2. Preliminary results

We work over the field of complex numbers. Varieties are always projective unless stated otherwise. We refer to [Reference Kollár and MoriKM98] for the notion of terminality and the basic notions on the minimal model program.

2.1 Group actions

We recall here some fundamental results on algebraic actions on varieties.

Definition 2.1. Let Y be a variety and let G be an algebraic group. We say that G acts rationally on Y if there exists a birational map

\[\mu\colon G\times Y\dashrightarrow G\times Y,\quad (g,y)\mapsto(g,\mu(g,y)),\]

that restricts to an isomorphism $U\to V$ on dense open subsets $U,V\subseteq G\times Y$ , whose projections onto G are surjective, and such that $\mu(gh,\cdot)=\mu(g,\cdot)\circ\mu(h,\cdot)$ for any $g,h\in G$ . If, moreover, the kernel of the induced homomorphism $G \to \text{Bir}(Y),\ g \mapsto \mu_g$ is trivial, i.e. if G acts faithfully on Y, then G is called an algebraic subgroup of $\text{Bir}(X)$ .

The algebraic subgroup G of $\text{Bir}(X)$ is called maximal if it is maximal with respect to the inclusion among the algebraic subgroups of $\text{Bir}(X)$ .

Notice that, if W is a rational variety and $\psi\colon W\dashrightarrow Y$ a birational map and $G\subseteq\text{Aut}(W)$ an algebraic group, then $G\times Y\to Y$ , $(g,y)\mapsto (g,\psi g\psi^{-1}(y))$ is a rational action of G on Y and G is an algebraic subgroup of $\text{Bir}(Y)$ ; conjugating G by $\psi$ embeds G into $\text{Bir}(Y)$ .

On the other hand, if G is a connected subgroup of $\text{Bir}(Y)$ acting rationally on Y, by the Weil regularisation theorem [Reference WeilWei55] there is a birational model of Y on which G acts regularly.

Remark 2.2. By [Reference BlancBla09], any algebraic subgroup of $\text{Bir}(\mathbb P^2)$ is contained in a maximal algebraic subgroup. Nevertheless, there are infinite increasing sequences of algebraic subgroups, see [Reference Fong and ZikasFZ23, Remark 2.8].

Remark 2.3. It is natural to ask if $\text{Bir}(\mathbb P^n)$ itself can be endowed with a structure of an (ind-)algebraic group. We know this is not possible, thanks to the work [Reference Blanc and FurterBF13] (see also [Reference Blanc, Fanelli and TerpereauBFT23, Section 2.5] for the construction of the functor $\frak{Bir}_{\mathbb P^n}$ ).

We also recall the following two classical facts on regular actions. The first follows from [Reference BrionBri96, Proposition 2, p. 8].

Lemma 2.4. Let G be an algebraic group acting regularly on a projective variety X. Let $n=\max\{\dim(G\cdot x)\mid x\in X\}$ be the maximal dimension of an orbit of G. Then, the set $\{x\in X\mid \dim (G\cdot x)<n\}$ is a closed subset of X. In particular, the union of orbits of dimension n is a dense open G-invariant subset of X.

The second is Blanchard’s lemma [Reference Brion, Samuel and UmaBSU13, Proposition 4.2.1].

Lemma 2.5. Let $f\colon X \to Y$ be a proper morphism between varieties such that $f_*(\mathcal{O}_X)=\mathcal{O}_Y$ . If a connected algebraic group G acts regularly on X, then there exists a unique regular action of G on Y such that f is G-equivariant.

By Blanchard’s lemma, there is a homomorphism $\mathrm{Aut}^{\circ}(X)\to\mathrm{Aut}^{\circ}(Y)$ . We set $\mathrm{Aut}^{\circ}(X)_Y$ as the kernel of this homomorphism.

2.2 Chow varieties

We refer to [Reference KollárKol96] for a presentation of Chow varieties, and here we introduce the notation and briefly recall some results.

Let X be a normal projective variety and G a connected group acting regularly on X. Let $\textrm{Chow}(X)$ be the Chow variety of X. We recall that $\textrm{Chow}(X)$ has countably many irreducible components (cf. [Reference KollárKol96, Theorem I.3.21(3)]).

If Z is a subvariety of X we denote by [Z] the corresponding point of $\textrm{Chow}(X)$ . If $\mathcal W$ is an irreducible subvariety of $\textrm{Chow}(X)$ , we denote by $\mathcal U \subseteq \mathcal W \times X$ the universal family defined as

$$\mathcal U=\{([Z],x)\in\mathcal W \times X\vert\; x\in Z \}.$$

Denote by $e\colon \mathcal U \to X$ and $u\colon \mathcal U \to\mathcal W$ the natural morphisms. We notice that $u^{-1}[Z]\cong Z$ for every cycle Z in $\mathcal W$ .

Moreover, if $\mathcal W_0\subseteq \mathcal W$ , then the universal family over $\mathcal W_0$ is $\mathcal U_0=u^{-1}\mathcal W_0$ : we will refer to it as the restriction of the universal family over $\mathcal W_0$ .

Moreover, G acts on every irreducible component of $\textrm{Chow}(X)$ : indeed, G preserves every irreducible component as those are countable. If [Z] is a subvariety of X and $g\in G$ , then the natural action is given by $g\cdot [Z]=[g(Z)]$ .

2.3 Ruled varieties

This section contains some definitions and facts relating to ruled varieties. We give first some definitions.

Definition 2.6. Let $\pi\colon X \to B$ be a morphism between normal projective varieties. One says that $\pi$ is:

1. (i) a $\mathbb{P}^1$ -fibration if its general fibre is a smooth rational curve;

2. (ii) a birationally trivial $\mathbb{P}^1$ -fibration if its generic fibre is isomorphic to $\mathbb{P}^1_{\mathbb{C}(B)}$ .

Let $\pi\colon X \to B$ be a $\mathbb{P}^1$ -fibration. Then one says that $\pi$ is:

1. (i) a standard conic bundle if X and B are smooth and $\rho(X/B)=1$ ;

2. (ii) an embedded conic bundle if there is a rank-3 vector bundle $\mathcal E$ on B and an embedding $X\hookrightarrow \mathbb P_B(\mathcal E)$ such that $\pi$ is the restriction of the natural morphism $\rho\colon \mathbb P_B(\mathcal E)\to B$ and X restricted to any fibre of $\rho$ is a conic.

