Bounding the volumes of singular weak log del Pezzo surfaces

TL;DR

This paper establishes an optimal upper bound for the anti-canonical volume of epsilon-log canonical weak log del Pezzo surfaces, exploring their geometric properties and providing explicit examples.

Contribution

It introduces the first sharp volume bound for these surfaces and analyzes the relationship with their Picard number, including explicit constructions.

Findings

Optimal upper bound for anti-canonical volume established

Relation between volume bound and Picard number analyzed

Examples of smooth weak log del Pezzo surfaces constructed

Abstract

We give an optimal upper bound for the anti-canonical volume of an $\epsilon$-lc weak log del Pezzo surface. Moreover, we consider the relation between the bound of the volume and the Picard number of the minimal resolution of the surface. Furthermore we consider blowing up several points on a Hirzebruch surface in general position and give some examples of smooth weak log del Pezzo surfaces.