Automorphisms of del Pezzo surfaces in odd characteristic

We complete the classification of automorphism groups of del Pezzo surfaces over algebraically closed fields of odd positive characteristic.


INTRODUCTION
A del Pezzo surface of degree d is a smooth projective algebraic surface X with ample anticanonical class −K X satisfying K 2 X = d. It is known that 1 ≤ d ≤ 9, and X ∼ = P 2 if d = 9. A del Pezzo surface of degree 8 is isomorphic to a smooth quadric surface or to the blow-up of the projective plane in one point. All other del Pezzo surfaces of degree d ≤ 7, are isomorphic to the blow-up of 9 − d points in P 2 in general position. Any non-degenerate smooth linearly normal surface in P d of degree d ≥ 3 is isomorphic to an anti-canonically embedded del Pezzo surface.
The group Aut(X) of automorphisms of a del Pezzo surface X is a finite group if d ≤ 5 and a smooth algebraic group of positive dimension if d ≥ 6.
The classification of automorphism groups of del Pezzo surfaces over an algebraically closed field of characteristic 0 has been known for more than a hundred years. A modern exposition and the history can be found in [7]. It is used in the classification of conjugacy classes of finite subgroups of the Cremona group of the projective plane in [6]. The knowledge about possible groups of automorphisms of del Pezzo surfaces over algebraically closed fields of positive characteristic is essential for the extension of this classification to positive characteristic. Partial results in this direction can be found in [8] and [9], and the classification of possible automorphism groups of cubic and quartic del Pezzo surfaces was accomplished in [10]. For the convenience of the reader, we recall this classification in Table 5 and Table 6 in the Appendix.
Any rational surface X with a group G of automorphisms of order prime to the characteristic can be lifted to characteristic zero together with an action of G on the lift [16]. This reduces the classification of automorphism groups in positive characteristic p to the classification of the groups of automorphisms of order divisible by p, which is the main part of this article, and 1 also to the analysis of pairs (X, G) in characteristic zero that admit a good reduction modulo p, see Section 4.5 and Section 5.5. The classification of the groups of automorphisms of del Pezzo surfaces of degree larger than 4 is rather easy, see Section 3. As the classification of automorphism groups of del Pezzo surfaces of degree 4 and 3 was achieved in [10], we will complete the classification of automorphisms of del Pezzo surfaces by concentrating on the task of classifying automorphism groups of del Pezzo surfaces of degree 2 and 1.
The cases of odd and even characteristic are drastically different and require different tools. This is due to the fact that, in odd characteristic, the classification of automorphism groups of del Pezzo surfaces of degree 2 and 1 essentially coincides with the classification of groups of projective automorphisms of smooth plane quartic curves or smooth genus 4 curves of degree 6 lying on a singular quadric. In the case of even characteristic no such relations exist. This is why we separate the classification into two parts. The present paper deals with the case of odd characteristic and the subsequent paper [11] will deal with the case of characteristic two. Our main result can be summarized as follows.
Theorem. A finite group G is realized as the automorphism group Aut(X) of a del Pezzo surface X of degree 2 (resp. 1) over an algebraically closed field k of characteristic char(k) = 2 if and only if G is listed in   Table 8) also gives the conjugacy classes in W (E 7 ) (resp. W (E 8 )) of all elements of Aut(X) for all del Pezzo surfaces X of degree 2 (resp. degree 1). Here, we use Carter's notation for these conjugacy classes, see [3] and Section 2. The knowledge of these conjugacy classes is important in order to understand the conjugacy relation between the various automorphism groups of del Pezzo surfaces inside the Cremona group of rank 2.
Acknowledgements: The authors would like to thank Claudia Stadlmayr for helpful comments on a first draft of this article.

NOTATION
We will use the following group-theoretical notations for the finite groups we encounter in the article. Throughout this article, p is a prime and q is a power of p.
• C n is the cyclic group of order n.
• S n and A n are the symmetric and alternating groups on n letters.
• Q 8 is the quaternion group of order 8.
• D 2n is the dihedral group of order 2n.
• For odd n, O n (q) is the subgroup of GL n (q) that preserves a non-degenerate quadratic form F . • For even n, O + n (q) (resp. O − n (q)) is the subgroup of GL n (q) that preserves a nondegenerate quadratic form F of Witt defect 0 (resp. 1).
• SO ± n (q) is the subgroup of O ± n (q) of elements with determinant 1. • PSO ± n (q) is the quotient of SO ± n (q) by its center. • Sp 2n (q) is the subgroup of SL q (2n) preserving the standard symplectic form on F 2n q . Its order is q n 2 (q 2n−1 − 1) · · · (q 2 − 1).
• PSU n (q 2 ) = SU n (q 2 )/C, where C is a cyclic group of order (q + 1, n) of diagonal Hermitian matrices. The simple group PSU n (q 2 ) is denoted by U n (q) in [4]. • H 3 (3) is the Heisenberg group of 3 × 3 upper triangular matrices with entries in F 3 . • A.B is a group that contains a normal subgroup A with quotient group B.
• A : B is the semi-direct product A ⋊ B.

PRELIMINARIES
2.1. The anti-canonical map. The main references for the known facts about del Pezzo surfaces to which we refer without proofs are [5] and [7]. Throughout, we are working over an algebraically closed field of characteristic p = 2.
The linear system | − K X | on a del Pezzo surface X of degree d is of dimension d. It is base-point-free if d = 1 and defines an embedding The image is a non-degenerate smooth surface of degree d in P d . If d = 9, the embedding coincides with the third Veronese map X ∼ = P 2 ֒→ P 9 .
A del Pezzo surface of degree d = 8 is isomorphic to a minimal ruled surface F 0 or F 1 . In all other cases, the image of j is a projection of a Veronese surface from a linear span of 9 − d points on the surface. This classical fact, due to del Pezzo, can be restated as the fact that any del Pezzo surface of degree d ≥ 3 is isomorphic to a smooth quadric or obtained as the blow-up of 9 − d points p 1 , . . . , p 9−d in the plane. The latter description of del Pezzo surfaces extends to del Pezzo surfaces of the remaining degrees d = 1 and 2. Here, the points p 1 , . . . , p 9−d must be in general position.
For the following description of del Pezzo surfaces of degrees 1 and 2, we assume p = 2.
If d = 2, the anti-canonical linear system | − K X | is of dimension 2. It defines a finite morphism of degree 2 f : X → P 2 with branch locus a smooth quartic curve. This allows us to view X as a hypersurface of degree 4 in the weighted projective space P(1, 1, 1, 2). Its equation is and its branch locus is the smooth plane quartic curve C = V (f 4 (x 0 , x 1 , x 2 )). If d = 1, the linear system | − K X | is a pencil with one base point o. The linear system | − 2K X | is of dimension 3. It defines a finite morphism of degree 2 where Q is an irreducible quadratic cone. The cover is branched along a smooth curve of genus 4 cut out by a cubic. This allows us to consider X as a hypersurface of degree 6 in the weighted projective space P(1, 1, 2, 3) given by the equation This equation can be rewritten in the form where a k is a binary form of degree k.
The double cover f extends to a double cover ) is a minimal rational ruled surface. In the natural basis (f, e) of Pic(F 2 ) with f 2 = 0, e 2 = −2, the branch curve of f ′ is the union of a smooth member of the linear system |6f + 3e| and the exceptional section identified with e. Equation (2) can be viewed as the Weierstrass model of the Jacobian elliptic fibration Y → P 1 defined by the proper transform of the anti-canonical pencil | − K X |.

