# In we studied normal rational varieties X of complexity

2021-04-30

In  we studied normal, rational -varieties X of complexity one, where the latter means that X comes with an effective torus action such that holds. We showed that for affine X with and at most log terminal singularities, the iteration of Cox rings is possible. In the present article, we characterize all varieties X with a torus action of complexity one that admit iteration of Cox rings. First consider the case . In order to have finitely generated divisor class group, X must be rational and then the Cox ring of X is of the form , with a polynomial ring in variables and modulo the ideal I generated by the trinomial relations with . For each exponent vector set . The Cox ring R is factorial if and only if the are pairwise coprime; see [3, Thm. 1.1]. We say that R is hyperplatonic if holds. After reordering decreasingly, the latter condition precisely means that holds for all and is a platonic triple, i.e., a triple of the form
We turn to the case . Here, and finite generation of the divisor class group of X force . In this situation, we obtain the following simple characterization.
As a consequence of the two theorems above, we obtain the following structural result, generalizing [1, Thm. 3], but using analogous ideas for the proof.
On our way of proving Theorem 1.1, we give in Proposition 2.6 an explicit description of the Cox ring of a variety SpecR for a hyperplatonic ring R. This allows to describe the possible Cox ring iteration chains more in detail. After reordering the numbers associated with R decreasingly, we call the basic platonic triple of R.
We will work in the notation of , , where the Cox ring of a rational T-variety of complexity one is encoded by a pair of defining matrices. Let us briefly recall the precise definitions we need from ; note that the setting will be slightly more flexible than the informal one given in the introduction.
By the results of ,  the rings are normal complete intersections, admit only constant homogeneous units and we have unique factorization in the multiplicative monoid of -homogeneous elements of . Moreover, suitably downgrading the rings leads to the Cox rings of the normal rational T-varieties X of complexity one with , see , , . In order to iterate a Cox ring , it TP-0903 mg is necessary that has finitely generated divisor class group. The latter turns out to be equivalent to rationality of . From [1, Cor. 5.8], we infer the following rationality criterion.
Observe that if is rational, then one can always achieve that is gcd-ordered by suitably reordering . This does not affect the -graded algebra up to isomorphy.
We are ready for the main ingredient of the proof of Theorem 1.1, the explicit description of the iterated Cox ring.
The defining property of a hyperplatonic ring is . Thus, for any such ring we find a (unique) platonic triple with pairwise different and all with u different from equal one. We call the basic platonic triple (bpt) of .
As a first step we relate the total coordinate space of a rational variety with torus action of complexity one admitting non-constant invariant functions to the total coordinate space of one with only constant invariant functions; see Corollary 3.4. This allows us to characterize rationality of the total coordinate space using previous results; see Corollary 3.5. Then we determine in a similar manner as before, the iterated Cox ring; see Proposition 3.7. This finally allows us to prove Theorem 1.2. We begin with recalling the necessary notions from .
Following  we call a ring arising from Construction 3.1 of Type 1 and a ring as in Construction 2.1 of Type 2. According to , the suitable downgradings of the rings of Type 1 yield precisely the Cox rings of the normal rational -varieties X of complexity one with .
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