Gbirational superrigidity of Del Pezzo surfaces of degree 2 and 3
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Abstract
Any minimal Del Pezzo Gsurface S of degree smaller than 3 is Gbirationally rigid. We classify those which are Gbirationally superrigid, and for those which fail to be so, we describe the equations of a set of generators for the infinite group \(\mathrm{Bir}^G(S)\) of Gbirational automorphisms.
Keywords
Birational rigidity Del Pezzo surfaces Cubic surfaces Bertini involution Geiser involutionMathematics Subject Classification
14E071 Introduction
The group of birational automorphisms of \(\mathbb {P}^2(\mathbb {C})\) is classically known as Cremona group, denoted \(\mathrm{Cr}_2(\mathbb {C})\). The classification of its finite subgroups up to conjugacy rose the interest of many classical authors and it has been completed in [5]. In this paper, we refine the description of the conjugacy class of some special finite subgroups.
The key reduction step in the classification consists in associating to any finite subgroup G of \(\mathrm{Cr}_2(\mathbb {C})\) a group of automorphisms of a rational surface, isomorphic to G, see [5, Section 3.4]. Via a Gequivariant version of Mori theory, one can suppose that the surface is minimal with respect to the Gaction. Here, we concentrate our attention to those finite subgroups of \(\mathrm{Cr}_2(\mathbb {C})\) which act minimally by automorphisms on Del Pezzo surfaces S of degree 2 and 3. In particular, when the normaliser of G is not generated by automorphisms of the Del Pezzo surface, i.e., the surface S is not Gbirationally superrigid, we describe explicitly the generators of the normaliser.
The main properties investigated in this paper are described in the following definitions.
Definition 1.1
Let \((S, \rho )\) be a minimal Del Pezzo Gsurface. Then \((S, \rho )\) is Gbirationally rigid if there is no Gbirational map from S to any other minimal Gsurface. Equivalently, if \(S'\) is any minimal Gsurface and \(\varphi :S \dashrightarrow S'\) is any Gbirational map, then S is Gisomorphic to \(S'\), not necessarily via \(\varphi \). More precisely, there exists a Gbirational automorphism \(\sigma :S \dashrightarrow S\) such that Open image in new window is a Gbiregular map.
Definition 1.2
The minimal Del Pezzo Gsurface \((S, \rho )\) is Gbirationally superrigid if it is Gbirationally rigid and in addition, in the notation of Definition 1.1, any Gbirational map \(\varphi :S \dashrightarrow S'\) is biregular. In particular, the group of Gbiregular automorphisms coincides with the group of Gbirational automorphisms, i.e., \(\mathrm{Aut}^G(S)=\mathrm{Bir}^G(S)\).
A classical theorem by Segre [9] and Manin [8] establishes that nonsingular cubic surfaces of Picard number 1 defined over a nonalgebraically closed field are birationally rigid. In analogy with this arithmetic case, Dolgachev and Iskoviskikh showed in [5, Section 7.3] that minimal Del Pezzo Gsurfaces of degree smaller than 3 are Gbirationally rigid. In this paper we determine which minimal Del Pezzo Gsurfaces of degree 2 and 3 are Gbirationally superrigid. When the Gsurface is not Gbirationally superrigid, we describe the generators of the group of Gbirational automorphisms \(\mathrm{Bir}^G(S)\), or equivalently the normaliser of the corresponding subgroup G in \(\mathrm{Cr}_2(\mathbb {C})\). Here, we collect our main results, adopting the notation of [5]:
Theorem 1.3
Let G be a noncyclic group and S be a minimal Del Pezzo Gsurface of degree 3. Then S is Gbirationally superrigid, unless G is isomorphic to the symmetric group \(S_3\) and S is not the Fermat cubic surface.
 (i)
\(S_3\) if S is of type V, VIII;
 (ii)
Open image in new window if S is of type VI;
 (iii)
Open image in new window if S is of type III, IV.
For the proof, see Sect. 4.1.
Theorem 1.4
 (i)
a cyclic group of order 3 of type \(3A_2\). The group \(\mathrm{Bir}^G(S)\) is (infinitely) generated by the Geiser involutions whose base points lie on the unique Gfixed nonsingular cubic curve and by a subgroup of \(\mathrm{Aut}(S)\) isomorphic to Open image in new window , if S is the Fermat cubic surface, or by \(\mathrm{Aut}(S)\) itself otherwise.
 (ii)
a cyclic group of order 6 of type \(E_6(a_2)\). The group \(\mathrm{Bir}^G(S)\) is (infinitely) generated by three Geiser involutions, the Bertini involutions whose base points lie on a Ginvariant nonsingular cubic curve C and by a subgroup of \(\mathrm{Aut}(S)\) isomorphic to Open image in new window , if S is the Fermat cubic surface, or by \(\mathrm{Aut}(S)\) itself otherwise.
 (iii)
a cyclic group of order 9 of type \(E_6(a_1)\). The group \(\mathrm{Bir}^G(S)\) is finitely generated by three Geiser involutions whose base loci are coplanar and by a subgroup of \(\mathrm{Aut}(S)\) isomorphic to the dihedral group \(D_{18}\).
 (iv)
a cyclic group of order 12 of type \(E_6\). The group \(\mathrm{Bir}^G(S)\) is finitely generated by G, by a Bertini involution and by a Geiser involution whose base loci are aligned.
For the proof, see Sect. 4.2.
Theorem 1.5
Let G be a noncyclic group and S be a minimal Del Pezzo Gsurface of degree 2. Then S is Gbirationally superrigid.
For the proof, see Sect. 5.1.
Theorem 1.6
 (i)
a group of order 2 of type \(A_1^7\);
 (ii)
a group of order 6 of types \(E_7(a_4), A_5 + A_1, D_6(a_2)+A_1\);
 (iii)
a group of order 14 of type \(E_7(a_1)\);
 (iv)
a group of order 18 of type \(E_7\).
 (v)
a cyclic group of order 4 of type \(2A_3+A_1\). The group \(\mathrm{Bir}^G(S)\) is generated by infinitely many Bertini involutions whose base loci lie in the unique Gfixed nonsingular curve of genus one and by a subgroup of \(\mathrm{Aut}(S)\) isomorphic to Open image in new window , if S is of type II, or by \(\mathrm{Aut}(S)\) itself otherwise.
 (vi)
a cyclic group of order 12 of type \(E_7(a_2)\). The group \(\mathrm{Bir}^G(S)\) is generated by two Bertini involutions and by a subgroup of \(\mathrm{Aut}(S)\) isomorphic to Open image in new window .
For the proof, see Sect. 5.2.
Corollary 1.7
Let G be a cyclic group and S be a minimal Del Pezzo Gsurface of degree smaller than 3. Then, S is Gbirationally superrigid if and only if the group \(\mathrm{Bir}^G(S)\) of Gbirational automorphisms is finite.
Proof
It is an immediate corollary of Theorems 1.4 and 1.6. In particular, see Lemmas 4.9, 4.11 and 5.3. The authors are not aware of a proof that does not rely on the above classification.\(\square \)
Minimal Del Pezzo Gsurfaces of degree 3 which are not Gbirationally superrigid
Type of G  G  Type of S  Equation of S  \(\mathrm{Aut}^G(S)\)  Geiser invol.  Bertini invol. 

