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On the connectedness of the automorphism group of an affine toric variety V. V. Kikteva Faculty of Computer Science, National Research University Higher School of Economics, Moscow, Russia
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    § 1. IntroductionLet $\mathbb{K}$ be an algebraically closed field of characteristic zero and $X$ be an algebraic variety over $\mathbb{K}$. We denote the group of regular automorphisms of the variety $X$ by $\operatorname{Aut}(X)$. In general, $\operatorname{Aut}(X)$ is not an algebraic group. However, for subgroups of $\operatorname{Aut}(X)$ connectedness can be defined. This concept was introduced in [1]; also see [2]. Let $S$ be an irreducible affine algebraic variety. Then any map $S \to \operatorname{Aut}(X)$, $s \mapsto \varphi_s$, defines a family $\{\varphi_s\}_{s \in S}$ in the automorphism group of $X$, which is parameterized by the variety $S$. A family is called algebraic if the map $S \times X \to X$, given by the rule $(s, x) \mapsto \varphi_s(x)$, is a morphism of algebraic varieties. Let $G$ be a subgroup of $\operatorname{Aut}(X)$. If for every element $g \in G$ there exists an algebraic family $\{\varphi_s\}_{s \in S}$ that contains $g$ and the identity automorphism, then $G$ is called a connected subgroup of $\operatorname{Aut}(X)$. The neutral component $\operatorname{Aut}(X)^0$ of the automorphism group of $X$ is the subgroup generated by the elements of all algebraic families containing the identity automorphism. It follows from [3] that the automorphism group of a nondegenerate affine toric variety of dimension at least two is infinite-dimensional, and therefore it is not an algebraic group. In contrast to the affine case, the automorphism group of a complete toric variety is an affine algebraic group. The automorphism groups of complete simplicial toric varieties were studied in [4] and [5]. Note that Corollary 4.7 in [4] contains a description of the neutral components and component groups of automorphism groups for complete simplicial toric varieties. There are examples of affine toric varieties with connected and disconnected automorphism groups. It was shown in [6], Lemma 4, and [2], Theorem 6, that for any positive integer number $n$ the automorphism group of the $n$-dimensional affine space is connected, that is, $\operatorname{Aut}(\mathbb{A}^n) = \operatorname{Aut}(\mathbb{A}^n)^0$. An example of an affine toric variety with disconnected automorphism group is the algebraic torus $T = (\mathbb{K}^{\times})^n$. It is well known that the automorphism group of the torus $T$ is isomorphic to $\mathrm{GL}_n(\mathbb{Z}) \rightthreetimes (\mathbb{K}^{\times})^n$ and is disconnected. These considerations lead naturally to the question whether the automorphism group of an affine toric variety is connected. The aim of this paper is to establish a criterion for the automorphism group of an affine toric variety to be connected. It is shown that the automorphism group of a degenerate affine toric variety is disconnected, and the automorphism group of a nondegenerate affine toric variety is connected if and only if there are no nontrivial automorphisms of the divisor class group that permute the classes of prime divisors invariant under the action of the acting torus. The necessary definitions are given in § 2. A criterion for the automorphism group of an affine toric variety to be connected is proved in § 3: see Theorem 1 and Corollary 1. Section 4 describes the component group of the automorphism group of a nondegenerate affine toric variety. It is shown that this component group is finite. It is remarkable that the description of the component group of the automorphism group is analogous to the description of the component group in the case of complete simplicial toric varieties. Section 5 contains an application of the connectedness criterion to the case of toric surfaces: see Proposition 3. Section 6 is dedicated to examples illustrating the results obtained. AcknowledgementsThe author is grateful to her scientific advisor Sergey Gaifullin and to Ivan Arzhantsev for their constant attention to this work. § 2. Preliminaries2.1. Toric varietiesRecall some facts on toric varieties. More detailed information and proofs can be found in [7] and [8]. A normal irreducible algebraic variety $X$ is called toric if it contains an algebraic torus $T = (\mathbb{K}^{\times})^n$ as a dense open subset in the Zariski topology, and the action of the torus on itself can be extended to a regular action of $T$ on the whole of $X$. Let $X$ be an affine toric variety with an acting torus $T$. We denote the lattice of one-parameter subgroups $\lambda\colon \mathbb{K}^{\times} \to T$ by $N$ and let $M = \operatorname{Hom}(N, \mathbb{Z})$ denote the dual lattice. We associate the lattice $M$ with the lattice of characters $\chi\colon T \to \mathbb{K}^{\times}$, where the pairing $N \times M \to \mathbb{Z}$ is given by
$$
\begin{equation*}
(\lambda, \chi) \to \langle\lambda, \chi\rangle, \quad \text{where } c^{\langle\lambda, \chi\rangle} =\chi(\lambda(c)) \text{ for } c\in \mathbb{K}^{\times}.
\end{equation*}
\notag
$$
Recall the correspondence between affine toric varieties and rational polyhedral cones. Consider a polyhedral cone $\sigma$ in the rational vector space $N_{\mathbb{Q}} = N \otimes_{\mathbb{Z}} \mathbb{Q}$. The dual cone $\sigma^{\vee}$ in the space $M_{\mathbb{Q}} = M \otimes_{\mathbb{Z}} \mathbb{Q}$ is defined by
$$
\begin{equation*}
\sigma^{\vee}=\{ m\in M_{\mathbb{Q}}\mid \langle u,m\rangle\geqslant 0 \ \forall\, u\in \sigma\}.
\end{equation*}
\notag
$$
The variety $X_{\sigma} = \operatorname{Spec}(\mathbb{K}[\sigma^{\vee} \cap M])$ is toric, and any affine toric variety can be constructed in this way. $T$-orbits on the variety $X_\sigma$ correspond to faces of the cone $\sigma$. In particular, with each ray of the cone $\sigma$ one can associate a prime $T$-invariant divisor on the variety $X_\sigma$, which is the closure of the corresponding $T$-orbit. A vector is called primitive if it is the shortest integer vector on its ray. If the cone $\sigma$ contains $r$ rays with primitive vectors $v_1, \dots, v_r$, then the corresponding prime $T$-invariant divisors are denoted by $D_1, \dots, D_r$. A toric variety is said to be nondegenerate if it has only constant invertible regular functions. A toric variety $X$ is nondegenerate if and only if it cannot be represented as a direct product of some toric variety and an algebraic torus. This condition is equivalent to the cone $\sigma$ corresponding to the variety $X$ being full-dimensional. In §§ 2.2 and 2.3 we assume $X$ to be a nondegenerate affine toric variety corresponding to the cone $\sigma$. 2.2. The divisor class groupWe denote by $\operatorname{WDiv}(X)$ the group of Weil divisors on a normal algebraic variety $X$ and by $\operatorname{PDiv}(X)$ the subgroup of principal divisors, that is,
$$
\begin{equation*}
\operatorname{PDiv}(X)=\{\operatorname{div}(f)\mid f\in\mathbb{K}(X)^{\times}\},
\end{equation*}
\notag
$$
where $\operatorname{div}(f)$ denotes the divisor of the rational function $f$. The divisor class group $\operatorname{Cl}(X)$ of the variety $X$ is defined as the quotient group of the group of Weil divisors by the subgroup of principal divisors:
$$
\begin{equation*}
\operatorname{Cl}(X)=\operatorname{WDiv}(X)/\operatorname{PDiv}(X)
\end{equation*}
\notag
$$
(see [9], Ch. III, § 1). We denote by $\operatorname{WDiv}_{T}(X)$ the subgroup of Weil divisors invariant under the action of $T$. It is freely generated by the $T$-invariant prime divisors $D_1, \dots, D_r$. For each element $m \in M$ let $\chi^m$ denote the corresponding character of the torus. According to Theorem 4.1.3 in [7], there exists an exact sequence
$$
\begin{equation*}
M\to \operatorname{WDiv}_{T}(X)\to \operatorname{Cl}(X)\to 0,
\end{equation*}
\notag
$$
where the first map is given by $m \mapsto \operatorname{div}(\chi^m)$, and the second map takes the $T$-invariant divisor $D$ to its class $[D] \in \operatorname{Cl}(X)$. By Proposition 4.1.2 in [7] we have
$$
\begin{equation*}
\operatorname{div}(\chi^m)=\sum_{i=1}^r\langle v_i, m\rangle D_i
\end{equation*}
\notag
$$
in the notation of § 2.1. Thus,
$$
\begin{equation*}
\begin{aligned} \, \operatorname{Cl}(X) &\simeq \langle D_1,\dots,D_r \rangle / \langle \operatorname{div}(\chi^{e_j})\mid j=1,\dots,n \rangle \\ &=\langle D_1,\dots,D_r \rangle/\biggl\langle \sum_{i=1}^r v_{ij}D_i\Bigm|j=1,\dots,n \biggr\rangle, \end{aligned}
\end{equation*}
\notag
$$
where $v_{i1}, \dots, v_{ij}$ are the coordinates of the vector $v_i$ in the basis of the lattice $N$, dual to the basis $e_1, \dots, e_n$ of the lattice $M$. In particular, it follows that the class group of an affine toric variety is finitely generated. Automorphisms in $\operatorname{Aut}(X)$ act naturally on the set of prime divisors and, consequently, on the group of Weil divisors. Principal divisors are mapped to principal ones under this action, and thus we have an action of $\operatorname{Aut}(X)$ on the class group $\operatorname{Cl}(X)$. Therefore, there exists a homomorphism
$$
\begin{equation*}
\widetilde{\alpha}\colon \operatorname{Aut}(X)\to \operatorname{Aut}(\operatorname{Cl}(X)).
