Proof. Step 1. We start with the case $n=3$ .

Let $L=k(\sqrt {a}, \sqrt {b})$ be a biquadratic extension. Embed $\mathbb G_{\mathrm m,k}$ into $R_{L/k}(\mathbb G_{\mathrm {m},L})$ and consider the quotient $T=R_{L/k}(\mathbb G_{\mathrm {m},L})/\mathbb G_{{\mathrm {m}},k}$ . The torus T is k-rational, being an open subset of

$$ \begin{align*}R_{L/k}(\mathbb A^1_{L})/\mathbb G_{{\mathrm{m}},k}=\mathbb A^4_{k}/\mathbb G_{{\mathrm{m}},k}=\mathbb P^3_{k}.\end{align*} $$

The character module $\hat T$ is isomorphic to $\ker \left [\mathbb Z[\Pi ]\to \mathbb Z\right ]$ , the augmentation ideal of $\mathbb Z[\Pi ]$ , see [Reference Voskresenskiĭ39, Section 4.8].

It is convenient to use a matrix representation of $\hat T$ . Fix a pair $\rho , \sigma $ of generators of $\Pi $ , a basis of $\mathbb Z^3$ , and an isomorphism of modules $\hat T$ and $\mathbb Z^3$ . Consider the subgroup of $\operatorname {{\mathrm {GL}}}(3,\mathbb Z)$ corresponding to the $\Pi $ -action on $\hat T$ . It is conjugate to

(2) $$ \begin{align} \left< \left(\begin{matrix} 0 & 0 & 1 \\ -1 & -1 & -1 \\ 1 & 0 & 0\end{matrix} \right), \left(\begin{matrix} -1 & -1 & -1 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} \right) \right>. \end{align} $$

Step 2. Define a birational involution g of T sending t to $t^{-1}$ . More formally, we define the action of g on $\hat T$ (written additively) as sign changing, so that the corresponding matrix is $-I_3$ . Denote $G=\left <g\right>$ .

Embed T into a smooth projective model X as an open subset. Recall (see Section 2) that such an X exists even when $\operatorname {{\mathrm {char}}} (k)> 0$ . Moreover, note that such an embedding can be made G-equivariant. Indeed, in the setup of Theorem 3.7, replace $\Gamma =\mathrm {Gal}(\bar k/k)$ with its finite quotient $\Pi =\mathrm {Gal} (L/k)$ and define $W=G\times \Pi $ . Since the G-action on the algebraic variety T comes from its action by automorphisms of the lattice $\hat T$ , we can use [Reference Colliot-Thélène, Harari and Skorobogatov14, Th. 1] to obtain a W-equivariant, smooth, projective fan from which we obtain the needed G-variety X.

Step 3. Consider now the W-module $\mathrm {Pic}(\overline X)$ .

Looking at the finite subgroup of $\operatorname {{\mathrm {GL}}}(3,\mathbb Z)$ corresponding to the W-action on $\hat T$ , we see from (2) and the definition of the action of g that it is conjugate to

(3) $$ \begin{align} \left<\left(\begin{matrix} 0 & 0 & 1 \\ -1 & -1 & -1 \\ 1 & 0 & 0\end{matrix} \right), \left(\begin{matrix} -1 & -1 & -1 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} \right) , \left(\begin{matrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{matrix} \right) \right>. \end{align} $$

Looking at the list of finite subgroups of $\operatorname {{\mathrm {GL}}}(3,\mathbb Z)$ corresponding to tori that are not stably rational [Reference Kunyavskiĭ26, Theorem 1], we see that subgroup (3) appears on that list where it is denoted by $W_2$ . According to [Reference Kunyavskiĭ26, Section 4], the flasque W-module $F=\mathrm {Pic} (\overline X)$ is not invertible (and hence not stably permutation).

See [Reference Hoshi and Yamasaki21, Section 7] for a computer verification of this result; see [Reference Cheltsov, Tschinkel and Zhang12, Section 5] for an explicit geometric description of the toric variety X.

