Families of embeddings of the alternating group of rank $5$ into the Cremona group

I. Krylov Igor Krylov
Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea
[email protected]

Abstract.

I study embeddings of the alternating group of rank five into the Cremona group of rank three. I find all embeddings induced by $\mathcal{A}_{5}\mathds{Q}$ -del Pezzo fibrations and I study their conjugacy. As an application, I show that there is a series of continuous families of pairwise non-conjugate embeddings of alternating group of rank five into $\operatorname{Cr}_{3}(\mathds{C})$ .

1. Introduction

The Cremona group $\operatorname{Cr}_{n}$ of rank $n$ is a group of birational transformations of $\mathds{P}^{n}$ . It is natural to study the group by studying its subgroups and finite subgroups in particular. The classification of finite subgroups in $\operatorname{Cr}_{2}$ up to conjugacy is almost complete [7]. It is not feasible to achieve a classification for $\operatorname{Cr}_{3}$ , nevertheless we can say something about its finite subgroups, for example we know that $\operatorname{Cr}_{n}$ is Jordan [16].

If we limit the group types, for example to $p$ -subgroups ([17]) or simple non-abelian, then the problem becomes more manageable. The motivating problem for me is the following.

Problem 1.1.

Classify the embeddings of finite simple non-abelian subgroups of $\operatorname{Cr}_{3}$ up to conjugacy.

The isomorphism types of simple non-abelian groups were classified in [13, Theorem 1.3], the possibilities are: $\mathcal{A}_{5}$ , $\mathcal{A}_{6}$ , $\mathcal{A}_{7}$ , $\operatorname{PSL}_{2}(7)$ , $\operatorname{SL}_{2}(8)$ , and $\operatorname{PSp}_{4}(3)$ . In this paper I study the families of subgroups of $\operatorname{Cr}_{3}$ .

Question 1.2.

Let $G$ be a group. Is there a continuous family of embeddings $G\hookrightarrow\operatorname{Cr}_{n}$ which are not pairwise conjugate to each other?

In general, one expects that the asnwer is positive for small groups and negative for big groups. For example there are huge families of pairwise non-conjugate embeddings of $\mathds{Z}/2\mathds{Z}$ induced by Bertini and Geiser involutions [15]. Also it was recently shown in [1] that there is a continuous family of pairwise non-conjugate embeddings of $\mathcal{S}_{4}$ into $\operatorname{Cr}_{3}$ .

On the other hand, finite non-abelian groups are big, thus one expects that there should be no continuous families of pairwise non-conjugate embeddings of these groups. Indeed, there are only six simple non-abelian subgroups of $\operatorname{Cr}_{2}$ up to conjugacy: three subgroups isomorphic to $\mathcal{A}_{5}$ , two subgroups isomorphic to $\operatorname{PSL}_{2}(7)$ , and one subgroup isomorphic to $\mathcal{A}_{6}$ . In dimension three we know by [13, Theorem 1.5] that there are only finitely many embeddings for the three largest finite simple non-abelian subgroups. I study the embeddings of the smallest finite simple non-abelian subgroup: $\mathcal{A}_{5}$ .

Theorem 1.3.

For each $k\geqslant 2$ there is a $2k-3$ -dimensional family of embeddings $\mathcal{A}_{5}\hookrightarrow\operatorname{Cr}_{3}$ which are not pairwise conjugate to each other.

The remaining cases are $\mathcal{A}_{6}$ and $\operatorname{PSL}_{2}(7)$ . Some progress toward classification the embeddings of these groups has been made in [3] and [11], so far the results agree with the following expectation.

Conjecture 1.4.

The embeddings into $\operatorname{Cr}_{3}$ of $\mathcal{A}_{6}$ and $\operatorname{PSL}_{2}(7)$ up to conjugation form discrete families.

1.1. $G\mathds{Q}$ -Mori fiber spaces

The study of rational group actions on $\mathds{P}^{3}$ may be replaced with the study of regular group actions on suitable rational varieties.

Definition 1.5.

I say that $\pi\colon Y\to Z$ (or simply $Y/Z$ if $\pi$ is understood) is a $G\mathds{Q}$ -Mori fiber space if

•

$\dim Z<\dim Y$ and $\pi$ has connected fibers,
•

$G$ -action on $Y$ is faithful and $Z$ admits a $G$ -action such that the map $\pi$ is $G$ -equivariant,
•

$Y$ is terminal and $G\mathds{Q}$ -factorial, that is $G$ -invariant divisors on $Y$ are $\mathds{Q}$ -factorial,
•

$-K_{Y}$ is $\pi$ -ample and the relative $G$ -invariant Picard rank is $\rho^{G}(Y/Z)=1$ .

When $Y$ is Gorenstein I omit $\mathds{Q}$ and say that $Y/Z$ is a $G$ -Mori fiber space. If $\dim Y-\dim Z=2$ , then I say that $Y/Z$ is a $G\mathds{Q}$ -del Pezzo fibration.

By [14, Proposition 1.2] for a rational action of $G$ on a rationally connected variety $X$ there is a $G\mathds{Q}$ -Mori fiber space $\pi\colon Y\to Z$ such that $X$ is $G$ -equivariantly birational to $Y$ , hence the $G$ -action on $Y$ induces a conjugate embedding of $G$ into $\operatorname{Bir}(X)$ . The classification of subgroups of $\operatorname{Cr}_{n}$ up to conjugation is equivalent to the classification of the rational $G\mathds{Q}$ -Mori fiber spaces up to $G$ -equivariant birational equivalence.

Now I give examples of $\mathcal{A}_{5}$ -del Pezzo fibrations.

Example 1.6.

The projective plane with $\mathcal{A}_{5}$ -action induced by a non-trivial three-dimensional representation $W_{3}$ is an $\mathcal{A}_{5}$ -del Pezzo surface. A del Pezzo surface $S_{5}$ of degree $5$ satisfies $\mathcal{A}_{5}\subset\operatorname{Aut}S_{5}$ and is $\mathcal{A}_{5}$ -minimal, hence it is an $\mathcal{A}_{5}$ -del Pezzo surface.

Let $S$ be a $G$ -del Pezzo surface, then $S\times\mathds{P}^{1}/\mathds{P}^{1}$ is a $G$ -del Pezzo fibration. It follows that $\mathds{P}^{2}\times\mathds{P}^{1}/\mathds{P}^{1}$ and $S_{5}\times\mathds{P}^{1}/\mathds{P}^{1}$ are $\mathcal{A}_{5}$ -del Pezzo fibrations.

Example 1.7.

Let $W_{3}$ and $W_{3}^{\prime}$ be the irreducible $3$ -dimensional representations of $\mathcal{A}_{5}$ . I consider $\mathcal{A}_{5}$ -linearized vector bundles defined as

\mathcal{E}_{n}=\big{(}W_{3}\times\mathds{P}^{1}\big{)}\oplus\mathcal{O}_{\mathds{P}^{1}}(-n)\quad\text{and}\quad\mathcal{E}_{n}^{\prime}=\big{(}W_{3}^{\prime}\times\mathds{P}^{1}\big{)}\oplus\mathcal{O}_{\mathds{P}^{1}}(-n).

Let $T_{n}=\mathds{P}(\mathcal{E}_{n})$ and $T_{n}^{\prime}=\mathds{P}(\mathcal{E}_{n}^{\prime})$ and denote by $\pi_{T}$ and $\pi_{T^{\prime}}$ the corresponding projections onto $\mathds{P}^{1}$ .

I introduce coordinates on $T_{n}$ (resp. $T_{n}^{\prime}$ ) as follows. Let $u,v$ be the coordinates on the base $\mathds{P}^{1}$ , $x,y,z$ be the coordinates on $W_{3}$ (resp. $W_{3}^{\prime}$ ), and let $w$ be the coordinate on the fiber of $\mathcal{O}_{\mathds{P}^{1}}(-n)\to\mathds{P}^{1}$ . The degrees of the coordinates are given by

\left(\begin{array}[]{ccccccc}u&v&x&y&z&w\\ 0&0&1&1&1&1\\ 1&1&0&0&0&-n\end{array}\right)

There is a unique $\mathcal{A}_{5}$ -invariant conic $\Delta$ on $\mathds{P}(W_{3})$ (resp. $\mathds{P}(W_{3}^{\prime})$ ). Up to a change of coordinates on $W_{3}$ (res. $W_{3}^{\prime})$ I may assume that its equation is $xz-y^{2}=0$ . Let $X_{n}$ (resp. $X_{n}^{\prime}$ ) be the hypersurface in $T_{n}$ (resp. $T_{n}^{\prime}$ ) given by the equation

a_{2n}(u,v)w^{2}=xz-y^{2},

where $\deg a_{2n}=2n$ . Observe that this equation is $\mathcal{A}_{5}$ -invariant, thus $X_{n}$ and $X_{n}^{\prime}$ admit a faithful $\mathcal{A}_{5}$ -action. The $\mathcal{A}_{5}$ -varieties $X_{n}$ and $X_{n}^{\prime}$ are smooth if and only if $a_{2n}$ has no multiple factors. If $l(u,v)$ is a multiple factor of $a_{2n}$ , then $X_{n}$ and $X_{n}^{\prime}$ have a $cA_{1}$ -singularity at $x=y=z=l(u,v)=0$ . We have $\rho(X_{n})=\rho(X_{n}^{\prime})=2$ if $a_{2n}$ is not a square (Lemma 3.4). The restriction $\pi$ (resp. $\pi^{\prime}$ ) of $\pi_{T}$ (resp. $\pi_{T}^{\prime}$ ) to $X_{n}$ (resp. $X_{n}^{\prime}$ ) induces the structure of $\mathcal{A}_{5}$ -del Pezzo fibration on $X_{n}$ (resp. $X_{n}^{\prime}$ ).

Note that the varieties $X_{1}$ and $X_{1}^{\prime}$ are unique while for $n\geqslant 2$ the families of varieties $X_{n}$ and $X_{n}^{\prime}$ are of dimension $2n-3$ .

I show that these are essentially the only $\mathcal{A}_{5}\mathds{Q}$ -del Pezzo fibrations with the trivial action on the base.

Theorem 1.8.

Let $\pi:V\to\mathds{P}^{1}$ be an $\mathcal{A}_{5}\mathds{Q}$ -del Pezzo fibration and suppose $\mathcal{A}_{5}$ acts trivially on the base. Then one of the following holds:

(1)

The general fiber of $\pi$ is $\mathds{P}^{2}$ and $V$ is $\mathcal{A}_{5}$ -equivariantly birational to $\mathds{P}^{2}\times\mathds{P}^{1}$ with $\mathcal{A}_{5}$ acting only on the first factor;
(2)

$V\cong_{\mathcal{A}_{5}}S_{5}\times\mathds{P}^{1}$ , where $S_{5}$ is the del Pezzo surface of degree $5$ and $\mathcal{A}_{5}$ acts only on the first factor;
(3)

The general fiber of $\pi$ is a quadric and $V$ is $\mathcal{A}_{5}$ -equivariantly birational to a smooth $X_{n}$ .

The varieties from the case (3) are inducing families of embeddings of $\mathcal{A}_{5}$ into $\operatorname{Cr}_{3}$ . In order to show that these embeddings are not pairwise conjugate I use the notion of $G$ -equivariant birational superrigidity.

