On iteration of Cox rings

TL;DR

This paper characterizes varieties with a torus action of complexity one that can undergo multiple Cox ring iterations, advancing understanding of their algebraic structure.

Contribution

It provides a complete characterization of varieties allowing Cox ring iteration within the specified class, a novel contribution to algebraic geometry.

Findings

Identifies all varieties with complexity one torus action permitting Cox ring iteration

Establishes criteria for the iteration process to be possible

Enhances understanding of the structure of Cox rings in algebraic varieties

Abstract

We characterize all varieties with a torus action of complexity one that admit iteration of Cox rings.

Equations

$$R_1=\underset{K}{⨁}\Gamma (X,\mathcal{O}(D)).$$

$$X_p\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$T_0^{l_0}+T_1^{l_1}+T_2^{l_2},{\theta }_1{T}_1^{l_1}+T_2^{l_2}+T_3^{l_3},\dots ,{\theta }_{r-2}{T}_{r-2}^{l_{r-2}}+T_{r-1}^{l_{r-1}}+T_r^{l_r},$$

$$(5,3,2),(4,3,2),(3,3,2),(x,2,2),(x,y,1),x,y\in {\mathbb{Z}}_{\ge 1}.$$

$$T_i^{l_i}:=T_{i1}^{l_{i1}}\cdots T_{i{n}_i}^{l_{i{n}_i}}\in \mathbb{C}[T_{ij},S_k;0\le i\le r,1\le j\le n_i,1\le k\le m].$$

$$g_{i}\ :=\ \det\left[\begin{array}[]{lll}T_{i}^{l_{i}}&T_{i+1}^{l_{i+1}}&T_{i+2}^{l_{i+2}}\\ a_{i}&a_{i+1}&a_{i+2}\end{array}\right]\ \in\ {\mathbb{C}}[T_{ij},S_{k}].$$

$$P_{0}\ :=\ \left[\begin{array}[]{ccccccc}-l_{0}&l_{1}&&0&0&\ldots&0\\ \vdots&\vdots&\ddots&\vdots&\vdots&&\vdots\\ -l_{0}&0&&l_{r}&0&\ldots&0\end{array}\right].$$

$$Q:{\mathbb{Z}}^{n+m}\to K_0:={\mathbb{Z}}^{n+m}/\mathrm{im}(P_0^{\ast }).$$

$$\mathrm{deg}(T_{ij}):=Q(e_{ij})\in K_0,\mathrm{deg}(S_k):=Q(e_k)\in K_0.$$

$$R(A,P_0):=\mathbb{C}[T_{ij},S_k]/⟨g_0,\dots ,g_{r-2}⟩.$$

$$P_{1}\ :=\ \left[\begin{array}[]{ccccccccc}\frac{-1}{\gcd(\mathfrak{l}_{0},\mathfrak{l}_{1})}l_{0}&\frac{1}{\gcd(\mathfrak{l}_{0},\mathfrak{l}_{1})}l_{1}&0&\dots&&0&0&\dots&0\\[5.0pt] \frac{-1}{\gcd(\mathfrak{l}_{0},\mathfrak{l}_{2})}l_{0}&0&\frac{1}{\gcd(\mathfrak{l}_{0},\mathfrak{l}_{2})}l_{2}&0&&0&&&\\ -l_{0}&0&&l_{3}&&0&\vdots&&\vdots\\ \vdots&&&\vdots&\ddots&\vdots&&&\\ -l_{0}&0&\dots&0&&l_{r}&0&\dots&0\end{array}\right].$$

$$K_0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\left[\begin{array}[]{cccccc}\frac{-1}{\gcd(\mathfrak{l}_{0},\mathfrak{l}_{1})}\mathfrak{l}_{0}&\frac{1}{\gcd(\mathfrak{l}_{0},\mathfrak{l}_{1})}\mathfrak{l}_{1}&0&\dots&&0\\[5.0pt] \frac{-1}{\gcd(\mathfrak{l}_{0},\mathfrak{l}_{2})}\mathfrak{l}_{0}&0&\frac{1}{\gcd(\mathfrak{l}_{0},\mathfrak{l}_{2})}\mathfrak{l}_{2}&0&&0\\ -\mathfrak{l}_{0}&0&&\mathfrak{l}_{3}&&0\\ \vdots&&&\vdots&\ddots&\vdots\\ -\mathfrak{l}_{0}&0&\dots&0&&\mathfrak{l}_{r}\\ \end{array}\right]$$

$$\frac{1}{\gcd ({\mathfrak{l}}_0,{\mathfrak{l}}_1)\gcd ({\mathfrak{l}}_0,{\mathfrak{l}}_2)}{\mathfrak{l}}_0\cdots {\mathfrak{l}}_{i-1}\cdot {\mathfrak{l}}_{i+1}\cdots {\mathfrak{l}}_r,\text{ where }i=0,\dots ,r.$$

$$\frac{{\mathfrak{l}}_0{\mathfrak{l}}_2}{\gcd ({\mathfrak{l}}_0,{\mathfrak{l}}_1)\gcd ({\mathfrak{l}}_0,{\mathfrak{l}}_2)},\frac{{\mathfrak{l}}_1{\mathfrak{l}}_2}{\gcd ({\mathfrak{l}}_0,{\mathfrak{l}}_1)\gcd ({\mathfrak{l}}_0,{\mathfrak{l}}_2)},\frac{{\mathfrak{l}}_0{\mathfrak{l}}_1}{\gcd ({\mathfrak{l}}_0,{\mathfrak{l}}_1)\gcd ({\mathfrak{l}}_0,{\mathfrak{l}}_2)}.$$