2.2.
Weyl groups. An automorphism of a del Pezzo surface X of degree d acts naturally on the Picard group Pic(X) of isomorphism classes of invertible sheaves on X, or divisor classes modulo linear equivalence. Assume that X = F 0 , so that X ∼ = Bl p 1 ,...,p 9−d (P 2 ). The group Pic(X) is a free abelian group generated by the divisor classes e i of the exceptional curves E i over the points p i and the divisor class e 0 such that |e 0 | defines the blowing down morphism X → P 2 . The basis (e 0 , e 1 , . . . , e 9−d ) is called a geometric basis of Pic(X). It depends on the isomorphism X ∼ = Bl p 1 ,...,p 9−d (P 2 ) since different point sets may lead to isomorphic blow-ups. If we fix one geometric basis, passing to another geometric basis defines an integer matrix of rank 10 − d.
The set of all such matrices forms a subgroup W of GL 10−d (Z).
The intersection product Pic(X)×Pic(X) → Z is a symmetric bilinear form (x, y) → x·y on Pic(X). It equips Pic(X) with the structure of a quadratic lattice. Since e 2 0 = 1, e 2 i = −1, i = 0, the signature of the corresponding real quadratic space Pic(X) R := Pic(X) ⊗ R is equal to (1, 9 − d). A geometric basis is an orthonormal basis for this inner product. This shows that the group W is a subgroup of the orthogonal group O(Pic(X)) of isometries of the quadratic lattice Pic(X).
The known behavior of the canonical class under a blow-up implies that and this formula is true for any geometric basis. This shows that K X is invariant with respect to the action of the group W , and hence W can be identified with a subgroup of the orthogonal group O(K ⊥ X ) of the orthogonal complement to ZK X in Pic(X). The sublattice K ⊥ X ⊆ Pic(X) has a basis α 0 = e 0 − e 1 − e 2 − e 3 , α i = e i − e i+1 . i = 1, . . . , 8 − d.
(4) Assume d ≤ 6. Computing the Gram matrix of this basis, we find that K ⊥ X is isomorphic to the root lattice E N of rank N = 9 − d of a simple Lie algebra of type in the cases d = 6, 5, 4, 3, 2, 1, respectively. The basis (4) is a basis of simple roots. The subgroup of O(E N ) generated by the reflections in simple roots is the Weyl group W (E N ) of the root lattice. Since any reflection in α i extends to an isometry of the lattice I 1,N generated by e 0 , . . . , e 9−d , the group W (E N ) can be identified with the stabilizer subgroup O(I 1,N ) k N of the vector k N given in (3). The homomorphism , it is also surjective. If N = 7, 8, the image is a direct summand with complement of order 2 generated by the symmetry of the Dynkin diagram of the root lattice E N [7, 8.2].
Returning to our geometric situation, we see that the natural homomorphism Since the only automorphism of P 2 that fixes four points in general position is the identity, we have the following theorem [ In particular, via ρ, we can consider Aut(X) as a subgroup of W (E N ). We recall the wellknown structure of the relevant Weyl groups.
Theorem 2.2. The Weyl group of E N with 3 ≤ N ≤ 8 can be described as follows: is the orthogonal group of the quadratic space F 8 2 equipped with the quadratic form It turns out that, even in arbitrary characteristic, not all cyclic subgroups of W (E N ) are realized by automorphisms of del Pezzo surfaces for N ≥ 3. To determine which ones are realized, we will use the classification of conjugacy classes of elements w of the Weyl groups. This classification can be found in [3]. Table 3 gives the list of conjugacy classes of all Weyl groups in Carter's notation. According to Carter, these conjugacy classes are indexed by certain graphs that we call Carter graphs. The subscript in the name of a Carter graph indicates the number of vertices of this graph.
Recall that a root lattice R is the orthogonal sum of root lattices of irreducible root systems of types A n , D n , E 6 , E 7 , and E 8 . If the name of a Carter graph is the name of a Dynkin diagram, say associated to the orthogonal sum of irreducible root lattices R 1 , . . . , R k , then an element w of the corresponding conjugacy class in W (E N ) preserves a sublattice of E N isomorphic to R 1 ⊥ . . . ⊥ R k . Moreover, this w acts on each summand as the Coxeter element of W (R i ) and as the identity on the orthogonal complement of R 1 ⊥ . . . ⊥ R k . For example, the notation A i 1 + · · · + A i k (we write kA n if i 1 = · · · = i k = n) for a conjugacy class of an element w ∈ W (E N ) means that w leaves invariant a sublattice of E N isomorphic to the orthogonal sum A i 1 ⊥ . . . ⊥ A i k of root lattices of type A is , and it acts on each summand as an element of order The Carter graphs that contain cycles are named Γ(a i ), where Γ is the name of a Dynkin diagram. These graphs correspond to certain conjugacy classes in W (Γ) that do not leave any non-trivial sublattices invariant. We refer the reader to [3, Table 2] for a detailed description of these graphs.
The characteristic polynomials given in Table 3 allow us to compute the trace tr 2 (w) of w ∈ W (E N ) acting on the étale l-adic cohomology H 2 (X, Q l ) ∼ = Pic(X) Q l . Since w acts trivially on H 0 (X, Q l ) and H 4 (X, Q l ), and H i (X, Q l ) = 0 for odd i, we can use the Lefschetz fixed point formula Lef(g * ) = 2 + tr 2 (ρ(g)) = e(X g ) to compute the trace from the Euler-Poincaré characteristic of the fixed locus X g of a tame automorphism g ∈ Aut(X).
If p divides the order of g, we could instead use Saito's generalized Lefschetz fixed-point formula [15]. However, we are able to avoid an application of this rather technical tool and compute the traces by using geometric methods. This will allow us to determine the conjugacy classes of wild elements ρ(g) ∈ W (E N ). Remark 2.3. One can also derive the description of the conjugacy classes of elements of the Weyl groups W (E 9−d ) with d = 1, 2, 3 from [4]. The disadvantage of the notation there is that it is not uniform for different Weyl groups. The conjugacy classes in the Weyl group W (E 6 ) (resp. W (E 7 ), resp. W (E 8 )) can be read off from the notation of the conjugacy classes in the group S 4 (3) (resp. S 6 (2), resp. O + 8 (2)) in [4]. For example, a translation from Carter's notation to ATLAS notation for the group W (E 6 ) can be found in [10, Table 6].
2.3. The Geiser and Bertini involutions. It follows from the description of a del Pezzo surface of degree 2 (resp. 1) as a double cover that Aut(X) contains a central element γ (resp. β) realized by the negation of x 3 (resp. y). It is called the Geiser involution (resp. the Bertini involution) of X.
The image of γ (resp. β) under ρ in (5) is the unique non-trivial central element w 0 of W (E N ) for N = 7, 8. It is the unique element of W (E N ) of maximal length when written as the reduced product of reflections in simple roots. It acts as −id E N if N = 7, 8.

2.4.
Wild finite subgroups of Aut(P 1 ) and Aut(P 2 ). A finite subgroup of order n of an algebraic group over a field k of characteristic p > 0 is called wild (resp. tame or p-regular) if p|n (resp. (n, p) = 1). An element of finite order is wild (tame) if it generates a wild (tame) cyclic group.
The following proposition follows immediately from Theorem 2.1 and Theorem 2.2.
Proposition 2.4. Let g be a wild automorphism of order p of a del Pezzo surface of degree 1 or 2. Then p ∈ {2, 3, 5, 7}.
We will show later that p = 5, 7 (resp. p = 7) in the case of del Pezzo surfaces of degree 2 (resp. 1). It is known that there are no wild automorphisms of del Pezzo surfaces of odd order p 2 [8, Theorem 8].
The image of a wild group G of automorphisms of a del Pezzo surface of degree 2 under the homomorphism ψ is a wild subgroup of Aut(P 2 ). If G is a wild group of automorphisms of a del Pezzo surface of degree 1, then G acts on the pencil | − K X | and, if the image of a wild element of G is non-trivial, it defines a wild subgroup of Aut(P 1 ).
It is known that any finite subgroup G of GL n (k) is isomorphic to a finite subgroup of GL n (q) for some q = p m . The classification of wild finite subgroups of Aut(P 1 ) ∼ = PGL 2 (k) can be found in [18,Chapter 3,§6]. We summarize it here for the convenience of the reader.
Theorem 2.5. Let G be a proper wild subgroup of PGL 2 (k). Then G is isomorphic to one of the following groups.
(1) The group G ξ,A of affine transformations x → ξ t x + a, where a is an element of a finite subgroup A of the additive group of k containing 1 and ξ is a root of unity such that ξA = A.
The classification of wild subgroups of Aut(P 2 ) ∼ = PGL 3 (k) ∼ = PSL 3 (k) is also known. It follows from the classification of conjugacy classes of subgroups of the groups PSL 3 (q) [1], [13]. Again, we summarize it here for the convenience of the reader.
Theorem 2.6. Assume p > 2. Let G be a finite subgroup of PGL 3 (k). Then G is conjugate to one of the following groups.

DEL PEZZO SURFACES OF DEGREE ≥ 3
For the convenience of the reader, we recall here the classification of automorphisms groups of del Pezzo surfaces of degree at least 3.