Definition 1.9.

Let $\pi_{X}\colon X\to S$ and $\pi_{Y}\colon Y\to Z$ be $G\mathds{Q}$ -Mori fiber spaces. A $G$ -equivariant birational map $\chi\colon X\dasharrow Y$ is called square if it fits into a commutative diagram

where $g$ is birational and, in addition, the induced map on the generic fibers $\chi_{\eta}\colon X_{\eta}\dasharrow Y_{\eta}$ is isomorphism of $G$ -varieties. In this case I say that $X/S$ and $Y/Z$ are $G$ -equivariantly square-birational.

Definition 1.10.

I say that $\pi_{X}\colon X\to S$ is $G$ -equivariantly superrigid if for any $G\mathds{Q}$ -Mori fiber space $\pi_{Y}\colon Y\to Z$ any $G$ -equivariant birational map $\chi\colon X\to Y$ is $G$ -equivariantly square birational.

Theorem 1.11.

Suppose $X_{n}$ (resp. $X_{n}^{\prime}$ ) is smooth and $n\geqslant 2$ , then $X_{n}$ (resp. $X_{n}^{\prime}$ ) is $\mathcal{A}_{5}$ -equivariantly birationally superrigid.

Corollary 1.12.

Suppose $X_{n}$ , $X_{k}$ , $X_{m}^{\prime}$ , and $X_{l}^{\prime}$ are smooth and $n,m,k,l\geqslant 2$ , then:

(1)

The varieties $X_{n}$ and $X_{m}^{\prime}$ are not $\mathcal{A}_{5}$ -equivariantly birational to $\mathds{P}^{2}\times\mathds{P}^{1}$ and $S_{5}\times\mathds{P}^{1}$ ;
(2)

The variety $X_{n}$ is not $\mathcal{A}_{5}$ -equivariantly birational to $X_{m}^{\prime}$ ;
(3)

The variety $X_{n}$ is $\mathcal{A}_{5}$ -equivariantly birational to $X_{k}$ if and only if $X_{n}$ is $\mathcal{A}_{5}$ -equivariantly isomorphic to $X_{k}$ and the same holds for $X_{m}^{\prime}$ and $X_{l}^{\prime}$ ;
(4)

For $n\geqslant 2$ the family of $\mathcal{A}_{5}$ -del Pezzo fibrations $X_{n}$ (or $X_{m}^{\prime})$ induces a $2n-3$ -dimensional family (resp. $2m-3$ -dimensional) of pairwise non-conjugate embeddings $\mathcal{A}_{5}\hookrightarrow\operatorname{Cr}_{3}$ .

The assertion (4) is Theorem 1.3.

Acknowledgments.

The author would like to thank Andrey Trepalin and Constantin Shramov for pointing out how to work with del Pezzo surfaces of degree $5$ over non-closed fields. The author was supported by KIAS individual grant MG069801 at Korea Institute for Advanced Study.

Notations and conventions

I work over $\mathds{C}$ unless stated otherwise. All varieties are considered to be projective and normal. I denote the symmetric and alternating groups of rank $n$ by $\mathcal{S}_{n}$ and $\mathcal{A}_{n}$ respectively. I denote the del Pezzo surface of degree $5$ by $S_{5}$ and Clebsch cubic by $S_{3}$ . I denote the linear equivalence of divisors by $\sim$ , the numerical equivalence of cycles by $\equiv$ , and $\mathcal{A}_{5}$ -equivariant biregular equivalence by $\cong_{\mathcal{A}_{5}}$ . Given a curve $\Delta$ I denote its orbit by $\overline{\Delta}$ . By a slight abuse of notations I denote the curve $\sum_{C\in\overline{\Delta}}C$ by $\overline{\Delta}$ as well.

2. The represenatives of the $\mathcal{A}_{5}$ -equivariant birational classes of $\mathcal{A}_{5}\mathds{Q}$ -del Pezzo fibrations with the trivial action on the base

2.1. Representations of $\mathcal{A}_{5}$ and its central extension

Recall that $\mathcal{A}_{5}$ has the following irreducible representations: $I=I_{1}$ , $W_{3}$ , $W_{3}^{\prime}$ , $W_{4}$ , and $W_{5}$ (see [4, Sections 5.2 and 5.6], for details). The lower index is the dimension of the representation. The action of $\mathcal{A}_{5}$ on $\mathds{P}^{1}$ is induced by irreducible representations of the central extension $2.\mathcal{A}_{5}$ : $U_{2}$ and $U_{2}^{\prime}$ .

I recall some facts about the action of $\mathcal{A}_{5}$ on smooth curves and surfaces

Lemma 2.1 ( [4, Lemmas 5.1.3, 5.1.4 and Section 5.2]).

Let $S$ be a smooth surface and $C$ be a smooth curve admitting a non-trivial action of $\mathcal{A}_{5}$ .

(1)

Let $\Sigma$ be an $\mathcal{A}_{5}$ -orbit on $C$ , then $\lvert\Sigma\rvert=12$ , $20$ , or $60$ .
(2)

Let $\Sigma$ be an $\mathcal{A}_{5}$ -orbit on $S$ , then $\lvert\Sigma\rvert\geqslant 5$ .

Proposition 2.2 ([4, Chapter 6]).

Let $\pi:X\to\mathds{P}^{1}$ be an $\mathcal{A}_{5}\mathds{Q}$ -del Pezzo fibration and let $F$ be a general fiber of $\pi$ . Then one of the following holds

(1)

$F\cong_{\mathcal{A}_{5}}\mathds{P}(W_{3})$ or $F\cong_{\mathcal{A}_{5}}\mathds{P}(W_{3}^{\prime})$ ,
(2)

$F\cong_{\mathcal{A}_{5}}S_{5}$ ,
(3)

$F\cong_{\mathcal{A}_{5}}S_{3}$ ,
(4a)

$F\cong_{\mathcal{A}_{5}}\mathds{P}(U_{2})\times\mathds{P}(U_{2})$ or $F\cong_{\mathcal{A}_{5}}\mathds{P}(U_{2}^{\prime})\times\mathds{P}(U_{2}^{\prime})$ , in this case I say $F$ is a quadric with a diagonal $\mathcal{A}_{5}$ -action,
(4b)

$F\cong_{\mathcal{A}_{5}}\mathds{P}(U_{2})\times\mathds{P}(U_{2}^{\prime})$ , this case I say that $F$ is a quadric with a twisted diagonal $\mathcal{A}_{5}$ -action,
(4c)

$F\cong_{\mathcal{A}_{5}}\mathds{P}(U_{2})\times\mathds{P}(I\oplus I)$ or $F\cong_{\mathcal{A}_{5}}\mathds{P}(U_{2}^{\prime})\times\mathds{P}(I\oplus I)$ , in this case I say that $F$ is a quadric with a one-factor $\mathcal{A}_{5}$ -action.

Knowing the induced $\mathcal{A}_{5}$ -action on $\mathds{P}^{3}$ is very useful for studying cases (4a), (4b), and (4c).

Lemma 2.3.

Let $F$ be a quadric with an action of $\mathcal{A}_{5}$ , let $V$ be the $4$ -dimensional representation dual to $H^{0}(F,-\frac{1}{2}K_{F})$ , and let $g:F\hookrightarrow\mathds{P}(V)$ be the $\mathcal{A}_{5}$ -equivariant embedding induced by $\lvert-\frac{1}{2}K_{F}\rvert$ . Then

(1)

the $\mathcal{A}_{5}$ -action on $F$ is diagonal if and only if $V\cong_{\mathcal{A}_{5}}W_{3}\oplus I$ or $V\cong_{\mathcal{A}_{5}}W_{3}^{\prime}\oplus I$ ;
(2)

the $\mathcal{A}_{5}$ -action on $F$ is twisted diagonal if and only if $V\cong_{\mathcal{A}_{5}}W_{4}$ ;
(3)

the $\mathcal{A}_{5}$ -action on $F$ is a one-factor action if and only if $\mathds{P}(V)\cong_{\mathcal{A}_{5}}\mathds{P}(U_{2}\oplus U_{2})$ , or $\mathds{P}(V)\cong_{\mathcal{A}_{5}}\mathds{P}(U_{2}^{\prime}\oplus U_{2}^{\prime})$ .

Proof.

Assertion (1) is equivalent to [4, Lemma 6.3.3, (i)].

Clearly, $\mathds{P}(V)$ has no invariant lines and planes if and only if $F$ has no invariant divisors of degree $(1,1)$ or $(0,1)$ . It follows that the action on $F$ is twisted diagonal if and only if $V\cong_{\mathcal{A}_{5}}W_{4}$ .

Suppose $F\cong_{\mathcal{A}_{5}}\mathds{P}(U_{2})\times\mathds{P}(I\oplus I)$ . Let $L_{1}$ and $L_{2}$ be the curves of bi-degree $(0,1)$ on $F$ . Then $L_{1}\cong_{\mathcal{A}_{5}}L_{2}\cong_{\mathcal{A}_{5}}\mathds{P}(U_{2})$ . Since $g(L_{1})$ and $g(L_{2})$ are $\mathcal{A}_{5}$ -invariant skew lines we see that $V\cong U_{2}\oplus U_{2}$ . The converse follows from the assertions (1) and (2). ∎

2.2. The general fiber is $\mathds{P}^{2}$

Lemma 2.4.

Let $\pi\colon X\to\mathds{P}^{1}$ be an $\mathcal{A}_{5}\mathds{Q}$ -del Pezzo fibration of degree $9$ and suppose $\mathcal{A}_{5}$ acts trivially on the base. Then $X$ is $\mathcal{A}_{5}$ -equivariantly birational to $\mathds{P}^{2}\times\mathds{P}^{1}$ with the trivial action of $\mathcal{A}_{5}$ on $\mathds{P}^{1}$ .

Proof.

Let $X_{\eta}$ be the generic fiber of $\pi$ . It is a form of $\mathds{P}^{2}$ and has a point, thus it is $\mathds{P}^{2}$ . The isomorphism $X_{\eta}\cong\mathds{P}^{2}$ induces the $\mathcal{A}_{5}$ -equivariant birational map to $\mathds{P}^{2}\times\mathds{P}^{1}$ . ∎

Note that $\mathds{P}^{2}\times\mathds{P}^{1}$ is not the only $\mathcal{A}_{5}\mathds{Q}$ -del Pezzo fibration of degree $9$ with a trivial action on the base.

Example 2.5.

Let $\Delta$ be the $\mathcal{A}_{5}$ -invariant curve of degree $2$ in the fiber $F\subset\mathds{P}^{2}\times\mathds{P}^{1}$ over a point $P\in\mathds{P}^{1}$ . Then I may blow up $\mathds{P}^{2}\times\mathds{P}^{1}$ at $\Delta$ and then contract the proper transform of $F$ to acquire a variety $V_{1}$ with a $\frac{1}{2}(1,1,1)$ -singularity. The composition $\mathds{P}^{2}\times\mathds{P}^{1}\dasharrow V_{1}$ of these maps is an elementary $\mathcal{A}_{5}$ -equivariant Sarkisov link of $\mathcal{A}_{5}\mathds{Q}$ -del Pezzo fibrations. The new fiber over $P$ is isomorphic to $\mathds{P}(1,1,4)$ .

