Equivariant birational geometry of quintic del Pezzo surface

Abstract

We prove that there are exactly two G-minimal surfaces which are G-birational to the quintic del Pezzo surface, where . These surfaces are the quintic del Pezzo surface itself and the surface .

1 Introduction

The study of finite subgroups of the Cremona group is classical, but the first serious treatment has been done by Igor V. Dolgachev and Vasily A. Iskovskikh at the beginning of this century, starting with Iskovskikh’s paper [11]. In their seminal work [7] all finite subgroups of the Cremona group $\mathrm{Cr}_2(\mathbb {C})$ are classified up to isomorphism. In the section “What is left” in [7] it is stated that not all conjugacy classes of $\mathrm{Cr}_2(\mathbb {C})$ are known and that a finer description of the the conjugacy classes would be desirable.

Let us recall from [11] that two subgroups of the Cremona group given by the biregular actions of a finite group G on two rational surfaces are conjugate if there exists a G-birational map . By general theory such a map can be factorised into elementary links [9]. In this paper we will contribute to the open questions from [7] by proving:

Theorem 1.1

Let $S_5$ be the smooth del Pezzo surface of degree 5, and let be a subgroup of order 20 in . Then $\mathrm{Pic}^{G_{20}}(S_5)=\mathbb {Z}$ and

• 1. $S_5$ is not $G_{20}$-birational to any conic bundle,

• 2. there exists a unique G-minimal del Pezzo surface which is $G_{20}$-birational to $S_5$, that is ,

• 3. the group of $G_{20}$-birational automorphisms is given by .

Here $C_n$ is a cyclic group of order n. It should be noticed that there are no G-conic fibrations birational to $S_5$.

In the notation of [1] we can say that $S_5$ is $G_{20}$-solid.

Remark 1.2

In the proof of Theorem 1.1 we will also see that the only smooth del Pezzo surfaces G-birational to $S_5$ are and the Clebsch cubic surface. But the latter is not $G_{20}$-minimal, i.e., $\mathrm{Pic}^{G_{20}}(\widetilde{S}) \ne \mathbb {Z}$. Indeed we will show in Remark 5.2, that its $G_{20}$-invariant Picard group is $\mathbb {Z}^2$.

Throughout this paper we assume all varieties to be complex and projective. For all notation in birational geometry, such as G-biregular, we use the conventions introduced in [7].

2 G-Sarkisov links

We will dedicate this section to the introduction of the notion of G-Sarkisov links where G is a finite group. For simplicity we will only consider the dimension 2 here. For a more detailed study see [5]. This language will allow us to state Theorem 1.1 in a more precise and technical way. We will firstly define a G-Mori fibre space.

Definition 2.1

A 2-dimensional G-Mori fibre space is

• DP: a smooth G-minimal del Pezzo surface S, i.e., $\mathrm{Pic}^G(S)=\mathbb {Z}$;

• CB: a G-conic bundle, i.e., a G-equivariant morphism , where S is a smooth surface and the general fibre of $\pi$ is such that $\mathrm{Pic}^G(S)=\mathbb {Z}^2$.

The main result about 2-dimensional G-Sarkisov link is the following:

Theorem 2.2

([5]) Let $S,S'$ be 2-dimensional G-Mori fibre spaces and let $\chi :S \dashrightarrow S'$ be a non-biregular G-birational map. Then $\chi$ is a composition of elementary links known as G-Sarkisov links.

There are five different G-Sarkisov links of dimension 2 which are described below. The first type is given by

where S and $S'$ are G-minimal del Pezzo surfaces and $\alpha$ and $\beta$ are blow-ups of G-orbits in S and $S'$ respectively. The second type is given by

where S is a G-minimal del Pezzo surface, $\alpha$ is a blow-up of a G-orbit and $\beta$ is a G-conic bundle. The third type is given by

where $S'$ is a G-minimal del Pezzo surface, $\beta$ is a blow-up of a G-orbit and $\alpha$ is a G-conic bundle. We shall notice that this is the inverse link of type (II). The fourth type is given by

where $\alpha$ and $\beta$ are G-conic bundles. Finally, the fifth type is given by

where S and $S'$ are not G-minimal del Pezzo surfaces and $\alpha$ and $\beta$ are blow-ups of G-orbits in S and $S'$ respectively. Additionally, $\pi$ and $\pi '$ are G-conic bundles and we call the whole link an elementary transformation of G-conic bundles (see [9]). This diagram commutes.