\(3A_2\)  3  I  \(t_0^3 + t_1^3 + t_2^3 + t_3^3\)  \(\infty \)  0  
\(3A_2\)  3  III  \(t_0^3 + t_1^3 + t_2^3 + t_3^3 + 6at_1t_2t_3\)  \(\infty \)  0  
\(20a^3+8a^6 =1\)  
\(3A_2\)  3  IV  \(t_0^3 + t_1^3 + t_2^3 + t_3^3 + 6at_1t_2t_3\)  \(\infty \)  0  
\(20a^3+8a^6 \ne 1, \> 8a^3 \ne 1 \)  
\(aa^4\ne 1 \)  
\(E_6(a_2)\)  6  I  \(t_0^3 + t_1^3 + t_2^3 + t_3^3\)  3  \(\infty \)  
\(E_6(a_2)\)  6  III  \(t_0^3 + t_1^3 + t_2^3 + t_3^3 +6at_1t_2t_3\)  3  \(\infty \)  
\(20a^3+8a^6 =1\)  
\(E_6(a_2)\)  6  IV  \(t_0^3 + t_1^3 + t_2^3 + t_3^3 +6at_1t_2t_3\)  3  \(\infty \)  
\(20a^3+8a^6 \ne 1, \> 8a^3 \ne 1 \)  
\(aa^4\ne 1 \)  
\(E_6(a_1)\)  9  I  \(t^2_3t_1 + t_1^2t_2 + t_2^2t_3 + t_0^3\)  \(D_{18}\)  3  0 
\(E_6\)  12  III  \(t^2_3t_1 + t_2^2t_3 + t_0^3 + t_1^3\)  12  1  1 
\(S_3\)  VI  2  0  
\(a \ne 0\)  
\(S_3\)  III–IV  \(t_0^3 + t_1^3 + t_2^3 + t_3^3 + t_0t_1(at_2+bt_3)\)  3  0  
V–VIII  \( a^3\ne b^3\ne 0\),  \(S_3\) 
Minimal Del Pezzo Gsurfaces of degree 2 which are not Gbirationally superrigid
Type of G  G  Type of S  Equation of S  \(\mathrm{Aut}^G(S)\)  Bertini invol. 