Note that the W-module $\mathrm {Pic} (\overline X)$ is flasque and coflasque, see [Reference Kunyavskiĭ27, Section 3], [Reference Hoshi and Yamasaki21]; this allowed the use of the torus $T'$ in [Reference Kunyavskiĭ27, Example 3.7] for constructing an example of a toric variety violating a local-to-global principle for rationality in the absence of the Brauer obstruction.

According to Theorem 3.7, G is not stably projectively k-linearizable.

Step 4.

Let us now choose L such that all decomposition groups $\Pi _v$ are cyclic (hence of order 1 or 2). Such a choice is always possible, see [Reference Neukirch, Schmidt and Wingberg32, Corollary 9.2.3]. Thus each $k_v$ -torus $T_v=T\times _{k}k_v$ splits by a quadratic extension $K:=k_v(\sqrt {d})$ . Then $T_v$ is a direct product of tori of dimension 1 or 2 [Reference Voskresenskiĭ39, Chapter 2, §5, Theorem 4]. By Remark 3.4, it is enough to show that the action of G on the 1- and 2-dimensional direct factors of $T_v$ is linearizable.

(Note that even in the case $n=2$ , the linearizability of a birational involution is not automatic, even if it is linearizable after a quadratic base change, see, e.g., [Reference Cheltsov, Mangolte, Yasinsky and Zimmermann11, Section 8].)

For the one-dimensional split torus $\mathbb G_{{\textrm {m}},k_v}$ the needed property is immediate: it equivariantly embeds into $\mathbb P^1$ where g acts by swapping the coordinates, and the projective representation $G\to \operatorname {{\mathrm {PGL}}}_2(k_v)$ corresponding to this action lifts to a linear representation.

Let now $T_v$ be the norm one torus $R_{K/k_v}^1{\mathbb G}_{{\textrm {m},K}}$ with the G-action given by $g(t)=t^{-1}$ . Let $\gamma $ denote the generator of $\mathrm {Gal} (K/k_v)$ . Then $T_v$ can be represented by the equation $N(x)=x\bar x=1$ , where $\bar x=\gamma (x)$ , and hence the action of g coincides with the action of $\gamma $ .

One can view $T_v$ as a smooth affine conic $C\ \mathbf \subset\ \mathbb A^2$ given by $x^2-dy^2=1$ ; it has a G-invariant rational point $P=(1,0)$ . The smooth equivariant compactification $\overline C \subset \mathbf P^2$ given by $X^2-dY^2=Z^2$ has a G-invariant rational point $\overline P=(1:0:1)$ . We have an isomorphism $f\colon \overline C \overset {\cong }{\to }\mathbf P^1$ , with the G-action induced by the G-action on $\overline C$ and G-invariant rational point $f(\overline P)$ . This regular action gives rise to a projective representation $G\to \operatorname {{\mathrm {PGL}}}(2,k_v)$ . The presence of a G-invariant rational point implies that the image of g is conjugate to $\left (\begin {matrix} 1 & 0 \\ 0 & -1\end {matrix} \right )$ (modulo scalar matrices). Hence this representation can be lifted to a linear representation $G\to \operatorname {{\mathrm {GL}}}(2,k_v)$ .

To finish the proof of the case $n=3$ of the theorem, we have to consider two-dimensional tori appearing in a direct decomposition of $T_v$ . Such a torus must be isomorphic either to a product of two one-dimensional tori (when Remark 3.4 is applicable), or to a torus of the form $S=R_{L/k_v}\mathbb G_{\textrm {m},L}$ , where $L/k_v$ is a quadratic field extension.

We first embed S into a $k_v$ -quadric $Q=R_{L/k_v}\mathbb P^1_L$ as an open subset, with the G-action on Q induced by the G-action on $\mathbb G_{\textrm {m},k_v}$ (recall that Weil’s restriction functor is compatible with group actions, see [Reference Brion7, Section 5]). More explicitly, the open set $x^2-dy^2 \neq 0$ in $\mathbb A^2$ can be identified with the open set $zt\neq 0$ in the quadric $x^2-dy^2+zt=0$ .

