The aim of this paper is to give a geometric characterization of the finite generation of the Cox rings of anticanonical rational surfaces. This characterization is encoded in the finite generation of the effective monoid. Furthermore, we prove that in the case of a smooth projective rational surface having a negative multiple of its canonical divisor with only two linearly independent global sections (e.g., an elliptic rational surface), the finite generation is equivalent to the fact that there are only a finite number of smooth projective rational curves of self-intersection −1. The ground field is assumed to be algebraically closed of arbitrary characteristic.
This paper is devoted to determine the geometry of a class of smooth projective rational surfaces whose minimal models are the Hirzebruch ones, concretely, are obtained as the blow-up 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 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 to 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.
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