\(2A_3 + A_1\)  4  II  \(t_3^2 + t_2^4 + t_0^4 + t_1^4\)  \(\infty \)  
\(2A_3 + A_1\)  4  III  \(t_3^2 + t_2^4 + t_0^4 + 2\sqrt{3}it_0^2t_1^2 + t_1^4\)  \(\infty \)  
\(2A_3 + A_1\)  4  V  \(t_3^2 + t_2^4 + t_0^4 + at_0^2t_1^2 + t_1^4\)  \(\infty \)  
\( a^2 \ne 0,12,4,36\)  
\(E_7(a_2)\)  12  III  \(t_3^2 + t_0^4 + t_1^4 + t_0t_2^3\)  2 
The structure of the paper is as follows: in Sect. 3 we rewrite in full detail the proof of the Gequivariant version of the abovementioned Segre–Manin theorem, see Theorem 3.1. Note that the statement is essentially proved in [5, Corollary 7.11]. Building on this result, we classify the minimal Del Pezzo Gsurfaces of degree 3 and 2 which are not Gbirationally superrigid in Sects. 4 and 5 respectively.
2 Preliminaries
A pair Open image in new window is canonical if Open image in new window for any exceptional divisor E and for any \(f:\widetilde{S} \rightarrow S\) birational morphism. A pair Open image in new window is called log Calabi–Yau if Open image in new window .
 A Del Pezzo surface S of degree 1 is a nonsingular hypersurface of degree 6 in the weighted projective space \(\mathbb {P}(1,1,2,3)\), embedded via the third pluricanonical linear system Open image in new window . Via the linear system Open image in new window , S can be realised as a double cover of the singular quadric \(\mathbb {P}(1,1,2)\) branched along a nonsingular sextic curve. In particular, since the double cover is canonical, its deck transformation \(\tau \) is a central element in the group of automorphisms \(\mathrm{Aut}(S)\), see also [5, Section 6.7].
 A Del Pezzo surface S of degree 2 is a nonsingular hypersurface of degree 4 in the weighted projective space \(\mathbb {P}(1,1,1,2)\), embedded via the second pluricanonical linear system Open image in new window . Via the canonical map, S can be realised as a double cover of \(\mathbb {P}^2\) branched along a nonsingular quartic curve. In particular, since the double cover is canonical, its deck transformation \(\tau \) is a central element in the group of automorphisms \(\mathrm{Aut}(S)\), see also [5, Section 6.6].
 A Del Pezzo surface S of degree 3 is a nonsingular hypersurface of degree 3 in the projective space \(\mathbb {P}^3\), embedded via the anticanonical linear system Open image in new window .
3 Gequivariant Segre–Manin theorem
In this section we present the proof, essentially due to Dolgachev and Iskoviskikh, of the following Gequivariant version of a classical arithmetic theorem by Segre [9] and Manin [8] (cf. also [2, Section 1.5]).
Theorem 3.1
(Gequivariant Segre–Manin theorem [5, Section 7.3]) Every minimal Del Pezzo Gsurface S of degree \(d \leqslant 3\) is Gbirationally rigid.
The main ingredients of the proof are Noether–Fano inequalities, which in modern language recast the failure of birational superrigidity in terms of the existence of a noncanonical log Calabi–Yau pair.
Theorem 3.2

if \(S'\) is a Del Pezzo Gsurface, then Open image in new window is the strict transform via \(\varphi ^{1}\) of the linear system H, where H is a very ample multiple of \(K_{S'}\);

if Open image in new window is a Gconic bundle and H is a very ample Ginvariant divisor of C, then Open image in new window is the strict transform via \(\varphi ^{1}\) of the linear system \(\psi ^*(H)\).
 (i)
If \(S'\) is a Del Pezzo surface and Open image in new window is canonical, then \(\varphi \) is biregular.
 (ii)
If \(S'\) is a conic bundle, then Open image in new window is not canonical.
Proof of Theorem 3.1
Let \(\varphi :S \dashrightarrow S'\) be a Gbirational nonbiregular map to a minimal Gsurface \(S'\). In order to prove that S is Gbirationally rigid we need to exhibit a Gbirational map \(\sigma :S\dashrightarrow S\) such that Open image in new window is a Gbiregular map.

