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 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, §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 Gsurface. 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 for some g ′ ∈ G. Then, a minimal G-surface is a G-surface with the property that any birational G-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 Gbirational 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 one defined over a non-algebraically closed field are birationally rigid. In analogy with this arithmetic case, Dolgachev and Iskoviskikh showed in [5, §7.3] that minimal Del Pezzo G-surfaces of degree smaller than 3 are G-birationally rigid. In this paper we determine which minimal Del Pezzo Gsurfaces 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 birational G-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 Gsurface 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: (1) S 3 if S is of type V, VIII; (2) S 3 × 2 if S is of type VI; (3) S 3 × 3 if S is of type III, IV.
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.
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: (1) G is 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 Gfixed 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. (2) G is 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.
(3) G is 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 . (4) G is 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.
Theorem 1.5. Let G be a non-cyclic group and S be a minimal Del Pezzo Gsurface of degree 2. Then S is G-birationally superrigid.
Theorem 1.6. Let G be a cyclic group and S be a minimal Del Pezzo G-surface of degree 2. Then S is G-birationally superrigid if and only if G is one of the following: (1) group of order 2 of type A 7 1 ; (2) group of order 6 of types E 7 (a 4 ), A 5 + A 1 , D 6 (a 2 ) + A 1 ; (3) group of order 14 of type E 7 (a 1 ); (4) 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: (1) G is a cyclic group of order 4 of type 2A 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. (2) G is 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.
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 birational G-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). The types of the G-surfaces appearing in Theorem 1.3, 1.4 and 1.6 are described in full details in Lemma 4.5, Proposition 4.7 and Proposition 5.2. For convenience, we summarise the contents of Theorem 1.3, 1.4 and 1.6 in Table 1 and 2.
The structure of the paper is as follows: in §3 we rewrite in full details 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 §4 and §5 respectively. Table 1. Minimal Del Pezzo surface G-surface of degree 3 which are not G-birationally superrigid.

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 coefficient are smaller than 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 ofS there exists a coefficient a(E i , S, D + M), called discrepancy, such that the following relation holds: 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 diagrams 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].
(1) 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, §6.7.].
(2) 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, §6.6.].
(3) 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

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]. 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. (1) if S ′ is a Del Pezzo surface and (S, λM) is canonical, then ϕ is biregular. (2) if S ′ is a conic bundle, then (S, λM) is not canonical.
Proof of Theorem 3.1. Let ϕ : S 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 (1) (log Calabi-Yau) K S + λM ∼ Q 0; (2) (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 m := mult p M > n for all points p ∈ O.
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 surfaceS 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 Let a, b, c, d be integers such that τ * (H) ∼ aH + bE and τ * (E) ∼ cH + dE, where H := −π * K S is the pullback of the ample generator of Pic G (S). Then, we obtain that Since τ preserve the canonical class KS ∼ −H + E, we obtain also that a − c = 1, so that 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 k1 σ −1 1 (M)) 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 ks ϕ −1 s (M)), with k s < k s−1 , is not canonical. However, if s > n, then the mobile linear system ϕ −1 s (M) 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.
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 In the latter case, since the pair 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 Proof. By the Nakai-Moishezon criterion for amplitude [7, Theorem 1.2.23], it is enough to check that

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.  (1) p does not lie in the exceptional locus of g; (2) the strict transforml of the line l passing through q i and q j does not contain p; (3) 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 curvesl 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 two lie on invariant lines passing through a fixed point for the action of G on S. Proof. Denote by q 1 and q 2 the points in the orbit of length two and by l q1q2 the line passing through those points in P 3 . Note that the line l q1q2 is G-invariant and it is not contained in S. Differently, it could be contracted, violating the minimality of G.
Moreover, the line l q1q2 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 q1q2 would intersect S with multiplicity at least 4, while S has degree 3. This implies that the invariant line l q1q2 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: (1) find G-fixed points and orbits of length two aligned with them, see Lemma 4.3; (2) if the conditions of Lemma 4.2 do not hold for these G-orbits, then S is G-birationally 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. 4.1. G-birational superrigidity for non-cyclic group. 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 where ǫ 3 is a primitive third root of unity. The stabiliser of the point p 0 is 3 2 ⋊K 4 ≃ 6 × S 3 , where K 4 is the non-normal Klein subgroup of S 4 generated by (12) and (34) and 3 2 is generated by σ and ρ.
Type a = b = 0. The surface S 00 is the Fermat cubic surface. The point 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.
Type a 3 = b 3 = 0. Up to a linear change of coordinates, we can suppose that a = b = 0. The group G is isomorphic to 2 × S 3 or S 3 , where S 3 is generated by σρ 2 and (12), and 2 is generated by (34), see [4, Theorem 8.1. Case 3.2.]. Hence, the only fixed point is the Eckardt point p 0 and the only invariant line through p 0 is l 1 . Note that the surface S ab is of type VI in the sense of [5, 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.
