---
abstract: 'We produce the family of Calabi-Yau hypersurfaces $X_{n}$ of $({\mathbb P}^{1})^{n+1}$ in higher dimension whose inertia group contains non commutative free groups. This is completely different from Takahashi’s result [@ta98] for Calabi-Yau hypersurfaces $M_{n}$ of ${\mathbb P}^{n+1}$.'
address:
- ' (Masakatsu Hayashi) Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyamacho 1-1, Toyonaka, Osaka 560-0043, Japan '
- ' (Taro Hayashi) Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyamacho 1-1, Toyonaka, Osaka 560-0043, Japan '
author:
- Masakatsu Hayashi and Taro Hayashi
title: 'Calabi-Yau hypersurfaces in the direct product of ${\mathbb P}^{1}$ and inertia groups'
---

Introduction
============

Throughout this paper, we work over ${\mathbb C}$. Given an algebraic variety $X$, it is natural to consider its birational automorphisms $\varphi {\colon}X \dashrightarrow X$. The set of these birational automorphisms forms a group ${\operatorname{Bir}}(X)$ with respect to the composition. When $X$ is a projective space ${\mathbb P}^{n}$ or equivalently an $n$-dimensional rational variety, this group is called the Cremona group. In higher dimensional case ($n \geq 3$), though many elements of the Cremona group have been described, its whole structure is little known.

Let $V$ be an $(n+1)$-dimensional smooth projective rational manifold. In this paper, we treat subgroups called the “inertia group" (defined below ) of some hypersurface $X \subset V$ originated in [@gi94]. It consists of those elements of the Cremona group that act on $X$ as identity.

In Section \[cyn\], we mention the result (Theorem \[tak\]) of Takahashi [@ta98] about the smooth Calabi-Yau hypersurfaces $M_{n}$ of ${\mathbb P}^{n+1}$ of degree $n+2$ (that is, $M_{n}$ is a hypersurface such that it is simply connected, there is no holomorphic $k$-form on $M_{n}$ for $0<k<n$, and there is a nowhere vanishing holomorphic $n$-form $\omega_{M_{n}}$). It turns out that the inertia group of $M_{n}$ is trivial (Theorem \[intro2\]). Takahashi’s result (Theorem \[tak\]) is proved by using the “Noether-Fano inequality". It is the useful result that tells us when two Mori fiber spaces are isomorphic. Theorem \[intro2\] is a direct consequence of Takahashi’s result.

In Section \[cy1n\], we consider Calabi-Yau hypersurfaces $$X_{n} = (2, 2, \ldots , 2) \subset ({\mathbb P}^{1})^{n+1}.$$ Let $${\operatorname{UC}}(N) {\coloneqq}\overbrace{{\mathbb Z}/2{\mathbb Z}* {\mathbb Z}/2{\mathbb Z}* \cdots * {\mathbb Z}/2{\mathbb Z}}^{N} = \operatorname*{\raisebox{-0.8ex}{\scalebox{2.5}{$\ast$}}}_{i=1}^{N}\langle t_{i}\rangle$$ be the *universal Coxeter group* of rank $N$ where ${\mathbb Z}/2{\mathbb Z}$ is the cyclic group of order 2. There is no non-trivial relation between its $N$ natural generators $t_{i}$. Let $$p_{i} {\colon}X_{n} \to ({\mathbb P}^{1})^{n}\ \ \ (i=1, \ldots , n+1)$$ be the natural projections which are obtained by forgetting the $i$-th factor of $({\mathbb P}^{1})^{n+1}$. Then, the $n+1$ projections $p_{i}$ are generically finite morphism of degree 2. Thus, for each index $i$, there is a birational transformation $$\iota_{i} {\colon}X_{n} \dashrightarrow X_{n}$$ that permutes the two points of general fibers of $p_{i}$ and this provides a group homomorphism $$\Phi {\colon}{\operatorname{UC}}(n+1) \to {\operatorname{Bir}}(X_{n}).$$

From now, we set $P(n+1) {\coloneqq}({\mathbb P}^{1})^{n+1}$. Cantat-Oguiso proved the following theorem in [@co11].

$($[@co11 Theorem 1.3 (2)]$)$\[iota\] Let $X_{n}$ be a generic hypersurface of multidegree $(2,2,\ldots,2)$ in $P(n+1)$ with $n \geq 3$. Then the morphism $\Phi$ that maps each generator $t_{j}$ of ${\operatorname{UC}}(n+1)$ to the involution $\iota_{j}$ of $X_{n}$ is an isomorphism from ${\operatorname{UC}}(n+1)$ to ${\operatorname{Bir}}(X_{n})$.

Here “generic” means $X_{n}$ belongs to the complement of some countable union of proper closed subvarieties of the complete linear system $\big| (2, 2, \ldots , 2)\big|$.

