Projective surfaces with many nodes

Abstract

We prove that a smooth projective complex surface $X$, not necessarily minimal, contains $h^{1,1} (X) - 1$ disjoint $(-2)$-curves if and only if $X$ is isomorphic to a relatively minimal ruled rational surface $\mathbf{F}_2$ or $\mathbf{P}^2$ or a fake projective plane.

We also describe smooth projective complex surfaces $X$ with $h^{1,1} (X) - 2$ disjoint $(-2)$-curves.