G-birational superrigidity of Del Pezzo surfaces of degree 2 and 3

Any minimal Del Pezzo G-surface S of degree smaller than 3 is G-birationally rigid. We classify those which are G-birationally superrigid, and for those which fail to be so, we describe the equations of a set of generators for the infinite group BirG(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Bir}^G(S)$$\end{document} of G-birational automorphisms.


Introduction
The group of birational automorphisms of P 2 (C) is classically known as Cremona group, denoted Cr 2 (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 Cr 2 (C) a group of automorphisms of a rational surface, isomorphic to G, see [5,Section 3.4]. Via a G-equivariant version of Mori theory, one can suppose that the surface is minimal with respect to the G-action. Here, we concentrate our attention to those finite subgroups of Cr 2 (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 G-birationally superrigid, we describe explicitly the generators of the normaliser.
In order to formulate our main results, we recall the definition of minimal G-surface. Let (S, ρ) be a G-surface, i.e., a nonsingular surface S defined over C, endowed with the action of a finite group of automorphisms ρ : G → Aut(S). Given two G-surfaces (S, ρ) and (S , ρ ), we say that a rational map ϕ : S S is G-rational if for any g ∈ G the following diagram commutes: Then, a minimal G-surface S is a G-surface with the property that any G-birational morphism S → S is an isomorphism. Equivalently, it is the output of a G-equivariant minimal model program, and as in the non-equivariant case, if S is rational, it is either a Del Pezzo surface with Pic G (S) Z, i.e., −K S is ample, or a conic bundle with Pic G (S) Z 2 (cf. [5,Theorem 3.8]).
The main properties investigated in this paper are described in the following definitions. Definition 1.1 Let (S, ρ) be a minimal Del Pezzo G-surface. Then (S, ρ) is Gbirationally rigid if there is no G-birational map from S to any other minimal G-surface. Equivalently, if S is any minimal G-surface and ϕ : S S is any G-birational map, then S is G-isomorphic to S , not necessarily via ϕ. More precisely, there exists a G-birational automorphism σ : S S such that ϕ • σ is a G-biregular map.

Definition 1.2 The minimal Del Pezzo G-surface (S, ρ) is G-birationally superrigid
if it is G-birationally rigid and in addition, in the notation of Definition 1.1, any G-birational map ϕ : S S is biregular. In particular, the group of Gbiregular automorphisms coincides with the group of G-birational automorphisms, i.e., Aut G (S) = Bir G (S).
A classical theorem by Segre [9] and Manin [8] establishes that nonsingular cubic surfaces of Picard number 1 defined over a non-algebraically closed field are birationally rigid. In analogy with this arithmetic case, Dolgachev and Iskoviskikh showed in [5,Section 7.3] that minimal Del Pezzo G-surfaces of degree smaller than 3 are Gbirationally rigid. In this paper we determine which minimal Del Pezzo G-surfaces of degree 2 and 3 are G-birationally superrigid. When the G-surface is not G-birationally superrigid, we describe the generators of the group of G-birational automorphisms Bir G (S), or equivalently the normaliser of the corresponding subgroup G in Cr 2 (C). Here, we collect our main results, adopting the notation of [5]: Theorem 1. 3 Let G be a non-cyclic group and S be a minimal Del Pezzo G-surface of degree 3. Then S is G-birationally superrigid, unless G is isomorphic to the symmetric group S 3 and S is not the Fermat cubic surface.
In this case, the group Bir G (S) is generated by two or three Geiser involutions whose base points lie on the unique G-fixed line and by a subgroup of Aut(S) isomorphic to: The group Bir G (S) of the very general non-G-birationally superrigid minimal Del Pezzo G-surface of degree 3 with G S 3 is not finite.
For the proof, see Sect. 4.1.