The notion of G-Sarkisov links is a good way to replace the technical result of the Noether–Fano inequality (see [6, 10]).

Remark 2.3

It follows from the definition of G-links that $\widehat{S}$ is a del Pezzo surface if S is a del Pezzo surface. Thus in the links of type (I), (II), (III) and (IV), the surface $\widehat{S}$ is a del Pezzo surface.

Using the notion of G-Sarkisov links we are able to restate Theorem 1.1.

Theorem 2.4

Let $S_5$ be the smooth del Pezzo surface of degree 5, and let be a subgroup of order 20 in . Then $\mathrm{Pic}^{G_{20}}(S_5)=\mathbb {Z}$ and the following assertions hold:

• 1. There exists a unique $G_{20}$-Sarkisov link that starts at $S_5$. It is given by

  

  (1)

  where $\pi$ is the blow-up of the unique $G_{20}$-orbit of length 2 in $S_5$, $\sigma$ is a blow-up of one of two $G_{20}$ orbits of length 5 and $\widetilde{S}$ is the Clebsch cubic surface.

• 2. Let be equipped with the $G_{20}$-action coming from (1). Then the only $G_{20}$-Sarkisov links starting from are the inverse of (1) and

  

  (2)

  where $\gamma$ is the blow-up of another $G_{20}$-orbit of length 5, $\widetilde{S}$ is the Clebsch cubic surface and $\delta =\pi$ is the blow-up of the unique $G_{20}$-orbit of length 2 in $S_5$.

Combining (1) and (2) yields a non-biregular $G_{20}$-birational map .

These links were constructed and described numerically by Dolgachev and Iskovskikh in [7, Proposition 7.13] but for our purposes we reconstruct them here and will fill in the details for these links.

3 Motivation

In this section, we want to motivate Theorem 1.1. There are various different starting points to investigate conjugacy in the Cremona group. We decided to start our research on del Pezzo surfaces. Those surfaces have been introduced by Pasquale del Pezzo in the late 18th century and since then various ways of studying them have been encountered.

For our purposes we will understand a del Pezzo surface of degree d, denoted by $S_d$, as the blow-up of in $9-d$ points in general position. To start our investigation of conjugacy classes of the Cremona group we need to introduce the notion of G-birational (super-) rigidity.

Definition 3.1

Let S be a smooth del Pezzo surface and be a finite group such that $\mathrm{Pic}^G(S)=\mathbb {Z}$. We say S is G-birationally rigid whenever

• (a) if S is G-birational to any G-minimal del Pezzo surface $S'$, then $S'$ is G-biregular to S, and

• (b) S is not G-birational to any G-conic bundle.

Condition (a) is equivalent to saying that for any birational G-map $\chi :S \dashrightarrow S'$, where $S'$ is a G-minimal del Pezzo surface, there exists a G-birational automorphism $\theta :S \dashrightarrow S$ such that $\chi {\circ } \theta$ is a G-isomorphism.

Definition 3.1 means, that the only G-Sarkisov links starting in S are of the form

where $\alpha$ and $\beta$ are blow-ups of G-orbits.

Definition 3.2

Let S be a smooth del Pezzo surface and be a finite group such that $\mathrm{Pic}^G(S)=\mathbb {Z}$. The surface S is G-birationally superrigid if it is G-birationally rigid and $\text {Bir}^G(S)=\mathrm{Aut}^G(S)$.

Definition 3.2 means that there are no G-Sarkisov links starting at S. With these definitions in hand we are able to state.

Theorem 3.3

([7]) Let S be a smooth del Pezzo surface of degree d, that is $K_S^2=d$, and let be such that $\mathrm{Pic}^G(S)=\mathbb {Z}$. Then the following assertions hold:

• (a) If S does not contain a G-orbit of length less then d, then S is G-birationally superrigid.

• (b) If S does not contain a G-orbit of length less then $d-2$, then S is G-birationally rigid.

Sketched Proof Assume $\widehat{S}$ is a smooth del Pezzo surface. Then by Remark 2.3, $\alpha :S \rightarrow \widehat{S}$ is a blow-up of a G-orbit of length less then d, because $K_{\widehat{S}}>0$. This proves (a).