$$\gcd ({\mathfrak{l}}_0{\mathfrak{l}}_2,{\mathfrak{l}}_0{\mathfrak{l}}_1,{\mathfrak{l}}_1{\mathfrak{l}}_2)=\gcd ({\mathfrak{l}}_0\mathfrak{l},{\mathfrak{l}}_1{\mathfrak{l}}_2)=\gcd ({\mathfrak{l}}_0,{\mathfrak{l}}_1)\gcd ({\mathfrak{l}}_0,{\mathfrak{l}}_2).$$

$$n_{i,1},\dots ,n_{i,c(i)}:=n_i,l_{ij,1},\dots ,l_{ij,c(i)}:=\gcd ((P_1{)}_{1,ij},\dots ,(P_1{)}_{r,ij}).$$

$$T_i^{l_i}:=T_{i1}^{l_{i1}}\cdots T_{i{n}_i}^{l_{i{n}_i}}\in \mathbb{C}[T_{ij},S_k;1\le i\le r,1\le j\le n_i,1\le k\le m].$$

$$g_i:=T_i^{l_i}-T_{i+1}^{l_{i+1}}-(a_{i+1}-a_i)\in \mathbb{C}[T_{ij},S_k].$$

$$P_{0}\ :=\ \left[\begin{array}[]{cccccc}l_{1}&&0&0&\ldots&0\\ \vdots&\ddots&\vdots&\vdots&&\vdots\\ 0&&l_{r}&0&\ldots&0\end{array}\right].$$

$$R(A,P_0):=\mathbb{C}[T_{ij},S_k]/⟨g_1,\dots ,g_{r-1}⟩.$$

$$\tilde{A}\ :=\ \left[\begin{array}[]{rrrr}-1&a_{1}&\dots&a_{r}\\ 0&1&\dots&1\end{array}\right],\qquad\tilde{P}_{0}\ :=\ \left[L_{0},P_{0}\right].$$

$$R(\tilde{A},{\tilde{P}}_0{)}_{{\tilde{T}}_{01}}\to R(A,P_0)[T_{01},T_{01}{]}^{-1},{\tilde{T}}_{01}\mapsto T_{01},{\tilde{T}}_{ij}\mapsto T_{ij}{T}_{01}^{\frac{\ell }{{\mathfrak{l}}_i}{\alpha }_{ij}}.$$

$$\psi :\mathbb{C}[{\tilde{T}}_{ij},{\tilde{S}}_k{]}_{{\tilde{T}}_{01}}\to \mathbb{C}[T_{ij},S_k][T_{01},T_{01}{]}^{-1},{\tilde{T}}_{01}\mapsto T_{01},{\tilde{T}}_{ij}\mapsto T_{ij}{T}_{01}^{\frac{\ell }{{\mathfrak{l}}_i}{\alpha }_{ij}},{\tilde{S}}_k\mapsto S_k.$$

$$\mathrm{deg}(T_{01})=(1,0)\in \mathbb{Z}\times K_0,\mathrm{deg}(T_{ij})=(0,e_{ij}+\mathrm{im}(P_0^{\ast }))\in \mathbb{Z}\times K_0.$$

$${\mathbb{Z}}^{n+1+m}\to {\tilde{K}}_0,{\mathbb{Z}}^{n+1+m}\to \mathbb{Z}\times K_0,$$

$$\tilde{g}_{i}\ :=\ \det\left[\begin{array}[]{ccc}\tilde{T}_{0}^{\ell}&T_{i}^{l_{i}}&T_{i+1}^{l_{i+1}}\\ -1&a_{i}&a_{i+1}\\ 0&1&1\end{array}\right].$$

$$n_{i,1},\dots ,n_{i,c(i)}:=n_i,l_{i,1},\dots ,l_{i,c(i)}:=\frac{1}{{\mathfrak{l}}_i}{l}_i.$$

$$\left[\begin{array}[]{cccccc}\frac{1}{\mathfrak{l}_{1}}l_{1}&\dots&0&0&\dots&0\\ \vdots&\ddots&\vdots&\vdots&&\vdots\\ 0&\dots&\frac{1}{\mathfrak{l}_{r}}l_{r}&0&\dots&0\end{array}\right].$$

Full text

1. Introduction

We consider normal algebraic varieties $X$ defined over the field ${\mathbb{C}}$ of complex numbers. If $X$ has finitely generated divisor class group $K$ and only constant invertible global regular functions, then one defines the $K$-graded Cox ring $R_{1}$ of $X$ as follows, see [2] for details:

[TABLE]

If the Cox ring $R_{1}$ is a finitely generated ${\mathbb{C}}$-algebra, then one has the total coordinate space $X_{1}:={\rm Spec}\,R_{1}$. We say that $X$ admits iteration of Cox rings if there is a chain

[TABLE]

dominated by a factorial variety $X_{p}$ where in each step, $X_{i+1}$ is the total coordinate space of $X_{i}$ and $H_{i}={\rm Spec}\,{\mathbb{C}}[K_{i}]$ the characteristic quasitorus of $X_{i}$, having the divisor class group $K_{i}$ of $X_{i}$ as its character group. Note that if the divisor class group $K$ of $X$ is torsion free, then $R_{1}$ is a unique factorization domain and iteration of Cox rings is trivially possible. As soon as $K$ has torsion, it may happen that during the iteration process a total coordinate space with non-finitely generated divisor class group pops up and thus there is no chain of total coordinate spaces as above, see [1, Rem. 5.12].