To show that the G-action on Q is $k_v$ -linearizable, we use the following lemma. Its proof (for the case $k=\mathbb C)$ and the subsequent remark were communicated to me by Ivan Cheltsov. The case of general k requires certain adjustments (suggested by Jean-Louis Colliot-Thélène).

Proof. Projecting from the G-fixed k-point, we get a k-birational map $Q\dashrightarrow \mathbb P_k^{n-1}$ . This gives rise to a projective k-representation $G\to \operatorname {{\mathrm {PGL}}}_n$ . Consider the degree n covering $\mathrm {SL}_n\to \operatorname {{\mathrm {PGL}}}_n$ . The long sequence of cohomology yields an exact sequence of groups

(4) $$ \begin{align} 1 \to \mu_{n}(k) \to \mathrm{SL}_{n}(k) \to \operatorname{{\mathrm{PGL}}}_{n}(k) \to k^*/k^{*n} \end{align} $$

where $\mu _n$ is the k-group scheme of $n^{\textrm {th}}$ roots of unity. Sequence (4) induces short exact sequences

$$ \begin{align*}1 \to \mu_{n}(k) \to \mathrm{SL}_{n}(k) \to \mathrm{SL}_{n}(k)/\mu_{n}(k) \to 1 \end{align*} $$

and

$$ \begin{align*}1 \to \mathrm{SL}_{n}(k)/\mu_{n}(k) \to \operatorname{{\mathrm{PGL}}}_{n}(k) \to k^*/k^{*n}. \end{align*} $$

Since we have an embedding $G \hookrightarrow \operatorname {{\mathrm {PGL}}}_{n}(k)$ and the order of G is prime to n, we conclude that G embeds into $\mathrm {SL}_{n}(k)/\mu _{n}(k)$ . By pull-back, we obtain a group extension

(5) $$ \begin{align} 1 \to \mu_{n}(k) \to G' \to G \to 1 \end{align} $$

where $G'$ is a subgroup of $\mathrm {SL}_{n}(k)$ and $\mu _{n}(k)$ is a central abelian subgroup of $G'$ of order prime to the order of G. Such extensions are classified by the group $H^2(G,\mu _{n}(k))$ , which is killed by the order of G and the order of $\mu _{n}(k)$ and is therefore zero.

Hence, extension (5) admits a section, a group homomorphism $G \to G'$ , whence a required embedding $G\hookrightarrow \mathrm {SL}_{n}(k)$ .

Thus the case $n=3$ is settled.

Step 5. To treat the case of $G\subset \operatorname {{\mathrm {Cr}}}_{l}(k)$ , $l>3$ , one can take the involution acting on the first three variables as g considered above and trivially on all the remaining variables. The resulting involution on $T\times \mathbb G_{\textrm {m}}^{l-3}$ extends to $X_l:=X\times \mathbb P^{l-3}$ with the three-dimensional X as above and the trivial action of G on the projective space. Therefore the W-module $\mathrm {Pic} (\overline X_l)$ is isomorphic to $\mathrm {Pic} (\overline X) \oplus \mathbb Z^{l-3}$ and is thus not stably permutation, as $\mathrm {Pic} (\overline X)$ , which proves that G is not stably projectively k-linearizable. The local linearizability is proved in exactly the same way as in the case $n=3$ .

This finishes the proof of Theorem 3.10.

Proof. Let $m \in X$ be a closed point. Since X is quasiprojective, the finitely many points $g.m$ are contained in an affine open set U. The intersection of the affine open sets $gU$ is a G-stable open set which is affine (see [37, Lemma 26.21.7]), and contains m.

Proof. There is a G-action on the graph $\Gamma \subset U \times Y $ of the morphism $U\to Y$ . The projection map $\Gamma \to U$ is an isomorphism. Let $Z \subset X \times\, Y$ be the closure of this graph. It is geometrically integral. The action of G extends to $Z \subset X \times Y$ .

