Negative curves of small genus on surfaces

TL;DR

This paper establishes an effective bound on the number of certain negative self-intersection curves on algebraic surfaces, linking algebraic geometry with hyperbolic geometry and spherical codes.

Contribution

It introduces a novel bound involving the Neron--Severi rank and relates negative curves to hyperbolic kissing numbers, bridging algebraic geometry and hyperbolic geometry.

Findings

Provides an explicit bound based on the Neron--Severi rank

Relates negative curves to hyperbolic kissing number

Establishes bounds connecting hyperbolic and Euclidean kissing numbers

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 $\rho(X)$ 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 \'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.