Example 2.6.

Consider a toric variety $R_{n}$ such that $\operatorname{Cox}R_{n}=\mathds{C}[u,v,x,y,z,w]$ , where the irrelevant ideal $I=\langle u,v\rangle\cap\langle x,y,z,w\rangle$ and the grading given by

\left(\begin{array}[]{ccccccc}u&v&x&y&z&w\\ 0&0&1&1&1&2\\ 1&1&0&0&0&-n\end{array}\right)

I assume that $\mathcal{A}_{5}$ -action on $R_{n}$ comes from the identification of $\mathds{C}^{3}_{x,y,z}\cong W_{3}$ . Suppose that the equation of the $\mathcal{A}_{5}$ -invariant quadric on $W_{3}$ is $xz-y^{2}=0$ . Then consider an $\mathcal{A}_{5}$ -invariant hypersurface $V_{n}$ in $R_{n}$ given by the equation

a_{n}(u,v)w=xz-y^{2}.

There is a natural $\mathcal{A}_{5}$ -equivariant projection $\pi_{R}\colon R_{n}\to\mathds{P}^{1}_{u,v}$ which is a $\mathds{P}(1,1,1,2)$ -bundle. The restriction $\pi=\pi_{R}|_{V_{n}}\colon V_{n}\to\mathds{P}^{1}_{u,v}$ is an $\mathcal{A}_{5}\mathds{Q}$ -del Pezzo fibration of degree $9$ with the trivial action on the base. The variety $V_{n}$ has $2$ -Goreinstein singularities at $x=y=z=a_{n}(u,v)=0$ . For every simple root of $a_{n}(u,v)$ there is a $\frac{1}{2}(1,1,1)$ -singularity and for a root of multiplicity $k$ there is a $cA_{1}/\mu_{2}$ -singularity, where $\mu_{2}$ is the cyclic group of order $2$ .

Question 2.7.

Let $\pi\colon X\to\mathds{P}^{1}$ be an $\mathcal{A}_{5}\mathds{Q}$ -del Pezzo fibration and suppose that the general fiber of $\pi$ is $\mathds{P}^{2}$ . Suppose $\mathcal{A}_{5}$ acts trivially on the base. Is it true that $X\cong V_{n}$ for some $a_{n}(u,v)$ ?

2.3. The general fiber is $S_{5}$

Proposition 2.8.

Let $\pi:X\to\mathds{P}^{1}$ be an $A_{5}\mathds{Q}$ -del Pezzo fibration of degree $5$ . Then it is $\mathcal{A}_{5}$ -equivariantly birational to $S_{5}\times\mathds{P}^{1}$ .

Proof.

Consider the generic fiber $X_{\eta}$ of $\pi$ . It is a del Pezzo surface of degree $5$ over $\mathds{C}(t)$ which admits an action of $\mathcal{A}_{5}$ . Recall, that a del Pezzo surface of degree $5$ over an algebraically closed field is unique and it admits a faithful $\mathcal{A}_{5}$ -action. Hence the $\mathcal{A}_{5}$ -action on $\operatorname{Pic}(\overline{X_{\eta}})$ is faithful and induces an embedding $\mathcal{A}_{5}\hookrightarrow\operatorname{W}(A_{4})\cong\mathcal{S}_{5}$ , where $A_{4}$ is the root lattice.

Let $\Gamma=\operatorname{Gal}(\overline{\mathds{C}(t)}/\mathds{C}(t))$ , then the action of $\Gamma$ and $\mathcal{A}_{5}$ on $\overline{X}_{\eta}$ commutes. It follows that $\Gamma$ commutes with $\operatorname{W}(A_{4})=\mathcal{S}_{5}$ , therefore all $(-1)$ -curves of $\overline{X}_{\eta}$ are defined over $\mathds{C}(t)$ . Thus $X_{\eta}$ is acquired by a blow up of $\mathds{P}^{2}$ at $4$ points defined over $\mathds{C}(t)$ . In particular a del Pezzo surface of degree $5$ with an $\mathcal{A}_{5}$ -action is unique.

On the other hand the generic fiber of the projection $S_{5}\times\mathds{P}^{1}\to\mathds{P}^{1}$ is a del Pezzo surface of degree $5$ with an action of $\mathcal{A}_{5}$ , hence isomorphic to $X_{\eta}$ . The isomorphism induces the $\mathcal{A}_{5}$ -equivariant birational map $X\dasharrow S_{5}\times\mathds{P}^{1}$ .

∎

Unlike the degree $9$ case, one can show that there are no other $\mathcal{A}_{5}\mathds{Q}$ -del Pezzo fibrations of degree $5$ .

Lemma 2.9.

The threefold $S_{5}\times\mathds{P}^{1}$ is the unique $\mathcal{A}_{5}\mathds{Q}$ -del Pezzo fibration of degree $5$ .

Proof.

It is well known that $\operatorname{lct}_{\mathcal{A}_{5}}S_{5}\geqslant 1$ , thus the statement follows from [2, Theorem 1.5] (see [11, Theorem 3.6] for $G$ -invariant version). ∎

I expect that $S_{5}\times\mathds{P}^{1}$ is the unique $\mathcal{A}_{5}\mathds{Q}$ -del Pezzo fibration in its birational class but it is $\mathcal{A}_{5}$ -equivariantly birational to other $\mathcal{A}_{5}$ -Mori fiber spaces. We can easily construct many $\mathds{P}^{1}$ -bundles $\mathcal{A}_{5}$ -equivariantly birational to $S_{5}\times\mathds{P}^{1}$ by blowing up orbits of fibers of the projection onto $S_{5}$ .

Question 2.10.

Does there exist an $\mathcal{A}_{5}\mathds{Q}$ -Mori fiber space which is not $\mathcal{A}_{5}$ -equivariantly square birational to $S_{5}\times\mathds{P}^{1}/\mathds{P}^{1}$ or $S_{5}\times\mathds{P}^{1}/S_{5}$ but is $\mathcal{A}_{5}$ -equivariantly birational to $S_{5}\times\mathds{P}^{1}$ ?

2.4. The general fiber is $S_{3}$

Lemma 2.11.

Let $X_{\eta}$ be a cubic over $\mathds{C}(t)$ admitting a faithful $\mathcal{A}_{5}$ -action, then $\rho^{\mathcal{A}_{5}}(X_{\eta})=2$ . In particular, the $\mathcal{A}_{5}\mathds{Q}$ -del Pezzo fibrations of degree $3$ do not exist.

Proof.

There is the unique $\mathcal{A}_{5}$ -invariant cubic $S_{3}$ in $\mathds{P}^{3}$ : Clebsh cubic. It follows that there is a unique $\mathcal{A}_{5}$ -invariant cubic $X_{\eta}$ in $\mathds{P}^{3}_{\mathds{C}(t)}$ . I may realize $X_{\eta}$ as the generic fiber of the projection $S_{3}\times\mathds{P}^{1}\to\mathds{P}^{1}$ . Thus $\rho^{\mathcal{A}_{5}}(X_{\eta})=2$ . ∎

2.5. The general fiber is a quadric

Recall that there are three types of $\mathcal{A}_{5}$ -action on $\mathds{P}^{1}\times\mathds{P}^{1}$ as defined in Proposition 2.2: diagonal, twisted diagonal, and one-factor action. The $\mathcal{A}_{5}$ -quadric fibration exist only for the former type.

Lemma 2.12.

Let $\pi:X\to\mathds{P}^{1}$ be an $\mathcal{A}_{5}$ -equivariant map such that the $\mathcal{A}_{5}$ -action on a general fiber $F$ of $\pi$ is either twisted diagonal or a one-factor action. Let $X_{\eta}$ be the generic fiber of $\pi$ , then $\rho^{\mathcal{A}_{5}}(X_{\eta})=2$ .

Proof.

Suppose the $\mathcal{A}_{5}$ -action on $F$ is twisted diagonal, then $F\subset\mathds{P}(W_{4})$ by Lemma 2.3. Since $\operatorname{Sym}^{2}(W_{4})\cong W_{5}\oplus W_{4}\oplus I$ , there is the unique $\mathcal{A}_{5}$ -invariant quadric $S_{\eta}$ in $\mathds{P}(W_{4}\otimes\mathds{C}(t))$ . On the other hand, $S_{\eta}$ is also the generic fiber of the projection

\mathds{P}(U_{2})\times\mathds{P}(U_{2}^{\prime})\times\mathds{P}(I\oplus I)\to\mathds{P}(I\oplus I),

thus we see that $\rho^{\mathcal{A}_{5}}(S_{\eta})=2$ .

Suppose the $\mathcal{A}_{5}$ -action on $F$ is a one-factor action corresponding to the representation $U_{2}$ (or $U_{2}^{\prime}$ ) of $2.\mathcal{A}_{5}$ . Similarly to the twisted diagonal case I only need to show that there is a unique $\mathcal{A}_{5}$ -invariant quadric in $\mathds{P}(U_{2}\oplus U_{2})$ (resp. $\mathds{P}(U_{2}^{\prime}\oplus U_{2}^{\prime})$ ). There are $\mathcal{A}_{5}$ -invariant skew lines $Z_{1}$ and $Z_{2}$ and any $\mathcal{A}_{5}$ -invariant quadric must contain both $Z_{1}$ and $Z_{2}$ since there are no orbits of length $\leqslant 2$ on $\mathds{P}^{1}$ . On the other hand, the space of quadrics containing a pair of $\mathcal{A}_{5}$ -invariant skew lines is isomorphic to $\mathds{P}(W)$ for some $4$ -dimensional representation $W$ of $\mathcal{A}_{5}$ or $2.\mathcal{A}_{5}$ . The image on $\mathds{P}(V)$ of the quadric with a one-factor action corresponds to an $\mathcal{A}_{5}$ -fixed point on $\mathds{P}(W)$ . On the other hand, the family of reducible quadrics containing $Z_{1}$ and $Z_{2}$ is of dimension $3$ and is $\mathcal{A}_{5}$ -invariant. It follows that $W\cong_{\mathcal{A}_{5}}W_{3}\oplus I$ or $W\cong_{\mathcal{A}_{5}}W_{3}^{\prime}\oplus I$ . Thus the $\mathcal{A}_{5}$ -invariant quadric on $\mathds{P}(U_{2}\oplus U_{2})$ (resp. $\mathds{P}(U_{2}^{\prime}\oplus U_{2}^{\prime})$ ) is unique. ∎

Recall the construction of the $\mathcal{A}_{5}\mathds{Q}$ -quadric fibrations $X_{n}$ and $X_{n}^{\prime}$ from Example 1.7.

Proposition 2.13.

Let $\pi_{V}\colon V\to\mathds{P}^{1}$ be an $\mathcal{A}_{5}\mathds{Q}$ -quadric fibration. Suppose the action on a general fiber $F$ of $\pi_{V}$ is diagonal. Then $V$ is $\mathcal{A}_{5}$ -equivariantly birational to a smooth $X_{n}$ or $X_{n}^{\prime}$ for some $a_{2n}(u,v)$ .

Proof.