Remark 2.7.

1. (i) We notice that a birationally trivial $\mathbb{P}^1$ -fibration is a $\mathbb{P}^1$ -fibration, and that $\mathbb{P}^1$ -fibrations are also called conic bundles.

2. (ii) Moreover, a fibration is birationally trivial if and only if its general fibre is isomorphic to $\mathbb P^1$ and it admits a birational section.

3. (iii) By [Reference SarkisovSar82, § 1.5], every standard conic bundle is embedded. If G is an algebraic group acting regularly on a standard conic bundle, then the embedding is equivariant.

We will often consider projective bundles of relative dimension 1, i.e. $\mathbb{P}^1$ -bundles, which are projectivisations of locally free sheaves. Let V be a projective variety. If $\mathcal E\to V$ is a rank-r vector bundle, we denote by $\mathbb P(\mathcal E)$ or $\mathbb P_V(\mathcal E)$ the Grothendieck projectivisation in $\mathcal E$

$$\mathbb P_V(\mathcal E)=\textrm{Proj}(\textrm{Sym}(\mathcal E)),$$

together with the natural morphism $\pi\colon \mathbb P(\mathcal E)\to V$ .

Remark 2.8.

1. (i) In particular, a surjection $\mathcal E^{\vee}\to\mathcal Q^{\vee}$ determines an embedding $\mathbb P(\mathcal Q)\to \mathbb P(\mathcal E)$ such that $\mathcal{O}_{\mathbb P(\mathcal Q)}(1)\sim \mathcal{O}_{\mathbb P(\mathcal E)}(1)\vert_{\mathbb P(\mathcal Q)}$ .

2. (ii) By the Noether–Enriques theorem, a smooth $\mathbb P^1$ -fibration over a curve is a $\mathbb P^1$ -bundle.

3. (iii) If Y and Z are smooth, then a $\mathbb{P}^1$ -bundle $g\colon Y\to Z$ is a standard conic fibration.

We conclude the section with a result that will be useful in the proof of the main theorem.

Proposition 2.9. Let $g\colon Y\to Z$ be a birationally trivial $\mathbb{P}^1$ -fibration. Then there is a smooth variety $\widetilde Z$ , a $\mathbb P^1$ -bundle $\tilde g\colon \widetilde Y\to\widetilde Z$ and a diagram

such that all the maps are $\mathrm{Aut}^{\circ}(Y)$ -equivariant and the horizontal arrows are birational.

Proof. Let $U\subseteq Z$ be the maximal open set such that g admits a section over U. Then U is left invariant by $\mathrm{Aut}^{\circ}(Y)$ . Let $Y_U=g^{-1}(U)$ . The section over U gives a relatively very ample line bundle and thus a morphism $U\to \textrm{ Grass}(\mathbb P^1,\mathbb P^N)$ for some N. Let $\mathcal U\subseteq \textrm{Grass}(\mathbb P^1,\mathbb P^N)\times \mathbb P^N$ be the universal family. Let X be the Zariski closure of the image of U in $\textrm{Grass}(\mathbb P^1,\mathbb P^N)$ . We have a morphism $Y_U\to \mathcal U$ and an isomorphism $Y_U\cong U\times_{\textrm{Grass}(\mathbb P^1,\mathbb P^N)} \mathcal U$ . Let W be the Zariski closure of the image of $Y_U$ in $\mathcal U$ . The group $\mathrm{Aut}^{\circ}(Y)$ acts on X and W, making all the morphisms equivariant. We take $\widetilde Z$ an $\mathrm{Aut}^{\circ}(Y)$ -equivariant projective smooth resolution of the indeterminacy of $Z \dashrightarrow X$ and $\widetilde Y=\widetilde Z\times_X W$ . Then $\tilde g\colon \widetilde Y\to \widetilde Z$ is the sought $\mathbb P^1$ -bundle.

2.4 Rationally connected and non-rational threefolds

We will also need the following statements on rationally connected irrational threefolds.

Proposition 2.10. Let X be a rationally connected non-rational threefold. Then $\mathrm{Aut}^{\circ}(X)$ is trivial.

Proof. Assume by contradiction that $\mathrm{Aut}^{\circ}(X)$ is non-trivial. Since X is rationally connected, $\mathrm{Aut}^{\circ}(X)$ is linear and thus contains a 1-parameter subgroup $\Gamma$ . By [Reference Blanc, Fanelli and TerpereauBFT23, Proposition 2.5.1], there is an open set X’ of X which is of the form $\Gamma\times U$ . Since X is rationally connected, any compactification of U is rationally connected. Since it is a surface, it is also rational. Thus X is birational to $\Gamma\times U$ , which is in turn birational to $\mathbb P^1\times\mathbb P^2$ , a contradiction.

Remark 2.11. More generally, one can prove that $\text{Bir}(X)$ contains no connected algebraic subgroups, when X is a rationally connected, non-rational threefold [Reference Blanc, Fanelli and TerpereauBFT23, Corollary 2.5.9].

Remark 2.12. Let X’ be a rationally connected threefold. Assume X’ has a conic bundle structure $X'\to S$ . Then, there is a birational model X of X’ with a fibration $c\colon X\to \mathbb P^2$ with general fibre $\mathbb P^1$ and sitting in the following diagram.

Indeed, since X’ is rationally connected, the surface S is rational. Let $\mathbb P^2\dashrightarrow S$ be a birational morphism, and let $X\to X'\times\mathbb P^2$ be a resolution of the indeterminacies of the induced map $X'\dashrightarrow \mathbb P^2$ . Then the induced morphism $X\to \mathbb P^2$ is the required morphism.

Moreover, if X’ is not rational, then the generic fibre of c is not $\mathbb P^1_{\mathbb C(\mathbb P^2)}$ .