If there is a G-orbit of length $d-1$, the blow-up of this orbit is $\widehat{S}=S_1$, the del Pezzo surface of degree 1, so we can use the Bertini involution there. Similarly, if there exists a G-orbit of length $d-2$, the del Pezzo surface of degree 2, we can blow up this orbit to obtain $\widehat{S}=S_2$, and we can use the Geiser involution. This proves (b). $\square$

From Theorem 3.3 we can immediately deduce the following corollary.

Corollary 3.4

([7]) Let S be a smooth del Pezzo surface of degree $d <3$, and let be a finite group such that $\mathrm{Pic}^G(S)=\mathbb {Z}$. If S is of degree 1, then S is G-birationally superrigid. If S is of degree 2 or 3, then S is G-birationally rigid.

This result is known for quite some time and was implicitly proven by Segre in 1943 and Manin in 1962. For proofs of Theorem 3.3 and Corollary 3.4 see [7, Section 7.1]. The proof of Theorem 3.3 easily implies

Theorem 3.5

([7]) Let S be a smooth del Pezzo surface of degree 4, and let be a finite group such that $\mathrm{Pic}^G(S)=\mathbb {Z}$. Then

• 1. if there are no G-fixed points, then S is G-birational rigid,

• 2. if there exists a G-fixed point, then there exists a G-Sarkisov link

  

  where $\alpha$ is the blow-up of a G-orbit, $\widetilde{S}$ is a smooth cubic surface and $\beta$ is a conic bundle.

In this paper, we are mostly interested in G-birational rigid del Pezzo surfaces or those which are close to them. By close we mean that these are del Pezzo surfaces which are not G-birational to any conic bundle (in the language of [1] these are G-solid del Pezzo surfaces).

Following Corollary  3.4 and Theorem 3.5 we will investigate links starting from the smooth del Pezzo surface of degree 5, which we will call $S_5$, in this paper. It is well known that

the symmetric group of five elements. A proof is provided in [4] . If we want $S_5$ to be a G-minimal surface (i.e., $\mathrm{Pic}^G(S_5)=\mathbb {Z}$), we require G to be one of the following (see [7, Theorem 6.4]):

• the symmetric group $\mathfrak {S}_5$ of five elements of order 120;

• the alternating group $\mathfrak {A}_5$ of five elements of order 60;

• the semidirect product of order 20;

• the dihedral group $D_{10}$ of order 10;

• the cyclic group $C_5$ of order 5.

For $\mathfrak {S}_5$ and $\mathfrak {A}_5$ the quintic del Pezzo surface is G-birationally superrigid (see [2]). For $C_5$ there exists a G-birational map from $S_5$ to (see [3]) such that $C_5$ has a fixed point there (see Lemma 4.1). The construction of this map can be generalised for $D_{10}$ which is done in Corollary 4.2. Hence these groups are better addressed when studying the G-equivariant birational geometry of . This has been done in [12]. We shall also notice that $S_5$ is not G-solid in this case.

In this paper we will therefore focus on the group as a subgroup of , which is also known as the general affine group of degree 1 over the field with five elements, denoted by .

4 The quintic del Pezzo surface

In the proof of Theorem 2.4 we will investigate the existence of $G_{20}$-equivariant birational maps between quintic del Pezzo surface, denoted by $S_5$, and the surface . First we need to understand the action of $G_{20}$ on $S_5$. To do this we use the following result from [3].

Lemma 4.1

([3]) There is a $C_5$-birational map $\phi$ (i.e., a $C_5$-Sarkisov link) between $S_5$ and given by the $C_5$-commutative diagram

Here $\alpha$ is the blow-up of a $C_5$-fixed point in $S_5$, and $\beta$ is the blow-up of five points in which form a $C_5$-orbit. $S_4$ is a quartic del Pezzo surface.

Proof

For the proof we will start with and invert the link ($\blacktriangle$). Consider $C_5$ as a subgroup of . There exists a $C_5$-equivariant Veronese embedding which defines a faithful action of $C_5$ on such that there exists a $C_5$-invariant conic (that is the image of ). Thus we can blow up the $C_5$-orbit of length 5 on this conic to obtain the quartic del Pezzo surface, denoted by $S_4$.

If we contract the the proper transform of K there we get the unique quintic del Pezzo surface. Since is unique up to conjugation the composition of the two described maps yields the desired link $\phi$.