Let $X \subset V$ be a projective variety. The *decomposition group* of $X$ is the group $$\begin{aligned}
{\operatorname{Dec}}(V, X) {\coloneqq}\{f \in {\operatorname{Bir}}(V)\ |\ f(X) =X \text{ and } f|_{X} \in {\operatorname{Bir}}(X) \}.\end{aligned}$$ The *inertia group* of $X$ is the group $$\begin{aligned}
\label{inertia}
{\operatorname{Ine}}(V, X) {\coloneqq}\{f \in {\operatorname{Dec}}(V, X)\ |\ f|_{X} = {\operatorname{id}}_{X}\}.\end{aligned}$$

Then it is natural to consider the following question:

\[qu\] Is the sequence $$\begin{aligned}
\label{se}
1 \longrightarrow {\operatorname{Ine}}(V, X) \longrightarrow {\operatorname{Dec}}(V, X) \overset{\gamma}{\longrightarrow} {\operatorname{Bir}}(X) \longrightarrow 1\end{aligned}$$ exact, i.e., is $\gamma$ surjective?

Note that, in general, this sequence is not exact, i.e., $\gamma$ is not surjective (see Remark \[k3\]). When the sequence is exact, the group ${\operatorname{Ine}}(V, X)$ measures how many ways one can extend ${\operatorname{Bir}}(X)$ to the birational automorphisms of the ambient space $V$.

Our main result is following theorem, answering a question asked by Ludmil Katzarkov:

\[intro\] Let $X_{n} \subset P(n+1)$ be a smooth hypersurface of multidegree $(2, 2, \ldots, 2)$ and $n \geq 3$. Then:

-   $\gamma {\colon}{\operatorname{Dec}}(P(n+1), X_{n}) \to {\operatorname{Bir}}(X_{n})$ is surjective, in particular Question $\ref{qu}$ is affirmative for $X_{n}$.

-   If, in addition, $X_{n}$ is generic, there are $n+1$ elements $\rho_{i}$ $(1 \leq i \leq n+1)$ of ${\operatorname{Ine}}(P(n+1), X_{n})$ such that $$\langle \rho_{1}, \rho_{2}, \ldots , \rho_{n+1} \rangle \simeq \underbrace{{\mathbb Z}* {\mathbb Z}* \cdots * {\mathbb Z}}_{n+1} \subset {\operatorname{Ine}}(P(n+1), X_{n}).$$ In particular, ${\operatorname{Ine}}(P(n+1), X_{n})$ is an infinite non-commutative group.

Our proof of Theorem \[intro\] is based on an explicit computation of elementary flavour.

We also consider another type of Calabi-Yau manifolds, namely smooth hypersurfaces of degree $n+2$ in ${\mathbb P}^{n+1}$ and obtain the following result:

\[intro2\] Suppose $n \geq 3$. Let $M_{n} = (n+2) \subset {\mathbb P}^{n+1}$ be a smooth hypersurface of degree $n+2$. Then Question $\ref{qu}$ is also affirmative for $M_{n}$. More precisely:

-   ${\operatorname{Dec}}({\mathbb P}^{n+1}, M_{n}) = \{ f \in {\operatorname{PGL}}(n+2, {\mathbb C}) = {\operatorname{Aut}}({\mathbb P}^{n+1})\ |\ f(M_{n}) = M_{n}\}$.

-   ${\operatorname{Ine}}({\mathbb P}^{n+1}, M_{n}) = \{{\operatorname{id}}_{{\mathbb P}^{n+1}}\}$, and $\gamma {\colon}{\operatorname{Dec}}({\mathbb P}^{n+1}, M_{n}) \overset{\simeq}{\longrightarrow} {\operatorname{Bir}}(M_{n}) = {\operatorname{Aut}}(M_{n})$.

It is interesting that the inertia groups of $X_{n} \subset P(n+1) = ({\mathbb P}^{1})^{n+1}$ and $M_{n} \subset {\mathbb P}^{n+1}$ have completely different structures though both $X_{n}$ and $M_{n}$ are Calabi-Yau hypersurfaces in rational Fano manifolds.

\[k3\] There is a smooth quartic $K3$ surface $M_{2} \subset {\mathbb P}^{3}$ such that $\gamma$ is not surjective (see [@og13 Theorem 1.2 (2)]). In particular, Theorem \[intro2\] is not true for $n = 2$.

Preliminaries
=============

In this section, we prepare some definitions and properties of birational geometry and introduce the Cremona group.

Divisors and singularities
--------------------------

Let $X$ be a projective variety. A *prime divisor* on $X$ is an irreducible subvariety of codimension one, and a *divisor* (resp. *${\mathbb Q}$-divisor* or *${\mathbb R}$-divisor*) on $X$ is a formal linear combination $D = \sum d_{i}D_{i}$ of prime divisors where $d_{i} \in {\mathbb Z}$ (resp. ${\mathbb Q}$ or ${\mathbb R}$). A divisor $D$ is called *effective* if $d_{i}$ $\geq$ 0 for every $i$ and denote $D \geq 0$. The closed set $\bigcup_{i}D_{i}$ of the union of prime divisors is called the *support* of $D$ and denote Supp$(D)$. A ${\mathbb Q}$-divisor $D$ is called *${\mathbb Q}$-Cartier* if, for some $0 \neq m \in {\mathbb Z}$, $mD$ is a Cartier divisor (i.e. a divisor whose divisorial sheaf ${\mathcal O}_{X}(mD)$ is an invertible sheaf), and $X$ is called ${\mathbb Q}$-*factorial* if every divisor is ${\mathbb Q}$-Cartier.