3. Automorphisms of $\mathbb P^1$ -bundles

In the first part of this section, we show that invariant sections of projective bundles induce equivariant elementary transformations of $\mathbb P^1$ -bundles (Lemma 3.1 and Lemma 3.2) and we recall the description of the automorphism group of projective bundles (Lemma 3.3). Recall that, for a line bundle $\mathcal L\to V$ and a divisor D on V, we write $\mathcal L^{\vee}(-D):=\mathcal{L}^{\vee}\otimes\mathcal O_V(-D)$ , and $\mathcal L^{\vee}|_{D}:=\mathcal L^{\vee}\otimes \mathcal O_D$ . Notice that $\mathcal L^{\vee}(-D)=(\mathcal L(D))^{\vee}$ .

Lemma 3.1. Let V be a smooth variety, $\mathcal E\to V$ a rank-2 vector bundle and $\pi\colon\mathbb P(\mathcal E)\to V$ be the induced $\mathbb P^1$ -bundle. Let $V_0$ be a section of $\pi$ defined by a surjective morphism $\mathcal E^{\vee}\to\mathcal L^{\vee}$ . Let $D_1$ be a smooth effective irreducible divisor in V. Then the following hold.

1. (i) The sheaf $\mathcal E_1^{\vee}$ equal to the kernel of the surjection $\mathcal E^{\vee}\to\mathcal L^{\vee}\vert_{D_1}$ is a rank-2 vector bundle on V.

2. (ii) More precisely, if $\mathcal E^{\vee}$ is an extension

   $$0\to\mathcal M^{\vee}\to\mathcal E^{\vee}\to\mathcal L^{\vee}\to 0,$$
   then $\mathcal E_1^{\vee}$ is an extension
   $$0\to\mathcal M^{\vee}\to\mathcal E_1^{\vee}\to\mathcal L^{\vee}(-D_1)\to 0.$$

3. (iii) There is an induced birational map $\psi\colon\mathbb P(\mathcal E)\dashrightarrow \mathbb P(\mathcal E_1) $ which factors as $\eta_2\circ\eta_1^{-1}$ , where $\eta_1\colon W\to \mathbb P(\mathcal E)$ is the blow up of the subvariety of $ \mathbb P(\mathcal E)$ defined by $\mathcal E^{\vee}\to\mathcal L^{\vee}\vert_{D_1}$ and $\eta_2\colon W\to \mathbb P(\mathcal E_1)$ is the contraction of the strict transform of $\pi^{-1}(D_1)$ in W. In particular, $\psi$ is a link.

Proof. We have a diagram of exact sequences.

Since $\mathcal M^{\vee}=\ker(\alpha)$ and $\mathcal E^{\vee}_1=\ker(\alpha_1\circ\alpha)$ , we have an injection $\mathcal M^{\vee}\hookrightarrow\mathcal E^{\vee}_1$ . Moreover, $\alpha\beta(\mathcal E^{\vee}_1)$ is sent to zero by $\alpha_2$ , therefore $\alpha\beta(\mathcal E^{\vee}_1)$ is contained in $\mathcal L^{\vee}(-D_1)$ . Via a diagram chase one can prove that the sequence

$$0\to\mathcal M^{\vee}\to\mathcal E_1^{\vee}\to\mathcal L^{\vee}(-D_1)\to 0$$

is exact. Since $\mathcal M^{\vee}$ and $\mathcal L^{\vee}$ have constant rank, the rank of $\mathcal E_1^{\vee}$ is constant as well and we have proved (i) and (ii).

As for (iii), let U be a trivialising set for $\mathcal E^{\vee}$ and $\mathcal E^{\vee}_1$ . There is a $2\times 2$ matrix M representing the inclusion $\mathcal E^{\vee}_1\vert_U\to \mathcal E^{\vee}\vert_U$ . If $(e_1,e_2)$ and $(e_1, f_2)$ are local frames for $\mathcal E^{\vee}$ and $\mathcal E^{\vee}_1$ over U such that $e_1$ is a local frame for $\mathcal M^{\vee}$ , then the matrix has the form

$$M=\left(\!\! \begin{array}{cc} a_{1,1}& a_{1,2}\\ 0& a_{2,2} \end{array}\!\! \right)\!,$$

where $a_{1,1}\in H^0(U,\mathcal O_V^*)$ , $a_{2,2}\in H^0(U,\mathcal O_V(D_1))$ , $a_{1,2}\in H^0(U,\mathcal O_V)$ . The induced map between $\pi^{-1}(U)=U\times\mathbb P^1$ and $\pi_1^{-1}(U)=U\times\mathbb P^1$ is defined by the action of the transposed of M. Thus we have

$$\psi(z,[x_0:x_1])=(z,[a_{1,1}x_0:a_{1,2}x_0+a_{2,2}x_1]).$$

Without loss of generality we can assume that $a_{1,1}=1$ and multiply by $b=a_{2,2}^{-1}\in H^0(U,\mathcal O_V(-D_1))$ . The section b is a local equation for $D_1$ because $\mathcal E^{\vee}/\mathcal E^{\vee}_1=\mathcal L^{\vee}\vert_{D_1}$ is supported on $D_1$ . We can assume that there are local analytic coordinates $z=(z_1,\ldots,z_k)$ in U such that $D_1\cap U=\{z_1=0\}$ . Therefore, there are a regular function f(z) on U and a constant c such that $\psi(z,[x_0:x_1])=(z,[c z_1 x_0:z_1 f(z) x_0+x_1]).$ The indeterminacy locus is thus $D_1\times\{[1:0]\}$ . We consider the chart $x_0\neq 0$ , set $s=x_1/x_0$ and blow up the ideal $(z_1, s)$ . The blow up is

$$W=\{(z_1,\ldots,z_k,s),[u:v]\vert\;z_1v-su=0\}.$$

In the chart $u\neq 0$ we have $s=z_1 v/u$ . Thus, on W we extend $\psi$ to a morphism by $\tilde\psi((z_1,\ldots,z_k,s),[u:v])=(z,[c u : f(z)u +v])$ . This proves (iii).

We give a criterion for the existence of a section fixed by the automorphisms. Recall that, for a $\mathbb P^1$ -bundle $P\to V$ , we denote by $\mathrm{Aut}^{\circ}(P)_V$ the subgroup of $\mathrm{Aut}^{\circ}(P)$ acting trivially on V.