(2) 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. Type 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 biregular G-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].
Proof. Due to [5,Theorem 6.14], the group Aut(S ab ) is one of the following. Type III. Aut(S ab ) ≃ H 3 (3) 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 .
Let S ⊂ P 3 (t0:t1:t2:t3) ×C 2 (a,b) be the hypersurface given by the equation {F ab = 0}, see equation (1)  Proof. Let ∆ be the diagonal in S × f S and Γ (ϕp 2 •ϕp 1 ) n be the graph of the composition (ϕ p2 • ϕ p1 ) n in S × f S. There is an induced projection morphism . Note that the locus of surfaces S ab with infinite Bir G (S ab ) contains C 2 (a,b) \ n C n . Therefore, if there exists (a 0 , b 0 ) ∈ C 2 (a,b) such that Bir G (S a0b0 ) 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 ) equals to (1, 1). To this aim, recall the following facts: (1) for any p ∈ S ab the point ϕ pi (p) is aligned with p i and p; (2) the involutions ϕ p1 and ϕ p2 fix the pencil of cubic curves (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, ϕ p2 • ϕ p1 (p) = p + 2p 1 . 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 p2 • ϕ p1 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 group.
In this section, we discuss the birational superrigidity of minimal cubic surface 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 ǫ n a primitive n-th root of unity.
Proposition 4.7. [5, Corollary 6.11] Let S = V (F ) be a nonsingular cubic surface, endowed with a minimal action of a cyclic group G of automorphisms, generated by g. Then, one can choose coordinates in such a way that g and F are given in the following list.
Type 3A 2 . G fixes the nonsingular cubic curve C = {t 3 = t 3 0 + t 3 1 + t 3 2 + αt 0 t 1 t 2 = 0}. S is not G-birationally superrigid and the group Bir G (S) is generated by biregular G-automorphisms 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 G-automorphisms. 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. §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 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 (0:0:0:1) of lines through (0 : 0 : 0 : 1) intersecting the line l 2 or to the pencil P (0:0:1:0) of lines through (0 : 0 : 1 : 0) intersecting the line l 2 . These pencils span respectively the planes t 2 = 0 and t 3 = 0. Orbits of length two lie on invariant lines, neither on l 1 (since it is tangent to the Eckardt point (0 : 0 : 1 : 0), thus l 1 ∩ S = {p}), nor on a line through P (0:0:0:1) (since the group G modulo the stabiliser of the plane t 2 = 0 acts on it as a cyclic group of order 3). On the other hand, the nonsingular cubic curve is covered by orbits of length two, 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 biregular G-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). 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 4 distinct characters. In particular, the points Type A 5 + A 1 . The only fixed point in S is the Eckardt point p 3 . In the following table, we list all the invariant lines and the orbits that they cut on S.
Note that the conic the only orbit of length two 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 p1 S ∩ S is an irreducible cubic curve. Indeed, T p1 S ∩ S = {t 2 = t 2 3 t 1 + t 3 0 = 0}. The invariant lines l p1p2 , l p2p3 and l p1p3 intersect S in two fixed points, one of them necessarily with multiplicity 2. The invariant lines l p0pi , with i = 1, 2, 3, are principal tangent lines at the singular point of the cuspidal cubic curves T pi S∩S. We conclude that S is not G-birationally superrigid and the group Bir G (S) is finitely generated by biregular G-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. Proof. It is enough to prove that the composition ϕ p2 • ϕ p1 has infinite order. To this aim, recall the following facts: (1) for any p ∈ S the point ϕ pi (p) is aligned with p i and p; (2) the involutions ϕ p1 and ϕ p2 fix the pencil of cubic curves (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, ϕ p2 • ϕ p1 (p) = p − 3p 2 . One can check (use MAGMA) that for a suitable choice of (λ : µ) (e.g. (1 : 1)), the point p 2 is not a torsion point. This implies that ϕ p2 • ϕ p1 has infinite order.