\end{equation*}
\notag
$$
Given an automorphism $\varphi \in \operatorname{Aut}(X)$, the automorphism $\widetilde{\alpha}(\varphi) \in \operatorname{Aut}(\operatorname{Cl}(X))$ acts as follows:
$$
\begin{equation*}
\widetilde{\alpha}(\varphi)\colon [D]\mapsto [\varphi(D)].
\end{equation*}
\notag
$$
Consider the antihomomorphism
$$
\begin{equation*}
\alpha\colon \operatorname{Aut}(X)\to \operatorname{Aut}(\operatorname{Cl}(X)), \qquad \varphi\mapsto \widetilde{\alpha}(\varphi^{-1}).
\end{equation*}
\notag
$$
Note that
$$
\begin{equation}
\operatorname{Ker}\alpha=\operatorname{Ker}\widetilde{\alpha}
\end{equation}
\tag{2.1}
$$
because
$$
\begin{equation*}
\begin{aligned} \, &\widetilde{\alpha}(\varphi)=\operatorname{id}_{\operatorname{Cl}(X)} \quad\Longleftrightarrow\quad [D]=[\varphi(D)]\ \forall\, D\in \operatorname{WDiv}(X) \\ &\qquad \quad\Longleftrightarrow\quad [D]=[\varphi^{-1}(D)]\ \forall\, D\in \operatorname{WDiv}(X) \quad\Longleftrightarrow\quad \alpha(\varphi)=\operatorname{id}_{\operatorname{Cl}(X)}. \end{aligned}
\end{equation*}
\notag
$$
Moreover,
$$
\begin{equation}
\widetilde{\alpha}(\operatorname{Aut}(X))=\alpha(\operatorname{Aut}(X)).
\end{equation}
\tag{2.2}
$$
Indeed,
$$
\begin{equation*}
\xi\in \widetilde{\alpha}(\operatorname{Aut}(X)) \quad\Longleftrightarrow\quad \xi^{-1}\in \widetilde{\alpha}(\operatorname{Aut}(X)) \quad\Longleftrightarrow\quad \xi\in \alpha(\operatorname{Aut}(X)).
\end{equation*}
\notag
$$
In [10], Lemma 2.2, it was proved that for a nondegenerate affine toric variety $X$ the component $\operatorname{Aut}(X)^0$ is contained in the kernel of the action of the automorphism group of $X$ on the class group of $X$. In Proposition 2 we show that
$$
\begin{equation*}
\operatorname{Aut}(X)^0=\operatorname{Ker}\widetilde{\alpha}=\operatorname{Ker}\alpha.
\end{equation*}
\notag
$$
2.3. Cox ringsCox rings were first introduced in [4]. For detailed information the reader is also referred to [11]. Recall the construction of the Cox ring for a normal algebraic variety $X$ with only constant invertible regular functions and finitely generated divisor class group. For a Weil divisor $D$ on $X$ consider the vector space
$$
\begin{equation*}
L(X, D):=\{f\in \mathbb{K}(X)^{\times}\mid \operatorname{div}(f)+D \geqslant 0 \} \cup \{0\}.
\end{equation*}
\notag
$$
For a subgroup $K \subseteq \operatorname{WDiv}(X)$ consider the $K$-graded $\mathbb{K}[X]$-algebra
$$
\begin{equation*}
S_K:=\bigoplus_{D\in K}S_D, \quad\text{where } S_D=L(X,D).
\end{equation*}
\notag
$$
Multiplication in $S_K$ is defined on homogeneous elements as follows. If $f_1 \in S_{D_1}$ and $f_2 \in S_{D_2}$, then their product in $S_K$ is the product $f_1f_2$ in $\mathbb{K}(X)$ regarded as an element of $S_{D_1 + D_2}$. For arbitrary elements of $S_K$ multiplication is defined by distributivity. In the group of Weil divisors we can choose a free finitely generated subgroup $K$, which maps surjectively onto the divisor class group after taking the quotient by the subgroup of principal divisors. Consider the group homomorphism
$$
\begin{equation*}
\chi\colon K \cap\operatorname{PDiv}(X)\to \mathbb{K}(X)^{\times}
\end{equation*}
\notag
$$
such that Let $I$ denote the ideal of $S_K$ generated by the elements $1 - \chi(E)$ for all Weil divisors $E \in K \cap \operatorname{PDiv}(X)$, where $1$ is a homogeneous element of degree $0$ and the element $\chi(E)$ is homogeneous and has degree $-E$. The Cox ring is defined as the quotient ring $R(X) := S_K / I$. This ring is graded by the divisor class group of $X$: where $R(X)_0 = \mathbb{K}[X]$. It is known that the Cox ring of the variety $X$ is independent of the choice of the subgroup $K$ and the homomorphism $\chi$ up to an isomorphism of $\operatorname{Cl}(X)$-graded rings.The Néron–Severi quasi-torus for the variety $X$ is a quasi-torus $N(X)$ whose character group is isomorphic to $\operatorname{Cl}(X)$. The quasi-torus $N(X)$ acts on $R(X)$ by automorphisms, and the $R(X)_u$ are the weight subspaces for this action, that is, $N$ acts on $R(X)_u$ by multiplication by the corresponding character. Thus, $\operatorname{Cl}(X)$-homogeneous components of $R(X)$ are invariant under the action of elements of $N(X)$. It was proved in [3], Theorem 5.1, that for an irreducible normal affine variety with finitely generated divisor class group and only constant invertible regular functions there exists an exact sequence
$$
\begin{equation}
1\to N(X) \to \widetilde{\operatorname{Aut}}(R(X)) \xrightarrow{\beta} \operatorname{Aut}(X) \to 1,
\end{equation}
\tag{2.3}
$$
where $\widetilde{\operatorname{Aut}}(R(X))$ denotes the set of the automorphisms of the Cox ring that normalize the $\operatorname{Cl}(X)$-grading:
$$
\begin{equation*}
\begin{aligned} \, \widetilde{\operatorname{Aut}}(R(X)) &:=\bigl\{\varphi \in \operatorname{Aut}(R(X))\mid \exists\, \varphi_0 \in \operatorname{Aut}(\operatorname{Cl}(X))\colon \\ &\qquad\qquad \varphi(R(X)_u)=R(X)_{\varphi_0(u)}\ \forall\, u\in \operatorname{Cl}(X)\bigr\}. \end{aligned}
\end{equation*}
\notag
$$
Since automorphisms in $\widetilde{\operatorname{Aut}}(R(X))$ normalize the $\operatorname{Cl}(X)$-grading, the component $R(X)_0$ is an invariant subset for any $\psi \in \widetilde{\operatorname{Aut}}(R(X))$. Therefore, the restriction $\psi|_{R(X)_0}$ is well defined. The antihomomorphism $\beta$ is defined by the following rule: $\beta(\psi) = \varphi$ if that is, for any $x \in X$ and $f \in \mathbb{K}[X]$. By the definition of the group $\widetilde{\operatorname{Aut}}(R(X))$, an automorphism of the Cox ring that normalizes the $\operatorname{Cl}(X)$-grading can be associated with an automorphism of the class group. Thus, we have a group homomorphism
$$
\begin{equation*}
\gamma\colon \widetilde{\operatorname{Aut}}(R(X)) \to \operatorname{Aut}(\operatorname{Cl}(X)).
