Serge Cantat

Julie Déserti
email: julie.deserti@math.cnrs.fr

Junyi Xie
email: junyi.xie@univ-rennes1.fr

Université Côte

THREE CHAPTERS ON CREMONA GROUPS

This article is made of three independent parts, the three of them concerning the Cremona group in 2 variables:

1.-In the first part, we answer a question of Igor Dolgachev, which is related to the following problem: given a birational transformation f :

k , and a linear projective transformation A ∈ PGL 3 (k), when is A • f regularizable ? In other words, when do there exist a surface X and a birational map ϕ :

, and the answer is negative (Theorem 1.5).

2.-In the second part, we look at the degree deg( f ) of the formulas defining f ∈ Bir(P 2 k ), and at the sequence n → deg( f n ). We show that there is no constraint on the oscillations of this function, that is on the sequence

for small values of n: given any bounded interval [0, N], the oscillations on that interval are arbitrary (see Theorem 2.1).

3.-The third part studies the degree of pencils of curves which are invariant by a birational transformation. In the most interesting case (i.e. when f is a Halphen or Jonquières twist), we prove that this degree is bounded by a function of deg( f ). We derive corollaries on the structure of conjugacy classes, and their properties with respect to the Zariski topology of Bir(P 2 k ) (see Theorems 4.2 and 4.5 for instance, and § 5).

These three themes are related together by the properties of degrees of birational maps and their iterates and in particular to the question: how do the sequence (deg( f n )) and its asymptotic behavior vary when f moves in a family of birational transformations of the plane ? A second feature of this article is

that we address the problem of finding examples which are defined over number fields (i.e. over Q) instead of C (see Theorems 1.9 and 2.12 for instance).

Acknowledgements.-Thanks to Vincent Guirardel, Stéphane Lamy, Anne Lonjou, and Christian Urech for interesting discussions.

DEGREES AND REGULARIZATION

1.1. Regularization and degree growth. Let M be a smooth, irreducible projective variety of dimension m, defined over an algebraically closed field k. A birational transformation f : M M is regularizable if there exist a smooth projective variety V and a birational map ϕ : V M such that

f V := ϕ -1 • f • ϕ (1.1)
is an automorphism of V .

Let H be a polarization of M. Let k be an integer between 0 and m. The degree of f in codimension k with respect to the polarization H is the positive integer given by the intersection product

deg H,k ( f ) = ( f * (H k )) • (H m-k ). (1.2)
When M is the projective space P m k , we shall choose the polarization O(1) given by hyperplanes. For simplicity, we denote by deg k ( f ) the degree when the polarization is clear or irrelevant, and by deg( f ) the degree in codimension 1. Say that the sequence (u n ) dominates (v n ) if |v n | ≤ A|u n | + B for some constants A and B, and that (u n ) and (v n ) have the same growth type if each one dominates the other: this defines an equivalence relation on sequences of real numbers. Then, the growth type of the sequence (deg k ( f n )) is a conjugacy invariant and does not depend on the choice of the polarization:

(deg H,k ( f n )) and (deg H ′ ,k ( f n V )) are equivalent sequences if f V is conjugate to f via a birational map ϕ : V
M and H ′ is a polarization of V . In particular, the k-th dynamical degree

λ k ( f ) = lim n→+∞ deg H,k ( f n ) 1/n (1.3)
is invariant under conjugacy and does not depend on the choice of the polarization (see [START_REF] Nguyen | Degrees of iterates of rational transformations of projective varieties[END_REF][START_REF] Truong | Relative dynamical degrees of correspondances over fields of arbitrary characteristic[END_REF]).

1.1.1. Bounded degrees. The degree sequence (deg H,k ( f n )) is bounded for at least one codimension 1 ≤ k ≤ m -1 if and only if it is bounded for all of them; in that case, a theorem of Weil asserts that f is regularizable ( [START_REF] Kraft | Regularization of rational group actions[END_REF]).

1.1.2. Exponential degree growth. There are also birational transformations of the projective space P m k which are regularizable and have an exponential degree growth (see [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF] for instance). Such a transformation is not conjugate to an automorphism of P m k because the growth type is a conjugacy invariant. Proposition 1.1. Let f be a birational transformation of a smooth, irreducible, projective variety M. If, for some 1 ≤ k ≤ dim(M) -1, the dynamical degree λ k ( f ) is an integer > 1, then f is not regularizable.

Proof. If f V is an automorphism of a smooth projective variety which is conjugate to f via a birational map ϕ : V M, then λ k ( f ) = λ k ( f V ) is the largest eigenvalue of ( f V ) * on the space N k (X ) of numerical classes of codimension k cycles. This property follows from the invariance of λ k under conjugacy, the existence of a salient, convex and ( f V ) * -invariant cone in N k (V ; R), and the Perron-Frobenius theorem (see [START_REF] Nguyen | Degrees of iterates of rational transformations of projective varieties[END_REF]).

The abelian group N k (V ; Z) determines an ( f V ) * -invariant lattice in N k (V ; R). On the other hand, if g is an isomorphism of a lattice Z k and λ is an eigenvalue of g, then either λ = ±1 or λ is not an integer; indeed, if λ is an integer, the eigenspace corresponding to the eigenvalue λ is defined over Z, hence it intersects Z k on a submodule L ≃ Z r with 1 ≤ r ≤ k: since g |L : L → L is at the same time an invertible endomorphism and the multiplication by λ, we obtain λ r = det(g |L ) = ±1. These two remarks conclude the proof.

As a corollary, if f is a birational transformation of the projective space P m k with (i) deg( f ) > 1 and (ii) deg( f n ) = deg( f ) n for all n ≥ 1, then f is not regularizable.

Before proving this result, note that there are examples of such birational transformations in all dimensions m ≥ 2 (see [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF][START_REF] Déserti | Degree growth of polynomial automorphisms and birational maps: some examples[END_REF]).

Proof. If f were regularizable, there would be an automorphism of a projective variety V whose action on N k (V ) would satisfy ( f * V ) n ≃ αn ℓ ; this means that all eigenvalues of f * V have modulus one (hence are roots of 1), and some iterate ( f * V ) ℓ is a non-trivial unipotent matrix whose larger Jordan block has size k + 1. But f * V preserves a salient, open, convex cone (see [START_REF] Nguyen | Degrees of iterates of rational transformations of projective varieties[END_REF]), and the existence of such an invariant cone is not compatible with a maximal Jordan block of even size ( [START_REF] Marquis | Around groups in Hilbert geometry[END_REF]).

1.2. Semi-continuity and dimension 2. Let Bir d (P m k ) be the algebraic variety of all birational transformations of P m k of degree d. When d is large these varieties have many distinct components, of various dimensions. This is shown in [START_REF] Bisi | On plane Cremona transformations of fixed degree[END_REF] when m = 2; in dimension m ≥ 3 it is better to look at the subvarieties obtained by fixing the m -1 degrees deg 1 ( f ), . . ., deg m-1 ( f ), but even these varieties may have many components.

The group Aut(P m k ) = PGL m+1 (k) acts by left translations, by right translations, and by conjugacy on Bir d (P m k ). Since this group is connected, these actions preserve each connected component.

Theorem 1.3 (Junyi Xie, [START_REF] Xie | Periodic points of birational transformations on projective surfaces[END_REF]). Let k be a field, and let d be a positive integer.

(1) In Bir d (P 2 k ) the subsets Z(d; λ) = f ∈ Bir d (P 2 ) | λ 1 ( f ) ≤ λ are Zariski closed.

(2) For every f ∈ Bir d (P 2 k ), the set B( f

; λ) = {A ∈ PGL 3 (k) | λ 1 (A • f ) ≤ λ} is a Zariski closed subset of PGL 3 (k).
(3) When λ < d, Z(d; λ) is a nowhere dense subset of Bir d (P 2 k ) and B( f ; λ) is a proper subset of PGL 3 (k) (for every d > 1 and f ∈ Bir d (P 2 k )). Note that the first assertion says that λ 1 is a lower-semicontinuous function for the Zariski topology.

Proof. The first statement follows from the semi-continuity result obtained in [START_REF] Xie | Periodic points of birational transformations on projective surfaces[END_REF], Theorem 1.5. The second is a direct consequence of the first, because the action of PGL 3 (k) on Bir d (P 2 k ) is algebraic. The third assertion is a consequence of Theorem 1.5 and of its proof (see pages 917 and 918 of [START_REF] Xie | Periodic points of birational transformations on projective surfaces[END_REF]). Fix d ≥ 2, and denote by λ 1 (d) < λ 2 (d) < . . . < λ j (d) . . . the sequence of dynamical degrees which are realized by birational maps of the plane of degree d. This is a countable sequence (see [START_REF] Urech | Remarks on the degree growth of birational transformations[END_REF][START_REF] Blanc | Dynamical degrees of birational transformations of projective surfaces[END_REF]). Choose an element g of Bir d (P 2 k ).

The Zariski closed subsets B(g; λ i (d)) form a countable family of proper Zariski closed subsets of PGL 3 (k). Thus, Theorem 1.3 and Proposition 1.1 provide the following result.

Corollary 1.4. Assume that k is not countable. If f is a birational map of the plane of degree d ≥ 2, then there exists an automorphism A of the plane such that λ 1 (A • f ) = d (and A • f is 1-stable, see [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF]). In particular, A • f is not regularizable.

1.3. Dolgachev's question. This is related to the following question, asked by Dolgachev to the second author of this paper:

Dolgachev's question.-Set Reg( f ) := {A ∈ Aut(P m k ) | A • f is regularizable} . (1.4)
Does there exist a birational transformation f of P m k such that deg( f ) > 1 and Reg( f ) is equal to Aut(P m k ) ? This question depends on m and k. The previous corollary treats the case m = 2 when k is uncountable. We shall now treat two more cases: In Section 1.4 we consider higher dimensional spaces (m > 2), and in Section 1.5 we consider the first example of a countable field, namely m = 2 and k = Q.

1.4. Higher dimension. We now answer Dolgachev's question by extending Corollary 1.4 to higher dimensional projective spaces.

If f is a birational transformation of P m k , we denote by Ind( f ) its indeterminacy set, and by Exc( f ) its exceptional locus: it is the union of all proper Zariski closed subsets W of P m k \ Ind( f ) whose strict transform f • (W ) satisfies dim( f • (W )) < dim(W ). The exceptional locus is a non-empty union of hypersurfaces, except when f is an automorphism of P m k . Consider two birational transformations f and g of P m k . Then, deg( f • g) ≤ deg( f ) deg(g) and this inequality is strict if and only if one can find a hypersurface W ⊂ P m k which is contracted by f into the indeterminacy locus Ind(g). Thus, deg( f n ) = deg( f ) n , unless there is an irreducible hypersurface D in the exceptional locus Exc( f ) which is ultimately mapped in Ind( f ) by some iterate f n of f (with n > 0). Theorem 1.5. Let f be a birational transformation of P m k of degree d ≥ 2. The set of automorphisms A in PGL m+1 (k) such that deg((A • f ) n ) = (deg(A • f )) n for some n > 0 is a countable union of proper Zariski closed subsets of PGL m+1 (k).

To prove this result, we present a simple variation on Xie's argument for the third assertion of Theorem 1.3; the proof of Theorem 1.9 below is even closer to Xie's original strategy.

Proof. Let D 1 , . . ., D ℓ be the irreducible components of codimension 1 in Exc( f ). Let C j,0 = f • D j be the strict transform of D j . The subsets C j,0 are Zariski closed, of codimension ≥ 2. For A in PGL m+1 (k), set g A = A • f . Then, the exceptional and indeterminacy loci satisfy Exc(g A ) = Exc( f ), Exc(g -1 A ) = A(Exc( f -1 )), Ind(g A ) = Ind( f ), and Ind(g -1 A ) = A(Ind( f -1 )). For every index 1 ≤ j ≤ ℓ, and every integer n ≥ 0, consider the subset F( j; n) ⊂ PGL m+1 (k) of automorphisms A such that A(C j,0 ) is not contained in Ind( f ), and then g A (A(C j,0 )) is not contained in Ind( f ), up to g n A (A(C j,0 )) which is not contained in Ind( f ) either. This set F( j, n) is Zariski open in PGL m+1 (k), and to prove the theorem our goal is to show that all these Zariski open subsets are non-empty. This follows from Lemma 1.6 below.

Lemma 1.6. Let W and Q be non-empty, irreducible, Zariski closed subsets of

P m k , with Ind( f ) ⊂ W = P m k .
Let n ≥ 0 be an integer. There exists an element A of

PGL m+1 (k) such that A(Q) is not contained in W , (A • f )(A(Q)) is not contained in W , . . ., (A • f ) n (A(Q)) is not contained in W .
Proof. We may assume that W is a hypersurface. Start with n = 0. We look for A such that A(Q) is not contained in W . The set of such matrices A is Zariski open and non-empty, because W has codimension 1 and the action of PGL m+1 (k) is transitive. Now, assume that the property is not satisfied for n = 1. This means that among the elements

A in PGL m+1 (k) such that A(Q) is not contained in W , all of them satisfy (A • f )(A(Q)) ⊂ W ; fixing an equation P of W , we obtain P((A • f )A(z)) = 0 (∀z ∈ Q \ A -1 Ind( f )). (1.5)
This equation makes sense on the set of points

(z, A) in Q × GL m+1 (k) such that A(z) / ∈ Ind( f )
and, more generally, on the set of points (z, A) in Q × Mat m+1 (k) such that z is not in the kernel of the linear map A and A(z) is not in Ind( f ). Now, take such a pair (z, A) with A of rank one. We choose A such that (1) its image projects P m k to a point q A outside W (in particular, q A is not in Ind( f )) and (2) the projection K A ⊂ P m k of its kernel does not contain f (q A ) and does not contain Q.

Then, for every point

z ∈ Q \ K A we obtain (A • f )(A(z)) = A( f (q A )) = q A / ∈ W and P(A • f )(A(z)) = 0.
This provides a contradiction with Equation (1.5).

Thus, the set of automorphisms A for which A(Q) and

(A • f )(A(Q)) are not contained in W is generic in PGL m+1 (k).
The same argument proves the lemma by recursion (choosing the same matrix of rank one).

Corollary 1.7. Let k be an uncountable, algebraically closed field. Let f be a birational transformation of P m k of degree d ≥ 2. There exists an automorphism A in PGL m+1 (k) such that A • f is not regularizable.

Remark 1.8. Let f be a birational transformation of a projective variety M. Assume that f contracts an irreducible subvariety D 0 of V onto a Zariski closed subset C 0 (i.e. C 0 = f • (D 0 ) and dim(C 0 ) < dim(D 0 )). Then, assume that the positive orbit of C 0 , namely

C 1 = f • (C 0 ), . . ., C n+1 = f • (C n ) is
made of subsets which are never contained in Ind( f ). Suppose, in the opposite direction, that the preimages D -n-1 = f • (D -n ) are never contracted. Then, the C n and the D -n form two sequences of pairwise distinct algebraic subsets; and if the union of the D -n is Zariski dense in M (this is automatic if they have codimension 1), the transformation f is not regularizable. Indeed, if ϕ : V M is a birational map, one can find large integers N > 1 such that D -N is not contracted and C N is not blown-up by ϕ -1 . Then,

f V = ϕ • f • ϕ -1 cannot be an automorphism because f 2N V contracts ϕ -1 (D -N ) onto ϕ -1 (C N ).
This argument provides the following extension of Corollary 1.7: Let k be an uncountable, algebraically closed field. Let f be a birational transformation of a homogeneous variety M = G/H. Assume that Exc( f ) is not empty. Then there is an element A of G such that A • f is not regularizable. 1.5. Dolgachev's question over Q. Theorem 1.9. Let f be a birational transformation of P m which is defined over the field Q, i.e. the formulas defining f have coefficients in Q. Then, there exists

an element A of PGL m+1 (Q) such that deg((A • f ) n ) = deg( f ) n for all n ≥ 1.
Theorem 1.5 does not imply directly this result because a countable union of proper Zariski closed subsets of PGL m+1 (C) can contain PGL m+1 (Q).

Proof. If the degree of f is equal to 1, we can just take A = Id, so we assume deg( f ) ≥ 2.

The exceptional locus of Exc( f ) is a non-empty hypersurface, defined by an equation J(x 0 , . . ., x m ) = 0 with integer coefficients; let J = ∏ s i=1 J i be the decomposition of J into irreducible factors over Q, i.e. as elements of Q[x 0 , . . . , x m ], and let (W i ) s i=1 be the corresponding irreducible components of Exc( f ). We are looking for an element A of PGL m+1 (Q) such that A • f and its iterates (A • f ) n do not map W i into Ind( f ), for all i = 1, . . . s and all n ≥ 1. Note that the W i may be reducible over Q; but if the strict transform of W i by (A • f ) n is not contained in Ind( f ), then none of its irreducible components is mapped into Ind( f ) (because A • f , Ind( f ), and W i are all defined over Q).