3.1.
Del Pezzo surfaces of degree ≥ 5. The computation of automorphism groups of del Pezzo surfaces of degree ≥ 5 is characteristic free and can be found, for example, in [7, Chapter 8]: 3.1.1. Degree 9. If d = 9, then X ∼ = P 2 and |K X | = |O P 2 (3)|. The embedding ν : P 2 ֒→ P 9 coincides with a Veronese embedding. The group of automorphisms of X coincides with the projective linear group PGL 3 (k).
If X ∼ = P 1 ×P 1 , the anti-canonical map is given by the linear system |−K X | = |O P 1 ×P 1 (2, 2)|. It coincides with the Segre-Veronese map and embeds P 1 × P 1 into P 8 . The group Aut 0 (X) coincides with PGL 2 (k) × PGL 2 (k), and the quotient group Aut(X)/ Aut 0 (X) is of order 2, generated by switching the factors of the product P 1 × P 1 .
If X ∼ = F 1 = Bl p 1 (P 2 ), the anti-canonical system is | − K X | = |O P 2 (3) − p 1 |. The anticanonical linear system defines a map that embeds X into P 8 . Its image is equal to the projection of the Veronese surface of degree 9 from a point lying on it. The group Aut(X) is isomorphic to the stabilizer subgroup of {p 1 } in PGL 3 (k) = Aut(P 2 ). It is a solvable algebraic subgroup of PGL 3 (k) of dimension 6.
3.1.3. Degree 7. If d = 7, then X ∼ = Bl p 1 ,p 2 (P 2 ) and | − K X | = |O P 2 (3) − p 1 − p 2 |. The anti-canonical linear system defines a map that embeds X into P 7 . The image is the projection of the Veronese surface of degree 9 from two points lying on it. The group Aut(X) is isomorphic to the stabilizer subgroup of {p 1 , p 2 } in PGL 3 (k) = Aut(P 2 ). It is a solvable algebraic subgroup of PGL 3 (k) of dimension 4. The quotient group Aut(X)/ Aut 0 (X) is of order 2. It is generated by a projective involution that switches the points p 1 , p 2 .
3.1.4. Degree 6. If d = 6, then X ∼ = Bl p 1 ,p 2 ,p 3 (P 2 ) and |−K X | = |O P 2 (3)−p 1 −p 2 −p 3 |, where the points p 1 , p 2 , p 3 are not collinear. The group Aut 0 (X) is isomorphic to the two-dimensional torus G 2 m,k . In appropriate projective coordinates where The quotient group is isomorphic to S 3 × 2. It is generated by projective transformations which permute the points and the standard quadratic Cremona transformation ( In this case Aut 0 (X) = {1} and Aut 0 (X) ∼ = S 5 . The group is generated by its subgroups of projective transformations that permute the four points and also a quadratic Cremona transformation of order 5 where we use projective coordinates such that p 1 , p 2 , p 3 are as in the previous case, and p 4 = [1, 1, 1].

3.2.
Quartic del Pezzo surfaces. This is the first case where the classification of automorphism groups depends on the characteristic p of the ground field. We assume p = 2 and refer the reader to [11] for the case of p = 2.
A quartic del Pezzo surface X is isomorphic to the blow-up Bl p 1 ,...,p 5 (P 2 ), where no three points are collinear. The anti-canonical linear system |−K X | = |O P 2 (3)−p 1 −p 2 −p 3 −p 4 −p 5 | embeds X into P 4 as a complete intersection of two quadrics.
If p = 0, the classification of automorphism groups can be found in [7, 8.6] and if p = 0, it was accomplished in [10, §3]. Since p = 2, the two quadrics can be simultaneously diagonalized so that X is given by equations where the binary form has no multiple roots. The automorphism group contains a normal subgroup H, isomorphic to 2 4 and generated by the transformations (t 0 , . . . , t 4 ) → (±t 0 , . . . , ±t 4 ). The quotient group Aut(X)/H is isomorphic to the subgroup G of PGL 2 (k) ∼ = Aut(P 1 ) that leaves the set V (∆) invariant. The classification is summarized in Table 5 in the Appendix. There, the first column refers to the values of the parameters a and b in Equation (8) above. The conjugacy classes of elements of Aut(X) can be obtained by combining [10, Table 2] and [3, Table 5].

Cubic surfaces.
A del Pezzo surface of degree 3 is isomorphic to the blow-up of 6 points p 1 , . . . , p 6 , where no three points are collinear and not all of them lie on a conic. The anti- Its image is a cubic surface.
As we remarked in the introduction, the classification of automorphism groups of smooth cubic surfaces in characteristic 0 has been known essentially since the 19th century (see the most recent exposition in [7, 9.5]). In the case of positive characteristic it can be found in [10]. We present the results of this classification in Table 6 in the Appendix. Our table is obtained by translating [10, Table 7] from ATLAS to Carter notation. We also give the name of the corresponding type in [7, Table 9.6] for the convenience of the reader.

DEL PEZZO SURFACES OF DEGREE 2
In this section, we are concerned with the classification of automorphism groups of del Pezzo surfaces of degree 2. Recall from Section 2.1 that every del Pezzo surface X of degree 2 is a double cover of P 2 branched over a smooth quartic C. Moreover, we have the product decomposition Aut(X) ∼ = 2 × Aut(C), which follows from the corresponding decomposition of W (E 7 ) (see Theorem 2.2), so classifying automorphism groups of del Pezzo surfaces of degree 2 is equivalent to classifying automorphism groups of smooth plane quartic curves.  Proof. Since p is odd, g leaves invariant the plane quartic C. There are two possible Jordan forms for a cyclic linear automorphism of P 2 of order p, namely (x 0 , x 1 , Since g(C) = C, and the cyclic group (g) admits no non-trivial characters, the equation of C is g-invariant.
By [19,Section 10.3], the invariant ring in the first case is defines a quartic which is singular at [1, 0, 0]. Therefore, we must have p = 3. Proof. By Lemma 4.1, this holds if p = 5. For p = 5, we can lift the curve to characteristic 0 together with its automorphism of order 5. But in characteristic 0, there are no smooth quartics with automorphisms of order 5, a contradiction.
Since we also know that Aut(C) does not contain automorphisms of order p 2 , we obtain the following.
Corollary 4.3. Let G be a finite group that acts faithfully on a smooth plane quartic. Then, the following hold.
Proof. If G is wild, then it contains C p , so p = 3 by Lemma 4.1. Since G embeds into W (E 7 ) via ρ and the only prime divisors of |W (E 7 )| are 2, 3, 5, and 7 by Theorem 2.2, the same holds for |G|. By Corollary 4.2, the divisor 5 does not occur, and the claim follows.
The restrictions |G| obtained thus far allow us to show that the list of groups in Theorem 2.6 that can leave a smooth plane quartic invariant is relatively short. (1) Only if p = 3: The group PSU 3 (9).
Proof. In the following, we study the items in Theorem 2.6 and exclude most of them using Corollary 4.3. Recall from Theorem 2.2 that and that, by Corollary 4.3, the only prime divisors of |G| are 2, 3, and 7.
Since p = 3, we can diagonalize this subgroup. It is elementary to check that there are no smooth quartics that are invariant under a diagonal action of 3 2 .
In the following sections, we analyze these four types of groups in detail and check which of them are realized as automorphism groups of smooth plane quartics. We will focus on the wild groups first and treat the tame groups afterwards in Section 4.5.
4.2. The groups PSU 3 (9) and L 2 (7). It is well-known that, over C, the Klein quartic curve defined by the equation is the unique (up to isomorphism) smooth plane quartic curve with group of automorphisms of maximal possible order equal to 168. Its group of automorphisms is isomorphic to the simple group L 2 (7). Less known is that it is isomorphic to two members of the pencil of plane quartic curves corresponding to the parameters t = 3 2 (−1 ± √ −7) (see [7, 6.5.2]). The change of coordinates realizing this isomorphism is given by where a = 1 + ζ 7 + ζ 2 7 + ζ 3 + ζ 5 7 , b = ζ 2 7 + ζ 7 and ζ 7 is a primitive 7-th root of unity [12, p.56]. In our case, when p = 3, we can still use this transformation and obtain an isomorphism between the Klein curve (9) and the Fermat quartic given by the equation Since automorphisms of order 7 are tame, the proof of [7, Lemma 6.5.1] shows the following.
Lemma 4.5. Let C be a smooth plane quartic with an automorphism of order 7. Then, C is isomorphic to the Klein quartic Let C be a smooth plane quartic. Then, the following are equivalent: (1) C 7 ⊆ Aut(C).
(7) C can be defined by the equation If p = 3, then Theorem 4.4 shows that L 2 (7) is maximal among all subgroups of PGL 3 (k) preserving a smooth quartic, hence (2) ⇒ (4). The converse is clear.
Remark 4.7. The Fermat quartic curve in characteristic 3 can be characterized among smooth plane quartics by the property that it has infinitely many inflection points, or equivalently, its Hessian is identically zero [14, Proposition 3.7].