In [1] we studied normal, rational, ${\mathbb{T}}$-varieties $X$ of complexity one, where the latter means that $X$ comes with an effective torus action ${\mathbb{T}}\times X\to X$ such that $\dim({\mathbb{T}})=\dim(X)-1$ holds. We showed that for affine $X$ with $\Gamma(X,\mathcal{O})^{{\mathbb{T}}}={\mathbb{C}}$ 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 $\Gamma(X,\mathcal{O})^{{\mathbb{T}}}={\mathbb{C}}$. In order to have finitely generated divisor class group, $X$ must be rational and then the Cox ring of $X$ is of the form $R={\mathbb{C}}[T_{ij},S_{k}]/I$, with a polynomial ring ${\mathbb{C}}[T_{ij},S_{k}]$ in variables $T_{ij}$ and $S_{k}$ modulo the ideal $I$ generated by the trinomial relations

[TABLE]

with $T_{i}^{l_{i}}=T_{i1}^{l_{i1}}\cdots T_{in_{i}}^{l_{in_{i}}}$. For each exponent vector $l_{i}$ set $\mathfrak{l}_{i}:=\gcd(l_{i1},\ldots,l_{in_{i}})$. We say that $R$ is hyperplatonic if $\mathfrak{l}_{0}^{-1}+\ldots+\mathfrak{l}_{r}^{-1}>r-1$ holds. After reordering $\mathfrak{l}_{0},\ldots,\mathfrak{l}_{r}$ decreasingly, the latter condition precisely means that $\mathfrak{l}_{i}=1$ holds for all $i\geq 3$ and $(\mathfrak{l}_{0},\mathfrak{l}_{1},\mathfrak{l}_{2})$ is a platonic triple, i.e., a triple of the form

[TABLE]

Theorem 1.1**.**

Let $X$ be a normal ${\mathbb{T}}$-variety of complexity one with $\Gamma(X,\mathcal{O})^{{\mathbb{T}}}={\mathbb{C}}$. Then the following statements are equivalent.

1. (i)

The variety $X$ admits iteration of Cox rings. 2. (ii)

The variety $X$ is rational with hyperplatonic Cox ring.

We turn to the case $\Gamma(X,\mathcal{O})^{{\mathbb{T}}}\neq{\mathbb{C}}$. Here, $\mathcal{O}(X)^{*}={\mathbb{C}}^{*}$ and finite generation of the divisor class group of $X$ force $\Gamma(X,\mathcal{O})^{{\mathbb{T}}}={\mathbb{C}}[T]$. In this situation, we obtain the following simple characterization.

Theorem 1.2**.**

Let $X$ be a normal ${\mathbb{T}}$-variety of complexity one with $\Gamma(X,\mathcal{O})^{{\mathbb{T}}}\neq{\mathbb{C}}$. Then $X$ admits Cox ring iteration if and only if $X$ and its total coordinate space are rational. Moreover, if the latter holds, then the Cox ring iteration stops after at most one step.

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.

Corollary 1.3**.**

Let $X$ be a normal, rational variety with a torus action of complexity one admitting iteration of Cox rings. Then $X$ is a quotient $X=X^{\prime}\mathbin{\mathchoice{/\mkern-6.0mu/}{/\mkern-6.0mu/}{/\mkern-5.0mu/}{/\mkern-5.0mu/}}G$ of a factorial affine variety $X^{\prime}:={\rm Spec}(R^{\prime})$, where $R^{\prime}$ is a factorial ring and $G$ is a solvable reductive group.

On our way of proving Theorem 1.1, we give in Proposition 2.6 an explicit description of the Cox ring of a variety ${\rm Spec}\,R$ for a hyperplatonic ring $R$. This allows to describe the possible Cox ring iteration chains more in detail. After reordering the numbers $\mathfrak{l}_{0},\ldots,\mathfrak{l}_{r}$ associated with $R$ decreasingly, we call $(\mathfrak{l}_{0},\mathfrak{l}_{1},\mathfrak{l}_{2})$ the basic platonic triple of $R$.

Corollary 1.4**.**

The possible sequences of basic platonic triples arising from Cox ring iterations of normal, rational varieties with a torus action of complexity one and hyperplatonic Cox ring are the following:

1. (i)

$(1,1,1)\rightarrow(2,2,2)\rightarrow(3,3,2)\rightarrow(4,3,2)$, 2. (ii)

$(1,1,1)\rightarrow(x,x,1)\rightarrow(2x,2,2)$, 3. (iii)

$(1,1,1)\rightarrow(x,x,1)\rightarrow(x,2,2)$, 4. (iv)

$(\mathfrak{l}_{01}^{-1}\mathfrak{l}_{0},\mathfrak{l}_{01}^{-1}\mathfrak{l}_{1},1)\rightarrow(\mathfrak{l}_{0},\mathfrak{l}_{1},1)$, where $\mathfrak{l}_{01}:=\gcd(\mathfrak{l}_{0},\mathfrak{l}_{1})>1$.