In more elementary terms, we may say that five points $P_1,\ldots ,P_5$ in general position in $\mathbb {P}^2$ always lie on a unique conic K. Then the group $C_5$ fixes two points $A_1$, $A_2$ on a conic [4] (i.e., the line through these two points is $C_5$-invariant). Additionally, it fixes a point which does not lie on the conic. The blow-up $\alpha$ of $P_1, \ldots , P_5$ does not affect B, neither does the contraction $\beta$. Thus there is a point

$$\begin{aligned} Q_2 = \phi ^{-1}(B) \in S_5, \end{aligned}$$

which is fixed by $C_5$. We know that $\alpha ^{-1}(K)$ is a $\beta$-exceptional curve in $S_4$. After the contraction $\beta$, we have

$$\begin{aligned} \phi ^{-1}(A_1)=\phi ^{-1}(A_2)=Q_1, \end{aligned}$$

which is another fixed point of $C_5$ in $S_5$. Thus we know that for there exist two $C_5$-fixed points $Q_1$ and $Q_2$. We shall mention that all other orbits are of length 5. $\square$

From the proof of Lemma 4.1 we can easily deduce.

Corollary 4.2

There is a $D_{10}$-birational map $\phi$ (i.e., a $D_{10}$-Sarkisov link) between $S_5$ and corresponding to the $D_{10}$-commutative diagram ($\blacktriangle$).

Proof

In the same way as in the proof of Lemma 4.1 we can construct the inverse link of ($\blacktriangle$). Furthermore the action of lifts to an action on . Then we can use the same argument as before.

In the notation of the proof of Lemma 4.1 we may say that the action of $D_{10}$ on interchanges the points $A_1$ and $A_2$ but fixes the point B. Thus we can use the same link $\phi$ as in Lemma 4.1 and by the same argument as above $D_{10}$ fixes points $Q_1$ and $Q_2$ in $S_5$. $\square$

We are now in the position to investigate orbits of small length $r < 5$ of the $G_{20}$-action on $S_5$.

We want to proceed in a similar way as in [11] which means that we need to classify all G-orbits of length . Then we will concentrate on those orbits of which the points are in general position, because this is a necessary condition for the existence of links starting from the surface $S_5$.

Remark 4.3

We say that points of an orbit are in general position if the blow-up of $S_5$ in this orbit is a del Pezzo surface again.

Lemma 4.4

There is a unique $G_{20}$-orbit of length $r <5$ on $S_5$. It is the orbit of length $r=2$ consisting of the points $Q_1$ and $Q_2$.

Proof

Let us consider all possible lengths for orbits.

$r=1$: Such an orbit does not exist. Assume it does. By Lemma 4.1 and Corollary 4.2, this point can only be $Q_1$, because if all of $G_{20}$ fixes it, the normal subgroups $C_5$ and $D_{10}$ fix it in particular. Hence the link ($\blacktriangle$) yields $G_{20}$-equivariant link from $S_5$ to . This means that $G_{20}$ acts on and preserves the conic K. This implies that $G_{20}$ acts faithfully on , but this is clearly a contradiction. Hence no orbit of length $r=1$ exists.

$r=2$: $\{ Q_1,Q_2\}$ is such an orbit. We know that $G_{20}$ has $D_{10}$ as a normal subgroup. If we consider the action of $D_{10}$ on $S_5$, then Corollary 4.2 tells us that there is indeed a unique orbit of length 2 which is the orbit $\{Q_1,Q_2 \}$.

$r=3$: Such an orbit does not exist because $3\not \mid 20 = |G_{20}|$, which is required by the orbit-stabilizer theorem.

$r=4$: Such an orbit does not exist. If there were such an orbit the stabilizer would satisfy but we know that $C_5$ actually fixes the same points as $D_{10}$ by Theorem 4.2 and hence the stabilizer would actually be $D_{10}$ which cannot give an orbit of length 4. $\square$

Lemma 4.4 implies that the only possible $G_{20}$-Sarkisov link starting from $S_5$ consists of a blow-up of the described orbit of length $r=2$.

Lemma 4.5

The blow-up of $Q_1$ and $Q_2$ in $S_5$ yields a smooth del Pezzo surface $\widetilde{S}$.