(log Calabi–Yau) Open image in new window ;

(not canonical singularities) the pair Open image in new window is not canonical.
Step 3 (Geiser and Bertini involution). By hypothesis, the degree of S is at most 3 and we are left with few possibilities:
In all the cases, the Noether–Fano inequalities (cf. Lemma 3.4) force Open image in new window with \(k_1 <n\).
Step 4 (inductive step). By Theorem 3.2, either Open image in new window is Gbiregular or the pair Open image in new window is not canonical. In the latter case, we can repeat the above arguments and construct a sequence of Bertini or Geiser Ginvolutions \(\sigma _1, \ldots , \sigma _s\) on S such that Open image in new window is nonbiregular and again, by Theorem 3.2, the mobile pair Open image in new window , with \(k_s < k_{s1}\), is not canonical. However, if \(s>n\), then the mobile linear system Open image in new window would not be \(\mathbb {Q}\)linearly equivalent to an effective divisor, which is a contradiction. Hence, there exists an integer s such that \(\varphi _s\) is Gbiregular. We conclude that S is Gbirationally rigid.\(\square \)
Corollary 3.3
Let S be a minimal Del Pezzo Gsurface of degree 3 (resp. 2). Then, every Gbirational map is a composition of a Gbiregular map, Geiser and/or Bertini involutions (resp. a Gbiregular map and Bertini involutions).
We now prove the lemmas used in the proof of Theorem 3.1.
Lemma 3.4
Proof
Lemma 3.5
Let S be a minimal Del Pezzo Gsurface of degree d. If \(\varphi :S \dashrightarrow S'\) is a nonbiregular Gbirational map, then the Gorbit O defined in Lemma 3.4 has length O strictly smaller than d.
Proof
Remark 3.6
Lemma 3.5 implies immediately that any Del Pezzo Gsurface of degree 1 is Gbirationally superrigid, see also [5, Corollary 7.11].
Lemma 3.7
Let S be a minimal Del Pezzo Gsurface of degree d and Open image in new window be a mobile linear system on S such that Open image in new window . Let Open image in new window be a Gequivariant blowup of S at a Gorbit O defined in Lemma 3.4. Then, \(S'\) is a Del Pezzo surface, i.e. \(K_{S'}\) is ample.
Proof

\((K_{S'}. C)>0\) for any curve \(C \subset S'\);

\(K_{S'}^2>0\).
4 Gbirational superrigidity of cubic surfaces
Let G be a finite group of automorphisms acting effectively on a minimal Del Pezzo surface of degree 3. It is well known that any Del Pezzo surface of degree 3 is a nonsingular cubic surface embedded in Open image in new window via the canonical embedding and every automorphism of S lifts to an automorphism of \(\mathbb {P}^3\). The 4dimensional vector space V is a Grepresentation, unique up to scaling by a character of G.
The content of this section is the proof of Theorem 1.4. Proposition 4.1 is one of the main ingredients of the proof.
Proposition 4.1
A minimal Del Pezzo Gsurface S of degree 3 is not Gbirationally superrigid if and only if it admits either Gequivariant Geiser or Bertini involutions. This is equivalent to the existence on S of a Gfixed point, not lying on a line, or a Gorbit of length 2, not lying on a line or a conic in S and such that no tangent space of one point contains the other.
Proof
This is a corollary of Theorem 3.1 and Corollary 3.3. The second statement follows from Lemma 4.2.\(\square \)
Lemma 4.2
 (i)
A point p in S is the base locus of a Geiser involution if and only if no line contained in S passes through p.
 (ii)The points \(\{p_1, p_2\}\) in S are the base locus of a Bertini involution if and only if
 (a)
there is no line in S passing through \(p_1\) or \(p_2\);
 (b)
there is no conic contained in S passing through \(p_1\) and \(p_2\);
 (c)
\(p_i\) is not contained in the tangent space of S at Open image in new window , \(i \ne j\).
 (a)
Proof

no three of them lie on a line;

no six of them lie on a conic;

no eight of them lie on a nodal or cuspidal cubic with one of them at the singular point.

p does not lie in the exceptional locus of g;

the strict transform \(\widetilde{l}\) of the line l passing through \(q_i\) and Open image in new window does not contain p;

the strict transform \(\tilde{c}\) of a conic c passing through five of the points \(q_i\) does not contain p.
In the following Lemma 4.3, we show that orbits of length 2 lie on invariant lines passing through a fixed point for the action of G on S.
Lemma 4.3
Let S be a minimal cubic Gsurface admitting an orbit of length 2, then G fixes a point in S.
Proof
Denote by \(q_1\) and \(q_2\) the points in the orbit of length 2 and by \(l_{q_1q_2}\) the line passing through those points in \(\mathbb {P}^3\). Note that the line \(l_{q_1q_2}\) is Ginvariant and it is not contained in S. Differently, it could be contracted, violating the minimality of G.
Moreover, the line \(l_{q_1q_2}\) intersects S with multiplicity 1 at \(q_1\) and \(q_2\). Otherwise, if the multiplicity at one of the two points is \(\geqslant 2\), then so it is at the other point due to the group action. However, this is a contradiction, since \(l_{q_1q_2}\) would intersect S with multiplicity at least 4, while S has degree 3. This implies that the invariant line \(l_{q_1q_2}\) intersects S in a third point, thus fixed by the action of G.\(\square \)
Remark 4.4
Notice that if \(\{p_1, p_2 \}\) is a Gorbit then condition (c) in Lemma 4.2 always holds, since otherwise the line between \(p_1\) and \(p_2\) is bitangent to S.
4.1 Gbirational superrigidity for noncyclic groups
Type\(a=b=0\). The surface \(S_{00}\) is the Fermat cubic surface. The points \(p_i\) are Eckardt points. No orbit can be the base locus of a Geiser or a Bertini involution. By Theorem 3.1, we conclude that \(S_{00}\) is Gbirationally superrigid.