Theorem 1.4 Let G be a cyclic group and S be a minimal Del Pezzo G-surface of degree 3. Then S is G-birationally superrigid if and only if G is of order 6 of type A 5 + A 1 . More precisely, if S is not G-birationally superrigid, then G is isomorphic to one of the following:
(i) a cyclic group of order 3 of type 3A 2 . The group Bir G (S) is (infinitely) generated by the Geiser involutions whose base points lie on the unique G-fixed nonsingular cubic curve and by a subgroup of Aut(S) isomorphic to 3 3 S 3 , if S is the Fermat cubic surface, or by Aut(S) itself otherwise. (ii) a cyclic group of order 6 of type E 6 (a 2 ). The group Bir G (S) is (infinitely) generated by three Geiser involutions, the Bertini involutions whose base points lie on a G-invariant nonsingular cubic curve C and by a subgroup of Aut(S) isomorphic to 3 3 × 2, if S is the Fermat cubic surface, or by Aut(S) itself otherwise. (iii) a cyclic group of order 9 of type E 6 (a 1 ). The group Bir G (S) is finitely generated by three Geiser involutions whose base loci are coplanar and by a subgroup of Aut(S) isomorphic to the dihedral group D 18 . (iv) a cyclic group of order 12 of type E 6 . The group Bir G (S) is finitely generated by G, by a Bertini involution and by a Geiser involution whose base loci are aligned. (i) a group of order 2 of type A 7 1 ; (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 .
Moreover, if S is not G-birationally superrigid, then G is isomorphic to one of the following: (v) a cyclic group of order 4 of type 2 A 3 + A 1 . The group Bir G (S) is generated by infinitely many Bertini involutions whose base loci lie in the unique G-fixed nonsingular curve of genus one and by a subgroup of Aut(S) isomorphic to 2× 4 2 2, if S is of type II, or by Aut(S) itself otherwise. (vi) a cyclic group of order 12 of type E 7 (a 2 ). The group Bir G (S) is generated by two Bertini involutions and by a subgroup of Aut(S) isomorphic to 2 × 12.
For the proof, see Sect. 5.2.

Corollary 1.7 Let G be a cyclic group and S be a minimal Del Pezzo G-surface of degree smaller than 3. Then, S is G-birationally superrigid if and only if the group Bir G (S) of G-birational 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.
In the paper we also provide explicit equations for the listed Del Pezzo surfaces S and the generators of the group Bir G (S), unless it coincides with Aut G (S).  Tables 1 and 2. The structure of the paper is as follows: in Sect. 3 we rewrite in full detail the proof of the G-equivariant version of the above-mentioned 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 G-surfaces of degree 3 and 2 which are not G-birationally superrigid in Sects. 4 and 5 respectively.

Preliminaries
Let S be a nonsingular surface. A linear system M on S is mobile if its fixed locus does not contain any divisorial component. The pair (S, D+M) is the datum of a nonsingular surface S, a Q-divisor D whose coefficients are at most 1 and a mobile linear system M, or equivalently one of its general members. Let α : S → S be a birational morphism. For each prime divisor E i of S there exists a coefficient a(E i , S, D + M), called discrepancy, such that the following relation holds: In particular, observe that the multiplicity mult p (M) of M at a point p ∈ S equals 1 − a(E, S, M), where E is the exceptional divisor of the blow-up of S at p. Bertini invol. Table 2 Minimal Del Pezzo G-surfaces of degree 2 which are not G-birationally superrigid Let G be a finite group of automorphisms acting effectively on a surface S. In the introduction we have already recalled the definition of a G-rational map. This concept must not be confused with that of a G-equivariant map, i.e., a birational map which makes the following diagram commute: The degree d of a Del Pezzo surface S is defined to be the self-intersection number of the canonical class K S , in symbols d . . = K 2 S . We briefly recall some properties of Del Pezzo surfaces of degree 3, see for instance [6, Chapter III, Theorem 3.5].
• A Del Pezzo surface S of degree 1 is a nonsingular hypersurface of degree 6 in the weighted projective space P(1, 1, 2, 3), embedded via the third pluricanonical linear system |− 3K S |. Via the linear system |− 2K S |, S can be realised as a double cover of the singular quadric P(1, 1, 2) branched along a nonsingular sextic curve. In particular, since the double cover is canonical, its deck transformation τ is a central element in the group of automorphisms 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 P(1, 1, 1, 2), embedded via the second pluricanonical linear system |− 2K S |. Via the canonical map, S can be realised as a double cover of P 2 branched along a nonsingular quartic curve. In particular, since the double cover is canonical, its deck transformation τ is a central element in the group of automorphisms 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 P 3 , embedded via the anticanonical linear system |− K S |.

