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

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 $a\le b$ such that the cone of curves of $X$ is finite polyhedral and minimally generated when $\delta \ge a$, and the Cox ring of $X$ is finitely generated whenever $\delta \ge b$. The integers $a$ and $b$ depend only on a combinatorial object (a graph decorated with arrows) representing the strict transforms of the exceptional divisors, their intersections and those with the fibers and special section of ${\mathbb{F}}_{\delta }$.