Note that, since the regular local ring is the unique factorization domain, every divisor automatically becomes the Cartier divisor on the smooth variety.

Let $f {\colon}X \dashrightarrow Y$ be a birational map between normal projective varieties, $D$ a prime divisor, and $U$ the domain of definition of $f$; that is, the maximal subset of $X$ such that there exists a morphism $f {\colon}U \to Y$. Then ${\operatorname{codim}}(X\setminus U) \geq 2$ and $D \cap U \neq \emptyset$, the image $(f|_{U})(D \cap U)$ is a locally closed subvariety of $Y$. If the closure of that image is a prime divisor of $Y$, we call it the *strict transform* of $D$ (also called the *proper transform* or *birational transform*) and denote $f_{*}D$. We define $f_{*}D = 0$ if the codimension of the image $(f|_{U})(D \cap U)$ is $\geq$ 2 in $Y$.

We can also define the strict transform $f_{*}Z$ for subvariety $Z$ of large codimension; if $Z \cap U \neq \emptyset$ and dimension of the image $(f|_{U})(Z \cap U)$ is equal to $\dim Z$, then we define $f_{*}Z$ as the closure of that image, otherwise $f_{*}Z$ = 0.

Let $(X, D)$ is a *log pair* which is a pair of a normal projective variety $X$ and a ${\mathbb R}$-divisor $D \geq 0$. For a log pair $(X, D)$, it is more natural to consider a *log canonical divisor* $K_{X} + D$ instead of a canonical divisor $K_{X}$.

A projective birational morphism $g {\colon}Y \to X$ is a *log resolution* of the pair $(X, D)$ if $Y$ is smooth, ${\operatorname{Exc}}(g)$ is a divisor, and $g_{*}^{-1}(D) \cup {\operatorname{Exc}}(g)$ has simple normal crossing support (i.e. each components is a smooth divisor and all components meet transversely) where ${\operatorname{Exc}}(g)$ is an exceptional set of $g$, and a divisor $over$ $X$ is a divisor $E$ on some smooth variety $Y$ endowed with a proper birational morphism $g {\colon}Y \to X$.

If we write $$K_{Y} + \Gamma + \sum E_{i} = g^{*}(K_{X}+D) + \sum a_{E_{i}}(X, D)E_{i},$$ where $\Gamma$ is the strict transform of $D$ and $E_{i}$ runs through all prime exceptional divisors, then the numbers $a_{E_{i}}(X, D)$ is called the *discrepancies of $(X, D)$ along $E_{i}$*. The *discrepancy of* $(X, D)$ is given by $${\operatorname{discrep}}(X, D) {\coloneqq}\inf\{ a_{E_{i}}(X, D)\ |\ E_{i} \text{ is a prime exceptional divisor over } X\}.$$ The discrepancy $a_{E_{i}}(X, D)$ along $E_{i}$ is independent of the choice of birational maps $g$ and only depends on $E_{i}$.

Let us denote ${\operatorname{discrep}}(X, D) = a_{E}$. A pair $(X, D)$ is *log canonical* (resp. *Kawamata log terminal* ($klt$)) if $a_{E} \geq 0$ (resp. $a_{E} > 0$). A pair $(X, D)$ is *canonical* (resp. *terminal*) if $a_{E} \geq 1$ (resp. $a_{E} > 1$).

Cremona groups
--------------

Let $n$ be a positive integer. The *Cremona group* ${\operatorname{Cr}}(n)$ is the group of automorphisms of ${\mathbb C}(X_{1}, \ldots, X_{n})$, the ${\mathbb C}$-algebra of rational functions in $n$ independent variables.

Given $n$ rational functions $F_{i} \in {\mathbb C}(X_{1}, \ldots, X_{n})$, $1 \leq i \leq n$, there is a unique endomorphism of this algebra maps $X_{i}$ onto $F_{i}$ and this is an automorphism if and only if the rational transformation $f$ defined by $f(X_{1}, \ldots, X_{n}) = (F_{1}, \ldots, F_{n})$ is a birational transformation of the affine space ${\mathbb A}^{n}$. Compactifying ${\mathbb A}^{n}$, we get $${\operatorname{Cr}}(n) = {\operatorname{Bir}}({\mathbb A}^{n}) = {\operatorname{Bir}}({\mathbb P}^{n})$$ where Bir$(X)$ denotes the group of all birational transformations of $X$.

In the end of this section, we define two subgroups in ${\operatorname{Cr}}(n)$ introduced by Gizatullin [@gi94].