G-equivariant Segre-Manin theorem
In this section we present the proof, essentially due to Dolgachev and Iskoviskikh, of the following G-equivariant version of a classical arithmetic theorem by Segre [9] and Manin [8] (cf. also [2, Section 1.5]). 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 non-canonical log Calabi-Yau pair. S be a G-birational non-biregular map to a minimal G-surface S . In order to prove that S is G-birationally rigid we need to exhibit a G-birational map σ : S S such that ϕ • σ is a G-biregular map.
Step 1 (non-canonical log Calabi-Yau pair). By Theorem 3.2, the existence of ϕ is equivalent to the existence of a mobile G-invariant linear system M on S such that • (log Calabi-Yau) K S + λM ∼ Q 0; • (not canonical singularities) the pair (S, λM) is not canonical.
Since K S generates Pic G (S) in degree 3, we can suppose λ = 1 n for some n ∈ N.
Step 2 (orbit of length 3). The proof of Lemma 3.4 implies that there exists a G-orbit O contained in the non-canonical locus of the log Calabi-Yau pair S, 1 n M such that Lemma 3.5 grants that the length of O is strictly less than the degree d of S.
Step 3 (Geiser and Bertini involution). By hypothesis, the degree of S is at most 3 and we are left with few possibilities: Case 1. O consists of a single G-fixed point p and the degree of S is either 2 or 3. Let π : S → S be the blow-up of S at p with exceptional divisor E. Then, the surface S is a Del Pezzo surface of degree 1 or 2 if S has degree 2 or 3 respectively (cf. Lemma 3.7), and it is endowed with a G-action via pullback of the G-action on S. These surfaces are endowed with a central G-invariant biregular involution τ , which descends to a G-birational non-biregular involution σ 1 on S, named Bertini or Geiser involution respectively. The defined G-birational maps are collected in the following diagram: Let a, b, c, d be integers such that τ * (H ) ∼ a H + bE and τ * (E) ∼ cH + d E, where H . . = −π * K S is the pullback of the ample generator of Pic G (S). Then, we obtain Note in particular that c > 0, because E is not τ -invariant and so τ * E is not contracted by π : by the ampleness of −K S , we obtain Since τ preserves the canonical class K S ∼ − H + E, we obtain also that a − c = 1, so that Case 2. O consists of two points p 1 and p 2 and the degree of S is 3. Analogously, the blow-up of S at p 1 and p 2 is a Del Pezzo surface of degree 1 endowed with a G-equivariant involution which descends to a non-biregular Bertini involution on S, denoted σ 1 .
In all the cases, the Noether-Fano inequalities (cf. Lemma 3.4) force σ −1 Step 4 (inductive step). By Theorem 3.2, either ϕ • σ 1 is G-biregular or the pair S, 1 is not canonical. In the latter case, we can repeat the above arguments and construct a sequence of Bertini or Geiser G-involutions σ 1 , . . . , σ s on S such that ϕ s . . = ϕ • σ 1 • · · · •σ s is non-biregular and again, by Theorem 3.2, the mobile pair S, 1 would not be Q-linearly equivalent to an effective divisor, which is a contradiction. Hence, there exists an integer s such that ϕ s is G-biregular. We conclude that S is G-birationally rigid.