The fiber $F$ embeds into $\mathds{P}(W_{3}\oplus I)$ or $\mathds{P}(W_{3}^{\prime}\oplus I)$ by Lemma 2.3. The two cases are analogous, suppose the former, then

X_{\eta}\hookrightarrow\mathds{P}\big{(}(W_{3}\oplus I)\otimes\mathds{C}(t)\big{)}.

Let $x,y,z$ be the coordinates on $W_{3}$ and let $w$ be the coordinate on $I$ . Then, up to a change of coordinates on $W_{3}$ , every $\mathcal{A}_{5}$ -invariant quadric in $\mathds{P}(W_{3}\oplus I)$ has the equation

\lambda(t)w^{2}=\mu(t)(xz-y^{2}).

It follows that up to a change of coordinates in $\mathds{P}\big{(}(W_{3}\oplus I)\otimes\mathds{C}(t)\big{)}$ the equation of $X_{\eta}$ is

b(t)w^{2}=xz-y^{2},

where $b(t)$ has no multiple roots.

Set $a(u,v)$ to be the homogeneous polynomial of degree $2n$ without multiple factors satisfying $a(t,1)=b(t)$ . Let $X_{n}$ be the hypersurface in $T_{n}$ given by

a(u,v)w^{2}=xz-y^{2}.

Then the general fiber of $X_{n}\to\mathds{P}^{1}$ is $\mathcal{A}_{5}$ -equivariantly isomorphic to $X_{\eta}$ . This isomorphism of the general fibers induces the $\mathcal{A}_{5}$ -equivariant birational map $V\dasharrow X_{n}$ . ∎

This completes the proof of Theorem 1.8.

3. Rigidity results

This section is devoted to proving Theorem 1.11. First, I present some elementary results on the geometry of varieties $X_{n}$ . Next I recall Noether-Fano method and the notion of maximal centers. Then I show that the only possible maximal centers are the points on the $\mathcal{A}_{5}$ -invariant curve of degree $2$ . At last, I use the technique of supermaximal singularities [18] to prove $\mathcal{A}_{5}$ -equivariant birational superrigidity of $X_{n}$ and $X_{n}^{\prime}$ for $n\geqslant 2$ .

3.1. The geometry of $X_{n}$

Note that $T_{n}\cong T_{n}^{\prime}$ and for a fixed $a_{2n}$ we have $X_{n}\cong X_{n}^{\prime}$ . These isomorphisms are not $\mathcal{A}_{5}$ -equivariant. From now on I work with $T_{n}$ and $X_{n}$ , the proofs for $X_{n}^{\prime}$ are identical. In this section I do not assume that $a_{2n}(u,v)$ has no multiple factors. In that case $X_{n}$ is no longer smooth. Indeed, for each multiple linear factor $l(u,v)$ of $a_{2n}(u,v)$ the point $(x=y=z=l(u,v)=0)$ is a $cA_{1}$ -singularity.

Denote by $H_{T}$ the divisor class of $(x=0)$ and by $F_{T}$ the divisor class of $(u=0)$ on $T_{n}$ . Denote $H=H_{T}|_{X_{n}}$ and $F=F_{T}|_{X_{n}}$ . Let $s_{T}\in A^{3}(T_{n})$ and $s\in A^{2}(X_{n})$ be the classes of the curve $x=y=w=0$ . Let $f_{T}\in A^{3}(T_{n})$ and $f\in A^{2}(X_{n})$ be the classes of the curve $x=y=u=0$ . Using the fact that

(x=y=z=w=0)=\varnothing\quad\text{and}\quad(x=y=z=u=0)=\operatorname{pt}

I compute the intersections on $T_{n}$ .

Lemma 3.1.

The following holds for $T_{n}$ :

(1)

The classes $s_{T}$ and $f_{T}$ generate the cone of effective curves on $T_{n}$ ;
(2)

$H_{T}^{2}\cdot F_{T}\equiv f_{T}$ ;
(3)

$H_{T}^{3}\equiv s_{T}+nf_{T}$ ;
(4)

$H_{T}^{3}\cdot F_{T}=1$ ;
(5)

$H_{T}^{4}=n$ .

Corollary 3.2.

The following holds for $X_{n}$ :

(1)

$H\cdot F\equiv 2f$ ;
(2)

$H^{2}\equiv 2s+2nf$ ;
(3)

$H^{2}\cdot F=2$ ;
(4)

$H^{3}=2n$ ;
(5)

$K_{X_{n}}\sim-2H+(n-2)F$ ;
(6)

$K_{X_{n}}^{2}\equiv 8s+24f-8nf$ .

In order to better understand a singular $X_{n}$ , it is useful to know how is it related to a smooth $X_{m}$ .

Lemma 3.3.

Let variety $X_{n}$ correspond to $a_{2n}(u,v)$ and $X_{n+1}$ to $u^{2}a_{2n}(u,v)$ . Denote the fiber $u=0$ by $F$ . Let $\sigma\colon\widetilde{X}\to X_{n}$ be the blow up of $X_{n}$ along the curve $u=w=0$ . Then there is a map $\psi\colon\widetilde{X}\to X_{n+1}$ contracting $\sigma^{-1}F$ to a $cA_{1}$ singularity $x=y=z=u=0$ . The composition $\varphi_{u}=\psi\circ\sigma^{-1}$ is an $\mathcal{A}_{5}$ -equivariant elementary Sarkisov link.

Proof.

Elementary calculations. ∎

Thus we see that any singular $X_{n}$ is $\mathcal{A}_{5}$ -equivariantly square birational to a smooth $X_{m}$ with $m=n-k$ for some $k>0$ .

Lemma 3.4.

Suppose $a_{2n}(u,v)$ is not a square, then

(1)

$\operatorname{Pic}X_{n}\cong\mathds{Z}\cdot H\oplus\mathds{Z}\cdot F$ ,
(2)

The classes $s$ and $f$ generate the cone of effective curves $\operatorname{NE}(X_{n})$ .

Proof.

The assetion (1) holds since $\operatorname{Pic}X_{n}\cong\operatorname{Pic}X_{\eta}\oplus\operatorname{Pic}\mathds{P}^{1}$ and $\operatorname{Pic}X_{\eta}$ is generated by the hyperplane section if and only if $a_{2n}(u,v)$ is not a complete square.

The assertion (2) follows from (1) and Lemma 3.1 (1) for smooth $X_{n}$ by Poincare duality. Any $X_{n}$ is related to some smooth $X_{m}$ by a sequence of elementary Sarkisov links described in Lemma 3.3. The elementary links preserve the dimension of $\operatorname{NE}(X_{n})$ , hence the assertion holds by Lemma 3.1 (1). ∎

Lemma 3.5.

Suppose $n\geqslant 2$ , then $X_{n}$ satisfies the $K$ -condition, that is $-K_{X_{n}}$ is not in the interior of the cone of movable divisors.

Proof.

The linear system $\lvert H\rvert$ defines a divisorial contraction $\sigma\colon X_{n}\to Y_{n}$ , where $Y_{n}$ is a hypersurface in $\mathds{P}(1,1,n,n,n)$ given by the equation

a_{2n}(u,v)=xz-y^{2}.

Thus the cone of movable divisors of $X_{n}$ is generated by $H$ and $F$ which implies the statement of the lemma by Corollary 3.2 (5). ∎

3.2. Noether-Fano method

For definitions of canonical singularities of pairs we refer the reader to [10, pages 16-17] and [5, Definition 2.1]. Let $\pi:X\to\mathds{P}^{1}$ be an $\mathcal{A}_{5}$ -del Pezzo fibration.

Suppose we are given a birational map $\chi\colon X\dasharrow Y$ to a Mori fiber space $\pi_{Y}\colon Y\to Z$ . Let $\mathcal{M}_{Y}$ be a very ample complete linear system on $Y$ . I say $\mathcal{M}=\chi^{-1}\mathcal{M}_{Y}$ is a mobile linear system associated to $\chi$ . There are numbers $\lambda\in\mathds{Q}_{+}$ and $l\in\mathds{Q}$ such that $\lambda\mathcal{M}\sim-K_{X}+lF$ . The Noether-Fano inequality is the essential result used to prove birational rigidity-type results.

Theorem 3.6 (Noether-Fano inequality, [6, Theorem 4.2]).

Suppose $l>0$ and $(X,\lambda\mathcal{M})$ has canonical singularities, then $\chi$ is isomorphism.

Let $E$ be a divisorial valuation of $\mathds{C}(X)$ . If $(E,X,\lambda\mathcal{M})<0$ , then I say that $E$ is a maximal singularity of the pair $(X,\lambda\mathcal{M})$ . I call the center $Z$ of $E$ on $X$ a maximal center of the pair $(X,\lambda\mathcal{M})$ .

Now I examine, which subvarieties $Z\subset X$ can be maximal centers.

Set $X=X_{n}$ for some $a_{2n}(u,v)$ that is not a square and let $\chi$ , $\mathcal{M}$ , $\lambda$ , $l$ be as above.

Let $C$ be an irreducible curve on $X$ , I say $C$ is horizontal if $\pi(C)=\mathds{P}^{1}$ and vertical if $\pi(C)$ is a point. I say that a curve $C$ is horizontal (resp. vertical) if every irreducible component of $C$ is horizontal (resp. vertical).

Let $C\subset X$ be a horizontal or a vertical curve. I define its degree as follows

\deg(C)=\begin{cases}-K_{X}\cdot C/2,&\text{if }C\text{ is vertical}\\ C\cdot F,&\text{if }C\text{ is horizontal}\end{cases}

Lemma 3.7.

Suppose $(X,\lambda\mathcal{M})$ is not canonical at a curve $\Delta$ , then $\Delta$ is an $\mathcal{A}_{5}$ -invariant vertical curve of degree $2$ .

Proof.

Let $\overline{\Delta}$ be the $\mathcal{A}_{5}$ -orbit of $\Delta$ . The pair $(X,\lambda\mathcal{M})$ is not canonical at a curve $\Delta$ if and only if $\operatorname{mult}_{\Delta}\lambda\mathcal{M}>1$ and hence if and only if $\operatorname{mult}_{\overline{\Delta}}\lambda\mathcal{M}>1$ .

Suppose $\Delta$ is horizontal, let $F$ be a general fiber of $\pi$ and let $D_{1},D_{2}\in\mathcal{M}$ be general divisors. Then the set-theoretic intersection

\Sigma=F\cap\overline{\Delta}\subset F\cap D_{1}\cap D_{2}

is a union of orbits on $F$ . Hence

8=F\cdot\lambda D_{1}\cdot\lambda D_{2}>\lambda^{2}F\cdot\overline{\Delta}\geqslant\big{|}\Sigma\big{|}\geqslant 12,

a contradiction.

Suppose $\overline{\Delta}\subset F$ and let $D$ be general in $\mathcal{M}$ . Let $H_{F}=H|_{F}$ , then $D|_{F}\sim 2H_{F}$ and $\operatorname{ord}_{\overline{\Delta}}D|_{F}>1$ . It follows that $\deg\overline{\Delta}\leqslant 4$ and [4, Lemma 6.4.4] implies that $\overline{\Delta}=\Delta$ is the $\mathcal{A}_{5}$ -invariant curve of degree $2$ .

