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Platonic Surfaces

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Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

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Abstract

We define the notion of Platonic surfaces. These are anticanonical smooth projective rational surfaces defined over any fixed algebraically closed field of arbitrary characteristic and having the projective plane as a minimal model with very nice geometric properties. We prove that their Cox rings are finitely generated. In particular, they are extremal and their effective monoids are finitely generated. Thus, these Platonic surfaces are built from points of the projective plane which are in good position. It is worth noting that not only their Picard number may be big but also an anticanonical divisor may have a very large number of irreducible components.

Dedicated to Professor Antonio Campillo-López ( http://www.singacom.uva.es/campillo/index.html ) on the occasion of his 65th birthday

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Acknowledgements

We are very grateful to the anonymous Referees for their comments and suggestions regarding this work. This research paper was partially supported by a grant from the group GNSAGA of INdAM, and another one from the Coordinación de Investigación Científica de la Universidad Michoacana de San Nicolás de Hidalgo during 2017 and 2018. De La Rosa-Navarro was supported by “Programa para el Desarrollo Profesional Docente, para el Tipo Superior” under the Grant Number UABC-PTC-558. Frías-Medina acknowledges the financial support of “Fondo Institucional de Fomento Regional para el Desarrollo Científico, Tecnológico y de Innovación”, FORDECYT 265667, during 2017.

Author information

Authors and Affiliations

  1. Facultad de Ciencias, Universidad Autónoma de Baja California, Ensenada, Baja California, México

    Brenda Leticia De La Rosa-Navarro

  2. Università Mediterranea di Reggio Calabria, Reggio Calabria, Italy

    Gioia Failla

  3. Instituto de Matemáticas, Universidad Nacional Autónoma de México. Área de la Investigación Científica. Circuito Exterior, Ciudad Universitaria, Coyoacán, Ciudad de México, Mexico

    Juan Bosco Frías-Medina

  4. Unidad Académica de Matemáticas, Universidad Autónoma de Zacatecas, Zacatecas, Zac, México

    Juan Bosco Frías-Medina

  5. Instituto de Física y Matemáticas (IFM), Universidad Michoacana de San Nicolás de Hidalgo (UMSNH), Morelia, Michoacán, México

    Mustapha Lahyane

  6. Dipartimento di Scienze Matematiche e Informatiche. Scienze Fisiche e Scienze della Terra, Università di Messina, Messina, Italy

    Rosanna Utano

Authors
  1. Brenda Leticia De La Rosa-Navarro
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  2. Gioia Failla
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  3. Juan Bosco Frías-Medina
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  4. Mustapha Lahyane
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  5. Rosanna Utano
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Corresponding author

Correspondence to Mustapha Lahyane .

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Editors and Affiliations

  1. Fachbereich Mathematik, Universität Kaiserslautern, Kaiserslautern, Germany

    Gert-Martin Greuel

  2. Departamento de Álgebra, Instituto de Matemáticas (IMUS), Universidad de Sevilla, Sevilla, Spain

    Luis Narváez Macarro

  3. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain

    Sebastià Xambó-Descamps

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De La Rosa-Navarro, B.L., Failla, G., Frías-Medina, J.B., Lahyane, M., Utano, R. (2018). Platonic Surfaces. In: Greuel, GM., Narváez Macarro, L., Xambó-Descamps, S. (eds) Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics. Springer, Cham. https://doi.org/10.1007/978-3-319-96827-8_12

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