Let D i be the intersection of Ind( f ) with W i : it is a Zariski closed subset of positive codimension in W i . There exists a line L i , defined over Q, such that (L i ∩ W i )(Q) is non-empty and L i does not intersect Ind( f )( 1 ); in particular, L i does not intersect D i . Restricting J i to the line L i , we get a homogeneous polynomial in one variable, with integer coefficients, that we denote P i . By a theorem of Frobenius, the density of prime numbers p such that P i splits into linear factors is positive (see [START_REF] Stevenhagen | Chebotarëv and his density theorem[END_REF] and []).

We shall now work modulo such a prime p. We can write f and its inverse f -1 in homogeneous coordinates, with homogeneous formulas whose coefficients are integers. Denote by f the reduction of (the formulas defining) f modulo p; we shall always choose p large enough to be sure that f is a birational map, of degree equal to the degree of f , and with inverse equal to f -1 . We denote by L i the reduction of L i modulo p, etc. For p large, the line L i does not intersect Ind( f ).

So, fix such a large prime p, and pick an F p -point x of L i ∩W i , corresponding to one of the roots of P i . This point is not on Ind( f ), hence f is regular on an open neighborhood U of x; set U i = U ∩ W i . Then f maps U i onto an open subset of a component of Ind( f ). Since SL m+1 (F p ) acts transitively on P m (F p ) there is an element A i ∈ SL m+1 (F p ) such that A i ( f (x)) = x. Then A i • f satisfies the following property: for every integer n ≥ 0, (A i • f ) n is regular at x and

(A i • f ) n (x) = x. This implies that the strict transform of W i by (A i • f ) n is not contained in Ind( f ).
We do such a construction for each of the s irreducible components W i of Exc( f ). This gives a finite sequence of prime numbers p i and matrices A i ∈ SL m+1 (F p i ). By the weak approximation theorem, there is a matrix A ∈ SL m+1 (Z)

1 Take a point a ∈ P m (Q) outside W i ∪ Ind( f ), and consider the projective space P m-1 of lines through a; the linear projection that maps a point b = a to the line (ab) provides a ramified cover π : W i → P m-1 . The codimension of π(Ind( f )) is positive because dim(Ind( f )) ≤ m -2, and this implies that there are points of P m-1 (Q) in the complement of π(Ind( f )). This corresponds to a line L i with the desired properties. such that A ≡ A i mod (p i ) for every i = 1, . . ., s. Set g A = A • f , and pick an irreducible component W i of Exc( f ). Then, (g A ) n never maps W i into Ind(g A ) = Ind( f ), because otherwise its reduction modulo p i would map W i into Ind( f ). This shows that no component of Exc(g A ) is mapped into Ind(g A ) and, according to [39, § 1.4]

OSCILLATIONS

Let N denote the set of non-negative integers, and N * the set of positive integers.

Theorem 2.1. Let D : N * → N be a function which vanishes identically on the complement of a finite set. Let k be an uncountable, algebraically closed field. There exist a birational transformation h :

P 2 k P 2 k and an integer d ≥ 1 such that deg(h n ) = d -D(n) for all n ≥ 1.
When k has characteristic 0, one may find h in Bir(P 2 Q ). This means that deg(h n ) oscillates arbitrarily around its asymptotic value d, at least on the support of D. Since deg(h n ) = d for large values of n, h is an elliptic transformation; in fact, we shall construct h as a conjugate g • A • g -1 of a linear projective transformation A ∈ Aut(P 2 k ) by a Jonquières transformation g. The first part of this theorem is proved in Sections 2.1 to 2.4; the last assertion is a corollary of Theorem 2.12 from Section 2.5.

2.1.

Preliminaries. Recall that the Picard-Manin space is generated by the class e 0 of the polarization O(1) (i.e. the class of a line in P 2 ) and by the classes e(p) of the exceptional divisors E p which are obtained by blowing up proper or infinitely near points p of the plane.

Let A be an element of Aut(P 2 k ) = PGL 3 (k). It acts by automorphisms on P 2 (k); for every point p ∈ P 2 (k), A lifts to an isomorphism from the blow-up of P 2 at p to the blow-up at A(p), sending the exceptional divisor E p to E A(p) . Thus, its action on the Picard-Manin space satisfies

A • e 0 = e 0 and A • e(p) = e(A(p)) (2.1)
for every proper or infinitely near point p of P 2 (k) (see [START_REF] Cantat | The Cremona group[END_REF]). Let g be a birational transformation of the plane, and let h be the birational

transformation h = g • A • g -1 . The sequence of degrees (deg(h n )) n is given by deg(h n ) = (g • A n • g -1 ) • e 0 |e 0 = A n • ((g -1 ) • e 0 )|g -1 • e 0 . (2.2) Write g -1 • e 0 = deg(g)e 0 - ℓ ∑ i=1 a i e(p i ) (2.3)
for a finite set of base points p i and multiplicities a i > 0; we assume for simplicity that there is no infinitely near point involved in this formula:

(Hyp. 1) Each e(p i ) is the class of a proper point p i ∈ P 2 (k).

In particular, the p i are exactly the indeterminacy points of g, and the a i are their multiplicities. Then, (A n • g -1 ) • e 0 = deg(g)e 0 -∑ ℓ i=1 a i e(A(p i )) and we obtain

deg(h n ) = deg(g) 2 - ℓ ∑ i, j=1 a i a j δ(A n (p i ), p j ) (2.4)
where δ(•, •) is the Kronecker symbol. For instance, if A n (p i ) = p j for all i, j, and n ≥ 1, then deg(h n ) = deg(g) 2 for all n ≥ 1. Now, let us organize the points p i with respect to the different orbits of A. This means that we index the points p i in such a way that there exist a sequence of integers k 0 = 0 < k 1 < • • • < k m = ℓ and a sequence of points q 1 , q 2 , . . ., q m ∈ P 2 (k) satisfying (Orb. 1) the points p k j-1 +1 , . . ., p k j lie on the orbit of q j under the action of A (∀ j ∈ {1, . . . , m}). (Orb. 2) the points q j are in pairwise distinct orbits.

The subset F j := {p k j-1 +1 , . . . , p k j } is therefore a finite part of the orbit A Z (q j ); it coincides with the intersection of A Z (q j ) with Ind(g). Now, assume that (Hyp. 2) The multiplicity function p i → a i is constant, and equal to some positive integer a j , on each F j .

Then, the formula for the degree of

h n reads deg(h n ) = deg(g) 2 - m ∑ j=1 a j 2 |A n (F j ) ∩ F j |. (2.5)
To describe the sequence n → |A n (F j ) ∩ F j |, assume that (Hyp. 3) None of the q j is a periodic point of A.

Parameterizing each of those orbits by ϕ j : n → A n (q j ), F j corresponds to a finite subset G j = ϕ -1 j (F j ) of Z, and A is conjugate to the translation T : n → n + 1 along the orbit of q j . Then,

|A n (F j ) ∩ F j | = |T n (G j ) ∩ G j | =: ol j (n) (2.6)
where ol j (n) is, by definition, the overlapping length of G j and T n (G j ). For instance, if |G j | = 1, then ol j (n) is equal to 1 if n = 0 and vanishes for n = 0. If G j = {a, b} and ba = s > 0, then ol j (n) is equal to 2 when n = 0, to 1 when n = s, and vanishes on Z \ {0, s}; in that case, the restriction of ol j to N * is the Dirac mass on s. Thus, we obtain Lemma 2.2. Every function D : N * → N with finite support S is a sum of |S| functions ol j (•) for subsets G j ⊂ N * with exactly two elements each.

Now, adding the assumption

(Hyp. 4) If |F j | ≥ 2, then a j = 1, one gets deg(h n ) = deg(g) 2 - m ∑ j=1 ol j (n) (2.7)
and ol j (n) = 0 when a j = 1 (resp. when |F j | = 1).

To prove Theorem 2.1, we first apply Lemma 2.2 to write D = ∑ m j=1 ol j (n) for some finite subsets G j ⊂ N * with exactly two elements each. Then, we shall construct an element g ∈ Bir(P 2 k ) with a unique indeterminacy point q 0 of multiplicity > 1, and m indeterminacy points p j of multiplicity 1. All we have to do is to choose g in such a way that the four hypotheses (Hyp. 1) to (Hyp. 4) are satisfied.

Jonquières transformations.

Let N be a positive integer. Given a distinguished point p 0 ∈ P 2 (k) (also denoted q 0 in what follows), and 2N points p 1 , . . ., p 2N ∈ P 2 (k), we consider the following incidence properties. (Jon. 1) The finite set {p 0 , p 1 , . . . , p 2N } does not contain three colinear points. (Jon. k) A plane curve of degree k which contains q 0 with multiplicity k -1 contains at most 2k of the remaining points p i (1

≤ i ≤ 2N).
Note that in the second property, we denote p 0 by q 0 to distinguish it from the other points. For example, condition (Jon. 3) means that there is no cubic curve which passes through q 0 with multiplicity 2 and contains 7 of the remaining points p 1 , . . ., p 2N . As we shall see, this is a generic property on the points p i . Condition (Jon. k) is empty when k > N.

Theorem 2.3 (see [START_REF] Blanc | Dynamical degrees of birational transformations of projective surfaces[END_REF]). If the conditions (Jon. 1) and (Jon. k) are satisfied for all 2 ≤ k ≤ N, there exists a birational transformation g : P 2 k P 2 k such that (1) g preserves the pencil of lines though q 0 ; (2) g has degree N + 1;

(3) the base points of g are p 0 = q 0 , with multiplicity N, and the points p 1 , . . ., p 2N , all with multiplicity 1; (4) g -1

• e 0 = (N + 1)e 0 -Ne(q 0 ) -∑ 2N j=1 e(p j ).

The transformations provided by this theorem are "Jonquières transformations", because they preserve a pencil of lines.

Remark 2. [START_REF] Blanc | Groupes de Cremona, connexité et simplicité[END_REF]. In what follows, we shall repeatedly use that the vector space of polynomial functions of degree k in 3 variables has dimension

Dim(k) = k + 2 2 = (k + 2)(k + 1) 2 = Dim(k -1) + k + 1,
and that the polynomial equations corresponding to curves passing through a given point q 0 with multiplicity (k -1) form a subspace of dimension

dim(k) = 2k + 1.
Also, note that two curves C and D in this system intersect in q 0 with multiplicity (k -1) 2 , and in 2k -1 extra points (counted with multiplicity).

2.3.

The construction. To prove Theorem 2.1 directly, we adopt the strategy developped in Section 2.1. First write D = ∑ m j=1 ol j for some subsets G j = {a, a + s j } of N * with 2 elements each. Fix an algebraically closed field k and an infinite order element A in PGL 3 (k). We want to construct m + 1 points q 0 , q 1 , . . ., q m such that (Prop. 1) the points q 0 , q 1 , . . ., q m are on infinite, pairwise distinct orbits of A; (Prop. 2) the (2m + 1) points p 0 = q 0 and p 2 j-j = q j , p 2 j = A s j (q j ) for 1 ≤ j ≤ m satisfy the conditions (Jon. 1) and (Jon. k) for all 2 ≤ k ≤ m.

Then, Theorem 2.3 provides a Jonquières transformation g of degree m + 1 with base points p 0 = q 0 of multiplicity m and p j of multiplicity 1 for 1 ≤ j ≤ 2m. And the birational transformation h = g • A • g -1 satisfies the conclusion of Theorem 2.1 (with d = (m + 1) 2 ). In what follows, A is fixed; the point q 0 is also fixed, and is chosen so that its orbit is infinite. We shall prove that a generic choice of the points q j satisfy (Prop. 1), and (Prop. 2).

A word of warning: we use the word "generic" in an unconventional way. We say that the q j are generic if (q j ) m j=0 is chosen in the complement of countably many proper, Zariski closed subsets W i of (P 2 k ) m+1 that depend only on A. A more appropriate terminology would be "very general", but we use "generic" and "generically" for short.

A generic choice of the q i , i ≥ 1, satisfy (Prop. 1), so we analyze only (Prop. 2). For that purpose, we prove recursively that a generic choice satisfies (Jon. k), adding progressively new points q m ; the recursion is on m (for each k). We start with condition (Jon. 1).

2.3.1. Condition (Jon. 1). Here, we prove that a generic choice of the q i , i ≥ 1 satisfies (Jon. 1).

First, choose q 1 in such a way that its orbit (under the action of A) is infinite and does not contain q 0 ; a generic choice of q 1 works. The extra condition imposed by (Jon.1) is that q 0 , q 1 , and A s 1 (q 1 ) are not colinear. But if those three points were colinear for a generic choice of q 1 , A s 1 would preserve the pencil of lines through q 0 , A s 1 (q 0 ) would be equal to q 0 , and q 0 would have a finite orbit. Thus, (Jon. 1) is satisfied for p 0 = q 0 , p 1 = q 1 , and p 2 = A s 1 (q 1 ) as soon as q 1 is chosen generically.

Choosing q 2 generically, its orbit is infinite, and disjoint from the orbits of q 0 and q 1 . We want to show that no three of the points p 0 = q 0 , p 1 = q 1 , p 2 = A s 1 (q 1 ), p 3 = q 2 , p 4 = A s 2 (q 2 ) are colinear. First, we choose q 2 in the complement of the lines connecting two of the points p 0 , p 1 , p 2 and their images by A -s 2 . Then, if three of those points are colinear, this means that there is a point p j with 0 ≤ j ≤ 2 such that p j , q 2 , and A s 2 (q 2 ) are colinear. Since q 2 is generic, this would imply that A s 2 preserves the pencil of lines through p j . But then p j would have a finite orbit, in contradiction with the choice of q 0 and q 1 .

The same argument works for the points q j , j ≥ 3: one just needs to choose q j generically in order to have an infinite orbit, to avoid the orbits of the previous points p i (0 ≤ i ≤ 2 j -2), and to avoid the lines connected two of the p i (0 ≤ i ≤ 2 j -2) and their images by A -s j . This proves that (Jon. 1) is satisfied for a generic choice. Thus, we may assume that it is satisfied in what follows.

2.3.2. Condition (Jon. 2). To warm up, we prove that a generic choice of the q i satisfies (Jon. 2), i.e. for conic curves. We already know it satisfies (Jon. 1). Thus, we want to prove that, for a generic choice of the q i , a conic that contains q 0 with multiplicity 1 does not contain more than 4 of the p i . First, we show that the following property is generic: (Uniq. 2) Five points (q 0 , p i 1 , . . . , p i 4 ) with i j ∈ {1, . . ., 2m} are always contained in a unique conic curve, and this curve is irreducible.

Proof. Fix four indices i j , and consider a conic curve C containing q 0 , p i 1 , . . ., and p i 4 ). From Remark 2.4, Dim(2) = 6 and there is at least one such curve C. If there is a second one D, we get two distinct conics with at least 5 common points, so that C and D share a common irreducible component. But a reducible conic containing (q 0 , p i 1 , . . ., p i 4 ) is made of two lines, and we obtain a contradiction with (Jon. 1).

Now, let us prove that (Jon. 2) is generically satisfied. As in Section 2.3.1, we do a recursion on the number m of points q i . This condition is obviously satisfied for m ≤ 2, since then at most five points are involved. Assume now that property (Jon. 2) is satisfied for a generic choice of (m -1) points. Let q m be an extra generic point. Consider the set of conics C containing q 0 and four points among the p i , for i ≤ 2(m -1), and their images by A -s m (C); by (Uniq. 2) this set is finite; since q m is generic, we may assume that it is not on one of these conics. It remains to show that, for any choice of three indices i 1 , i 2 , i 3 ≤ 2(m -1), the unique conic through (q 0 , p i 1 , p i 2 , p i 3 , q m ) does not contain A s m (q m ). Assume the contrary. Since q m is chosen generically, then A s m would fix the pencil of conics through the base points (q 0 , p i 1 , p i 2 , p i 3 ); it would permute these base-points, and the orbit of q 0 would not be infinite, a contradiction.

