Platonic Harbourne-Hirschowitz Rational Surfaces
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Platonic Harbourne-Hirschowitz Rational Surfaces Brenda Leticia De La Rosa-Navarro, Juan Bosco Fr´ıas-Medina and Mustapha Lahyane Abstract. The aim of this work was to study the finite generation of the effective monoid and Cox ring of a Platonic Harbourne-Hirschowitz rational surface with an anticanonical divisor not reduced which contains some exceptional curves as irreducible components. Such surfaces are obtained as the blow up of the n-Hirzebruch surface at any number of points lying in the union of the negative section and n + 2 different fibers. Moreover, the procedure that ensures the finite generation of the effective monoid provides a technique for explicit computation of the minimal generating set for such monoid in concrete cases. As an application, we present explicitly the minimal generating set for the effective monoid of some surfaces which are obtained by considering a degenerate cubic consisting in three lines intersecting at one point in the projective plane and blowing-up the singular point and some ordinary and infinitely near points. The base field of our surfaces is assumed to be algebraically closed of arbitrary characteristic. Mathematics Subject Classification. Primary 14J26, 14C20, 14C22; Secondary 14C17, 14Q20. Keywords. Cox rings, Rational surfaces, Effective monoid, Hirzebruch surfaces.
1. Introduction In [17], Hu and Keel introduced the Cox ring for projective varieties defined over an algebraically closed field as a generalization of the homogeneous coordinate ring introduced by Cox in [2]. Indeed, for a projective variety X over J.B. Fr´ıas-Medina acknowledges the financial support of Fondo Institucional de Fomento Regional para el Desarrollo Cient´ıfico, Tecnol´ ogico y de Innovaci´ on, FORDECYT 265667, during 2018 and of “Programa de Becas Posdoctorales 2019” de la Direcci´ on General de Asuntos del Personal Acad´emico, UNAM. M. Lahyane acknowledges a partial support from Coordinaci´ on de la Investigaci´ on Cient´ıfica de la Universidad Michoacana de San Nicol´ as de Hidalgo (UMSNH) during 2019 and 2020.
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an algebraically closed field k such that the linear equivalence is equal to the numerical equivalence, the Cox ring of X is defined by Cox(X) = H 0 (X, Ln1 1 ⊗ · · · ⊗ Lnt t ), (n1 ,...,nt )∈Zt
where L1 , . . . , Lt form a basis for the Picard group Pic(X) of X. One of the main problems in the study of Cox rings is to determine if it is finitely generated as k-algebra, in fact, the importance of such property for Q-factorial higher dimensional projective varieties is that the Minimal Model Program can be carried out for any divisor on such varieties (see [17, Proposition 2.9, p. 342]). The finite generation of the Cox ring is a nontrivial problem and this k-algebra may fail to be finitely generated even in the two-dimensional case, for example, the Cox ring of the surface obtained as blow up of the projective plane P2 in at least nine points in general position is not finitely generated because contains an infini
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