Contents

1. 1 Introduction
2. 2 Proof of Theorem 1.1
3. 3 Proof of Theorem 1.2

2. Proof of Theorem 1.1

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.

Construction 2.1**.**

Fix integers $r,n>0$, $m\geq 0$ and a partition $n=n_{0}+\ldots+n_{r}$. For every $i=0,\ldots,r$, fix a tuple $l_{i}\in{\mathbb{Z}}_{>0}^{n_{i}}$ and define a monomial

[TABLE]

We will also write ${\mathbb{C}}[T_{ij},S_{k}]$ for the above polynomial ring. Let $A:=(a_{0},\ldots,a_{r})$ be a $2\times(r+1)$ matrix with pairwise linearly independent columns $a_{i}\in{\mathbb{C}}^{2}$. For every $i=0,\dots,r-2$ we define

[TABLE]

We build up an $r\times(n+m)$ matrix from the exponent vectors $l_{0},\ldots,l_{r}$ of these polynomials:

[TABLE]

Denote by $P_{0}^{*}$ the transpose of $P_{0}$ and consider the projection

[TABLE]

Denote by $e_{ij},e_{k}\in{\mathbb{Z}}^{n+m}$ the canonical basis vectors corresponding to the variables $T_{ij}$, $S_{k}$. Define a $K_{0}$-grading on ${\mathbb{C}}[T_{ij},S_{k}]$ by setting

[TABLE]

This is the finest possible grading of ${\mathbb{C}}[T_{ij},S_{k}]$ leaving the variables and the $g_{i}$ homogeneous. In particular, we have a $K_{0}$-graded factor algebra

[TABLE]

By the results of [3, 5] the rings $R(A,P_{0})$ are normal complete intersections, admit only constant homogeneous units and we have unique factorization in the multiplicative monoid of $K_{0}$-homogeneous elements of $R(A,P_{0})$. Moreover, suitably downgrading the rings $R(A,P_{0})$ leads to the Cox rings of the normal rational $T$-varieties $X$ of complexity one with $\Gamma(X,\mathcal{O})^{\mathbb{T}}={\mathbb{C}}$, see [4, 3, 5].

In order to iterate a Cox ring $R(A,P_{0})$, it is necessary that ${\rm Spec}\,R(A,P_{0})$ has finitely generated divisor class group. The latter turns out to be equivalent to rationality of ${\rm Spec}\,R(A,P_{0})$. From [1, Cor. 5.8], we infer the following rationality criterion.

Remark 2.2**.**

Let $R(A,P_{0})$ be as in Construction 2.1 and set $\mathfrak{l}_{i}:=\gcd(l_{i1},\ldots,l_{in_{i}})$. Then ${\rm Spec}\,R(A,P_{0})$ is rational if and only if one of the following conditions holds:

1. (i)

We have $\gcd(\mathfrak{l}_{i},\mathfrak{l}_{j})=1$ for all $0\leq i<j\leq r$, in other words, $R(A,P_{0})$ is factorial. 2. (ii)

There are $0\leq i<j\leq r$ with $\gcd(\mathfrak{l}_{i},\mathfrak{l}_{j})>1$ and $\gcd(\mathfrak{l}_{u},\mathfrak{l}_{v})=1$ whenever $v\not\in\{i,j\}$. 3. (iii)

There are $0\leq i<j<k\leq r$ with $\gcd(\mathfrak{l}_{i},\mathfrak{l}_{j})=\gcd(\mathfrak{l}_{i},\mathfrak{l}_{k})=\gcd(\mathfrak{l}_{j},\mathfrak{l}_{k})=2$ and $\gcd(\mathfrak{l}_{u},\mathfrak{l}_{v})=1$ whenever $v\not\in\{i,j,k\}$.

Definition 2.3**.**

Let $R(A,P_{0})$ be as in Construction 2.1 such that ${\rm Spec}\,R(A,P_{0})$ is rational. We say that $P_{0}$ is $\gcd$-ordered if it satisfies the following two properties

1. (i)

$\gcd(\mathfrak{l}_{i},\mathfrak{l}_{j})=1$ for all $i=0,\ldots,r$ and $j=3,\ldots,r$, 2. (ii)

$\gcd(\mathfrak{l}_{1},\mathfrak{l}_{2})=\gcd(\mathfrak{l}_{0},\mathfrak{l}_{1},\mathfrak{l}_{2})$.

Observe that if ${\rm Spec}\,R(A,P_{0})$ is rational, then one can always achieve that $P_{0}$ is $\gcd$-ordered by suitably reordering $l_{0},\ldots,l_{r}$. This does not affect the $K_{0}$-graded algebra $R(A,P_{0})$ up to isomorphy.

Lemma 2.4**.**

Let $R(A,P_{0})$ be as in Construction 2.1 such that ${\rm Spec}\,R(A,P_{0})$ is rational and $P_{0}$ is $\gcd$-ordered. Then, with $K_{0}={\mathbb{Z}}^{n+m}/{\rm im}(P_{0}^{*})$, the kernel of ${\mathbb{Z}}^{n+m}\to K_{0}/K_{0}^{\mathrm{tors}}$ is generated by the rows of

[TABLE]

Proof.