4.3.
Wild subgroups of (k × ) 2 : S 3 . In this section, we classify the wild subgroups of (k × ) 2 : S 3 that leave invariant smooth plane quartics. So, we have to classify smooth quartics that are invariant under a cyclic permutation of coordinates in characteristic p = 3.
The element g belongs to the conjugacy class of Type Proof. Computing the equation of C is straightforward. To compute the conjugacy class of g, we can argue as follows: Since C has 28 bitangents and g has order 3, the automorphism of the del Pezzo surface X induced by g preserves a (−1)-curve, hence g is induced by an automorphism g ′ of a cubic surface and X is obtained by blowing up a fixed point of g ′ . As g has only one fixed point on P 2 , the automorphism g ′ also has only one fixed point. By [10,Theorem 10.4] this implies that the conjugacy class of g ′ is of Type 2A 2 .
It turns out that every smooth quartic which is invariant under a cyclic permutation of coordiantes is projectively equivalent to one that is invariant under arbitrary permutations of coordinates. We remark that this happens in characteristic 0 as well (see [7, Proof of Theorem 6.5.2]).
which is conjugate to the group S 3 of permutation matrices. In particular, C admits an equation of the form It suffices to prove that the statement holds for a general C in the family described in Lemma 4.8. With notation as in that lemma, the quartics where b = c are invariant under S 3 . The centralizer Z of g in GL 3 (k) consists of matrices of the form with α + β + γ = 0, and the centralizer of the full group S 3 is the subgroup where β = γ. So, if we conjugate the equations where b = c with elements of Z with β = γ, we obtain a general C in the family. Hence, S 3 ⊆ Aut(C) holds for a general C, hence for every C, in the family of Lemma 4.8. Now, the Fermat quartic (11) is a member of the family of quartics that occur in Lemma 4.9. The following lemma gives us a criterion to detect it. Proof. Computing the Hessian of this curve, we find that it is equal to zero if and only if c = d = 0. We conclude using Remark 4.7.
Lemma 4.11. Let C be a smooth plane quartic in characteristic p = 3. Assume that Aut(C) is conjugate to a subgroup of (k × ) 2 : S 3 and S 3 Aut(C). Then, C can be defined by an equation of the form with a ∈ k \ F 3 . Conversely, a quartic C defined by an equation of this form satisfies Aut(C) ∼ = 2 2 : Proof. We assume that Aut(C) ∼ = H : S 3 , where H is non-trivial and acts diagonally, and S 3 is the group of permutation matrices. Thus, we may assume that C is given by an equation as in Lemma 4.9. The polynomial is not semi-invariant under any diagonal automorphism, hence d = 0. By Lemma 4.10, we have c = 0 and we can assume c = 1 by rescaling the coordinates. A diagonal automorphism for which x 2 0 x 2 1 + x 2 1 x 2 2 + x 2 2 x 2 0 is semi-invariant has order 2 and since H is non-trivial and C 3 ⊆ Aut(C), we deduce that H ∼ = 2 2 . But then the polynomial This shows that C is given by the stated equation and a ∈ F 3 follows from the Jacobian criterion.
It remains to show that Aut(C) = 2 2 : S 3 ∼ = S 4 . By Lemma 4.10, Corollary 4.6, and Theorem 4.4, it suffices to show that Aut(C) is not conjugate to a subgroup of k 2 : GL 2 (k). If it were, then it would have to act through GL 2 (k), since S 4 does not admit any non-trivial normal subgroup of order a power of 3. In particular, the fixed locus of an element of order 3 in S 4 would be a line and this is not the case in our situation. Hence, Aut(C) ∼ = S 4 .

