Let X be a rational surface obtained by blowing up at a configuration \(\mathscr {C}\) 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 \(\mathbb {F}_\delta \) .