The arguments are similar as for [1, Cor. 6.3]. The row lattice of $P_{0}$ is a sublattice of finite index of that of $P_{1}$ and thus there is a commutative diagram

[TABLE]

We have to show, that ${\mathbb{Z}}^{n+m}/{\rm im}(P_{1}^{*})$ is torsion free. Suitable elementary column operations on $P_{1}$ reduce the problem to showing that for the $r\times(r+1)$ matrix

[TABLE]

the $r$-th determinantal divisor and therefore the product of the invariant factors equals one. Up to sign, the $r\times r$ minors of the above matrix are

[TABLE]

Suppose that some prime $p$ divides all these minors. Then $p\nmid\mathfrak{l}_{j}$ holds for all $j\geq 3$, because otherwise we find an $i\neq j$ with $p\mid\mathfrak{l}_{i}$, contradicting $\gcd$-orderedness of $P_{0}$. Thus, $p$ divides each of the numbers

[TABLE]

By the assumption of the lemma, $\mathfrak{l}:=\gcd(\mathfrak{l}_{1},\mathfrak{l}_{2})$ equals $\gcd(\mathfrak{l}_{0},\mathfrak{l}_{1},\mathfrak{l}_{2})$. Consequently, we obtain

[TABLE]

We conclude $p=1$; a contradiction. Being the greatest common divisor of the above minors, the $r$-th determinantal divisor equals one. ∎

Lemma 2.5**.**

Let $R(A,P_{0})$ be as in Construction 2.1 and $X:={\rm Spec}\,R(A,P_{0})$ be rational. Assume that $P_{0}$ is $\gcd$-ordered. Then the number $c(i)$ of irreducible components of $V(X,T_{ij})$ is given by

[TABLE]

Proof.

The assertion is a direct consequence of [1, Lemma 6.4]. ∎

We are ready for the main ingredience of the proof of Theorem 1.1, the explicit description of the iterated Cox ring.

Proposition 2.6**.**

Let $R(A,P_{0})$ be non-factorial with ${\rm Spec}\,R(A,P_{0})$ rational. Assume that $P_{0}$ is $\gcd$-ordered and let $P_{1}$ be as in Lemma 2.4. Define numbers ${n^{\prime}:=c(0)n_{0}+\ldots+c(r)n_{r}}$ and

[TABLE]

Then the vectors $l_{i,\alpha}:=(l_{i1,\alpha},\ldots,l_{in_{i},\alpha})\in{\mathbb{Z}}^{n_{i,\alpha}}$ build up an $r^{\prime}\times(n^{\prime}+m)$ matrix $P_{0}^{\prime}$. With a suitable matrix $A^{\prime}$, the affine variety ${\rm Spec}\,R(A^{\prime},P^{\prime}_{0})$ is the total coordinate space of the affine variety ${\rm Spec}\,R(A,P_{0})$.

Proof.

The idea is to work with the action of the torus $H_{0}^{0}:={\rm Spec}\,{\mathbb{C}}[K_{0}/K_{0}^{\rm tors}]$ on $X:={\rm Spec}\,R(A,P_{0})$ and to use the description of the Cox ring of a variety with torus action provided in [4]. For this, one has to look at the exceptional fibers of the map $\pi\colon X_{0}\to Y$, where $X_{0}\subseteq X$ is the set of points with at most finite $H_{0}^{0}$-isotropy and the curve $Y$ is the separation of $X_{0}/H_{0}^{0}$. Following the lines of the proof of [1, Prop. 6.6], one uses Lemma 2.5 to determine the number of components for each fiber of $\pi$ and Lemma 2.4 to determine the order of the general (finite) $H_{0}^{0}$-isotropy groups on each component. The rest is application of [4]. ∎

If $R(A,P_{0})$ is a hyperplatonic ring, then $\mathfrak{l}_{0}^{-1}+\ldots+\mathfrak{l}_{0}^{-1}\geq r-1$ holds. Thus, we find a (unique) platonic triple $(\mathfrak{l}_{i},\mathfrak{l}_{j},\mathfrak{l}_{k})$ with $i,j,k$ pairwise different and all $\mathfrak{l}_{u}$ with $u$ different from $i,j,k$ equal one. We call $(\mathfrak{l}_{i},\mathfrak{l}_{k},\mathfrak{l}_{k})$ the basic platonic triple (bpt) of $R(A,P_{0})$.

Remark 2.7**.**

Let $R(A,P_{0})$ be non-factorial and hyperplatonic with basic platonic triple $(\mathfrak{l}_{0},\mathfrak{l}_{1},\mathfrak{l}_{2})$. Then Remark 2.2 ensures that $X:={\rm Spec}\,R(A,P_{0})$ is rational. Moreover, Lemma 2.5 and Proposition 2.6 yield that the exponent vectors of the defining relations of the Cox ring $R(A^{\prime},P^{\prime}_{0})$ of $X$ are computed in terms of the exponent vectors $l_{0},\ldots,l_{r}$ of $R(A,P_{0})$ according to the table below, where “$a\times l_{i}$” means that the vector $l_{i}$ shows up $a$ times:

[TABLE]

Lemma 2.8**.**

Let $R(A,P_{0})$, arising from Construction 2.1, be non-factorial and assume that $X:={\rm Spec}\,R(A,P_{0})$ is rational. If the total coordinate space of $X$ is rational as well, then $\mathfrak{l}_{i}>1$ holds for at most three $0\leq i\leq r$.

Proof.

