Jérémy Blanc
email: jeremy.blanc@unibas.ch

Andrea Fanelli
email: andrea.fanelli.1@u-bordeaux.fr

Ronan Terpereau
email: ronan.terpereau@u-bourgogne.fr

Automorphisms of P 1 -bundles over rational surfaces

Keywords: Rational threefolds, vector bundles, P 1 -bundles, automorphisms, birational maps, algebraic groups, classification 2020 Mathematics Subject Classification. 14E07, 14M17, 14L30, 14J60, 14D20; 14M20, 14J30

In this paper, we provide the complete classification of P 1 -bundles over smooth projective rational surfaces whose neutral component of the automorphism group is maximal. Our results hold over any algebraically closed field of characteristic zero.

In this article, we work over an algebraically closed field k of characteristic zero (even if most of the partial results we obtain along the way are valid over any algebraically closed field k, as we explain at the beginning of each section). We study P 1 -bundles X → S (always assumed to be locally trivial for the Zariski topology), where S is a smooth projective rational surface (the smoothness condition is actually not a strong restriction; see § 2.2), and study the group scheme Aut(X) of automorphisms of X. This group scheme can have infinitely many components, but the connected component of the identity Aut • (X) is a connected algebraic group; see [START_REF] Matsumura | Representability of group functors, and automorphisms of algebraic schemes[END_REF]. The aim of this article is to study the pairs (X, Aut • (X)) and to classify these, up to birational conjugation. In particular, we describe the geometry of the pairs arising and the restriction of X → S to natural curves of S.

Our motivation for this study comes from the classification of connected algebraic subgroups of the Cremona group Bir(P 3 ), stated by Enriques and Fano in [START_REF] Enriques | Sui gruppi continui di trasformazioni Cremoniane dello spazio[END_REF] and achieved by Umemura over the field k = C in a series of four papers [Ume80, Ume82a, Ume82b, Ume85] , using analytic methods. As explained in [START_REF] Mukai | Minimal rational threefolds[END_REF][START_REF]Minimal rational threefolds[END_REF], these groups act on some minimal rational threefolds, and it turns out that most of the threefolds obtained are P 1 -bundles over rational surfaces. Our plan is to give a shorter geometric proof of the classification of Umemura, and this article is the first step in this direction (the second step is in [START_REF] Blanc | Connected Algebraic Groups Acting on three-dimensional Mori Fibrations[END_REF]). Instead of starting with the group action and trying to find threefolds where the group acts, we will directly study the possible varieties and their symmetries, then reduce to some simple varieties, and in the end compute the neutral component of their automorphism groups.

Our approach should be seen as the analogue of the following way to understand the classification of connected algebraic subgroups of the Cremona group Bir(P 2 ). This classification was initiated by Enriques in [START_REF] Enriques | Sui gruppi continui di trasformazioni cremoniane nel piano[END_REF] and can nowadays be easily recovered via the classification of smooth projective rational surfaces, as we now explain. One can conjugate any connected algebraic subgroup of Bir(P 2 ) to a group of automorphisms of a smooth projective rational surface S. Contracting all (-1)-curves of S, we can moreover assume that S is a minimal surface, i.e. that S is isomorphic to the projective plane P 2 or a Hirzebruch surface F a , with a ≥ 0, a 1; see [START_REF] Beauville | Complex algebraic surfaces, Translated from French[END_REF] for a general survey on surfaces. One then checks that the groups of automorphisms obtained are maximal, as these have no orbit of finite size: this forbids the existence of equivariant birational maps towards other smooth projective rational surfaces. Every connected algebraic subgroup of Bir(P 2 ) is thus contained in a maximal connected algebraic subgroup of Bir(P 2 ), which is conjugate to the group Aut • (S), where S is a minimal smooth rational surface. See also [START_REF] Blanc | Sous-groupes algébriques du groupe de Cremona[END_REF] for the classification of the (not necessarily connected) maximal algebraic subgroups of Bir(P 2 ) with a similar approach.

The aim of our classification is then to proceed as in the case of surfaces and study minimal threefolds. There are many more such varieties in dimension 3 than in dimension 2, and not all of them yield maximal algebraic subgroups of Bir(P 3 ). Moreover, some maximal connected algebraic subgroups of Bir(P 3 ) are realised by infinitely many minimal threefolds. Our aim is then to understand the geometry of the minimal threefolds obtained in this way and the equivariant birational maps between them.

Another motivation consists in unifying some well-known results (most of them over the field of complex numbers, using topological arguments) on P 1 -bundles over minimal rational surfaces, i.e. the projective plane P 2 and the Hirzebruch surfaces F a , and to obtain these results from the perspective of automorphisms groups. See for instance Corollaries 3.3.3 and 4.2.2 and Remark 4.3.2.

We also provide moduli spaces parametrising the P 1 -bundles X → F a over Hirzebruch surfaces having no jumping fibre (see Corollary 3.3.8), i.e. the P 1 -bundles such that all fibres of the natural morphism X → P 1 (induced by the structure morphism F a → P 1 ) are Hirzebruch surfaces F b for a fixed b not depending on the fibre. These P 1 -bundles can also be described using exact sequences of vector bundles (see Corollary 3.3.3). The action of Aut • (F a ) on the moduli spaces, depending on some natural numerical invariants (see Definition 1.4.1), is described explicitly; see § 3.4, and more precisely Corollary 3.4.6.

Our approach uses very basic tools and is aimed to be easy to follow by any interested reader, not necessarily expert. Most of the time we give proofs which do not use any known results, and we make a reference to the literature when we re-prove a known fact.

Summary of the classification

Definition 1.2.1. Let π : X → S and π : X → S be two P 1 -bundles over two smooth projective rational surfaces S and S . A birational map ϕ : X X is said to be (1) a square birational map (resp. square isomorphism/square automorphism) if there exists a birational map η : S S such that π ϕ = ηπ (and if ϕ is resp. an isomorphism/automorphism)

X ϕ G G π X π S η G G S ;
we say in these cases that ϕ is above η;

(2) a birational map (resp. isomorphism/automorphism) of P 1 -bundles if S = S and π ϕ = π (and if ϕ is resp. an isomorphism/automorphism)

X ϕ G G π A A X ; π u u S (3) Aut • (X)-equivariant if ϕ Aut • (X)ϕ -1 ⊂ Aut • (X ) (or, equivalently, if ϕ Aut • (X)ϕ -1 ⊂ Aut(X )).
As the definition depends on π and π , and not only on X and X , we will often write ϕ : (X, π) (X , π ) and say that (X, π) and (X , π ) are, respectively, square birational/square isomorphic/birational P 1 -bundles/isomorphic P 1 -bundles if ϕ satisfies the corresponding condition.

Remark 1.2.2. In the above definition, every element of Aut • (X) is a square automorphism (see Lemma 2.1.1) but not necessarily a birational map of P 1 -bundles.

Definition 1.2.3. Let π : X → S be a P 1 -bundle over a smooth projective rational surface S. We say that Aut • (X) is maximal if for each Aut • (X)-equivariant square birational map ϕ : (X, π) (X , π ), we have ϕ Aut • (X)ϕ -1 = Aut • (X ). If we moreover have (X , π ) (X, π) (resp. ϕ is an isomorphism of P 1 -bundles) for each such ϕ, we say that the P 1 -bundle (X, π) is stiff (resp. superstiff ).

Remark 1.2.4. This definition depends on X and π, and not only on X. For instance, taking X = P 1 × F 1 and two standard P 1 -bundle structures π : X → P 1 × P 1 and π : X → F 1 , Aut • (X) is maximal with respect to π but not with respect to π . (The pairs (X, π) and (X, π ) correspond, respectively, to F 1,0 0 and F 0,0 1 , see Definition 3.1.1, so this observation follows from Theorem A).

Remark 1.2.5. To the best of our knowledge, the notion of stiff/superstiff is new. It is analogous to the notion of equivariant birational rigidity/superrigidity for Mori fibre spaces but is not equivalent since here we only consider P 1 -bundles. Moreover, birational rigidity for Mori fibre spaces is always up to squares, while stiffness also detects these birational maps; see [START_REF] Corti | Singularities of linear systems and 3-fold birational geometry[END_REF] and [START_REF] Pukhlikov | Birationally rigid varieties[END_REF] to know more about the notions of rigidity and superrigidity for Mori fibre spaces.

The next statement, which summarises most of our work, is our main result (the notation is explained after Theorem B).

Theorem A. Let π : X → S be a P 1 -bundle over a smooth projective rational surface S. Then, there exists an Aut • (X)-equivariant square birational map (X, π) (X , π ) such that Aut • (X ) is maximal. Moreover, the group Aut • (X) is maximal if and only if (X, π) is square isomorphic to one of the following:

(a) a decomposable P 1 -bundle F b,c a -→ F a with a, b ≥ 0, a 1, c ∈ Z, c ≤ 0 if b = 0,
and where a = 0 or b = c = 0 or -a < c < ab; (b) a decomposable P 1 -bundle P b -→ P 2 for some b ≥ 0; (c) an Umemura P 1 -bundle U b,c a -→ F a for some a, b ≥ 1, c ≥ 2, with cab < 2 if a ≥ 2, and cab < 1 if a = 1; (d) a Schwarzenberger P 1 -bundle S b -→ P 2 for some b ≥ 1; or (e) a P 1 -bundle V b -→ P 2 for some b ≥ 2.

