Full exceptional collections for anticanonical log del Pezzo surfaces

TL;DR

This paper constructs explicit full exceptional collections for certain log del Pezzo surfaces with cyclic quotient singularities, linking homological mirror symmetry, McKay correspondence, and classifications of degree two del Pezzo surfaces.

Contribution

It provides a new explicit construction of full exceptional collections for canonical stacks of log del Pezzo surfaces with non-Gorenstein singularities, using McKay correspondence and minimal resolutions.

Findings

Constructed explicit full exceptional collections for the surfaces.

Classified degree two del Pezzo surfaces with generalized Eckardt points.

Extended the adjoints of the Ishii-Ueda functor to certain subgroup actions.

Abstract

Motivated by homological mirror symmetry, this paper constructs explicit full exceptional collections for the canonical stacks associated with the series of log del Pezzo surfaces constructed by Johnson and Koll\'ar. These surfaces have cyclic quotient, non-Gorenstein, singularities. The construction involves both the $\mathrm{GL}(2,\mathbb{C})$ McKay correspondence, and the study of the minimal resolutions of the surfaces, which are birational to degree two del Pezzo surfaces. We show that a degree two del Pezzo surface arises in this way if and only if it admits a generalized Eckardt point, and in the course of the paper we classify the blow-ups of $\mathbb{P}^2$ giving rise to them. Our result on the adjoints of the functor of Ishii-Ueda applies to any finite small subgroup of $\mathrm{GL}(2,\mathbb{C})$.