To sum up, properties (Jon. 1), (Jon. 2) and (Uniq. 2) are satisfied for a generic choice of the points q i . 2.3.3. Condition (Jon. k). Now, we prove by induction that (Jon. k) is generically satisfied for all k ≤ m. As above, the points p i will be p 0 = q 0 , and p 2i-1 = q i , p 2i = A s i (q i+1 ) for i ≥ 1. Our recursion will simultaneously prove that (Jon. k) and the following condition (Uniq. k) are satisfied for a generic choice of the q i . (Uniq. k) For every choice of 2k points (p i 1 , . . ., p i 2k ) with i j ∈ {1, . . . , 2m}, there is a unique plane curve of degree k containing q 0 with multiplicity k -1 and (p i 1 , . . . , p i 2k ) with multiplicity ≥ 1, and this curve is irreducible.

First, we show that the properties (Jon. ℓ) for ℓ ≤ k imply (Uniq. k + 1). For this, consider the system of curves of degree k +1 with multiplicity k at q 0 . By Remark 2.4, their equations form a vector space of dimension dim(k + 1) = 2k + 3. Now, fix a finite subset P of the p i with 2(k + 1) elements, and consider the subsystem of curves containing P. There is at least one such curve C because 2k + 3 -2(k + 1) = 1. If there were a second curve C ′ of this type, then, counted with multiplicities, C ∩ C ′ would contain at least k 2 + 2(k + 1) = (k + 1) 2 + 1 points. This is larger than deg(C) deg(C ′ ). So, either C is unique, or C is a reducible curve containing P and passing through q 0 with multiplicity k. Then, one of the connected components would contradict (Jon. ℓ), for some ℓ ≤ k.

Indeed, decompose C into connected components C = C 1 + • • • + C s of degree d i , with d 1 + • • • d s = k + 1. If C i is not a line, its multiplicity in q 0 is bounded by mult(C i ; q 0 ) ≤ d i -1.
Thus, at most one of these components has degree ≥ 2. If C is a union of k + 1 lines, we obtain a contradiction with (Jon. 1).

If C = L 1 + • • • L s-1 + C s with deg(C s ) ≥ 2,
and none of the lines L i contains three points of P, then C s has degree d s = k + 1 -s, contains q 0 with multiplicity d s -1 = ks, and contains 2k + 2 -s points p i ∈ P. From (Jon. k + 1 -s) we know that such a curve does not exist. Now, we derive (Jon. k + 1) from (Uniq. k + 1). We do a recursion on the number m of points. There is nothing to prove for m ≤ k. Now, assume that (Jon. k + 1) is generically satisfied for m points; the problem is to add one more point q m+1 and verify that (Jon. k + 1) is satisfied. Apply (Uniq. k + 1): if P is a finite subset of the p i with |P| = 2(k + 1), there is a unique curve C P of degree k + 1 containing P and passing through q 0 with multiplicity k + 1. Since q m+1 is chosen generically, we may assume that it is not on one of the curves C P or A -s m+1 (C P ) for any P ⊂ {p 1 , . . . p 2m } with |P| = 2(k + 1). Now, (Jon. k + 1) fails if and only if there is a finite subset Q ⊂ {p 0 , . . . p 2m } with 2k + 1 elements such that the linear system of curves of degree k + 1 containing q 0 with multiplicity k and Q with multiplicity 1 is A s m+1 -invariant. By (Uniq. k + 1), this linear system is a pencil with a finite set of base points containing Q; then, the orbit of the points p j ∈ Q would be finite, contradicting (Jon. 1).

Conclusion.

Let us come back to the proof of Theorem 2.1. What did we do in Section 2.3 ? We fixed the function D. Then we proved that, if k is an algebraically closed field, and if A ∈ PGL 3 (k) is an element of infinite order, then a very general choice of the points q 0 , q 1 , . . ., q m satisfy (Prop. 1) and (Prop. 2). As a consequence, if k is uncountable, such a (m + 1)-tuple does exist, and there is a Jonquières transformation g such that the birational transformation h = g • A • g -1 satisfies the conclusion of Theorem 2.1.

2.5.

On the realization of bounded degree sequences. In this section, our main goal is to prove that a bounded sequence of degrees (deg( f n )) n≥0 which is realizable by at least one birational map f ∈ Bir(P 2 C ) is also realizable by some birational map g ∈ Bir(P 2 Q ); we also use this opportunity to show how to combine the adelic topology with p-adic arguments.

2.5.1. The adelic topology. In this section, we make use of the adelic topology introduced in Section 3 of [START_REF] Xie | The existence of Zariski dense orbits for endomorphisms of projective surfaces (with an appendix in collaboration with Thomas Tucker[END_REF]; here we gather some of its properties.

Let k be an algebraically closed extension of Q of finite transcendence degree. Let X be a variety over k. The adelic topology is a topology on X (k), which is defined by considering all possible embeddings of k in C and C p , for all primes p. Assume, for instance, that X is defined over the prime field

Q. If ι : k → C p (resp. C) is an embedding, it induces an embedding Φ ι : X (k) → X (C p ) (resp. X (C)); then,
given any open subset U of X (C p ) for the p-adic (resp. euclidean) topology, the union ∪ ι Φ -1 ι (U ) for all embeddings ι : k → C p is, by definition, an open subset of X (k) in the adelic topology (see [START_REF] Xie | The existence of Zariski dense orbits for endomorphisms of projective surfaces (with an appendix in collaboration with Thomas Tucker[END_REF]). The adelic topology has the following basic properties.

(1) It is stronger than the Zariski topology. If dim(X ) ≥ 1, there are non- (5) The irreducible components of X (k) in the Zariski topology coincide with the irreducible components of X (k) in the adelic topology. (6) Let K be any subfield of k such that (a) K = k, (b) K is finitely generated over Q and (c) X is defined over K. Endow the Galois group Gal (k/K) with its profinite topology. Then the action

empty, adelic, open subsets U and U ′ of X (k) such that U \ U ′ is Zariski dense in X . (2) It is T 1 , that is for every pair of distinct points x, y ∈ X (k) there are adelic open subsets U, V of X (k) such that x ∈ U, y ∈ U and y ∈ V , x ∈ V .
Gal (k/K) × X (k) → X (k)
is continuous with respect to the adelic topology.

In particular, when X is irreducible, the intersection of finitely many nonempty adelic open subsets of X (k) is nonempty. We say that a property P "holds for an adelic general point" if there exists an adelic dense open subset U of X (k), such that P holds for all points in U.

2.5.2. Sequences of incidence times. Let k be a field. Let X be a variety, f : X → X be an automorphism, and Z ⊂ X be a subvariety, all of them defined over k.

For every x ∈ X , set

N(x, f , Z) = {n ∈ Z ; f n (x) ∈ Z}. (2.8)
This is the set of times n ∈ Z for which the incidence f n (x) ∈ Z occurs.

Theorem 2.5. Let k be an algebraically closed extension of Q of finite transcendence degree. Let X be a variety, f : X → X be an automorphism, and Z ⊂ X be a subvariety, all three defined over k. Let V be any irreducible subvariety of X and let η be the generic point of V . Then N(x, f , Z) = N(η, f , Z) holds for an adelic general point x ∈ V (k).

In the following corollary, we keep the notations and hypotheses of this theorem. Recall that the constructible topology on X is a topology for which constructible sets form a basis of open sets; for instance, if η is the generic point of an irreducible subvariety V of X , any open set containing η contains a dense Zariski open subset of V ; and any constructible set is both open and closed (see [START_REF] Grothendieck | Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas[END_REF], Section (1.9) and in particular (1.9.13)).

Corollary 2.6. Let χ : X → Z be a function which is continuous with respect to the constructible topology of X . Then

χ( f n (x)) = χ( f n (η)) for all n ∈ Z for an adelic general point x of V (k).
Proof. Since X is compact with respect to the constructible topology, χ(X ) is compact and therefore finite. For r ∈ χ(X ), χ -1 (r) is both open and closed in the constructible topology; since it is compact, it is a finite union

χ -1 (r) = m r i=1 (Z r i \ S r i ) (2.9)
where the Z r i and S r i are closed subvarieties of X . Let B be the set of subvarieties

{Z r i , S r i ; r ∈ χ(X ), i = 1, . . ., m r }. For every x ∈ X , the sequence (χ( f n (x))) n∈Z is determined by the sets N(x, f , Z), for Z ∈ B.
By Theorem 2.5, for every Z ∈ B, there exists a nonempty adelic open subset

U Z of V (k) such that for every x ∈ U Z , N(x, f , Z) = N(η, f , Z). Then ∩ Z∈B U Z is nonempty and every x in this intersection satisfies N(x, f , Z) = N(η, f , Z) for each Z ∈ B. For such a point, χ( f n (x)) = χ( f n (η)) for all n ∈ Z.
Corollary 2.7. Let k be an algebraically closed field of characteristic zero. Let X be a variety and f : X → X be an automorphism of X , both defined over k. Let χ : X → Z be a function which is continuous with respect to the constructible topology of X . Let V be an irreducible subvariety of X and η the generic point of V . Then there exists a point x ∈ V (k) such that χ( f n (x)) = χ( f n (η)) for all n ∈ Z.

Proof. There exists a subfield k 0 of k which is algebraically closed and of finite transcendence degree over Q such that X , f , χ and V are defined over k 0 .

Replacing k by k 0 , the conclusion follows from Corollary 2.6.

Remark 2.8. The proofs of Theorem 2.5 and its corollaries hold when f is a non-ramified endomorphism, if we consider only the forward orbit of x.

2.6. Proof of Theorem 2.5. After replacing X by the Zariski closure of the set { f n (η)} n∈Z , we may assume that X is smooth at η. We leave the proof of the following lemma to the reader.

Lemma 2.9. Let Z and Z ′ be subvarieties of X and let l be a nonzero integer.

Then we have

(1) N(x, f , Z ∩ Z ′ ) = N(x, f , Z) ∩ N(x, f , Z ′ ); (2) N(x, f , Z ∪ Z ′ ) = N(x, f , Z) ∪ N(x, f , Z ′ ); (3) N(x, f l , Z) = -N(x, f -l , Z); (4) N(x, f , Z) = ∪ |l|-1 i=1 (|l| × N(x, f |l| , f -i (Z)) + i); (5) N( f l (x), f , Z) = N(x, f , Z) -l.
By (4) of Lemma 2.9, we may assume X to be irreducible, and by ( 1) and (2), we may assume Z to be an irreducible hypersurface.

Since X is smooth at η, we can pick a point b ∈ V (k) such that X and V are smooth at b. Let K be a finitely generated extension of Q such that K = k and X , Z, V , f , and b are all defined over K. There exists a projective variety

X K → Spec (K) such that X = X K × Spec (K) Spec (k) and f K : X K → X K such that f = f K × Spec (K) Id.
There exist a subring R of K that is finitely generated over Z such that Frac (R) = K, and a model π : X R → Spec (R) which is projective over Spec (R) and whose generic fiber is X K . Shrinking Spec R, we may assume that

(i) all fibers of X R are absolutely irreducible of dimension d = dim X , (ii) f extends to an automorphism f R : X R → X R over R, and, denoting by Z R , V R , b R the Zariski closures of Z, V , b in X R , (iii) all fibers of Z R and V R are absolutely irreducible, (iv) b R is a section of V R → Spec R, (v) Spec R is smooth, and Z R and X R are smooth along b R .
Lemma 2.10 (see [START_REF] Lech | A note on recurring series[END_REF][START_REF] Bell | A generalised Skolem-Mahler-Lech theorem for affine varieties[END_REF]). Let L be a finitely generated extension of Q and let B be any finite subset of L. The set of primes p for which there is an embedding L → Q p that maps B into Z p has positive density2 among the set of all primes.

Denote by C •

p the closed unit ball in C p , and by C

(Z p ) → Spec (R). Set X C p := X R × Spec (R) Spec (C p ), and f C p := f R × Spec (R)
Id. All fibers X y , for y ∈ Spec (R), are absolutely irreducible and of dimension d; hence, the special fiber

X F p of X C • p → Spec (C • p ) is absolutely irreducible. Denote by b F p ∈ X F p the specialization of b R . It is a smooth point of X C • p . There exists m ≥ 1 such that ( f | X F p ) m (x) = x and (d f m | X F p ) x = Id. Let U be the open subset of X (C p ) consisting of points whose specialization is y; then U ≃ (C •• p ) d .
We have f m (U ) = U , and the orbit of every point of U is welldefined. The restriction of f m to U is an analytic automorphism

f m | U : (x 1 , . . . , x d ) → (F 1 , . . . , F d ) (2.10)
where the F n are analytic functions on U ; more precisely, F n (x) = x n + ∑ I a n I x I with coefficients a n I ∈ C •• p ∩Z p that converge towards 0 as the length of the multiindex I goes to ∞. Then, there exists l ∈ Q + , such that

f m | U = Id mod p 2/l .
(2.11)

Set W := {(x 1 , . . ., x d ) ∈ U ; |x i | ≤ p -1/l } ≃ (C • p ) d . Then W is invariant by f m and f m | W = Id mod p 1/l . (2.12)
After replacing m by some of its multiples km, we get

f m | W = Id mod p. (2.13)
By (4) of Lemma 2.9, we may replace f by f m and suppose m = 1; by (2), we may assume that Z is irreducible. By [37, Theorem 1], there exists an analytic action Φ :

C • p ×W → W of (C • p , +) on V such that for every n ≥ Z, Φ(n, •) = f n | W (•). Denote by π 1 : C • p ×W → C •
:= Φ -1 (Z) is an analytic hypersurface of C • p × W .
Changing W in a smaller polydisc, we may assume that every irreducible component

Y i , i = 1, . . ., l, of Y is smooth and π 2 | Y i : Y i → W is surjective. Shrinking W again, we may assume that all restrictions π 2 | Y i : Y i → W are isomorphisms. Define φ i : W → C • p by φ i := π 1 • π 2 | -1 Y i . Permuting the indices, we suppose that φ i is non-constant for 1 ≤ i ≤ s and φ i is constant for s + 1 ≤ i ≤ l. If x is a point of W ∩V (k), then N(x, f , Z) = {φ i (x)| i = 1, . . . , l} ∩ Z ⊆ {φ i (x)| i = 1, . . ., l} ∩ Z p (2.14) and N(η, f , Z) = {φ i | i = s + 1, . . . , l} ∩ Z, (2.15)
because φ i is a constant (an element of C • p ) if and only if i ≥ s + 1. Now, note the following: if i ≤ s, φ i is not constant and is therefore an open mapping; since Z p is a nowhere dense and closed subset of

C p , φ -1 i (C p \ Z p
) is open and dense in W . Thus, the set

U ′ := ∩ s i=1 φ -1 i (C p \ Z p
) is open and dense in W , and for x ∈ U ′ ∩V (k) we get N(x, f , Z) = N(η, f , Z). By [START_REF] Xie | The existence of Zariski dense orbits for endomorphisms of projective surfaces (with an appendix in collaboration with Thomas Tucker[END_REF]Remark 3.11], the same equality holds for an adelic general point x ∈ V (k). This concludes the proof.

2.6.1. Bounded degree sequences. Let m ≥ 0 be an integer, and k be an algebraically closed field. Denote by R(k, m) ⊂ (N) Z the set of sequences (deg( f n )) n∈Z for all birational transformations f of P m (k). Denote by R b (k, m) the subset of bounded sequences (deg(

f n )) n∈Z in R(k, m). Remark 2.11. The sequence (deg( f n )) n∈Z is bounded if and only if (deg( f n )) n∈Z ≥0 is bounded. Theorem 2.12. If k is an algebraically closed field of characteristic zero, then R b (k, m) = R b (Q, m).
The last assertion of Theorem 2.1 follows from this statement.

Proof of Theorem 2.12. Since k has characteristic zero and is algebraically closed, we may view it as an extension of

Q. In particular, R b (Q, m) ⊂ R b (k, m). Let f : P m k P m
k be a birational transformation for which the degree sequence

(deg( f n )) n∈Z is bounded. We want to show that this sequence is in R b (Q, m).
There exists a Q-subalgebra A of k such that A is finitely generated over Q and f is defined over A. Then we may assume k = Frac A. We view

B := Spec A (2.16)
as a variety over Q, and we denote by η its generic point. There is a birational transformation F : P m B P m B over B whose restriction to the generic fiber is f .

B(Q) such that deg(F n b ) = deg( f n ) for all n ∈ Z.
Since (deg f n ) is bounded, we may apply Weil's regularization theorem (see [13, Theorem 2.5] for a statement and references, and [START_REF] Weil | On algebraic groups of transformations[END_REF][START_REF] Huckleberry | Actions of groups of birationally extendible automorphisms[END_REF][START_REF] Zaitsev | Regularization of birational group operations in the sense of Weil[END_REF][START_REF] Kraft | Regularization of rational group actions[END_REF] for proofs).

H X := π -1 • F • π : X → X (2.17)
is an automorphism on the generic fiber X := X η , where η denotes the generic point of B. After shrinking B, we may assume that B is smooth, π is birational on all fibers and H X is an automorphism over B.