Proof. The projection $\mathcal{E}^{\vee}\to\mathcal L_2^{\vee}$ induces a section $V_0\to \mathbb P(\mathcal E)$ such that $\mathcal O_{\mathbb P(\mathcal E)}(1)\vert_{V_0}\sim \mathcal L_2^{\vee}$ (see Remark 2.8). Vice versa, any section $V_0\to \mathbb P(\mathcal E)$ with $\mathcal O_{\mathbb P(\mathcal E)}(1)\vert_{V_0}\sim \mathcal L_2^{\vee}$ corresponds to a surjective morphism $\mathcal{E}^{\vee}\to\mathcal L_2^{\vee}$ . Now

\begin{align*} \textrm{Hom}(\mathcal E^{\vee}, \mathcal L_2^{\vee})&=H^0(V, \mathcal L_2^{\vee}\otimes \mathcal E) \\ &=H^0(V, \mathcal L_1\otimes \mathcal L_2^{\vee})\oplus H^0(V, \mathcal L_2\otimes \mathcal L_2^{\vee})=\mathbb C, \end{align*}

where the last equality follows from the hypothesis on $\mathcal L_1, \mathcal L_2$ . It follows that $V_0$ is unique with the property that the restriction of $\mathcal{O}_{\mathbb P(\mathcal E)}(1)$ to it is $\mathcal L_2^{\vee}$ and thus is preserved by the automorphism group.

We recall the description of the automorphism group of a projective bundle of relative dimension 1.

Proof. We essentially follow [Reference MaruyamaMar71, 90–92]. Let $V=\cup V_i$ be a trivialising cover for $\mathbb P(\mathcal{E})$ . For the morphism $\phi\colon \mathcal{L}\hookrightarrow\mathcal{E}$ induced by the surjection $\phi^{\vee}\colon \mathcal{E}^{\vee}\to\mathcal{L}^{\vee}$ , the image $\operatorname{Im}\phi$ coincides with the annihilator of $\ker(\phi^{\vee})$ , which is a hyperplane. By hypothesis, $V_0=\mathbb P(\operatorname{Im}\phi)$ is fixed by $\mathrm{Aut}^{\circ}(\mathbb P(\mathcal{E}))_V$ . If $(v,[x_0:x_1])$ are local coordinates above $V_i$ , we can suppose that $V_0$ is given by $x_0=0$ . Therefore, an automorphism $\varphi\in\mathrm{Aut}^{\circ}(\mathbb P(\mathcal{E}))_V$ is given by

\[\varphi_i:=\varphi|_{V_i}=\Big(\begin{matrix} \alpha_{i}& s_i\\0&1\end{matrix}\Big)\in\mathrm{PGL}_2(\mathcal{O}_V(V_i)).\]

Moreover, the transition functions $\{g_{ij}\}_{i,j}$ of $\mathbb P(\mathcal{E})$ are given by

\[g_{ij}:=\Big(\begin{matrix} a_{ij}& c_{ij}\\ 0&1\end{matrix}\Big)\in\mathrm{PGL}_2(O_V(V_i\cap V_j)),\]

where the $\{a_{ij}\}_{ij}={b_{ij}}/{d_{ij}}$ and $b_{ij}$ are the transition functions of $\ker(\phi^{\vee})$ . Notice that the $\varphi_i$ glue to $\varphi\in\mathrm{Aut}^{\circ}(\mathbb P(\mathcal{E}))_V$ if and only if $g_{ij}\varphi_j=\varphi_ig_{ij}$ for all i,j, which is equivalent to

\[\alpha_ia_{ij}=a_{ij}\alpha_j,\quad a_{ij}s_j+c_{ij}=\alpha_jc_{ij}+s_i \quad\text{for all}\,\,i,j.\]

The first condition is equivalent to $\alpha_i=\alpha_j=:\alpha \in \Gamma(V,\mathcal{O}_V^*)=\mathbb{G}_m$ for all i,j. The second then becomes $s_ja_{ij}-s_i=c_{ij}(\alpha-1)$ .

Suppose that $\alpha\neq1$ . Then, conjugating $g_{ij}$ as follows:

\[\Big(\begin{matrix} 1& {s_i}/{(\alpha-1)}\\ 0 &1\end{matrix}\Big)\Big(\begin{matrix} a_{ij}& c_{ij}\\ 0&1\end{matrix}\Big)\Big(\begin{matrix} 1& -{s_j}/{(\alpha-1)}\\ 0 &1\end{matrix}\Big)=\Big(\begin{matrix} a_{ij}& 0\\ 0&1\end{matrix}\Big),\]

yields that $\mathbb P(\mathcal{E})$ is decomposable, i.e. $\mathbb P(\mathcal{E})\simeq \mathbb P(\ker(\phi^{\vee})\oplus \mathcal{L}')$ for some subbundle $\mathcal{L}'\subset\mathcal{E}$ . We can then assume that $c_{ij}=0$ and obtain that $s_i=a_{ij}s_j$ . Recall that $a_{ij}={b_{ij}}/{d_{ij}}$ , where $b_{ij}$ and $d_{ij}$ are respectively the transition functions of $\ker(\phi^{\vee})$ and $\mathcal{L}'$ . The $\{d_{ij}^{-1}\}$ define the line bundle $\det\mathcal{E}\otimes \ker(\phi^{\vee})$ , so the $s_i$ glue into a section $s\in\Gamma(V,\det\mathcal{E}\otimes \ker(\phi^{\vee})^{\otimes2})$ . This yields (i).

If $\alpha=1$ , then $s_i=a_{ij}s_j$ and again the $s_i$ glue to a section $s\in\Gamma(V,\det\mathcal{E}\otimes \ker(\phi^{\vee})^{\otimes2})$ and we obtain (ii).

1. (iii) If $\Gamma$ is non-trivial, then it is a non-trivial unipotent group and it has therefore at most one fixed point on a general fibre of the $\mathbb P^1$ -bundle $\pi$ .

