nLab hyperbolic curves

Let $k$ be an algebraically closed field. Let $X$ be a smooth one-dimensional variety over $k$. Such a curve has a unique projective model $\overline X$, and $X=\overline X\smallsetminus \{x_1,\dots,x_r\}$ for some finite collection $x_1,\dots,x_r$ of points in $\overline X$. Let $g$ be the genus of $\overline X$. We say that $X$ is hyperbolic if $2 g-2+r \gt 0$¹.

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1. Mochizuki, Shinichi. Correspondences on hyperbolic curves ↩