Computing $α$-invariants of singular del Pezzo surfaces

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.