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

Abstract Let X be a rational surface obtained by blowing up at a configuration $$\mathscr {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> of infinitely near points over a Hirzebruch surface $$\mathbb {F}_\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>δ</mml:mi> </mml:msub> </mml:math> . We prove that there exist two positive integers $$a \le b$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> </mml:math> such that the cone of curves of X is finite polyhedral and minimally generated when $$\delta \ge a$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>≥</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> </mml:math> , and the Cox ring of X is finitely generated whenever $$\delta \ge b$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>≥</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> </mml:math> . 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 $$\mathbb {F}_\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>δ</mml:mi> </mml:msub> </mml:math> .