Smoothable del Pezzo surfaces with quotient singularities

TL;DR

This paper classifies del Pezzo surfaces with quotient singularities and Picard rank 1 that can be smoothed via Q-Gorenstein deformations, identifying infinite toric families and sporadic cases.

Contribution

It provides a complete classification of such surfaces, linking them to solutions of Markov-type equations and describing their deformation relationships.

Findings

14 infinite toric families identified

Surfaces correspond to solutions of Markov-type equations

Finite list of sporadic surfaces included

Abstract

We give a complete classification of del Pezzo surfaces with quotient singularities and Picard rank 1 which admit a Q-Gorenstein smoothing. There are 14 infinite families of toric examples. The surfaces in each family correspond to solutions of a Markov-type equation. The remaining surfaces are obtained as deformations of the toric surfaces or belong to a finite list of sporadic surfaces.