Suppose $F$ is singular, then it is a cone over $\mathds{P}^{1}$ with the $\mathcal{A}_{5}$ -action inherited from $\mathds{P}^{1}$ . Let $C$ be the unique $\mathcal{A}_{5}$ -invariant curve of degree $2$ on $F$ . Suppose $\Delta\neq C$ and denote $\Delta\sim kH_{F}$ . Set $\Sigma=\Delta\cap C$ , it is a union of $\mathcal{A}_{5}$ -orbits on $C$ hence $\lvert\Sigma\rvert\geqslant 12$ by Lemma 2.1. It follows that $\Delta\cdot C\geqslant 12$ and $k\geqslant 6$ . On the other hand, $D|_{F}\sim 2H_{F}$ , a contradiction. ∎

Lemma 3.8.

Let $P$ be an $\mathcal{A}_{5}$ -invariant point and suppose $X$ is smooth at $P$ . Then pair $(X,\lambda\mathcal{M})$ is canonical at $P$ .

Proof.

Suppose $(X,\lambda\mathcal{M})$ is not canonical at $P$ and let $E_{\infty}$ be the divisorial valuation over $X$ such that $a(E_{\infty},X,\lambda\mathcal{M})<0$ .

First, observe that a fiber $F$ of $\pi$ containing $P$ is a quadratic cone and $P$ is its vertex. Let $\sigma\colon\widetilde{X}\to X$ be the blow up at $P$ and let $E$ be the exceptional divisor of $\sigma$ . Let $L$ be a general line through $P$ , then for general $D\in\mathcal{M}$

\operatorname{mult}_{P}\lambda D\leqslant L\cdot\lambda D=2.

It follows that $a(E,X,\lambda\mathcal{M})\geqslant 0$ , hence the center $B$ of $E_{\infty}$ on $\widetilde{X}$ is a point or a curve on $E$ .

Note that the action of $\mathcal{A}_{5}$ on $E$ is non-trivial. Indeed, the point $P$ up to a change of coordinates on $\mathds{P}^{1}_{u,v}$ has the equations $u=x=y=z=0$ and the local equation of $X$ near $P$ is $u=0$ , thus $E\cong_{\mathcal{A}_{5}}\mathds{P}(W_{3})$ . Denote $\widetilde{\lambda}\mathcal{M}_{E}=(\sigma^{-1}\mathcal{M})|_{E}$ , then $\operatorname{ord}_{B}\lambda\widetilde{\mathcal{M}}_{E}>1$ and $\deg\widetilde{\mathcal{M}}_{E}=\operatorname{mult}_{P}\lambda\mathcal{M}\leqslant 2$ . On the other hand, if $B$ is a curve, then $\deg\overline{B}\geqslant 2$ , a contradiction.

Suppose $B$ is a point and let $\overline{B}$ be the $\mathcal{A}_{5}$ -orbit of $B$ . Then $\big{|}\overline{B}\big{|}\geqslant 6$ and there are $4$ points $P_{1},P_{2},P_{3},P_{4}\in\overline{B}\subset E\cong\mathds{P}^{2}$ in general position. I claim that

(1)

\sum_{i=1}^{4}\operatorname{mult}_{P_{i}}C\leqslant 2\deg C

for any curve $C\subset E$ . Indeed, denote by $L_{ij}$ the line on $E$ passing through $P_{i}$ and $P_{j}$ . Decomposing $C=C^{\prime}+\sum\alpha_{ij}L_{ij}$ and counting multiplicities I conclude the inequality (1). Thus

4<4\operatorname{mult}_{B}\lambda\mathcal{M}_{E}=\sum_{i=1}^{4}\operatorname{mult}_{P_{i}}\lambda\mathcal{M}_{E}\leqslant 4,

a contradiction. ∎

It is possible that the pair $(X,\lambda\mathcal{M})$ is not canonical at $\mathcal{A}_{5}$ -invariant curves of degree $2$ . On the other hand, the elementary Sarkisov links originating from them are described in Lemma 3.3, these links are $\mathcal{A}_{5}$ -equivariant square birational maps. Using these links I acquire a new pair which is canonical at all curves.

Let $b(u,v)$ be a polynomial of degree $k$ . Then there is the associated map $\varphi_{b}\colon X\dasharrow X_{b}$ , where $X_{b}$ is a hypersurface in $T_{n+k}$ given by the equation $a(u,v)(b(u,v))^{2}w^{2}=xz-y^{2}$ . The map $\varphi_{b}$ is the composition of elementary links described in Lemma 3.3. Denote $\mathcal{M}_{b}=\varphi_{b}^{-1}\mathcal{M}$ .

Proposition 3.9.

Suppose $X$ is smooth, then there is $b(u,v)$ such that the pair $(X_{b},\lambda\mathcal{M}_{b})$ is canonical at curves and $\mathcal{A}_{5}$ -invariant points.

Proof.

I prove the proposition by playing the two-ray game. Suppose $(X,\lambda\mathcal{M})$ is not canonical at a curve $\Delta_{1}$ . By Lemma 3.7 the curve $\Delta_{1}$ is of degree $2$ and is $\mathcal{A}_{5}$ -invariant, hence its equations are $l_{1}(u,v)=w=0$ for some linear $l_{1}(u,v)$ . The elementary Sarkisov link starting at $\Delta_{1}$ is the map $\varphi_{l_{1}}$ . Let $\mathcal{M}_{l_{1}}=\varphi_{l_{1}}^{-1}\mathcal{M}$ . If the pair $(X_{l_{1}},\mathcal{M}_{l_{1}})$ is canonical at curves, then I am done. Otherwise there is a curve $\Delta_{2}$ and an elementary Sarkisov link $\varphi_{l_{2}}\colon X_{l_{1}}\dasharrow X_{l_{1}l_{2}}$ , and I repeat the process as many times as required. The process terminates by [6, Theorem 6.1] and I set $b=l_{1}l_{2}\dots l_{k}$ . The pair $(X_{b},\lambda\mathcal{M}_{b})$ is canonical at curves by construction and Lemma 3.7.

The pair $(X_{b},\lambda\mathcal{M}_{b})$ is canonical at smooth $\mathcal{A}_{5}$ -invariant points by Lemma 3.8. I will now show that the pair $(X_{b},\lambda\mathcal{M}_{b})$ is canonical at singular $\mathcal{A}_{5}$ -invariant points as well.

Recall that by [9, Theorem 1.1] if the pair $(X_{b},\lambda\mathcal{M}_{b})$ is not canonical at a $cA_{1}$ -point $P$ with the local equation

xz-y^{2}-u^{N}=0,

then either $a(E_{s,t},X_{b},\lambda\mathcal{M}_{b})<0$ , where $E_{s,t}$ is the exceptional divisor of a $(s,t,2t-s,1)$ -weighted blow up at $P$ for some coprime $s\leqslant t\leqslant N/2$ or $N=3$ and $a(E_{sp},X_{b},\lambda\mathcal{M}_{b})<0$ , where $E_{sp}$ is the exceptional divisor of a $(1,3,5,2)$ -weighted blow up if $N=3$ .

Note that $a(E_{1,1},X_{b},\lambda\mathcal{M}_{b})>0$ by construction. Thus, if $N=2$ , we are done. We proceed by induction. If $N=3$ , then $a(E_{sp},X_{b},\lambda\mathcal{M}_{b})<0$ and the pair $(X_{b/u},\lambda\mathcal{M}_{b/u})$ is not canonical at the $\mathcal{A}_{5}$ -invariant point in the fiber $u=0$ , which contradicts Lemma 3.8.

Similarly if $N\geqslant 4$ , then $a(E_{s,t},X_{b},\lambda\mathcal{M}_{b})\geqslant 0$ for $s\geqslant 2$ . Indeed, otherwise the pair $(X_{b/u},\lambda\mathcal{M}_{b/u})$ is not canonical at the $\mathcal{A}_{5}$ -invariant point in the fiber $u=0$ , which contradicts the assumption of induction. Thus I may assume that $s=1$ and $t>1$ . Let $\sigma\colon\widetilde{X}_{b}\to X$ be the blow up at $P$ and let $\widetilde{\mathcal{M}}_{b}=\sigma^{-1}\mathcal{M}_{b}$ . Then the pair $(\widetilde{X}_{n},\lambda\widetilde{\mathcal{M}}_{b})$ is not canonical at a line $L$ on the exceptional divisor $E$ of $\sigma$ . Hence, it is not canonical at each line in the orbit of $L$ . It follows that $\deg\widetilde{\mathcal{M}}_{b}|_{E}>6$ since the length of the orbit of $L$ is at least $12$ , which contradicts $a(E,X_{b},\mathcal{M}_{b})>0$ . ∎

It remains to show that the points that are not fixed by the $\mathcal{A}_{5}$ -action cannot be maximal centers.

3.2.1. Orbits of points as non-canonical centers

Let $P$ be a maximal center of the pair $(X,\lambda\mathcal{M})$ . I have already shown, that $P$ is a point which is not fixed by $\mathcal{A}_{5}$ -action. In this subsection I show that $P$ must lie on an $\mathcal{A}_{5}$ -invariant curve of degree $2$ .

Denote by $F$ the fiber containing $P$ and denote by $\overline{P}$ the $\mathcal{A}_{5}$ -orbit of $P$ , then $(X,\lambda\mathcal{M})$ is not canonical at any point $P_{i}\in\overline{P}$ . Let $\Delta$ be the $\mathcal{A}_{5}$ -invariant curve of degree $2$ in the fiber containing $P$ .

Lemma 3.10.

Suppose $P\not\in\Delta$ , then one of the following holds:

(1)

There is an irreducible curve $\Gamma$ of degree $4$ and distinct points $P_{1},\dots,P_{8}\in\overline{P}\cap\Gamma$ such that $\Gamma$ is smooth at $P_{1},\dots,P_{8}$ ;
(2)

There are smooth disintct irreducible curves $\Gamma_{1},\Gamma_{2}$ of degree $2$ , distinct points $P_{1},\dots,P_{4}\in\overline{P}\cap\Gamma_{1}$ , and distinct points $P_{5},\dots,P_{8}\in\overline{P}\cap\Gamma_{2}$ ;
(3)

The fiber $F$ is smooth, there is a smooth irreducible curve $\Gamma$ of bi-degree $(2,1)$ and there are distinct points $P_{1},\dots,P_{7}\in\overline{P}\cap\Gamma$ .
(4)

The fiber $F$ is singular, there are disinct smooth irreducible curves $\Gamma_{1},\Gamma_{2}$ of degree $3$ , there are distinct points $P_{1},\dots,P_{7}\in\Gamma_{1}\cap\overline{P}$ , and there are distinct points $P_{1}^{\prime},\dots,P_{7}^{\prime}\in\Gamma_{2}\cap\overline{P}$ .

Proof.

For any $8$ points on $F$ there is a unique quadric section passing through them. Let $C$ be the quadric section through $P_{1},\dots,P_{8}$ , note that $\deg C=4$ .

First, suppose $C$ is reducible, then $C$ has at most $3$ components. If $C$ has $3$ components, then there is a conic $C$ containing at least $6$ points among $P_{1},\dots,P_{8}$ . There is $g\in\mathcal{A}_{5}$ such that $gC\neq gC$ , thus I set $\Gamma_{1}=C$ and $\Gamma_{2}=gC$ .