Denote by Aut B (X ) the automorphism group scheme of X over B. Then H X can be viewed as a section of Aut B (X )(B). We view Aut(X ) as the generic fiber of Aut B (X ) and denote by H the element of Aut(X ) induced by H X .

Since

(deg( f n )) is bounded, the closure G of {H n ; n ∈ Z} in Aut(X )
is a commutative subgroup of Aut(X ) of finite type. After shrinking B, we may assume that the closure G of G in Aut B (X ), is a group scheme of finite type. We have H ∈ G(B).

Denote by φ : G → B the structure morphism. Denote by

L H : G → G the automorphism x → H φ(x) •
x, where • is the product for the group structure. View G as a variety over Q. The function δ : G → N defined by

δ : x → deg(F φ(x) ) = deg((π • H χ • π -1 ) φ(x) )
is lower-semi continuous with respect to the Zariski topology; in particular, δ is continuous with respect to the constructible topology. View (the image of the section) H as a subvariety of G and denote by α its generic point. Then, by construction, deg(

f n ) = δ(L n H (α)) for all n ∈ Z. All we need to show is the existence of a point x ∈ F(Q) such that δ(L n F (α)) = δ(L n F (x)
) for all n: this follows from Corollary 2.7.

AUTOMORPHISMS OF HALPHEN SURFACES

In this section we study Halphen surfaces, their automorphisms, and conjugacy classes in the group Aut(X ). Our main goal is Theorem 3.14. Sections 3.1 and 3.2 summarize and rephrase classical results concerning Halphen surfaces, their Néron-Severi lattice, and their group of automorphisms: we refer to [START_REF] Dolgachev | Rational surfaces with a pencil of elliptic curves[END_REF][START_REF] Cantat | Rational surfaces with a large group of automorphisms[END_REF][START_REF] Gizatullin | Rational G-surfaces[END_REF][START_REF] Grivaux | Parabolic automorphisms of projective surfaces (after M. H. Gizatullin)[END_REF][START_REF] Iskovskikh | Algebraic surfaces[END_REF] for proofs of these results. This is applied in Sections 3. Halphen pencils (see [START_REF] Dolgachev | Rational surfaces with a pencil of elliptic curves[END_REF][START_REF] Cantat | Rational surfaces with a large group of automorphisms[END_REF]). A Halphen pencil p of index m is a pencil of plane curves of degree 3m, with 9 base points of multiplicity ≥ m and no fixed component. We shall denote by p 1 , p 2 , . . ., p 9 the nine base points (some of the p i may be infinitely near). Blowing up them, we get a rational surface X with Picard number 10, together with a birational morphism ε : X → P 2 k ; the pencil p determines a fibration π : X → P 1 k , with fibers of arithmetic genus 1. Going back to P 2 k , π corresponds to a rational function π = π • ε -1 : P 2 k P 1 k whose fibers are the members of the pencil p. When m > 1, π has a unique multiple fiber, of multiplicity m; it corresponds to a cubic curve with multiplicity m in the pencil p (see [START_REF] Gizatullin | Rational G-surfaces[END_REF]).

In what follows, p will always denote a Halphen pencil, and X the associated Halphen surface.

Warning. In characteristic 2 or 3, the fibration π may be quasi-elliptic, all fibers being singular curves with a node or a cusp. By convention, we shall always assume that the general fibers are smooth (so π is a genus 1 fibration whose generic fiber is smooth; we exclude quasi-elliptic fibrations: see Section 4.1.1 which explains why we do so).

3.1.2. Automorphisms of the pencil (see [START_REF] Gizatullin | Rational G-surfaces[END_REF][START_REF] Grivaux | Parabolic automorphisms of projective surfaces (after M. H. Gizatullin)[END_REF][START_REF] Iskovskikh | Algebraic surfaces[END_REF]). Let Bir(P 2 k ; p) be the group of birational transformations of the plane preserving the pencil p. After conjugacy by the blow-up ε : X → P 2 k , Bir(P 2 k ; p) becomes a group of birational transformations of X that permutes the fibers of π. The fibration π is relatively minimal, which amounts to say that there is no exceptional divisor of the first kind in the fibers of π; this implies that Bir(P 2 k ; p) is contained in Aut(X ). Moreover, the members of the linear system | -mK X | are given by the fibers of π; hence, Aut(X ) permutes the fibers of π, and we conclude that

ε -1 • Bir(P 2 k ; p) • ε = Aut(X ) (3.1)
and there is a homomorphism τ :

Aut(X ) → Aut(P 1 k ) such that π • f = τ( f ) • π (∀ f ∈ Aut(X )). (3.2)
Equivalently, there is a homomorphism τ : Bir(P 2 ; p) → Aut(P 1 ) such that π • f = τ( f ) • π for every f ∈ Bir(P 2 ; p).

3.2.

The Néron-Severi group and its geometry.

3.2.1. Picard group (see [START_REF] Cantat | The Cremona group[END_REF][START_REF] Cantat | Rational surfaces with a large group of automorphisms[END_REF][START_REF] Grivaux | Parabolic automorphisms of projective surfaces (after M. H. Gizatullin)[END_REF]). The Picard group of X is discrete, and coincides with the Néron-Severi group NS(X ). A basis is given by the class e 0 of a line (pulled back to X by ε) and the nine classes e i corresponding to the (total transform of the) exceptional divisors of the blow-ups of the p i . The intersection form satisfies e 2 0 = 1, e i • e j = 0 if i = j, and e 2 i = -1 if i ≥ 1. So, with this basis, the lattice NS(X ) is isometric to Z 1,9 . We set NS(X ; R) := NS(X ) ⊗ Z R and view NS(X ) as a lattice of NS(X ; R). The canonical class of X is the vector

k X = -3e 0 + e 1 + • • • + e 9 . The anticanonical class ξ := -k X = 3e 0 -e 1 -• • • -e 9 (3.3)
will play an important role; the class of a general fiber of π is equal to mξ.

3.2.2.

The spaces ξ ⊥ , P and Q. The orthogonal complement ξ ⊥ of ξ for the intersection form is a subgroup of NS(X ) of rank 9 that contains ξ; we denote by ξ ⊥ R the real vector subspace of dimension 9 in NS(X ; R) spanned by ξ ⊥ . Denote by P the quotient space

ξ ⊥ R / ξ R , where ξ R = R • ξ is the subspace of ξ ⊥ R
spanned by ξ, and by q P : ξ ⊥ R → P the projection. The intersection form induces a negative definite quadratic form on P; since ξ is primitive, ξ ⊥ /ξ is isomorphic to a lattice P(Z) ⊂ P. We denote by • P the euclidean norm given by this quadratic form:

q P (w) P = (-w • w) 1/2 , ∀w ∈ ξ ⊥ R . (3.4)
Let Q 0 be the affine space

Q 0 = e 0 + ξ ⊥ R ⊂ NS(X ; R); (3.5) then Q 0 (Z) = e 0 + ξ ⊥ = Q 0 ∩ NS(X ) is a lattice in Q 0 . Denote by Q the quotient of Q 0 by the (R, +)-action x → x + tξ, t ∈ R, and by q Q : Q 0 → Q the quotient morphism; set e = q Q (e 0 ) and Q(Z) = q Q (Q 0 (Z)). (3.6)
Then Q(Z) = e + ξ ⊥ /ξ is a lattice of Q. Observe that the composition of the translation w ∈ ξ ⊥ R → w+e 0 and the projection q Q induces a natural isomorphism w ∈ P → w + e ∈ Q. Using this isomorphism to transport the metric

• P on Q, we get a natural metric • Q : for every v 1 , v 2 ∈ Q and w i ∈ q -1 Q (v i ), i = 1, 2, we have v 1 -v 2 2 Q = -(w 1 -w 2 ) 2 (3.7)
where the square, on the right hand side, denotes self-intersection.

3.2.3. Horospheres (see [START_REF] Cantat | The Cremona group[END_REF][START_REF] Cantat | Elements generating a proper normal subgroup of the Cremona group[END_REF]). Let H denote the half-hyperboloid

H = {u ∈ NS(X ; R) ; u 2 = 1 and u • e 0 > 0}. (3.8)
With the riemannian metric induced by (the opposite of) the intersection form, H is a model of the hyperbolic space of dimension 9. The isotropic vector ξ determines a point of the boundary ∂H. By definition, the set

Hor = {u ∈ H ; u • ξ = 3} = {u ∈ Q 0 ; u 2 = 1} (3.9)
is the horosphere centered at ξ that contains e 0 . We have

Hor = {u = e 0 + v ∈ NS(X ; R) ; 2e 0 • v + v 2 = 0, and v • ξ = 0}. (3.10)
Observe that the restriction q Q | Hor : Hor → Q is a homeomorphism. In what follows, we denote its inverse by

hor := (q Q | Hor ) -1 . (3.11)
Remark 3.1. The projection q P induces a linear isometry ξ ⊥ ∩ e ⊥ 0 → P. If we compose it with the isomorphism w → e + w from P to Q, and then apply hor, we get the following parametrization of the horosphere Hor:

w ∈ ξ ⊥ ∩ e ⊥ 0 → hor(e + q P (w)) = e 0 + w - w 2 6 ξ. (3.12) 3.2.4. Isometries fixing ξ. If ϕ is an isometry of NS(X ; R) we set deg(ϕ) = e 0 • ϕ(e 0 ) (3.13)
and call it the degree of ϕ. When ϕ = f * for some f ∈ Aut(X ), then deg(ϕ) coincides with the degree of the birational transformation ε • f • ε -1 ; we denote this number by deg( f ) or by deg( f * ) and call it the degree of f . If ϕ fixes ξ, then ϕ preserves H, the horosphere Hor, and the affine subspace

Q 0 ; it induces an affine isometry ϕ Q of Q such that hor -1 • ϕ| Hor • hor = ϕ Q . ( 3.14)
Since Hor generates NS(X ; R), the isometry ϕ Q determines ϕ uniquely:

Lemma 3.2. The homomorphism ϕ → ϕ Q , from the stabilizer of ξ in Isom (NS(X ; R))
to the group of affine isometries of Q, is faithful.

The group Aut(X ) preserves the anti-canonical class ξ, so every element f ∈ Aut(X ) determines such an affine isometry f Q of Q as well as a linear isometry f P of P. The lattices Q(Z) and P(Z) are invariant by these isometries. Hence, Aut(X ) is an extension of a discrete group of affine isometries of Q by the kernel of the representation f → f * of Aut(X ) on NS(X ).

For every v ∈ Q, set d(v) = e 0 • hor(v). Equations (3.7) and (3.12) imply

d(v) = e 0 • hor(v) = 1 + 1 2 v -e 2 Q . (3.15)
Then for every f ∈ Aut(X ), we get deg( f

) = d( f Q (e)).
3.2.5. Reducible fibers, and the subspace R(X ) (see [START_REF] Gizatullin | Rational G-surfaces[END_REF][START_REF] Grivaux | Parabolic automorphisms of projective surfaces (after M. H. Gizatullin)[END_REF]). Consider the finite set Irr(X ) ⊂ NS(X ) of classes [C i ] of irreducible components of fibers of π; for instance, mξ ∈ Irr(X ) (take a regular fiber); if m > 1 and F = mF 0 is the unique multiple fiber, we get [F 0 ] = ξ ∈ Irr(X ), so ξ ∈ Irr(X ) whatever the value of m. Taking two disjoint fibers, we obtain

[C i ] • ξ = 0 for every [C i ] ∈ Irr(X ). Let R(X ) ⊂ NS(X ) (resp. R(X ) R ⊂ NS(X ; R)) be the linear span of Irr(X ).
If F is a fiber, and

F = ∑ µ(F)
i=1 a i C i is its decomposition into irreducible components -with µ(F) the number of such components -, then there is a linear relation with positive coefficients

[F] = µ(F) ∑ i=1 a i [C i ] = mξ (3.16)
between the elements ξ and [C i ] of Irr(X ). It turns out that all linear relations between the elements of Irr(X ) are linear combinations of the relations given in Equation (3.16), as F describes the set of reducible fibers. So, (1) R(X ) is a free abelian group; its rank is equal to 1

+ ∑ F (µ(F) -1)
, where the sum is over all reducible fibers;

(2) ξ ∈ R(X ) ⊆ ξ ⊥ .
The genus formula and the equality ξ = -k X show that there are only three possibilities for the elements c of Irr(X ): either c = ξ, or c = mξ (in these cases c 2 = 0), or c is the class of a smooth rational curve of self-intersection c 2 = -2. In fact, by the Hodge index theorem, an element v of R(X ) satisfies v 2 = 0 if and only if it is contained in Zξ. This gives (3) every class c ∈ Irr(X ) satisfies c 2 ∈ {0, -2}, with c 2 = 0 if and only if c = ξ or mξ. Then, using an Euler characteristic argument and Kodaira's classification of singular fibers, one can show that there are at most 12 singular fibers, each of them containing at most 10 irreducible components (see [START_REF] Cossec | Progress in Mathematics[END_REF]). If C i is such an irreducible component, either C i comes from a component of the pencil p, and Lemma 3.3. Let m be a positive integer. Consider the set of all Halphen surfaces X of index m: there are only finitely many possibilities for the subset Irr(X ) ⊂ Z 1,9 = NS(X ) and for its linear spans R(X ) and R(X ) R .

1 ≤ deg([C i ]) := [C i ] • e 0 ≤ 3m
Here, NS(X ) is identified with Z 1,9 by our choice of basis e 0 , . . ., e 9 , as in Section 3.2.1.

Proof. According to Properties (2) and (4),

Irr(X ) ⊂ {ξ, mξ} ∪ {v ∈ NS(X ) ; v 2 = -2 and v • e 0 ≤ B 1 (m)}.
(3.17

)
This set is finite, bounded by a uniform constant that depends on m but not on X (or p). The lemma follows.

3.2.6. The cylinder Q + . Now, let c be an element of Irr(X )

⊆ ξ ⊥ . Since c• ξ = 0, the linear map w ∈ NS(X ; R) → w • c ∈ R induces an affine function on Q: if v = q Q (w)
for some w in Q 0 , then we set v • c = w • c and this does not depend on the choice of w. Set

Q(c ⊥ ) = {u ∈ Q ; u • c = 0} and Q(c + ) = {u ∈ Q ; u • c ≥ 0}. (3.18) If c ∈ Z >0 ξ then Q(c ⊥ ) = / 0 and Q(c + ) = Q. If c ∈ Irr(X ) satisfies c 2 = -2, then Q(c ⊥ ) is a hyperplane of Q and Q(c + ) is one of the two closed half-spaces bounded by Q(c ⊥ ).
We want to describe

Q + = c∈Irr(X) Q(c + ). (3.19)
Since c is represented by an effective curve, the point e = q Q (e 0 ) is in Q + . Consider the spaces A 0 ⊂ ξ ⊥ R and A ⊂ P defined by

A 0 = R(X ) ⊥ R and A = A 0 / ξ R . (3.20)
They act respectively by translations on Q 0 and Q, and we denote by q Q/A : Q → Q/A the affine quotient map associated to this action. Note that every fiber of

π is irreducible if and only if R(X ) R = ξ R , if and only if A = ξ ⊥ R / ξ R ,

if and only if A acts transitively on Q, if and only if

Q/A = 0.
Lemma 3.4. The set Q + contains e and is invariant under the action of A by translations. Its projection q Q/A (Q + ) is a compact polytope D(X ) in Q/A; the number of faces of D(X ) is uniformly bounded (by a constant B 0 that does not depend on X and m).

Proof. The first assertion follows from the inclusion A 0 ⊂ R(X ) ⊥ R . For the second, we shall use the following fact. Let E be a finite dimensional vector space, and let (L i ) 1≤i≤s be a finite generating subset of the dual space E ∨ . Let A i , 1 ≤ i ≤ s, be real numbers, and let C ⊂ E be the convex set determined by the affine inequalities

ℓ i (x) := L i (x) + A i ≥ 0. (3.21)
Assume that ∑ i a i ℓ i = 0 for some positive real numbers a i . Then C is compact or empty. In our case, the ℓ i are given by the inequalities

[C i ] • x ≥ 0 in Q/A.
Consider a reducible fiber F of π, and decompose it into irreducible components

F = ∑ µ(F) i=1 a i C i , with [C i ] ∈ R(X ).
All a i are > 0, and

µ(F) ∑ i=1 a i ([C i ] • v) = mξ • v = 0 (3.22)
for every v in Q. Since D(X ) contains e it is no empty, and we conclude that D(X ) is compact. Since Q + is the intersection of the half spaces Q(c + ) and ♯Irr(X ) ≤ B 0 is uniformly bounded (see § 3.2.5, Property (4)), D(X ) has at most B 0 faces. Denote by Cent(D(X )) the center of mass of D(X ) (for the Lebesgue measure on Q/A): Cent(D(X )) depends only on the affine structure of Q/A. Set

Ax = q -1 Q/A (Cent(D(X ))) ⊂ Q; (3.23)
it is an affine subspace of Q of dimension dim Ax = dim A = 9 -rank(R(X )).