Proof

We need to prove that $-K_{\widetilde{S}}$ is ample. This is equivalent to saying that $Q_1$ and $Q_2$ neither lie on the $(-1)$-curves nor in an exceptional conic in $S_5$. We prove this by contradiction. For this we will consider different cases.

• We first prove that there are no $(-1)$-curves containing $Q_1$ or $Q_2$. Assume $Q_1$ lies on one of the 10 exceptional curves in $S_5$. Clearly $Q_2$ needs to lie on such a curve as well. If they lie on two different exceptional curves these two are interchanged by the group action of $G_{20}$. This contradicts the fact that $\mathrm{Pic}^{G_{20}}(S_5)=\mathbb {Z}$.

  Similarly, we may assume that $Q_1$ lies on one of the intersections of two exceptional curves. Again this contradicts $\mathrm{Pic}^{G_{20}}(S_5)=\mathbb {Z}$. So indeed $Q_1$ and $Q_2$ do not lie on the $(-1)$-curves in $S_5$ which proves that the blow-up of these two points yields another del Pezzo surface.

• It remains to show that $Q_1$ and $Q_2$ are not contained in an exceptional conic $S_5$. There are five classes of conic in $S_5$ and each of them has self-intersection . Going through all these cases in detail one can show that $Q_1$ and $Q_2$ either lie on one line which we ruled out previously or they cannot lie on one conic. Due to heavy computational work we omit the different cases at this point. $\square$

The resulting surface of this blow-up will have degree $5-2=3$, so it is a cubic surface. The only smooth cubic surface with a $G_{20}$-action is the Clebsch cubic surface (this was proved in [8]) which we will investigate in the next section.

5 The Clebsch cubic surface

Theorem 2.4 states that the only G-Sarkisov links starting from the quintic del Pezzo surface $S_5$ are of the form

From Lemma 4.4 we know that $\pi$ is the blow-up of the unique $G_{20}$-orbit $\{Q_1, Q_2\}$ of length 2. Hence $\widetilde{S}$ is the Clebsch cubic surface, which is defined as follows:

Definition 5.1

The Clebsch cubic surface, denoted by $\widetilde{S}$, is a cubic given by two defining equations in :

$$\begin{aligned} {\left\{ \begin{array}{ll} \,x_0+x_1+x_2+x_3+x_4=0;\\ \,x_0^3+x_1^3+x_2^3+x_3^3+x_4^3=0. \end{array}\right. } \end{aligned}$$

Remark 5.2

In [7] it is shown that $\mathrm{Pic}^{G_{20}}(\widetilde{S})\ne \mathbb {Z}$. The link ($\blacklozenge$) proves that in fact, $\mathrm{Pic}^{G_{20}}(\widetilde{S})=\mathbb {Z}^2$.

Now it is well known that the automorphism group of the Clebsch cubic surface is $\mathfrak {S}_5$. Thus the action of $G_{20}$ can be described very explicitly by understanding $G_{20}$ as a subgroup of $\mathfrak {S}_5$ acting by permutation on the coordinates of this surface.

We know that all representations of $G_{20}$ are conjugate to each other and thus we will use a generation by $\sigma _{(12345)}$ and $\sigma _{(2354)}$, where we use the notation introduced in [4]. Considering orbits of length 4 on $\widetilde{S}$ we obtain

Lemma 5.3

There is a unique orbit of length 4 of the $G_{20}$-action on the Clebsch cubic surface given by the points

with $\zeta$ being a primitive fifth root of unity.

Proof

An orbit of length 4 has the stabilizer ${\text {Stab}}_G = K \cong C_5$ which is isomorphic to the group generated by $\sigma _{(12345)}$, which is the unique subgroup of $G_{20}$ isomorphic to $C_5$. It has exactly the fixed points as stated in Lemma 5.3, which are obtained by straightforward calculations. It is easy to verify that these four points lie indeed on $\widetilde{S}$ and form an orbit of length 4. $\square$

The orbits of length 5 are a bit more sophisticated.