Remark 1.2.6. We keep the notation of Theorem A:

• Aut • (F b,c a ) is described in Lemma 3.1.5 and Remark 3.1.6; • Aut • (P b ) is described in Lemma 4.1.2 and Remark 4.1.3;

• Aut • (U b,c
a ) is described in Lemma 3.6.3 and Remark 3.6.4; • Aut • (S b ) is described in Lemma 4.2.5(2); and • Aut • (V b ) is described in Lemma 5.5.1(3)-(4).

Contrary to the dimension 2 case, there are many P 1 -bundles X → S with maximal Aut • (X) which are birationally conjugate. This means that the P 1 -bundles of Theorem A are not always stiff. The next result describes all the possible links between such P 1 -bundles. The construction of such links is detailed in § 5. In the other cases, the equivariant square birational maps between the P 1 -bundles of Theorem A are given by compositions of square isomorphisms of P 1 -bundles and of birational maps appearing in the following list:

(1) for all integers a, b ≥ 0, c ∈ Z with a 1, -a < c < 0, an infinite sequence of equivariant birational maps of P 1 -bundles

F b,c a F b+1,c+a a • • • F b+n,c+an a • • • ;
(2) for all integers a, b ≥ 1 with (a, b) (1, 1), an infinite sequence of equivariant birational maps of

P 1 -bundles U b,2 a U b+1,2+a a • • • U b+n,2+an a • • • ;
(3) for each b ≥ 2, a birational involution S b S b ; (4) for each b ≥ 2, the equivariant birational morphisms U b,2 1 → V b obtained by contracting the preimage of the (-1)-curve of F 1 onto the fibre of a point of P 2 in V b .

In Theorems A and B, decomposable P 1 -bundles are simply the projectivisations of decomposable rank 2 vector bundles. The P 1 -bundles over F a and P 2 are particularly easy to describe (see § § 3.1 and 4.1). The Schwarzenberger P 1 -bundles over P 2 are projectivisations of the classical rank 2 vector bundles of the same name (see below). We describe the notation here. Definition 1.2.7 (Schwarzenberger P 1 -bundles). Let b ≥ -1 be an integer, and let κ : P 1 × P 1 → P 2 be the (2 : 1)-cover defined by κ :

P 1 × P 1 → P 2 [y 0 : y 1 ], [z 0 : z 1 ] → [y 0 z 0 : y 0 z 1 + y 1 z 0 : y 1 z 1 ],
whose branch locus is the diagonal ∆ ⊂ P 1 × P 1 and whose ramification locus is the smooth conic Γ = {[X : Y : Z] | Y 2 = 4XZ} ⊂ P 2 . The b-th Schwarzenberger P 1 -bundle S b → P 2 is the P 1 -bundle defined by

S b = P κ * O P 1 ×P 1 (-b -1, 0) → P 2 .
Note that S b is the projectivisation of the classical Schwarzenberger vector bundle κ * O P 1 ×P 1 (-b -1, 0) introduced in [START_REF] Schwarzenberger | Vector bundles on the projective plane[END_REF]. Moreover, there is a natural family of lines of P 2 whose preimage by S b → P 2 is isomorphic to F b for each b ≥ 0 (by Lemma 4.2.5(1)). This explains the shift in the notation.

The families of Umemura P 1 -bundles U b,c a , introduced by Umemura in [Ume88, § 10], need a bit more notation to be described. This is done in § 3.6. The P 1 -bundles V b 1 → P 2 is a family of P 1 -bundles such that Aut • (V b ) is maximal and birationally conjugate to certain Aut • (U b,c 1 ), but with an action on P 2 fixing a point; see Lemma 5.5.1 for the precise relation between the two families.

Remark 1.2.8. Let us mention that the P 1 -bundles that appear in Theorem A are toric varieties (families (a) and (b)) or minimal smooth compactifications of SL 2 /H with H ⊆ SL 2 a finite subgroup (families (c), (d), and (e) and certain members of families (a) and (b)). Here we call minimal smooth compactification of SL 2 /H an SL 2 -equivariant embedding SL 2 /H → X such that X is a smooth projective variety and any SL 2 -equivariant birational morphism X → X , with X smooth, is an isomorphism; these SL 2 -threefolds were studied and classified by Nakano in [START_REF] Nakano | On equivariant completions of 3-dimensional homogeneous spaces of SL(2, C), Japan[END_REF] (over C) as an application of Mori theory. The finite subgroups of SL 2 (C) are well known: there are the cyclic groups, the binary dihedral groups, and the binary polyhedral groups. When H is non-cyclic, SL 2 /H admits a unique minimal smooth compactification, namely the Fano threefolds V 5 , V MU 22 , and the smooth quadric Q 3 for the binary polyhedral groups (see [Nak89, Proposition 2.5]), and the Schwarzenberger P 1 -bundles or P 2 for the binary dihedral groups (see [Nak89, Theorem 2.8]). But when H is cyclic, SL 2 /H admits several non-isomorphic minimal smooth compactifications including the P 1 -bundles corresponding to families (c) and (e) in Theorem A (see [START_REF] Nakano | On equivariant completions of 3-dimensional homogeneous spaces of SL(2, C), Japan[END_REF]Theorem 2.31] for the complete list).

Comparison with Umemura's classification

Each of the maximal connected algebraic subgroups of Bir(P 3 C ) given in [Ume85, Theorem 2.1] acts on some Mori fibre spaces (as explained in [START_REF]Minimal rational threefolds[END_REF]). The cases (P1), (P2), (E1), (E2) correspond to Fano threefolds, the cases ( J10), ( J12) to Del Pezzo fibrations (P 2 -bundles and quadric fibrations), and all other cases ( J1)-( J9), ( J11) correspond to P 1 -bundles. We now explain to which of the cases (a)-(d) of Theorem A these correspond.

Note that ( J1) is Aut • (P 2 × P 1 ) and is thus given by P 0 → P 2 . ( J2) is Aut • (P 1 × P 1 × P 1 ) and is thus given by F 0,0 0

F b,c a = P(O F a ⊕ O F a (-bs a + cf )) → F a (
→ F 0 . ( J3) is Aut • (F m × P 1 ), with m ≥ 2, which is given either by F 0,0 m → F m or by F 0,m 0 → F 0 , which is square isomorphic to F m,0 0 → F 0 (but not isomorphic as a P 1 -bundle). ( J4) is Aut • (PGL 3 /B)
(S b ), with b = n -1 ≥ 3, by Remark 4.2.7. ( J6) is Aut • (L m,n ), where L m,n → F 0 is a P 1 -bundle of bidegree (m, n), with m ≥ 2, n ≤ -2 (see [Ume85, Theorem 2.1( J6)] and [Ume88, § 5]). It thus corresponds to F m,-n 0 → F 0 , given in (a). ( J7) is equal to Aut • (J m ) = Aut • (J m ), with m ≥ 2, where the P 1 -bundle J m → P 2 defined in [Ume88, § 6] is a compactification of J m of [Ume85, Theorem 2.1( J7)], isomorphic to P b → P 2 with m = b, given in (b). ( J8) is Aut • (L m,n ), where L m,n → F 0 is a P 1 -bundle of bidegree (m, n), with m ≥ n ≥ 1 (see [Ume85, Theorem 2.1( J8)] and [Ume88, § 7]). It thus corresponds to F m,n 0 → F 0 , which is square isomorphic to F n,m 0 → F 0 , both given in (a). ( J9) is Aut • (F m,n ), for some integers m > n ≥ 2 (see [Ume85, Theorem 2.1( J9)]), and is also equal to Aut • (F k m,n ) for each integer k ≥ m/n , where F k m,n = P(O F n ⊕O F n (-ks -n -mf )) = P(O F n ⊕O F n (-ks n +(nk - m)f )) = F k,nk-m n → F n , given in (a). ( J11) is Aut • (E l m )
, where m = 1, l ≥ 3 or m, l ≥ 2 (see [Ume85, Theorem 2.1( J11)]). Then E l m admits a family of compactifications given in [Ume88, § 10] and depending on a parameter j ≥ l; these compactifications correspond to the Umemura bundles U b,c a → F a , with a = m, b = j, c = (jl)m + 2. Let us note that the family (e) in Theorem A was overlooked in the work of Umemura. These have to correspond to maximal algebraic subgroups of Bir(P 3 ) and should appear in [Ume88, § 10].

Some of the elements of our list do not appear in [START_REF]On the maximal connected algebraic subgroups of the Cremona group. II[END_REF][START_REF]Minimal rational threefolds[END_REF]. This is sometimes because these do not correspond to maximal algebraic subgroups of Bir(P 3 ), as we can embed the groups in larger groups of automorphisms of Mori fibre spaces, or it is because they are equivalent to other elements of the list, by a birational map not preserving the P 1 -bundle structure. This is for instance the case of S 2 → P 2 , which is Aut(S 2 )-equivariantly birational to P 3 , or of F 0,1 0 F 1 × P 1 → P 1 × P 1 , which is not maximal as it is equivariantly birational to P 2 × P 1 . There are many other cases, which are studied in [START_REF] Blanc | Connected Algebraic Groups Acting on three-dimensional Mori Fibrations[END_REF].

Overview of the proof

Starting with a P 1 -bundle π : X → Ŝ over a smooth projective rational surface Ŝ, there is a birational morphism η : Ŝ → S, where S is a Hirzebruch surface F a or the projective plane P 2 . Applying Lemma 2.3.2, we obtain a square birational map ψ : ( X, π) (X, π), unique up to isomorphism. Moreover, ψ is Aut • (X)equivariant. We then need to study P 1 -bundles over Hirzebruch surfaces or over the projective plane.