We may assume that $P_{0}$ is $\gcd$-ordered. Then Proposition 2.6 provides us with the exponent vectors of the Cox ring $R(A^{\prime},P_{0}^{\prime})$ of $X$. As $R(A,P_{0})$ is rational and non-factorial, Remark 2.2 leaves us with the following two cases.

Case 1. We have $\gcd(\mathfrak{l}_{0},\mathfrak{l}_{1})>1$ and $\gcd(\mathfrak{l}_{i},\mathfrak{l}_{j})=1$ whenever $j\geq 2$. This means in particular $\mathfrak{l}_{0},\mathfrak{l}_{1}>1$. Assume that there are $2\leq i<j\leq r$ with $\mathfrak{l}_{i},\mathfrak{l}_{j}>1$. According to Proposition 2.6, we find $c(i)$ times the exponent vector $l_{i}$ and $c(j)$ times the exponent vector $l_{j}$ in $P_{0}^{\prime}$. Lemma 2.5 tells us $c(j)=c(i)=\gcd(\mathfrak{l}_{0},\mathfrak{l}_{1})>1$. Thus, for the first two copies of $l_{i}$ and $l_{j}$, we obtain $\gcd(l_{i,1},l_{i,2})=\mathfrak{l}_{i}>1$ and $\gcd(l_{j,1},l_{j,2})=\mathfrak{l}_{j}>1$ respectively. Remark 2.2 shows that ${\rm Spec}\,R(A^{\prime},P_{0}^{\prime})$ is not rational; a contradiction.

Case 2. We have $\gcd(\mathfrak{l}_{0},\mathfrak{l}_{1})=\gcd(\mathfrak{l}_{0},\mathfrak{l}_{2})=\gcd(\mathfrak{l}_{1},\mathfrak{l}_{2})=2$. Assume that there is an index $3\leq i\leq r$ with $\mathfrak{l}_{i}>1$. Proposition 2.6 and Lemma 2.5 yield that the exponent vector $l_{i}$ occurs $c(k)=4$ times in the matrix $P_{0}^{\prime}$. As in the previous case we conclude via Remark 2.2 that the total coordinate space ${\rm Spec}\,R(A^{\prime},P_{0}^{\prime})$ is not rational; a contradiction. ∎

Proof of Theorem 1.1.

We prove “(ii)$\Rightarrow$(i)”. Then $X$ is a rational and has a hyperplatonic ring $R(A,P_{0})$ provided by Construction 2.1 as its Cox ring. If $R(A,P_{0})$ is factorial, then there is nothing to show. So, let $R(A,P_{0})$ be non-factorial. We may assume that $P_{0}$ is $\gcd$-ordered. Then $(\mathfrak{l}_{0},\mathfrak{l}_{1},\mathfrak{l}_{2})$ is the basic platonic triple of $R(A,P_{0})$. From Remark 2.7 we infer that $X_{1}:={\rm Spec}\,R(A,P_{0})$ is rational with hyperplatonic Cox ring $R(A^{\prime},P_{0}^{\prime})$. So, we can pass to $X_{2}:=R(A^{\prime},P_{0}^{\prime})$ and so forth. The table of possible basic platonic triples given in Remark 2.7 shows that the iteration process terminates at a factorial ring.

We prove “(i)$\Rightarrow$(ii)”. Since $X$ has a Cox ring, $X$ must have finitely generated divisor class group. As for any ${\mathbb{T}}$-variety of complexity one, the latter is equivalent to $X$ being rational. The Cox ring of $X$ is a ring $R(A,P_{0})$ as provided by Construction 2.1. If $R(A,P_{0})$ is factorial, then we are done. So, let $R(A,P_{0})$ be non-factorial. Then we may assume that $P_{0}$ is $\gcd$-ordered and, moreover, $\mathfrak{l}_{01}\neq 1$. Since $X_{1}={\rm Spec}\,R(A,P_{0})$ has a Cox ring $R(A^{\prime},P_{0}^{\prime})$, it must be rational. By Lemma 2.8 we have $\mathfrak{l}_{j}=1$ whenever $j\geq 3$ holds. Remark 2.2 leaves us with the following cases.

Case 1. We have $\mathfrak{l}_{01}:=\gcd(\mathfrak{l}_{0},\mathfrak{l}_{1})>1$ and $\gcd(\mathfrak{l}_{i},\mathfrak{l}_{j})=1$ whenever $j\geq 2$ holds. Then we may assume $\mathfrak{l}_{0}\geq\mathfrak{l}_{1}$.

1.1. Consider the case $\mathfrak{l}_{01}>3$. By Lemma 2.5, the exponent vector $l_{2}$ occurs $\mathfrak{l}_{01}$ times in the defining relations of the Cox ring $R(A^{\prime},P_{0}^{\prime})$ of $X_{1}$. Since ${\rm Spec}\,R(A^{\prime},P_{0}^{\prime})$ is rational, Remark 2.2 yields $\mathfrak{l}_{2}=1$. We conclude that $(\mathfrak{l}_{0},\mathfrak{l}_{1},\mathfrak{l}_{2})$ is platonic.

1.2. Assume $\mathfrak{l}_{01}=3$. Then $l_{2}$ occurs $3$ times as exponent vector in the defining relations of $R(A^{\prime},P_{0}^{\prime})$. Remark 2.2 shows $\mathfrak{l}_{2}\leq 2$. Thus, $(\mathfrak{l}_{0},\mathfrak{l}_{1},\mathfrak{l}_{2})$ is platonic.