Lemma 5.4

There are three orbits of length 5 of the $G_{20}$-action on the Clebsch cubic surface given by:

Proof

An orbit of length 5 has the stabilizer in $G_{20}$. There are five subgroups of $G_{20}$ which are isomorphic to $C_4$. Let $H \cong C_4$ be the subgroup generated by $\sigma _{(2354)}$. Then H fixes four points in with which are

whereas the $R_4$ does not lie on $\widetilde{S}$ because the cubes of the coordinates do not sum to zero. Again it is easy to verify that the points $(R_1,\ldots ,R_4)$ are indeed fixed points. Acting by an element of order 5, we obtain fixed points corresponding to the action of $\sigma _{(12345)}$ on the coordinates of $R_i$. Thus we deduce, that there are three orbits of length 5 on $\widetilde{S}$ as stated in Lemma 5.4. $\square$

We shall notice that $R_2$ and $R_3$ lie on the line $x_1+x_4=x_2+x_3=0$. Generalising this we make the following important observation.

Corollary 5.5

The points and respectively lie on one of the 27 real lines on the Clebsch cubic surfaces. These five resulting lines in the link are:

• (i) $L_1:x_1+x_4=x_2+x_3=0$ through $U_1$ and $W_1$.

• (ii) $L_2:x_0+x_2=x_3+x_4=0$ through $U_2$ and $W_2$.

• (iii) $L_3:x_0+x_4=x_1+x_3=0$ through $U_3$ and $W_3$.

• (iv) $L_4:x_0+x_1=x_2+x_4=0$ through $U_4$ and $W_4$.

• (v) $L_5:x_0+x_3=x_1+x_2=0$ through $U_5$ and $W_5$.

It is easy to see that these five lines are disjoint.

Proof

This is an easy exercise of calculating the lines through each pair of points and comparing it with the lines on the Clebsch cubic, which are well known. $\square$

Lemmas 5.3 and 5.4 allow us to state the main result for this section.

Proposition 5.6

Let $\widetilde{S}$ be the Clebsch cubic surface. Then the $G_{20}$-orbits of length $r < 8$ on $\widetilde{S}$ are:

• (a) The unique orbit described in Lemma 5.3 of length 4.

• (b) The three orbits and described in Lemma 5.4 of length 5.

Proof

The orbit-stabilizer theorem tells us immediately that orbits of length $r=6$ or $r=7$ cannot exist. It remains to show that there are no orbits of length 1 or 2 on $\widetilde{S}$. This follows directly from our description of the orbits but we include computational explanation, too. An orbit of length 1 would have the whole group $G_{20}$ as its stabilizer. We see immediately that this is not possible because the subgroups K and H generated by $\sigma _{(12345)}$ and $\sigma _{(2354)}$ have completely different fixed points.

By a similar argument, there cannot be any orbits of length 2. These would have the subgroup $F \cong D_{10}$ generated by $\sigma _{(12345)}$ and $\sigma _{(25)(34)}$ as its stabilizer. Again it is easy to verify that F has $K\cong C_5$ as a subgroup. On the other hand F has the group generated isomorphic to $C_2$ generated by $\sigma _{(25)(34)}$ which is a subgroup of H as a subgroup.

But we have seen that H and K do not have any common fixed points. Hence F cannot have fixed points which means that there does not exist an orbit of length 2. $\square$

Remark 5.7

Proposition 5.6 supports the statement of Lemma 4.4. For the unique orbit of length 4 each pair of points lies on one of 27 real lines on the Clebsch cubic. Hence after contracting two of them to obtain $S_5$, we are left with an orbit of length 2.

An orbit of length 4 in $S_5$ would lift to a different orbit of length 4 in $\widetilde{S}$, but for the given reason this cannot be , which means that there do not exist orbits of length 4 in the quintic del Pezzo surface.

Given Corollary 5.5, we may consider the contraction of these five lines.

Proposition 5.8

The contraction of the five lines $L_1,\ldots ,L_5$ described in Corollary 5.5 yields the surface and this is the only other contraction that can be conducted apart from the inverse of the blow-up from $S_5$.

Proof

We know that $\widetilde{S}$ is a del Pezzo surface, so $-K_{\widetilde{S}}$ is ample. Remark 2.3 tells us that the resulting surface of the described contraction will be a del Pezzo surface of degree $3+5=8$, so it can only be or $\mathbb {F}_1$, but $\mathrm{Pic}^{G_{20}}(\mathbb {F}_1) \ne \mathbb {Z}$, which we require.