\end{equation*}
\notag
$$
It was proved in [4] that for a nondegenerate toric variety $X = X_\sigma$ the Cox ring is isomorphic to the polynomial ring in $r$ variables over the field $\mathbb{K}$, where $r$ denotes the number of the rays of the cone $\sigma$: where the $T_i$ are homogeneous with respect to the $\operatorname{Cl}(X)$-grading and ${\deg(T_i) \!=\! [D_i]}$.§ 3. Criterion for an automorphism group to be connectedProposition 1. Let $X$ be a degenerate affine toric variety with acting torus $T$. Then the automorphism group of $X$ is disconnected. Proof. Since $X$ is a degenerate affine toric variety, it can be decomposed into a direct product $X = Y \times \widetilde{T}$, and for the torus $T$ we have $T = \overline{T} \times \widetilde{T}$, where $\overline{T}$ and $\widetilde{T}$ are algebraic tori and $Y$ is a nondegenerate affine toric variety with acting torus $\overline{T}$. Thus, we have the following equality:
$$
\begin{equation}
\mathbb{K}[X]=\mathbb{K}[Y] \otimes \mathbb{K}[\widetilde{T}]=\mathbb{K}[Y] \otimes \mathbb{K}[t_1,t_1^{-1},\dots,t_q,t_q^{-1}],
\end{equation}
\tag{3.1}
$$
where $t_1, \dots, t_q$ are the coordinate functions on the torus $\widetilde{T}$. Take an automorphism $\varphi \in \operatorname{Aut}(X)$. Then $\varphi^*$ is an automorphism of the algebra (3.1). We show that $\varphi^*$ acts as follows on the coordinate functions $y_1, \dots, y_p$ of the variety $Y$ and $t_1, \dots, t_q$ of the torus $\widetilde{T}$:
$$
\begin{equation}
\varphi^*\colon y_i\mapsto \varphi^*(y_i), \quad i=1,\dots,p; \qquad t_j\mapsto \nu^* (t_j), \quad j=1,\dots,q,
\end{equation}
\tag{3.2}
$$
for some automorphism $\nu^*$ of the algebra $\mathbb{K}[\widetilde{T}]$. Indeed, under the action of an automorphism invertible functions, in particular, $t_i$, are mapped to invertible ones, and the algebra $\mathbb{K}[Y]$ does not contain invertible functions other than constants. Since invertible elements of the algebra (3.1) are Laurent monomials of the variables $t_1, \dots, t_q$, the elements $\nu^*(t_j)$ do not depend on $y_1, \dots, y_p$.
Suppose that the group $\operatorname{Aut}(X)$ is connected. Let us show that this implies the connectedness of $\operatorname{Aut}(\widetilde{T})$, which is a contradiction. We choose an automorphism $\psi \in \operatorname{Aut}(\widetilde{T})$ and construct an automorphism $\varphi \in \operatorname{Aut}(X)$ such that for $\varphi^*$ the following holds:
$$
\begin{equation*}
\varphi^*\colon y_i\mapsto y_i, \quad i=1,\dots,p; \qquad t_j\mapsto \psi^*(t_j), \quad j=1,\dots,q.
\end{equation*}
\notag
$$
From the connectedness of $\operatorname{Aut}(X)$ it follows that $\varphi$ can be included in some algebraic family $\{\varphi_s\}_{s \in S}$ containing the identity automorphism. Using (3.2) we see that each automorphism $\varphi_s^*$ has the form
$$
\begin{equation*}
\varphi_s^*\colon y_i\mapsto \varphi_s^*(y_i), \quad i=1,\dots,p; \qquad t_j\mapsto \nu_s^* (t_j), \quad j=1,\dots,q,
\end{equation*}
\notag
$$
where $\nu_s^*$ is an automorphism of the algebra $\mathbb{K}[\widetilde{T}]$. Therefore, $\{ \nu_s \}_{s \in S}$ is an algebraic family containing the automorphism $\psi$ and the identity automorphism of the torus $\widetilde{T}$.
Proposition 1 is proved. Now assume that $X$ is a nondegenerate affine toric variety. For such varieties the Cox ring $R(X)$ is well defined; see § 2. Recall that the antihomomorphisms
$$
\begin{equation*}
\alpha\colon \operatorname{Aut}(X)\to\operatorname{Aut}(\operatorname{Cl}(X))
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\beta\colon \widetilde{\operatorname{Aut}}(R(X))\to \operatorname{Aut}(X)
\end{equation*}
\notag
$$
and the homomorphism
$$
\begin{equation*}
\gamma\colon \widetilde{\operatorname{Aut}}(R(X))\to \operatorname{Aut}(\operatorname{Cl}(X))
\end{equation*}
\notag
$$
were introduced in the same section. Lemma 1. For the maps $\alpha$, $\beta$ and $\gamma$ the equality $\alpha \circ \beta = \gamma$ holds. Proof. Let $\varphi \in \operatorname{Aut}(X)$. Following [3], we construct an automorphism from $\beta^{-1}(\varphi)$. Let us show that $\varphi^*$ extends to a map for any Weil divisor $D \in \operatorname{WDiv}(X)$. Indeed, for any $x \in X$ and $f \in \mathbb{K}(X)^{\times}$. Therefore,
$$
\begin{equation}
\operatorname{div}(\varphi^*(f))=\varphi^{-1}(\operatorname{div}(f)),
\end{equation}
\tag{3.4}
$$
and the chain of equivalences
$$
\begin{equation*}
\begin{aligned} \, &f\in L(X,D) \quad\Longleftrightarrow\quad \operatorname{div}(f)+D\geqslant 0 \quad\Longleftrightarrow\quad \varphi^{-1}(\operatorname{div}(f))+\varphi^{-1}(D)\geqslant 0 \\ &\qquad \stackrel{(3.4)}{\Longleftrightarrow}\quad \operatorname{div}(\varphi^*(f))+\varphi^{-1}(D)\geqslant 0 \quad\Longleftrightarrow\quad \varphi^*(f)\in L(X,\varphi^{-1}(D)) \end{aligned}
\end{equation*}
\notag
$$
holds for any $f \in \mathbb{K}(X)^{\times}$ and $D \in \operatorname{WDiv}(X)$. Thus, (3.3) is proved. Consequently,
$$
\begin{equation*}
\varphi^*\colon S_K=\bigoplus_{D\in K}L(X,D)\to S_{\varphi^{-1}(K)}=\bigoplus_{D\in\varphi^{-1}(K)}L(X,D).
\end{equation*}
\notag
$$
Let us prove that $\varphi^*$ defines consistently an automorphism of $R(X)$. The image of $1 - \chi(E)$ under the map $\varphi^*$ is $1 - \varphi^*(\chi(E))$, where $1$ is a homogeneous element of degree $0$ and $\varphi^*(\chi(E))$ has degree $-\varphi^{-1}(E)$ by (3.3). We define the group homomorphism
$$
\begin{equation*}
\chi'=\varphi^* \circ \chi \circ \varphi\colon \operatorname{PDiv}(X)\cap \varphi^{-1}(K)\to\mathbb{K}(X)^{\times}.
\end{equation*}
\notag
$$
The homomorphism $\chi'$ constructed satisfies the equality $\operatorname{div}(\chi'(D)) = D$ for each Weil divisor $D \in \operatorname{PDiv}(X) \cap \varphi^{-1}(K)$. Indeed,
$$
\begin{equation*}
\operatorname{div}(\chi'(D))=\operatorname{div}(\varphi^*\circ\chi\circ\varphi(D))\stackrel{(3.4)}{=} \varphi^{-1}(\operatorname{div}(\chi\circ\varphi(D)))=\varphi^{-1}(\varphi(D))=D
\end{equation*}
\notag
$$
for each principal Weil divisor $D$ in $\varphi^{-1}(K)$. Thus, $\varphi^*$ maps the ideal $I$ to the ideal of the ring $S_{\varphi^{-1}(K)}$ generated by the elements $1 - \chi'(D)$ for all Weil divisors $D$ in $\operatorname{PDiv}(X) \cap \varphi^{-1}(K)$, where $1$ is a homogeneous element of degree $0$ and the element $\chi'(D)$ is homogeneous and has degree $-D$. Therefore, $\varphi^*$ is a homomorphism that maps the Cox ring constructed using the homomorphism $\chi$ and subgroup $K$ to the Cox ring constructed using the homomorphism $\chi'$ and subgroup $\varphi^{-1}(K)$. The Cox ring does not depend on the choice of a homomorphism and a subgroup in the group of Weil divisors satisfying the conditions from § 2.3. Moreover, a homomorphism of the Cox ring to itself obtained from the automorphism $\varphi^{-1}$ by the same construction is the inverse of $\varphi^*$. Hence, $\varphi^*$ is an automorphism of the Cox ring.
