Abstract
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.
Highlights
Ask if it is a finitely generated C-algebra
X is called a Mori Dream Space (MDS) due to Hu and Keel [29], who showed that this property is equivalent to a good behaviour with respect to the minimal model program, whence the name
Examples of MDS include toric [14] and more general spherical [12] varieties as well as certain ones with an action of a torus of lower dimension, see [26]. It was shown in [10] that varieties of Fano type - i.e. projective varieties admitting an effective Q-divisor Δ such that −(K X + Δ) is ample and (X, Δ) is Kawamata log terminal - are MDS
Summary
(i) is essential in the proof of Theorem 1 as it allows to reduce to the case of finite Cl(X ) while (ii) allows to pass from quasitorus quotients to finite Galois covers in the proof of Theorem 3. In terms of iteration of Cox rings, this generalizes to the following: Theorem 4 Let · · · → X4 → X3 → X2 → X1 be a chain of quotient presentations. Note that if for example X1 has finite iteration of Cox rings with non-MDS master Cox ring. Due to [5, Thm. 4.2.1.4], the characteristic space X → X is a universal reductive abelian (possibly nondiscrete) ’cover’ of X in the sense that it factors through any quotient presentation Y → X. This can be generalized for MDS with finite iteration of Cox rings and factorial master Cox ring in the following way. If X has non-MDS master Cox ring, difficulties as with Theorem 4 arise
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