Open image in new window . The conic Open image in new window passes through the lengthtwo orbit \(\{p_1, p_2\}\). By Lemma 4.2 and Theorem 3.1, we conclude that \(S_{ab}\) is Gbirationally superrigid.

\(G \simeq S_3\). The fixed points \(p_1\) and \(p_2\) are not Eckardt points. Therefore, \(S_{ab}\) is not Gbirationally superrigid and the group \(\mathrm{Bir}^G(S_{ab})\) is generated by \(\mathrm{Aut}(S_{ab})\) and the two Geiser involutions with base locus \(p_1\) and \(p_2\) respectively. The equations of these Geiser involutions and the infinitude of the group \(\mathrm{Bir}^G(S_{ab})\) for the very general surface \(S_{ab}\) are discussed in the following paragraphs.
Lemma 4.5
The normaliser of G in \(\mathrm{Aut}(S_{ab})\), denoted \(N_{\mathrm{Aut}(S_{ab})}(G)\), is isomorphic to Open image in new window , if \(S_{ab}\) is of type III or IV, or to \(S_3\), if \(S_{ab}\) is of type V or VIII.
Proof
 Type III.
Open image in new window , where \(H_3(3)\) is the Heisenberg group of unipotent Open image in new window matrices over the finite field \(\mathbb {F}_3\), see Sect. 4.2, Type \(E_6\), for explicit generators. The generator of 4 conjugates the nonconjugate subgroups of type \(S_3\) in Open image in new window , see [5, Theorem 6.14, Type III]. We conclude that \(N_{H_3(3)\rtimes 4}(S_3)=N_{H_3(3)\rtimes 2}(S_3)\).
 Type IV.
Open image in new window . It contains two nonconjugate subgroups isomorphic to \(S_3\), normalized by the subgroups isomorphic to Open image in new window obtained from the previous ones by adding the central element, see [5, Theorem 6.14, Type III].
 Type V.
\(\mathrm{Aut}(S_{ab}) \simeq S_4\). Any subgroup isomorphic to \(S_3\) is a nonnormal maximal subgroup of \(S_4\).
 Type VIII.
\(\mathrm{Aut}(S_{ab}) \simeq S_3\). \(\square \)
Let G be again the group of biregular automorphisms acting minimally on \(S_{ab}\) with a fixed point \(p_0\) and isomorphic to \(S_3\). The following lemma establishes the infinitude of the group of Gbirational automorphisms \(\mathrm{Bir}^G(S_{ab})\) for the very general surface \(S_{ab}\).
Let Open image in new window be the hypersurface given by the equation Open image in new window , see equation (1), and Open image in new window be the divisor Open image in new window in Open image in new window (equivalently Open image in new window ). Denote by Open image in new window the family of cubic surfaces \(S_{ab}\) and by Open image in new window that of surfaces \(S_{ab}\) with the property that \(a=b\) (equivalently \(a=\epsilon ^i_3 b\)). The Geiser involutions \(\varphi _{p_i}\) on \(S_{ab}\) glue together to birational involutions of Open image in new window and Open image in new window respectively, as their equations are polynomial in (a, b).
Lemma 4.6
The group \(\mathrm{Bir}^G(S_{ab})\) is not a finite group for the very general surface \(S_{ab}\) in Open image in new window and in Open image in new window .
Proof

