Minimality and mutation-equivalence of polygons

TL;DR

This paper introduces a minimality concept for Fano polygons and provides an algorithm to classify their mutation-equivalence classes, advancing the classification of orbifold del Pezzo surfaces via mirror symmetry.

Contribution

It establishes finiteness results for Fano polygons with fixed singularity content and presents an algorithm for mutation-equivalence classification.

Findings

Finitely many Fano polygons with given singularity content up to mutation.

Algorithm for determining mutation-equivalence classes.

Classification of polygons related to specific surface types.

Abstract

We introduce a concept of minimality for Fano polygons. We show that, up to mutation, there are only finitely many Fano polygons with given singularity content, and give an algorithm to determine the mutation-equivalence classes of such polygons. This is a key step in a program to classify orbifold del Pezzo surfaces using mirror symmetry. As an application, we classify all Fano polygons such that the corresponding toric surface is qG-deformation-equivalent to either (i) a smooth surface; or (ii) a surface with only singularities of type 1/3(1,1).