Proposition 3.5. The convex set Q + is a cylinder with base D(X ) and axis Ax. The action of Aut(X ) on Q preserves Q + and Ax.

Proof. The group Aut(X ) permutes the elements of Irr(X ), so it preserves Q + , and on Q/A it preserves the polytope D(X ); since it acts by affine automorphisms, it preserves its center of mass.

For every f ∈ Aut(X ), denote by f Q/A its action on Q/A and by f Ax its action on Ax. Denote by p Ax : Q → Ax the orthogonal projection. Since f Q is an isometry, we have p Ax

• f A = f Ax • p Ax .
Since D(X ) is compact, there exists r ≥ 0, such that for every x ∈ Q + , p Ax (x) -x Q ≤ r. This r only depends on Irr(X ). Then for every x, y ∈ Q + , we have

x -

y 2 Q ≤ p Ax (x) -p Ax (y) 2 Q +4r 2 . (3.24)
3.3. The subgroup Aut t (X ).

3.3.1. Vertical translations (see [START_REF] Cantat | Rational surfaces with a large group of automorphisms[END_REF][START_REF] Grivaux | Parabolic automorphisms of projective surfaces (after M. H. Gizatullin)[END_REF]). The generic fiber X of π is a genus 1 curve over the field k(P 1 k ). If we take an element u in Pic(X ) = NS(X ) and restrict it to the generic fiber, we get an element of Pic(X); this restriction has degree 0 if and only if u is in ξ ⊥ . This defines a homomorphism Tr : ξ ⊥ → Pic 0 (X)(k(P 1 k )). It is surjective, and its kernel coincides with the subgroup R(X ) of NS(X ). So, we get an isomorphism of abelian groups

tr : ξ ⊥ /R(X ) → Pic 0 (X)(k(P 1 k )). (3.25)
(we use the notation tr after taking the quotient by R(X )). Now, start with an element q of Pic 0 (X)(k(P 1 k )). On a general fiber X t of X , with t ∈ P 1 k , we get a point q(t) ∈ Pic 0 (X t ), and then a translation z → z + q(t) of the elliptic curve X t . This gives a birational map T (q) of X acting by translations along the general fibers; since the fibration π : X → P 1 k is relatively minimal, this birational map is in fact an automorphism (see § 3.1.2). So, we obtain a homomorphism

T : Pic 0 (X)(k(P 1 k )) → Aut(X ). (3.26)
In what follows, ι : ξ ⊥ /R(X ) → Aut t (X ) will denote the composition T • tr, as well as T • Tr, with the same notation since this should not induce any confusion. Also, recall that τ : Aut(X ) → Aut(P 1 k ) is defined in § 3.1.2. Proposition 3.6. The homomorphism T satisfies the following properties.

(1) T is injective.

(2) Its image is the normal subgroup Aut t (X ) ⊂ Aut(X ) of automorphisms f such that (i) f preserves each fiber of π, i.e. τ( f ) = Id P 1 k ; (ii) if X t is a smooth fiber, then f acts as a translation on it.

(3) If α ∈ NS(X ) and γ ∈ ξ ⊥ , then ι(γ) * α = α + (α • ξ)γ mod R(X ) (i.e. (T (Tr(γ))) * α = α + (α • ξ)γ mod R(X )).
Proof. The homomorphism T is injective because q(t) = 0 for at least one t if q ∈ Pic 0 (X)(k(P 1 k )) \ {0}. Its image is contained in the normal subgroup Aut t (X ); it coincides with it, because if f is an element of Aut t (X ), and X t is any smooth fiber, one can set q(t) = f (x) -x for any x ∈ X t , and this defines an element of Pic 0 (X)(k(P 1 k )) for which T (q) = f . So, we only need to prove [START_REF] Blanc | Conjugacy classes of affine automorphisms of K n and linear automorphisms of P n in the Cremona groups[END_REF]. By linearity, we may assume that α is represented by an irreducible curve V ⊂ X . Denote by V η the fiber of V at the generic point η of P 1 . Observe that V η is a zero cycle in X of degree (α • ξ). Since ι(γ) is an element of Aut t (X ), it fixes every fiber X t of π and permutes its irreducible components. We can therefore assume

V η = / 0; then ι(γ) * V is the Zariski closure of (ι(γ))| X ) * V η .
Pick a divisor D on X which represents γ and denote by D η its generic fiber. Then Tr(γ) is represented by D η . On X η , ι(γ)| X η is the translation by Tr(γ); thus, (ι(γ)| X η ) * V η and V η +(α • ξ)D η are linearly equivalent as divisors of Pic(X η ), and of Pic(X). In other words, there exists a rational function

h in k(P 1 )(X) = k(X ) such that Div(h)| X = (V η + (α • ξ)D η ) -(ι(γ)| X ) * V η , (3.27)
where Div(h) is the divisor associated to h. Taking Zariski closures in X , we get

Div(h) = (V + (α • ξ)γ) -(ι(γ)| X ) * V +W (3.28)
where W is a vertical divisor i.e. W ∩ X = / 0. It follows that ι(γ) * α = α + (α • ξ)γ mod R(X ), which concludes the proof.

3.3.2.

Action of ξ ⊥ on Ax. Denote by ι : ξ ⊥ /R(X ) → Aut(NS(X )/R(X )) the induced action.

Lemma 3.7. The homomorphism ι is injective. For every f ∈ Aut(X ) and γ in ξ ⊥ (resp. in ξ ⊥ /R(X )), we have ι(

f * γ) = f -1 • ι(γ) • f .
Proof. Let γ be in the kernel of ι. Apply Property (3) of Proposition 3.6 to the class α = e 0e 1e 2 ; since α • ξ = 1, we obtain γ = 0 mod R(X ). This proves that ι is injective. If f is an element of Aut(X ), then f * fixes ξ and it stabilizes ξ ⊥ and R(X ). Denote by f * the action of f on NS(X )/R(X ). For all γ ∈ ξ ⊥ /R(X ) and α ∈ NS(X )/R(X ), we obtain the following equalities in NS(X )/R(X )

ι( f * γ)(α) = α + (α • ξ) f * γ (3.29) = f * ( f * ) -1 α + (( f * ) -1 α • ξ) γ (3.30) = ( f * • ι(γ))(( f * ) -1 α) (3.31)
because f * preserves intersections and fixes ξ. Thus, ι(

f * γ) = f * • ι(γ) • ( f * ) -1 . Then ι( f * γ) = f -1 • ι(γ) • f , so Aut t (X )
is a normal subgroup and ι is injective.

Denote by q P the quotient map P → ξ ⊥ /R(X ) R , and by q

A : A → ξ ⊥ /R(X ) R its restriction to A = R(X ) ⊥ / ξ R . By the Hodge index theorem, R(X ) R ∩ R(X ) ⊥ R = ξ R . (3.32)
So q A is an isomorphism. For γ ∈ ξ ⊥ , denote by a(γ) the unique element of A such that q A (a(γ)) = q P (q P (γ)). Alternatively a(γ) is the orthogonal projection of q P (γ) onto A ⊂ P. The image of the linear map a is A(Z), a lattice in A.

We want to understand the action of ξ ⊥ on Ax given by γ → ι(γ) Ax . Set Ax 0 = q -1 Q (Ax) ⊂ Q 0 . Since every element α of Ax 0 satisfies α • ξ = 3, the third property of Proposition 3.6 gives

ι(γ) * α ∈ (α + 3γ + R(X ) R ) ∩ Ax 0 (∀α ∈ Ax 0 , ∀γ ∈ ξ ⊥ ); (3.33) since Ax 0 = α + R(X ) ⊥ R , we get ι(γ) * α -α ∈ (3γ + R(X ) R ) ∩ R(X ) ⊥ R (∀α ∈ Ax 0 , ∀γ ∈ ξ ⊥ ). (3.34)
Lemma 3.8.

(1) for γ ∈ ξ ⊥ , ι(γ) Ax : Ax → Ax is the translation v → v + 3a(γ);

(2) ι(ξ ⊥ ) Ax is a discrete and co-compact group of translations of Ax;

(3) this group ι(ξ ⊥ ) Ax depends only on Irr(X ) ⊂ ξ ⊥ . (It does not depend on other geometric features of X .)

Proof. Pick any α in Ax 0 . The Equation (3.34) gives q A (q P (ι(γ) * αα)) = 3q P (q P (γ)). Thus, ι(γ) * αα = 3a(γ) modulo Zξ. The three properties follow from this equation.

3.4.

Automorphisms and conjugation. The group Aut(X ) acts on Q by affine isometries for the (euclidean) metric induced by the intersection form. In Q, we have

• the axis Ax, an affine subspace of dimension dim R(X ) ⊥ ;

• the convex cylinder Q + : it is equal to the pre-image of the compact polytope D(X ) ⊂ Q/Ax by q Q/A . The diameter of D(X ) is bounded by some constant r > 0 that depends only on m (not on the geometry of X , see Equation (3.24)). • the projection e of e 0 , an element of Q + ; we shall denote by o the orthogonal projection of e on Ax (the projection of o in D(X ) coincides with the center of mass of D(X ), see Section 3.2.6). The group Aut(X ) preserves Q + and Ax. In what follows, we view Q as a vector space, whose origin is o; then, every element f of Aut(X ) gives rise to an affine isometry

f Q (v) = L f (v) + t( f ) (3.35) where L f is the linear part of f (fixing o) and t( f ) is the translation part. Note that t( f ) = f Q (o) is an element of the lattice Ax∩Q(Z) ⊂ Ax. Since f Q is an isometry and preserves the lattice Q(Z), we have L f ∈ O(Q(Z)) := O(Q) ∩ GL (Q(Z)), where O(Q) is the orthogonal group of (Q, • Q ) and GL (Q(Z)) is the subgroup of GL (Q) that preserves Q(Z). Since O(Q(Z)) is a finite subgroup of GL (Q(Z))
and the rank of Q(Z) is 9, independently of m and X , we get the following lemma.

Lemma 3.9. The group of all linear parts {L f ; f ∈ Aut(X )} is finite: its cardinality is uniformly bounded, by an integer that does not depend on X .

(v) = L f | Ax (v) + t( f ) where L f | Ax is the restriction of L f to Ax. Then L f | Ax ∈ O(Q(Z); Ax)| Ax . Lemma 3.10. There is a constant B 2 (m) ≥ 1 such that -B 2 (m)(1+ t( f ) Q ) ≤ deg( f ) - 1 2 t( f ) 2 Q ≤ B 2 (m)(1+ t( f ) Q ) for all f ∈ Aut(X ). In particular, there is a constant B 3 (m) ≥ 1 such that B 3 (m) -1 deg( f ) ≤ ( t( f ) Q +1) 2 ≤ B 3 (m)deg( f ).
Proof. The degree deg( f ) is equal to the intersection number e 0 • f * e 0 . Since f * e = L f (e) + t( f ), Equation (3.15) gives

deg( f ) = 1 + 1 2 (L f -Id Q )(e) + t( f ) 2 Q . (3.36)
The first estimate follows because L f is an isometry, and the length of e (i.e. the length eo Q in Q) is bounded by a constant that depends only on m. To derive the second inequalities from the first, we set

C 1 (m) = sup x≥0 (x + 1) 2 max{1, 1 2 x 2 -B 2 (m)(1 + x)} ≥ 1. (3.37) Then C 1 (m) deg( f ) ≥ C 1 (m) max{1, 1 2 t( f ) 2 Q -B 2 (m)(1+ t( f ) Q )} (3.38) ≥ ( t( f ) Q +1) 2 . (3.39)
And with C 2 (m) = max{B 2 (m), 1/2}, we get

deg( f ) ≤ 1 2 t( f ) 2 Q +B 2 (m)(1+ t( f ) Q ) ≤ C 2 (m)(1+ t( f ) Q ) 2 . (3.40)
Setting B 3 (m)(m) := max{C 1 (m),C 2 (m)}, we conclude the proof.

The group Aut t (X ) is a finitely generated abelian group. Denote by Aut t tor (X ) its torsion subgroup and set Aut t f r (X ) := Aut t (X )/Aut t tor (X ).

Then Aut t f r (X ) is isomorphic to Z d for d = dim R (ξ ⊥ R /R(X ) R ). Example 3.11. Consider the elliptic curve E = C/Z[j],
where j is a primitive cubic root of unity, and then the abelian surface A = E × E. The group GL 2 (Z[j]) induces a group of automorphisms of A; let η be the automorphism defined by η(x, y) = (jx, jy), and let G ≃ Z/3Z be the subgroup of Aut(A) generated by η. Let X be a minimal resolution of X 0 = A/G. Then X is a smooth rational surface. Since G is contained in the center of GL 2 (Z[j]), the group GL 2 (Z[j]) acts on X 0 and on X by automorphisms. The singularities of X 0 correspond to the points of A which are fixed by some non-trivial element in G; the fixed points of G form a subgroup K of A and the action of K on A/G by translations induces an action on X by automorphisms of finite order. Now, consider the fibration π : X → P 1 k induced by the first projection π 1 : A → E. This is an example of a (blow up of a) Halphen fibration. With respect to this fibration, elements of K of type (x, y) → (x, y + s) determine elements of Aut t tor (X ) and elements of GL 2 (Z[j]) acting by (x, y) → (x, y + ax) determine elements of Aut t (X ) of infinite order. Remark 3.12. Let ρ 3 : GL (NS(X ; Z)) → GL (NS(X ; Z/3Z)) be the homomorphism given by reduction modulo 3. Its kernel is a torsion free subgroup of index ≤ |GL 10 (Z/3Z)| = (3 10 -1)(3 10 -3) • • • (3 10 -3 9 ). So, if f ∈ Aut(X ) is elliptic, ( f * ) k = Id for some k ≤ 3 100 . Now, assume that f ∈ Aut t (X ) has finite order, choose such a power k, and pick a multisection E of π with negative selfintersection; such a curve exists, for instance among the 9 exceptional divisors blown-down by ε : X → P 2 k , and we may even assume that E intersects a general fiber of π in at most m points. Since E is determined by its class [E], f k fixes E and f km! fixes each of the points given by E in the general fibers of π. Since a general fiber is an elliptic curve in which f km! has a fixed point, we conclude that the order of f is at most 24 × 3 100 × m!. Of course, a more detailed analysis would give a much better uniform upper bound. By Lemma 3.9, there is a constant N ≥ 1 that does not depend on X , such that for every h ∈ Aut t (X ),

L N h = Id. Set Aut t N (X ) := ι(Nξ ⊥ ) ⊂ Aut t (X ). (3.41)
The group Aut t N (X ) has finite index in Aut t (X ). By Lemma 3.7, the action of Aut t (X ) on NS(X ) is faithful, hence so is its action on Q. This implies that Aut t N (X ) is torsion free, because Aut t N (X ) acts by translation on Q, and a translation of finite order is the identity. So Aut t N (X ) is isomorphic to

Z d for d = dim R (ξ ⊥ R /R(X ) R ). Since Aut t (X ) is a normal subgroup of Aut(X ), Aut t N (X ) = {h N ; h ∈ Aut t (X )
} is also normal in Aut(X ). Lemma 3.13. There is a constant B 4 (m) ≥ 1 such that, for every f ∈ Aut(X ), there exists h ∈ Aut t N (X ) with L h = Id and deg(h

• f ) ≤ B 4 (m). Proof. The action of Aut t (X ) = ι(ξ ⊥ ) on Ax is co-compact. Since Aut t N (X ) is of finite index in Aut t (X ), the action of Aut t N (X ) is also co-compact. So the Voronoi cell {u ∈ Ax ; dist Q (u, o) ≤ dist(u, ι(γ) Q o), ∀ι(γ) ∈ Aut t N (X )
} is a compact fundamental domain; let r(m) be the radius of this domain. There is an h in

Aut t N (X ) such that t(h • f ) = h( f (o)
) has length at most r(m). The conclusion follows from Lemma 3.10. Theorem 3.14. Let X be a Halphen surface of index m. Let f and g be two automorphisms of X . If f and g are in the same conjugacy class of Aut(X ), there exists an element h of Aut(X ) such that

h • f • h -1 = g and deg(h) := (e 0 • h * e 0 ) ≤ A(m)(deg( f ) + deg(g)),
for some constant A(m) that depends only on m.