If $C$ is non-reduced, then simiarly to the previous case I set $\Gamma_{1}=C/2$ and $\Gamma_{2}=g\Gamma_{1}$ for some $g$ such that $g\Gamma_{1}\neq\Gamma_{1}$ .

Suppose $C=C_{1}+C_{2}$ where $C_{1},C_{2}$ are irreducible. If $\deg C_{1}=\deg C_{2}=2$ , then I set $\Gamma_{i}=C_{i}$ . If $\deg C_{1}=1$ and $\deg C_{2}=3$ , then $C_{1}$ contains at most one points among $P_{1},\dots P_{8}$ . Thus I may assume that $P_{1},\dots,P_{7}\in C_{2}$ . If $F$ is smooth, then it is the situation (3). For singular $F$ set $\Gamma_{1}=C_{2}$ and $\Gamma_{2}=g\Gamma_{1}$ for some $g$ such that $g\Gamma_{1}\neq\Gamma_{1}$ .

Pick a point $P_{9}\in\overline{P}$ distinct from $P_{1},\dots,P_{8}$ . Let $C_{i}$ be the quadric section through points $P_{1},\dots,P_{i-1},P_{i+1},\dots,P_{9}$ . I may assume that all $C_{i}$ are irreducible, otherwise one of the previous cases applies. I may also assume that every $C_{i}$ is singular at one of the points $P_{j}$ , otherwise I am done. Note that $C_{i}$ is singular at one point at most, hence if all $C_{i}$ coincide, I am also done. Since $C_{9}\cdot C_{i}=8$ the curve $C_{i}$ must be singular at $P_{9}$ for $i\neq 9$ . But then $C_{1}\cdot C_{2}\geqslant 10$ , a contradiction. ∎

Corollary 3.11.

Suppose $(X,\lambda\mathcal{M})$ is not caninical at a point $P$ , then $P\in\Delta$ .

Proof.

The proof is the case by case analysis for the curves $\Gamma$ , $\Gamma_{1}$ , and $\Gamma_{2}$ from Lemma 3.10. The case (1) is analogous to the case (3) and the case (4) is analogous the case (2).

Suppose there is a curve $\Gamma$ as in the case (3). Let $D$ be a general divisor in $\mathcal{M}$ and denote $D_{F}=D|_{F}$ . Then $\operatorname{mult}_{P_{i}}\lambda D_{F}>1$ for $i=1,\dots,7$ . I decompose $\lambda D_{F}=\alpha\Gamma+C$ , where $\Gamma\not\subset\operatorname{Supp}C$ and $\alpha\leqslant 1$ since bi-degree of $\lambda D_{F}$ equals $(2,2)$ . Thus

7<\sum_{i=1}^{7}\operatorname{mult}_{P_{i}}\lambda D_{F}\leqslant 7\alpha+C\cdot\Gamma=7\alpha+6-6\alpha=6+\alpha\leqslant 7,

a contradiction.

Suppose there are curves $\Gamma_{1},\Gamma_{2}$ as in the case (2). Let $D$ be a general divisor in $\mathcal{M}$ and denote $D_{F}=D|_{F}$ . Then $\operatorname{mult}_{P_{i}}\lambda D_{F}>1$ for $i=1,\dots,8$ . I decompose $\lambda D_{F}=\alpha_{1}\Gamma_{1}+\alpha_{2}\Gamma_{2}+C$ , where $\Gamma_{1},\Gamma_{2}\not\subset\operatorname{Supp}C$ and $\alpha_{1}+\alpha_{2}\leqslant 2$ since $\deg\lambda D_{F}=4$ .

At most two points among $P_{1},P_{2},P_{3},P_{4}$ coincide with points among $P_{5},P_{6},P_{7},P_{8}$ since $\Gamma_{1}\cdot\Gamma_{2}=2$ . Thus after renumbering points $P_{i}$ I may assume that

	$\displaystyle\Gamma_{1}\cap\Big{(}\{P_{5},P_{6},P_{7},P_{8}\}\setminus\{P_{1},P_{2},P_{3},P_{4}\}\Big{)}=\varnothing\quad\text{and}$
	$\displaystyle\Gamma_{2}\cap\Big{(}\{P_{1},P_{2},P_{3},P_{4}\}\setminus\{P_{5},P_{6},P_{7},P_{8}\}\Big{)}=\varnothing.$

Thus

8<\sum_{i=1}^{4}\operatorname{mult}_{P_{i}}\lambda D_{F}+\sum_{i=5}^{8}\operatorname{mult}_{P_{i}}\lambda D_{F}\leqslant 4\alpha_{1}+C\cdot\Gamma_{1}+4\alpha_{2}+C\cdot\Gamma_{2}=4+2\alpha_{1}+2\alpha_{2}\leqslant 8,

a contradiction. ∎

3.3. Supermaximal singularities

To finish the proof of $\mathcal{A}_{5}$ -equivarian birational superrigidity I use the technique of supermaximal singularities. It has been introduced in [18] for proving birational rigidity of del Pezzo fibrations of degrees $1$ , $2$ , and $3$ .

First, I require a stronger version of Noether-Fano inequality.

Proposition 3.12 ([12, Proposition 2.7]).

Let $\pi:X\to\mathds{P}^{1}$ be a del Pezzo fibration. Suppose that we are given a non-square birational map $f\colon X\dasharrow Y$ to a Mori fiber space $\pi_{Y}\colon Y\to Z$ and let $\mathcal{M}$ be a movable linear system associated to $f$ . Define numbers $\lambda\in\mathds{Q}_{+}$ and $l\in\mathds{Q}$ by the equivalence $\lambda\mathcal{M}+K_{X}\sim lF$ . Suppose in addition that the pair $(X,\lambda\mathcal{M})$ is canonical at curves on $X$ and $l\geqslant 0$ . Then there exist points $Q_{1},\dots,Q_{k}$ of $X$ contained in distinct $\pi$ -fibers and positive rational numbers $\gamma_{1},\dots,\gamma_{k}$ with the following properties:

•

$(X,\lambda\mathcal{M}-\sum\gamma_{j}F_{j})$ is not canonical at $Q_{1},\dots,Q_{k}$ , where $F_{i}$ is the $\pi$ -fiber containing $\Gamma_{i}$ .
•

$\sum_{j=1}^{k}\gamma_{j}>l$ .

Let $D_{1},D_{2}$ be general divisors in $\mathcal{M}$ and denote their scheme theoretic intersection $Z=D_{1}\cap D_{2}$ . I may decompose $Z$ into the horizontal and the vertical parts $Z=Z^{h}+Z^{v}$ . I may further decompose

Z^{v}=\sum Z^{v}_{i},\text{ where }\operatorname{Supp}Z_{i}^{v}\subset F_{i}.

Lemma 3.13.

Suppose $n\geqslant 2$ , then

	$\displaystyle\lambda^{2}\deg Z^{h}$	$\displaystyle=8$
	$\displaystyle\lambda^{2}\deg Z^{v}$	$\displaystyle\leqslant 8l+8$

Proof.

I compute

\lambda^{2}D_{1}D_{2}\equiv(-K_{X}+lF)^{2}\equiv K_{X}^{2}-2lF\cdot K_{X}.

By Corollary 3.2

-2lF\cdot K_{X}\equiv 8lf

and

K_{X}^{2}\equiv 8s+24f-8nf

These equivalences imply the statement of the lemma. ∎

Corollary 3.14.

There is a fiber $F_{j}$ and a divisorial valuation $E_{\infty}$ of $\mathds{C}(X)$ such that $\deg\lambda^{2}Z^{v}_{j}\leqslant 8+8\gamma_{i}$ , $a(E_{\infty},X,\lambda\mathcal{M}-\gamma_{j}F_{j})<0$ , and the center $Q_{j}$ of $E_{\infty}$ is a point on $F_{j}$ .

The valuation $E_{\infty}$ is called supermaximal singularity. By the previous section $Q_{j}\in\Delta$ , where $\Delta$ is the $\mathcal{A}_{5}$ -invariant curve of degree $2$ on $F_{j}$ . From now on I denote $F_{j}$ by $F$ , $\gamma_{i}$ by $\gamma$ , $Q_{j}$ by $P$ , and $Z^{v}_{i}$ by $Z_{v}$ .

3.4. Pukhlikov’s inequality

Consider the tower of blow ups realizing $E_{\infty}$

(2)

X_{N}\xrightarrow{\sigma_{N}}\dots\xrightarrow{\sigma_{2}}X_{1}\xrightarrow{\sigma_{1}}X_{0}=X,

that is $\sigma_{i}$ is the blow up of $X_{i-1}$ at the center $B_{i-1}$ of $E_{\infty}$ on $X_{i-1}$ , $E_{i}$ is the exceptional divisor of $\sigma_{i}$ , and $E_{N}=E_{\infty}$ as divisorial valuations of $\mathds{C}(X)$ .

Let $A$ be an object on $X_{i}$ , then I denote its proper transform on $X_{j}$ , $j>i$ by $A^{(j)}$ . Denote

	$\displaystyle K=\min\{i\mid B_{i}\text{ is not a point}\},$
	$\displaystyle K^{\prime}=\min\{i\mid B_{i}\not\in F^{(i)}\}\cup{K},$
	$\displaystyle K^{\prime\prime}=\min\{i\mid B_{i}\not\in\Delta^{(i)}\},$

clearly $K\geqslant K^{\prime}\geqslant K^{\prime\prime}$ .

There is an oriented graph associated to $(\ref{tower})$ . It consists of $N$ vertices $v_{i}$ , and there is an edge $v_{i}\to v_{j}$ if $i>j$ and $B_{i-1}\subset E_{j}^{(i-1)}$ . Denote by $p_{i}$ the number of paths from $v_{N}$ to $v_{i}$ and set $p_{N}=1$ . Also set

\displaystyle\Sigma_{0}=\sum_{i=1}^{K}p_{i},\quad\Sigma_{0}^{\prime}=\sum_{i=1}^{K^{\prime}}p_{i},\quad\Sigma_{0}^{\prime\prime}=\sum_{i=1}^{K^{\prime\prime}}p_{i},\quad\Sigma_{1}=\sum_{i=K+1}^{N}p_{i}.

Denote $\nu_{i}=\operatorname{mult}_{B_{i-1}}\lambda\mathcal{M}^{(i-1)}$ , then non-canonity of the pair $(X,\lambda\mathcal{M}-\gamma F)$ is equivalent to

(3)

\sum_{i=1}^{N}p_{i}\nu_{i}>2\Sigma_{0}+\Sigma_{1}+\gamma\Sigma_{0}^{\prime}.

Theorem 3.15 ([18, Proposition 4.2] Pukhlikov’s inequality).

\lambda^{2}\Big{(}\sum_{i=1}^{K}p_{i}\operatorname{mult}_{P_{i-1}}Z_{h}^{(i-1)}+\sum_{i=1}^{K^{\prime}}p_{i}\operatorname{mult}_{P_{i-1}}Z_{v}^{(i-1)}\Big{)}\geqslant 4\Sigma_{0}+4\gamma\Sigma_{0}^{\prime}+\frac{(\Sigma_{1}-\gamma\Sigma_{0}^{\prime})^{2}}{\Sigma_{0}+\Sigma_{1}}.