In Remark 5.2 we have seen that $\mathrm{Pic}^{G_{20}}(\widetilde{S})=\mathbb {Z}^2$. From this we conclude that there are two external rays in the Mori cone. We have shown that one consists of two lines and the other one of five. These are the only possible contraction of $\widetilde{S}$. $\square$

Proposition 5.8 allows us to state the following lemma about the link ($\blacklozenge$) which we introduced at the beginning of this section.

Lemma 5.9

Considering the desired link ($\blacklozenge$) from $S_5$ to , we know

• 1. $\pi$ is the contraction of two disjoint lines $E_1, E_2$ in the Clebsch cubic surface (respectively the blow-up of $Q_1$ and $Q_2$ in $S_5$).

• 2. $\sigma$ is the contraction of five disjoint lines $F_1,\ldots ,F_5$ in the Clebsch cubic surface (respectively the blow-up of five points in ).

• 3. The following equations hold for the exceptional divisors:

  

  where $-K_{S_5}$ is the anticanonical divisor of $S_5$, $E_1+E_2$ are the two $(-1)$-curves of the blow-up of $Q_1$ and $Q_2$, H is a divisor of bidegree (1, 1) on and are the $(-1)$-curves of the blow-up of five points in .

Lemma 5.9 implies that $\pi (F_i)$ is a smooth twisted cubic curve in $S_5$ and $E_1$ and $E_2$ are smooth twisted cubics of bidegree (2, 1) and (1, 2) respectively.

6 The surface

The $G_{20}$-action on cannot be understood in a way which is as simple as in Sect. 4 or Sect. 5. For that reason, we will use our previous observations to analyse the $G_{20}$-orbits on .

Lemma 6.1

There is a unique $G_{20}$-orbit of length 4 in . The four points are given by the intersections $F_{11} \cap F_{21}, F_{11} \cap F_{22}, F_{12} \cap F_{21}$ and $F_{12} \cap F_{22}$ of the four rulings $F_{11}, F_{12}, F_{21}$ and $F_{22}$.

Proof

The orbit of length 4 described in Lemma 5.3 lies away from the lines of contraction, thus it has an embedding in .

Generally, four points need eight different lines to describe them. A $G_{20}$-orbit of length 4 has the stabilizer $C_5$. But we know that acts on the rulings of which are copies of . Hence the $C_5$-action cannot split over each of the four points (i.e., interchanging the two lines) but fixes the rulings.

For this reason, the four points on the $G_{20}$-orbit need to lie on the four intersections of four copies of (i.e., the rulings of ), because otherwise it would not be an orbit of length 4. Hence all four points in this orbit lie on two rulings in . $\square$

Remark 6.2

We could prove Lemma 6.1 in a different way by considering the orbit of length 4 in the Clebsch cubic surface and considering their configuration there. We can show merely computationally that there exist four conics each passing through exactly one of the four points and not intersecting the $(-1)$-curves. Considering the blow-up $\sigma$ we obtain Lemma 6.1 for the four points on .

As in Sect. 5 the orbits of length 5 are a bit more difficult.

Lemma 6.3

There are exactly two $G_{20}$-orbits and of length 5 in .

Proof

We will now identify as a quadric in $\mathbb {P}^3$. We know that has a natural embedding (Segre) into $\mathbb {P}^3$. Similar to the Clebsch cubic we can understand $\mathbb {P}^3$ as a hyperplane in $\mathbb {P}^4$ with . Now let be the quadric given by

$G_{20}$ acts on by permutations of coordinates in a similar way as described in the previous section for the Clebsch cubic.

We found the orbits of length 5 explicitly on $\widetilde{S}$. Observe that the orbits and lie in , whereas does not. Hence we may assume and .

Additionally, we can check computationally that the points of each of the orbits lie in general position (i.e., no two on a line in and no four on a plane). Hence we can indeed consider the blow-up of each of these orbits which will yield the Clebsch cubic surface and together with Lemma 5.9 shows that this blow-up is indeed the inverse link of the described contraction.

Furthermore, we see that the orbits and are essentially the same orbits, only permuted by complex conjugation. In fact these two orbits are interchanged by an automorphism of a quadric which is proven in Theorem 7.3. $\square$

Now we can finally state the last proposition we need for the proof of Theorem 1.1.

Proposition 6.4

The only $G_{20}$-orbits of length $r < 8$ on are:

• (a) The unique orbit described in Lemma 6.1 of length 4.

