JournalsrmiVol. 31, No. 4pp. 1131–1140

A geometric criterion for the finite generation of the Cox rings of projective surfaces

  • Brenda Leticia De La Rosa Navarro

    Universidad Autónoma de Baja California, Ensenada, Mexico

  • Juan Bosco Frías Medina

    Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico

  • Mustapha Lahyane

    Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico

  • Israel Moreno Mejía

    Universidad Nacional Autónoma de México, México, D.F., Mexico

  • Osvaldo Osuna Castro

    Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico
A geometric criterion for the finite generation of the Cox rings of projective surfaces cover
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Abstract

The aim of this paper is to give a geometric characterization of the finite generation of the Cox rings of anticanonical rational surfaces. This characterization is encoded in the finite generation of the effective monoid. Furthermore, we prove that in the case of a smooth projective rational surface having a negative multiple of its canonical divisor with only two linearly independent global sections (e.g., an elliptic rational surface), the finite generation is equivalent to the fact that there are only a finite number of smooth projective rational curves of self-intersection −1. The ground field is assumed to be algebraically closed of arbitrary characteristic.

A correction to this paper is available.

Cite this article

Brenda Leticia De La Rosa Navarro, Juan Bosco Frías Medina, Mustapha Lahyane, Israel Moreno Mejía, Osvaldo Osuna Castro, A geometric criterion for the finite generation of the Cox rings of projective surfaces. Rev. Mat. Iberoam. 31 (2015), no. 4, pp. 1131–1140

DOI 10.4171/RMI/878