4.4.
Wild subgroups of k 2 : GL 2 (k). We already saw one conjugacy class of wild automorphisms in Lemma 4.8. The other conjugacy class is studied in the following lemma.
with f i homogeneous of degree i. The element g belongs to the conjugacy class of type 3A 2 .
Proof. The equation of C follows from the structure of the invariant ring of g (see the proof of Lemma 4.1). The closed subscheme of fixed points of g is V (x 1 ). Our automorphism leaves the bitangent line given by the equation f 1 (x 1 , x 2 ) = 0 invariant. Since g is of order 3, it also leaves each irreducible component of the preimage of V (f 1 ) in X invariant. Thus, blowing down one of the components of this preimage, we obtain an automorphism of order 3 of a cubic surface that fixes a plane section (the preimage of V (x 1 )) pointwise. It follows from [10,Theorem 10.4] that this automorphism belongs to the conjugacy class 3A 2 in W (E 6 ). Thus, the corresponding automorphism of X is in the conjugacy class 3A 2 in W (E 7 ).
Lemma 4.13. Let G be a wild subgroup of k 2 : GL 2 (k). If 9 | |G| and G preserves a smooth plane quartic C, then C is given by an equation of the form for some c ∈ k. Any plane quartic curve C with such an equation is smooth. If c = 0, then Aut(C) ∼ = PSU 3 (9), and if c = 0, then Aut(C) ∼ = H 3 (3) : 2.
Proof. Since G contains no elements of order 9 [8, Theorem 8], it must contain two commuting elements g 1 , g 2 of order 3 that generate a subgroup K of order 9. Since K preserves a bitangent, it acts faithfully on a cubic surface, hence, by [10], it embeds into the Heisenberg group H 3 (3).
Since K is abelian of order 9 and H 3 (3) is a non-split extension of 3 2 by its center, the group K contains the center of H 3 (3), hence it contains elements of conjugacy class 3A 2 in W (E 6 ). By [10], there are at most two elements of conjugacy class 3A 2 in an automorphism group of a cubic surface, so the other elements of K are of conjugacy class 2A 2 . Back on C, this means that K contains automorphisms of both conjugacy classes 2A 2 and 3A 2 . Without loss of generality, we may assume that the g i are upper triangular matrices, that g 1 is of class 2A 2 , and that g 2 is of class 3A 2 . Up to conjugation by an upper triangular matrix, we may assume that g 1 acts as x 0 → x 0 + x 1 , as x 1 → x 1 + x 2 , or as x 0 → x 0 + x 2 . In the first two cases, the centralizer of g 1 in the group of strict upper triangular matrices consists of automorphisms of class 3A 2 , hence we are in the third case. Analogously to Lemma 4.12, we deduce that C is given by an equation of the form After conjugating by a suitable upper triangular matrix, we may assume that the automorphism g 2 acts as (x 0 , x 1 , x 2 ) → (x 0 + x 1 , x 1 + µx 2 , x 2 ) for some µ = 0. Since C is invariant under g 2 , it is given by an equation of the form , and then scale a to 1. These substitutions commute with g 1 , so C is still invariant under g 1 , but they change the shape of g 2 . However, the equation is now in the simpler form x 2 ) with λ 9 − λ 3 c + λ = 0 and µ 3 − µ + λ 12 + cλ 6 = 0. These substitutions form a group isomorphic to H 3 (3). We also see that C is invariant under the involution τ : It is known that a subgroup of Sp 6 (2) = W (E 7 )/ ±id isomorphic to H 3 (3) : 2 is conjugate to a subgroup of H 3 (3) : 8 which is realized as a maximal subgroup of PSU 3 (9) [4, p.14]. So, if Aut(C) = PSU 3 (9), then H 3 (3) is normal in Aut(C) and Aut(C) contains an element τ ′ whose square is τ . In particular, τ ′ acts on the fixed locus V (x 2 ) of g 1 and this action is non-trivial, since τ acts non-trivially on V (x 2 ). Since Aut(C) fixes the point V (x 2 ) ∩ C = [1, 0, 0], Theorem 2.5 shows that τ ′ generates a subgroup of k × of order 4 which normalizes Hence, in the following, we may assume that |H| = 3. There are three conjugacy classes of elements of order 3 in k 2 : GL 2 (k), namely In each case, the group G is contained in the normalizer N i of g i in k 2 : GL 2 (k). We have In each case, the centralizer of g i has index 2 and is obtained by setting γ = 1, β = 1, or γ = 1, respectively. The element g 2 is conjugate in PGL 3 (k) to g 1 via a cyclic permutation of coordinates and this conjugation maps N 2 into N 1 , so we only have to study the cases where G contains g 1 or g 3 . If G leaves a smooth quartic invariant, then |G| = 2 n · 3 for some n ≥ 0. We settled the cases where n = 0 in Lemma 4.12 and Lemma 4.8. If n ≥ 1, then G contains an involution. Next, we will study involutions in N 1 and N 3 .
Assume that G contains g 1 and leaves invariant a smooth quartic C. Every involution in N 1 is conjugate to one where α = β = 0. Then, it lies in the group of lower triangular matrices in GL 2 (k), and thus, after conjugating by a lower triangular matrix, we may assume that it acts diagonally. Thus, we have the three representatives In N 1 , the elements τ 1 , τ 2 , τ 3 represent the conjugacy class of involutions with parameters (γ, ε) = (−1, 1), (1, −1), and (−1, −1), respectively. Lemma 4.14. Let C be a smooth plane quartic in characteristic p = 3. Assume that C is invariant under G ⊆ k : GL 2 (k), G contains g 1 and 9 ∤ |G|. If Aut(C) contains an involution, then C is one of the quartics of Lemma 4.13.
Proof. In Lemma 4.12, we have determined the g 1 -invariant quartics. Determining the equations for the quartics which are additionally τ i -invariant is straightforward. Now, it remains to show that every τ i -invariant C occurs in the family of Lemma 4.13.
Assume G contains τ 1 . Then, either the equation of C is τ 1 -anti-invariant and then C contains the line V (x 1 ), or C is given by an equation of the form . This curve is singular at the point [α, 0, 1] with aα 3 + d = 0.
Assume G contains τ 2 . Then, either C is reducible or given by an equation of the form Rescaling coordinates, we may assume a = 1 and then we can use a substitution of the form x 0 → x 0 + αx 1 to set b = 0. After rescaling x 2 and swapping x 1 and x 2 , we obtain the normal form of Lemma 4.13. Finally, assume that G contains τ 3 . Then, In both cases, C is singular.
Finally, assume that G contains g 3 and 9 ∤ |G|. The unique conjugacy class of involutions in N 3 is represented by τ : (x 0 , x 1 , x 2 ) → (x 0 , −x 1 + x 2 , x 2 ). All elements of N 3 which are not in this conjugacy class have order 1 or 3, hence G ⊆ S 3 with equality if and only if G is conjugate to g 3 , τ , which, in turn, is conjugate in GL 3 (k) to the group of permutation matrices. Hence, this case is reduced to Lemma 4.9.
4.5. Tame automorphism groups of smooth plane quartics. As we mentioned in the introduction, the pair (X, Aut(X)) can be lifted to characteristic zero if Aut(X) is tame. Since X does not admit global vector fields, specialization of automorphisms is injective, hence the lift X ′ of X to characteristic 0 has the same automorphism group as X.
In particular, this means that the list of tame groups that can be realized as automorphism groups of del Pezzo surfaces of degree 2 in positive characteristic is contained in the list of [7, Theorem 6.5.2]. To finish the classification of tame groups, we have to show that the equations in this list have smooth reduction modulo p and that the automorphism group of the reduction cannot be larger for a general choice of parameters. In the following, we explain how to do this for the first five groups in the list of [7, Theorem 6.5.2].
• L 2 (7): This group is tame if p = 3, 7. It is realized by the Klein quartic in every such characteristic. • 4 2 : S 3 : This group is tame if p = 3. Its order is 96, so by Theorem 4.4, it is maximal among all groups that can occur as automorphism groups of smooth plane quartics in these characteristics. It is realized by the Fermat quartic.  Table 6.1] is smooth if p = 3, so this group is realized. • S 4 : This group is tame if p = 3. The equation given in [7, Table 6.1] defines a 1dimensional family of smooth quartics with an action of this group in any such characteristic. Since all smooth quartics with larger automorphism group are unique, a generic choice of parameters will yield a smooth quartic with automorphism group S 4 . • 4.2 2 : This group is tame. Again, the equation given in [7, Table 6.1] defines a 1dimensional family of smooth plane quartics with an action of this group in any characteristic. By the above and Table 1, all smooth quartics whose automorphism group contains 4.2 2 are isolated, hence a generic member of this 1-dimensional family has Aut(C) = 4.2 2 . The other cases work similarly and are left to the reader. We conclude that all groups in the list of [7, Theorem 6.5.2] occur as the full automorphism group of some smooth plane quartic in characteristic p if p does not divide the order of the group. 4.6. Wild automorphism groups of smooth plane quartics. Here, we summarize the classification of wild groups of automorphisms of smooth plane quartics by collecting the results of the previous sections. We use the notation of [7, Table 6.1] for wild groups in characteristic p that also occur in characteristic 0. After a suitable change of coordinates, Equation (16) is the reduction modulo 3 of the family of Type VIII in [7, Table 6.1], hence we call it Type VIII. The Klein quartic in characteristic 3 is the reduction modulo 3 of the three quartics of type I, II, and III in [7, Table 6.1]. Now, our classification of wild groups can be summarized as follows: Type Aut(C) Order Equation # Parameters Conditions

Conjugacy classes.
As explained in Section 2, the conjugacy class of a tame automorphism g of a del Pezzo surface X can be determined using the Lefschetz fixed point formula. Let g be the composition of g with the Geiser involution. Then, by [7, Lemma 6.5.1], the pair (e(X g ), e(Xḡ)) depends only on the Jordan form of the automorphism of P 2 induced by g and g, hence so does the pair of conjugacy classes of g and g. By Lemma 4.8 and Lemma 4.12, the pair of conjugacy classes of g and g is also uniquely determined by the Jordan form in P 2 if g has order 3, and it is not hard to extend this to the case where g is wild of higher order. We summarize the translation between Jordan forms and pairs of conjugacy classes in Carter notation in Table 4 in the Appendix. In that table, we give the orders of g andḡ in the first column, the Jordan form of the induced automorphism of P 2 in the second column, the conjugacy classes of g/ḡ in the third column, and the traces of the actions of g andḡ on E 7 in the last column.
Using this dictionary, one can determine the conjugacy classes in W (E 7 ) of all automorphisms of del Pezzo surfaces of degree 2. The result is summarized in Table 7 in the Appendix. In that table, we give the name of the relevant family of del Pezzo surfaces in the first column, following [7, Table 6.1]. In the second column, we note the characteristics in which the family occurs. The third and fourth columns give the group Aut(X) and its order. The remaining columns give the number of elements of a given Carter conjugacy class in Aut(X).