I apply this inequality for each valuation in the orbit of $E$ to show a contradiction.

Denote the valuations in the $\mathcal{A}_{5}$ -orbit of $E_{\infty}$ by $E_{k,\infty}$ , in particular $E_{1,\infty}=E_{\infty}=E_{N}$ as valuations, and let $\theta$ be the length of the orbit. Recall that $a(E_{k,_{\infty}},X,\lambda\mathcal{M}-\gamma F)<0$ for $k=1,\dots,\theta$ . Thus there is a tower (2) for each valuation $k=1,\dots,\theta$ . The graphs for the towers are identical for each $E_{k,N}$ , thus the invariants $K$ , $K^{\prime}$ , $K^{\prime\prime}$ , $N$ , $p_{i}$ , $\Sigma_{0}$ , $\Sigma_{0}^{\prime}$ , $\Sigma_{0}^{\prime\prime}$ , and $\Sigma_{1}$ are the same for each $E_{k,N}$ as well.

From now on suppose that $\sigma_{i}$ is the blow up of $X_{i-1}$ at the centers of $E_{k,N}$ . Denote the center of $E_{k,N}$ on $X_{i}$ by $B_{k,i}$ and the exceptional divisor of $\sigma_{i}$ over $B_{k,i-1}$ by $E_{k,i}$ . Note that $\mathcal{M}$ is $\mathcal{A}_{5}$ -invariant, therefore $\operatorname{mult}_{B_{k,i}}\mathcal{M}^{(i)}=\operatorname{mult}_{B_{j,i}}\mathcal{M}^{(i)}=\nu_{i+1}$ for any $1\leqslant j,k\leqslant\theta$ , $0\leqslant i\leqslant N-1$ . Applying the Pukhlikov’s inequality to $E_{k,N}$ and taking a sum I get the following inequality.

Corollary 3.16.

(4)

\lambda^{2}\sum_{k=1}^{\theta}\Big{(}\sum_{i=1}^{K}p_{i}\operatorname{mult}_{B_{k,i-1}}Z_{h}^{(i-1)}+\sum_{i=1}^{K^{\prime}}p_{i}\operatorname{mult}_{B_{k,i-1}}Z_{v}^{(i-1)}\Big{)}\geqslant\theta\big{(}4\Sigma_{0}+4\gamma\Sigma_{0}^{\prime}+\frac{(\Sigma_{1}-\gamma\Sigma_{0}^{\prime})^{2}}{\Sigma_{0}+\Sigma_{1}}\big{)}.

To find a bound for the left-hand side, I decompose $\lambda^{2}Z_{v}=\xi\Delta+C$ , where $\Delta\not\in\operatorname{Supp}C$ .

Lemma 3.17.

The following inequalities hold

(1)

$\lambda^{2}\sum_{k=1}^{\theta}\sum_{i=1}^{K}p_{i}\operatorname{mult}_{B_{k,i-1}}Z_{h}^{(i-1)}\leqslant 8\Sigma_{0},$
(2)

$\sum_{k=1}^{\theta}\operatorname{mult}_{B_{k,0}}C\leqslant 8+8\gamma-2\xi,$

in particular, $\xi\leqslant 4+4\gamma$ .

Proof.

The inequality

\lambda^{2}\sum_{k=1}^{\theta}\operatorname{mult}_{B_{k,0}}Z_{h}\leqslant\lambda^{2}Z_{h}\cdot F=8.

implies (1).

Assertion (2) follows from

\sum_{k=1}^{\theta}\operatorname{mult}_{B_{k,0}}C\leqslant C\cdot\Delta=\deg C\leqslant 8+8\gamma-2\xi.

∎

Proposition 3.18.

The following inequality holds:

8\Sigma_{0}>\theta\frac{(\Sigma_{1}-\gamma\Sigma_{0}^{\prime})^{2}}{\Sigma_{0}+\Sigma_{1}}.

Proof.

Combining Lemma 3.17 with (4) I get

(5)

8\Sigma_{0}+\theta\xi\Sigma_{0}^{\prime\prime}+(8+8\gamma-2\xi)\Sigma_{0}^{\prime}>\theta\Big{(}4\Sigma_{0}+4\gamma\Sigma_{0}^{\prime}+\frac{(\Sigma_{1}-\gamma\Sigma_{0}^{\prime})^{2}}{\Sigma_{0}+\Sigma_{1}}\Big{)}.

Observe that

\theta\Sigma_{0}^{\prime\prime}>2\Sigma_{0}^{\prime}.

Indeed, otherwise (5) becomes

8\Sigma_{0}+(8+8\gamma)\Sigma_{0}^{\prime}>4\theta\Sigma_{0}+4\theta\gamma\Sigma_{0}^{\prime}\geqslant 48\Sigma_{0}+48\gamma\Sigma_{0}^{\prime},

a contradiction.

I apply the bound $\xi\leqslant 4+4\gamma$ to (5) to get the inequality

8\Sigma_{0}+(4+4\gamma)\theta\Sigma_{0}^{\prime\prime}>4\theta\Sigma_{0}+4\theta\gamma\Sigma_{0}^{\prime}+\theta\frac{(\Sigma_{1}-\gamma\Sigma_{0}^{\prime})^{2}}{\Sigma_{0}+\Sigma_{1}}.

Since $\Sigma_{0}^{\prime}\geqslant\Sigma_{0}^{\prime\prime}$ I can further simplify

8\Sigma_{0}+4\theta\Sigma_{0}^{\prime\prime}>4\theta\Sigma_{0}+\theta\frac{(\Sigma_{1}-\gamma\Sigma_{0}^{\prime})^{2}}{\Sigma_{0}+\Sigma_{1}},

which implies the statement of the proposition. ∎

I use a different technique to get a bound contradicting this one.

3.5. The technique of restricted multiplicities

The technique I am using is inspired by [8, Proof of Theorem 3.1].

Consider the linear system $\mathcal{S}\subset\lvert\gamma H+F\rvert$ of divisors containing $\Delta$ . Let $S$ be general in $\mathcal{S}$ and let $D$ be general in $\mathcal{M}$ . Denote $D_{S}=D|_{S}$ and $\operatorname{mult}_{\Gamma}D=\alpha$ , then $\operatorname{ord}_{\Gamma}D_{S}=\alpha$ . Indeed, any $S\in\mathcal{S}$ has the equation $a(u,v)w+up(x,y,z,w)=$ for some linear $p$ and $a$ . Thus $\operatorname{ord}_{\Gamma}S_{1}\cap S_{2}=1$ for general $S_{1},S_{2}\in\mathcal{S}$ . It follows that $D_{S}=\alpha\Gamma+D_{S}^{\prime}$ , where $\Gamma\not\subset\operatorname{Supp}D_{S}^{\prime}$ .

Lemma 3.19.

Let $B$ be a point or a smooth curve on a smooth threefold $X$ . Let $S$ be a surface on $X$ , suppose $\widetilde{B}=B\cap S$ is a point and suppose $S$ is smooth at $\widetilde{B}$ . Let $\sigma\colon Y\to X$ be the blow up at $B$ , let $E$ be its exceptional divisor, let $S_{Y}=\sigma^{-1}S$ and let $e=E\cap S_{Y}$ . Let $D$ be an effective divisor on $X$ and let $D_{Y}=\sigma^{-1}D$ , then

\operatorname{mult}_{\widetilde{B}}D|_{S}=\operatorname{mult}_{B}D+\operatorname{ord}_{e}D_{Y}|_{S_{Y}}.

Proof.

Elementary calculations in local coordinates. ∎

Note that $\operatorname{mult}_{B_{k,i-1}}D_{S}^{(i-1)}=\operatorname{mult}_{B_{j,i-1}}D_{S}^{(i-1)}$ for general $D$ and $S$ and $1\leqslant j,k\leqslant\theta$ . Denote $\widetilde{\nu}_{i}=\lambda\operatorname{mult}_{B_{k,i-1}}D_{S}^{(i-1)}$ , then one can bound $\nu_{i}$ using $\widetilde{\nu}_{i}$ .

Lemma 3.20.

For any $i\leqslant K^{\prime\prime}$ we have

\nu_{1}+\dots+\nu_{i}\leqslant\widetilde{\nu}_{1}+\dots+\widetilde{\nu}_{i}.

Proof.

Let $e_{k,i}=S^{(i)}\cap E_{k,i}$ and denote $m_{i}=\operatorname{ord}_{e_{k,i}}\lambda D^{(i)}|_{S^{(i)}}$ , then

\lambda D^{(i)}|_{S^{(i)}}=\lambda D_{S}^{(i)}+m_{1}e_{k,1}^{(i)}+\dots+m_{i-1}e_{k,i-1}^{(i)}+m_{i}e_{k,i}.

Observe that $B_{k,i}\not\in E_{k,i-1}^{(i)}$ for $i<K^{\prime\prime}$ since $\Delta$ is smooth. In particular $B_{k,i}\not\in e_{k,j}^{(i)}$ for $j<i$ , thus

\operatorname{mult}_{B_{i}}\lambda D^{(i)}|_{S^{(i)}}=\widetilde{\nu}_{i+1}+m_{i}\quad\text{for}~{}i\geqslant 1.

On the other hand by Lemma 3.19

\operatorname{mult}_{B_{i}}\lambda D^{(i)}|_{S^{(i)}}=\nu_{i+1}+m_{i+1}.

Thus I get the equalities

	$\displaystyle\widetilde{\nu}_{1}=\nu_{1}+m_{1}$
	$\displaystyle\widetilde{\nu}_{2}+m_{1}=\nu_{2}+m_{2}$
	$\displaystyle\quad\quad\quad\dots\dots$
	$\displaystyle\widetilde{\nu}_{K^{\prime\prime}}+m_{K^{\prime\prime}-1}=\nu_{K^{\prime\prime}}+m_{K^{\prime\prime}},$

which together imply the statement of the lemma. ∎

Proposition 3.21.

There is a bound

\nu_{1}+\dots+\nu_{K^{\prime\prime}}\leqslant K^{\prime\prime}+\frac{4}{\theta},

in particular

(6)

p_{1}\nu_{1}+\dots+p_{N}\nu_{N}\leqslant(\Sigma_{0}+\Sigma_{1})(1+\frac{4}{\theta K^{\prime\prime}}),

Proof.

The curve $\Gamma$ is smooth at $P_{i}$ , thus

\displaystyle p_{j}=p_{1}\quad\text{and}\quad\widetilde{\nu}_{j}=\lambda\alpha+\lambda\operatorname{mult}_{\widetilde{B}_{i,j-1}}(D_{S}^{\prime})^{(j-1)}\quad\text{for}~{}j\leqslant K^{\prime\prime}

On the other hand $S\cap F=\Gamma$ , therefore

\lambda\sum_{i=1}^{2\gamma}\sum_{j=1}^{L}\operatorname{mult}_{\widetilde{B}_{i,j}}(D_{S}^{\prime})^{(j)}\leqslant\lambda D_{S}\cdot F=\lambda D\cdot F\cdot S=4.

