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 ^[2] ; Israel Moreno Mejía ^[3] ; Osvaldo Osuna Castro ^[2]
1. [1] Universidad Autónoma de Baja California
  
  Universidad Autónoma de Baja California
  
  México
2. [2] Universidad Michoacana de San Nicolás de Hidalgo
  
  Universidad Michoacana de San Nicolás de Hidalgo
  
  México
3. [3] Universidad Nacional Autónoma de México
  
  Universidad Nacional Autónoma de México
  
  México
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Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 31, Nº 4, 2015, págs. 1131-1140
Idioma: inglés
DOI: 10.4171/RMI/878
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Resumen
- 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.