for any \(p\in S_{ab}\) the point \(\varphi _{p_i}(p)\) is aligned with \(p_i\) and p;
 the involutions \(\varphi _{p_1}\) and \(\varphi _{p_2}\) fix the pencil of cubic curves
The same proof holds for Open image in new window since \(S_{11} \subset \mathscr {S}'\).\(\square \)
Open Question
Is the group \(\mathrm{Bir}^G(S_{ab})\) not finite for any \((a,b)\ne (0,0)\)?
4.2 Gbirational superrigidity for cyclic groups
In this section, we discuss the birational superrigidity of minimal cubic surfaces endowed with the action of a finite cyclic group G. Dolgachev and Iskovskikh classified these groups in [5]. For the convenience of the reader, we recall their result.
Here and in the following we denote by \(\epsilon _n\) a primitive nth root of unity.
Proposition 4.7
 (i)\(3A_2\), order 3, Open image in new window ,$$\begin{aligned} F=t^3_0+t^3_1+t^3_2+t^3_3+\alpha t_0 t_1 t_2; \end{aligned}$$
 (ii)\(E_6(a_2)\), order 6, Open image in new window ,$$\begin{aligned} F=t^3_0+t^3_1+t^3_3+t_2^2(\alpha t_0+t_1); \end{aligned}$$
 (iii)\(A_5 + A_1\), order 6, Open image in new window ,$$\begin{aligned} F=t^2_3t_1+t_0^3+t_1^3+t_2^3+\lambda t_0 t_1 t_2; \end{aligned}$$
 (iv)\(E_6(a_1)\), order 9, Open image in new window ,$$\begin{aligned} F=t^2_3t_1+t^2_1t_2+t^2_2t_3+t^3_0; \end{aligned}$$
 (v)\(E_6\), order 12, Open image in new window ,$$\begin{aligned} F=t^2_3t_1+t^2_2t_3+t^3_0+t^3_1. \end{aligned}$$
We proceed with an analysis case by case.
The normaliser \(N_{\mathrm{Aut}(S)}(G)\) of G in \(\mathrm{Aut}(S)\) is the group \(\mathrm{Aut}^G(S)\) of biregular Gautomorphisms. If C is equianharmonic, i.e., it has an automorphism of order 6, then S is the Fermat cubic surface and Open image in new window (cf. Sect. 4.1): the normaliser \(N_{\mathrm{Aut}(S)}(G)\) is isomorphic to Open image in new window . Otherwise, g is a central element of \(\mathrm{Aut}(S)\), which is isomorphic to Open image in new window or \(H_3(3){\rtimes }2\), where \(H_3(3)\) is the Heisenberg group of unipotent Open image in new window matrices over the finite field \(\mathbb {F}_3\) (cubic surfaces of type III or IV; see [5, Table 4]). Then, the group \(\mathrm{Aut}^G(S)\) coincides with \(\mathrm{Aut}(S)\).
We conclude that S is not Gbirationally superrigid and that the group \(\mathrm{Bir}^G(S)\) is generated by Gbiregular automorphisms of S, three Geiser involutions with base loci contained in \(l_2\cap S\), and infinitely many Bertini involutions, whose base locus points lie on the nonsingular cubic curve given by \(t_3=0\). We complete the list of generators, computing the normaliser \(N_{\mathrm{Aut}(S)}(G)\) of G in \(\mathrm{Aut}(S)\).
Lemma 4.8
Proof
Note that S is a cyclic cover of degree 3 of \(\mathbb {P}^2\) branched along a nonsingular cubic curve C, and G is generated by \(g_1g_2\), where \(g_1\) is the deck transformation of the cover and \(g_2\) is the lift of the involution on C.
If S is not the Fermat cubic surface, then S is a surface of type III or IV [5, Table 4] and \(\mathrm{Aut}(S)\) is a central extension of \(\mathrm{Aut}(C)\) via \(g_1\). Therefore, \(N_{\mathrm{Aut}(S)}(G)\) is a central extension of \(N_{\mathrm{Aut}(C)}(g_2)\) via \(g_1\), but since \(g_2\) is central in \(\mathrm{Aut}(C)\), we conclude that \(N_{\mathrm{Aut}(S)}(G)=\mathrm{Aut}(S)\), or equivalently that Open image in new window .\(\square \)
Types\(A_5 + A_1\), \(E_6(a_1)\)and\(E_6\). In the last few cases, i.e., \(A_5 + A_1\), \(E_6(a_1)\) and \(E_6\), the group G acts on \(\mathbb {P}^3\) by means of four distinct characters. In particular, the points Open image in new window and Open image in new window are the only fixed points in \(\mathbb {P}^3\). The only invariant lines are those interpolating pairs of points Open image in new window , where \(i \ne j\), shortly written Open image in new window . Note that eventual orbits of length 2 lie on Open image in new window .
Invariant lines Open image in new window  Orbits in Open image in new window  