We complete the list of generators of the group Bir G (S), describing the group of biregular G-automorphisms of S. Note first that via the following change of coordinates (s 0 :s 1 : s 2 : 9t 0 : t 1 + t 2 + t 3 : ǫ 9 (t 1 + ǫ 6 9 t 2 + ǫ 3 9 t 3 ) : ǫ 2 9 (t 1 + ǫ 3 9 t 2 + ǫ 6 9 t 3 )), we can suppose that S is given by the equation  Proof. Recall that the automorphism group of a Fermat cubic is the group 3 3 ⋊ S 4 . Let G ′ be the image of G in S 4 , generated by the permutation (234), and K := G ∩ 3 3 , generated by h(s 0 : s 1 : s 2 : s 3 ) = (s 0 : ǫ 3 s 1 : ǫ 3 s 2 : ǫ 3 s 3 ). The image of N Aut(S) (G) is contained in N S4 ((234)), which is generated by (234) and (23) and isomorphic to S 3 . Therefore, N Aut(S) (G) is a subgroup of 3 3 ⋊ S 3 and admits a subgroup homomorphic to S 3 . The kernel of the projection N Aut(S) (G) → S 3 is 3 3 ∩ N Aut(S) (G) = K. Indeed, the conjugation of g via an element σ a0 ρ a1 θ a2 ∈ 3 3 is c σ a 0 ρ a 1 θ a 2 (g)(s 0 : s 1 : s 2 : Type E 6 . In the following tables, we list fixed points and invariant lines and the orbits that they cut on S.

Fixed points
T pi S T pi S ∩ S Eckardt point fixed Eckardt point p 2 and fixed point p 3 Observe that the hypothesis of Lemma 4.2.
(2) holds for the orbit {q 1 , q 2 }. Indeed, the set {q 1 , q 2 } is the only orbit of length two and q i are not Eckardt points, since T qi S ∩ S = {t 1 ± it 3 = t 2 2 t 3 + t 3 0 = 0} 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 biregular G-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 point q 1 and q 2 is the deck transformation of the double cover ψ : S → P 3 , (t 0 : t 1 : t 2 : t 3 ) → (t 2 1 + t 2 3 : t 2 0 : t 0 t 2 : t 2 2 ), 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 p3 • ϕ q1q2 has infinite order. Note that: (1) for any p ∈ S the point ϕ p3 (p) is aligned with p 3 and p; (2) for any p = p 3 , the points ϕ q1q2 (p) and p belong to a conic contained in the plane Π q1q2p , spanned by q 1 , q 2 and p, and tangent to S ∩ Π q1q2p at q 1 and q 2 ; (3) the involutions ϕ p3 and ϕ q1q2 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, ϕ p3 • ϕ q1q2 (p) = p − 3p 3 . 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 p3 • ϕ q1q2 has infinite order.
We complete the list of generators of the group Bir G (S), observing that the only biregular G-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 generated by the image ofg 1 andg 2 , is non-trivial. 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 This implies that N Aut(S) (G) ∩ H 3 (3)/[H 3 (3), H 3 (3)] = 1, 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 an 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 four. 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 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. 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-upS 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)-curve, 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 . 5.1. G-birational superrigidity for non-cyclic group. 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]. We conclude that S is G-birationally superrigid by Theorem 3.1 and Lemma 5.1 and concludes the proof of Theorem 1.5. It remains to analyse cyclic groups. 5.2. G-birational 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 ǫ n is a primitive n-th root of the unit and F i is a polynomial of degree i.
Proposition 5.2. [5, Section 6.6.] Let S = V (F ) be a Del Pezzo surface of degree 2, endowed with a minimal action of a cyclic group G of automorphisms, generated by g. Then, one can choose coordinates in such a way that g and F are given in the following list.
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 Gbirationally superrigid.
Type 2A 3 + A 1 . The curve S ∩ {t 2 = 0} is fixed by the action of G. It is the preimage of the line l = {t 2 = 0} under the double cover ν. The intersection of l with the branched quartic C = {t 4 2 + F 4 (t 0 , t 1 ) = 0} 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 normalizer 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 subject to the following relations τ g 2 τ = g 2 τ g 1 τ = g −1 1 g −1 2 = g 3 1 g 3 2 . In particular, the group G is generated by g = γg 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 [5, Theorem 6.17, Type III]), where 2 is generated by γ(t 0 : t 1 : t 2 : t 3 ) = (t 0 : t 1 : t 2 : −t 3 ) and 4A 4 is a central extension of the alternating group A 4 generated by g 1 (t 0 : t 1 : t 2 : t 3 ) = (t 1 : t 0 : t 2 : −t 3 ), Since c is central in Aut(S) and g = γc, we conclude that N Aut(S) (G) = Aut(S).
(3) if a = 0, ±2 √ 3i, then Aut(S) ≃ 2×AS 16 , where AS 16 is a non-abelian group of order 16 isomorphic to 2 × 4 ⋊ 2 (c.f. [5,Tables 1 & 6]). The generator 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.
The automorphism group of S is Aut(S) = 2 × 4A 4 , see [5,  Let G ′ be the image of G in A 4 under the composition of quotient homomorphisms 2 × 4A 4 → 4A 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 2×4A4 (G) is contained in N A4 (3). Moreover, notice that N A4 (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 × 4A 4 is a central extension of a central extension of A 4 , one obtains N 2×4A4 (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.