1.3. Let $\mathfrak{l}_{01}=2$. If $\mathfrak{l}_{0}=\mathfrak{l}_{1}=2$ holds, then $(\mathfrak{l}_{0},\mathfrak{l}_{1},\mathfrak{l}_{2})$ is a platonic triple for any $\mathfrak{l}_{2}$. So, assume $\mathfrak{l}_{0}>\mathfrak{l}_{1}\geq 2$. As we are in Case 1, the number $\mathfrak{l}_{2}$ must be odd. If $\mathfrak{l}_{2}=1$ holds, then $(\mathfrak{l}_{0},\mathfrak{l}_{1},\mathfrak{l}_{2})$ is a platonic triple. By Proposition 2.6 and Lemma 2.5, we find the exponent vectors $1/2\,l_{0}$ and $1/2\,l_{1}$ as well as twice $l_{2}$ in $P_{0}^{\prime}$. Since $X_{1}={\rm Spec}\,R(A^{\prime},P_{0}^{\prime})$ is rational and $\mathfrak{l}_{0}>\mathfrak{l}_{1}$ holds, Lemma 2.8 shows $\mathfrak{l}_{1}=2$ and the triple of non-trivial gcd’s of exponent vectors of $P_{0}^{\prime}$ is $(\mathfrak{l}_{0}/2,\mathfrak{l}_{2},\mathfrak{l}_{2})$. After gcd-ordering $P_{0}^{\prime}$, we can apply Case 1.1 and with $\mathfrak{l}_{0}/2>1$ we obtain $\mathfrak{l}_{0}=4$ and $\mathfrak{l}_{2}=3$. In particular, $(\mathfrak{l}_{0},\mathfrak{l}_{1},\mathfrak{l}_{2})$ is platonic.

Case 2: We have $\gcd(\mathfrak{l}_{0},\mathfrak{l}_{1})=\gcd(\mathfrak{l}_{0},\mathfrak{l}_{2})=\gcd(\mathfrak{l}_{1},\mathfrak{l}_{2})=2$. Then we may assume $\mathfrak{l}_{0}\geq\mathfrak{l}_{1}\geq\mathfrak{l}_{2}$. Proposition 2.6 and Lemma 2.5 tell us that each of the exponent vectors $1/2\,l_{0}$, $1/2\,l_{1}$ and $1/2\,l_{2}$ occurs twice in $P_{0}^{\prime}$. Since ${\rm Spec}\,R(A^{\prime},P_{0}^{\prime})$ is rational, Lemma 2.8 yields $\mathfrak{l}_{1}=\mathfrak{l}_{2}=2$. Thus, $(\mathfrak{l}_{0},\mathfrak{l}_{1},\mathfrak{l}_{2})$ is platonic. ∎

3. Proof of Theorem 1.2

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].

Construction 3.1**.**

Fix integers $r,n>0$, $m\geq 0$ and a partition $n=n_{1}+\ldots+n_{r}$. For each $1\leq i\leq r$, fix a tuple $l_{i}\in{\mathbb{Z}}_{>0}^{n_{i}}$ and define a monomial

[TABLE]

Let $A:=(a_{1},\ldots,a_{r})$ be a list of pairwise different elements of ${\mathbb{C}}$. Define for every $i=1,\dots,r-1$ a polynomial

[TABLE]

We build up an $r\times(n+m)$ matrix from the exponent vectors $l_{1},\ldots,l_{r}$ of these polynomials:

[TABLE]

Similar to the case in Construction 2.1 the matrix $P_{0}$ defines a grading of the group $K_{0}:={\mathbb{Z}}^{n+m}/{\rm im}(P_{0}^{*})$ on the ring

[TABLE]

Following [5] we call a ring $R(A,P_{0})$ arising from Construction 3.1 of Type 1 and a ring $R(A,P_{0})$ as in Construction 2.1 of Type 2. According to [5], the suitable downgradings of the rings $R(A,P_{0})$ of Type 1 yield precisely the Cox rings of the normal rational ${\mathbb{T}}$-varieties $X$ of complexity one with $\Gamma(X,\mathcal{O})^{{\mathbb{T}}}={\mathbb{C}}[T]$.

Construction 3.2**.**

Consider a ring $R(A,P_{0})$ of Type 1. Set $\mathfrak{l}_{i}:=\gcd(l_{i1},\ldots,l_{in_{i}})$ and $\ell:={\rm lcm}(\mathfrak{l}_{1},\ldots,\mathfrak{l}_{r})$. Then, writing $L_{0}$ for the column vector $-(\ell,\ldots,\ell)\in{\mathbb{Z}}^{r}$, we obtain a ring $R(\tilde{A},\tilde{P}_{0})$ of Type 2 with defining matrices

[TABLE]

Proposition 3.3**.**

Let $R(A,P_{0})$ be a ring of Type 1 and $R(\tilde{A},\tilde{P}_{0})$ the associated ring of Type 2 obtained via Construction 3.2. Fix $\alpha_{ij}\in{\mathbb{Z}}$ with $\mathfrak{l}_{i}=\alpha_{i1}l_{i1}+\ldots+\alpha_{in_{i}}l_{in_{i}}$. Then one obtains an isomorphism of graded ${\mathbb{C}}$-algebras

[TABLE]

Proof.

