ON ELEMENTS OF ORDER p^{s} IN THE PLANE CREMONA GROUP OVER A FIELD OF CHARACTERISTIC p

IGOR V. DOLGACHEV

Dedicated to Vasily Iskovskikh on his 70th birthday

Abstract. We show that the plane Cremona group over a field of characteristic p > 0 does not contain elements of power of p larger than 2. We also describe conjugacy classes of elements of order p^{2}.

1. Introduction

The classification of conjugacy classes of elements of finite order ℓ in the plane Cremona group Cr_{2}(k) over an algebraically closed field k of characteristic 0 has been known for more than a century. The possible orders of elements not conjugate to a projective transformation are 2, …, 10, 12, 14, 15, 18, 20, 24 and 30 (see [3] and historic references there). Much less is known in the case when k is of positive characteristic p and the order is divisible by p.

In this note we prove the following Main Theorem.

Theorem 1. Let k be a field of characteristic p > 0. Then the group Cr_{2}(k) does not contain elements of order p^{s} with s > 2.

We will also describe conjugacy classes of elements of order p^{2} over algebraically closed field of characteristic p > 0.

I thank J.-P. Serre for asking about the existence of elements of order 8 in Cr_{2}(k) over a field of characteristic 2. The question had initiated the present paper.

It is a great pleasure to dedicate this paper to Vasya Iskovskikh, a long-time friend and one of the main contributers to the modern development of the theory of Cremona transformations.

2. Conic bundles

It is clear that in the proof of Main Theorem, we may assume that k is an algebraically closed field of characteristic p > 0. On several occasions I refer to [3] where the ground field was assumed to be the field of complex numbers. The proofs of the facts which I will use extend to our case.

Let σ ∈ Cr_{2}(k) be of order p^{s}. A standard argument (see [3]) shows that σ acts biregularly on one of the following rational surfaces X

(i) X has a structure of a conic bundle f : X → ℙ_{k}^{1} with m ≥ 0 singular fibres,

(ii) X is a Del Pezzo surface of degree d.

Moreover, we may assume that X is σ-minimal, i.e. Pic(X)^{σ} is of rank 2 in the first case and of rank 1 in the second case. This is equivalent to that any σ-equivariant birational morphism X → X′ must be an isomorphism. When X is σ-minimal, we say that σ acts minimally on X.

We start from the first case. Recall the following well-known fact.

Lemma 2. Let σ be an element of order p^{s} in Aut(ℙ_{k}^{r}). Then s ≤ log_{p}(r + 1).

Proof. Let A ∈ GL_{r+1}(k) represent σ and A^{p^{s}} = cI_{r+1} for some constant c. Multiplying A by c^{\frac{1}{p^{s}}} we may assume that A^{p^{s}} = 1. Since k^{∗} does not contain non-trivial p^{s}-th roots of unity, we can reduce A to the Jordan form with 1 at the diagonal. Obviously A^{p^{s}} = I_{r+1} + (A − I_{r+1})^{p^{s}}. Since, for any Jordan block-matrix J with zeros at the diagonal we have J^{r+1} = 0, we get A^{p^{s}} = 0 as soon as p^{s} ≥ r + 1. The assertion follows. □

Corollary 3. Let f : X → ℙ_{k}^{1} be a conic bundle and σ be an automorphism of X of order p^{s} preserving the conic bundle. Then s ≤ 2.

Proof. Let ḡ be the image of σ in the automorphism group of the base of the fibration. By the previous lemma σ̄^{p} = 1. Thus σ^{p} acts identically on the base and hence acts on the general fibre of f. The latter is isomorphic to the projective line over the functional field of the base. Applying the lemma again we obtain that σ^{p^{2}} = 1. □

This checks the theorem in the case of a conic bundle. Let us give a closer look at elements of order p^{2}.

Theorem 4. Let σ be a minimal automorphism of order p^{2} of a conic bundle X → ℙ_{k}^{1}. Then p = 2.

ON ELEMENTS OF ORDER p^{n} 3

Proof. Let m = K_{X}^{2} − 8 be the number of singular fibres of the conic bundle. Assume first that m = 0, i.e. π : X → ℙ_{k}^{1} is a minimal ruled surface F_{n}. If n = 1, the surface is not σ-minimal. If n = 0, the automorphism group of F_{0} ≅ ℙ_{k}^{1} × ℙ_{k}^{1} preserving one of the rulings is isomorphic to Aut(ℙ_{k}^{1})×Aut(ℙ_{k}^{1}). It does not contain elements of order p^{2}.

