Platonic Harbourne-Hirschowitz Rational Surfaces

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.