Corollary 3.3 Let S be a minimal Del Pezzo G-surface of degree 3 (resp. 2). Then, every G-birational map is a composition of a G-biregular map, Geiser and/or Bertini involutions (resp. a G-biregular map and Bertini involutions).
We now prove the lemmas used in the proof of Theorem 3.1.

Lemma 3.4 Let S be a G-surface and (S, M) be a G-pair, i.e., M is a G-invariant mobile linear system. If (S, M) is not canonical, then there exists a G-orbit O in S such that
i.e., the multiplicity of each point of O on M is greater than 1.
Proof Let α : S → S be a G-equivariant log resolution of the pair (S, M). This means that α is a G-equivariant birational morphism such that the fixed locus of the pullback linear system α * (M) has simple normal crossing. We prove the statement by induction on the number s of G-equivariant blow-ups through which α factors. If α is the blow-up of S at a single G-orbit O with exceptional divisor E, then Let O be the centre of the blow-up α 1 with exceptional divisor E 1 . Then, either mult O (M) > 1, or a(E 1 , S, M) 0. In the latter case, since the pair (S 1 , (

Lemma 3.5 Let S be a minimal Del Pezzo G-surface of degree d. If ϕ : S S is a non-biregular G-birational map, then the G-orbit O defined in Lemma 3.4 has length |O| strictly smaller than d.
Proof Let M be the linear system defined in Theorem 3.2. Consider C 1 and C 2 two general G-invariant Q-divisors of M ∼ Q − nK S . For any G-orbit O defined in Lemma 3.4, the following sequence of inequalities holds: Remark 3.6 Lemma 3.5 implies immediately that any Del Pezzo G-surface of degree 1 is G-birationally superrigid, see also [5,Corollary 7.11].

Lemma 3.7 Let S be a minimal Del Pezzo G-surface of degree d and M be a mobile linear system on S such that K S
Proof By the Nakai-Moishezon criterion for amplitude [7, Theorem 1.2.23], it is enough to check that Note that In particular, we obtain that for any curve since M is an ample linear system and because of Lemma 3.4. If C = E, then Finally, K 2 S = K 2 S − |O| > 0, by Lemma 3.5.

G-birational 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 P 3 = P(V ) via the canonical embedding and every automorphism of S lifts to an automorphism of P 3 . The 4-dimensional vector space V is a G-representation, 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.  • p does not lie in the exceptional locus of g; • the strict transform l of the line l passing through q i and q j does not contain p; • the strict transformc of a conic c passing through five of the points q i does not contain p.
Equivalently, we require that no (−1)-curve contains p. Indeed, the g-exceptional lines and the curves l andc are all the 27 (−1)-curves in S.
In order to construct a Bertini involution, we need to check in addition that the strict transforms of a singular cubic curve s containing all the points q i does not contain both p 1 and p 2 . Suppose on the contrary that such a curves exists. We distinguish two cases: either q i is a singular point of s or one of the p i , say p 1 , is a singular point ofs. In the former case,s is a conic. Indeed, it is a nonsingular rational curve with Vice versa, if the points { p 1 , p 2 } lie on a nonsingular conic c in S, then wherec is the strict transform of c via f , and F 1 and F 2 are the f -exceptional divisors. Thus, S is not a Del Pezzo surface. In the latter case,s is an anticanonical divisor, hence a hyperplane section singular at p i . In particular, the tangent plane at p 1 contains both the points p 1 and p 2 .
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 G-surface 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 1 q 2 the line passing through those points in P 3 . Note that the line l q 1 q 2 is G-invariant and it is not contained in S. Differently, it could be contracted, violating the minimality of G. Moreover, the line l q 1 q 2 intersects S with multiplicity 1 at q 1 and q 2 . Otherwise, if the multiplicity at one of the two points is 2, then so it is at the other point due to the group action. However, this is a contradiction, since l q 1 q 2 would intersect S with multiplicity at least 4, while S has degree 3. This implies that the invariant line l q 1 q 2 intersects S in a third point, thus fixed by the action of G.
Our strategy to show that a nonsingular cubic surface is G-birationally superrigid is the following: • find G-fixed points and orbits of length 2 aligned with them, see Lemma 4.3; • if the conditions of Lemma 4.2 do not hold for these G-orbits, then S is Gbirationally superrigid.
In view of the latter, recall that a point of intersection of three lines on a cubic surface is called Eckardt point. It is just the case to mention that a point p is an Eckardt point if and only if the intersection of its tangent space to S and S itself is the union of three lines passing through p.

