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Let $X$ be a rational surface obtained by blowing up at a configuration$\mathcal{C}$ of infinitely near points over a Hirzebruch surface$\mathbb{F}_\delta$. We prove that there exist two positive integers $a \leq b$such that the cone of curves of $X$ is finite polyhedral and minimallygenerated when $\delta \geq a$, and the Cox ring of $X$ is finitely generatedwhenever $\delta \geq b$. The integers $a$ and $b$ depend only on acombinatorial object (a graph decorated with arrows) representing the stricttransforms of the exceptional divisors, their intersections and those with thefibers and special section of $\mathbb{F}_\delta$.

The cone of curves and the Cox ring of rational surfaces over Hirzebruch surfaces