Thus it has been shown that the set $\beta^{-1}(\varphi)$ contains the automorphism $\varphi^*$, which maps the Cox ring component of degree $[D]$ to the component of degree $[\varphi^{-1}(D)]$ with respect to the $\operatorname{Cl}(X)$-grading. It remains to note that since the sequence (2.3) is exact, any other automorphism in $\beta^{-1}(\varphi)$ differs from $\varphi^*$ by an automorphism corresponding to the action of some element of the Néron–Severi quasi-torus. Homogeneous components of $R(X)$ are preserved by the action of $N(X)$. Consequently, for any automorphism $\psi^* \in \widetilde{\operatorname{Aut}}(R(X))$ and any Weil divisor $D$ we have
$$
\begin{equation*}
\gamma(\psi^*)\colon [D]\mapsto [\beta(\psi^*)^{-1}(D)]=\alpha\circ\beta(\psi^*)([D]).
\end{equation*}
\notag
$$
Thus, Lemma 1 is proved. Consider the kernel of the homomorphism $\gamma$. It consists precisely of those automorphisms of the Cox ring that preserve the $\operatorname{Cl}(X)$-grading. With any element $g^* \in \operatorname{Ker}\gamma$ one can associate an automorphism
$$
\begin{equation*}
g\in\operatorname{Aut}(\operatorname{Spec}(R(X))) = \operatorname{Aut}(\mathbb{A}^r).
\end{equation*}
\notag
$$
We set
$$
\begin{equation*}
G:=\{ g\in \operatorname{Aut}(\mathbb{A}^r)\mid g^*\in\operatorname{Ker}\gamma\}\subseteq \operatorname{Aut}(\mathbb{A}^r).
\end{equation*}
\notag
$$
Proof. Using the methods of Lemma 4 in [6] and Theorem 6 in [2] we show that each automorphism in $G$ is a composition of automorphisms from some subgroups $A$ and $H$, which are connected subgroups of $G$. This implies the connectedness of $G$ in $\operatorname{Aut}(\mathbb{A}^r)$.
Recall that the Cox ring of the toric variety $X$ is a polynomial ring with a $\operatorname{Cl}(X)$-grading:
$$
\begin{equation*}
R(X)=\mathbb{K}[T_1,\dots,T_r], \quad\text{where } \operatorname{deg}(T_i)=[D_i]\in\operatorname{Cl}(X).
\end{equation*}
\notag
$$
For any automorphism $\varphi^* \in\operatorname{Aut}(R(X))$ consider the homomorphism $l(\varphi^*)$ of the algebra $R(X)$ into itself constructed as follows. Suppose that $\varphi^*$ acts on the variables according to the formula
$$
\begin{equation}
\varphi^*\colon T_i \mapsto F_{i0}+F_{i1}(T_{1},\dots,T_{r})+\dots+F_{im}(T_{1},\dots,T_{r}),
\end{equation}
\tag{3.5}
$$
where $F_{ij}(T_{1},\dots,T_{r})$ is a form of degree $j$ in $T_1,\dots,T_r$. Then we set
$$
\begin{equation*}
l(\varphi^*)\colon T_i\mapsto F_{i0}+F_{i1}(T_{1},\dots,T_{r})
\end{equation*}
\notag
$$
and extend it to $R(X)=\mathbb{K}[T_1,\dots,T_r]$ by linearity and multiplicativity.
Note that
$$
\begin{equation*}
l(\varphi^* \circ \psi^*)=l(\varphi^*)\circ l(\psi^*)
\end{equation*}
\notag
$$
for any $\varphi^*,\psi^*\in\operatorname{Aut}(R(X))$. Hence
$$
\begin{equation*}
\operatorname{id}_{R(X)}=l(\varphi^* \circ (\varphi^*)^{-1})=l(\varphi^*)\circ l((\varphi^*)^{-1}),
\end{equation*}
\notag
$$
which means that $l(\varphi^*)^{-1}=l((\varphi^*)^{-1})$ and $l(\varphi^*)$ is invertible. Therefore, $l(\varphi^*)$ is an automorphism of the algebra $R(X)$ for any $\varphi^*\in \operatorname{Aut}(R(X))$.
Further, consider the set
$$
\begin{equation*}
A^*:= \{l(\varphi^*)\mid \varphi^*\in\operatorname{Ker}\gamma \}.
\end{equation*}
\notag
$$
Let us check that if an automorphism $\varphi^*$ normalizes the $\operatorname{Cl}(X)$-grading, then $l(\varphi^*)$ also normalizes it. To do this we show that for any element $g \in R(X)$ we have
$$
\begin{equation}
\operatorname{deg}(\varphi^*(g))=\operatorname{deg}(l(\varphi^*)(g)).
\end{equation}
\tag{3.6}
$$
Then we can take $\gamma(\varphi^*)$ as $l(\varphi^*)_0$ for $l(\varphi^*)$ in the definition of $\widetilde{\operatorname{Aut}}(R(X))$. For $T_1,\dots,T_r$ equality (3.6) is satisfied because all terms in (3.5) are homogeneous and have degree $\gamma(\varphi^*)([D_i])$. For products of homogeneous elements and sums of homogeneous elements of the same degree equality (3.6) is obtained by the linearity and multiplicativity of $l(\varphi^*)$.
Moreover, $\gamma(\varphi^*)=\gamma(l(\varphi^*))$. Therefore, for any $\varphi^*\in\operatorname{Ker}\gamma$ the automorphism $l(\varphi^*)$ is also contained in the kernel of $\gamma$. It is straightforward to verify that $A^*$ is a subgroup of $\operatorname{Ker}\gamma$. Consider the subgroup Let us prove that $A$ is connected. Suppose that there exist precisely $k$ distinct elements $d_1,\dots,d_k$ among $[D_1],\dots,[D_r]$ and for each $i=1,\dots,k$ there are precisely $n_i$ variables $T_j$ of degree $d_i$, that is, $r=n_1+\dots+n_k$. Let us introduce new notation for the indices of $T_j$. Namely, let
$$
\begin{equation*}
\begin{gathered} \, \operatorname{deg}(T_{11})=\dots=\operatorname{deg}(T_{1n_1})=d_1, \\ \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots \\ \operatorname{deg}(T_{k1})=\dots=\operatorname{deg}(T_{kn_k})=d_k. \end{gathered}
\end{equation*}
\notag
$$
If there is no zero element among $d_1,\dots,d_k$, then the group
$$
\begin{equation}
A\simeq A^* = \operatorname{GL}_{n_1}(\mathbb{K})\times\dots\times \operatorname{GL}_{n_k}(\mathbb{K})
\end{equation}
\tag{3.7}
$$
is connected. If $d_i=0\in\operatorname{Cl}(X)$, then the corresponding factor $\mathrm{GL}_{n_i}(\mathbb{K})$ in the product (3.7) is replaced by $\mathrm{GL}_{n_i}(\mathbb{K}) \rightthreetimes \mathbb{K}^{n_i}$. In this case $A$ is also connected.
By definition set
$$
\begin{equation*}
H^*:=\{ l(\varphi^*)^{-1}\circ \varphi^*\mid \varphi^*\in\operatorname{Ker}\gamma\} =\{\varphi^*\in\operatorname{Ker}\gamma\mid l(\varphi^*)=\operatorname{id}_{R(X)} \}.
\end{equation*}
\notag
$$
Note that $H^*$ is a subgroup of $\operatorname{Ker}\gamma$.
We consider the subgroup and prove that it is connected. Fix any automorphism $h\in H$ and the corresponding automorphism $h^*\in H^*$. The linear part of $h^*$ is the identity automorphism. Therefore, $h^*$ is of the form
$$
\begin{equation*}
h^*(T_i)=T_i+ \sum_{j=2}^{m_i} h_{ij}, \qquad i=1,\dots,r,
\end{equation*}
\notag
$$
for some integers $m_i\in\mathbb{N}_{\geqslant 2}$, where each $h_{ij}$ is either zero or a homogeneous form of degree $j$ in $T_1,\dots, T_r$. Here homogeneity is understood in the sense of the standard grading $\deg(T_j)=1\in \mathbb{Z}$ for $j=1,\dots,r$.
For any element $t\in \mathbb{K}^{\times}$ we denote by $\xi_{t}^*\in A^*$ the automorphism acting on the variables as follows: Let then
$$
\begin{equation*}
h^*_t(T_i)=T_i+\sum_{j=2}^{m_i}t^{j-1}h_{ij}, \qquad i=1,\dots,r.
\end{equation*}
\notag
$$
Set $h^*_0:=\mathrm{id}_{R(X)}$; then $\{h_t\}_{t\in\mathbb{K}}\subseteq H$ is an algebraic family containing $h=h_1$ and $\mathrm{id}_{\mathbb{A}^r}=h_0$. Consequently, the subgroup $H$ is connected.
