On the effective, nef, and semi-ample monoids of blowups of Hirzebruch surfaces at collinear points

This paper is devoted to determine the geometry of a class of smooth projective rational surfaces whose minimal models are the Hirzebruch ones; concretely, they are obtained as the blowup of a Hirzebruch surface at collinear points. Explicit descriptions of their effective monoids are given, and we present a decomposition for every effective class. Such decomposition is used to confirm the effectiveness of some divisor classes when the Riemann–Roch theorem does not give information about their effectiveness. Furthermore, we study the nef divisor classes on such surfaces. We provide an explicit description for their nef monoids, and, moreover, we present a decomposition for every nef class. On the other hand, we prove that these surfaces satisfy the anticanonical orthogonal property. As a consequence, the surfaces are Harbourne–Hirschowitz and their Cox rings are finitely generated. Finally, we prove that the complete linear system associated with any nef divisor is base-point-free; thus, the semi-ample and nef monoids coincide. The base field is assumed to be algebraically closed of arbitrary characteristic.