DEL PEZZO SURFACES OF DEGREE 1
In this section, we classify automorphism groups of del Pezzo surfaces X of degree 1. Recall that X is a double cover of the quadratic cone P(1, 1, 2) ⊆ P 3 branched over a sextic curve C. Moreover, we have Aut(X) ∼ = 2. Aut(C), but, in contrast to the case of degree 2, this central extension is not necessarily split, so the classification of automorphism groups of del Pezzo surfaces of degree 1 is more complicated than the classification of automorphism groups of possible branch curves C.
5.1. The elliptic pencil. As we have already observed in Section 2, the blow up of the unique base point o of a del Pezzo surface X of degree 1 has the structure of an elliptic surface π : Y → P 1 with Weierstrass equation We have Since −K X is ample, the fibration has no reducible fibers. If p = 3, and the absolute invariant j(F η ) of the generic fiber F η of π (considered as an elliptic curve over the field k(P 1 ) of rational functions on the base of the fibration) is equal to zero, we may assume that a 2 = 0. In this case a 4 = 0 (otherwise F η is singular) and a 6 = 0 (otherwise the surface X has singular points over V (a 4 )).
Obviously, any g ∈ Aut(X) fixes the point o and hence lifts to an automorphismg of Y that preserves the corresponding section S. Conversely, any automorphism of Y leaving invariant the section S descends to X. Thus We may viewφ as the natural action of Aut(X) on the base of the fibration π.
In the following, we collect some preliminary restrictions on the groups H and P . The group P is a subgroup of PGL 2 (k), so we can use the classification of finite subgroups of PGL 2 (k) to study it. The classification of finite tame subgroups of PGL 2 (k) coincides with the well-known classification of finite subgroups of PGL 2 (C): The finite subgroups of PGL 2 (C) are exactly the polyhedral groups, that is, the groups C n , D n , A 4 , S 4 , and A 5 . The classification of wild subgroups of PGL 2 (k) is contained in Theorem 2.5.
The group H is an extension H 0 : C n , where C n is a cyclic subgroup of order n prime to p. Since p is odd, C n contains the Bertini involution, so 2 | n. We can also describe H as the automorphism group of the generic fiber F η of the elliptic fibration. The structure of this group is well-known (see e.g. [17, Appendix A]) and we have the following normal forms: Proof. The first three cases are well-known, and H acts as (x, y, t 0 , t 1 ) → (u 2 x, u 3 y, t 0 , t 1 ), where u n = 1 with n = 2, 4, 6, respectively. In Case (iv), the automorphism group of the geometric generic fiber of π is the alternating group A 4 . To show that H 0 ∈ {1, C 3 } and H/H 0 = C 2 , it thus suffices to show that no automorphism of order 4 can descend to F η . Assume that F η admits an automorphism of order 4. Its fixed locus on X is a 2-torsion section of the elliptic pencil π : Y → P 1 , which, after a suitable change of the x-coordinate, we may assume to be given by x = y = 0. In other words, X admits a Weierstrass equation of the form This equation is singular over the roots of a 4 , so X is not a del Pezzo surface.
By  Proof. By Theorem 2.2, the prime divisors of the order of W (E 8 ) are 2, 3, 5, and 7. So, it suffices to exclude p = 7. An automorphism g of order 7 acts on the set of 120 tritangent planes preserving one of them. It fixes each of the (−1)-curves in the preimage pair. Blowing one of them down, we obtain that g descends to a wild automorphism of a del Pezzo surface of degree 2. However, by Corollary 4.6, we know that there are no wild automorphisms of order 7 on del Pezzo surfaces of degree 2.
As in the case of degree 2, the restrictions obtained so far allow us to give a preliminary list of possible automorphism groups of del Pezzo surfaces of degree 1. Proof. By Lemma 5.1, we have G ∼ = H : P , where H is a cyclic group and P ⊆ PGL 2 (k).
If G is tame, then it lifts to characteristic 0, so P embeds into PGL 2 (C), hence it is a polyhedral group. If G is wild, then p = 3, 5 by Lemma 5.2. The description of the possible P follows from Theorem 2.5.
In the following sections, we first classify the wild automorphism groups of del Pezzo surfaces of degree 1 in characteristic 5 and 3. Afterwards, we study the tame automorphism groups using the approach of Section 4.5 in Section 5.5.

5.3.
Wild automorphism groups in characteristic 5. In this section, we classify wild automorphism groups of del Pezzo surfaces of degree 1 in characteristic p = 5.
Theorem 5.4. Let X be a del Pezzo surface of degree 1 in characteristic p = 5. Assume that G = Aut(X) is wild. Then, one of the following cases occurs.
(i) G ∼ = 2.D 10 and X is given by Proof. Let g be a wild automorphism of order 5. Since g ∈ H = Ker(φ) by Lemma 5.1, it acts non-trivially on the base of the elliptic fibration. As g has one fixed point in P 1 , we may assume this point to be [0, 1], and that g acts by (t 0 , t 1 ) → (t 0 , t 0 + t 1 ). The roots of a 4 coincide with the roots of the j-invariant, hence they are preserved by g. But a 4 has degree 4, so a 4 = c 1 t 4 0 for some c 1 ∈ k. Then, the discriminant of the elliptic fibration is given by ∆ = 4a 3 4 + 27a 2 6 , so a 6 is preserved by g as well and t 2 0 ∤ a 6 , for otherwise ∆ would vanish with multiplicity at least 3 at [0, 1] and the fiber over this point would be reducible. Thus, choosing coordinates such that a 6 has a root at [1, 0], we may assume that In other words, the surface X admits the equation If c = 0, then j(F η ) ∈ {0, 1728}, hence H ∼ = C 2 . The group P =φ(Aut(X)) ⊆ Aut(P 1 ) preserves V (a 4 ) = [0, 1] and the set V (a 6 ) = P 1 (F 5 ), hence P ⊆ C 5 : C 4 . Since c = 0, the discriminant ∆ has a double root at [0, 1] and 10 simple roots. Now, P cannot be C 5 : C 4 , since the square of an automorphism of order 4 would fix V (a 4 ) and at least two simple roots of ∆, hence all of P 1 . However, P contains an involution lifting to X as an automorphism of order 4 given by (x, y, t 0 , t 1 ) → (−x, λy, t 0 , −t 1 ) with λ 4 = 1. Hence, in this case, Aut(X) is the binary dihedral group of order 20.