Recall that $\mathcal{M}$ is canonical at $\Gamma$ by Proposition 3.9, that is $\lambda\alpha\leqslant 1$ . Putting the bounds together I get the first inequality. It follows that $\nu_{i}\leqslant 1+\frac{4}{\theta K^{\prime\prime}}$ for $i>K^{\prime\prime}$ which implies the second inequality in the lemma. ∎

Corollary 3.22.

Let $P$ be a point which is not $\mathcal{A}_{5}$ -fixed. Then there is no supermaximal singularity at $P$ .

Proof.

I combine (6) with (3) to get a lower bound on $\Sigma_{1}$ :

\gamma\Sigma_{0}^{\prime}+2\Sigma_{0}+\Sigma_{1}<(\Sigma_{0}+\Sigma_{1})(1+\frac{4}{\theta K^{\prime\prime}}),

equivalently

\theta K^{\prime\prime}\gamma\Sigma_{0}+(\theta-4)K^{\prime\prime}\Sigma_{0}<4\Sigma_{1}.

In particular, since $K^{\prime\prime}\geqslant 1$ and $\theta\geqslant 12$ I get

\Sigma_{1}>2\Sigma_{0}+3\gamma\Sigma_{0}^{\prime}.

Applying this bound to Proposition 3.18 I get

8\Sigma_{0}-4\theta(\Sigma_{0}-\Sigma^{\prime\prime})>12\frac{4\Sigma_{0}^{2}+8\gamma\Sigma_{0}\Sigma_{0}^{\prime}}{3\Sigma_{0}+3\gamma\Sigma_{0}^{\prime}}>8\Sigma_{0},

a contradiction. ∎

At last I am ready to prove Theorem 1.11.

Proof of Theorem 1.11.

Let $\chi\colon X_{n}\dasharrow Y$ be an $\mathcal{A}_{5}$ -equivariant birational map to a $\mathcal{A}_{5}\mathds{Q}$ -Mori fiber space $Y/Z$ and let $\mathcal{M}$ be the mobile $\mathcal{A}_{5}$ -invariant linear system associated $\chi$ . Define the numbers $\lambda$ and $l$ by the equivalence $\lambda\mathcal{M}+K_{X_{n}}\sim lF$ . By Lemma 3.9 there is another birational model $\pi_{b}\colon X_{b}\to\mathds{P}^{1}$ and an $\mathcal{A}_{5}$ -equivariant square birational map $\varphi_{b}\colon X_{n}\dasharrow X_{b}$ such that the corresponding pair $(X_{b},\lambda\mathcal{M}_{b})$ is canonical at curves and $\mathcal{A}_{5}$ -fixed points. By Corollary 3.22 the pair $(X_{b},\lambda\mathcal{M}_{b})$ is also canonical at points not fixed by $\mathcal{A}_{5}$ -action.

By Proposition 3.12 and Corollary 3.14 if the map $\chi\circ\varphi_{b}^{-1}\colon X_{b}\to Y$ is not $\mathcal{A}_{5}$ -equivariantly square, then there is a supermaximal singularity $E_{\infty}$ . But by Corollary 3.22 there are no supermaximal singularities. It follows that $\chi\circ\varphi_{b}^{-1}$ is $\mathcal{A}_{5}$ -equivariantly square and hence $\chi$ is $\mathcal{A}_{5}$ -equivariantly square. ∎

4. The $\mathcal{A}_{5}$ -equivariant birational geometry of $X_{1}$

In this section I describe some $\mathcal{A}_{5}$ -Mori fiber spaces $\mathcal{A}_{5}$ -equivariantly birational to the variety $X_{1}$ .

Let $Q\subset\mathds{P}(W_{3}\oplus I\oplus I)$ be a smooth $\mathcal{A}_{5}$ -invariant quadric. Let $x,y,z$ be the coordinates on $W_{3}$ and $u,v$ be the coordinates on $I\oplus I$ . Let $\Gamma$ be the $\mathcal{A}_{5}$ -invariant point curve of degree $2$ , then it has equations $(xz-y^{2}=u=v=0)$ after a change of coordinates on $W_{3}$ . Thus the blow up of $Q$ at $\Gamma$ is $\sigma_{\Gamma}\colon X_{1}\to Q$ . On the other hand $Q$ is a quadric with an $\mathcal{A}_{5}$ -invariant point, hence it is $\mathcal{A}_{5}$ -equivariantly birational to $\mathds{P}(W_{3}\oplus I)$ .

Consider the blow up at the $\mathcal{A}_{5}$ -invariant point $\sigma_{Y}\colon Y_{1}\to\mathds{P}(W_{3}\oplus I)$ , where

Y_{1}=\mathds{P}(\mathcal{O}_{\mathds{P}(W_{3})}\oplus\mathcal{O}_{\mathds{P}(W_{3})}(-1)).

The $\mathds{P}^{1}$ -bundle $\tau\colon Y_{1}\to\mathds{P}(W_{3})$ has many $\mathcal{A}_{5}$ -equivariantly square birational to it models. For example, an $\mathcal{A}_{5}$ -invariant conic on the exceptional divisor of $\sigma_{Y}$ induces the $\mathcal{A}_{5}$ -equivariant elementary Sarkisov link $Y_{1}\dasharrow Y_{3}$ , where

Y_{3}=\mathds{P}(\mathcal{O}_{\mathds{P}(W_{3})}\oplus\mathcal{O}_{\mathds{P}(W_{3})}(-3)).

Similarly, $Y_{1}$ is $\mathcal{A}_{5}$ -equivariantly birational to $Y_{2k+1}$ for any $k$ . Alternatively, we can take a fiber $f$ of $\tau$ and the blow up $\sigma\colon\widetilde{Y}\to Y_{1}$ at the orbit of $f$ , it is easy to see that $\widetilde{Y}$ admits an $\mathcal{A}_{5}$ -equivariant $\mathds{P}^{1}$ -bundle.

As we can see, $X_{1}$ has a rich $\mathcal{A}_{5}$ -equivariant birational geometry with the following $\mathcal{A}_{5}$ -Mori fiber spaces structures:

(1)

$\mathcal{A}_{5}$ -Fano variety $Q\in\mathds{P}(W_{3}\oplus I\oplus I)$ ,
(2)

$\mathcal{A}_{5}$ -Fano variety $\mathds{P}(W_{3}\oplus I)$ ,
(3)

$\mathcal{A}_{5}$ -del Pezzo fibration $\pi\colon X_{1}\to\mathds{P}^{1}$ ,
(4)

$\mathcal{A}_{5}$ -conic bundle $\tau\colon Y_{1}\to\mathds{P}(W_{3})$ .

I believe that these are the only $\mathcal{A}_{5}$ -Mori fiber space structures.

Suppose $\chi\colon Q\to Y$ is a birational $\mathcal{A}_{5}$ -equivariant map to a $\mathcal{A}_{5}\mathds{Q}$ -Mori fiber space $\pi_{Y}\colon Y\to Z$ . Let $\mathcal{M}_{Q}$ be the associated mobile linear system. If $(Q,\lambda_{Q}\mathcal{M}_{Q})$ is not canonical at the $\mathcal{A}_{5}$ -invariant conic, then the map $\chi$ factors through $X_{1}$ . Let $\mathcal{M}$ be the corresponding mobile linear system on $X_{1}$ . Elementary calculations show that $\lambda\mathcal{M}\sim-K_{X_{1}}+lF$ , where $l>0$ . I believe that the results of Section 3 can be refined to show that $Y/Z$ is $\mathcal{A}_{5}$ -equivariantly square birational to $X_{1}/\mathds{P}^{1}$ .

Question 4.1.

Does there exist an $\mathcal{A}_{5}\mathds{Q}$ -Mori fiber space $\mathcal{A}_{5}$ -equivariantly birational to $Q$ which is not $\mathcal{A}_{5}$ -equivariantly square birational to $X_{1}/\mathds{P}^{1}$ , $Y_{1}/\mathds{P}(W_{3})$ , $Q$ , or $\mathds{P}(W_{3}\oplus I)$ ?

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Families of embeddings of the alternating group of rank 55 into the Cremona group

Abstract.

1. Introduction

Problem 1.1.

Question 1.2.

Theorem 1.3.

Conjecture 1.4.

1.1. G​ℚG\mathds{Q}-Mori fiber spaces

Definition 1.5.

Example 1.6.

Example 1.7.

Theorem 1.8.

Definition 1.9.

Definition 1.10.

Theorem 1.11.

Corollary 1.12.

Acknowledgments.

Notations and conventions

2. The represenatives of the 𝒜5\mathcal{A}_{5}-equivariant birational classes of 𝒜5​ℚ\mathcal{A}_{5}\mathds{Q}-del Pezzo fibrations with the trivial action on the base

2.1. Representations of 𝒜5\mathcal{A}_{5} and its central extension

Lemma 2.1 ( [4, Lemmas 5.1.3, 5.1.4 and Section 5.2]).

Proposition 2.2 ([4, Chapter 6]).

Lemma 2.3.

Proof.

2.2. The general fiber is ℙ2\mathds{P}^{2}

Lemma 2.4.

Proof.

Example 2.5.

Example 2.6.

Question 2.7.

2.3. The general fiber is S5S_{5}

Proposition 2.8.

Proof.

Lemma 2.9.

Proof.

Question 2.10.

2.4. The general fiber is S3S_{3}

Lemma 2.11.

Proof.

2.5. The general fiber is a quadric

Lemma 2.12.

Proof.

Proposition 2.13.

Proof.

3. Rigidity results

3.1. The geometry of XnX_{n}

Lemma 3.1.

Corollary 3.2.

Lemma 3.3.

Proof.

Lemma 3.4.

Proof.

Lemma 3.5.

Proof.

3.2. Noether-Fano method

Theorem 3.6 (Noether-Fano inequality, [6, Theorem 4.2]).

Lemma 3.7.

Proof.

Lemma 3.8.

Proof.

Proposition 3.9.

Proof.

3.2.1. Orbits of points as non-canonical centers

Lemma 3.10.

Proof.

Corollary 3.11.

Proof.

3.3. Supermaximal singularities

Proposition 3.12 ([12, Proposition 2.7]).

Lemma 3.13.

Proof.

Corollary 3.14.

3.4. Pukhlikov’s inequality

Theorem 3.15 ([18, Proposition 4.2] Pukhlikov’s inequality).

Corollary 3.16.

Lemma 3.17.

Proof.

Proposition 3.18.

Proof.

3.5. The technique of restricted multiplicities

Families of embeddings of the alternating group of rank $5$ into the Cremona group

1.1. $G\mathds{Q}$ -Mori fiber spaces

2. The represenatives of the $\mathcal{A}_{5}$ -equivariant birational classes of $\mathcal{A}_{5}\mathds{Q}$ -del Pezzo fibrations with the trivial action on the base

2.1. Representations of $\mathcal{A}_{5}$ and its central extension

2.2. The general fiber is $\mathds{P}^{2}$

2.3. The general fiber is $S_{5}$

2.4. The general fiber is $S_{3}$

3.1. The geometry of $X_{n}$

4. The $\mathcal{A}_{5}$ -equivariant birational geometry of $X_{1}$