So we may assume that n ≥ 2. The automorphism group Aut(X) of the surface F_{n} is well-known (see [3]). By blowing down the exceptional section, we obtain that Aut(X) is isomorphic to the group of automorphisms of the weighted projective plane ℙ(1, 1, n) with coordinates t_{0}, t_{1} of degree 1 and coordinate t_{2} of degree n. Any automorphism g of ℙ(1, 1, n) can be given by the formula

σ : (t_{0}, t_{1}, t_{2}) ↦ (at_{0} + bt_{1}, ct_{0} + dt_{1}, et_{2} + f_{n}(t_{0}, t_{1})),

where f_{n} is a binary form of degree n. In our case we can change the coordinates to assume that a = b = d = 1, c = 0. By iterating, we get e^{p^{s}} = 1, hence e = 1. Also

σ^{p} : (t_{0}, t_{1}, t_{2}) = (t_{0}, t_{1}, t_{2} + ∑_{j=0}^{p−1} f_{n}(t_{0} + jt_{1}, t_{1}).

Let σ̄ be the transformation (t_{0}, t_{1}) ↦ (t_{0} + t_{1}, t_{1}). Since ∑_{i=0}^{p−1} σ̄^{i} = 0, we get that the sum in above is equal to zero, hence σ^{p} = 1. Thus there are no automorphisms of order p^{2}.

Assume now that m > 0, i.e. X is obtained from a minimal ruled surface F_{n} by blowing up m points. If n > 0, the proper transform of the exceptional section of F_{n} is a section of the conic bundle with negative self-intersection. If n = 0, the proper transform of a section of F_{0} passing through a point we blow up, is a section with negative self-intersection. So, in any case we have a section of the conic bundle with negative self-intersection. It intersects a component of a singular fibre at its nonsingular point. Since X is σ-minimal, σ cannot fix this component, so σ(E) ≠ E. By Lemma 2, σ^{p} acts identically on the base of the conic bundle. Since p > 2, σ^{p} cannot switch components of singular fibres, hence it must act identically on Pic(X). Since an irreducible curve with negative self-intersection does not move in a linear system, σ^{p} fixes E and σ(E). But in characteristic p > 0 an automorphism of order p of a general fibre has only one fixed point. This shows that σ^{p} = 1 if p > 2. □

Example 1. Recall that Cr_{2}(k) contains a subgroup of de Jonquières transformations of the form (x, y) ↦ (\frac{αx+β}{γx+δ} , \frac{a(x)y+b(x)}{c(x)y+d(x)}). Each element of finite order in this subgroup is realized as an automorphism of a conic bundle. Assume p = 2. Without loss of generality we may assume that x ↦ x + 1.

Let a(x) = d(x) = x^{2^{n}+1} + 1. We have

a(x + 1) = (x + 1)^{2^{n}+1} + 1 = 1 + (x + 1)(x + 1)^{2^{n}} = x(x^{2n} + x^{2^{n}−1} + 1).

Let b(x) = (x^{2n} + x^{2^{n}−1} + 1), c(x) = x + 1, so that

a(x)a(x + 1) + b(x)c(x + 1) = a(x)a(x + 1) + b(x + 1)c(x) = 0.

With this choice, we have σ^{2} : (x, y) ↦ (x, R(x)/y), where

R(x) = \frac{a(x+1)b(x)+a(x)b(x+1)}{a(x)c(x+1)+a(x+1)c(x)} = \frac{a(x)a(x+1)^{2}/x+a(x)^{2}a(x+1)/(x+1)}{a(x)x+a(x+1)(x+1)}

= \frac{a(x)a(x + 1)}{x(x + 1)} = P (x) := (x^{2^{n}} + x^{2^{n}−1} + … + 1)(x^{2^{n}} + x^{2^{n}−1} + 1).

For n > 1, the polynomial P (x) has no multiple roots. It is known that the de Jonquières involution (x, y) ↦ (x, P (x)/y) is realized as a minimal automorphism of a conic bundle with the number m of singular fibres equal to the degree of P (x). On the other hand, it is known that for m ≥ 8 a minimal automorphism of such a conic bundle is not conjugate to neither a projective automorphism, nor a minimal automorphism of a Del Pezzo surface, nor a minimal automorphism of a conic bundle with number of singular fibres different from m (see Corollary 7.11 in [3]). Thus we have constructed a countable set of conjugacy classes of elements of order 4 in Cr_{2}(k).