Let $V$ be an $(n+1)$-dimensional smooth projective rational manifold and $X \subset V$ a projective variety. The *decomposition group* of $X$ is the group $${\operatorname{Dec}}(V, X) {\coloneqq}\{f \in {\operatorname{Bir}}(V)\ |\ f(X) =X \text{ and } f|_{X} \in {\operatorname{Bir}}(X) \}.$$ The *inertia group* of $X$ is the group $${\operatorname{Ine}}(V, X) {\coloneqq}\{f \in {\operatorname{Dec}}(V, X)\ |\ f|_{X} = {\operatorname{id}}_{X}\}.$$

The decomposition group is also denoted by Bir$(V, X)$. By the definition, the correspondence $$\gamma {\colon}{\operatorname{Dec}}(V, X) \ni f \mapsto f|_{X} \in {\operatorname{Bir}}(X)$$ defines the exact sequence: $$\begin{aligned}
\label{seq}
1 \longrightarrow {\operatorname{Ine}}(V, X) = \ker \gamma \longrightarrow {\operatorname{Dec}}(V, X) \overset{\gamma}{\longrightarrow} {\operatorname{Bir}}(X).\end{aligned}$$

So, it is natural to consider the following question (which is same as Question \[qu\]) asked by Ludmil Katzarkov:

\[qexact\] Is the sequence $$\begin{aligned}
\label{exact}
1 \longrightarrow {\operatorname{Ine}}(V, X) \longrightarrow {\operatorname{Dec}}(V, X) \overset{\gamma}{\longrightarrow} {\operatorname{Bir}}(X) \longrightarrow 1\end{aligned}$$ exact, i.e., is $\gamma$ surjective?

In general, the above sequence is not exact, i.e., $\gamma$ is not surjective. In fact, there is a smooth quartic $K3$ surface $M_{2} \subset {\mathbb P}^{3}$ such that $\gamma$ is not surjective ([@og13 Theorem 1.2 (2)]).

Calabi-Yau hypersurface in ${\mathbb P}^{n+1}$ {#cyn}
==============================================

Our goal, in this section, is to prove Theorem \[intro2\] (i.e. Theorem \[ta\]). Before that, we introduce the result of Takahashi [@ta98].

Let $X$ be a normal ${\mathbb Q}$-factorial projective variety. The 1*-cycle* is a formal linear combination $C = \sum a_{i}C_{i}$ of proper curves $C_{i} \subset X$ which are irreducible and reduced. By the theorem of the base of Néron-Severi (see [@kl66]), the whole numerical equivalent class of 1-cycle with real coefficients becomes the finite dimensional ${\mathbb R}$-vector space and denotes $N_{1}(X)$. The dimension of $N_{1}(X)$ or its dual $N^{1}(X)$ with respect to the intersection form is called the *Picard number* and denote $\rho(X)$.

$($[@ta98 Theorem 2.3]$)$\[tak\] Let $X$ be a Fano manifold $($i.e. a manifold whose anti-canonical divisor $-K_{X}$ is ample,$)$ with $\dim X \geq 3$ and $\rho(X) = 1$, $S \in |-K_{X}|$ a smooth hypersurface with ${\operatorname{Pic}}(X) \to {\operatorname{Pic}}(S)$ surjective. Let $\Phi {\colon}X \dashrightarrow X'$ be a birational map to a ${\mathbb Q}$-factorial terminal variety $X'$ with $\rho(X') = 1$ which is not an isomorphism, and $S' = \Phi_{*}S$. Then $K_{X'} + S'$ is ample.

This theorem is proved by using the *Noether-Fano inequality* which is one of the most important tools in birational geometry, which gives a precise bound on the singularities of indeterminacies of a birational map and some conditions when it becomes isomorphism.

This inequality is essentially due to [@im71], and Corti proved the general case of an arbitrary Mori fiber space of dimension three [@co95]. It was extended in all dimensions in [@ta95], [@bm97], [@is01], and [@df02], (see also [@ma02]). In particular, a log generalized version obtained independently in [@bm97], [@ta95] is used for the proof of Theorem \[tak\].

After that, we consider $n$-dimensional *Calabi-Yau manifold* $X$ in this paper. It is a projective manifold which is simply connected, $$H^{0}(X, \Omega_{X}^{i}) = 0\ \ \ (0<i<\dim X = n),\ \ \textrm{and \ } H^{0}(X, \Omega_{X}^{n}) = {\mathbb C}\omega_{X},$$ where $\omega_{X}$ is a nowhere vanishing holomorphic $n$-form.

The following theorem is a consequence of the Theorem \[tak\], which is same as Theorem \[intro2\]. This provides an example of the Calabi-Yau hypersurface $M_{n}$ whose inertia group consists of only identity transformation.

\[ta\] Suppose $n \geq 3$. Let $M_{n} = (n+2) \subset {\mathbb P}^{n+1}$ be a smooth hypersurface of degree $n+2$. Then $M_{n}$ is a Calabi-Yau manifold of dimension $n$ and Question $\ref{qexact}$ is affirmative for $M_{n}$. More precisely:

-   ${\operatorname{Dec}}({\mathbb P}^{n+1}, M_{n}) = \{ f \in {\operatorname{PGL}}(n+2, {\mathbb C}) = {\operatorname{Aut}}({\mathbb P}^{n+1})\ |\ f(M_{n}) = M_{n}\}$.

-   ${\operatorname{Ine}}({\mathbb P}^{n+1}, M_{n}) = \{{\operatorname{id}}_{{\mathbb P}^{n+1}}\}$, and $\gamma {\colon}{\operatorname{Dec}}({\mathbb P}^{n+1}, M_{n}) \overset{\simeq}{\longrightarrow} {\operatorname{Bir}}(M_{n}) = {\operatorname{Aut}}(M_{n})$.