2. (iv) If V is rationally connected and irrational, then $\mathrm{Aut}^{\circ}(V)$ is trivial by Proposition 2.10 and hence $\mathrm{Aut}^{\circ}(\mathbb P(\mathcal{E}))\simeq \mathrm{Aut}^{\circ}(\mathbb P(\mathcal{E}))_V$ .

3. (v) This follows from (iii) and (iv).

We conclude the section with the following proposition.

4. A family of projective bundles over a rationally connected non-rational threefold

We introduce the projective bundle $\mathcal P_n$ , a main player in this article, and prove Proposition 4.4 below that characterises $\mathrm{Aut}^{\circ}(\mathcal P_n)$ -equivariant birational maps starting from $\mathcal P_n$ .

Let us recall the properties of $\mathrm{Aut}^{\circ}(\mathcal P_n)$ .

The main goal of this section is to prove the following statement.

Proposition 4.4. Let $n\geq2$ and let $\Phi\colon\mathcal P_n\dashrightarrow W$ be a birational $\mathrm{Aut}^{\circ}(\mathcal P_n)$ -equivariant map. Then the following hold:

1. (i) $\mathrm{Aut}^{\circ}(\mathcal P_n)$ is a normal subgroup of $\mathrm{Aut}^{\circ}(W)$ ;

2. (ii) there are smooth varieties Y,Z and a fibration $Y\to Z$ with generic fibre $\mathbb P^1_{\mathbb C(Z)}$ and a birational $\mathrm{Aut}^{\circ}(W)$ -equivariant map $\eta\colon W\dashrightarrow Y$ and a birational map $\varphi\colon X\dashrightarrow Z$ fitting into the following commutative diagram.

To prove Proposition 4.4, let us fix the following notation and construction. For properties and notations regarding $\mathrm{Chow}(X)$ we refer to § 2.2.

Construction 4.5. Notation as in Proposition 4.4. Since $\Phi$ is $\mathrm{Aut}^{\circ}(\mathcal{P}_n)$ -equivariant, the group $\mathrm{Aut}^{\circ}(\mathcal P_n)$ acts on W. The general orbit has dimension 1, therefore, by Lemma 2.4, the orbits of the action of $\mathrm{Aut}^{\circ}(\mathcal P_n)$ on W have dimension 0 or 1. Let $\mathcal K_0\subseteq \textrm{Chow}(W)$ be the subvariety parametrising the closures of general orbits of $\mathrm{Aut}^{\circ}(\mathcal P_n)$ and $u_0\colon \mathcal U_0\to \mathcal K_0$ the restriction of the universal family. Let $\mathcal K\supseteq \mathcal K_0$ be the smallest $\mathrm{Aut}^{\circ}(W)$ -invariant closed set in $\textrm{Chow}(W)$ containing $\mathcal K_0$ and $u\colon \mathcal U\to \mathcal K$ the restriction of the universal family.

There are evaluation morphisms $e\colon \mathcal U\to W$ and $e_0=e\vert_{\mathcal U_0}\colon \mathcal U_0\to W$ . Those are $\mathrm{Aut}^{\circ}(W)$ -equivariant and $\mathrm{Aut}^{\circ}(\mathcal P_n)$ -equivariant, respectively. By Lemma 4.3, the conic bundle $\pi\colon \mathcal{P}_n\to X$ has a unique $\mathrm{Aut}^{\circ}(\mathcal P_n)$ -invariant section, which we denote by $X_0$ . As recalled in § 2.2, this implies that $e_0$ is birational.

In the following lemmas we follow the notation from Construction 4.5 and Proposition 4.4.

Suppose that $\mathcal K_0\neq \mathcal K$ and let M be the pullback by $\pi\colon \mathcal{P}_n\to X$ of an ample divisor on X. Let $[\Gamma]\in\mathcal K\setminus \mathcal K_0$ and $[\Gamma_0]\in\mathcal K_0$ be classes of curves $\Gamma,\Gamma_0$ on W such that $\Gamma_0\subset\Phi(\pi^{-1}(U))$ and such that $\Gamma$ is not in the exceptional locus of $\Phi^{-1}$ .

We denote by $(p,q)\colon\hat W\to \mathcal{P}_n\times W$ an $\mathrm{Aut}^{\circ}(\mathcal{P}_n)$ -equivariant resolution of $\Phi$ . We denote by $\hat{X}_0$ the strict transform in $\widehat W$ of $X_0$ . Let $\widehat\Gamma_0$ be the strict transform of $\Gamma_0$ in $\widehat{W}$ and let $\widehat\Gamma$ be an irreducible curve in $\hat W$ such that $q_*(\widehat\Gamma)=\Gamma$ . Notice that $\widehat\Gamma$ is not contracted by $\pi\circ p$ , because $[\Gamma]\notin\mathcal K_0$ .

Claim 4.8. We have

\[p^*M\cdot\widehat\Gamma_0=p^*M\cdot\widehat\Gamma+p^*M\cdot C\]

for some curve C in $\hat W$ .

Assuming the claim, we finish the proof. The left-hand side is zero, because $[\Gamma_0]\in\mathcal K_0$ , while $p^*M\cdot\widehat\Gamma>0$ and $p^*M\cdot C\geq0$ . This is impossible, so it follows that $\mathcal K_0=\mathcal K$ .

We are left with the proof of Claim 4.8. Let $(a,b)\colon\widehat{\mathcal{U}}\to \mathcal{U}\times\widehat W$ be a resolution of the indeterminacies of $\mathcal{U}\dashrightarrow\widehat W$ and let $\hat u\colon \widehat{\mathcal{U}}\to \mathcal{K}$ be the induced fibration. Let $\mathcal{C}$ be an irreducible curve in $\mathcal{K}$ such that $[\Gamma],[\Gamma_0]\in \mathcal{C}$ . Let S be the component of dimension 2 of $\hat u^{-1}(\mathcal{C})$ surjecting onto $\mathcal{C}$ . Then $b_*\hat u^*[\Gamma_0]=\widehat \Gamma_0$ and there is an effective curve C such that $b_*\hat u^*[\Gamma]=\widehat \Gamma +C$ . The claim follows as $b_*\hat u^*[\Gamma_0]\equiv b_*\hat u^*[\Gamma]$ .