3. Del Pezzo surfaces of degree ≥ 3

Now we consider the case when σ is an automorphism of order p^{s} of a Del Pezzo surface X of degree d := K_{X}^{2} ≥ 4.

If d = 9, X = ℙ_{k}^{2} and by Lemma 2 we get s ≤ 2. All elements of order p^{2} are conjugate in Aut(ℙ_{k}^{2}).

If d = 8, then X ≅ ℙ_{k}^{1} × ℙ_{k}^{1} because the ruled surface F_{1} is not σ-minimal. We know that Aut(F_{0}) contains a subgroup of index 2 isomorphic to Aut(ℙ_{k}^{1}) × Aut(ℙ_{k}^{1}). Applying Lemma 2 we obtain s = 1 if p ≠ 2, and s ≤ 2 otherwise. The automorphism of X given in affine coordinates by (x, y) ↦ (y + 1, x) is of order 4.

If d = 7, the surface is not σ-minimal since it is obtained by blowing up two points in ℙ_{k}^{2}, the proper transform of the line joining the points is a σ-invariant (−1)-curve.

Assume d = 6. Then Aut(X) is isomorphic to the semi-direct product T ⋊ G, where T ≅ k^{∗}^{2} is a 2-dimensional torus and G is a dihedral group D_{12} ≅ (ℤ/2ℤ) × S_{3}. Since T does not contain elements of order p and D_{12} does not contain elements of order p^{s}, s > 1, we obtain that the only possibility is s = 1 and p = 2, 3.

Assume d = 5. It is known that Aut(X) acts faithfully on the Picard group of X of a Del Pezzo surface of degree ≤ 5. Via this action it becomes isomorphic to a subgroup of the Weyl group W (A_{4}) ≅ S_{5}. Thus s = 1 unless p = 2 and s = 2. The group W (A_{4}) acts on K_{X}^{⊥} ≅ ℤ^{4} via its standard irreducible representation on {(a_{1}, …, a_{5}) ∈ ℤ^{5} : a_{1} + … + a_{5} = 0}. A cyclic permutation of order 4 has a fixed vector. This shows that X is not σ-minimal.

Assume d = 4. In this case Aut(X) is isomorphic to a subgroup of the Weyl group W (D_{5}) ≅ (ℤ/2ℤ)^{4} ⋊ S_{5}. Thus an automorphism of order p^{s} with s > 1 may exist only if p = 2.

It is known that X is isomorphic to the blow-up of 5 points p_{1}, …, p_{5} in the plane, no three among them are collinear. The surface admits 5 pairs (|C_{i}|, |C′_{i}|) of pencils of conics in the anti-canonical embedding X ↪ ℙ_{k}^{4}. The pencil |C_{i}| is the proper transform of the pencil of lines through the point p_{i} and the pencil |C′_{i}| is the proper transform of the pencil of conics through the points p_{j}, j ≠ i. Since C_{i} + C′_{i} ∼ −K_{X}, the Weil group permutes the 5 pairs of the divisor classes [C_{i}], [C_{i}]′ and switches [C_{i}] with [C′_{i}] in even pairs of them (see [3], Proposition 6.6). It is known that the anti-canonical linear system | − K_{X}| maps X isomorphically onto the intersection of two quadrics in ℙ_{k}^{4}. Under the multiplication map |C_{i}| × |C′_{i}| → | − K_{X}|, the two pencils generate a hyperplane H_{i} in | − K_{X}| and the map f_{i} × f′_{i} : X → ℙ_{k}^{1} × ℙ_{k}^{1} defined by the two pencils is equal to the composition of the anti-canonical map and the projection from the point h_{i} ∈ | − K_{X}|^{∗} corresponding to the hyperplane H_{i}. Since the image of X under this projection is a nonsingular quadric, we see that the center of the projection lies on a singular quadric Q_{i} of corank 1 in the pencil Q of quadrics containing X. Conversely, every such quadric defines a degree 2 map f : X → ℙ_{k}^{1}×ℙ_{k}^{1}, and the pre-images of the ruling define a pair of pencils of conics on X. Thus we see that the pencil of quadrics Q contains exactly five singular quadrics. Any automorphism σ of X acts on the pencil Q leaving the set of five quadrics invariant. Its square σ^{2} acts identically on the pencil and hence leaves invariant all pairs of conic pencils. Since the divisor classes [C_{i}] together with K_{X} generate Pic(X), we obtain that σ^{4} acts identically on Pic(X), hence it is the identity.