The projection map $q\colon Z \to X$ is birational. Over U, the map $q = q^{-1}(U) \to U$ is an isomorphism.

Let $\tilde {Z}$ be the normalization of Z. Since G acts on Z, it acts on $\tilde {Z}$ . Indeed, via Lemma 4.1 this statement is reduced to the following statement. Suppose we have an inclusion of integral commutative rings $A \subset B$ with the same field of fractions K, with B normal and integral over A. Suppose we have an action of G on A, hence on K. If we are given $g\in G$ and $b \in B$ satisfying an equation $b^n+ \sum _{i=0}^{n-1} a_{i}b^{i}=0$ , then $(gb)^n+ \sum _{i=0}^{n-1} a_{i}(gb)^{i}=0$ . Thus $gb \in B$ .

One then takes V to be the open set of normal points of U.

If X is normal and $Y/k$ is projective, then the given map $U \to Y$ extends to a G-morphism from an open set $U \subset X$ that contains all codimension 1 points of X.

Proof. Once equipped with Proposition 4.3, one may copy the proof of [Reference Colliot-Thélène and Sansuc16, Prop. 2.A.1, p. 461/462] (a proof due to Laurent Moret-Bailly) word for word. One gets isomorphisms of W-modules

$$ \begin{align*}\mathrm{Pic}(X) \oplus P_{1} \simeq CH^1(\tilde{Z}),\end{align*} $$

$$ \begin{align*}\mathrm{Pic}(Y) \oplus P_{2} \simeq CH^1(\tilde{Z}),\end{align*} $$

where $CH^1(\tilde {Z})$ is the Chow group of codimension 1 cycles modulo rational equivalence on the normal variety $\tilde {Z}$ of Proposition 4.3, and where $P_{1}$ and $P_{2}$ are permutation modules. This gives a W-isomorphism

$$ \begin{align*}\mathrm{Pic}(X) \oplus P_{1} \simeq \mathrm{Pic}(Y) \oplus P_{2}.\\[-34pt] \end{align*} $$

Proof. The action of G on $\mathrm {Pic}(\mathbb P(V_i)) = \mathbb Z$ ( $i=1,2$ ) is the trivial one. We have natural G-isomorphisms

$$ \begin{align*}\mathrm{Pic}(X \times \mathbb P(V_1)) \simeq \mathrm{Pic}(X) \oplus \mathrm{Pic}(\mathbb P(V_1)) = \mathrm{Pic}(X) \oplus \mathbb Z,\end{align*} $$

and similarly for $\mathrm {Pic}(Y \times \mathbb P(V_2))$ . It remains to go over to $X_K$ , $Y_K$ , and apply Proposition 4.4.

Proof. We will use the no-name lemma in the following form [Reference Chernousov, Gille and Reichstein13, Lemma 4.4]:

To prove the proposition, we can assume $X = \mathbb P^n_L$ with a G-action induced from a linear G-representation on $\mathbb A^{n+1}_L$ . Denote $d = [L:k]$ . We have an induced G-equivariant rational map

$$ \begin{align*}\mathbb A_k^{(n+1)d} = R_{L/k}\mathbb A_L^{n+1} \dashrightarrow R_{L/k}Tot(\mathcal O_{\mathbb P_L^n}(-1)) \to R_{L/k}\mathbb P_L^n. \end{align*} $$

Here $R_{L/k}Tot(\mathcal O_{\mathbb P_L^n}(-1))$ is a G-linearized vector bundle over $R_{L/k}\mathbb P_L^n$ . (Weil’s descent of a G-linearizable vector bundle is a G-linearizable vector bundle, see [Reference Brion6, Lemma 2.2].) Thus, by Lemma 6.5 we have a G-birational map

$$ \begin{align*}R_{L/k}Tot(\mathcal O_{\mathbb P_L^n}(-1)) \dashrightarrow R_{L/k}\mathbb P_L^n \times \mathbb A_k^d. \end{align*} $$

Therefore $R_{L/k}\mathbb P_L^n$ is stably k-linearizable.