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On the effective, nef, and semi-ample monoids of blowups of Hirzebruch surfaces at collinear points
Part of:
Computational aspects in algebraic geometry
Cycles and subschemes
Surfaces and higher-dimensional varieties
Published online by Cambridge University Press: 11 April 2023
Abstract
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.
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Footnotes
Dedicated to Professor Brian Harbourne on the occasion of his 65th birthday
Juan Bosco Frías-Medina is supported by the “Programa de Estancias Posdoctorales por México Convocatoria 2022 de CONACYT,” and Mustapha Lahyane acknowledges a partial support from the Coordinación de la Investigación Científica de la Universidad Michoacana de San Nicolás de Hidalgo (UMSNH) during 2022.
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