Section 3 concerns the case of P 1 -bundles over Hirzebruch surfaces π : X → F a , with a ≥ 0. We denote by τ a : F a → P 1 a P 1 -bundle structure on F a , and we study the surfaces S p = (τ a π) -1 (p) with p ∈ P 1 . The situation is described by Proposition 3.2.2; we explain it now. There is an integer b ≥ 0 such that S p F b for a general p ∈ P 1 . If this holds for all points of P 1 , we in fact have a F b -bundle τ a π : X → P 1 : we say that there is no jumping fibre. If a special point p ∈ P 1 is such that S p F b for some b b, then b > b and we can blow up the exceptional curve of S p F b and contract the strict transform of S p , in an Aut • (X)-equivariant way. After finitely many steps, we reduce to the case where there is no jumping fibre.

We then associate to any P 1 -bundle π : X → F a with no jumping fibres two integers b, c ∈ Z such that b ≥ 0, and c ≤ 0 if b = 0. The integer b is the one such that τ a π : X → P 1 is a F b -bundle, and the integer c can be seen either with the transition function of π : X → F a or using exact sequences associated to the rank 2 vector bundle corresponding to π, with the following definition (Corollary 3.2.3 shows the equivalence between the two points of view).

Definition 1.4.1. Let a, b, c ∈ Z be such that a, b ≥ 0 and c ≤ 0 if b = 0. We say that a P 1 -bundle π : X → F a has numerical invariants (a, b, c) if it is the projectivisation of a rank 2 vector bundle E which fits in a short exact sequence

0 → O F a → E → O F a (-bs a + cf ) → 0,
where f , s a ⊂ F a are a fibre and a section of self-intersection a of τ a : F a → P 1 . In Proposition 3.3.1, we show that a P 1 -bundle π : X → F a with numerical invariants (a, b, c) has no jumping fibre and that every P 1 -bundle π : X → F a with no jumping fibre has numerical invariants (a, b, c), for some uniquely determined integers b, c ∈ Z, with b ≥ 0, and c ≤ 0 if b = 0. The above numerical invariants are thus really invariant under isomorphisms.

We then prove (see Corollary 3.3.7) that every P 1 -bundle X → F a with numerical invariants (a, b, c) is decomposable if b = 0 or c < 2, and construct a moduli space M b,c a , which is isomorphic to a projective space (see Remark 3.3.9), parametrising the non-decomposable P 1 -bundles X → F a with numerical invariants (a, b, c) when b ≥ 1 and c ≥ 2 (see Corollary 3.3.8).

The group Aut • (F a ) acts naturally on this moduli space, via an algebraic action, which is detailed in § 3.4. Using this action, we are able to describe the geometry of P 1 -bundles X → F a (see Proposition 3.7.4). We prove in particular that if no surface S p (with the notation above) is invariant by Aut • (X), then π is isomorphic to a decomposable P 1 -bundle, to an Umemura P 1 -bundle, or to a P 1 -bundle Ŝb → P 1 ×P 1 , which is obtained by pulling back the Schwarzenberger bundle S b → P 2 via the double cover κ : P 1 × P 1 → P 2 defined above (see Lemma 4.2.4); the last two cases correspond to natural elements of the moduli spaces M b,c a fixed by Aut • (F a ).

In the case Ŝb → P 1 × P 1 and in the case where a fibre S p is invariant, we can reduce to the case of decomposable P 1 -bundles over F a (again by Proposition 3.7.4).

Section 4 concerns P 1 -bundles over P 2 . Despite the fact that the geometry of such bundles is quite rich and complicated (see e.g. [START_REF] Okonek | Vector bundles on complex projective spaces[END_REF] for an overview when k = C), our approach allows us to give a quite simple proof in this case, using the work done in § 3 for P 1 -bundles over Hirzebruch surfaces. We first study the Schwarzenberger P 1 -bundles S b → P 2 (see Lemmas 4.2.1, 4.2.4, and 4.2.5 and Corollary 4.2.2). We then take a P 1 -bundle π : X → P 2 and denote by H ⊂ Aut(P 2 ) the image of Aut • (X). If H fixes a point, we blow up the fibre of this point and reduce our study to the case of P 1 -bundles over F 1 . Otherwise, using the structure of Aut(P 2 ), we see that either H = Aut(P 2 ) = PGL 3 or H ⊂ Aut(P 2 , C) = {g ∈ Aut(P 2 ) | g(C) = C}, for some smooth curve C which is a line or a conic (see Lemma 4.3.3). The case where C is a line cannot happen (see Proposition 4.3.4), and the case where C is a conic corresponds to the Schwarzenberger case; this is proven by using the double cover κ : P 1 × P 1 → P 2 of Definition 1.2.7 and the results on P 1 -bundles over Hirzebruch surfaces. The case where H = PGL 3 corresponds to the decomposable P 1 -bundles over P 2 or to the special Schwarzenberger P 1 -bundle S 1 , isomorphic to the projectivised tangent bundle P(T P 2 ). Again, this is proven by a reduction to the case of P 1 -bundles of F 1 (studied in § 3) by blowing up a point of P 2 .

In § 5, we prove Theorems A and B, which achieves our classification. Once we have reduced our study to decomposable P 1 -bundles over F a or P 2 or to Umemura or Schwarzenberger P 1 -bundles (see Proposition 5.1.1), we study birational maps of P 1 -bundles between elements of these four families, which are in fact obtained by elementary links centred at invariant curves (see Lemma 5.3.1), square isomorphisms, and special contractions from P 1 -bundles over F 1 to P 1 -bundles over P 2 (see Remark 5.2.1). The study of the possible links is made on each family, by describing the possible invariant curves. These are naturally contained in the preimage of invariant curves of the surface S over which we take the P 1 -bundles and are most of the time obtained by a negative curve in the fibres of smooth rational curves.

Acknowledgments.

We are grateful to the anonymous referee for the careful reading of a previous version of this paper and for their helpful comments. We would like to thank Michel Brion, Paolo Cascini, Ciro Ciliberto, Daniele Faenzi, Enrica Floris, Anne-Sophie Kaloghiros, Massimiliano Mella, Lucy Moser-Jauslin, Boris Pasquier, Francesco Russo, and Jean Vallès for interesting discussions related to this work. Thanks to Mateusz Michalek for indicating the proof of the combinatoric Lemma 3.4.4 to us. Thanks also to the MathOverflow community for its help regarding the proofs of Lemmas 3.7.2 and 4.3.3.

Preliminaries

All the results in § 2 are valid over an algebraically closed field k of arbitrary characteristic.

Blanchard's lemma

We recall a result due to Blanchard [Bla56, § I.1] in the setting of complex geometry, whose proof has been adapted to the setting of algebraic geometry by Brion, Samuel, and Uma.

Lemma 2.1.1 (cf. [BSU13, Proposition 4.2.1]). Let f : X → Y be a proper morphism between algebraic varieties such that f * (O X ) = O Y .
If a connected algebraic group G acts on X, then there exists a unique action of G on Y such that f is G-equivariant.

Resolution of indeterminacies

The next result implies that if π : X → S is a P 1 -bundle over a singular surface, then there exists an Aut • (X)-equivariant square birational map (X, π) (X , π ), with π : X → S a P 1 -bundle over a smooth surface. Therefore, it is enough to consider the P 1 -bundles over smooth surfaces to determine all the maximal automorphism groups Aut • (X) in the sense of Definition 1.2.3. Lemma 2.2.1. Let π : X → S be a P 1 -bundle over a singular projective surface S, and let G ⊂ Aut • (X) be a connected algebraic subgroup. Then there exist a P 1 -bundle π : X → S , equipped with a G-action, and G-equivariant birational morphisms η : S → S and η : X → X such that η is a resolution of singularities and the diagram below is cartesian; in particular, η-

1 G η is a subgroup of Aut • (X ). X η G G π X π S η G G S
Proof. By Lemma 2.1.1, the group G acts biregularly on S. Since G is connected, we can solve the singularities of S in a G-equivariant way by repeatedly alternating between normalization and blowing-up of the singular points (see for instance [START_REF] Artin | Lipman's proof of resolution of singularities for surfaces[END_REF]). We denote by η : S → S a G-equivariant resolution of singularities of S and define π : X := X × S S → S to be the pull-back of π along η : S → S. The pull-back of a P 1 -bundle along a morphism of schemes is again a P 1 -bundle (indeed, all the P 1 -bundles that we consider are Zariski locally trivial). Also, since S → S is G-equivariant, G acts on X , and η : X → X is G-equivariant. The last statement follows from the fact that η, and thus η, is birational.

In the following, we will only consider the P 1 -bundles over smooth surfaces; in particular, we will not describe the P 1 -bundles X → S over singular surfaces such that Aut • (X) is maximal. There are many such P 1 -bundles, obtained from those of Theorem A by contracting the negative curve on a Hirzebruch surface onto a singular point. The obtained threefolds have, however, singularities which are of codimension 2, so it is natural to avoid them when working in birational geometry with the classical minimal model program (where varieties have terminal singularities). We also recall that a P n -bundle over a smooth variety is isomorphic to the projectivisation of a rank n + 1 vector bundle over this variety (see e.g. [Har77, Exercise II.7.10(c)]); this well-known fact will be used implicitly throughout the rest of the article.

