Negative curves of small genus on surfaces

Abstract

Let $X$ be an irreducible smooth geometrically integral projective surface over a field. In this paper we give an effective bound in terms of the Neron--Severi rank $ρ(X)$ of $X$ for the number of irreducible curves $C$ on $X$ with negative self-intersection and geometric genus less than $b_1(X)/4$, where $b_1(X)$ is the first étale Betti number of $X$. The proof involves a hyperbolic analog of the theory of spherical codes. More specifically, we relate these curves to the hyperbolic kissing number, and then prove upper and lower bounds for the hyperbolic kissing number in terms of the classical Euclidean kissing number.

v4 Significant rewrite of the previous version with new more general results and strengthening of previous results; v5 Added several new results; v6 Complete overhaul from previous versions; v7 Final version to appear in Mathematische Zeitschrift