By construction, $R(\tilde{A},\tilde{P}_{0})$ is a factor algebra of ${\mathbb{C}}[\tilde{T}_{ij},\tilde{S}_{k}]$ and $R(A,P_{0})$ of ${\mathbb{C}}[T_{ij},S_{k}]$. We have an isomorphism of ${\mathbb{C}}$-algebras

[TABLE]

Observe $\psi(\tilde{T}_{i}^{l_{i}})=T_{01}^{\ell}T_{i}^{l_{i}}$. We claim that $\psi$ is compatible with the gradings by $\tilde{K}_{0}$ on the l.h.s. and by ${\mathbb{Z}}\times K_{0}$ on the r.h.s., where the latter grading is given by

[TABLE]

Indeed, because of $\psi(\tilde{T}_{01}^{-\ell}\tilde{T}_{i}^{l_{i}})=T_{i}^{l_{i}}$, the kernels of the respective downgrading maps

[TABLE]

generated by the rows $\tilde{P}_{0}$ and $P_{0}$, correspond to each other under $\psi$. The defining ideal of $R(\tilde{A},\tilde{P}_{0})$ is generated by the polynomials $\tilde{g}_{1},\ldots,\tilde{g}_{r-1}$, where

[TABLE]

The above isomorphism sends $\tilde{g}_{i}$ to $T_{0}^{\ell}g_{i}$, where the $g_{i}$ are the generators of the defining ideal of $R(A,P_{0})$, and thus induces the desired isomorphism. ∎

Corollary 3.4**.**

Let $X:={\rm Spec}\,R(A,P_{0})$ be the affine variety arising from a ring of Type 1 and $\tilde{X}:={\rm Spec}\,R(\tilde{A},\tilde{P}_{0})$ the one arising from the associated ring of Type 2. Then $X\times{\mathbb{C}}^{*}$ is isomorphic to the principal open subset $\tilde{X}_{\tilde{T}_{01}}\subseteq\tilde{X}$. In particular, $X$ is rational if and only if $\tilde{X}$ is so.

Corollary 3.5**.**

Let $R(A,P_{0})$ be a ring of Type 1. Then $X={\rm Spec}\,R(A,P_{0})$ is rational if and only if one of the following conditions holds:

1. (i)

One has $\mathfrak{l}_{i}=1$ for all $1\leq i\leq r$, in other words, $R(A,P_{0})$ is factorial. 2. (ii)

There is exactly one $1\leq i\leq r$ with $\mathfrak{l}_{i}>1$. 3. (iii)

There are $1\leq i<j\leq r$ with $\mathfrak{l}_{i}=\mathfrak{l}_{j}=2$ and $\mathfrak{l}_{u}=1$ whenever $u\notin\left\{i,j\right\}$

Proof.

Combine Corollary 3.4 with the rationality criterion Remark 2.2. ∎

Lemma 3.6**.**

Let $R(A,P_{0})$ be of Type 1 with $X:={\rm Spec}\,R(A,P_{0})$ rational and assume that $(\mathfrak{l}_{1},\dots,\mathfrak{l}_{r})$ is decreasingly ordered. Then the number $c(i)$ of irreducible components of $V(X,T_{ij})$ is given as

[TABLE]

Proof.

Due to Corollary 3.4, we can realize $X\times{\mathbb{C}}^{*}$ as a principal open subset of the associated variety $\tilde{X}$ of Type 2. Then the irreducible components of $V(X,T_{ij})\times{\mathbb{C}}^{*}$ are in one-to-one correspondence with the irreducible components $X\cap V(\tilde{X},\tilde{T}_{ij})$. The assertions follows. ∎

Proposition 3.7**.**

Let $R(A,P_{0})$ be non-factorial of Type 1 with ${\rm Spec}\,R(A,P_{0})$ rational and $(\mathfrak{l}_{1},\ldots,\mathfrak{l}_{r})$ decreasingly ordered. Define numbers ${n^{\prime}:=c(1)n_{1}+\ldots+c(r)n_{r}}$ and

[TABLE]

Then the vectors $l_{i,\alpha}\in{\mathbb{Z}}^{n_{i,\alpha}}$ build up an $r^{\prime}\times(n^{\prime}+m)$ matrix $P_{0}^{\prime}$. With a suitable matrix $A^{\prime}$ the affine variety ${\rm Spec}\,R(A^{\prime},P^{\prime}_{0})$ is the total coordinate space of the affine variety ${\rm Spec}\,R(A,P_{0})$.

Proof.

First observe that the kernel of ${\mathbb{Z}}^{n+m}\rightarrow K_{0}/K_{0}^{\mathrm{tors}}$ is generated by the rows of the following $r\times(n+m)$ matrix:

[TABLE]

Now one determines the Cox ring of $X={\rm Spec}\,R(A,P_{0})$ in the same manner as in the proof of [1, Prop. 6.6] by exchanging the matrix $P_{1}$ used there by the matrix above and applying Lemma 3.6. ∎

Proof of Theorem 1.2.

If $R(A,P_{0})$ is rational of Type 1, then Proposition 3.7 shows that the Cox ring of ${\rm Spec}\,R(A,P_{0})$ is factorial. Thus, Cox ring iteration is possible for $X$ if and only if the total coordinate space of $X$ is rational. Moreover, if the latter holds then the Cox ring iteration ends with at most one step. ∎