Proof. The group Aut(X ) acts by conjugacy on Aut t (X ) and on Aut t N (X ); if g is in Aut(X ), we denote by A g the endomorphism t → g • t • g -1 of Aut t (X ).

Start with a conjugacy c 0 • f • c -1 0 = g, for some c 0 ∈ Aut(X ). According to Lemma 3.13 there is an element

s 0 ∈ Aut t N (X ) such that c 0 = s 0 • c 1 with deg(c 1 ) ≤ B 4 (m). Then c 1 • f • c -1
1 and g have the same class in the quotient Aut(X )/Aut t N (X ), so that

c 1 • f • c -1 1 = s ′ • g for some s ′ ∈ Aut t N (X ) (indeed for s ′ = [s -1 0 , g]).

Let us use additive notations for Aut t (X ). The conjugacy equation

h • f • h -1 = g, with h = s • c 1 , is now equivalent to s ′ = (A g -Id)(s)
, and we know that there is at least one solution s 0 .

Lemma 3.15. There is s

∈ Aut t N (X ) such that s ′ = (A g -Id)(s) and t(s) Q ≤ B 5 (m) t(s ′ ) Q
for some constant B 5 (m) that depends only on m.

Note that the translation part s

′ Q (o) = t(s ′ ) = t(c 1 • f • c -1 1 • g -1 ) satisfies t(s ′ ) Q ≤ A 1 (m)( t(g) Q + t( f ) Q ) + A 2 (m) (3.42)
for some constants A 1 (m) and A 2 (m) that depend only on m; one can take

A 1 (m) = M 4 with M = max{ L h ; h ∈ Aut(X )} and A 2 (m) = 2A 1 (m) t(c 1 ) Q . Then by Lemma 3.15, we have t(s) Q + 1 ≤ A 3 (m)( t( f ) Q + t(g) Q + 2) where A 3 (m) = B 5 (m)(A 1 (m) + A 2 (m)) + 1. By Lemma 3.10, deg(h) ≤ deg(c 1 ) deg(s) (3.43) ≤ B 4 (m) deg(s) (3.44) ≤ B 4 (m)A 3 (m) 2 (( t( f ) Q + 1) + ( t(g ′ ) Q + 1)) 2 (3.45) ≤ 2B 4 (m)A 3 (m)B 3 (m)(deg( f ) + deg(g)), (3.46)
which concludes the proof.

Proof of Lemma 3.15. Lemma 3.8 shows that Aut t N (X ) acts on Ax faithfully by translations. We may identify Aut t N (X ) to the orbit of o in Ax via the map

h ∈ Aut t N (X ) → h Q (o) = t(h).
The automorphism g acts also on Ax, by g Q (v) = L g (v) + t(g), and the equation (A g -Id)(s) = s ′ corresponds to the equivalent equation

(L g | Ax -Id)t(s) = t(s ′ ).
(3.47)

If E = (e 1 , . . . , e d ) is a basis of Aut t N (X ), denote by • E the euclidean norm d ∑ i=1 a i e i E = ( d ∑ i=1 a 2 i ) 1/2 . (3.48)
There is a constant C(E) ≥ 1 such that for every s ∈ Aut t N (X ), we have

C(E) -1 t(s) Q ≤ s E ≤ C(E) t(s) Q . (3.49)
For every endomorphism Φ : Aut t N (X ) → Aut t N (X ), there are two basis

E 1 (Φ) = (u 1 , . . . , u d ), E 2 (Φ) = (v 1 , . . . , v d ) of Aut t N (X ), a sequence of integers (a 1 , . . . , a d ) ∈ Z d and an index l ≤ d such that • Φ(u i ) = a i v i for all 1 ≤ i ≤ d; • a i = 0 if and only if i > l. Then for every v ∈ Φ(Aut t N (X )), we may write v = ∑ l i=1 a i x i v i , where x i ∈ Z; equivalently, v = Φ(∑ l i=1 x i u i ). This shows that for every v ∈ Φ(Aut t N (X )), there is u ∈ Φ -1 (v) such that u E 1 (Φ) ≤ v E 2 (Φ) . Setting C(Φ) := C(E 1 )C(E 2 ), every v ∈ Φ(Aut t N (X )) is the image of an element u ∈ Aut t N (X ) such that t(u) Q ≤ C(Φ) t(v) Q . (3.50)
We apply this remark to Φ g := A g -Id. For every m, there are only finitely many possibilities for Ax and L g , so there are only finitely many possible L g | Ax ∈ O(Q(Z); Ax)| Ax . Thus, there are only finitely many possible values for the constant C(Φ g ), and the proof follows from Inequality 3.50.

HALPHEN TWISTS, THEIR INVARIANT PENCILS, AND THEIR LIMITS

In this section we study invariant pencils of Jonquières and Halphen twists. In particular, we prove that for every integer d ≥ 1, there exists an integer pen(d) with the following property: if f is a Halphen twist or a Jonquières twist of degree at most d, the unique f -invariant pencil is of degree at most pen(d). This may be considered as a positive answer to the Poincaré problem of bounding the degree of first integrals, but for birational twists instead of algebraic foliations (see [START_REF] Cerveau | Holomorphic foliations in CP(2) having an invariant algebraic curve[END_REF][START_REF] Neto | Some examples for the Poincaré and Painlevé problems[END_REF][START_REF] Pereira | On the Poincaré problem for foliations of general type[END_REF]). Then, we show how this degree bound leads to constructibility results for conjugacy classes of Halphen twists. ( f m )) m≥0 grows quadratically, like c( f )m 2 for some constant c( f ) > 0 (see [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF][START_REF] Boucksom | Degree growth of meromorphic surface maps[END_REF][START_REF] Blanc | Degree growth of birational maps of the plane[END_REF]). It is a Jonquières twist if and only if deg( f m ) grows like c( f )m for some c( f ) > 0. And it is elliptic if deg( f m ) is bounded. If f is not a twist or an elliptic element, then its dynamical degree

λ( f ) := lim m→+∞ deg( f m ) 1/m ( 4.1)
is larger than 1 and, in fact, larger than the Lehmer number (see [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF], and [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF][START_REF] Blanc | Dynamical degrees of birational transformations of projective surfaces[END_REF]).

Halphen twists have been studied by Gizatullin in [START_REF] Gizatullin | Rational G-surfaces[END_REF]. He proved the following result, for which we also refer to [17, Theorem 4.6],

Theorem 4.1. There is an integer B H ≤ 8! with the following property. Let k be an algebraically closed field. Let f ∈ Bir(P 2 k ) be a Halphen twist. Then, up to conjugacy, f preserves a Halphen pencil p (of some index m ≥ 1), and:

(H.1) this Halphen pencil provides an elliptic fibration of the corresponding Halphen surface X , (H.2) the kernel of the homomorphism Aut(X ) → GL (NS(X )) is finite, of order ≤ B H , (H.3) f does not preserve any other pencil of curves.

"Up to conjugacy" means that there is a birational map α :

P 2 k P 2
k such that α -1 • f • α preserves such a Halphen pencil p; of course, f preserves the pencil α(p), and Property (H.3) means that there is no other invariant pencil. The first item means that the fibration π : X → P 1 k induced by p is not quasi-elliptic (see Section 3.1.1 above and [START_REF] Cantat | Elements generating a proper normal subgroup of the Cremona group[END_REF]). Of course, p, its index m, and the conjugacy α depend on f . Fix such a Halphen twist f , and after conjugacy, assume that f preserves a Halphen pencil p, as in Theorem 4.1. We can now apply the results of Section 3 to p. First, the group Bir(P 2 ; p) is conjugate to Aut(X ) by the birational morphism ε : X → P 2 k ; in particular f is conjugate to an automorphism of X . Second, Lemma 3.10 and Property (H.2) imply that (H.4) The group of vertical translations Aut t (X ) is a normal subgroup of Aut(X ) of index ≤ I(m) for some positive integer I(m) that depends only on m. Some positive iterate of f is contained in this group.

The fifth property we shall need is the existence of an integer J(m) (depending only on m) such that:

(H.5) If h is an element of Aut(X ) of finite order, then its order divides J(m).

If h ∈ Aut(X ) is elliptic, then it has finite order (and thus h J(m) = Id).

To prove this last fact, we first apply (H.4): h I(m) is a vertical translation of X . From Section 3.3.1, the order of the torsion part of Pic 0 (X)(k(P 1 k )), hence of Aut t (X ), is bounded by some constant J 0 (m), because there are only finitely many possibilities for ξ ⊥ /R(X ). Thus if h has finite order, its order divides

J(m) = I(m) × J 0 (m). If h is elliptic, h * ∈ GL (NS(X )
) is an element of finite order. By (H.2), h itself has finite order and we conclude with the first assertion. 4.1.2. Jonquières twists. Similarly, (J.1) a Jonquières twist preserves a unique pencil of curves, and it is a pencil of rational curves; (J.2) after conjugacy, we can take this pencil to be given by the lines through the point [0 : 1 : 0] ∈ P 2 k . With coordinates (x, y) = [x : y : 1] on the affine part z = 0, any birational transformation f preserving this pencil can be written f (x, y) = (A(x), B x (y)) where A ∈ PGL 2 (k) and B x ∈ PGL 2 (k(x)) (note that such a transformation can be either elliptic or a Jonquières twist).

(P 2 k ) → deg( f m ) ∈ Z + (4.2)
is lower semi-continous for the Zariski topology; in other words, the sets {g ∈ Bir d (P 2 k ) ; deg(g m ) ≤ D} are Zariski closed: degrees of iterates can only decrease under specialization (see [START_REF] Xie | Periodic points of birational transformations on projective surfaces[END_REF]Lemma 4.1]). The dynamical degree f → λ( f ) is also lower semi-continous (see [START_REF] Xie | Periodic points of birational transformations on projective surfaces[END_REF]Theorem 4.3]). In particular, the set

Pog(d) ⊂ Bir d (P 2 k ) defined by Pog(d) = {g ∈ Bir d (P 2 k ) ; λ(g) = 1} (4.3)
is a Zariski closed subset of Bir d (P 2 k ); here, Pog stands for "polynomial growth", because an element of Pog has bounded, linear, or quadratic degree growth: Pog(d) is the disjoint union of the three subsets

Hal(d) = {g ∈ Bir d (P 2 k ) ; g is a Halphen twist}, (4.4
)

Jon(d) = {g ∈ Bir d (P 2 k ) ; g is a Jonquières twist}, (4.5
)

Ell(d) = {g ∈ Bir d (P 2 k ) ; g is elliptic}. (4.6)
If X is a Zariski closed subset of Bir d (P 2 k ), we denote by Hal X (d), Jon X (d), and Ell X (d) the respective intersections of X with these three sets.

Invariant pencils of Halphen twists.

Theorem 4.2. Let W be an irreducible subvariety of Bir d (P 2 k ). If Hal W (d) is Zariski dense in W , then there is an integer d W and a proper Zariski closed subset Z W ⊂ W such that every element g ∈ W \ Z W is a Halphen twist and

(1) the unique g-invariant pencil has degree ≤ d W ;

(2) there is a birational map α g : P 2 k P 2 k of degree ≤ d W that maps this pencil to a Halphen pencil of degree at most d W .

Proof. First, we view the generic element of W as a birational transformation f of P 2 defined over the function field K = k(W ); for every t ∈ W , the specialization f t of f is a birational map of P 2 k of degree d. Lemma 4.3. The birational transformation f of P 2 K is a Halphen twist. Proof of Lemma 4.3. If f ∈ Bir(P 2 K ) were loxodromic, with first dynamical degree λ, then by the semi-continuity Theorem 4.3 of [START_REF] Xie | Periodic points of birational transformations on projective surfaces[END_REF] there would exist a Zariski dense open W ′ ⊂ W such that for every t ∈ W ′ , f t would be loxodromic with λ( f t ) > (λ + 1)/2 > 1, and this would contradict our assumption.

So, f is either elliptic, or a Jonquières twist, or a Halphen twist. If it were elliptic, the degrees of its iterates would be bounded, and by semi-continuity of the degrees every specialization f t would be elliptic, contradicting the existence of Halphen twists f t in W .

Suppose now that f is a Jonquières twist. By semi-continuity of the degrees, for every t ∈ B, n ≥ 0, deg f n t ≤ deg f n ≤ Cn + D for some C, D > 0, contradicting the existence of Halphen twists f t in W . Thus, f ∈ Bir(P 2 K ) is a Halphen twist. Replacing K by some finite extension K ′ , there is a rational function π : P 2

K ′ P 1
K ′ and a birational map α : P 2 K ′ P 2 K ′ such that f preserves the pencil defined by π, and α maps this pencil to some Halphen pencil. According to Property (H.4), π • f ℓ = π for some integer ℓ ≥ 1 (with ℓ ≤ I(m), m being the index of the f -invariant pencil). Write K ′ = k(W ′ ), for some finite base change τ : W ′ → W . Let deg(π), deg(α), and δ be the degrees of π, α, and the Halphen pencil respectively. For s in the complement of a divisor D ′ of W ′ , π specializes to a rational function π s of degree deg(π) defining a pencil of genus 1, and the specialization α s is a birational transformation of P 2 k of degree deg(α) that maps the f τ(s) -invariant pencil to a Halphen pencil of degree δ. Thus, for t ∈ W \ τ(D ′ ), the birational transformation f t preserves a pencil of degree deg(π) by curves of genus 1. Since Jonquières twists preserve a unique pencil, made of rational curves (see § 4.1.2), f t can not be such a twist. If f t were elliptic, f k t = Id for some integer k ≤ 24 × 3 100 × m! (see Remark 3.12). But the subset of birational maps g ∈ W of order ≤ 24 × 3 100 × m! is a Zariski closed subset, and is a proper subset because W contains Halphen twists. Thus, for t in the complement of a proper Zariski closed subset of W , f t is a Halphen twist preserving a pencil of degree d W := deg(π).

Recall that a constructible set is a finite union of locally closed sets. ) is constructible in the Zariski topology, and one can find a finite stratification of Hal(d) such that, on each stratum, the degree of the f -invariant pencil is constant and there is a birational map α f of fixed degree mapping this pencil to some Halphen pencil p f of fixed degree.

Note that p f and α f depend on f and are not canonically defined; for instance α f can be composed with a birational transformation preserving p f , and p f can be changed into a projectively equivalent Halphen pencil.

Proof. Set V 0 = Pog(d), and consider the Zariski closure Hal(d) of Hal(d) in V 0 . Let (W i ) k
i=1 denote the irreducible components of maximal dimension of Hal(d). For each i ∈ {1, . . ., k}, Theorem 4.2 provides an effective divisor D i ⊂ W i and a positive integer d i such that all elements of W i \ D i are Halphen twists preserving a pencil of degree d i . Let V 1 be the union of all D i , together with all the irreducible components of Hal(d) of dimension < dim(W i ). The same argument applies to the closure of Hal

V 1 (d) in V 1 .
Thus, in at most dim Hal(d) steps, we obtain: Hal(d) is constructible and there is a stratification of Hal(d) in locally closed sets on which the degree of the invariant pencil is piecewise constant. The same argument, together with the construction of α given in the proof of Theorem 4.2, gives the last assertion of the corollary.

• f 0 = α • f • α -1 and g 0 = β • g • β -1
are Halphen twists preserving p. Note that f 0 and g 0 have degree at most d × hal(d) 2 and are conjugate by a birational map of the plane preserving the pencil p. Since Bir(P 2 ; p) coincides with Aut(X ; π), where X is the Halphen surface obtained by blowing up the 9 base points of p, the birational transformations f 0 and g 0 can be seen as automorphisms of X which are conjugate by an element h of Aut(X ). Then, the conclusion follows from Theorem 3.14, with

HalCo(d) = 2 × A(m) × d × hal(d) 2 .
This theorem implies that the conjugacy classes of Halphen twists are constructible subsets of the group Bir(P 2 k ) (endowed with its Zariski topology, as in [START_REF] Serre | Le groupe de Cremona et ses sous-groupes finis[END_REF], §1.6).

Corollary 4.6. The conjugacy class of a Halphen twist f in Bir(P 2 k ) is constructible.

