Gorensteinness and iteration of Cox rings for Fano type varieties

We show that finitely generated Cox rings are Gorenstein. This leads to a refined characterization of varieties of Fano type: they are exactly those projective varieties with Gorenstein canonical quasicone Cox ring. We then show that for varieties of Fano type and Kawamata log terminal quasicones X, iteration of Cox rings is finite with factorial master Cox ring. In particular, even if the class group has torsion, we can express such X as quotients of a factorial canonical quasicone by a solvable reductive group.

Examples of Non-finitely Generated Cox Rings

We bring examples of toric varieties blown up at a point in the torus that do not have finitely generated Cox rings. These examples are generalizations of our earlier work, where toric surfaces of Picard number 1 were studied. In this article we consider toric varieties of higher Picard number and higher dimension. In particular, we bring examples of weighted projective 3-spaces blown up at a point that do not have finitely generated Cox rings.

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

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.