Computing $\alpha$-invariants of singular del Pezzo surfaces

TL;DR

This paper introduces new local inequalities for divisors on surfaces and uses them to compute alpha-invariants of singular del Pezzo surfaces, establishing their Kähler-Einstein property for specific singularity types.

Contribution

It develops novel local inequalities for divisors and applies them to determine alpha-invariants of certain singular del Pezzo surfaces, advancing understanding of their geometric properties.

Findings

Alpha-invariants computed for specific singular del Pezzo surfaces.

Del Pezzo surfaces with certain A-type singularities are proven to be Kähler-Einstein.

New inequalities for divisors on surfaces are established.

Abstract

We prove new local inequality for divisors on surfaces and utilize it to compute $\alpha$-invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type $\mathbb{A}_{1}$, $\mathbb{A}_{2}$, $\mathbb{A}_{3}$, $\mathbb{A}_{4}$, $\mathbb{A}_{5}$ or $\mathbb{A}_{6}$ are K\"ahler-Einstein.