Proof. Let d be the degree of f . Theorem 4.5 implies that the intersection of the conjugacy class of f with Bir e (P 2 k ) is equal to

Bir e (P 2 k ) ∩ ψ • f • ψ -1 ; ψ ∈ Bir(P 2 k
) and deg(ψ) ≤ HalCo(max{d, e}) (4.7) and this is a constructible set because the group law is continuous for the Zariski topology (see [START_REF] Serre | Le groupe de Cremona et ses sous-groupes finis[END_REF][START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF]). ). If Jon W (d) is dense in W , then every element of W is either elliptic or a Jonquières twist. Moreover, there is an integer d W ≥ 1 and a proper Zariski closed subset Z W ⊂ W such that if g ∈ W \ Z W is a Jonquières twist, then

(1) the unique g-invariant pencil has degree ≤ d W ;

(2) there is a birational map α g : P 2 k P 2 k of degree ≤ d W that maps this pencil to the pencil of lines through a given point of P 2 k . The proof is the same as for Theorem 4.2, the main remark being that the generic element f of W , viewed as a birational transformation of P 2 K which is defined over the function field K = k(W ), is a Jonquières twist. Indeed, by lower semi-continuity of λ 1 , λ 1 ( f ) = 1; by Theorem 4.2, it cannot be a Halphen twist; and if f were elliptic, every element in W would elliptic by lower semi-continuity of degrees.

As a consequence, the degree of f n is bounded by bn + o(n) for some real number b > 0. Then, to prove that no element of W is a Halphen twist, assume by contradiction that f t is a Halphen twist for some parameter t ∈ W . By definition, deg( f n t ) ≥ an 2 -o(n 2 ) for some real number a > 0. Taking m large enough we get deg( f m ) < deg( f m t ), and this contradicts the semi-continuity of the degree map g → deg(g m ).

f n a (x, y) = x + na, y x x + 1 x + a x + a + 1 • • • x + (n -1)a x + (n -1)a + 1 . (4.8)
It follows that f a is a Jonquières twist except if some multiple of a is equal to 1. Thus, if the characteristic of k is 0, the set of parameters a such that f a is elliptic is a countable Zariski dense subset of k.

Example 4.9. Set f α (x, y) = (αx, q(x)y) where q ∈ k(x) and α ∈ k * . If q(x) = (x -1)/(xβ), then f α is a Jonquières twist if and only if β / ∈ α Z .

Theorem 4.10. Let f α be the birational transformation of the plane defined by f α (x, y) = (αx, q(x)y) where α ∈ k * is not a root of unity and q ∈ k(x) is not constant. Assume that the set α Q \ {1} does not contain any zero, pole or ratio z/z ′ of zeroes and/or poles of q. Then, for every integer m > 1 we have:

(1) f α is a Jonquières twist;

(2) f α is conjugate to g α,m = (αx, α m (αx -1)(xα m ) -1 q(x)y);

(3) there is no conjugacy between f α and g α,m of degree < m.

Note that the degrees of f α and g α,m are bounded by deg(q) + 1; so Theorem 4.5 has no analogue for Jonquières twists.

Let us prove Theorem 4.10. The n-th iterate of f α is the birational transformation f α (x, y) = (α n x, q n (x)y) with

q n (x) = n-1 ∏ j=0 q(α j x).
(4.9)

The degree of q n is n deg(q) except if there is a zero z 1 of q, a pole z 2 of q, and integers 0 ≤ j, k ≤ n such that (α j xz 1 )(α k xz 2 ) -1 is a constant. This happens if, and only if z 1 /α j is equal to z 2 /α k or, equivalently, if z 1 /z 2 is in α Z , which is excluded by hypothesis. Thus, f is a Jonquières twist.

Remark 4.11. We shall work in P 1 k × P 1 k and use affine coordinates (x, y). The indeterminacy points of f are the points (x, y) such that q(x) = 0 and y = ∞ or q(x) = ∞ and y = 0; they are located at the intersection of the horizontal lines y = 0 or ∞ and vertical lines x = z where z is a zero or a pole of q.

Let z be a zero of q. Then, f contracts the vertical line {x = z} onto the point (αz, 0). The forward orbit of this point is the sequence f n (αz, 0) = (α n+1 z, 0): since α n z = z ′ for all zeros and poles of f , the positive iterates of f are well defined along the orbit of (αz, 0). This remark holds also for poles of q and negative orbits of indeterminacy points. In particular, f is algebraically stable, as a birational map of P 1 × P 1 .

To prove that f is conjugate to g m , consider a birational map h m (x, y) = (x, s m (x)y) where s m will be specified in a few lines. Then,

h m • f • h -1 m (x, y) = αx, q(x) s m (αx) s m (x) y
and to obtain g α,m it suffices to choose s m (x) = (x -1)(xα) • • •(x -α m ). For the third assertion of the theorem, we start with the following lemma.

Lemma 4.12. Let u be a birational transformation of the plane that commutes to f α . There exists ℓ ∈ Z and β ∈ k * such that u • f ℓ α (x, y) = (x, βy). In other words, the centralizer of f α is the direct product of ( f α ) Z and the multiplicative group, acting on the second factor by (x, y) → (x, βy) for β ∈ k * .

Proof. Let u be an element of the centralizer of f α . Since f α preserves a unique pencil of curves, u is an element of the Jonquières group, and since its action on the x-variable must commute to x → αx, we deduce that u(x, y) = (εx, B x (y)) for some ε ∈ k * and B x ∈ PGL 2 (k(x)). Moreover, the sections y = 0 and y = ∞ of the fibration are f α -invariant, and every invariant section coincides with one of these two sections; indeed, if S(x) = (x, s(x)) is an f α -invariant section, then s(α n x) = q n (x)s(x), and if s is not one of the constants 0 or ∞ this contradicts deg(q n ) = n deg(q). Thus, u fixes or permutes those two sections, and we obtain u(x, y) = (εx, r(x)y ±1 ) (4.10)

for some rational function r. Since u commutes to f n α , we obtain r(α n x)q n (x) = r(x)q n (εx) or r(α n x)q n (x) -1 = r(x)q n (εx).

(

In both cases, we see that q n (x) and q n (εx) share many zeros and/or poles; in other words, we get equalities of type z/α m = z ′ /(εα ℓ ) where z and z ′ are pairs of zeros and/or poles. Iterating u, i.e. changing ε into ε k we can choose z = z ′ and we see that there are integers k ≥ 1 and ℓ ∈ Z such that ε k = α ℓ (with k ≤ 2 deg(q)). Then, we get a relation z/z ′ ∈ α Q , unless ε is an integral power of α. Thus, changing u into u • f ℓ α for some ℓ ∈ Z, we may assume that ε = 1. Now, writing u • f ℓ α (x, y) = (x, r ′ (x)u), the commutation reads r ′ (α n x) = r ′ (x) or r ′ (α n x) = r ′ (x)q n (x) 2 .

In the left-hand case, r ′ must be a constant because α is not a root of unity. And the right-hand case is excluded because the degree of q n is not bounded.

We can now conclude the proof of the theorem. Let h be a conjugacy between f α and g α,m , then h

• h -1
m is an element of the centralizer of f α . Thus, there is an integer n and an element β of k * such that

h = f n α • β • h m = (x, y) → (α n
x, q n (x)βs m (x)y). The degree of q n s m is equal to the sum deg(q n ) + deg(s m ) because the poles of q are distinct from the zeroes of s m (otherwise z/α k = α j for some pair of integers). Thus, the degree of h is larger than m.

LIMITS

In this section, we work over the field of complex numbers, and we endow the Cremona group Bir(P 2 C ) with the eulidean topology introduced in [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF]. A sequence of birational transformations f n : P 2 C P 2

C of degree at most D converges towards a birational transformation h if and only if the following equivalent properties are satisfied:

(A) One can find homogeneous polynomials (P, Q, R) and (P n , Q n , R n ) of the same degree d ≤ D such that h = [P : Q : R], f n = [P n : Q n : R n ] for all n, and (P n , Q n , R n ) converges towards (P, Q, R) in the vector space of triples of homogeneous polynomials (note that (P, Q, R), or the (P n , Q n , R n ) may have common factors);

(B) The graph of f n in P 2 (C) × P 2 (C) converges towards the union of the graph of h and a residual algebraic subset whose projection on each factor has dimension < 2;

(C) There is a non-empty open subset U of P 2 (C) (in the euclidean topology) on which h and the f n are regular and f n converges to h uniformly.

Our goal is to understand the possible changes of types that occur when taking limits; for instance, can we obtain any elliptic element h as a limit of loxodromic (resp. parabolic) elements f n ? More generally, C could be replaced by any local field, as in [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF].

5.1. Isometries of finite dimensional hyperbolic spaces. To illustrate this problem, consider a finite dimensional hyperbolic space H m ⊂ R 1+m , with its group of isometries PSO 1,m (R). There are three types of isometries, elliptic, parabolic, and loxodromic ones, and in the Lie group PSO 1,m (R)

(1) every parabolic element is a limit of loxodromic elements; if m is odd, every elliptic element is a limit of loxodromic element; (2) every parabolic element is a limit of elliptic elements;

(3) there are elliptic elements which are not limits of parabolic elements (resp. of loxodromic elements if m is odd).

If we consider sequences of pairwise conjugate isometries, the only change of type that may occur by taking limits is a sequence of parabolic elements converging to an elliptic one. In other words, elliptic and loxodromic elements have closed conjugacy classes, but parabolic elements don't. Our goal in this section is to look at this question for the Cremona group in two variables.

). (1) If λ( f n ) = 1 for all n, then λ( f ) = 1.
(2) If all f n are Jonquières twists or elliptic transformations, then f is not a Halphen twist.

Example 5.1. Consider the birational transformation

f b (x, y) = (x + 1, y × x x+b ). Then f b is elliptic if and only if b ∈ Z; moreover, f b is conjugate to f b ′ for any pair of integers (b, b ′ ) ∈ Z 2 (take a conjugacy h(x, y) = (x + b -b ′ , ys(x)) with s(x) = x(x + 1) • • • (x + b -b ′ -1)
). Now, if the field is not C but Q p , then we can take a sequence of integers a n in Z that converges in Z p towards a limit a ∞ ∈ Z p \ Z; this gives a sequence of pairwise conjugate elliptic elements that converges towards a Jonquières twist. (This phenomenon is not possible in the group of isometries of H m .) Example 5.2. Starting with the family f α from Theorem 4.10, but with α n a sequence of well chosen roots of unity, one obtains a similar phenomenon in Bir(P 2 C ) (instead of Bir(P 2 Q p )). 5.3. Limits of Halphen twists. Theorem 5.3. There is a sequence of positive integers hal(d) with the following property. Let h be a birational transformation of the plane P 2 C which is a limit of Halphen twists of degree ≤ d. Then h preserves a pencil of curves of degree ≤ hal(d), and the action of h on the base of the pencil has order ≤ 60. The construction by renormalization described in the next Section provides examples of Halphen twists converging to elliptic transformations: an element A of PGL 3 (C) preserving every member of a pencil of plane curves is a limit of pairwise conjugate Halphen twists.

Before starting the proof, recall that if X is a Halphen surface with a Halphen twist, then the image of τ : Aut(X ) → Aut(P 1 C ) contains at most 60 elements; this follows from the fact that the invariant fibration π : X → P 1 C has at least 3 and at most 12 singular fibers (see [START_REF] Cantat | Elements generating a proper normal subgroup of the Cremona group[END_REF]).

Proof. Assume that the twists f n converge toward h, and apply Theorem 4.2: taking a subsequence, we may assume that the f n all have degree d 0 ≤ d, their invariant pencils p n have degree D 0 ≤ Hal(d 0 ), and f k n preserves every member of p n for some fixed positive integer k ≤ 60. In other words, the linear system p n is the level set of a rational function π n : P 2 C P 1

C of degree D 0 and π n • f k n = π n . Each linear system p n corresponds to a line in the projective space P(H 0 (P 2 , O P 2 (D 0 ))); thus, taking a further subsequence we may assume that p n converges towards a linear system p ∞ . The line p ∞ determines a pencil of plane curves of degree D 0 , but this linear system may have fixed components; removing them, we get a pencil p ′ ∞ of degree ≤ D 0 . One can find a non-empty open subset U ⊂ P 2 (C) in the euclidean topology such that on U , and along a subsequence,

• the pencils p n converge towards p ′ ∞ ; • f k n converges towards h k . Since f k n preserves every curve of the pencil p n , we deduce that h k preserves every member of the pencil p ∞ .

Example 5.4. Consider a pencil of cubic curves with nine distinct base points p i in P 2 k . Given a point m in P 2 k , draw the line (p 1 m) and denote by m ′ the third intersection point of this line with the cubic of our pencil that contains m: the map m → σ 1 (m) = m ′ is a (Bertini) involution. Replacing p 1 by p 2 , we get a second involution and, for a very general pencil, the composition σ 1 • σ 2 is a Halphen twist that preserves our cubic pencil.

Let us play the same game with a highly degenerate cubic pencil, namely the pencil generated by C + H and C + V where C is the circle x 2 + y 2 = z 2 , H is the horizontal line y = 0, and V is the vertical line x = 0. Every member of this pencil of cubics decomposes as the union of a line through the origin and the circle C. Choose p 1 = [1 : 0 : 1] and p 2 = [0 : 1 : 1] as our distinguished base points. Then the composition σ 1 • σ 2 is a Jonquières twist preserving the pencil of lines through the origin. If the plane is parameterized by (s,t) → (st,t), then we see that this Jonquières twist is conjugate to

(s,t) → s, (s -1)t + 1 (s 2 + 1)t + s -1 . ( 5.1)
Now, if we consider a family of general cubic pencils converging towards this degenerate pencil, we obtain a sequence of Halphen twists converging to a Jonquières twist. As noted above, the opposite change of type -from Jonquières to Halphen-is not possible. 5.4. Renormalization. In this section, we adapt an argument already used (at least) by Blanc, Furter, and Maubach (see [START_REF] Blanc | Groupes de Cremona, connexité et simplicité[END_REF][START_REF] Blanc | Conjugacy classes of special automorphisms of the affine spaces[END_REF][START_REF] Furter | A characterization of semisimple plane polynomial automorphisms[END_REF]).

Consider a birational transformation f of P 2 C with a non-degenerate fixed point q. This means that f and f -1 are well defined at q and f (q) = q. After a linear change of coordinates, we may assume that q is the origin of the affine plane

A 2 C ⊂ P 2 C and that f (x, y) = P(x, y) Q(x, y) , R(x, y) S(x, y) (5.2)
in affine coordinates (x, y), where P, Q, R, and S are polynomial functions in x and y and the rational fractions P/Q and R/S are reduced. Since the origin is a non-degenerate fixed point of f , Q and S do not vanish at the origin, while P and R do; thus, we may assume that Q(0, 0) = S(0, 0) = 1; then, the degree 1 parts of P and R give the differential D f (0,0) . Conjugating f by the homothety m(x, y) = (tx,ty) with t ∈ C, and letting t go to 0, we obtain a family of pairwise conjugate birational transformations of the plane that converges towards the linear transformation D f (0,0) in the Cremona group. Every element of PGL 3 (C) is conjugate, in the Cremona group, to one of the following transformations of the affine plane (see [START_REF] Blanc | Conjugacy classes of affine automorphisms of K n and linear automorphisms of P n in the Cremona groups[END_REF][START_REF] Blanc | Degree growth of birational maps of the plane[END_REF]):

(a) g(x, y) = (ax, by) for some non-zero complex numbers a and b; (b) h(x, y) = (x + 1, by) for some non-zero complex number b.

In the first case, consider the following automorphism of the plane: f (x, y) = (ax + (y + x 2 ) 2 , by + bx 2 ).

(5.3)

This automorphism fixes the origin and its linear part coincides with g(x, y) = (ax, by). It is a loxodromic transformation, with dynamical degree 4. Thus, g is in the closure of the conjugacy class of f . In the second case, note that h is conjugate in the Cremona group to h ′ (x, y) = (b -1 (x + y), b -1 y), a linear transformation of the affine plane fixing the origin. Then, apply the same strategy with f ′ (x, y) = (b -1 (x + y + (y + (x + y) 2 ) 2 , b -1 (y + (x + y) 2 ) : we obtain that h is also in the closure of the conjugacy class of a loxodromic element.

Theorem 5.5. Let f be any element of PGL 3 (C), or any elliptic element of Bir(P 2 C ) of infinite order. Then, f is a limit of pairwise conjugate loxodromic elements (resp. Jonquières twists) in the Cremona group Bir(P 2 C ). Proof. If f is an elliptic element of infinite order, it is conjugate to an element of PGL 3 (C), so that we may assume f to be in PGL 3 (C). We already explained how the renormalization argument shows that f is a limit of pairwise conjugate loxodromic elements. To show that f is a limit of conjugate Jonquières twists, we only need to show that, given any pair of non-zero complex numbers (a, b), there is a Jonquières twist f fixing a point q, whose linear part D f q ∈ GL 2 (C) is The birational transformations f (x, y) = (a(1 + y)x, by) and f ′ (x, y) = (b(1 + y)(x + y), by) satisfy these properties at the fixed point q = (0, 0). This theorem does not prove that any elliptic element is a limit of pairwise conjugate loxodromic elements, because there are finite order elements in Bir(P 2 C ) which are not conjugate to any automorphism of P 2

C ; an example is given by the Bertini involution. Also, one could ask more on the loxodromic elements. For instance, is every element of PGL 3 (C) a limit of pairwise conjugate loxodromic regularizable transformations ?