Remark 1. Another proof of non-existence of an automorphism of order 8 on a Del Pezzo surface of degree 4 was suggested by J.-P. Serre. It is known that an element of order 8 in W (D_{5}) has trace equal to −1 in the root lattice. Since the latter is isomorphic to K_{X}^{⊥}, the automorphism of order 8 has trace 0 in Pic(X) and hence in the second cohomology group with ℓ-adic coefficients. Thus the Lefschetz number of σ is equal to 2, and hence, by the Lefschetz-fixed-point formula, σ has a fixed point. Blowing it up we get an automorphism of order 8 of a cubic surface. Since any automorphism of a cubic surface is the restriction of an automorphism of ℙ_{k}^{3}, applying Lemma 2 we find a contradiction.

Let us summarize what we have learnt.

Theorem 5. A Del Pezzo surface of degree ≥ 4 does not contain elements of order p^{3}. An automorphism of order p^{2} not conjugate to a projective automorphism in Cr_{2}(k) exists only if p = 2. It is minimally realized on X = ℙ_{k}^{1} × ℙ_{k}^{1} or on a Del Pezzo surface of degree 4.

Note that any automorphism of order 4 of ℙ_{k}^{1} × ℙ_{k}^{1} has a fixed point, and the projection from this fixed point makes it conjugate to a projective transformation.

Assume now that d = 3, i.e. X is a cubic surface embedded in ℙ_{k}^{3} by the anti-canonical linear system |−K_{X}|. In this case Aut(X) is isomorphic to a subgroup of the Weil group W (E_{6}) of a simple root lattice of type E_{6}. By Corollary 6.11 from [3], all elements of order p^{s}, s > 1, in W (E_{6}) have an invariant vector in the lattice E_{6} ≅ K_{X}^{⊥} unless p^{s} = 9. Thus we have to consider the existence of an automorphism σ of order 9 of a cubic surface over a field of characteristic p = 3.

The following argument was suggested to me by J.-P. Serre. It follows from the classification of conjugacy classes of elements of W (E_{6}) that the trace of σ in its action in K_{X}^{⊥} is equal to 0. Thus the Lefschetz number of σ in in the ℓ-adic cohomology of X is equal to 3. This implies that σ has a fixed point x_{0}. Since σ acts trivially on | − K_{X} − x_{0}| ≅ ℙ_{k}^{2}, we obtain that it acts trivially on | − K_{X}| ≅ ℙ_{k}^{3}.

We have proved the following.

Theorem 6. A cubic surface does not admit minimal automorphisms of order p^{s} with s > 1.

4. Del Pezzo surfaces of degree 2

It is known (see [2]) that the linear system | − K_{X}| defines a degree 2 map f : X → ℙ_{k}^{2}. The map must be finite since −K_{X} is ample. It is also a separable map because otherwise X must be homeomorphic to ℙ_{k}^{2}, but comparing the l-adic Betti numbers we find this impossible. The cover f is a Galois cover with order 2 cyclic Galois group ⟨γ⟩. The automorphism γ of X is called the Geiser involution. For any divisor D we have

D + γ^{∗}(D) ∼ (D · K_{X})K_{X}.

This implies that γ^{∗} acts on K_{X}^{⊥} as the minus identity. The lattice K_{X}^{⊥} is isomorphic to the root lattice of type E_{7}, and the isometry γ^{∗} generates the center of the Weyl group W (E_{7}).

