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.

中文翻译：

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Platonic Harbourne-Hirschowitz有理曲面

这项工作的目的是研究柏拉图式Harbourne-Hirschowitz有理曲面的有效单面体和Cox环的有限生成，该曲面具有未减小的反经典除数，该除数包含一些作为不可约成分的异常曲线。这样的表面是在负截面和\（n + 2 \）的并集处的任意数量的点处的n -Hirzebruch表面的爆炸而获得的不同的纤维。而且，确保有效的类半体的有限生成的过程提供了一种在具体情况下显式计算此类类半体的最小生成集的技术。作为一种应用，我们明确提出了一些表面有效半凸面的最小生成集，这是通过考虑由三个直线组成的简并三次投影线在投影平面上的一个点相交并炸开奇异点以及一些普通且无限大而获得的。近点。假定表面的基场是代数封闭的任意特征。