Del Pezzo surfaces with a single 1/k(1,1) singularity

TL;DR

This paper studies del Pezzo surfaces with a specific type of singularity, showing they form a unified deformation family, constructing models and degenerations, and exploring their mirror symmetry properties.

Contribution

It introduces a unified cascade for del Pezzo surfaces with 1/k(1,1) singularities, constructs models and toric degenerations, and classifies certain cases with toric degenerations.

Findings

Surfaces with 1/k(1,1) singularities form a single deformation cascade.

Constructed models and toric degenerations in low codimension.

Classified surfaces with specific singularities admitting toric degenerations.

Abstract

Inspired by the recent progress by Coates-Corti-Kasprzyk et al. on Mirror Symmetry for del Pezzo surfaces, we show that for any positive integer k the deformation families of del Pezzo surfaces with a single 1/k(1,1) singularity (and no other singular points) fit into a single cascade. Additionally we construct models and toric degenerations of these surfaces embedded in toric varieties in codimension less than or equal two. Several of these directly generalise constructions of Reid-Suzuki (in the case k=3). We identify a root system in the Picard lattice, and in light of the work of Gross-Hacking-Keel, comment on Mirror Symmetry for each of these surfaces. Finally we classify all del Pezzo surfaces with certain combinations of 1/k(1,1) singularities for k=3,5,6 which admit a toric degeneration.