Harbourne–Hirschowitz surfaces whose anticanonical divisors consist only of three irreducible components

Abstract

In this paper, we provide new families of Harbourne–Hirschowitz surfaces whose effective monoids are finitely generated, and consequently, their Cox rings are finitely generated. Indeed, these properties are achieved by imposing some reasonable numerical conditions. Our method gives an efficient way of computing the minimal generating sets whenever the effective monoids are finitely generated. These surfaces are anticanonical ones having triangle anticanonical divisors consisting of smooth projective rational curves. Moreover, we present some families that do not satisfy the imposed numerical conditions but their effective monoids are still finitely generated by giving explicitly the minimal generating sets.