It remains to note that for any $\varphi^*\in \operatorname{Ker}\gamma$ we have
$$
\begin{equation*}
\varphi^*=l(\varphi^*)\circ l(\varphi^*)^{-1}\circ \varphi^*=a^*\circ h^*,
\end{equation*}
\notag
$$
where $a^*=l(\varphi^*)\in A^*$ and $h^*=l(\varphi^*)^{-1}\circ \varphi^*\in H^*$. Thus, for each $\varphi\in G$ there exists a factorization $\varphi=h\circ a$, where $h\in H$ and $a\in A$. Since the subgroups $A$ and $H$ are connected, so is the group $G$.
Lemma 2 is proved. Note that the antihomomorphism $\beta$ maps the kernel of $\gamma$ surjectively onto the kernel of $\alpha$. Indeed, if an element $f$ belongs to the kernel of $\alpha$, then its $\beta$-preimage exists and lies in the group $\widetilde{\operatorname{Aut}}(R(X))$ because $\beta$ is surjective. Then it follows from Lemma 1 that $\gamma(\beta^{-1}(f))=\alpha(f)=\mathrm{id}_{\operatorname{Cl}(X)}$ and the element $\beta^{-1}(f)$ belongs to $\operatorname{Ker}\gamma$. Proof. Recall that
$$
\begin{equation*}
\operatorname{Ker}(\operatorname{Aut}(X)\curvearrowright \operatorname{Cl}(X))=\operatorname{Ker}\widetilde{\alpha} \stackrel{(2.1)}{=} \operatorname{Ker}\alpha.
\end{equation*}
\notag
$$
Let us show that $\operatorname{Ker}\alpha$ is a connected subgroup of $\operatorname{Aut}(X)$. Fix an automorphism
$$
\begin{equation*}
\varphi\in \operatorname{Ker} \alpha\subseteq \operatorname{Aut}(X).
\end{equation*}
\notag
$$
By the surjectivity of $\beta$ there exists
$$
\begin{equation*}
\psi^*\in\operatorname{Ker}\gamma\subseteq \widetilde{\operatorname{Aut}}(R(X))
\end{equation*}
\notag
$$
such that $\beta(\psi^*)=\varphi$. This automorphism $\psi^*$ of the Cox ring corresponds to an automorphism
$$
\begin{equation*}
\psi\in G\subseteq \operatorname{Aut}(\operatorname{Spec}(R(X)))=\operatorname{Aut}(\mathbb{A}^r).
\end{equation*}
\notag
$$
The subgroup $G$ is connected in $\operatorname{Aut}(\mathbb{A}^r)$ by Lemma 2. Hence for some irreducible affine algebraic variety $S$ there exists an algebraic family $\Psi=\{\psi_{s}\}_{s\in S}\subseteq G$ containing $\psi$ and $\mathrm{id}_{\mathbb{A}^r}$. Since this family is algebraic, the map is a morphism of algebraic varieties.
For each element $\psi_s\in \Psi$ consider $\psi_s^*\in \operatorname{Aut}(R(X))$. Note that $\psi_s^*$ belongs to $\operatorname{Ker}\gamma$ because $\psi_s\in G$ for each $s\in S$. Therefore, the automorphism $\psi_s^*$ preserves the $\operatorname{Cl}(X)$-grading on $R(X)$, and the restriction $\psi_s^*|_{R(X)_0}$ is well defined. Set
$$
\begin{equation}
\varphi_s^*:=\psi_s^*|_{R(X)_0}=\psi_s^*|_{\mathbb{K}[X]}\in\operatorname{Aut}(\mathbb{K}[X]).
\end{equation}
\tag{3.8}
$$
The automorphism $\varphi_s^*\in \operatorname{Aut}(\mathbb{K}[X])$ corresponds to an automorphism $\varphi_s$ in $\operatorname{Aut}(X)$. Moreover, $\psi_s^*\in\operatorname{Ker}\gamma$ and $\varphi_s=\beta(\psi_s^*)$, hence $\varphi_s\in \operatorname{Ker}\alpha$. Thus, the set $\Phi=\{\varphi_s\}_{s\in S}$ is a family in $\operatorname{Ker}\alpha$, containing $\varphi$ and the identity automorphism of the variety $X$. It remains to prove that the family $\Phi$ is algebraic.
For the morphism $\xi$ defined above consider the homomorphism Note that
$$
\begin{equation}
(\xi^*(f))(s,z)=f(\xi(s,z))=f(\psi_s(z))=(\psi_s^*(f))(z)
\end{equation}
\tag{3.9}
$$
for any $s\in S$, $f\in R(X)$ and $z\in \operatorname{Spec}(R(X))=\mathbb{A}^r$.
Let $f\in R(X)_0$. Then by (3.9) we have as $\psi_s^*\in \operatorname{Ker}\gamma$ and therefore $\psi_s^*(f)\in R(X)_0$.Hence the homomorphism
$$
\begin{equation*}
\zeta^*:=\xi^*|_{R(X)_0}\colon R(X)_0\to \mathbb{K}[S]\otimes R(X)_0
\end{equation*}
\notag
$$
is well defined. Taking the equality $R(X)_0=\mathbb{K}[X]$ into account we obtain that the algebra homomorphism $\zeta^*$ corresponds to the morphism We show that $\zeta$ is a required morphism, that is, $\zeta(s,x)=\varphi_s(x)$. For any elements $s\in S$, $f\in\mathbb{K}[X]$ and $x\in X$ we have
$$
\begin{equation*}
\begin{aligned} \, f(\zeta(s,x)) &=(\zeta^*(f))(s,x)=(\xi^*(f))(s,x) \stackrel{(3.9)}{=} (\psi_s^*(f))(x) \\ &\!\!\stackrel{(3.8)}{=} (\varphi_s^*(f))(x)=f(\varphi_s(x)). \end{aligned}
\end{equation*}
\notag
$$
Therefore, the maximal ideals in $\mathbb{K}[X]$ corresponding to the points $\zeta(s,x)$ and $\varphi_s(x)$ coincide, that is, $\mathfrak{m}_{\zeta(s,x)}=\mathfrak{m}_{\varphi_s(x)}$, where $\mathfrak{m}_x=\{f\in\mathbb{K}[X]\mid f(x)=0\}$. Consequently, the equality $\zeta(s,x)=\varphi_s(x)$ holds, and the morphism $\zeta$ is the required one. Thus, we obtain that any automorphism $\varphi\in\operatorname{Ker}\alpha$ can be included in some algebraic family $\Phi\subseteq \operatorname{Ker}\alpha$ containing the identity automorphism of the variety $X$. Therefore, $\operatorname{Ker}\alpha$ is a connected subgroup of $\operatorname{Aut}(X)$. Hence
$$
\begin{equation*}
\operatorname{Ker}\alpha=\operatorname{Ker}(\operatorname{Aut}(X)\curvearrowright \operatorname{Cl}(X)) \subseteq \operatorname{Aut}(X)^0.
\end{equation*}
\notag
$$
The reverse inclusion follows from Lemma 2.2 in [10]. Proposition 2 is proved. Therefore, the diagram is commutative.Let us use the description of the neutral component obtained in Proposition 2 to prove the criterion for the automorphism group of a nondegenerate affine toric variety to be connected. Theorem 1. Let $X$ be a nondegenerate affine toric variety with acting torus $T=(\mathbb{K}^{\times})^n$. Then the following conditions are equivalent: (1) the automorphism group of $X$ is connected; (2) automorphisms of the Cox ring that normalize the $\operatorname{Cl}(X)$-grading preserve this grading, that is, $\widetilde{\operatorname{Aut}}(R(X))=\operatorname{Ker}\gamma$; (3) there is no linear operator $L\in\mathrm{GL}_n(\mathbb{Z})$, $L(\sigma)=\sigma$, such that $L(v_i)=v_j$ but $[D_i]\neq [D_j]$, where $v_i$ is the primitive vector on the $i$th ray of the cone $\sigma$ and $[D_i]$ is the class of the corresponding $T$-invariant prime divisor in the divisor class group. Note that another condition equivalent to the connectedness of the automorphism group is presented in Corollary 1. Proof of Theorem 1. Let us prove the equivalence of conditions (1) and (2). If
$$
\begin{equation*}
\widetilde{\operatorname{Aut}}(R(X))=\operatorname{Ker}\gamma,
\end{equation*}
\notag
$$
then by the diagram (3.10) we have $\operatorname{Aut}(X)=\operatorname{Ker}\alpha$. From Proposition 2 it follows that $\operatorname{Ker}\alpha=\operatorname{Aut}(X)^0$. Therefore, $\operatorname{Aut}(X)=\operatorname{Aut}(X)^0$, and the group $\operatorname{Aut}(X)$ is connected. Conversely, if there exists an automorphism then and the group $\operatorname{Aut}(X)$ is disconnected. So the equivalence of conditions (1) and (2) is proved.
