-smooth hypersurfaces in weighted projective spaces.

Surfaces are non-toric, broadening the class of studied examples.

Abstract

In this paper we construct a full exceptional collection of sheaves on some family of log Del Pezzo surfaces, viewed as a smooth stack. These surfaces can be embedded as a quasi-smooth hypersurfaces into a certain weighted projective space, and they are not toric.