Projective surfaces with many nodes

TL;DR

This paper characterizes complex projective surfaces with many disjoint (-2)-curves, extending previous results and classifying those with the maximum number of such curves.

Contribution

It generalizes prior work by classifying surfaces with a high number of disjoint (-2)-curves, including non-minimal cases, and describes surfaces with slightly fewer such curves.

Findings

Surfaces with h^{1,1}(X)-1 disjoint (-2)-curves are isomorphic to specific minimal rational surfaces or a fake projective plane.

Complete classification of surfaces with h^{1,1}(X)-2 disjoint (-2)-curves.

Extension of Dolgachev, Mendes Lopes, and Pardini's results to non-minimal surfaces.

Abstract

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.