It follows from the classification of conjugacy classes in W (E_{7}) that for any automorphism of order p^{s}, s > 1, the rank of Pic(X)^{σ} is greater than 1, unless p = s = 2. So, it suffices to consider the latter case. All such automorphisms form one conjugacy class (of type 2A_{3} + A_{1} in the notation from [3]). It follows from the description of degree 2 covers of smooth varieties (see [1], Chapter 0) that X is isomorphic to a surface ℙ(1, 1, 1, 2) given by an equation

u^{2} + a_{2}(x, y, z)u + a_{4}(x, y, z) = 0,

where a_{2}, a_{4} are homogeneous forms of degree 2 and 4. Since the anticanonical map is separable we have a_{2} ≠ 0. An automorphism σ of order 4 acts linearly in ℙ_{k}^{2} = | − K_{X}|^{∗} leaving the branch curve V (a_{2}) invariant. If V (a_{2}) is an irreducible conic, then σ^{2} is identical on the conic, and hence it is identical on ℙ_{k}^{2}. This implies that σ^{2} is the Geiser involution u ↦ u + a_{2}. However, the Weil group W (E_{7}) does not contain square roots of the Geiser involution. Suppose now that V (a_{2}) is reducible. If it is not a double line, we can choose projective coordinates x, y, z to assume that a_{2} = xy. Then σ^{2} must change z to z + ax + by and leave x, y unchanged. This forces a_{4} to be invariant with respect to this transformation. Writing

a_{4} = l_{0}z^{4} + z^{3}l_{1} + z^{2}l_{2} + zl_{3} + l_{4},

where l_{i} are binary forms in x, y, we find that l_{1} = 0. This implies that the point (x, y, z, u) = (0, 0, 1, 0) is a singular point on the surface. Thus σ^{2} must be the Geiser involution and we finish as in the previous case. Finally we may assume that the equation of X looks like u^{2} + x^{2}u + a_{4} = 0. In this case, σ^{∗}(x) = x and we may assume that σ acts on the variables x, y, z by x ↦ x, y ↦ y + x, z ↦ z + y. The polynomial a_{4}(x, y, z) must be invariant with respect to the coordinate change σ^{2} : (x, y, z) ↦ (x, y, z + x). It is easy to see that the ring of polynomials in x, z invariant with respect (x, z) ↦ (x, z + x) is generated by x and z(z + x). This implies that a_{4} can be written as a polynomial in z(z + x), x, y

a_{4} = cz^{2}(z + x)^{2} + z(z + x)g(x, y) + h(x, y).

It is immediate to check that the point (x, y, z, u) = (0, 0, 1, \sqrt{c}) is a singular point of the surface.

To sum up, a Del Pezzo surface of degree 2 does not contain minimal automorphisms of order p^{s}, s > 1.

5. Del Pezzo surfaces of degree 1

This is the most difficult and interesting case. The linear system | − 2K_{X}| defines a degree 2 map f : X → Q, where Q is a quadratic cone in ℙ_{k}^{3}. Again, since −K_{X} is ample, f is a finite map, and arguing as in the previous case we see that the map is separable. The Galois group of the cover is generated by an automorphism β of X known as the Bertini involution. For any divisor D we have

(1)  D + γ^{∗}(D) ∼ 2(D · K_{X})K_{X}.

This shows that β^{∗} acts as the minus identity on the lattice K_{X}^{⊥}. The lattice K_{X}^{⊥} is isomorphic to the root lattice of type E_{8}. The involution β^{∗} generates the center of the Weyl group W (E_{8}).

The automorphism group Aut(X) is a subgroup of W (E_{8}). Possible orders p^{s}, s > 1, of minimal automorphisms are 4 and 8 (see [3]).

So we assume p = 2. The linear system | − K_{X}| has one base point p_{0}. Blowing it up we obtain a fibration π : X′ → ℙ_{k}^{1} whose general fibre is an irreducible curve of arithmetic genus 1. Since −K_{X} is ample, all fibres are irreducible, and this implies that a general fibre is an elliptic curve (see [1]). Let S_{0} be the exceptional curve of the blow-up. It is a section of the elliptic fibration. We take it as the zero in the Mordell-Weil group of sections of π. The map f : X → Q extends to a degree 2 separable finite map f′ : X′ → F_{2}, where F_{2} is the minimal ruled surface with the exceptional section E satisfying E^{2} = −2. Its branch curve is equal to the union of E and a curve B from the divisor class 3f + e, where f is the class of a fibre and e = [E]. We have f′^{∗}(E) = 2S_{0}. The elliptic fibration on X′ is the pre-image of the ruling of F_{2}. We know that τ = σ^{2} acts identically on the base of the elliptic fibration. Since it also leaves invariant the section S_{0}, it defines an automorphism of the generic fibre considered as an abelian curve with zero section defined by S_{0}. If τ^{2} = 1, then τ is the negation automorphism, hence defines the Bertini transformation of the projective plane. Its image in the Weyl group W (E_{8}) generates the center. The group of automorphisms of an abelian curve in characteristic 2 is of order 2 if the absolute invariant of the curve is not equal to 0 or of order 24 otherwise. In the latter case it is isomorphic to Q_{8} ⋊ ℤ/3, where Q_{8} is the quaternion group with the center generated by the negation automorphism (see [4], Appendix A). Thus τ^{4} = 1 and the Weierstrass model of the generic fibre is

