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Platonic Surfaces

  • Brenda Leticia De La Rosa-Navarro
  • Gioia Failla
  • Juan Bosco Frías-Medina
  • Mustapha LahyaneEmail author
  • Rosanna Utano
Chapter

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.

Notes

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.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Brenda Leticia De La Rosa-Navarro
    • 1
  • Gioia Failla
    • 2
  • Juan Bosco Frías-Medina
    • 3
    • 4
  • Mustapha Lahyane
    • 5
    Email author
  • Rosanna Utano
    • 6
  1. 1.Facultad de CienciasUniversidad Autónoma de Baja CaliforniaEnsenadaMéxico
  2. 2.Università Mediterranea di Reggio CalabriaReggio CalabriaItaly
  3. 3.Instituto de MatemáticasUniversidad Nacional Autónoma de México. Área de la Investigación Científica. Circuito Exterior, Ciudad UniversitariaCoyoacánMexico
  4. 4.Unidad Académica de Matemáticas, Universidad Autónoma de ZacatecasZacatecasMéxico
  5. 5.Instituto de Física y Matemáticas (IFM)Universidad Michoacana de San Nicolás de Hidalgo (UMSNH)MoreliaMéxico
  6. 6.Dipartimento di Scienze Matematiche e Informatiche. Scienze Fisiche e Scienze della TerraUniversità di MessinaMessinaItaly

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