Remark 4.4
Notice that if { p 1 , p 2 } is a G-orbit then condition (c) in Lemma 4.2 always holds, since otherwise the line between p 1 and p 2 is bitangent to S.

G-birational superrigidity for non-cyclic groups
Suppose now that G is non-cyclic. Minimal non-cyclic finite groups acting effectively by automorphisms on cubic surfaces and fixing a point have been classified by Dolgachev and Duncan [4]. Any cubic surface endowed with an action of such a group is projectively equivalent to a surface S ab defined by  ). In particular, As the values of the parameters (a, b) vary, we have the following cases.
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 G-birationally superrigid.  Table 4] and the automorphism group of Aut(S ab ) is isomorphic to 2 × S 3 . We consider the cases G 2 × S 3 and G S 3 separately.
• G S 3 . The fixed points p 1 and p 2 are not Eckardt points. Therefore, S ab is not G-birationally superrigid and the group Bir G (S ab ) is generated by 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 Bir G (S ab ) for the very general surface S ab are discussed in the following paragraphs.
The only fixed points are p 0 , p 1 , p 2 . None of them is an Eckardt point and the only invariant line through p i is l 1 . Therefore, S ab is not G-birationally superrigid and the group Bir G (S ab ) is generated by G-biregular automorphisms of S ab and three Geiser involutions with base loci contained in l 1 ∩ S ab . The Geiser involutions are given by the equations We complete the list of generators, computing the normaliser N Aut(S ab ) (G) of G in Aut(S ab ). We adopt the surface type convention of [5]. 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 G-birational automorphisms Bir G (S ab ) for the very general surface S ab .

Lemma 4.5 The normaliser of G in
Let  (a, b).

Lemma 4.6 The group Bir G (S ab ) is not a finite group for the very general surface S ab in S and in S .
Proof Let be the diagonal in S × f S and (ϕ p 2 • ϕ p 1 ) n be the graph of the composition (ϕ p 2 • ϕ p 1 ) n in S × f S. There is an induced projection morphism Define the (closed) algebraic subset

Note that the locus of surfaces S ab with infinite Bir
is not finite, then C n is a proper closed subset of C 2 (a,b) and the lemma holds. We claim that Bir G (S 11 ) is not finite, i.e., we can choose (a 0 , b 0 ) equal to (1, 1). To this aim, recall the following facts: • for any p ∈ S ab the point ϕ p i ( p) is aligned with p i and p; • the involutions ϕ p 1 and ϕ p 2 fix the pencil of cubic curves Fix (λ:μ) ∈ P 1 (λ:μ) such that C (λ:μ) is nonsingular. Observe that the point p 0 is an inflection point of C (λ:μ) . Due to the previous facts, the following relations for the elliptic curve (C (λ:μ) , p 0 ) hold: In particular, One can check (use MAGMA) that for a suitable choice of (λ:μ) (e.g. (1:1)), the point p 1 is not a torsion point. This implies that ϕ p 2 • ϕ p 1 has infinite order in S 11 .
The same proof holds for S since S 11 ⊂ S .
Open Question Is the group Bir G (S ab ) not finite for any (a, b) = (0, 0)?

