Restrictions on the Singularity Content of a Fano Polygon

TL;DR

This paper investigates the limitations on the singularity configurations of Fano polygons, establishing bounds and showing certain singularity baskets cannot occur, thus advancing understanding of orbifold del Pezzo surfaces.

Contribution

It provides new bounds on the number of specific singularities in Fano polygons and rules out certain singularity baskets, enhancing classification efforts.

Findings

No Fano polygons without T-singularities exist with certain baskets.

Established upper bounds on the number of 1/R(1,1) singularities.

Certain singularity configurations are proven impossible for Fano polygons.

Abstract

We determine restrictions on the singularity content of a Fano polygon, or equivalently of certain orbifold del Pezzo surfaces. We establish bounds on the maximum number of 1/R(1,1) singularities in the basket of residual singularities. In particular, there are no Fano polygons without T-singularities and with a basket given by (i) {k x 1/R(1,1)} where k is a positive integer and R>4, or (ii) {1/R1(1,1), 1/R2(1,1), 1/R3(1,1)}.