It remains to prove that conditions (1) and (3) are equivalent. The existence of a linear operator $L\in\mathrm{GL}_n(\mathbb{Z})$, $L(\sigma)=\sigma$, such that for some natural numbers $i$ and $j$ we have $L(v_i)=v_j$ but $[D_i]\neq [D_j]$ is equivalent to the existence of a $T$-equivariant automorphism $\varphi\in \operatorname{Aut}(X)$ such that for some $i$ and $j$ we have $\varphi(D_i)=D_j$ but $[D_i]\neq [D_j]$; see [7], Theorem 3.3.4. Thus, $\varphi\notin \operatorname{Aut}(X)^0$, as $\varphi$ acts nontrivially on the divisor class group, and so the group $\operatorname{Aut}(X)$ is disconnected. Conversely, suppose that the automorphism group of $X$ is disconnected. We use Corollary 1, which is proved in § 4. Since the automorphism group of $X$ is disconnected, there exists a nontrivial automorphism $\varphi$ of the group $\operatorname{Cl}(X)$ permuting the elements $[D_1],\dots,[D_r]$ in accordance with some permutation $\tau\in S_r$. We fix some basis $e_1,\dots,e_n $of the lattice $M$. Suppose that the primitive vectors on the rays of the cone $\sigma$ have coordinates
$$
\begin{equation*}
v_i=\begin{pmatrix} v_{i1} \\ \vdots \\ v_{in} \end{pmatrix}, \qquad i=1,\dots,r,
\end{equation*}
\notag
$$
in the basis of the vector space $N_{\mathbb{Q}}$ dual to $e_1,\dots,e_n$. We denote by $V$ and $V_{\tau^{-1}}$ two matrices composed of the coordinates of these vectors:
$$
\begin{equation*}
V=\begin{pmatrix} v_1 & \dots & v_r\end{pmatrix}, \qquad V_{\tau^{-1}}=\begin{pmatrix} v_{\tau^{-1}(1)} & \dots & v_{\tau^{-1}(r)}\end{pmatrix}.
\end{equation*}
\notag
$$
It follows from § 2.2 that for the elements $[D_1],\dots,[D_r]$ the relations
$$
\begin{equation}
V \begin{pmatrix} [D_1] \\ \vdots \\ [D_r] \end{pmatrix} = \begin{pmatrix} [\operatorname{div}(\chi^{e_1})] \\ \vdots \\ [\operatorname{div}(\chi^{e_n})] \end{pmatrix} =\begin{pmatrix} 0 \\ \vdots \\ 0 \end{pmatrix}
\end{equation}
\tag{3.11}
$$
hold and any other relations on the elements $[D_1],\dots,[D_r]$ are linear combinations of the ones in (3.11), because the subgroup of $T$-invariant principal divisors is generated by the elements $\operatorname{div}(\chi^{e_1}),\dots,\operatorname{div}(\chi^{e_n})$. Since $\varphi$ is a group homomorphism, we have
$$
\begin{equation*}
V \begin{pmatrix} \varphi([D_1]) \\ \vdots \\ \varphi([D_r]) \end{pmatrix} = V \begin{pmatrix} [D_{\tau(1)}] \\ \vdots \\ [D_{\tau(r)}] \end{pmatrix} =V_{\tau^{-1}}\begin{pmatrix} [D_1] \\ \vdots \\ [D_r] \end{pmatrix}=\begin{pmatrix} 0 \\ \vdots \\ 0 \end{pmatrix}.
\end{equation*}
\notag
$$
By definition set
$$
\begin{equation*}
\begin{pmatrix} \widetilde{D_1} \\ \vdots \\ \widetilde{D_n} \end{pmatrix}:= V_{\tau^{-1}}\begin{pmatrix} D_1 \\ \vdots \\ D_r \end{pmatrix}.
\end{equation*}
\notag
$$
The divisors $\widetilde{D_1}, \dots, \widetilde{D_n}$ are $T$-invariant as linear combinations of $T$-invariant ones. Moreover, they are principal since their image in the divisor class group is zero. Therefore,
$$
\begin{equation*}
\begin{pmatrix} \widetilde{D_1} \\ \vdots \\ \widetilde{D_n} \end{pmatrix}= L\begin{pmatrix} \operatorname{div}(\chi^{e_1}) \\ \vdots \\ \operatorname{div}(\chi^{e_n}) \end{pmatrix}
\end{equation*}
\notag
$$
for some integer $n\times n$ matrix $L$. It remains to note that
$$
\begin{equation*}
\begin{pmatrix} \widetilde{D_1} \\ \vdots \\ \widetilde{D_n} \end{pmatrix}=V_{\tau^{-1}}\begin{pmatrix} D_1 \\ \vdots \\ D_r \end{pmatrix}= L\begin{pmatrix} \operatorname{div}(\chi^{e_1}) \\ \vdots \\ \operatorname{div}(\chi^{e_n}) \end{pmatrix} =LV\begin{pmatrix} D_1 \\ \vdots \\ D_r \end{pmatrix}
\end{equation*}
\notag
$$
and the elements $D_1,\dots,D_r$ are independent. Hence $V_{\tau^{-1}}=LV$ and the matrix $L$ is nondegenerate. The corresponding linear operator maps the vector $v_i$ to $v_{\tau^{-1}(i)}$ for $i=1,\dots,r$. Note that there exists an index $j$ such that since $\varphi$ is a nontrivial automorphism of the divisor class group and the elements $[D_1],\dots,[D_r]$ generate the class group.
Theorem 1 is proved. § 4. The component groupNow assume that $X$ is a nondegenerate affine toric variety corresponding to a rational polyhedral cone $\sigma$. The component group of the automorphism group of $X$ is the quotient group Denote by $\Sigma_D$ the set of maps of $\operatorname{Cl}(X)$ to itself that permute the elements $[D_1],\dots,[D_r]$, that is,
$$
\begin{equation*}
\Sigma_D:=\{\varphi\colon \operatorname{Cl}(X)\to \operatorname{Cl}(X)\mid \exists\,\tau\in S_r\colon \varphi([D_i])=[D_{\tau(i)}]\}.
\end{equation*}
\notag
$$
Each element of $\Sigma_D \cap \operatorname{Aut}(\operatorname{Cl}(X))$ corresponds to at least one permutation in $S_r$ according to its action on $[D_1],\dots,[D_r]$. Moreover, the permutations corresponding to different elements of $\Sigma_D \cap \operatorname{Aut}(\operatorname{Cl}(X))$ are distinct. Therefore, $|\Sigma_D \cap \operatorname{Aut}(\operatorname{Cl}(X))|\leqslant |S_r|=r!$ . Let us describe the component group of $\operatorname{Aut}(X)$ and prove that it is finite. Theorem 2. Let $X$ be a nondegenerate affine toric variety $X$. Then In particular, where $r$ is the number of rays of the cone $\sigma$. Proof. The first part of (4.1) follows from Proposition 2 and the fundamental theorem on homomorphisms:
$$
\begin{equation*}
\operatorname{Aut}(X)/\operatorname{Aut}(X)^0\simeq \operatorname{Aut}(X)/\operatorname{Ker}\widetilde{\alpha}\simeq \widetilde{\alpha}(\operatorname{Aut}(X)).
\end{equation*}
\notag
$$
Moreover, we have
$$
\begin{equation*}
\widetilde{\alpha}(\operatorname{Aut}(X)) \stackrel{(2.2)}{=} \alpha(\operatorname{Aut}(X)).
\end{equation*}
\notag
$$
By Lemma 1 the diagram (3.10) is commutative. Hence It remains to prove that If $\varphi$ is an automorphism of $\operatorname{Cl}(X)$ that permutes the elements $[D_1], \dots, [D_r]$ in accordance with some permutation $\tau$, then its preimage under $\gamma$ contains the automorphism ${T_i\mapsto T_{\tau(i)}}$. This map is an automorphism of the Cox ring and normalizes the $\operatorname{Cl}(X)$-grading.