\(t_0^3+t_1^3=0\)  orbit of length 3  
\(t_0^3+t^3_2=0\)  orbit of length 3  
\(t^3_0=0\)  fixed Eckardt point \(p_3\)  
\(t^3_1+t_2^3=0\)  orbit of length 3  
\(t_3^2t_1+t^3_1=0\)  fixed Eckardt point \(p_3\) and orbit of length 2 given by: Open image in new window Open image in new window  
\(t^3_2=0\)  fixed Eckardt point \(p_3\) 
Lemma 4.9
The group \(\mathrm{Bir}^G(S)\) is not a finite group.
Proof

for any \(p\in S\) the point \(\varphi _{p_i}(p)\) is aligned with \(p_i\) and p;
 the involutions \(\varphi _{p_1}\) and \(\varphi _{p_2}\) fix the pencil of cubic curves
Lemma 4.10
The normaliser of G in \(\mathrm{Aut}(S)\), denoted \(N_{\mathrm{Aut}(S)}(G)\), is isomorphic to the dihedral group \(D_{18}\).
Proof
Recall that the automorphism group of a Fermat cubic is the group Open image in new window . Let \(G'\) be the image of G in \(S_4\), generated by the permutation (234), and \(K{:=}G\cap 3^3\), generated by Open image in new window . The image of \(N_{\mathrm{Aut}(S)}(G)\) is contained in \(N_{S_4}((234))\), which is generated by (234) and (23) and isomorphic to \(S_3\). Therefore, \(N_{\mathrm{Aut}(S)}(G)\) is a subgroup of Open image in new window and admits a subgroup isomorphic to \(S_3\).
Fixed points  \(T_{p_i}S\)  \(T_{p_i}S \cap S\)  Eckardt point 

\(p_2\)  \(t_3=0\)  \(t^3_0+t^3_1=0\)  yes 
\(p_3\)  \(t_1=0\)  \(t^2_2 t_3+t^3_0=0\)  no 
Invariant lines Open image in new window  orbits in Open image in new window  

\(t_0^3+t_1^3=0\)  orbit of length 3  
\(t_0^3=0\)  fixed Eckardt point \(p_2\)  
\(t^3_0=0\)  fixed point \(p_3\)  
\(t^3_1=0\)  fixed Eckardt point \(p_2\)  
\(t_3^2t_1+t^3_1=0\)  fixed point \(p_3\) and orbit of length 2 given by: Open image in new window Open image in new window  
\(t^2_2t_3=0\)  fixed Eckardt point \(p_2\) and fixed point \(p_3\) 
Lemma 4.11
The group \(\mathrm{Bir}^G(S)\) is not a finite group.
Proof

for any \(p\in S\) the point \(\varphi _{p_3}(p)\) is aligned with \(p_3\) and p;

