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 Néron–Severi rank of X for the number of irreducible curves C on X with negative self-intersection and geometric genus less than $$b_1(X)/4$$ b 1 ( X ) / 4 , where $$b_1(X)$$ 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.

中文翻译：

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曲面上小属的负曲线

设 X 是场上一个光滑的几何不可约射影曲面。在本文中，我们根据 X 的 Néron-Severi 秩为 X 上具有负自交和几何属小于 $$b_1(X)/4$$ b 1 的不可约曲线 C 的数量给出了一个有效的上限( X ) / 4 ，其中 $$b_1(X)$$ b 1 ( X ) 是 X 的第一个 étale Betti 数。证明涉及球形码理论的双曲线模拟。更具体地说，我们将这些曲线与双曲接吻数联系起来，然后根据经典欧几里得接吻数证明双曲接吻数的上下界。