5.4.
Wild automorphism groups in characteristic 3. The case p = 3 is more difficult. If Aut(X) is a wild group, then so is H or P . Recall that X is given by an equation of the form The discriminant ∆ and j-invariant j of the associated elliptic fibration π are ∆ = −a 3 2 a 6 − a 2 2 a 2 4 − a 3 4 j = a 6 2 ∆ We treat the cases j(F η ) = 0 and j(F η ) = 0 separately. Theorem 5.5. Let X be a del Pezzo surface of degree 1 in characteristic p = 3 such that Aut(X) is wild. Assume that j(F η ) = 0. Then, X admits one of the following two Weierstrass equations: with a, b, c = 0 and ac = 1, or if X is given by Equation (22), then Aut(X) = C 6 .
(2) if X is given by Equation (23), then Aut(X) = C 2 × C 2 3 . Proof. Note that j(F η ) = 0 is equivalent to a 2 = 0. Moreover, H = C 2 , so if Aut(X) is wild, then P is wild. We may assume that P contains the substitution g : t 1 → t 1 + t 0 . Since, the roots of a 2 are the roots of the j-map, a 2 is g-invariant, hence, after rescaling x, y, we may assume that a 2 = t 2 0 . Note that, under this assumption on a 2 , substitutions of the form x → x + b 2 preserve a 2 and the class a 4 of a 4 modulo t 2 0 . Now, a suitable substitution in x allows us to eliminate the monomials t 4 0 , t 3 0 t 1 , and t 2 0 t 2 1 in a 4 , so we can write a 4 = at 0 t 3 1 + bt 4 1 . As a 4 is g-semi-invariant, we must have b = 0. After scaling t 0 , we may assume that a 4 = ǫt 0 t 3 1 , where ǫ ∈ {0, 1}, and that a 2 = at 2 0 for some a = 0. Since g lifts to X and g has odd order, there is an element of order 3 in Aut(X) that induces g. By [17, Appendix A, Proposition 1.2], this means that there exists b 2 ∈ k[t 0 , t 1 ] 2 such that the equation of X is preserved by (y, x, t 0 , t 1 ) → (y, x + b 2 , t 0 , t 1 + t 0 ). This b 2 satisfies a 4 (t 0 , t 1 ) = a 4 (t 0 , t 1 + t 0 ) − ab 2 t 2 0 (24) If ǫ = 0, Equation (24) shows that b 2 = 0 and comparing the coefficients of t 2 0 t 4 1 in Equation (25) shows that a 6 does not contain the monomial t 0 t 5 1 . Computing the partial derivatives, we see that this implies that X is singular over [0, 1].
We obtain the normal form If ad = 1, a substitution of the form (t 1 , x) → (t 1 + λt 0 , x + a −1 (λ 3 − λ)) can be used to set c = 0. This yields Equation (22) after renaming d to c. Computing the partial derivatives, we see that Equation (22) defines a smooth surface if and only if b, c = 0 and bc = 1. Moreover, note that a = 0 is equivalent to j(F η ) = 0. The discriminant of this equation is ). To compute Aut(X), note that the associated elliptic fibration π has a unique cuspidal fiber over [0, 1], so every element of P acts as h : t 1 → At 1 + Bt 0 . Comparing what happens to the coefficients of x 3 , t 0 x 2 , and t 0 t 3 1 x under such a substitution, we deduce that A = 1. In particular, ∆ must be h-invariant. The coefficient of Since a 2 − a 3 c = 0, this shows that B 3 = B, hence h is a multiple of g, so Aut(X) = C 6 .
If ad = 1, we can apply a substitution of the form ) for a suitable λ and then rescale t 0 and t 1 to obtain Equation (23). Note that the rescaling of t 0 and t 1 does not necessarily preserve our description of g. To calculate Aut(X), we use that, by the same argument as in the previous paragraph, every element of P acts as h : t 1 → t 1 + Bt 0 . Analogously to Equation (25), the condition that an automorphism with A = 1 lifts to X is that The only non-trivial condition in the second equation is for the coefficient of t 6 0 , where we get 0 = B 9 + aB 3 + bB.
Since b = 0, this equation has exactly 9 solutions, hence P = C 2 3 and Aut(X) = C 2 × C 2 3 . Theorem 5.6. Let X be a del Pezzo surface of degree 1 in characteristic p = 3 such that Aut(X) is wild. Assume that j(F η ) = 0.
(1) If H is tame, then H = C 2 and X admits a Weierstrass equation of the form  Proof. Recall that j(F η ) = 0 implies that X admits a Weierstrass equation of the form Case 1: H is tame.
Since Aut(X) is wild and H = C 2 by Lemma 5.1, the group P is wild. We may assume that P contains the substitution g : t 1 → t 1 + t 0 . The ring of invariants k[t 0 , t 1 ] (g) is generated by t 0 and t 1 (t 2 1 − t 2 0 ). Since ∆ = −a 3 4 , g preserves the set of roots of a 4 and since g is wild, it has only one fixed point on P 1 . On the other hand, by Lemma 5.1, a 4 is not a square. So, a 4 has 4 distinct roots, and we may assume that [0, 1] and [1,0] are among them, so that a 4 = t 0 t 1 (t 2 1 − t 2 0 ) and, after simplifying, a 6 = at 6 0 + bt 4 0 t 2 1 + ct 2 0 t 4 1 + dt 6 1 . Since g lifts to X, there exists b 2 = αt 2 0 + βt 0 t 1 + γt 2 1 ∈ k[t 0 , t 1 ] 2 such that b 3 2 + a 4 (t 0 , t 0 + t 1 )b 2 + a 6 (t 0 , t 0 + t 1 ) = a 6 (t 0 , t 1 ). Comparing the coefficients, we obtain This yields the normal form in the theorem. Taking partial derivatives, we find that the Weierstrass surface X is smooth if and only if a, b = 0.
It remains to determine the full group P . Since P acts faithfully on the set of four roots of ∆ = −a 3 4 , Theorem 2.5 shows that P ∈ {C 3 , 3 : C 2 , 3 : If P ∈ {3 : C 2 , 3 : C 4 , PGL 2 (F 3 )}, then P contains an involution that normalizes the subgroup generated by g and such that conjugation by this involution maps g to g −1 . Without loss of generality, we may assume that this involution is (t 0 , t 1 ) → (t 0 , −t 1 ). Then, an explicit computation shows that such an involution never lifts to X.
Case 2: H is wild, a 4 has more than one root.
Since H is wild, Lemma 5.1 shows that a 4 is a square. Since it has more than one root, we may assume that a 4 = −t 2 0 t 2 1 . A suitable substitution in x allows us to describe X by an equation of the following form: As in the previous case, computing the partial derivatives, we find that a, d = 0, so we can rescale d to 1. By Lemma 5.1, we have H = C 6 . It remains to determine P . After replacing elements of g by scalar multiples, we may assume that g * (a 4 ) = a 4 for every g ∈ P . Note that such a g either acts diagonally or anti-diagonally, hence g * (a 6 ) does not contain the monomials t 6 0 , t 3 0 t 3 1 , and t 6 1 . The condition that g can be lifted to an automorphism of X is that there exists b 2 = s 1 t 2 0 + s 2 t 0 t 1 + s 3 t 2 1 such that b 3 2 + a 4 b 2 + g * (a 6 ) = ∓a 6 .
Comparing the coefficient of t 6 0 , t 3 0 t 3 1 , and t 6 1 , we find that b 2 = st 0 t 1 with s(s 2 − 1) = 0. Now, assume first that g acts as (t 0 , t 1 ) → (αt 0 , α −1 t 1 ) with α 2 = 1. Comparing the coefficients in Equation (30), we obtain where the choice of sign is compatible with Equation (30). If b = 0 or c = 0, then α 2 = −1 and the sign on the right-hand side of Equation (30) is a minus, so that α −4 + 1 = 0 cannot hold. Hence, g can exist only if b = c = 0. In this case, the unique condition for the liftability of g is α 8 = 1, so the group of such g determines a cyclic subgroup of order 4 in PGL 2 (k). Moreover, note that if g has order 4, then every lift of g to X with b 2 = 0 has order 8.
Case 3: H is wild, a 4 has exactly one root.
Since H is wild, Lemma 5.1 shows that a 4 is a square. Since it has exactly one root, we may assume that a 4 = −t 4 0 . Since X is smooth, we must have t 2 0 ∤ a 6 . A suitable substitution in x allows us to describe X via an equation of the following form: Taking partial derivatives, we see that X is always smooth. Using a substitution of the form t 1 → t 1 + λt 0 , x → x + αt 2 0 + βt 0 t 1 , we may assume additionally that c = 0. By Lemma 5.1, we have H = C 6 . It remains to determine P .
If B = 1, then the transformation is the identity. If B = −1, then b = 0 and, conversely, if b = 0, the transformation with B = −1 lifts to X (as an automorphism of order 4). If B is a primitive 5-th root of unity, then a = b = 0 and again, the transformation lifts to X under these conditions. 5.5. Tame automorphism groups. We showed in Section 4.5 that all automorphism groups of del Pezzo surfaces of degree 2 in characteristic 0 appear as the tame automorphism groups of del Pezzo surfaces of degree 2 in positive characteristic. It turns out that this is not true anymore for del Pezzo surfaces of degree 1. If p = 3, 5, some of these tame groups appear only as proper subgroups of the full groups of automorphisms. If p = 3, the reason is that in their equations the coefficient a 2 is equal to zero and hence the j-invariant of the general member of the elliptic pencil vanishes. This implies that the group H 0 becomes a group of order 3 and the full group of automorphisms is larger.
We go through the list of automorphism groups of del Pezzo surfaces of degree 1 in characteristic 0 given in [7, Table 8.14] and find those of them which are tame in some characteristic p > 0.
• 3 × (SL 2 (3) : 2): This group is tame if p = 3. By Corollary 5.3, it is maximal among tame automorphism groups of del Pezzo surfaces of degree 1. For a suitable choice of coordinates, the unique surface with an action of this group is given by the equation  Table 8.14] defines a one-dimensional family of smooth del Pezzo surfaces with an action of this group if p = 3. Since the del Pezzo surfaces with a larger automorphism group are unique by [7, Table 8.14] and Theorem 5.4, a generic member of this family has automorphism group isomorphic to 6 × D 6 . • 30: This group is tame if p = 3, 5 and it is maximal among all possible automorphism groups. The equation given in [7, Table 8.14] defines a surface with an action of this group in characteristic p = 3, 5. • SL 2 (3), 2.D 12 , and 2 × 12: These groups are tame if p = 3 and the equations given in [7, Table 8.14] with general parameters define surfaces with this automorphism group if p = 3. • 3 × D 8 : This group is tame if p = 3. It appears in the list [6, Page 519,520], but it is missing in [6, Table 8] and [7, Table 8.14]. It is realized by the surfaces with Weierstrass equation ) for a general choice of a. This group occurs in every characteristic p = 3. We denote it by M in Table 8.