for any \(p \ne p_3\), the points \(\varphi _{q_1 q_2}(p)\) and p belong to a conic contained in the plane \(\Pi _{q_1 q_2 p}\), spanned by \(q_1,q_2\) and p, and tangent to \(S \cap \Pi _{q_1 q_2 p}\) at \(q_1\) and \(q_2\);
 the involutions \(\varphi _{p_3}\) and \(\varphi _{q_1 q_2}\) fix the pencil of cubic curves
Lemma 4.12
The group G is selfnormalising in \(\mathrm{Aut}(S)\), i.e., the normaliser of G in \(\mathrm{Aut}(S)\) is G itself.
Proof
The results of this section are summarised in Theorem 1.4.
5 Gbirational superrigidity of Del Pezzo surfaces of degree 2
As in the previous section, the proof of the Segre–Manin theorem (Theorem 3.1) implies that a minimal Del Pezzo Gsurface of degree 2 is not Gbirationally superrigid if and only if it admits a Gequivariant Bertini involution.
Lemma 5.1
Let S be a Del Pezzo surface of degree 2. Then, a point p is the base locus of a Bertini involution if and only if p lies neither on a \((1)\)curve nor on the ramification locus of the double cover \(\nu :S \rightarrow \mathbb {P}^2\).
Proof
The proof is analogous to that of Lemma 4.2. Recall that a Del Pezzo surface of degree 2 is a blowup of \(\mathbb {P}^2\) at points \(q_1, \ldots , q_7\) in general position, see [1, Exercise IV.8.(10).(a)]. We need to check that the blowup \(\widetilde{S}\) of S at p is a Del Pezzo surface, or equivalently that the seven points \(q_i\) and the image of p via the blowdown are in general position. We prove that if this is not the case, then p lies on a \((1)\)curve or on the ramification locus. Indeed, note that the strict transform of a line passing through two of the points \(q_i\) or that of a conic through five of them or that of a singular cubic curve through seven of them, with one of the \(q_i\) at the singular point, is a \((1)\)curve. Similarly, the strict transform of a singular cubic curve through all of the \(q_i\), singular at p, is an anticanonical divisor, hence the pullback of a line via \(\nu \). Since this curve is singular at p, then p lies on the ramification locus.
Conversely, if p lies on a \((1)\)curve, the canonical class of the blowup \(S'\) of S at p has trivial intersection with the strict transform of the line, hence \(K_{S'}\) is not ample. On the other hand, if p lies on the ramification locus, then the preimage of the tangent line to the branch locus via \(\nu \) is either an irreducible anticanonical divisor, singular only at p, i.e., the strict transform of a singular cubic passing through \(q_i\), or the union of two \((1)\)curves, if the line is bitangent to the branch locus.\(\square \)
Our strategy to identify birational superrigid Gsurfaces will then consist in finding the fixed points of the given Gaction and checking if these points lie on the ramification locus or on \((1)\)curves. Recall that \((1)\)curves on Del Pezzo surfaces of degree 2 are contained in the preimage of a bitangent line of the branched quartic in \(\mathbb {P}^2\).
5.1 Gbirational superrigidity for noncyclic groups
The minimal noncyclic groups G acting on S and fixing a point have been classified by Dolgachev and Duncan, the possible fixed points lie either on the ramification curve or they are the intersection of four \((1)\)curves, see cases 2A and 2B of [4, Theorem 1.1]. Therefore, S is Gbirationally superrigid by Theorem 3.1 and Lemma 5.1. This concludes the proof of Theorem 1.5. It remains to analyse cyclic groups.
5.2 Gbirational superrigidity for cyclic groups
We describe the fixed locus of minimal cyclic groups G according to Dolgachev and Iskoviskikh classification. As before, we stick to their notation. Recall in particular that \(\epsilon _n\) is a primitive nth root of the unit and \(F_i\) is a polynomial of degree i.
Proposition 5.2
 (i)
 (ii)
 (iii)
 (iv)\(A_5+A_1\), order 6, Open image in new window ,$$\begin{aligned} F = t_3^2 +t_2^3 t_0 + t_0^4 +t_1^4 + at_0^2 t_1^2; \end{aligned}$$
 (v)
 (vi)\(E_7(a_2)\), order 12, Open image in new window ,$$\begin{aligned} F = t_3^2 + t_0^4 + t_1^4 + t_0t_2^3; \end{aligned}$$
 (vii)\(E_7(a_1)\), order 14, Open image in new window ,$$\begin{aligned} F = t_3^2 + t_0^3t_1 + t_1^3 t_2 + t_2^3t_0 ; \end{aligned}$$
 (viii)\(E_7\), order 18, Open image in new window ,$$\begin{aligned} F = t_3^2 + t_0^4 + t_0t_1^3 + t_2^3t_1. \end{aligned}$$
We proceed with an analysis case by case.
Type\(A^7_1\). The generator g is the standard Geiser involution of the surface S leaving the ramification curve Open image in new window fixed. Hence, the surface is Gbirationally superrigid.
 if \(a = 0\), then Open image in new window (cf. [5, Theorem 6.17, Type II]), where 2 is generated by Open image in new window , the symmetric group \(S_3\) is generated by the transpositions and \(4^2\) is generated by subject to the following relations:In particular, the group G is generated by \(g=\gamma g_2\). Notice that \(\langle g_2 \rangle \) is central in Open image in new window and therefore it is central in Open image in new window . Since we conclude that Open image in new window .$$\begin{aligned} \tau g_2 \tau = g_2, \quad \tau g_1 \tau = g_1^{1}g_2^{1} = g_1^3g_2^3. \end{aligned}$$
 if \(a = \pm 2\sqrt{3}\,i \), then Open image in new window (cf. [5, Theorem 6.17, Type III]), where 2 is generated by Open image in new window and \(4A_4\) is a central extension of the alternating group \(A_4\) generated by Since c is central in \(\mathrm{Aut}(S)\) and \(g = \gamma c\), we conclude that \(N_{\mathrm{Aut}(S)}(G) = \mathrm{Aut}(S)\).

if \(a \ne 0, \pm 2\sqrt{3}\,i \), then Open image in new window , where \(AS_{16}\) is a nonabelian group of order 16 isomorphic to Open image in new window (cf. [5, Tables 1 & 6]). The generators of \(\mathrm{Aut}(S)\) coincide with that of the previous case with the exception of the generator \(g_3\). Hence, as in the previous case, g is a central element and \(N_{\mathrm{Aut}(S)}(G) = \mathrm{Aut}(S)\).
Lemma 5.3
The group \(\mathrm{Bir}^G(S)\) is not a finite group.
Proof
The cases above yield the proof of Theorem 1.6.
Notes
Acknowledgements
We would like to express our gratitude to Ivan Cheltsov for suggesting the problem at the summer school on Rationality, Stable Rationality and Birationally Rigidity of Complex Algebraic Varieties held in Udine in September 2017. Most of the lemmas in Sect. 3 were solved during the exercise classes in Udine. We would like to thank Fabio Bernasconi, Damián Gvirtz, Costya Shramov and Christian Urech for useful discussions. We would like to thank also our advisors Vladimir Guletskiĭ and Paolo Cascini. We are grateful to the anonymous referee for his/her helpful suggestions.
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