By Lefschetz hyperplane section theorem for $n \geq 3$, $\pi_{1}(M_{n}) \simeq \pi_{1}({\mathbb P}^{n+1}) = \{{\operatorname{id}}\}$, ${\operatorname{Pic}}(M_{n}) = {\mathbb Z}h$ where $h$ is the hyperplane class. By the adjunction formula, $$K_{M_{n}} = (K_{{\mathbb P}^{n+1}} + M_{n})|_{M_{n}} = -(n+2)h + (n+2)h = 0$$ in Pic$(M_{n})$.

By the exact sequence $$0 \longrightarrow {\mathcal O}_{{\mathbb P}^{n+1}}(-(n+2)) \longrightarrow {\mathcal O}_{{\mathbb P}^{n+1}} \longrightarrow {\mathcal O}_{M_{n}} \longrightarrow 0$$ and $$h^{k}({\mathcal O}_{{\mathbb P}^{n+1}}(-(n+2))) = 0\ \ \text{for}\ \ 1 \leq k \leq n,$$ $$H^{k}({\mathcal O}_{M_{n}}) \simeq H^{k}({\mathcal O}_{{\mathbb P}^{n+1}}) = 0\ \ \text{for}\ \ 1 \leq k \leq n-1.$$ Hence $H^{0}(\Omega^{k}_{M_{n}}) = 0$ for $1 \leq k \leq n-1$ by the Hodge symmetry. Hence $M_{n}$ is a Calabi-Yau manifold of dimension $n$.

By ${\operatorname{Pic}}(M_{n}) = {\mathbb Z}h$, there is no small projective contraction of $M_{n}$, in particular, $M_{n}$ has no flop. Thus by Kawamata [@ka08], we get ${\operatorname{Bir}}(M_{n}) = {\operatorname{Aut}}(M_{n})$, and $g^{*}h = h$ for $g \in {\operatorname{Aut}}(M_{n}) = {\operatorname{Bir}}(M_{n})$.

So we have $g = \tilde{g}|_{M_{n}}$ for some $\tilde{g} \in {\operatorname{PGL}}(n+1, {\mathbb C})$. Assume that $f \in {\operatorname{Dec}}({\mathbb P}^{n+1}, M_{n})$. Then $f_{*}(M_{n}) = M_{n}$ and $K_{{\mathbb P}^{n+1}} + M_{n} = 0$. Thus by Theorem \[tak\], $f \in {\operatorname{Aut}}({\mathbb P}^{n+1}) = {\operatorname{PGL}}(n+2, {\mathbb C})$. This proves (1) and the surjectivity of $\gamma$.

Let $f|_{M_{n}} = {\operatorname{id}}_{M_{n}}$ for $f \in {\operatorname{Dec}}({\mathbb P}^{n+1}, M_{n})$. Since $f \in {\operatorname{PGL}}(n+1, {\mathbb C})$ by (1) and $M_{n}$ generates ${\mathbb P}^{n+1}$, i.e., the projective hull of $M_{n}$ is ${\mathbb P}^{n+1}$, it follows that $f = {\operatorname{id}}_{{\mathbb P}^{n+1}}$ if $f|_{M_{n}} = {\operatorname{id}}_{M_{n}}$. Hence ${\operatorname{Ine}}({\mathbb P}^{n+1}, M_{n}) = \{{\operatorname{id}}_{{\mathbb P}^{n+1}}\}$, i.e., $\gamma$ is injective. So, $\gamma {\colon}{\operatorname{Dec}}({\mathbb P}^{n+1}, M_{n}) \overset{\simeq}{\longrightarrow} {\operatorname{Bir}}(M_{n}) = {\operatorname{Aut}}(M_{n})$.

Calabi-Yau hypersurface in $({\mathbb P}^{1})^{n+1}$ {#cy1n}
====================================================

As in above section, the Calabi-Yau hypersurface $M_{n}$ of ${\mathbb P}^{n+1}$ with $n \geq 3$ has only identical transformation as the element of its inertia group. However, there exist some Calabi-Yau hypersurfaces in the product of ${\mathbb P}^{1}$ which does not satisfy this property; as result (Theorem \[main\]) shows.

To simplify, we denote $$\begin{aligned}
P(n+1) &{\coloneqq}({\mathbb P}^{1})^{n+1} = {\mathbb P}^{1}_{1} \times {\mathbb P}^{1}_{2} \times \cdots \times {\mathbb P}^{1}_{n+1},\\
P(n+1)_{i} &{\coloneqq}{\mathbb P}^{1}_{1} \times \cdots \times {\mathbb P}^{1}_{i-1} \times {\mathbb P}^{1}_{i+1} \times \cdots \times {\mathbb P}^{1}_{n+1} \simeq P(n),\end{aligned}$$ and $$\begin{aligned}
p^{i} {\colon}P(n+1) &\to {\mathbb P}^{1}_{i} \simeq {\mathbb P}^{1},\\
p_{i} {\colon}P(n+1) &\to P(n+1)_{i}\end{aligned}$$ as the natural projection. Let $H_{i}$ be the divisor class of $(p^{i})^{*}({\mathcal O}_{{\mathbb P}^{1}}(1))$, then $P(n+1)$ is a Fano manifold of dimension $n+1$ and its canonical divisor has the form $\displaystyle{-K_{P(n+1)} = \sum^{n+1}_{i=1}2H_{i}}$. Therefore, by the adjunction formula, the generic hypersurface $X_{n} \subset P(n+1)$ has trivial canonical divisor if and only if it has multidegree $(2, 2, \ldots, 2)$. More strongly, for $n \geq 3$, $X_{n} = (2, 2, \ldots, 2)$ becomes a Calabi-Yau manifold of dimension $n$ and, for $n=2$, a $K3$ surface (i.e. 2-dimensional Calabi-Yau manifold). This is shown by the same method as in the proof of Theorem \[ta\].

