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  • In we studied normal rational varieties X of complexity

    2021-04-30

    In [1] 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 [3], [5], 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 [5]; note that the setting will be slightly more flexible than the informal one given in the introduction.
    By the results of [3], [5] 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 [4], [3], [5]. 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 [5].
    Following [5] 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 [5], the suitable downgradings of the rings of Type 1 yield precisely the Cox rings of the normal rational -varieties X of complexity one with .
    Introduction Prostaglandins are multifunctional compounds in mammals that are involved in inflammation, cell-cell signaling, regulation of immunity, and maintenance of tissue homeostasis [[1], [2], [3]]. Prostaglandins (PGs) are oxygenated lipid mediators formed from the ω6 essential fatty acid, arachidonic acid (AA). The committed step in PG biosynthesis is the conversion of AA to PG H2 (PGH2), catalyzed by either PG endoperoxide H synthase-1 or -2, commonly known as cyclooxygenase-1 (COX-1) and cyclooxygenase-2 (COX-2), respectively [4]. Two isoforms of COX have been identified: COX-1 is constitutively expressed in many cells and is involved in cell homeostasis, angiogenesis and cell-cell signaling; COX-2 is not expressed in normal condition however it is strongly expressed in inflammation [4].