G-birational 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 n a primitive n-th root of unity.
Type 3A 2 . G fixes the nonsingular cubic curve S is not G-birationally superrigid and the group Bir G (S) is generated by biregular Gautomorphisms of S and infinitely many Geiser involutions whose base locus points lie on the nonsingular cubic curve given by t 3 = 0. The normaliser N Aut(S) (G) of G in Aut(S) is the group 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 Aut(S) 3 3 S 4 (cf. Sect. 4.1): the normaliser N Aut(S) (G) is isomorphic to 3 3 S 3 . Otherwise, g is a central element of Aut(S), which is isomorphic to H 3 (3) 4 or H 3 (3) 2, where H 3 (3) is the Heisenberg group of unipotent 3 × 3-matrices over the finite field F 3 (cubic surfaces of type III or IV; see [5, Table 4

]). Then, the group Aut G (S) coincides with Aut(S).
Type E 6 (a 2 ). The line l 2 = {t 2 = 0, t 3 = 0} ⊆ P 3 is fixed. The intersection which are three cuspidal cubic curves (we can suppose without loss of generality that α 3 = 1, otherwise S would be singular). There is only one further isolated fixed point on S, namely (0 : 0 :1: 0), which is an Eckardt point and whose tangent space is given by the equation αt 0 + t 1 = 0. An invariant line, which is not l 1 = {t 0 = t 1 = 0}, belongs either to the pencil P is covered by orbits of length 2, since the group G modulo the stabiliser of the plane t 3 = 0 acts on it as a cyclic group of order 2. We conclude that S is not G-birationally superrigid and that the group Bir G (S) is generated by G-biregular automorphisms of S, three Geiser involutions with base loci contained in l 2 ∩ 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 Aut(S) (G) of G in Aut(S).

Lemma 4.8
Proof Note that S is a cyclic cover of degree 3 of P 2 branched along a nonsingular cubic curve C, and G is generated by g 1 g 2 , where g 1 is the deck transformation of the cover and g 2 is the lift of the involution on C.
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 P 3 by means of four distinct characters. In particular, the points Note that the conic contains the only orbit of length 2 and the only fixed point in S is contained in a line. We conclude that S is G-birationally superrigid.
Type E 6 (a 1 ). All the fixed points in S are the points p 1 , p 2 and p 3 . They are not Eckardt points: by cyclic permutation of the variable (t 1 , t 2 , t 3 ) it is enough to check that T p 1 S ∩ S is an irreducible cubic curve. Indeed, The invariant lines l p 1 p 2 , l p 2 p 3 and l p 1 p 3 intersect S in two fixed points, one of them necessarily with multiplicity 2. The invariant lines l p 0 p i , with i = 1, 2, 3, are principal tangent lines at the singular point of the cuspidal cubic curves T p i S ∩ S. We conclude that S is not G-birationally superrigid and the group Bir G (S) is finitely generated by G-biregular automorphisms of S and three Geiser involutions with base loci p 1 , p 2 and p 3 respectively. More explicitly, the Geiser involutions are given by Although finitely generated, Bir G (S) is not a finite group, as we show in the following lemma.

Lemma 4.9 The group Bir G (S) is not a finite group.
Proof It is enough to prove that the composition ϕ p 2 • ϕ p 1 has infinite order. To this aim, recall the following facts: • for any p ∈ S the point ϕ p i ( p) is aligned with p i and p; • the involutions ϕ p 1 and ϕ p 2 fix the pencil of cubic curves Fix (λ:μ) ∈ P 1 (λ:μ) such that C (λ:μ) is nonsingular and choose O an inflection point on C (λ:μ) . Due to the previous facts, the following relations for the elliptic curve (C (λ:μ) , O) hold: 2 p 2 + p 1 = 0; In particular, Invariant lines l p i p j l p i p j ∩ S orbits in l p i p j ∩ S Observe that the hypothesis of Lemma 4.2 (ii) holds for the orbit {q 1 , q 2 }. Indeed, the set {q 1 , q 2 } is the only orbit of length 2 and q i are not Eckardt points, since are cuspidal cubic curves. Moreover, the pencil of planes containing {q 1 , q 2 } does not cut any conic on S and q i is not contained in the tangent space of q j , for i = j, by Remark 4.4. We conclude that S is not G-birationally superrigid and the group Bir G (S) is generated by G-biregular automorphisms of S, a Bertini involution and a Geiser involution whose base loci are aligned: {q 1 , q 2 } and p 3 respectively. The Bertini involution with base points q 1 and q 2 is the deck transformation of the double cover ψ : S → P 3 , and it is given explicitly by The Geiser involution with base point p 3 can be written as