From now, $X_{n}$ is a generic hypersurface of $P(n+1)$ of multidegree $(2, 2, \ldots , 2)$ with $n \geq 3$. Let us write $P(n+1) = {\mathbb P}^{1}_{i} \times P(n+1)_{i}$. Let $[x_{i1} : x_{i2}]$ be the homogeneous coordinates of ${\mathbb P}^{1}_{i}$. Hereafter, we consider the affine locus and denote by $\displaystyle x_{i} = \frac{x_{i2}}{x_{i1}}$ the affine coordinates of ${\mathbb P}^{1}_{i}$ and by ${\bf z}_{i}$ that of $P(n+1)_{i}$. When we pay attention to $x_{i}$, $X_{n}$ can be written by following equation $$\begin{aligned}
\label{xn}
X_{n} = \{ F_{i,0}({\bf z}_{i})x_{i}^{2} + F_{i,1}({\bf z}_{i})x_{i} + F_{i,2}({\bf z}_{i}) = 0 \}\end{aligned}$$ where each $F_{i,j}({\bf z}_{i})$ $(j = 0, 1, 2)$ is a quadratic polynomial of ${\bf z}_{i}$. Now, we consider the two involutions of $P(n+1)$: $$\begin{aligned}
\tau_{i} {\colon}(x_{i}, {\bf z}_{i}) &\to \left(-x_{i}- \frac{F_{i,1}({\bf z}_{i})}{F_{i,0}({\bf z}_{i})}, {\bf z}_{i} \right)\label{tau}\\
\sigma_{i} {\colon}(x_{i}, {\bf z}_{i}) &\to \left(\frac{F_{i,2}({\bf z}_{i})}{x_{i} \cdot F_{i,0}({\bf z}_{i})}, {\bf z}_{i} \right).\label{sigma}\end{aligned}$$ Then $\tau_{i}|_{X_{n}} = \sigma_{i}|_{X_{n}} = \iota_{i}$ by definition of $\iota_{i}$ (cf. Theorem \[iota\]).

We get two birational automorphisms of $X_{n}$ $$\begin{aligned}
\rho_{i} = \sigma_{i} \circ \tau_{i} {\colon}(x_{i}, {\bf z}_{i}) &\to \left( \frac{F_{i,2}({\bf z}_{i})}{-x_{i} \cdot F_{i,0}({\bf z}_{i}) - F_{i,1}({\bf z}_{i})}, \  {\bf z}_{i} \right)\\
\rho'_{i} = \tau_{i} \circ \sigma_{i} {\colon}(x_{i}, {\bf z}_{i}) &\to \left( -\frac{x_{i} \cdot F_{i,1}({\bf z}_{i}) + F_{i,2}({\bf z}_{i})}{x_{i}\cdot F_{i,0}({\bf z}_{i})}, \  {\bf z}_{i} \right).\end{aligned}$$ Obviously, both $\rho_{i}$ and $\rho'_{i}$ are in Ine$(P(n+1), X_{n})$, map points not in $X_{n}$ to other points also not in $X_{n}$, and $\rho_{i}^{-1} = \rho'_{i}$ by $\tau_{i}^{2} = \sigma_{i}^{2} = {\operatorname{id}}_{P(n+1)}$.

\[order\] Each $\rho_{i}$ has infinite order.

By the definiton of $\rho_{i}$ and $\rho'_{i} = \rho_{i}^{-1}$, it suffices to show $$\begin{aligned}
{\begin{pmatrix}
0 & F_{i,2}\\
-F_{i,0} & -F_{i,1}
\end{pmatrix}}^{k}
\neq \alpha I\end{aligned}$$ for any $k \in {\mathbb Z}\setminus \{0\}$ where $I$ is an identity matrix and $\alpha \in {\mathbb C}^{\times}$. Their eigenvalues are $$\frac{-F_{i,1} \pm \sqrt{F_{i,1}^{2} - 4F_{i,0}F_{i,2}}}{2}.$$ Here $F_{i,1}^{2} - 4F_{i,0}F_{i,2} \neq 0$ as $X_{n}$ is general (for all $i$).

If $\begin{pmatrix}
0 & F_{i,2}\\
-F_{i,0} & -F_{i,1}
\end{pmatrix}^{k}
= \alpha I$ for some $k \in {\mathbb Z}\setminus \{0\}$ and $\alpha \in {\mathbb C}^{\times}$, then $$\left(\frac{-F_{i,1} + \sqrt{F_{i,1}^{2} - 4F_{i,0}F_{i,2}}}{-F_{i,1} - \sqrt{F_{i,1}^{2} - 4F_{i,0}F_{i,2}}}\right)^{k} = 1,$$ a contradiction to the assumption that $X_{n}$ is generic.

