Negative curves of small genus on surfaces

Abstract

Let X be a smooth geometrically irreducible projective surface over a field. In this paper we give an effective upper bound in terms of the Neron–Severi rank 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 etale 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.