Lemma 4.11 The group Bir G (S) is not a finite group.
Proof The proof is analogous to the one of Lemma 4.9. It is enough to prove that the composition ϕ p 3 • ϕ q 1 q 2 has infinite order. Note that: • for any p ∈ S the point ϕ p 3 ( p) is aligned with p 3 and p; • for any p = p 3 , the points ϕ q 1 q 2 ( p) and p belong to a conic contained in the plane q 1 q 2 p , spanned by q 1 , q 2 and p, and tangent to S ∩ q 1 q 2 p at q 1 and q 2 ; • the involutions ϕ p 3 and ϕ q 1 q 2 fix the pencil of cubic curves Fix (λ:μ) ∈ P 1 (λ:μ) such that C (λ:μ) is nonsingular and choose O an inflection point on C (λ:μ) . Due to the previous facts, the following relations for the elliptic curve (C (λ:μ) , O) hold: In particular, One can check (use MAGMA) that for a suitable choice of (λ:μ) (e.g. (1:1)), the point p 3 is not a torsion point. This implies that ϕ p 3 • ϕ q 1 q 2 has infinite order.
We complete the list of generators of the group Bir G (S), observing that the only Gbiregular automorphisms of S are the elements of G itself. Note that up to a change of coordinates [5, 6.5, Case 3, Type III], we can suppose that S is given by the equation

Lemma 4.12
The group G is self-normalising in Aut(S), i.e., the normaliser of G in , generated by the image ofg 1 andg 2 , is nontrivial. Note that the elementg 4 acts on H 3 (3) by conjugation via (g 1 ,g 2 ) → (g 2 2 ,g 1 ), see [5, Theorem 6.14, Type III]. As a result, we havẽ which yields a contradiction. We conclude that G is self-normalising in Aut(S).
The results of this section are summarised in Theorem 1.4.

G-birational superrigidity of Del Pezzo surfaces of degree 2
In this section we prove Theorem 1.6 and we classify the Del Pezzo G-surfaces of degree 2 which are not G-birationally superrigid. Recall that a Del Pezzo surface S of degree 2 is a double cover of P 2 branched over a nonsingular quartic curve. The surface S is a hypersurface of degree 4 in the weighted projective space P(1, 1, 1, 2) given by the equation where F 4 is a polynomial of degree 4. The covering map ν : S → P 2 is then given by the projection on the first three coordinates and the ramification curve R is the intersection of S with {t 3 = 0}. As in the previous section, the proof of the Segre-Manin theorem (Theorem 3.1) implies that a minimal Del Pezzo G-surface of degree 2 is not G-birationally superrigid if and only if it admits a G-equivariant 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
Proof The proof is analogous to that of Lemma 4.2. Recall that a Del Pezzo surface of degree 2 is a blow-up of P 2 at points q 1 , . . . , q 7 in general position, see [1,Exercise IV.8.(10).(a)]. We need to check that the blow-up 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 blow-down 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 ν. 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 blow-up 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 ν 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.
Our strategy to identify birational superrigid G-surfaces will then consist in finding the fixed points of the given G-action 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 P 2 .

