Abstract
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.
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The authors are grateful to the referees for their helpful suggestions that improved highly the style of the paper.
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Tendrement, à la mémoire de Gang XIAO. http://wims.unice.fr/xiao/xiao.html .
Research partially supported during 2015 and 2016 by the Coordinación de la Investigación Científica de la Universidad Michoacana de San Nicolás de Hidalgo, Morelia, México. The third author thanks Dr. Ricardo Becerril Bárcenas, the director of the Institute of Physics and Mathematics (IFM), as well as Dr. Medardo Serna González, the rector (chancellor) of the University of Michoacán (UMSNH) for a sabbatical leave. The second author acknowledges the financial support of Consejo Nacional de Ciencia y Tecnología under the Grant Number 339809. The first author gratefully thanks Dr. Juan Tapia Mercado, the director of the Faculty of Sciences of the Autonomous University of Baja California at Campus Ensenada, for giving her a leave permission in order to finish this collaboration at IFM-UMSNH.
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De La Rosa-Navarro, B.L., Frías-Medina, J.B. & Lahyane, M. Rational surfaces with finitely generated Cox rings and very high Picard numbers. RACSAM 111, 297–306 (2017). https://doi.org/10.1007/s13398-016-0296-0
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DOI: https://doi.org/10.1007/s13398-016-0296-0
Keywords
- Cox rings
- Rational surfaces
- Anticanonical Iitaka dimension
- Effective monoid
- Geometrically ruled surfaces
- Blowing-up