The descent lemma

The following simple observation will be often used later. It explains that P 1 -bundles π : X → S over smooth projective surfaces are uniquely determined by a description on an open subset with finite complement. This is for instance useful over P 2 or Hirzebruch surfaces, where then only the restriction of π to two open subsets isomorphic to A 2 is needed to describe the whole P 1 -bundle.

Lemma 2.3.1. Let S be a smooth projective surface, let Ω ⊂ S be a finite set, and let g ∈ Aut(S) be an automorphism that satisfies g(Ω) = Ω. Let π 1 : X 1 → S and π 2 : X 2 → S be two P 1 -bundles, and let ĝ : X 1 X 2 be a birational map that restricts to an isomorphism (π 1 ) -1 (S \ Ω) -→ (π 2 ) -1 (S \ Ω) and satisfies

π 2 ĝ = π 1 g. Then ĝ is an isomorphism of varieties X 1 -→ X 2 : X 1 ĝ G G π 1 X 2 π 2 S g G G S.
In particular, if g is the identity, then ĝ is an isomorphism of P 1 -bundles.

Proof. It suffices to take a point p 1 ∈ Ω 1 and to show that ĝ is a local isomorphism around every point of the curve (π 1 ) -1 (p 1 ) ⊂ X 1 . We denote by U 1 ⊂ S an open neighbourhood of p 1 and write U 2 = g(U 1 ) for its image, which is an open neighbourhood of p 2 = g(p 1 ) ∈ Ω. By shrinking U 1 , we can assume that π 1 and π 2 are trivial P 1 -bundles over U 1 and U 2 , respectively. The birational map ĝ is then described by

U 1 × P 1 U 2 × P 1 x, u v → g(x), M(x) • u v ,
where M ∈ GL 2 (k(S)). Since k(S) is the function field of O p 1 (S), which is a UFD (because S is smooth), we can multiplying M with an element of k(S) and obtain a matrix M whose entries are all in O p 1 (S) and do not have a common factor. Denote by f ∈ O p 1 (S) the determinant of M . If f does not vanish at p 1 , it is invertible in O p 1 (S), and ĝ yields an isomorphism U × P 1 → g(U ) × P 1 , where U ⊂ U corresponds to the open subset where f 0. This yields the result since p 1 ∈ U . It remains to show that f cannot vanish at p 1 . Indeed, otherwise the zero set of f yields a curve C of S passing through p 1 , and there is then a curve Ĉ ⊂ C with π 1 ( Ĉ) = C on which the map ĝ is not defined (since the matrix M is unique up to multiplication with an element of O p 1 (S) * and since at least one of the entries of M is not divisible by f ).

We can then prove the following descent lemma, already invoked in the introduction.

Lemma 2.3.2 (Descent lemma). Let η : Ŝ → S be a birational morphism between two smooth projective surfaces. Let U ⊂ S and Û ⊂ Ŝ be two maximal open subsets such that η induces an isomorphism Û -→ U and Ω = S \ U is finite, and let π : X → Ŝ be a P 1 -bundle.

Then, there exist a P 1 -bundle π : X → S and a birational map ψ : X X such that η π = πψ (ψ is a square birational map over η) and such that ψ induces an isomorphism π-1 ( Û ) -→ π -1 (U ). Moreover, ψ is unique, up to composition by an isomorphism of P 1 -bundles at the target, and ψ is Aut • ( X)-equivariant, which means that ψ Aut • ( X)ψ -1 is a subgroup of Aut • (X).

Proof. Writing G = Aut • ( X), Lemma 2.1.1 implies that π and η are G-equivariant for some unique biregular action of G on Ŝ and S.

If η is an isomorphism, everything is trivial. Otherwise, η is the blow-up of finitely many points, so η restricts to an isomorphism Û -→ U , where U ⊂ S is an open subset, Ω = S \ U is a finite set, and Û = η -1 (U ). We denote by j the inclusion U → S.

Let E → Ŝ be a rank 2 vector bundle such that P(E) X, and let E Û → Û be its restriction over Û . Then E Û → Û identifies with η * E Û → U , and we can consider the reflexive hull E = (j * (η * E Û )) ∨∨ of the coherent sheaf j * (η * E Û ) on S. By [Har80, Corollary 1.4], the reflexive sheaf E is locally free, and thus E → S is a rank 2 vector bundle that extends (uniquely) η * E Û . Writing X = P(E ), we obtain a P 1 -bundle X → S that extends (uniquely) the P 1 -bundle P(η * E Û ) → U .

Since η : Ŝ → S is G-equivariant and the open subsets Û and U are G-stable, the P 1 -bundle X → Ŝ is G-equivariantly birational to X U (= P(η * E Û )) → U . It remains to apply Lemma 2.3.1 to see that the G-action on X U extends to X.

Remark 2.3.3. If η is the blow-up of a point p ∈ S and E = η -1 (p) ⊂ Ŝ is the exceptional curve, the birational map ψ can be described as follows (according to [Mel02, § 5.7.4, p. 700]), depending on the surface Z = π-1 (E) F a : if a = 0, then ψ is a birational morphism whose restriction to Z P 1 × P 1 is the "other projection"; if a > 0, then ψ is given by the anti-flip of the exceptional curve of Z followed by the contraction of the strict transform of Z, isomorphic to the weighted projective plane P(1, 1, a), onto a smooth point.

Hirzebruch surfaces

In the following, we will always use the following coordinates for Hirzebruch surfaces, which are the analogue of the standard coordinates for P 2 = (A 3 \ {0})/G m . Definition 2.4.1. Let a ∈ Z. The a-th Hirzebruch surface F a is defined to be the quotient of (A 2 \ {0}) 2 by the action of (G m ) 2 given by

(G m ) 2 × (A 2 \ {0}) 2 → (A 2 \ {0}
) 2 (µ, ρ), (y 0 , y 1 , z 0 , z 1 ) → (µρ -a y 0 , µy 1 , ρz 0 , ρz 1 ).

The class of (y 0 , y 1 , z 0 , z 1 ) will be written [y 0 : y 1 ; z 0 : z 1 ]. The projection

τ a : F a → P 1 , [y 0 : y 1 ; z 0 : z 1 ] → [z 0 : z 1 ]
identifies F a with P(O P 1 (a) ⊕ O P 1 ) as a P 1 -bundle over P 1 . The disjoint sections s -a , s a ⊂ F a given by y 0 = 0 and y 1 = 0 have self-intersection -a and a, respectively. The fibres f ⊂ F a given by z 0 = 0 and z 1 = 0 are linearly equivalent and of self-intersection 0. We moreover get Pic(F a ) = Zf Zs -a = Zf Zs a since s a ∼ s -a + af .

Remark 2.4.2. The surface F a is naturally isomorphic to F -a via [y 0 : y 1 ; z 0 : z 1 ] → [y 1 : y 0 ; z 0 : z 1 ], so we will most of the time choose a ≥ 0.

Let us recall the classical structure of the automorphism groups of Hirzebruch surfaces. The description of Definition 2.4.1 allows one to present all automorphisms in a simple way.

Remark 2.4.3. If a = 0, then F a P 1 × P 1 and the natural action of (GL 2 ) 2 on (A 2 \ {0}) 2 yields a surjective group homomorphism (GL 2 ) 2 → Aut • (F 0 ) (PGL 2 ) 2 (see e.g. [BSU13, Corollary 4.2.7] for the last isomorphism). All automorphisms are then of the form [y 0 : y 1 ; z 0 : z 1 ] → [αy 0 + βy 1 : γy 0 + δy 1 ; α z 0 + β z 1 :

γ z 0 + δ z 1 ].
The action of Aut • (F 0 ) on F 0 is then homogeneous (one single orbit). Moreover, Aut(F 0 ) = Aut • (F 0 ) ι , with ι : [y 0 : y 1 ; z 0 : z 1 ] → [z 0 : z 1 ; y 0 : y 1 ].

Remark 2.4.4. If a ≥ 1, then the curve s -a given by y 0 = 0 is the unique section of negative self-intersection of F a → P 1 and is thus invariant. Denoting by k[z 0 , z 1 ] a ⊂ k[z 0 , z 1 ] the vector space of homogeneous polynomials of degree a, one gets an action of k[z 0 , z 1 ] a GL 2 on F a via [y 0 : y 1 ; z 0 : z 1 ] → [y 0 : y 1 + y 0 p(z 0 , z 1 ); αz 0 + βz 1 : γz 0 + δz 1 ],

which yields an exact sequence

1 → µ a → k[z 0 , z 1 ] a GL 2 → Aut(F a ) → 1,
where µ a ⊂ GL 2 is the cyclic group of homotheties α with α a = 1. To prove the surjectivity of the morphism k[z 0 , z 1 ] a GL 2 → Aut(F a ), it suffices to consider the elements acting trivially on the basis of the P 1 -bundle F a → P 1 (by Blanchard's lemma, since GL 2 itself acts transitively on the basis P 1 ). One then restricts such an automorphism to the two charts z 0 0 and z 1 0 and obtains two A 1 -automorphisms of P 1 × A 1 that fix y 0 = 0, i.e. two triangular elements of GL 2 (k[z 0 /z 1 ]) and GL 2 (k[z 1 /z 0 ]), and that must coincide on the intersection. A simple calculation then shows that it comes from an element of k[z 0 , z 1 ] a GL 2 as above. In particular, this implies that Aut(F a ) is connected. Moreover, Aut(F a ) = Aut • (F a ) acts on F a with two orbits, namely s -a and its complement.