G-birational superrigidity for non-cyclic groups
The minimal non-cyclic 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 G-birationally superrigid by Theorem 3.1 and Lemma 5.1. This concludes the proof of Theorem 1.5. It remains to analyse cyclic groups.
Type A 7 1 . The generator g is the standard Geiser involution of the surface S leaving the ramification curve {t 3 = F 4 (t 0 , t 1 , t 2 ) = 0} fixed. Hence, the surface is G-birationally superrigid.
is simply given by F 4 (t 0 , t 1 ) = 0. Notice that the polynomial F 4 has four distinct roots as C is nonsingular, hence there are four distinct intersection points and l is not a bitangent line of C. This implies that every point in the preimage of l is the base locus of a Bertini involution with the exception of the preimages of the four points of intersection with C and of the points of intersection with the bitangent lines of C. In other words, Bir G (S) is generated by G-automorphisms and infinitely many Bertini involutions, in particular S is not G-birationally superrigid.
To complete the list of generators of Bir G (S) it suffices to compute the normaliser N Aut(S) (G). Notice that up to a linear change of coordinates in the variables t 0 , t 1 , the equation F can be written as The automorphism group Aut(S) depends on the parameter a and in each case we compute the normaliser N Aut(S) (G) of G in Aut(S): • if a = 0, then Aut(S) 2 × (4 2 S 3 ) (cf. [5, Theorem 6.17, Type II]), where 2 is generated by γ (t 0 : t 1 : t 2 : t 3 ) = (t 0 : t 1 : t 2 :− t 3 ), 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 = γ g 2 . Notice that g 2 is central in 4 2 2 = τ, g 1 , g 2 and therefore it is central in 2 × 4 2 2. Since Since c is central in Aut(S) and g = γ c, we conclude that N Aut(S) (G) = Aut(S). • if a = 0, ±2 √ 3 i, then Aut(S) 2 × AS 16 , where AS 16 is a non-abelian group of order 16 isomorphic to 2× 4 2 (cf. [5,Tables 1 & 6]). The generators of 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 Aut(S) (G) = Aut(S). Type E 7 (a 4 ). The fixed locus is given by S ∩ {t 2 = t 3 = 0} and (0 : 0 :1: 0). In particular all fixed points lie on the ramification curve and therefore they do not give rise to Bertini involutions, thus S is G-birationally superrigid.
In other words, the lines tangent to C and passing through the possible q are given by − 2 · 3 3/4 i j t 1 + 3 · 2 1/3 k 3 t 2 = 0 , j = 1, 2, 3, 4, k = 1, 2, 3, which are pairwise distinct and intersect C in three distinct points each, and hence, are not bitangent lines. Therefore p 1 and p 2 are not in any (−1)-curve and it follows that S is not G-birationally superrigid. The Bertini involution with base point p 1 is the deck transformation of the map Proof The proof is analogous to the one of Lemmas 4.9 and 4.11. It is enough to prove that the composition ϕ p 1 • ϕ p 2 has infinite order. To this aim, note that the involutions ϕ p 1 and ϕ p 2 fix the pencil of curves of genus one C (λ:μ) = λt 1 − μt 2 = t 2 3 + t 4 0 + t 4 1 + t 0 t 3 2 = 0 .
The automorphism group of S is Aut(S) = 2 × 4 A 4 , see [5,  Let G be the image of G in A 4 under the composition of quotient homomorphisms 2 × 4 A 4 → 4 A 4 → A 4 . Notice G 3, since G is necessarily a cyclic group of A 4 whose order is a multiple of 3. It follows the image of N 2×4 A 4 (G) is contained in N A 4 (3). Moreover, notice that N A 4 (3) = 3, as there are no proper normal subgroups in A 4 containing 3 and 3 is not normal in A 4 . Finally, since 2 × 4 A 4 is a central extension of a central extension of A 4 , one obtains N 2×4 A 4 (12) = 2 × 12. The group Bir G (S) is generated by G, the standard Geiser involution γ and two Bertini involutions with base locus p 1 and p 2 respectively.
The cases above yield the proof of Theorem 1.6.