Conversely, the inclusion $\gamma(\widetilde{\operatorname{Aut}}(R(X)))\subseteq \operatorname{Aut}(\operatorname{Cl}(X))$ follows from the definition of the homomorphism $\gamma$. To show the inclusion $\gamma(\widetilde{\operatorname{Aut}}(R(X)))\subseteq \Sigma_D$, consider some ${\varphi\in \widetilde{\operatorname{Aut}}(R(X))}$. The Jacobian of $\varphi$ is a nonzero element of the field $\mathbb{K}$. Hence there exists a permutation $\tau\in S_r$ such that
$$
\begin{equation*}
\frac{\partial\varphi(T_1)}{\partial T_{\tau(1)}} \dotsb \frac{\partial\varphi(T_r)}{\partial T_{\tau(r)}}
\end{equation*}
\notag
$$
contains a nonzero element of $\mathbb{K}$ as a summand. Therefore, for each $i=1,\dots,r$, the element $\varphi(T_i)$ contains a nonzero linear term in $T_{\tau(i)}$:
$$
\begin{equation*}
\varphi(T_i)=c_i T_{\tau(i)}+\dotsb, \qquad i=1,\dots,r,
\end{equation*}
\notag
$$
for some nonzero $c_i \in \mathbb{K}$. Taking into account that $\deg(T_i)=[D_i]$ and the automorphism $\varphi$ maps homogeneous elements to homogeneous ones, we obtain The inclusion $\gamma(\widetilde{\operatorname{Aut}}(R(X)))\subseteq \Sigma_D$ is proved.
Inequality (4.2) follows from relation (4.1) we have proved and the fact that $|\Sigma_D \cap \operatorname{Aut}(\operatorname{Cl}(X))|\leqslant r!$ . Theorem 2 is proved. Remark 1. Let us observe that Corollary 4.7, (v), in [4] contains a description of the component group of the automorphism group of a complete simplicial toric variety. We fix some necessary notation in accordance with [4]. The rays of the fan $\Delta$ corresponding to the toric variety $X$ can be represented as the partition $\Delta_1\cup\dots\cup \Delta_s$, where the $T_j$ corresponding to rays in the same $\Delta_i$ have the same $\operatorname{Cl}(X)$-degree. We denote by $\operatorname{Aut}(N,\Delta)$ the automorphism group of the lattice $N$ that preserve the fan $\Delta$. Consider the subgroups $\Sigma_{\Delta_i}$ of $\operatorname{Aut}(N,\Delta)$ consisting of the automorphisms permuting the elements of $\Delta_i$ and fixing the other elements. For a complete simplicial toric variety $X$ Let us prove that for nondegenerate affine toric varieties the component group of the automorphism group has the same description. To do this, let us show that
$$
\begin{equation}
\operatorname{Aut}(\operatorname{Cl}(X))\cap \Sigma_D\simeq \operatorname{Aut}(N,\sigma) / \prod_{i=1}^s \Sigma_{\Delta_i}.
\end{equation}
\tag{4.3}
$$
Consider the group homomorphism
$$
\begin{equation*}
\kappa\colon \operatorname{Aut}(N,\sigma)\to \operatorname{Aut}(\operatorname{Cl}(X)).
\end{equation*}
\notag
$$
Each $\lambda\in \operatorname{Aut}(N,\sigma)$ corresponds to a $T$-equivariant automorphism $\varphi_{\lambda}\in\operatorname{Aut}(X)$ according to Theorem 3.3.4 in [7]. By definition set $\kappa(\lambda)=\widetilde{\alpha}(\varphi_\lambda)$. We see that
$$
\begin{equation*}
\kappa(\operatorname{Aut}(N,\sigma))\subseteq \widetilde{\alpha}(\operatorname{Aut}(X))=\operatorname{Aut}(\operatorname{Cl}(X))\cap \Sigma_D.
\end{equation*}
\notag
$$
In fact, equality holds because for any automorphism $\varphi\in\operatorname{Aut}(\operatorname{Cl}(X))$ such that $\varphi$ permutes the elements $[D_1],\dots,[D_r]$ in accordance with some permutation $\tau$, there exists a nondegenerate linear operator $L$ of the lattice $N$ that permutes the rays of the cone $\sigma$ in accordance with $\tau^{-1}$, as shown in the proof of Theorem 1. Consequently, $\kappa(L^{-1})=\varphi$, and the homomorphism $\kappa$ is surjective. The kernel of $\kappa$ consists of the automorphisms of the lattice $N$ that permute the rays of the cone $\sigma$ corresponding to equivalent prime divisors in the class group. Therefore, $\operatorname{Ker}\kappa=\prod_{i=1}^s \Sigma_{\Delta_i}$. By the fundamental theorem on homomorphisms we have (4.3). As a corollary to Theorem 2, we provide another condition equivalent to the connectedness of the automorphism group. Corollary 1. Let $X$ be a nondegenerate affine toric variety. Then the group $\operatorname{Aut}(X)$ is connected if and only if there are no nontrivial automorphisms of the group $\operatorname{Cl}(X)$ that permute the elements $[D_1],\dots,[D_r]$. § 5. Affine toric surfacesThe aim of this section is to apply the above results to the study of the connectedness of the automorphism group of a nondegenerate affine toric surface. Recall that the automorphism groups of such surfaces were described in [12], Theorem 4.2. Let $X$ be a nondegenerate affine toric surface. For an appropriate choice of a basis for $N$ we may assume that the rational polyhedral cone $\sigma^{\vee}$ corresponding to the surface has the form $\langle (1,0),(a,b)\rangle$, where $b > a \geqslant 0$ and $(a,b)$ is a primitive vector; see [8], pp. 32–33. Then the dual cone $\sigma$ is generated by the vectors $v_1=(0,1)$ and $v_2=(b,-a)$ (see Figure 1). Let us find the group $\operatorname{Cl}(X)$. Let the $T$-invariant prime divisors $D_1$ and $D_2$ correspond to the vectors $v_1$ and $v_2$, respectively. Then
$$
\begin{equation*}
\operatorname{Cl}(X)=\langle [D_1], [D_2] \rangle=\langle D_1,D_2 \rangle/ \langle \operatorname{div}(\chi^{(1,0)}),\operatorname{div}(\chi^{(0,1)}) \rangle.
\end{equation*}
\notag
$$
Moreover,
$$
\begin{equation*}
\operatorname{div}(\chi^{(1,0)}) =\langle v_1,(1,0)\rangle D_1+ \langle v_2,(1,0)\rangle D_2=bD_2
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\operatorname{div}(\chi^{(0,1)}) =\langle v_1,(0,1)\rangle D_1+ \langle v_2,(0,1)\rangle D_2=D_1-aD_2.
\end{equation*}
\notag
$$
Consequently,
$$
\begin{equation*}
\operatorname{Cl}(X)\simeq \mathbb{Z}_b, \qquad [D_1]=a\in \mathbb{Z}_b\quad\text{and}\quad [D_2]=1\in \mathbb{Z}_b.
\end{equation*}
\notag
$$
Proposition 3. Let $X$ be a nondegenerate affine toric surface corresponding to the cone $\sigma$ defined above. Then the group $\operatorname{Aut}(X)$ is connected if and only if one of the following three conditions hold: Moreover, if $\operatorname{Aut}(X)$ is disconnected, then Proof. We prove this statement using Corollary 1. If the group $\operatorname{Aut}(X)$ is disconnected, then there exists a nontrivial automorphism $\varphi$ of the group $\operatorname{Cl}(X)\simeq \mathbb{Z}/b\mathbb{Z}$ that permutes the elements $[D_1]=a\in\mathbb{Z}/b\mathbb{Z}$ and $[D_2]=1\in\mathbb{Z}/b\mathbb{Z}$. Consequently, $b\neq 1$, for otherwise the class group is trivial, and $a\neq 1$, for otherwise the classes of $D_1$ and $D_2$ coincide. Then the automorphism $\varphi$ acts as follows: Any automorphism of the group $\mathbb{Z}/b\mathbb{Z}$ acts by multiplication by some invertible element. Therefore, we have the system
$$
\begin{equation}
\begin{cases} a^2\equiv1\ (\operatorname{mod}b), \\ a\neq 1, \\ b\neq 1. \end{cases}
\end{equation}
\tag{5.2}
$$
Conversely, if conditions (5.2) are satisfied, then there is an automorphism of $\operatorname{Cl}(X)$ that permutes $[D_1]$ and $[D_2]$: it is defined by (5.1).
In the case when the automorphism group of $X$ is disconnected, we have
$$
\begin{equation*}
1< |\operatorname{Aut}(X)/\operatorname{Aut}(X)^0|\leqslant 2!
\end{equation*}
\notag
$$
by Theorem 2. Hence the component group consists of two elements and is isomorphic to $\mathbb{Z}/2\mathbb{Z}$.