Remark 2.4.5. It follows from Remarks 2.4.3 and 2.4.4 that for each a ≥ 0, the morphism τ a : F a → P 1 yields a surjective group homomorphism

Aut • (F a ) Aut(P 1 ) PGL 2 .
In particular, Aut(F a ) acts transitively on the set of fibres of τ a : F a → P 1 . We also recall the following easy observation, that we will need further on. Lemma 2.4.6. Let a ≥ 0, and let E → P 1 be a rank 2 vector bundle that fits into an exact sequence

0 → O P 1 ι → E → O P 1 (-a) → 0.
Then we have an isomorphism from P(E) → P 1 to F a → P 1 which sends the section corresponding to ι(O P 1 ) onto s -a .

Proof. We trivialise E on the two open subsets of P 1 given by {

[1 : z] | z ∈ A 1 } and {[z : 1] | z ∈ A 1 }, change
coordinates so that ι(O P 1 ) corresponds to x 0 = 0, and get a transition

A 2 × A 1 A 2 × A 1 (x 0 , x 1 ), z (z a x 0 , x 1 + f (z)), 1 z for some f ∈ k[z, z -1 ].
We can then compose at the source and target with some automorphisms of the form ((x 0 , x 1 ), z) → ((x 0 , y

1 + h i (z)), z) for polynomials h 1 , h 2 ∈ k[z]. This replaces f by f + h 1 (z) + h 2 (z -1 )z a .
Since a ≥ 0, we can thus replace f with 0. The projectivisation of E is then isomorphic to F a , by sending ([x 0 : x 1 ], z) ∈ P 1 × A 1 onto [x 0 : x 1 ; 1 : z] and [x 0 : x 1 ; z : 1] = [x 0 z a : x 1 ; 1 : z -1 ], on both charts.

P 1 -bundles over Hirzebruch surfaces

Most of the results in § 3 are valid over an algebraically closed field k of arbitrary characteristic. More precisely, § § 3.1-3.3 are valid in arbitrary characteristic, the results in § 3.4 are all valid in arbitrary characteristic, but the proof of Corollary 3.4.6 that we give is shortened a bit by using the characteristic zero assumption (see Remark 3.4.7); § § 3.5 and 3.6 are also valid in arbitrary characteristic different from 2, except Lemma 3.5.5(3), which fails if and only if char(k) divides b + 1 (see Remark 3.5.6). In § 3.7 we must assume that the ground field k has characteristic zero as Lemma 3.7.2 and Proposition 3.7.4 are false over any field of positive characteristic (see Remarks 3.7.3 and 3.7.5).

Decomposable P 1 -bundles over Hirzebruch surfaces

As for Hirzebruch surfaces F a (Definition 2.4.1), one can give global coordinates on decomposable P 1 -bundles over F a . Definition 3.1.1. Let a, b, c ∈ Z. We define F b,c a to be the quotient of (A 2 \ {0}) 3 by the action of (G m ) 3 given by

(G m ) 3 × (A 2 \ {0}) 3 → (A 2 \ {0}) 3 (λ, µ, ρ), (x 0 , x 1 , y 0 , y 1 , z 0 , z 1 ) → (λµ -b x 0 , λρ -c x 1 , µρ -a y 0 , µy 1 , ρz 0 , ρz 1 ).
The class of (x 0 , x 1 , y 0 , y 1 , z 0 , z 1 ) will be written [x 0 : x 1 ; y 0 :

y 1 ; z 0 : z 1 ]. The projection F b,c a → F a , [x 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ] → [y 0 : y 1 ; z 0 : z 1 ] identifies F b,c a with P O F a (bs a ) ⊕ O F a (cf ) = P O F a ⊕ O F a (-bs a + cf )
as a P 1 -bundle over F a , where s a , f ⊂ F a are given by y 1 = 0 and z 1 = 0.

Moreover, every fibre of the composed morphism F b,c a → F a → P 1 given by the z-projection is isomorphic to F b , and the restrictions of F b,c a on the curves s -a and s a given by y 0 = 0 and y 1 = 0 are isomorphic to F c and F c-ab , respectively.

As for Hirzebruch surfaces, one can reduce to the case a ≥ 0 without changing the isomorphism class, by exchanging y 0 and y 1 . We then observe that the exchange of x 0 and x 1 yields an isomorphism F b,c a -→ F -b,-c a . We will assume a, b ≥ 0 most of the time from here on. If b = 0, we can moreover assume c ≤ 0. Remark 3.1.2. Every decomposable P 1 -bundle over F a is isomorphic to F b,c a → F a for some b, c ∈ Z, b ≥ 0. We can moreover assume b ≥ 0, and c ≤ 0

if b = 0, since F b,c a F -b,-c a .
The P 1 -bundle F b,c a → F a has numerical invariants (a, b, c) (see Definition 1.4.1). As we will see later (Remark 3.3.2), these numerical invariants are indeed invariant under isomorphism. This will show that the isomorphism classes of decomposable P 1 -bundles over Hirzebruch surfaces are parametrised by these invariants.

Remark 3.1.3. All decomposable P 1 -bundles F b,c a are toric varieties, with an action given by the map [x 0 : x 1 ; y 0 :

y 1 ; z 0 : z 1 ] → [x 0 : αx 1 ; y 0 : βy 1 ; z 0 : γz 1 ]. Remark 3.1.4. We have two open embeddings F b × A 1 → F b,c a [x 0 : x 1 ; y 0 : y 1 ], z → [x 0 : x 1 ; y 0 : y 1 ; 1 : z] [x 0 : x 1 ; y 0 : y 1 ], z → [x 0 : x 1 ; y 0 : y 1 ; z : 1] over which the F b -bundle F b,c a → P 1 is trivial, with a transition function [x 0 : x 1 ; y 0 : y 1 ], z → [x 0 : x 1 z c ; y 0 z a : y 1 ], 1 z . Lemma 3.1.5. Let a, b ≥ 0 and c ∈ Z. The morphism π : F b,c a → F a yields a surjective group homomorphism ρ : Aut • (F b,c a ) Aut • (F a ).
Proof. The existence of ρ is given by Lemma 2.1.1. The fact that it is surjective can be seen by observing that every automorphism g ∈ Aut • (F a ) comes from an automorphism of (A 2 \ {0}) 2 (see Remarks 2.4.3 and 2.4.4), so we can extend the action to (A 3 \ {0}) 2 and then F b,c a by doing nothing on x 0 and x 1 .

Remark 3.1.6. For each i, j ∈ Z, we denote by k[y 0 , y 1 , z 0 , z 1 ] i,j ⊂ k[y 0 , y 1 , z 0 , z 1 ] the space of homogeneous polynomials of bidegree (i, j), where the variables y 0 , y 1 , z 0 , z 1 are of bidegree (1, -a), (1, 0), (0, 1), (0, 1). The group of automorphisms of P 1 -bundles of F b,c a identifies with the (connected) group

p 1,0,0 p 2,-b,c p 3,b,-c p 4,0,0 ∈ PGL 2 (k[y 0 , y 1 , z 0 , z 1 ]) p k,i,j ∈ k[y 0 , y 1 , z 0 , z 1 ] i,j for k = 1, . . . , 4.
whose action on F b,c a is as follows:

[x 0 : x 1 ; y 0 : y 1 : z 0 ; z 1 ] → [x 0 p 1 + x 1 p 2 : x 0 p 3 + x 1 p 4 ; y 0 : y 1 ; z 0 : z 1 ].
This can be seen directly from the global description of F b,c a in Definition 3.1.1 and by using trivialisations on open subsets isomorphic to A 2 .

Removal of jumping fibres

Removing a fibre into a Hirzebruch surface, we get an open subset isomorphic to A 1 × P 1 . It is then natural to study the P 1 -bundles over A 1 × P 1 in order to get a local description of the P 1 -bundles over Hirzebruch surfaces. Lemma 3.2.1. Let π : X → A 1 × P 1 be a P 1 -bundle, and let τ : A 1 × P 1 → A 1 be the first projection. Then, there exist an integer b ≥ 0 and a dense open subset U ⊂ A 1 (both uniquely determined by π) such that the following hold:

(1) The generic fibre of the morphism τπ : X → A 1 is isomorphic to the Hirzebruch surface F b .

(2) There exists a commutative diagram

(τπ) -1 (U ) G G π B B U × F b , pr 1 ×τ b t t U × P 1
where pr 1 × τ b sends (u, x) onto (u, τ b (x)) and τ b : F b → P 1 is the standard P 1 -bundle.