We also remark that Proposition \[order\] is also implicitly proved in Theorem \[main\].

Our main result is the following (which is same as Theorem \[intro\]):

\[main\] Let $X_{n} \subset P(n+1)$ be a smooth hypersurface of multidegree $(2, 2, \ldots, 2)$ and $n \geq 3$. Then:

-   $\gamma {\colon}{\operatorname{Dec}}(P(n+1), X_{n}) \to {\operatorname{Bir}}(X_{n})$ is surjective, in particular Question $\ref{qexact}$ is affirmative for $X_{n}$.

-   If, in addition, $X_{n}$ is generic, $n+1$ elements $\rho_{i} \in {\operatorname{Ine}}(P(n+1), X_{n})$ $(1 \leq i \leq n+1)$ satisfy $$\langle \rho_{1}, \rho_{2}, \ldots , \rho_{n+1} \rangle \simeq \underbrace{{\mathbb Z}* {\mathbb Z}* \cdots * {\mathbb Z}}_{n+1} \subset {\operatorname{Ine}}(P(n+1), X_{n}).$$ In particular, ${\operatorname{Ine}}(P(n+1), X_{n})$ is an infinite non-commutative group.

Let ${\operatorname{Ind}}(\rho)$ be the union of the indeterminacy loci of each $\rho_{i}$ and $\rho^{-1}_{i}$; that is, $\displaystyle {\operatorname{Ind}}(\rho) = \bigcup_{i=1}^{n+1}\big({\operatorname{Ind}}(\rho_{i}) \cup {\operatorname{Ind}}(\rho^{-1}_{i})\big)$ where ${\operatorname{Ind}}(\rho_{i})$ is the indeterminacy locus of $\rho_{i}$. Clearly, ${\operatorname{Ind}}(\rho)$ has codimension $\geq 2$ in $P(n+1)$.

Let us show Theorem \[main\] (1). Suppose $X_{n}$ is generic. For a general point $x \in P(n+1)_{i}$, the set $p_{i}^{-1}(x)$ consists of two points. When we put these two points $y$ and $y'$, then the correspondence $y \leftrightarrow y'$ defines a natural birational involutions of $X_{n}$, and this is the involution $\iota_{j}$. Then, by Cantat-Oguiso’s result [@co11 Theorem 3.3 (4)], ${\operatorname{Bir}}(X_{n})$ $(n\geq 3)$ coincides with the group $\langle \iota_{1}, \iota_{2}, \ldots , \iota_{n+1} \rangle \simeq  \underbrace{{\mathbb Z}/2{\mathbb Z}* {\mathbb Z}/2{\mathbb Z}* \cdots * {\mathbb Z}/2{\mathbb Z}}_{n+1}$.

Two involutions $\tau_{j}$ and $\sigma_{j}$ of $X_{n}$ which we construct in and are the extensions of the covering involutions $\iota_{j}$. Hence, $\tau_{j}|_{X_{n}} = \sigma_{j}|_{X_{n}} = \iota_{j}$. Thus $\gamma$ is surjective. Since automorphisms of $X_{n}$ come from that of total space $P(n+1)$, it holds the case that $X_{n}$ is not generic. This completes the proof of Theorem \[main\] (1).

Then, we show Theorem \[main\] (2). By Proposition \[order\], order of each $\rho_{i}$ is infinite. Thus it is sufficient to show that there is no non-trivial relation between its $n + 1$ elements $\rho_{i}$. We show by arguing by contradiction.

Suppose to the contrary that there is a non-trivial relation between $n+1$ elements $\rho_{i}$, that is, there exists some positive integer $N$ such that $$\begin{aligned}
\label{rho}
\rho_{i_{1}}^{n_{1}} \circ \rho_{i_{2}}^{n_{2}} \circ \cdots \circ \rho_{i_{l}}^{n_{l}} = {\operatorname{id}}_{P(n+1)}\end{aligned}$$ where $l$ is a positive integer, $n_{k} \in {\mathbb Z}\setminus\{0\}$ $(1\leq k \leq l)$, and each $\rho_{i_{k}}$ denotes one of the $n + 1$ elements $\rho_{i}$ $(1 \leq i \leq n+1)$ and satisfies $\rho_{i_{k}} \neq \rho_{i_{k+1}}$ $(0 \leq k \leq l-1)$. Put $N = |n_{1}| + \cdots + |n_{l}|$.

In the affine coordinates $(x_{i_{1}}, {\bf z}_{i_{1}})$ where $x_{i_{1}}$ is the affine coordinates of $i_{1}$-th factor ${\mathbb P}^{1}_{i_{1}}$, we can choose two distinct points $(\alpha_{1}, {\bf z}_{i_{1}})$ and $(\alpha_{2}, {\bf z}_{i_{1}})$, $\alpha_{1} \neq \alpha_{2}$, which are not included in both $X_{n}$ and ${\operatorname{Ind}}(\rho)$.