Proposition 3 is proved. Remark 2. We also provide a proof of the first part of Proposition 3 that uses Theorem 1. This approach can be useful in the case of varieties of higher dimension. Let us prove the first part of Proposition 3 using condition (2) in Theorem 1. If the automorphism group of $X$ is disconnected, then there exists an automorphism of the Cox ring that normalizes, but does not preserve the $\operatorname{Cl}(X)$-grading. We denote this automorphism by $\varphi$. Then the image of $\varphi$ under $\gamma$ is a nontrivial automorphism of the class group. Note that $b \neq 1$, for otherwise the class group is trivial and does not have nontrivial automorphisms. Any automorphism of the group $\mathbb{Z}/b\mathbb{Z}$ acts by multiplication by some invertible element of $\mathbb{Z}/b\mathbb{Z}$. Let $\gamma(\varphi)$ be multiplication by an invertible element $c \in \mathbb{Z}/b\mathbb{Z}$. Then $c$ and $b$ are coprime and ${c \neq 1}$. From [4] we obtain
$$
\begin{equation*}
\operatorname{deg}(T_1)=[D_1]=a\in \operatorname{Z}_b \quad\text{and}\quad \operatorname{deg}(T_2)=[D_2]=1\in \mathbb{Z}_b.
\end{equation*}
\notag
$$
Consequently, $\deg(\varphi(T_1))$ coincides with $ac\ (\operatorname{mod}b)$ in the group $\mathbb{Z}/b\mathbb{Z}$, and $\deg(\varphi(T_2))$ coincides with $c$. Moreover, the Jacobian of the automorphism $\varphi$ is a nonzero element of the field $\mathbb{K}$, so each of the elements $\varphi(T_1)$ and $\varphi(T_2)$ contains a term linear in $T_1$ or $T_2$. However, if $\varphi(T_i)$ contains a linear term in $T_i$, then $c$ is equal to $1$, and this is a contradiction. Thus,
$$
\begin{equation*}
\varphi(T_1) = k_1T_2+\dotsb \quad\text{and}\quad \varphi(T_2) = k_2T_1+\dotsb
\end{equation*}
\notag
$$
for some nonzero elements $k_1$ and $k_2$ of the field $\mathbb{K}$. Therefore,
$$
\begin{equation*}
\operatorname{deg}(\varphi(T_1)) = c\operatorname{deg}(T_1) = \operatorname{deg}(T_2)\quad\text{and}\quad \operatorname{deg}(\varphi(T_2)) = c\operatorname{deg}(T_2) = \operatorname{deg}(T_1).
\end{equation*}
\notag
$$
Consequently, we have
$$
\begin{equation*}
ac \equiv 1\ (\operatorname{mod}b), \qquad c = a\quad\text{and} \quad c\neq 1.
\end{equation*}
\notag
$$
Thus, the disconnectedness of the automorphism group of $X$ implies (5.2). Let us prove the converse. Suppose conditions (5.2) hold. Consider the automorphism $\varphi \in \widetilde{\operatorname{Aut}}(R(X))$ defined by
$$
\begin{equation*}
\varphi(T_1)=T_2 \quad\text{and}\quad \varphi(T_2)=T_1.
\end{equation*}
\notag
$$
The automorphism $\varphi$ normalizes the $\operatorname{Cl}(X)$-grading, as it permutes the homogeneous components of $R(X)$ in accordance with multiplication by $a$ in the divisor class group. At the same time, $\varphi$ does not preserve the $\operatorname{Cl}(X)$-grading since $a \neq 1$. Therefore, ${\widetilde{\operatorname{Aut}}(R(X)) \neq \operatorname{Ker}\gamma}$, and condition (2) in Theorem 1 implies that the automorphism group of $X$ is disconnected. Let us present a proof of the first part of Proposition 3 using condition (3) in Theorem 1. Suppose that the automorphism group of $X$ is disconnected. Then there exists a linear operator $L \in \mathrm{GL}_2(\mathbb{Z})$, $L(\sigma) = \sigma$, such that ${L(v_1) = v_2}$ and ${L(v_2) = v_1}$ but $[D_1] \neq [D_2]$. Note that it follows that $b \neq 1$, for otherwise the class group is trivial and does not have distinct elements. We have $a \neq 1$, for otherwise $[D_1] = [D_2]$. Furthermore, we observe that Thus, from the linearity of $L$ we obtain
$$
\begin{equation*}
L((1,0))=\frac{L(v_2)+aL(v_1)}{b}=\frac{(ab,1-a^2)}{b}=\biggl(a,\frac{1-a^2}{b}\biggr).
\end{equation*}
\notag
$$
Since $L \in \mathrm{GL}_2(\mathbb{Z})$, we have $\frac{1-a^2}{b} \in \mathbb{Z}$ and $a^2 \equiv 1\ (\mathrm{mod} \ b)$. Consequently, the disconnectedness of the automorphism group of $X$ implies (5.2). Conversely, suppose that conditions (5.2) are satisfied. Then to prove that $\operatorname{Aut}(X)$ is disconnected it suffices to find an operator $L \in \mathrm{GL}_2(\mathbb{Z})$ satisfying condition (3) in Theorem 1. One required map is the operator $\widetilde{L}$ defined at the basis vectors by Indeed, $\widetilde{L}$ permutes the vectors $v_1$ and $v_2$, while $[D_1] \neq [D_2]$.§ 6. ExamplesIn this section we provide some examples illustrating our results. Example 1. The automorphism group of the affine space $\mathbb{A}^n$ is connected by Theorem 6 in [2]. This fact also follows from Theorem 1, since the affine space is a factorial toric variety and the divisor class group of a factorial variety is trivial; see, for example, [7], Theorem 4.0.18, (b). Example 2. Consider the variety  Figure 2.The cones $\sigma_{1}^{\lor}$, $\sigma_{2}^{\lor}$ and $\sigma_{3}^{\lor}$ from Examples 2, 3 and 4. Example 3. Let Example 4. Consider Example 5. Consider the nondegenerate nonsimplicial affine toric variety The primitive vectors on the rays of the cone $\sigma_4$ dual to $\sigma_4^{\vee}$ are
$$
\begin{equation*}
v_1=\begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}, \qquad v_2=\begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}, \qquad v_3=\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \quad\text{and}\quad v_4=\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}.
\end{equation*}
\notag
$$
These vectors correspond to the $T=(\mathbb{K}^{\times})^3$-invariant prime divisors
$$
\begin{equation*}
\begin{gathered} \, D_1 =\{ x=w=0\}, \qquad D_2=\{x=z=0\}, \\ D_3=\{z=y=0\} \quad\text{and}\quad D_4=\{y=w=0\}. \end{gathered}
\end{equation*}
\notag
$$
Let us find the divisor class group of $X_4$. The group of $T$-invariant principal divisors on $X_4$ is generated by the following Weil divisors:
$$
\begin{equation*}
\begin{gathered} \, \operatorname{div}(\chi^{(0,0,1)})=\sum_{i=1}^{4}\langle v_i,(0,0,1) \rangle D_i=D_4-D_2, \\ \operatorname{div}(\chi^{(0,1,0)})=\sum_{i=1}^{4}\langle v_i,(0,1,0) \rangle D_i=D_3-D_1 \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\operatorname{div}(\chi^{(1,0,0)})=\sum_{i=1}^{4}\langle v_i,(1,0,0) \rangle D_i=D_1+D_2.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\operatorname{Cl}(X_4)\simeq \langle D_1,D_2,D_3,D_4 \rangle /(D_4-D_2,D_3-D_1,D_1+D_2)\simeq \langle [D_1] \rangle \simeq \mathbb{Z},
\end{equation*}
\notag
$$
where The group $\operatorname{Cl}(X_4)$ has a unique nontrivial automorphism $\varphi$, which permutes the elements of the set $\{[D_1],[D_2],[D_3],[D_4]\}$. It acts by multiplication by $-1$ in the group $\mathbb{Z}$. By Theorem 2 we have
$$
\begin{equation*}
\operatorname{Aut}(X_4)/\operatorname{Aut}(X_4)^0\simeq \mathbb{Z}_2.
\end{equation*}
\notag
$$
Note that by Proposition 2 the neutral component of the automorphism group of $X_4$ contains the automorphism
$$
\begin{equation*}
x\mapsto y, \qquad y\mapsto x, \qquad z\mapsto w, \qquad w\mapsto z
\end{equation*}
\notag
$$
and does not contain the automorphism
$$
\begin{equation*}
x\mapsto y, \qquad y\mapsto x, \qquad z\mapsto z, \qquad w\mapsto w.
\end{equation*}
\notag
$$
Example 6. Finally, let us present an example of an affine toric variety with noncommutative component group of the automorphism group. Let $\sigma_5$ be the cone in the three-dimensional rational vector space generated by the vectors 
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