Since $\mathrm{Aut}^{\circ}(\mathcal{P}_n)$ is not normal in $\mathrm{Aut}^{\circ}(W)$ , by Lemma 4.6 we have $\mathcal K_0\subsetneq\mathcal K$ . Then there is a 1-parameter subgroup $G\subset\mathrm{Aut}^{\circ}(W)$ with $G\not\subseteq\mathrm{Aut}^{\circ}(\mathcal{P}_n) $ and such that, for a general $g\in G$ , for a general $[\Gamma_0]\in\mathcal{K}_0$ , we have $[g\Gamma_0]\not\in\mathcal{K}_0$ . Let C be the strict transform of $g\Gamma_0$ in $\mathcal P_n$ . Lemma 4.9 implies that $C\cap X_0$ is non-empty. Let $H\subset\mathrm{Aut}^{\circ}(\mathcal{P}_n)$ be an additive 1-parameter subgroup, set $C_t:=tC$ , $t\in H$ , and consider the pencil $\{C_t\}_{t\in H}$ . The pencil defines a morphism $\mu\colon \mathbb A^1\times\mathbb P^1\to \mathcal{P}_n$ . Let F be the normalisation of the Zariski closure of $\mu(\mathbb A^1\times\mathbb P^1)$ and $n\colon F\to \mathcal P_n$ the induced morphism. The image $\pi(C)$ is a curve because $[g\Gamma_0]\not\in\mathcal{K}_0$ . Let $n\colon D\to \pi(C)$ be the normalisation of $\pi(C)$ . By abuse of notation, the strict transform of $C_t$ (respectively C) on $F$ is denoted by $C_t$ (respectively C) as well, as no confusion will arise.

Notice that F is smooth and that it is a $\mathbb P^1$ -bundle above D. Let $\theta\colon S\to F$ be a minimal resolution of the base locus of the pencil $\{C_t\}_{t\in H}$ . Then there is a conic fibration $u\colon S\to \mathbb P^1$ , whose fibres are the strict transforms $\overline{C_t}$ of $C_t$ , such that the following diagram commutes.

Notice that, since $\Gamma_0$ is rational, so is $C=g\Gamma_0$ and thus $D\simeq\mathbb P^1$ . By abuse of notation, the strict transform of $X_0$ on F will be denoted by $X_0$ as well, as no confusion will arise. Notice that H fixes $X_0$ pointwise as $H\subset\mathrm{Aut}^{\circ}(\mathcal{P}_n)$ , so all the points in $C\cap X_0$ are base points of the pencil $\{C_t\}_{t\in H}$ .

The H-action on F lifts to S and permutes non-trivially the fibres of u, since any fibre of $\pi$ intersects C in only finitely many points. By Blanchard’s lemma ([Reference BlanchardBla56], [Reference Brion, Samuel and UmaBSU13, Proposition 4.2.1]), H acts on $\mathbb P^1$ (base of the fibration u) and, since it is additive, it fixes exactly one point $[\infty]\in\mathbb P^1$ . Moreover, a curve E contained in the exceptional locus of $\theta$ is either contained in a fibre of u or it is a section of u, and the latter is the case if and only if E is the exceptional divisor of a point that is blown up last, or, equivalently, a $(-1)$ -curve.

Let t be such that $\overline C_t$ is irreducible. Thus we get

$$f\cdot C_t=\theta^*f\cdot\overline C_t=\bar f\cdot\overline C_t=1,$$

where the first equality is the projection formula, the second is because the pullback $\theta^*f$ coincides with the strict transform $\bar f$ , and the third because $\bar f$ is a section of u. This finishes the proof that $C_t$ is a section of $\pi\colon F\to D$ if $t\in H$ is general.

In particular, $C=C_0$ is a section of $\pi$ . It follows that every fibre of $\pi$ meets the base locus in F in at most one point. Since $C\cap X_0$ is non-empty, we pick a point $y_0\in C\cap X_0$ , we set $0=\pi(y_0)\in D$ and $f_0=\pi^{-1}(0)$ . Let $f_0=f_{t_0},f_{t_1},\dots,f_{t_l}$ be the fibres of $\pi$ meeting the base locus of the pencil $\{C_t\}_{t\in H}$ , and let $\overline{f_{t_i}}$ (respectively $\overline{f_0}$ ) be the strict transform of $f_{t_i}$ (respectively $f_0$ ) in S. Since $C_t$ is a section of $\pi$ , the $f_{t_i}$ are not contained in the union $\cup_{t\in H}C_t=\theta(\mathbb A^1\times\mathbb P^1)$ , and more precisely the intersection of $f_{t_i}$ with $\cup_{t\in H}C_t$ coincides with one point. Therefore each $f_{t_i}$ is contained in $\theta(u^{-1}[\infty])$ .

For each $i\geq0$ , denote by $E_{ij}$ the irreducible components of $\theta^{-1}(y_i)$ that are not sections of u and by $E_i^{\mathrm{sec}}$ the unique irreducible component that is a section of u. Since $\textrm{Exc}(\theta)$ is preserved by H, we have $\theta(E_{ij})=[\infty]$ for every i,j. Then

\[\theta^*C_0=\overline C_0+\sum a_{ij} E_{ij}+\sum_{i=0}^l a_i E_i^{\mathrm{sec}}\]

for some integers $a_{ij},a_i\geq0$ . Denote by $\overline C_{\infty}$ the fibre of u above $[\infty]$ . Since all $E_{ij}$ are contained in $\overline C_{\infty}$ and the $E_i^{\mathrm{sec}}$ are sections of u, we have

\[\overline C_{\infty}\cdot\theta^*C_0=C_t\cdot\theta^*C_0=\sum_{i=0}^l a_i\quad\text{for any $t\in H$}.\]