APPENDIX: INVARIANT PENCILS VIA HYPERBOLIC GEOMETRY

Here, working in characteristic 0, we give an alternative proof of Theorem 6.1. For every integer d ≥ 1, there exists an integer Hal(d) with the following property: if f is a Halphen twist of degree at most d, the unique finvariant pencil is of degree at most Hal(d). Let p be a Halphen pencil of index m (and degree 3m). As in Section 3, we denote by ξ = 3e 0e 1 -. . .e 9 the class of a plane cubic curve C such that mC is a member of p; the general member of p is a curve of degree 3m with class mξ in the Picard-Manin space. We shall say that ξ is the primitive class determined by p. Lemma 6.2. Let ε be a positive real number. There is a positive constant ∆ H (ε) with the following property. Let p be any Halphen pencil of index m, and let ξ be the associated primitive class. For every point u in the horosphere Hor ξ (ε) and every Halphen twist f preserving p, we have dist(u, f • (u)) ≥ m∆ H (ε).

To explain the content of this formula, note that the horosphere Hor ξ (ε) is an infinite dimensional euclidean space when endowed with the Riemannian metric coming from the hyperbolic space H. A Halphen twist fixing p fixes ξ, and acts on Hor ξ (ε) as a fixed point free isometry. The lemma asserts that the (euclidean) translation length of this isometry is uniformly bounded from below by a positive constant m∆ H (ε).

Proof. Let X be the Halphen surface associated to p; we may assume that X has an automorphism of infinite order (given by a Halphen twist f ).

First, we work in the Néron-Severi group NS(X ; R), where X → P 2 is the blow-up of the nine base points of p. The lattice NS(X ; Z) is isomorphic to Z 10 and the intersection form is the standard quadratic form of signature [START_REF] Bell | A generalised Skolem-Mahler-Lech theorem for affine varieties[END_REF][START_REF] Blanc | Degree growth of birational maps of the plane[END_REF] in the basis given by (e 0 , . . . , e 9 ).

In the group Aut(X ), consider the subgroup Aut t (X ; π) acting trivially on the base P 1 of the fibration π and by translation along the general fiber of π. According to [START_REF] Grivaux | Parabolic automorphisms of projective surfaces (after M. H. Gizatullin)[END_REF]Proposition 7.10]3 , the existence of a parabolic automorphism in Aut(X ) implies that the Halphen fibration π : X → P 1 has at least 3 and at most 12 singular fibers. The group Aut(X ) permutes the fibers, and acts on the base P 1 via a homomorphism τ : Aut(X ) → Aut(P 1 ); its image preserves the critical values of π, a finite set with between 3 and 12 points. Thus, τ(Aut(X )) is a finite group, with cardinality uniformly bounded (by 60). Thus, the kernel of τ has index ≤ 60, and since the group of translations in the group of automorphisms of a genus 1 curve has index ≤ 12 (in characteristic 0), we deduce that Aut t (X ) has index ≤ 360 in Aut(X ).

Let Isom(NS(X ; Z); ξ) be the subgroup of GL (NS(X ; Z)) preserving the intersection form and the class ξ. From [START_REF] Gizatullin | Rational G-surfaces[END_REF][START_REF] Cantat | Rational surfaces with a large group of automorphisms[END_REF][START_REF] Grivaux | Parabolic automorphisms of projective surfaces (after M. H. Gizatullin)[END_REF], we know that there is a discrete, free abelian subgroup E ≃ Z 8 in Isom(NS(X ; Z); ξ) such that the image of Aut t (X ; π) in GL (NS(X ; Z)) is contained in mE, where m is the index of the pencil (here, mE denotes the subgroup of E = Z 8 of index m 8 obtained as the image of the multiplication by m).

Fix a horosphere Hor X,ξ (ε) in the hyperbolic space H X ⊂ NS(X ; R); this horosphere is isometric to a euclidean affine space R 8 on which the group E acts as a discrete group of translations. Thus, there is a positive constant δ(ε) such that the translation length of every element in E \ {Id} along Hor X,ξ (ε) is larger than δ(ε; 1); hence, the translation length of Aut t (X ; π) in Hor X,ξ (ε) is larger than mδ(ε).

Since H X is an Aut(X )-invariant, totally geodesic subspace of H, the same lower bound applies to the action of Aut t (X ; π) on Hor ξ (ε) ⊂ H. Now, there is a positive constant ℓ ≤ 360 such that every Halphen twist f ∈ Aut(X ) satisfies f ℓ ∈ Aut t (X ; π). Thus, the translation length of f along Hor ξ (ε) ⊂ H is not less than m∆ H (ε) with ∆ H (ε) = δ H (ε)/360. Example 6.3. Consider the Hilbert space ℓ 2 (Z + ), with Hilbert basis (e i ) i≥1 . Let (η i ) i≥1 be a sequence of complex numbers of modulus 1, with η i = 1 for all i; let A be the unitary operator defined by A(e i ) = η i e i for each index i ≥ 1. Let T be the vector ∑ i (1/i)e i . The isometry I(z) = A(z) + T has a fixed point if and only if the vector

u = ∑ i 1 i(1 -η i ) e i (6.2)
is in ℓ 2 (Z + ). Choosing the η i so that ∑ i (i|η i -1|) -2 does not converge, we see that I has no fixed point, but its translation length vanishes (for every ε > 0 there is a point v with dist(I(v), v) < ε). Adding one more coordinate, one constructs from I a parabolic isometry of an infinite dimensional hyperbolic space for which the translation length vanishes on every invariant horosphere. Lemma 6.2 implies that this phenomenon does not appear among Halphen twists.

6.1.2. Translation length of Jonquières twists. Now, assume that f ∈ Bir(P 2 ) is a Jonquières twist, preserving the pencil of lines through a point q ∈ P 2 . Let e 1 denote the class of the exceptional divisor obtained by blowing up q 1 ; then, the class ξ = e 0e 1 of the Picard-Manin space is f -invariant and the horospheres Hor ξ (ε) are f -invariant too.

The euclidean metric of the horosphere which is induced by the hyperbolic metric satisfies (see [START_REF] Cantat | Distortion in Cremona groups[END_REF]) dist Hor ξ (ε) (u, v) = 2 sinh(dist H (u, v)/2); on the other hand, cosh(dist H (u, v)) = u • v. Now, apply these formulas to the points e 0 and f n • e 0 , which are both on the horosphere Hor ξ [START_REF] Bell | A generalised Skolem-Mahler-Lech theorem for affine varieties[END_REF]. Since e 0 • f n • e 0 = deg( f n ) ≃ µn for some constant µ > 0 (see [START_REF] Blanc | Degree growth of birational maps of the plane[END_REF]), we obtain dist Hor ξ (ε) (e 0 , f n • e 0 ) ≃ 2µn. Thus, instead of a linear drift, we obtain a drift of order √ n. This behavior does not appear for parabolic isometries of finite dimensional hyperbolic spaces H k . Now, consider the euclidean barycenter b n of the sequence of points (e 0 , f • e 0 , . . . , f n-1

• e 0 )
in the euclidean space Hor ξ (1). Since these n points are in the ball of radius A √ n for some A > 0, the distance between f • b n and b n is at most 2A/ √ n. Thus, f almost have a fixed point in Hor ξ (1) (but, of course, the sequence q n is not a Cauchy sequence in the Picard-Manin space). This shows that Lemma 6.2 does not hold for Jonquières twists. Thus, if ε is fixed and D is large, this distance goes to +∞.

Proof of Theorem 6.1. Consider a Halphen twist f of degree d. There is a Halphen pencil p of index m and a birational map ϕ such that ϕ maps p to the unique f -invariant pencil. Let ξ be the primitive isotropic class associated to p and let η be its image by ϕ • ; let D be the degree of η (i.e. e 0 • η = D). The twist f is conjugate to the twist g = ϕ -1 • f • ϕ that preserves p and ξ. The translation length of g along the horosphere Hor ξ (1) is bounded from below by m∆ H [START_REF] Bell | A generalised Skolem-Mahler-Lech theorem for affine varieties[END_REF]. The image of this horosphere by ϕ • is the horosphere Hor η (1). Thus, the translation length of f • along this horosphere is bounded from below by m∆ H (1); in particular, if w 0 is the projection of e 0 on Hor η (1), dist(w 0 , f • w 0 ) ≥ m∆ H [START_REF] Bell | A generalised Skolem-Mahler-Lech theorem for affine varieties[END_REF]. Now, the degree of f satisfies d = cosh(dist(e 0 , f • (e 0 ))). Consider the triangle with vertices e 0 , f • (e 0 ) and η; the points w 0 and f • (w 0 ) are on the two sides [ξ, e 0 ] and [ξ, f • (e 0 )] respectively. Since d is fixed, the distance dist(e 0 , f • (e 0 )) = argcosh(d) is fixed too; since triangles are thin, the distance between w 0 and f • (w 0 ) is bounded from above by argcosh(d). Since cosh(dist(e 0 , w 0 )) = D ≥ 3, we obtain a uniform upper bound (depending only on d) for both m and D. Thus, the degree mD of the pencil is bounded too.

( 3 )

 3 Morphisms between algebraic varieties over k are continuous in the adelic topology. (4) Étale morphisms are open with respect to the adelic topology.

p and π 2

 2 : C • p × W → W the projections onto the first and second factors. Observe that Y

For

  every point b ∈ B, denote by F b the restriction of F to the special fiber at b. All we need to show is the existence of a point b ∈

  3 and 3.4 to control conjugacy classes. 3.1. Halphen pencils and Halphen surfaces.

3. 1

 1 .1.

  , or C i comes from the blow-up of an infinitely near point, and deg([C i ]) := [C i ] • e 0 = 0. So, we obtain the existence of an integer B 1 (m), that depends only on the index m of the Halphen pencil, and an integer B 0 (that does not depend on X or m) such that (4) the set Irr(X ) has at most B 0 elements; (5) every class c ∈ Irr(X ) has degree at most B 1 (m), i.e. 0 ≤ c • e 0 ≤ B 1 (m).

  Moreover, denote by O(Q(Z); Ax) the subgroup of O(Q(Z)) that preserves Ax and O(Q(Z); Ax)| Ax ⊆ O(Ax) the image of O(Q(Z); Ax) under the restriction to Ax. Denote by f Ax the restriction of f to Ax. We have f Ax

4. 1 .

 1 Halphen and Jonquières twists.

4. 1 . 1 .

 11 Halphen twists. A birational transformation of the plane is a Halphen twist if and only if (deg

4. 1 . 3 .

 13 The set Pog(d). Let k be an algebraically closed field. For every integer d ≥ 1, denote by Bir d (P 2 k ) the algebraic variety of birational transformations of P 2 k of degree d. Recall Theorem 1.3: for every m ≥ 0, the function f ∈ Bir d

Corollary 4 . 4 .

 44 The set Hal(d) ⊂ Bir d (P 2 k

4. 3 .

 3 Conjugacy classes of Halphen twists. Theorem 4.5. For every integer d ≥ 1, there is a positive integer HalCo(d) that satisfies the following property. Let f and g be Halphen twists of degree ≤ d. If f is conjugate to g in Bir(P 2 k ), then there is a birational map ψ of degree ≤ HalCo(d) such that ψ • f • ψ -1 = g. Proof. Since f and g are conjugate, their invariant pencils are birationally equivalent to the same Halphen pencil p, the index of which will be denoted by m. From Corollary 4.4, there exist an integer hal(d) and two birational maps α and β in Bir(P 2 k ) such that • deg(α), deg(β), and deg(p) = 3m are bounded from above by hal(d);

4. 4 .

 4 Jonquières twists. We now consider the Poincaré and conjugacy problems for Jonquières twists.

4. 4 . 1 .

 41 Bounding the degree of the invariant pencil. Theorem 4.7. Let W be an irreducible subvariety of Bir d (P 2 k

4. 4 . 2 .

 42 Examples. On the other hand, the following examples show that Jon(d) does not behave as well as Hal(d). These examples, and those in § 5.2, are variations on examples given in[START_REF] Blanc | Algebraic elements of the Cremona groups[END_REF][START_REF] Blanc | On degenerations of plane Cremona transformations[END_REF][START_REF] Blanc | Degree growth of birational maps of the plane[END_REF][START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF]).

Example 4 . 8 .

 48 Consider the birational map f a (x, y) = (x + a, y x x+1 ), where a ∈ k is a parameter. In homogeneous coordinates, f a [x : y : z] = [(x + az)(x + z) : xy : (x + z)z], so deg( f a ) = 2 for every parameter a. The n-th iterate of f a is

6. 1 .

 1 Translation lengths along horospheres.6.1.1. Picard-Manin class. If ξ is an isotropic point of the Picard-Manin space, and ε is a positive real number, the horosphere Hor ξ (ε) is defined byHor ξ (ε) = {u ∈ H | u • ξ ≤ ε} .(6.1)These horospheres Hor ξ (ε) are invariant under the action of the stabilizer of ξ in the group of isometries of H; the stabilizer of Rξ permutes the horospheres Hor ξ (ε): an isometry ϕ of H that preserves Rξ and has translation length τ(ϕ) maps Hor ξ (ε) to Hor ξ (exp(±τ(ϕ))ε).

6. 2 .

 2 Images of Halphen pencils. Let η = De 0 -∑ i M i e i be a boundary point of H. Fix a positive ε such that the horoball surrounded by Hor η (ε) does not contain e 0 ; this amounts to the inequality ε ≤ D. The projection w 0 of e 0 on Hor ξ ′ (ε) is w 0 = αe 0 + βξ ′ (6.3) with α = ε/D, and β = (1/2D)(D/ε -ε/D). The distance from e 0 to the horosphere Hor η (ε) satisfies cosh dist(e 0 , w 0

  this implies deg(g n A ) = deg(g A ) n = deg( f ) n for all n ≥ 0. Notes.-One can extend the result, changing Q into any number field K. For this one only needs to use Chebotarev's theorem and weak approximation in number fields for the group SL m+1 . The results we need are in Platonov and Rapinchuk, [36, Thms 1.2 and 7.7].

  •• p the open unit ball. Since R is integral and finitely generated over Z, Lemma 2.10 provides infinitely many primes p ≥ 3 such that R embeds into Z p ⊆ C • p . This induces an embedding Spec

  5.2. Limits and types. Let us first list two basic remarks, which are consequences of the semi-continuity of dynamical degrees and Section 4.2. Let ( f n ) be a sequence of elements of Bir(P 2 C ) that converges towards f ∈ Bir(P 2 C

1.1.3. Polynomial degree growth. Another example is given by Halphen and Jonquières twists of surfaces. If X is a surface, and f is a birational transformation of X such that deg( f n ) ≃ αn 2 for some constant α > 0 (a Halphen twist), then f is regularizable, on a surface Y X which supports an f Y -invariant genus 1 fibration (see[START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF][START_REF] Gizatullin | Rational G-surfaces[END_REF][START_REF] Cantat | Rational surfaces with a large group of automorphisms[END_REF], and Section 4.1 below). If f is a birational transformation of a surface X , and deg( f n ) ≃ αn for some constant α > 0 (a Jonquières twist), then f preserves a unique pencil of rational curves, but f is not regularizable (see[START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF][START_REF] Blanc | Degree growth of birational maps of the plane[END_REF], and Section 4.1 below). Here is a statement that generalizes this last remark.

By positive density, we mean that the proportion of primes p among the first N primes that satisfy the statement is bounded from below by a positive number if N is large enough.

Here, we use our assumption that char(k) = 0, because this proof makes use of the classification of singular fibers of elliptic fibrations, together with a topological Euler characteristic argument. A more elaborate argument would be necessary to deal with surfaces in positive characteristic (see[START_REF] Lang | Configurations of singular fibres on rational elliptic surfaces in characteristic two[END_REF] for instance).

The author was partially supported by the ANR grant Fatou ANR-17-CE40-0002-01 and the ANR grant Foliage ANR-16-CE40-0008-01.

The author was partially supported by the ANR grant Fatou ANR-17-CE40-0002-01. 1 FAMILIES with deg k ( f n ) ≃ αn ℓ for some