K3 surfaces and log del Pezzo surfaces of index three

TL;DR

This paper classifies certain log del Pezzo surfaces of index three using automorphism classifications of K3 surfaces, identifying examples with specific properties and non-toric singularities.

Contribution

It provides a partial classification of log del Pezzo surfaces of index three, especially those with the 'Multiple Smooth Divisor Property', using novel methods involving automorphisms and resolutions.

Findings

Classified log del Pezzo surfaces with the 'Multiple Smooth Divisor Property'

Identified several surfaces with non-toric singularities of index three

Developed methods for right resolutions of quotient singularities

Abstract

We use classification of non-symplectic automorphisms of K3 surfaces to obtain a partial classification of log del Pezzo surfaces of index three. We can classify those with "Multiple Smooth Divisor Property", whose definition we will give. Our methods include the definition of right resolutions of quotient singularities of index three and some analysis of automorphism-stable elliptic fibrations on K3 surfaces. In particular we find several log del Pezzo surfaces of Picard number one with non-toric singularities of index three.