Extended Klein model and a bound on curves with negative self-intersection

TL;DR

This paper introduces an extended Klein model approach to establish an exponential bound on collections of negative self-intersection curves on algebraic surfaces, advancing understanding of their geometric constraints.

Contribution

It provides a novel proof technique using an extended Klein disc model to bound negative self-intersection curves on surfaces, improving previous bounds.

Findings

Bound on negative self-intersection curves is exponential in Picard number

Extended Klein model effectively analyzes hyperbolic geometric properties

New proof technique simplifies bounding arguments for algebraic surfaces

Abstract

Let $S$ be an irreducible smooth projective surface and $\mathcal{F}$ a collection of curves with negative self-intersection on $S$ such that no positive combination $aC_1 + bC_2$ is connected nef. In this paper, we provide an alternate proof that $\mathcal{F}$ is bounded by an exponential function of the Picard number $\rho(S)$ of $S$ using an extended version of the Klein disc model for hyperbolic space.