• (b) The two orbits and described in Lemma 6.3 of length 5 where the five points lie in general position.

Proof

It remains to show that there are no other orbits than the ones described in Lemmas 6.1 and 6.3.

Orbits of length 6, 7 or 8 cannot exist by the orbit-stabilizer theorem as $6,7,8 \not \mid 20=|G_{20}|$. Assume there is an orbit of length less than 4. Then an orbit of this length would also exist in the Clebsch cubic surface but Proposition 5.6 tells us, that they do not exist there. $\square$

7 Proof of Theorem 2.4

The link (1) in Theorem 2.4 is the only G-Sarkisov link starting from the quintic del Pezzo surface $S_5$. From Lemma 4.4 we know that $\pi$ is the blow-up of the unique $G_{20}$-orbit $\{Q_1, Q_2\}$ of length 2.

Now Corollary 5.5 tells us that there are five disjoint lines on the Clebsch cubic which we can contract to obtain and Proposition 6.4 says that we need to consider two different cases for birational maps starting from .

Lemma 7.1

(orbit of length 4) Let be the blow-up of the four points $P_1,\ldots , P_4$ in the orbit and let $E_1,\ldots ,E_4$ be the corresponding exceptional curves. Then the proper transforms of the described rulings are $(-2)$-curves. This means that the resulting surface $\widehat{S}$ is not a del Pezzo surface.

Proof

This follows immediately from Lemma 6.1. $\square$

Lemma 7.1 tells us that we cannot continue from $\mathbb {P}^1 {\times } \mathbb {P}^1$ to obtain a G-Sarkisov link by blowing up the orbit of length 4.

Lemma 7.2

Let be a blow-up of the five points $P_1,\ldots , P_5$ in the orbits or respectively. Then one of the following holds for $\tau$:

• (a) $\tau$ is the same as the blow-up $\sigma$ described in ($\blacklozenge$), so that .

• (b) $\tau$ is the same as the blow-up $\gamma$ described in diagram (2) of Theorem 2.4, so that is a $G_{20}$-birational map $S_5 \dashrightarrow S_5$.

Proof

It is clear that we can obtain case (a) if we blow up the five points in or , i.e., $\tau = \sigma$ in the link ($\blacklozenge$). We get back exactly the model of the Clebsch cubic we had before because the elements in or are the points we obtained by the contraction described in Lemma 5.9. For symmetric reasons, we may assume that these are the points in . Therefore as described in ($\blacklozenge$).

Proposition 6.4 tells us that the two orbits and are interchanged by an automorphism. Let us now consider the blow-up of the orbit , which is not the same blow-up as $\sigma$. Then we may contract the two $(-1)$-curves, $E_1$ and $E_2$, on the Clebsch cubic.

This gives us back $S_5$ because the smooth quintic del Pezzo surface is unique. This means that is a birational map $S_5 \dashrightarrow S_5$. We obtain that as shown in ($\clubsuit$). This is a birational map $S_5 \dashrightarrow S_5$ which is not biregular. $\square$

Lemmas 7.1 and 7.2 tell us that there is no $G_{20}$-equivariant link starting from , that leads to a different minimal surface than the quintic del Pezzo surface or itself. This together with Remark 5.2 finalises the proof of Theorem 2.4 and implies the first two parts of Theorem 1.1.

In Theorem 2.4 we additionally stated that $\mathrm{Bir}^{G_{20}}(S_5)$ is of order 40. In fact one can show that

Theorem 7.3

Let $S_5$ be the smooth del Pezzo surface of degree 5, and let be a subgroup of order 20 in . Then

$$\begin{aligned} \mathrm{Bir}^{G_{20}}(S_5)=G_{40}, \end{aligned}$$

where .

Proof

We need to find the normalizer . Obviously, it is enough to find $G_{40} \cap H$, where H is the subgroup of which preserves rulings. Certainly, $G_{40} \cap H$ lies inside the group $\mathrm{Norm}_{H} (D_{10})$. The normalizer of $D_{10}$ in is equal to $D_{20}$ and generated by $D_{10}$ and the involution . Thus $G_{40}\cap H$ lies inside the group $\langle D_{10}, a, b\rangle$, with

One can easily check that only ab normalizes the group $G_{20}$ and . $\square$

This proof was communicated to me by Artem Avilov and I thank him for thus completing the proof of Theorem 1.1.