• 20: This group is tame if p = 5. If p = 3, 5, the equation given in [7, Table 8.14] defines a surface with this automorphism group, since it is maximal among tame automorphism groups. If p = 3, this group does not occur. Indeed, its generator must act by the formula (t 0 , t 1 , x, y) → (t 0 , ζ 10 t 1 , −x, ζ 4 y) as in the case of characteristic 0, where ζ n is a primitive n-th root of unity. This easily implies that its equation y 2 +x 3 +a 2 x 2 +a 4 x+a 6 = 0 in characteristic 3 has the coefficient a 2 = 0, hence the j-invariant j(F η ) of the generic fiber of the elliptic fibration vanishes. But this increases the size of the subgroup H and the automorphism group becomes larger. It follows from Theorem 5.6 that the surface is from Case (3) (c) and the automorphism group is C 6 .C 10 .
• D 16 : This group is tame. If p = 3, the equation given in [7, Table 8.14] defines a 1-dimensional family of surfaces on which this group acts. Since all surfaces whose automorphism group strictly contains D 16 are unique, the generic member X of this family has Aut(X) ∼ = D 16 . The same argument as in the previous case shows that the equation of the surface X in characteristic 3 whose automorphism group contains D 16 has coefficient a 2 equal to zero. This increases the size of the subgroup H of Aut(X). The surface is from Case 2 (c) in Theorem 5.6. Its group of automorphism group is C 6 .D 8 .
• D 12 and 2 × 6: These groups are tame if p = 3 and the equations given in [7,Table 8.14] with general parameters define surfaces with this automorphism group if p = 3. • 10: This group is tame if p = 5. If p = 3, 5, the equation given in [7, Table 8.14] defines a 1-dimensional family of surfaces on which this group acts. Since the surfaces whose automorphism groups strictly contain C 10 are unique, the generic member X of this family realizes the group C 10 . If p = 3, the equation y 2 + x 3 + t 2 0 x 2 + at 0 (t 5 1 + t 5 0 ) = 0 defines a one-dimensional family of surfaces with an action of C 10 given by (t 0 , t 1 , x, y) → (t 0 , ζ 5 t 1 , x, −y). • Q 8 , 2 × 4, and D 8 : These groups are tame and the equations given in [7, Table 8.14] with general parameters define surfaces with these automorphism groups. • 6: This group is tame if p = 3 and the equations given in [7, Table 8.14] with general parameters define surfaces with this automorphism group. • 4 and 2 2 : These groups are tame. The equations given in [7, Table 8.14] define two at least 4-dimensional families of surfaces with an action of this group. Since the surfaces with larger automorphism group depend on less than three parameters, even in the wild case, a general member of these families will have the desired automorphism group. • 2: This is the general case that occurs in every characteristic. 5.6. Wild automorphism groups. Here, we summarize the classification of wild groups of automorphisms of del Pezzo surfaces of degree 1 by collecting the results of the previous sections. We use the notation of [7, Table 8.14] for wild groups in characteristic p that also occur in characteristic 0. Let us explain our notation for the remaining cases: The surfaces in Equation (19) are reductions modulo 5 of the surfaces of Type XIII in [7, Table  8.14] after a suitable change of coordinates, hence we call them Type XIII. Similarly, the surface given by Equation (20) is the reduction modulo 5 of the surfaces of Type I, II, and IV in [7, Table  8.14].
As for characteristic p = 3, consider first the surfaces given by Equation (23). Recall that they have automorphism group 2 × 3 2 . Since all the wild automorphisms of order 3 in this group fix a cuspidal curve on X, they are either of conjugacy class 4A 2 or they arise from a del Pezzo surface Y of degree 2 given by Equation (15) and then they are of conjugacy class 3A 2 . Now, comparing eigenvalues of the commuting automorphisms of order 3 on E 8 , one checks that the existence of an automorphism of class 4A 2 would force some other automorphism to be of type A 2 or 2A 2 . Hence, all elements in C 2 3 are of conjugacy class 3A 2 . The surfaces in Equation (22) together with their wild automorphisms are generalizations of Equation (23) and the surfaces in Equation (26) are specializations of Equation (22), so all the wild automorphisms in these families are of class 3A 2 . We call the surfaces given by Equations (22) and (26) with a = b Type XVIII, as in [7,Table 8,14]. Equation (23) has no analogue in characteristic 0, but it is a specialization of Type XVIII, so we call it XVIII'. A general equation of Type V in [7, Table 8.14] is smooth in characteristic 3, hence it defines a del Pezzo surface with an action of SL 2 (3) in characteristic 3. Therefore, we call the surfaces given by Equation (26) with a = b Type V as well.
The wild automorphisms of order 3 of the surfaces defined by Equations (27) fix the base of the elliptic fibration and act with a unique fixed point on a general fiber, hence they do not leave invariant any (−1)-curve on X. Therefore, they must be of conjugacy class 4A 2 . Hence, the general surface defined by these equations is the characteristic 3 analogue of Type XVII of [7, Table 8.14]. The subfamily with a 2 c 4 − b 4 = 0 has automorphism group 2 × 6 with C 3 acting trivially on the base of the elliptic fibration, so we call them Type XI. The surfaces given by Equation (27) with b = c = 0 are reductions modulo 3 of the surfaces of Type IX in [7, Table  8.14] and, after a suitable transformation, also of those of Type M.
The general surface given by Equation (28) admits a wild automorphism of conjugacy class 4A 2 , by the same argument as in the previous paragraph, so we denote these surfaces by XVII as well. The subfamily with b = 0 and a = 0 has no analogue in characteristic 0, but they lift together with the action of C 4 , so we denote them by XIX'. The surface given by Equation (28) with a = b = 0 lifts to characteristic 0 together with the action of C 20 , hence it is a reduction modulo 3 of the surface of Type VIII in [7, Table 8.14]. After a suitable change of coordinates, Type IV [7, Table 8.14] reduces to this surface as well, hence we name it Type IV/VIII 5.7. Conjugacy classes. As in the case of degree 2, the conjugacy classes of a tame automorphism g of a del Pezzo surface can be determined using the Lefschetz fixed point formula. The conjugacy classes of wild automorphisms can be determined using the description in the previous section of the corresponding del Pezzo surfaces as reductions modulo p of certain del Pezzo surfaces in characteristic 0, where the automorphism is tame. For the surfaces of Type XIX' and XVIII' that have no analogue in characteristic 0, we determined the conjugacy classes of the wild automorphisms of order 3 in the previous section. The remaining automorphisms of these surface are obtained by composing with the Bertini involution, and the conjugacy classes of these compositions can be easily determined using Table 3.
We give the classification of all conjugacy classes that can occur in Table 8. In that table, we give the name of the relevant family of del Pezzo surfaces in the first column, following [7, Table 8.14]. In the second column, we note the characteristics in which the family occurs. The third and fourth columns give the group Aut(X) and its order. The remaining columns give the number of elements of a given Carter conjugacy class in Aut(X).
Remark 5.7. As a concluding remark, we note that, while there are no conjugacy classes in the Weyl groups W (E N ) that are realized in characteristic p but not in characteristic 0, the tables in  the Appendix show that there are certain conjugacy classes that occur in characteristic 0 but do not occur in characteristic p if p ∈ {3, 5, 7}. If p = 7, the conjugacy classes A 6 and E 7 (a 1 ) do not occur, because the Klein quartic becomes singular in characteristic 7.
If p = 5, the conjugacy classes A 4 and E 8 (a 2 ) do not occur. In characteristic 0, the Clebsch cubic surface is the unique del Pezzo surface realizing the conjugacy class A 4 , and this surface becomes singular in characteristic 5. The conjugacy class of type E 8 (a 2 ) can only occur on del Pezzo surfaces of degree 1, where it acts as an automorphism of order 10 on the base of the elliptic pencil. Since PGL 2 (k) contains no element of order 10 if p = 5, this explains why E 8 (a 2 ) does not occur.
If p = 3, the conjugacy classes A 2 , A 2 + A 1 , A 2 + 2A 1 , A 5 + A 2 + A 1 , D 4 , D 4 + A 2 , 2D 4 , D 5 (a 1 ), E 6 + A 1 , E 6 (a 1 ), E 7 , and E 8 (a 1 ) do not occur in any degree and there are a couple of conjugacy classes that are not realized in all the degrees that they are realized in in characteristic 0. This is a consequence of several phenomena: Firstly, as above, certain special del Pezzo surfaces in characteristic 0 become singular in characteristic 3. Secondly, the group PGL 2 (k) contains no elements of order np with (p, n) = 1 and n ≥ 2. Thirdly, there are no wild automorphisms of order p 2 on del Pezzo surfaces in any degree by [8,Theorem 8]. And finally, the fixed locus of an automorphism g of order 3 on a del Pezzo surface X in characteristic 3 tends to be smaller and in a more special position than in characteristic 0, making it harder to produce blow-ups of X to which g lifts and which are still del Pezzo surfaces.