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Let $S$ be an irreducible smooth projective surface and $\mathcal{F}$ acollection of curves with negative self-intersection on $S$ such that nopositive combination $aC_1 + bC_2$ is connected nef. In this paper, we providean alternate proof that $\mathcal{F}$ is bounded by an exponential function ofthe Picard number $\rho(S)$ of $S$ using an extended version of the Klein discmodel for hyperbolic space.

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