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

Part of: Cycles and subschemes Computational aspects in algebraic geometry Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  11 April 2023

Brenda Leticia de la Rosa-Navarro Open the ORCID record for Brenda Leticia de la Rosa-Navarro [Opens in a new window] ,
Juan Bosco Frías-Medina Open the ORCID record for Juan Bosco Frías-Medina [Opens in a new window]  and
Mustapha Lahyane Open the ORCID record for Mustapha Lahyane [Opens in a new window]
Brenda Leticia de la Rosa-Navarro*
Affiliation:
Facultad de Ciencias, Universidad Autónoma de Baja California, Ensenada, Mexico
Juan Bosco Frías-Medina
Affiliation:
Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico e-mail:[email protected] [email protected]
Mustapha Lahyane
Affiliation:
Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico e-mail:[email protected] [email protected]
*
e-mail: [email protected]
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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.

Keywords

Cox ringsrational surfaceseffective monoidHirzebruch surfaces

MSC classification

Primary: 14J26: Rational and ruled surfaces 14C20: Divisors, linear systems, invertible sheaves 14C22: Picard groups
Secondary: 14C17: Intersection theory, characteristic classes, intersection multiplicities 14Q20: Effectivity, complexity
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 4 , December 2023 , pp. 1179 - 1193
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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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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