We also have

\[\overline C_{\infty}=\theta^*\bigg(\alpha X_0+\sum_{i=0}^l \beta_i f_{t_i}\bigg)-\sum_{i=0}^lb_i E_i^{\mathrm{sec}}+C'\]

for some integers $\alpha,\beta_i,b_i\geq0$ and some effective divisor C’ having no common component with $\sum E_i^{\mathrm{sec}}$ . In fact, $b_i\geq\beta_i\geq1$ for all $i\geq1$ , because $E_i^{\mathrm{sec}}\subseteq\mathrm{supp}(\theta^*f_{t_i})$ and $\bar f_{t_i}$ is contained in $\overline{C_{\infty}}$ for $i\geq1$ . Furthermore, $b_0\geq\beta_0+\alpha\geq2$ , because $E_0^{\mathrm{sec}}\subset\mathrm{supp}(\theta^*f_{0})\cap\mathrm{supp}(\theta^*X_0)$ and $\bar f_0$ is contained in $\overline{C_{\infty}}$ . We compute

\begin{align*}\sum_{i=0}^l a_i&=\bar C_{\infty}\cdot\theta^*C_0=\bar C_{\infty}\cdot\bigg(\sum_{i=0}^l a_i E_i^{\mathrm{sec}}\bigg)\\&=\bigg(\theta^*\bigg(\alpha X_0+\sum_{i=0}^l \beta_i f_{t_i}\bigg)-\sum_{i=0}^l b_i E_i^{\mathrm{sec}}+C'\bigg)\bigg(\sum_{i=0}^l a_i E_i^{\mathrm{sec}}\bigg)\\&\stackrel{(E_i^{\mathrm{sec}})^2=-1}=\sum_{i=1}^l b_ia_i+C'\cdot \bigg(\sum_{i=1}^l a_i E_i^{\mathrm{sec}}\bigg)\geq \sum_{i=0}^l b_ia_i\geq a_0+\sum_{i=0}^l a_i,\end{align*}

where the last inequality holds because $b_0\geq2$ and $b_i\geq1$ for $i\geq1$ . It follows that $a_0=0$ , which contradicts $f_0$ containing a base point of $\{C_t\}_{t\in H}$ . This proves that there is an $\mathrm{Aut}^{\circ}(W)$ -equivariant desingularisation of W which does not extract $X_0$ .

We denote by $(p,q)\colon\widehat W\to \mathcal{P}_n\times W$ an $\mathrm{Aut}^{\circ}(\mathcal{P}_n)$ -equivariant resolution of the indeterminacy of $\Phi$ . By Lemma 4.3(iii), the conic bundle $\pi\colon \mathcal{P}_n\to X$ has a unique $\mathrm{Aut}^{\circ}(\mathcal P_n)$ -invariant section $X_0\subset\mathcal{P}_n$ , and by $\widehat{X}_0$ we denote its strict transform in $\widehat W$ .

Let $(\bar{p},\bar{q})\colon\overline W\to \widehat W\times\widetilde W$ be a resolution of the indeterminacies of $\mu^{-1}q\colon \widehat{W}\dashrightarrow\widetilde W$ , such that $\bar q$ is a composition of blow ups of smooth centres.

Then the strict transform $\overline{X}_0$ in $\overline W$ is among the exceptional divisors of those blow ups. It follows that $X_0$ is birational to $\mathbb P^k\times Z$ with $k\in\{1,2,3\}$ and $\dim Z=3-k$ . Since $\dim Z\leq 2$ and Z is rationally connected, Z is rational and so is $X_0$ . This contradicts the hypothesis that X is not rational.

Therefore, $\mathcal K_0=\mathcal K$ , and hence $\mathrm{Aut}^{\circ}(\mathcal P_n)$ is normal in $\mathrm{Aut}^{\circ}(W)$ by Lemma 4.6. Let $\overline{\mathcal U}$ and $\overline{\mathcal K}$ be $\mathrm{Aut}^{\circ}(W)$ -equivariant compactifications of $\mathcal U$ and $\mathcal K$ such that there is a morphism $u\colon \overline{\mathcal U}\to \overline{\mathcal K}$ extending u. The map $e\colon \overline{\mathcal U}\dashrightarrow W$ is birational and $\mathrm{Aut}^{\circ}(W)$ -equivariant, see Construction 4.5. We set $Y=\overline{\mathcal U}$ and $Z=\overline{\mathcal K}$ .

Moreover, Z is birational to X because $\mathcal K_0=\mathcal K$ parametrises the 1-dimensional orbits of $\mathrm{Aut}^{\circ}(\mathcal P_n)$ .

5. $\mathrm{Aut}^{\circ}(\mathcal P_n)$ is not contained in a maximal subgroup of $\text{Bir}(\mathcal P_n)$

The aim of this section is to show in Theorem 5.2 that $\mathrm{Aut}^{\circ}(\mathcal P_n)$ is not contained in a maximal connected algebraic subgroup of $\text{Bir}(\mathcal P_n)$ if $n\geq2$ .

We have $\mathrm{Aut}^{\circ}(\mathbb P(\mathcal E))_V\simeq \Gamma\rtimes G$ , for some additive group $\Gamma$ which is non-trivial by our hypothesis on $\mathrm{Aut}^{\circ}(\mathbb P(\mathcal E))_V$ and $G=\mathbb G_m$ or $G=\{1\}$ , see Lemma 3.3. Let D be a very ample divisor on V such that:

1. – there is a smooth element $D_1\in |D|$ ;

2. – the line bundle $\mathcal M^{\vee}\otimes \mathcal L(D)$ is very ample, so that $\mathrm{Ext}^1_V(\mathcal L^{\vee}(- D),\mathcal M^{\vee})=H^1(\mathcal M^{\vee}\otimes \mathcal L( D))=0$ ; and

3. – $\dim \Gamma<\dim H^0 (V,\mathcal M^{\vee}\otimes \mathcal L( D)).$

Consider now the kernel $\mathcal E_1^{\vee}$ of the surjection $\mathcal E^{\vee}\to\mathcal L^{\vee}\vert_{D_1}$ . By Lemma 3.1(ii), the sheaf $\mathcal E_1^{\vee}$ is an extension of $\mathcal L^{\vee}(-D)$ and $\mathcal M^{\vee}$ and since $\mathrm{Ext}^1_V(\mathcal L^{\vee}(-D),\mathcal M^{\vee})=0$ we have $\mathcal E_1^{\vee}\cong \mathcal M^{\vee}\oplus \mathcal L^{\vee}(-D)$ . Therefore $\mathcal E_1$ is decomposable and $\mathcal E_1\cong \mathcal M\oplus \mathcal L(D)$ . By Lemma 3.2, the group $\mathrm{Aut}^{\circ}(\mathbb P(\mathcal E_1))_V$ fixes the section corresponding to $\phi^{\vee}\colon\mathcal E_1^{\vee}\to \mathcal L^{\vee}(-D)$ . We have $\ker(\phi^{\vee})=\mathcal M^{\vee}$ and $\det \mathcal E_1=\mathcal M\otimes \mathcal L(D)$ . Then by Lemma 3.3(i) $\mathrm{Aut}^{\circ}(\mathbb P(\mathcal E_1))_V\simeq H^0 (V,\mathcal M^{\vee}\otimes \mathcal L(D)) \rtimes \mathbb{G}_m$ .

