A boundedness result for toric log Del Pezzo surfaces

TL;DR

This paper establishes an upper bound on the Picard number of rational surfaces resolving singularities of toric log Del Pezzo surfaces, showing it is a quadratic polynomial in the index.

Contribution

It provides a new quadratic upper bound for the Picard number of such surfaces based on their index, advancing understanding of their geometric properties.

Findings

Upper bound for Picard number is quadratic in the index

Bound applies to rational surfaces resolving singularities

Enhances classification of toric log Del Pezzo surfaces

Abstract

In this paper we give an upper bound for the Picard number of the rational surfaces which resolve minimally the singularities of toric log Del Pezzo surfaces of given index $\ell$. This upper bound turns out to be a quadratic polynomial in the variable $\ell$.