y^{2} + a_{3}y + x^{3} + a_{4}x + a_{6} = 0.

In global terms the Weierstrass model of the elliptic fibration π : X′ → ℙ_{k}^{1} is a surface in ℙ(1, 1, 2, 3) given by the equation

y^{2} + a_{3}(u, v)y + x^{3} + a_{4}(u, v)x + a_{6}(u, v),

where a_{i} are binary forms of degree i. It is obtained by blowing down the section S_{0} to the point (u, v, x, y) = (0, 0, 1, 1) and is isomorphic to our Del Pezzo surface X. The image of the branch curve B is given by the equation a_{3}(u, v) = 0, i.e. B consists of three fibres of the fibration plus the section S_{0}. Since a general point of B is a 2-torsion point of a general fibre, we see that all nonsingular fibres of the elliptic fibration are supersingular elliptic curves (i.e. have no non-trivial 2torsion points). An automorphism of order 4 of X is defined by

(u, v, x, y) ↦ (u, v, x + s(u, v)^{2}, y + s(u, v)x + t(u, v)),

where s is binary forms of degree 1 and t is a binary form of degree 3 satisfying

(2)  a_{3} = s^{3}, t^{2} + a_{3}t + s^{6} + a_{4}s^{2} = 0.

In particular, it shows that a_{3} must be a cube, so we can change the coordinates (u, v) to assume that s = u, a_{3} = u^{3}. The second equality in (2) tells that t is divisible by u, so we can write it as t = uq for some binary form q of degree 2 satisfying q^{2} + u^{2}q + u^{4} + a_{4} = 0. Let α be a root of the equation x^{2} + x + 1 = 0 and b = q + αu^{2}. Then b satisfies a_{4} = b^{2} + u^{2}b and t = ub + αu^{3}. Conversely, any surface in ℙ(1, 1, 2, 3) with equation

(3)  y^{2} + u^{3}y + x^{3} + (b(u, v)^{2} + u^{2}b(u, v))x + a_{6}(u, v) = 0

where b is a quadratic form in (u, v) and the coefficient at uv^{5} in a_{6} is not zero (this implies that the surface is nonsingular) is a Del Pezzo surface of degree 1 admitting an automorphism of order 4

τ : (u, v, x, y) ↦ (u, v, x + u^{2}, y + ux + ub + αu^{3}).

Note that τ^{2} : (u, v, x, y) ↦ (u, v, x, y + u^{3}) coincides with the Bertini transformation.

Counting parameters it is easy to see that our surfaces depend on 4 parameters (3 coefficients in b and 7 coefficients in a_{6} modulo transformation a_{6} ↦ a_{6} + g^{2} + u^{3}g, (u, v) ↦ (u, c_{1}u + c_{2}v), where g is a binary form of degree 3).

Theorem 7. Let X be a Del Pezzo surface (3). Then it does not admit an automorphism of order 8.

Proof. Assume τ = σ^{2}. Since σ leaves invariant | − K_{X}|, it fixes its unique base point, and lifts to an automorphism of the elliptic surface X′ preserving the zero section S_{0}. Since the general fibre of the elliptic fibration f : X′ → ℙ_{k}^{1} has no automorphism of order 8, the transformation σ acts nontrivially on the base of the fibration. Note that the fibration has only one singular fibre F_{0} over (u, v) = (0, 1). It is a cuspidal cubic. The transformation σ leaves this fibre invariant and hence acts on ℙ_{k}^{1} by (u, v) ↦ (u, u + c). Since the restriction of σ to F_{0} has at least two fixed points: the cusp and the origin F_{0} ∩ S_{0}, it acts identically on F_{0} and freely on its complement X′ \ F_{0}.