(3) For each p ∈ A 1 \ U , the fibre (τπ) -1 (p) is isomorphic to the Hirzebruch surface F b+2 for some positive integer . (4) For each p ∈ A 1 \ U , we can blow up the exceptional section of (τπ) -1 (p) and contract the strict transform of (τπ) -1 (p); this replaces X by another P 1 -bundle X → P 1 × A 1 as above, with a new open subset U which is equal either to U or to U ∪ {p}. After finitely many such steps, we get the case where

U = A 1 , corresponding to a trivial F b -bundle. Proof. We choose two open subsets of V 0 , V 1 ⊂ A 1 × P 1 isomorphic to A 2 via ι 0 : A 2 -→ V 0 ⊂ A 1 × P 1 , (x, y) → x, [1 : y] ι 1 : A 2 -→ V 1 ⊂ A 1 × P 1 . (x, y) → x, [y : 1]
The restriction of π to the open subsets V 0 and V 1 yield P 1 -bundles over A 2 , which are then trivial, by the Quillen-Suslin theorem (see [Lan02, Theorem XXI.3.7]) combined with the fact that a P 1 -bundle over A 2 is isomorphic to the projectivisation of a rank 2 vector bundle over A 2 since A 2 is a smooth variety. This gives the existence of isomorphisms

ϕ i : π -1 (V i ) → A 2 × P 1 , for i = 0, 1, such that ι i pr 1 ϕ i = π, where pr 1 : A 2 × P 1 → A 2
is the first projection. The isomorphisms ϕ i are uniquely determined, up to composing at the target and the source with elements of PGL 2 (k[A 2 ]) ⊂ Aut(A 2 × P 1 ). We then write the transition function ϕ 1 (ϕ 0 ) -1 as

(x, y), [u : v] → (x, y -1 ), [α 11 (x, y)u + α 12 (x, y)v : α 21 (x, y)u + α 22 (x, y)v] ,
where

A = α 11 α 12 α 21 α 22 ∈ GL 2 (k[x, y, y -1 ]). Note that the P 1 -bundle π is determined by the equivalence class of A modulo A ∼ λMAM , where λ ∈ k[x, y, y -1 ] * = k * y Z , M ∈ GL 2 (k[x, y -1 ]), M ∈ GL 2 (k[x, y]).
In particular, we can multiply A with an element of k * and assume det(A) ∈ y Z .

Working over the field k(x), we get a P 1 -bundle over P 1 k(x) , which is therefore isomorphic to a Hirzebruch surface F b , with b ≥ 0 (this follows from the fact that a vector bundle on P 1 is decomposable over any field). This yields two matrices

B ∈ GL 2 (k(x)[y -1 ]) and C ∈ GL 2 (k(x)[y]) such that B -1 AC = D = y m 0 0 y n for some integers m, n ∈ Z with m -n = b, and yields part (1). Since det(B), det(C) ∈ k(x) * and det(A), det(D) ∈ y Z , we get det(A) = det(D) = y m+n , so det(B) = det(C) ∈ k(x) * .
Writing the equality AC = BD, we can multiply both B and C with the same element of k(x) * and assume

B ∈ Mat 2,2 (k[x, y -1 ]), C ∈ Mat 2,2 (k[x, y]), and (B(λ), C(λ)) (0, 0) for each λ ∈ k, where B(λ), C(λ) ∈ Mat 2,2 (k(y)) are obtained by replacing x by λ in B, C.
We denote by

Z ⊂ A 1 the zero set of det(B) = det(C) ∈ k[x].
On the open set U = P 1 \ Z, the matrices B and C correspond to automorphisms of U × A 1 on the two charts, respectively, so we have a trivial

F b -bundle on U . In particular, if Z = ∅, then τπ : X → A 1 is a trivial F b -bundle,
and the proof is over. We can thus prove the result by induction on the degree of the polynomial det(B).

Suppose that λ ∈ Z is such that the fibre (τπ) -1 (λ) is a Hirzebruch surface Fb. Hence, A(λ) corresponds to the transition function of Fb, which means that B-1 (a) If the first column of B(λ) is zero, then so is the first column of C(λ).

A(λ) C = y m 0 0 y ñ with B ∈ GL 2 (k[y -1 ]), C ∈ GL 2 (k[y]), m, ñ ∈ Z, and m-ñ = b ≥ 0. Computing the determinant yields m+n = m+ ñ. Writing = m-m = n-ñ, we then get b -b = ( m-ñ)-(m-n) = 2 . Replacing A, B, C with B-1 A C, ( B) -1 B, ( C) -1 C,
Writing ∆ = x-λ 0 0 1 , we get B = B ∆ and C = C ∆ for some B , C ∈ Mat 2,2 (k[x, y, y -1 ]).
(b) If < 0, then -b -= -b < 0, so the second column of B(λ) and C(λ) is zero since B(λ) ∈ Mat 2,2 (k[y -1 ]) and C(λ) ∈ Mat 2,2 (k[y]); we then apply (a). (c) If = b = 0, then B(λ) = C(λ) ∈ Mat 2,2 . There thus exists an R ∈ GL 2 such that the first column of B(λ)R is zero. We can replace B, C with BR, CR, since R commutes with D = y m • I = y n • I,
β 11 = 0 since det(B(λ)) = 0, hence B(λ) = 0 0 β 21 β 22 , so the first column of B(λ) • R is zero, with R = β 22 0 -β 21 1 ∈ GL 2 (k[y -1 ]). Writing R = D -1 RD = β 22 0 -β 21 y b 1 ∈ GL 2 (k[y]
), we can replace B and C with BR and CR and get case (a).

(e) The last case is when > 0, which implies that b = b + 2 ≥ b + 2 ≥ 2 and that β 11 = β 12 = 0.

After applying the steps above, we can assume that all elements of Z give rise to case (e). Writing U = A 1 \ Z, this yields parts (2) and (3).

It remains to show part (4), by studying more carefully case (e). Note that the fibre x = λ corresponds to the Hirzebruch surface Fb, with b ≥ 2, and a special section corresponds to u = 0 in the charts A 2 × P 1 .

The blow-up of the exceptional section, followed by the contraction of the strict transform of the surface Fb, corresponds locally to

A 2 × P 1 A 2 × P 1 ((x, y), [u : v]) → ((x, y), [u : (x -λ)v]).
This replaces the transition matrix A with A = ∆ -1 A∆, where ∆ = x-λ 0 0 1 :

A = α 11 α 12 α 21 α 22 , A = α 11 α 12 x-λ (x -λ)α 21 α 22 .
Note that the new transition A still belongs to GL 2 (k[x, y, y -1 ]) since A(λ) is diagonal, which implies that α 12 and α 21 are multiples of xλ. Moreover, the first line of B(λ) and C(λ) is zero, so we can write B = ∆B and C = ∆C for some B , C ∈ Mat 2,2 (k[x, y, y -1 ]). The blow-up replaces A by A , and we can replace B, C with B , C since

A C = (∆ -1 A∆)(∆ -1 C) = ∆ -1 BD = B D.
This process decreases the degree of det(B) = det(C); we get a trivial F b -bundle after finitely many steps.

As a consequence of Lemma 3.2.1, we get the following result. Proposition 3.2.2 (Removal of jumping fibres). Let a ≥ 0, and let π : X → F a be a P 1 -bundle. There exist an integer b ≥ 0 and a dense open subset of U ⊂ P 1 such that (τ a π) -1 (p) is a Hirzebruch surface F b for each p ∈ U . Moreover, we have:

(1) If U = P 1 , then τ a π : X → P 1 is an F b -bundle which is trivial on every affine open subset of P 1 . In this case, we say that π has no jumping fibre.

(2) If one fibre (τ a π) -1 ({p}) is isomorphic to F c for some c b, then cb is a positive even integer (we say that τ -1 a ({p}) is a jumping fibre), and the blow-up of the (unique) exceptional section of F c followed by the contraction of the strict transform of F c gives an Aut • (X)-equivariant birational map X X to another P 1 -bundle over F a . After finitely many such steps, one gets case (1).

Proof. For each p ∈ P 1 , we have a commutative diagram

A 1 × P 1 G G pr 1 F a \ τ -1 a ({p}) τ a A 1 G G P 1 \ {p}.
We can thus apply Lemma 3.2.1 on each affine subset P 1 \ {p} and get the result.

Another consequence of Lemma 3.2.1 is the following description.

Corollary 3.2.3. Let π : X → S = A 1 × P 1 be a P 1 -bundle, let τ : S → A 1 be the first projection, and let b ≥ 0.

The following conditions are equivalent:

(1) The fibre (τπ

) -1 (p) is a Hirzebruch surface F b for each p ∈ A 1 . (2) There exists a commutative diagram X G G π A A A 1 × F b , pr 1 ×τ b t t S = A 1 × P 1
where pr 1 × τ b sends (u, x) on (u, τ b (x)).

(3) The P 1 -bundle X → S is the projectivisation of a rank 2 vector bundle E which fits in a short exact sequence

0 → O S → E → O S (-bs) → 0,
where s is a fibre of pr 2 : S → P 1 (which satisfies Pic(S) = Zs).

Proof. The implication (1) ⇒ (2) is given by Proposition 3.2.2(1). To get (2) ⇒ (3), we observe that (2) yields an isomorphism between the P 1 -bundles X → A 1 × P 1 and A 1 × P O P 1 O P 1 (b) → A 1 × P 1 . To get (3) ⇒ (1), we restrict the exact sequence to each fibre of τ : S → A 1 and apply Lemma 2.4.6.

Moduli spaces of P 1 -bundles over F a with no jumping fibre

The following proposition associates to every P 1 -bundle over a Hirzebruch surface with no jumping fibre three unique invariants (a, b, c), called numerical invariants in Definition 1.4.1. The integer a is given by the Hirzebruch surface over which the P 1 -bundle is taken. The integer b is given by the generic fibre of the projection to P 1 , which is a Hirzebruch surface F b . The last integer c can be seen using exact sequences or using transition functions, as the following result explains. Proposition 3.3.1. Let a ≥ 0, and let π : X → F a be a P 1 -bundle.

