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

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.