Recall that X′ is obtained by blowing up 9 points p_{1}, …, p_{9} in ℙ_{k}^{2}, the base points of a pencil of cubic curves. We may assume that X is the blow-up of the first 8 points, and the exceptional curve over p_{9} is the zero section S_{0}. Let S be the exceptional curve over any other point. We know that β = σ^{4} is the Bertini involution of X. Applying formula (1), we obtain that S · β(S) = 3. Identifying β(S) and S with their pre-images in X′, we see that β(S) ⊕ S = S_{0} in the Mordell-Weil group of sections of π : X′ → ℙ_{k}^{1}. Thus S and β(S) meet at 2-torsion points of fibres. However, all nonsingular fibres of our fibration are supersingular elliptic curves, hence S and β(S) can meet only at a singular fibre F_{0}. Let Q ∈ F_{0} be the intersection point. The sections S and β(S) are tangent to each other at Q with multiplicity 3. Now consider the orbit of the pair (S, β(S)) under the cyclic group ⟨σ⟩. It consists of 4 pairs

(S, σ^{4}(S)), (σ(S), σ^{5}(S)), (σ^{2}(S), σ^{6}(S)), (σ^{3}(S), σ^{7}(S)).

Let D_{i} = σ^{i}(S) + σ^{i+4}(S), i = 1, 2, 3, 4. We have D_{1} + … + D_{4} ∼ −8K_{X},, hence for i ≠ j we have D_{i} · D_{j} = (64 − 16)/12 = 4. Let Y → X be the blow-up of Q. Since Q is a double point of each D_{i}, the proper transform D̄_{i} of each D_{i} in Y has self-intersection 0 and consists of two smooth rational curves intersecting at one point with multiplicity 2. Moreover, we have D̄_{i} · D̄_{j} = 0. Applying (1), we get D_{i} ∈ |−2K_{X}|. Since Q is a double point of D_{i}, we obtain D̄_{i} ∈ |−2K_{Y} |. The linear system | − 2K_{Y} | defines a fibration Y → ℙ_{k}^{1} with a curve of arithmetic genus 1 as a general fibre (an elliptic or a quasi-elliptic fibration). The curves D̄_{i} are singular fibres of Kodaira’s type III. The automorphism σ acts on the base of the fibration and the four special fibres form one orbit. But the action of σ on ℙ_{k}^{1} is of order 2 and this gives us a contradiction. □

6. Conjugacy classes of elements of order p^{2}

Assume that k is algebraically closed. As we have seen in the previous sections, an element of order p^{2} not conjugate to a projective transformation exists only for p = 2. It can be realized as a minimal automorphism of a conic bundles, or a Del Pezzo surfaces of degree 1 or 4. Del Pezzo surfaces of degree 1 are super-rigid, i.e. a minimal automorphism of such a surface could be conjugate only to a minimal automorphism of the same surface. A minimal automorphism of a Del Pezzo surface of degree 4 is conjugate to a minimal automorphism of a conic bundle with 5 singular fibres (see [3], §8).

Thus we have proved the following.

Theorem 8. An element of order p^{2} not conjugate to a projective transformation exists only if p = 2. Assume that k is algebraically closed. An element of order 4 is either conjugate to a projective transformation, or conjugate to an element realized by a minimal automorphism of a conic bundle, or a Del Pezzo surface of degree 1.

For the completeness sake let us add that elements of order p not conjugate to a projective transformations occur for any p. They can be realized as automorphisms of conic bundles, and if p = 2, 3, 5 as automorphisms of Del Pezzo surfaces.

References

1. F. Cossec and I. Dolgachev, Enriques surfaces. I, Progress in Mathematics, 76. Birkhuser Boston, Inc., Boston, MA, 1989.

2. M. Demazure, Surfaces de Del Pezzo, I-V, in “Séminaire sur les Singularités des Surfaces”, Ed. by M. Demazure, H. Pinkham and B. Teissier. Lecture Notes in Mathematics, 777. Springer, Berlin, 1980, pp. 21–69.

3. I. Dolgachev and V. Iskovskikh, Finite subgroups of the plane Cremona group, math. AG/0610595 (to appear in “Algebra, Arithmetic and Geometry, Manin’s Festschrift”, Birkhaüser. Boston).

4. J. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1992.

Department of Mathematics, University of Michigan, 525 E. University Av., Ann Arbor, Mi, 49109, USA

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