(1) For all integers b ≥ 0 and c ∈ Z, the following are equivalent:

(i ) The variety X is the gluing of two copies of

F b × A 1 along F b × A 1 \ {0} by the automorphism ν c,P ∈ Aut(F b × A 1 \ {0})
given by ν c,P : ([x 0 : x 1 ; y 0 :

y 1 ], z) → [x 0 : x 1 z c + x 0 P (y 0 , y 1 , z); y 0 z a : y 1 ], 1 z for some P ∈ k[y 0 , y 1 , z, z -1 ], homogeneous of degree b in y 0 , y 1 , such that π : X → F a sends ([x 0 : x 1 ; y 0 : y 1 ], z) ∈ F b × A 1 onto, respectively, [y 0 : y 1 ; 1 : z] ∈ F a and [y 0 : y 1 ; z : 1] ∈ F a on the two charts. (ii ) The P 1 -bundle π : X → F a is the projectivisation of a rank 2 vector bundle E which fits in a short exact sequence 0 → O F a → E → O F a (-bs a + cf ) → 0,
where f , s a ⊂ F a are given by y 1 = 0 and z 1 = 0. (2) If there exists a b ≥ 0 such that the preimage of each fibre of the P 1 -bundle τ a : F a → P 1 is isomorphic to F b (no jumping fibre), then there is an integer c ∈ Z such that the above properties are satisfied. If b > 0, the integer c is unique. If b = 0, then |c| is unique and π : X → F a is isomorphic to the decomposable bundles F 0,c a → F a and F 0,-c a → F a .

Proof. We first prove (i ) ⇒ (ii ). The section x 0 = 0 being invariant by the transition function, one can see X as the projectivisation of a rank 2 vector bundle E which fits in a short exact sequence

0 → O F a → E → O F a (βs a + γf ) → 0
for some integers β, γ ∈ Z. To compute these numbers, we take the two open subsets

U 0 , U 1 ⊂ F a isomorphic to A 2 via θ 0 : A 2 -→ U 1 ⊂ F a , (y, z) → [1 : y; 1 : z] θ 1 : A 2 -→ U 2 ⊂ F a (y, z) → [y : 1; z : 1]
and observe that the vector bundle has a transition function of the form

(x 0 , x 1 , y, z) → y b z -c x 0 , x 1 + x 0 P (1, y, z)z -c , z a y , 1 z , which yields β = -b and γ = c.
We then prove (2). It follows from Proposition 3.2.2 that the pull-back on X of the two open subsets V 0 , V 1 ⊂ F a given by θ 0 :

P 1 × A 1 -→ V 0 ⊂ F a , ([y 0 : y 1 ], z) → [y 0 : y 1 ; 1 : z] θ 1 : P 1 × A 1 -→ V 1 ⊂ F a ([y 0 : y 1 ], z) → [y 0 : y 1 ; z : 1] are isomorphic to A 1 × F b .
The transition function on F a being given by ([y 0 :

y 1 ], z) ([z a y 0 : y 1 ], z -1 ), we get a transition function ψ ∈ Aut(F b × A 1 \ {0}) of the form F b × A 1 F b × A 1 [x 0 : x 1 ; y 0 : y 1 ], z → [f 0 (x 0 , x 1 , y 0 , y 1 , z) : f 1 (x 0 , x 1 , y 0 , y 1 , z); z a y 0 : y 1 ], 1 z for some f 0 , f 1 ∈ k[x 0 , x 1 , y 0 , y 1 ][z, z -1 ].
The isomorphism class of π : X → F a (as in Definition 1.2.1) is then determined by ψ, up to composition at the source and target with automorphisms of the P 1 -bundle

F b × A 1 → P 1 × A 1 . If b = 0, then ψ is of the form ([x 0 : x 1 ; y 0 : y 1 ], z) → ([α(z)x 0 + β(z)x 1 ; γ(z)x 0 + δ(z)x 1 ; z a y 0 : y 1 ], z -1 ), where A = α(z) β(z) γ(z) δ(z) ∈ GL 2 (k[z, z -1 ]).
The isomorphism class of π : X → F a is then given by the matrix A, up to replacing A by λBAC, where

B ∈ GL 2 (k[z -1 ]), C ∈ GL 2 (k[z]
), and λ ∈ k * . The class of A modulo this replacement corresponds to a vector bundle over P 1 , which is therefore equivalent to a decomposable one, with a diagonal matrix A. We then get an integer c ∈ Z, unique up to sign, such that ψ is of the form

ψ : [x 0 : x 1 ; y 0 : y 1 ], z → [x 0 : x 1 z c ; y 0 z a : y 1 ], 1 z .
This shows that X → F a is isomorphic to F b,c a = F 0,c a → F a (see Remark 3.1.4) and also to

F 0,-c a → F a . If b > 0, then ψ is of the form ([x 0 : x 1 ; y 0 : y 1 ], z) → ([x 0 : µ(z)x 1 + x 0 P (y 0 , y 1 , z); z a y 0 : y 1 ], z -1 ), where µ(z) ∈ k[z, z -1 ] * = k * •z Z and P (y 0 , y 1 , z) is a homogeneous polynomial of degree b in y 0 , y 1 , with coefficients in k[z, z -1 ].
Applying a diagonal automorphism at the target, we can assume µ(z) = z c for some c ∈ Z. We then observe that c is unique since the transition function is determined up to automorphisms of

F b × A 1 → P 1 × A 1 at
the target and the source, which do not change c.

It remains to prove (ii ) ⇒ (i ). The inclusion O F a → E corresponds to a section of X → F a . We restrict the exact sequence to a fibre f and get

0 → O f → E f → O f (-b) → 0.
The corresponding section then needs to be the exceptional section of F b , unique if and only if b > 0 (see Lemma 2.4.6). The preimage of each fibre of the P 1 -bundle τ a : F a → P 1 is then isomorphic to F b (no jumping fibre). We then apply (2) and get (i ) for some unique integer c . The calculation made in the proof of (i ) ⇒ (ii ) implies that c = c since the inclusion O F a → E corresponds to the section x 0 = 0. Remark 3.3.2. Proposition 3.3.1 shows that two P 1 -bundles with different numerical invariants (see Definition 1.4.1) are not isomorphic. The numerical invariants defined in Definition 1.4.1, which correspond to the integers (a, b, c) of Proposition 3.3.1 (where c ≤ 0 when b = 0), are then really invariant under isomorphisms.

In particular, F b,c a → F a is the unique isomorphism class of decomposable P 1 -bundles with invariants (a, b, c). (This follows from Remark 3.1.2).

As a direct consequence of Proposition 3.3.1, we obtain the following corollary (which is well known over the field of complex numbers; see e.g. [ABM12, § 2.2]).

Corollary 3.3.3. Let a, b ≥ 0, and let E be a rank 2 vector bundle over F a . Then the following are equivalent:

(1) There exists an exact sequence

0 → O F a (ds -a + rf ) → E → O F a (d s -a + r f ) → 0 for some integers d, d , r, r such that b = d -d .
(2) The preimage by P(E) → F a of each fibre of the P 1 -bundle τ a :

F a → P 1 is isomorphic to F b . Moreover, the extension in (1) is unique if b > 0.
Proof. (1) ⇒ (2) The inclusion O F a (ds -a + rf ) → E corresponds to a section of P(E) → F a . We restrict the exact sequence to a fibre f and get

0 → O f (d) → E f → O f (d ) → 0, which yields the same P 1 -bundle as 0 → O f → E f → O f (-b) → 0.
The corresponding section then needs to be the exceptional section of F b (see Lemma 2.4.6), which is unique if b > 0.

(2) ⇒ (1) This follows from Proposition 3.3.1(2).

P ∈ k[y 0 , y 1 , z, z -1 ] b = {f ∈ k[y 0 , y 1 , z, z -1 ] homogeneous of degree b in y 0 , y 1 },
we denote by Z b,c,P a → F a the P 1 -bundle given by the gluing of two copies of

F b × A 1 along F b × A 1 \ {0} by the automorphism ν c,P ∈ Aut(F b × A 1 \ {0}) given by ν c,P : [x 0 : x 1 ; y 0 : y 1 ], z → [x 0 : x 1 z c + x 0 P (y 0 , y 1 , z); y 0 z a : y 1 ], 1 z , such that π : X → F a sends ([x 0 : x 1 ; y 0 : y 1 ], z) ∈ F b × A 1 onto,
∈ k * and Q 1 , Q 2 ∈ k[y 0 , y 1 , z] homogeneous of degree b in y 0 , y 1 such that P = λP + Q 1 (y 0 , y 1 , z)z c + Q 2 (y 0 z a , y 1 , z -1 ).
Proof. If b b , then the two P 1 -bundles are not isomorphic since the preimages of the fibres of the P 1 -bundle τ a : F a → P 1 are not isomorphic. We can thus assume b = b.

