Classification of Minimal Polygons with Specified Singularity Content

TL;DR

This paper introduces an efficient algorithm for classifying minimal polygons with specified singularity content, demonstrating its effectiveness through examples involving Fano polygons and their mutation-equivalence classes.

Contribution

The paper presents a novel algorithm for classifying minimal polygons with given singularity content and applies it to classify Fano polygons with fixed singularity baskets.

Findings

Successfully classified all mutation-equivalence classes of Fano polygons in two examples.

Demonstrated the efficiency of the new classification algorithm.

Confirmed finiteness of classes with given singularity content.

Abstract

It is known that there are only finitely many mutation-equivalence classes with a given singularity content, and each of these equivalence classes contains only finitely many minimal polygons. We describe an efficient algorithm to classify these minimal polygons. To illustrate this algorithm we compute all mutation-equivalence classes of Fano polygons with a fixed basket of singularities in two examples.