Rational surfaces with finitely generated Cox rings and very high Picard numbers

In this paper, we provide new families of smooth projective rational surfaces whose Cox rings are finitely generated. These surfaces are constructed by blowing-up points in Hirzebruch surfaces and may have very high Picard numbers. Such construction is not straightforward, and we achieve our results using the facts that these surfaces are extremal, and their effective monoids are finitely generated. Furthermore, we give an example illustrating the existence of rational surfaces which are not extremal. The base field of our varieties is assumed to be algebraically closed of arbitrary characteristic.