The paper is a generalization of a result of I. Dolgachev, M. Mendes Lopes, and R. Pardini. 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 minimal rational ruled surface ${ \bf F}_2$ or ${ \bf P}^2$ or a fake projective plane. We also describe smooth projective complex surfaces $X$ with $h^{ 1,1}(X)-2$ disjoint $(-2)$-curves.
Categories: math.AG