By a suitable projective linear coordinate change of ${\mathbb P}^{1}_{i_{1}}$, we can set $\alpha_{1} = 0$ and $\alpha_{2} = \infty$. When we pay attention to the $i_{1}$-th element $x_{i_{1}}$ of the new coordinates, we put same letters $F_{i_{1},j}({\bf z}_{i_{1}})$ for the definitional equation of $X_{n}$, that is, $X_{n}$ can be written by $$X_{n} = \{ F_{i_{1},0}({\bf z}_{i_{1}})x_{i_{1}}^{2} + F_{i_{1},1}({\bf z}_{i_{1}})x_{i_{1}} + F_{i_{1},2}({\bf z}_{i_{1}}) = 0 \}.$$ Here the two points $(0, {\bf z}_{i_{1}})$ and $(\infty, {\bf z}_{i_{1}})$ not included in $X_{n} \cup {\operatorname{Ind}}(\rho)$. From the assumption, both two equalities hold:

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We proceed by dividing into the following two cases.

[(i). The case where $n_{1} > 0$. Write $\rho_{i_{1}} \circ \rho_{i_{1}}^{n_{1}-1} \circ \rho_{i_{2}}^{n_{2}} \circ  \cdots \circ \rho_{i_{l}}^{n_{l}} = {\operatorname{id}}_{P(n+1)}$. ]{}

Let us denote $\rho_{i_{1}}^{n_{1}-1} \circ \cdots \circ \rho_{i_{l}}^{n_{l}}(0, {\bf z}_{i_{1}}) = (p, {\bf z}_{i_{1}}')$, then, by the definition of $\rho_{i_{1}}$, it maps $p$ to $0$. That is, the equation $F_{i_{1},2}({\bf z}'_{i_{1}}) = 0$ is satisfied. On the other hand, the intersection of $X_{n}$ and the hyperplane $(x_{i_{1}}=0)$ is written by $$X_{n} \cap (x_{i_{1}}=0) = \{F_{i_{1},2}({\bf z}_{i_{1}}) = 0\}.$$ This implies $(0, {\bf z}'_{i_{1}}) = \rho_{i_{1}}(p, {\bf z}'_{i_{1}}) = (0, {\bf z}_{i_{1}})$ is a point on $X_{n}$, a contradiction to the fact that $(0, {\bf z}_{i_{1}}) \notin X_{n}$.

[(ii). The case where $n_{1} < 0$. Write $\rho^{-1}_{i_{1}} \circ \rho_{i_{1}}^{n_{1}+1} \circ \rho_{i_{2}}^{n_{2}} \circ  \cdots \circ \rho_{i_{l}}^{n_{l}} = {\operatorname{id}}_{P(n+1)}$. ]{}

By using the assumption , we lead the contradiction by the same way as in (i). Precisely, we argue as follows.

Let us write $\displaystyle x_{i_{1}} = \frac{1}{y_{i_{1}}}$, then $(x_{i_{1}} = \infty, {\bf z}_{i_{1}}) = (y_{i_{1}} = 0, {\bf z}_{i_{1}})$ and $X_{n}$ and $\rho^{-1}_{i_{1}}$ can be written by $$X_{n} {\coloneqq}\{F_{i_{1},0}({\bf z}_{i_{1}}) + F_{i_{1},1}({\bf z}_{i_{1}})y_{i_{1}} + F_{i_{1},2}({\bf z}_{i_{1}})y_{i_{1}}^{2} = 0\},$$ $$\rho^{-1}_{i_{1}} {\colon}(y_{i_{1}}, {\bf z}_{i_{1}}) \to \left(\ -\frac{F_{i_{1},0}({\bf z}_{i_{1}})}{F_{i_{1},1}({\bf z}_{i_{1}}) + y_{i_{1}}\cdot F_{i_{1},2}({\bf z}_{i_{1}})},\ {\bf z}_{i_{1}} \right).$$

Let us denote $ \rho_{i_{1}}^{n_{1}+1} \circ \rho_{i_{2}}^{n_{2}} \circ  \cdots \circ \rho_{i_{l}}^{n_{l}} (y_{i_{1}} =0, {\bf z}_{i_{1}}) = (y_{i_{1}} = q, {\bf z}_{i_{1}}'')$, then $\rho^{-1}_{i_{1}}$ maps $q$ to $0$. That is, the equation $F_{i_{1},0}({\bf z}''_{i_{1}}) = 0$ is satisfied, but the intersection of $X_{n}$ and the hyperplane $(y_{i_{1}} = 0)$ is written by $$X_{n}\cap (y_{i_{1}} = 0) = \{F_{i_{1},0}({\bf z}_{i_{1}}) = 0\}.$$ This implies $(y_{i_{1}}=0, {\bf z}''_{i_{1}}) = \rho^{-1}_{i_{1}}(y_{i_{1}} = q, {\bf z}_{i_{1}}'') = (x_{i_{1}}=\infty, {\bf z}_{i_{1}})$ is a point on $X_{n}$; that is, $(x_{i_{1}}=\infty, {\bf z}_{i_{1}}) \in X_{n} \cap (x_{i_{1}}=\infty)$. This is contradiction.

From (i) and (ii), we can conclude that there does not exist such $N$. This completes the proof of Theorem \[main\] (2).

Note that, for the cases $n = 2$ and $1$, Theorem \[main\] (2) also holds though (1) does not hold.

[**Acknowledgements:** ]{} The authors would like to express their sincere gratitude to their supervisor Professor Keiji Oguiso who suggested this subject and has given much encouragement and invaluable and helpful advices.

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