Moreover, by the invariance of $V_0$ , the link or birational map $\psi\colon \mathbb P(\mathcal{E})\dashrightarrow\mathbb P(\mathcal{E}_1)$ obtained by Lemma 3.1(iii) is $\mathrm{Aut}^{\circ}(\mathbb P(\mathcal E))_V$ -equivariant. Since $\Gamma\subsetneq H^0 (V,\mathcal M^{\vee}\otimes \mathcal L(D))$ by assumption on D, we have $\mathrm{Aut}^{\circ}(\mathbb P(\mathcal E))_V\subsetneq \mathrm{Aut}^{\circ}(\mathbb P(\mathcal E_1))_V$ . This proves the claim.

We are now ready to prove the main theorem of this section.

By the Weil regularisation theorem [Reference WeilWei55], there is a variety W birational to $\mathcal P_n$ such that $H\subseteq \mathrm{Aut}^{\circ}(W)$ . By Proposition 4.4, there are smooth varieties Y,Z and a fibration $g\colon Y\to Z$ with generic fibre $\mathbb P^1_{\mathbb C(Z)}$ , a birational $\mathrm{Aut}^{\circ}(W)$ -equivariant map $W\dashrightarrow Y$ , and a birational map $X\dashrightarrow Z$ fitting into a commutative diagram.

By Proposition 2.9, we can assume that Z is smooth and g is a $\mathbb P^1$ -bundle.

We claim the following.

Assuming the claim, we finish the proof. Since Z is birational to X, by Proposition 2.10 we have $\mathrm{Aut}^{\circ}(Y)=\mathrm{Aut}^{\circ}(Y)_Z$ . Since $\mathrm{Aut}^{\circ}(\mathcal{P}_n)\subseteq\mathrm{Aut}^{\circ}(Y)$ , there is a non-trivial additive subgroup of $\mathrm{Aut}^{\circ}(Y)=\mathrm{Aut}^{\circ}(Y)_Z$ . By Claim 5.3, the group $\mathrm{Aut}^{\circ}(Y)$ fixes a rational section $Z_0$ of g. Since $\mathrm{Aut}^{\circ}(Y)$ acts trivially on Z, the base change $Y_0=Z_0\times_Z Y$ yields an $\mathrm{Aut}^{\circ}(Y)$ -equivariant birational morphism $Y_0\to Y$ . Thus $\mathrm{Aut}^{\circ}(Y)_Z=\mathrm{Aut}^{\circ}(Y)=\mathrm{Aut}^{\circ}(Y_0)=\mathrm{Aut}^{\circ}(Y_0)_{Z_0}$ . Moreover, $Y_0$ admits a section fixed pointwise by $\mathrm{Aut}^{\circ}(Y_0)$ . Therefore, by Proposition 5.1, there is a $\mathbb P^1$ -bundle $\mathbb P(\mathcal E_1)\to Z_{0}$ such that $\mathrm{Aut}^{\circ}(Y_{0})=\mathrm{Aut}^{\circ}(Y_{0})_{Z_{0}}\subsetneq \mathrm{Aut}^{\circ}(\mathbb P(\mathcal E_1))_{Z_{0}}=\mathrm{Aut}^{\circ}(\mathbb P(\mathcal E_1))$ . We may set $G=\mathrm{Aut}^{\circ}(\mathbb P(\mathcal E_1))$ .

We are left with the proof of Claim 5.3: by Proposition 3.5, if there is no $\mathrm{Aut}^{\circ}(Y)$ -equivariant rational section then $Y=Z\times\mathbb P^1$ . Then we would have $\mathrm{Aut}^{\circ}(Y)\cong \mathrm{PGL}_2(\mathbb{C})$ . But this contradicts the fact that $\mathrm{Aut}^{\circ}(Y)$ contains $\mathrm{Aut}^{\circ}(\mathcal{P}_n)$ , which has dimension at least 4 by Lemma 4.3(iii).

6. Proof of Main Theorem

We start with some preliminary lemmas on birational map from products with special properties.

Lemma 6.1. Let $X_1$ and $X_2$ be normal projective varieties such that $h^1(X_i,\mathcal O_{X_i})=0$ for $i=1,2$ and let $p_i\colon X_1\times X_2\to X_i$ be the projection onto $X_i$ . Then

\[\mathrm{Nef}(X_1\times X_2)=p_1^*\mathrm{Nef}(X_1)\oplus p_2^*\mathrm{Nef}(X_2).\]

Since q is a birational morphism, it is induced by a Cartier divisor D on $\hat P\times Y$ that is big and nef. By Lemma 6.1, we can write $D=p_{\hat P}^*D_1+p_Y^*D_2$ and $D_1\in\mathrm{Nef}(\hat P)$ and $D_2\in\mathrm{Nef}(Y)$ . Then q is of the form $q=(f_1,f_2)\colon \hat P\times Y\to P'\times Y'$ , where $f_1\colon\hat P\to P'$ is defined by $D_1$ and $f_2\colon Y\to Y_1$ is defined by $D_2$ , and both $f_1,f_2$ are birational since q is birational (so $D_1,D_2$ are nef and big). Since $\rho(Y)=1$ and $D_2$ is nef and big, it follows that $D_2$ is ample and hence that $f_2$ is an isomorphism.

We are now ready for the proof of our main result.