Revista Matemática Iberoamericana


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Volume 31, Issue 4, 2015, pp. 1131–1140
DOI: 10.4171/RMI/878

Published online: 2015-12-23

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

Brenda Leticia De La Rosa Navarro[1], Juan Bosco Frías Medina[2], Mustapha Lahyane[3], Israel Moreno Mejía[4] and Osvaldo Osuna Castro[5]

(1) Universidad Autónoma de Baja California, Ensenada, Mexico
(2) Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico
(3) Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico
(4) Universidad Nacional Autónoma de México, México, D.F., Mexico
(5) Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico

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.

Keywords: Cox rings, rational surfaces, effective monoid, nef monoid, extremal surfaces

De La Rosa Navarro Brenda Leticia, Frías Medina Juan Bosco, Lahyane Mustapha, Moreno Mejía Israel, Osuna Castro Osvaldo: A geometric criterion for the finite generation of the Cox rings of projective surfaces. Rev. Mat. Iberoamericana 31 (2015), 1131-1140. doi: 10.4171/RMI/878