Projective surfaces with many nodes

<!-- *** Custom HTML *** --> We prove that a smooth projective complex surface X, not necessarily minimal, contains h1,1 (X) - 1 disjoint $(-2)$-curves if and only if X is isomorphic to a relatively minimal ruled rational surface F₂ or P² or a fake projective plane. We also describe smooth projective complex surfaces X with h1,1 (X) - 2 disjoint $(-2)$-curves.