The two P 1 -bundles are obtained by gluing two copies of

F b × A 1 over F b × A 1 \ {0} by ν c,P , ν c ,P ∈ Aut(F b × A 1 \ {0}) (see Notation 3.3.4
). The P 1 -bundles are thus isomorphic if and only if ν c ,P = αν c,P β for some automorphisms α, β of the P 1 -bundle

F b × A 1 → P 1 × A 1 . Since b ≥ 1, such elements are of the form θ λ,Q : F b × A 1 -→ F b × A 1 [x 0 : x 1 ; y 0 : y 1 ], z → [x 0 : λx 1 + x 0 Q(y 0 , y 1 , z); y 0 : y 1 ], z , where λ ∈ k * and Q ∈ k[y 0 , y 1 , z] b . The composition θ λ 2 ,Q 2 ν c,P θ λ 1 ,Q 1 yields F b × A 1 \ {0} -→ F b × A 1 \ {0} [x 0 : x 1 ; y 0 : y 1 ], z → [x 0 : z c λ 1 λ 2 x 1 + x 0 P (y 0 , y 1 , z); z a y 0 : y 1 ], 1 z , with P (y 0 , y 1 , z) = λ 2 P (y 0 , y 1 , z) + λ 2 Q 1 (y 0 , y 1 , z)z c + Q 2 y 0 z a , y 1 , 1 z .
To get a transition function of the form ν c ,P , we then need c = c and λ 1 λ 2 = 1.

From now on we write

k[z] ≤r = {f ∈ k[z] | deg(f ) ≤ r} = k ⊕ kz ⊕ • • • ⊕ kz r . Corollary 3.3.7. Let a, b, c ∈ Z, with a, b ≥ 0 and with c ≤ 0 if b = 0.
(1) There is a unique isomorphism class of decomposable P 1 -bundles X → F a with numerical invariants

(a, b, c), represented by F b,c a → F a . (2) If b = 0 or c ≤ 1, every P 1 -bundle X → F a with numerical invariants (a, b, c) is decomposable, and thus isomorphic to F b,c a → F a . (3) If b ≥ 1 and c ≥ 2, every P 1 -bundle with numerical invariants (a, b, c) is isomorphic to Z b,c,P a → F a , where P (y 0 , y 1 , z) = b i=0 y 0 i y 1 b-i P i (z)z ai+1 and P i (z) ∈ k[z] ≤c-2-ai (hence P i = 0 if c < ai + 2), for i = 0, . . . , b.
The isomorphism class of the P 1 -bundle is determined by the class of P , up to scalar multiplication by an element of k * . The P 1 -bundle is decomposable if and only if P = 0.

Proof. Assertion (1) has already been proven (Remark 3.3.2). If b = 0, then Proposition 3.3.1 shows that every

P 1 -bundle over F a with numerical invariants (a, b, c) is isomorphic to F b,c a = F 0,c
a . We can thus assume b ≥ 1.

Proposition 3.3.1 shows that every P 1 -bundle X → F a with invariants (a, b, c) is isomorphic to Z b,c,P a → F a for some P ∈ k[y 0 , y 1 , z, z -1 ] b . Lemma 3.3.6 shows that the isomorphism class is inside the set of P 1 -bundles with invariants (a, b, c) and corresponds to an equivalence class on k[y 0 , y 1 , z, z -1 ] b , where P and P are equivalent if and only if

P = λP + Q 1 (y 0 , y 1 , z)z c + Q 2 (y 0 z a , y 1 , z -1 ), for λ ∈ k * and Q 1 , Q 2 ∈ k[y 0 , y 1 , z] b .
In particular, each equivalence class is given by an element

P (y 0 , y 1 , z) = b i=0 y 0 i y 1 b-i P i (z)z ai+1 ,
where

P i (z) ∈ k[z] is of degree at most c -ai -2 for i = 0, . . . ,
M b,c a = P       b i=0 y i 0 y b-i 1 • k[z] ≤c-2-ai       .
Proof. This follows from Corollary 3.3.7(3).

Remark 3.3.9. We have M b,c a P 1 2 (d+1)(2(c-1)-ad)-1 , where d is the biggest integer such that d ≤ b and ad ≤ c -2 (and is thus equal to b if ab ≤ c -2). Indeed, the dimension of the vector space k

[z] ≤c-2-ai is equal to c -1 -ai if ai ≤ c -2 and to 0 if ai > c -2. Hence, the dimension of b i=0 y i 0 y b-i 1 • k[z] ≤c-2-ai is equal to d i=0 (c -1 -ai) = 1 2 (d + 1)(2(c -1) -ad).

Action of Aut

• (F a ) on the moduli spaces M b,c a If π : X → F a is a P 1 -bundle with numerical invariants (a, b, c), then so is ϕ • π : X → F a for each ϕ ∈ Aut • (F a ). Indeed, the action of Aut • (F a ) does not change the exact sequence of Definition 1.4.1. This then gives a natural left-action of Aut • (F a ) on M b,c a , that we describe in this section; it will be very useful later, for the following reason: a point [p :

X → F a ] ∈ M b,c
a is fixed by g ∈ Aut • (F a ) if and only if g lifts to an automorphism of the variety X.

Since we have a group homomorphism GL 2 → Aut • (F a ) (see §2.4), we get an action of GL 2 on

M b,c a = P b i=0 y i 0 y b-i 1 • k[z] ≤c-2-ai (see Corollary 3.3.8
). We will show that this GL 2 -action coincides with the following one. Definition 3.4.1. Let r ≥ 0, and let us equip V = k 2 with the standard left-action of GL 2 . There is a unique left-action of GL 2 on k[z] ≤r making the following map GL 2 -equivariant:

V → k[z] ≤r (u, v) → r i=0 u i v r-i • z i .
Remark 3.4.2. Equipped with the GL 2 -action of Definition 3.4.1, the vector space k[z] ≤r identifies with the r-th symmetric power of the standard representation of GL 2 . In particular, k[z] ≤r is an irreducible GL 2 -representation (as we assumed k to be of characteristic zero).

We first need the following observations on the action of GL 2 on k[z] ≤r of Definition 3.4.1. (1)

If σ = 0 1 1 0 ∈ GL 2 , then σ (P ) = P (z -1 ) • z r . (2) If σ = α β 0 δ ∈ GL 2 , then P := σ (P ) ∈ k[z]
≤r is the unique polynomial that satisfies

(3.1) P (z) = α δ r (βz + α) • P δz βz + α ,
where we identify k[z] ≤r with k[z]/(z r+1 ) and compute the above equality in this ring.

Proof.

(1) The action of σ on V being (u, v) → (v, u), the action on P ∈ k[z] ≤r sends r i=0 a i z i onto r i=0 a r-i z i , which is exactly P → P (z -1 )z r .

(2) We first observe that Equation (3.1) uniquely determines P in terms of P since α, δ, and βz + α are invertible in k[z, z -1 ]/(z r+1 ) (because αδ 0). We then check that if Equation (3.1) is true for P , Q ∈ k[z] ≤r , then it is true for all µP + νQ with µ, ν ∈ k. We thus only need to check it for P = r i=0 u i v r-i • z i , where (u, v) ∈ k 2 . By the definition of the action, we get P = σ (P ) = r i=0 (αu + βv) i (δv) r-i • z i , which yields

(βz + α) r+1 • P (z) = r i=0 u i v r-i z i (βz + α) r+1 , α(βz+α) r δ r • P δz βz+α = α(βz+α) r δ r • r i=0 (αu + βv) i (δv) r-i • δz βz+α i = r i=0 α(αu + βv) i (βz + α) r-i v r-i z i .
Comparing the coefficients of z p for p = 0, . . . , r -1, we need to prove that

p i=0 u i v r-i r+1 p-i β p-i α r+1-(p-i) = p i=0 (αu + βv) i v r-i r-i p-i β p-i α r+1-p for each p = 0, . . . , r -1. Comparing the coefficients of u k v r-k , we get r+1 p-k β p-k α r+1-(p-k) = p i=k i k β i-k α k r-i
p-i β p-i α r+1-p , so the result follows from the next combinatoric lemma. Lemma 3.4.4. For all integers 0 ≤ k ≤ p ≤ r, we have

r + 1 p -k = p i=k i i -k r -i p -i .
Proof. The sides of the equations count the number of elements of the sets

S = {C ⊂ {0, . . . , r} | C contains p -k elements}, S = (i, A, B) i ∈ {k, . . . , p}, A ⊂ {0, . . . , i -1}, B ⊂ {i + 1, . . . , r}, A contains i -k elements, B contains p -i elements ,
so it remains to prove that the map S → S, (i, A, B) → A ∪ B is bijective. This corresponds to showing that for each C ∈ S, there exists a unique i ∈ {k, . . . , p} \ C such that C ∩ {0, . . . , i -1} contains ik elements. This is because the map

τ : {k, . . . , p} → N, i → (i -k) -|C ∩ {0, . . . , i -1}|
is non-decreasing, is increasing outside C, and satisfies τ(k) ≤ 0 ≤ τ(p).

Lemma 3.4.5. Let a ≥ 0, b ≥ 1, c ≥ 2. Let π : Z b,c,P a → F a be the P 1 -bundle induced by P (y 0 , y 1 , z) = b i=0 y i 0 y b-i 1 P i (z)z ai+1 with P i ∈ k[z] ≤c-2-ai (Notation 3.3.4). For each ϕ ∈ Aut • (F a ), the P 1 -bundle ϕπ is isomorphic to Z b,c, P a → F a , where P is defined by P (y 0 , y 1 , z) = b i=0 y i 0 y b-i 1 Pi (z)z ai+1 , with Pi ∈ k[z, z -1 ]
given as follows:

(1) If ϕ([y 0 : y 1 ; z 0 : z 1 ]) = [y 0 : y 1 ; αz 0 + βz 1 : γz 0 + δz 1 ] for some σ = α β γ δ ∈ GL 2 , then Pi = σ (P i ) for each i, where the action of GL 2 on the vector space k[z] ≤c-ai is that of Definition 3.4.1.

(2) If a = 0 and ϕ([y 0 : y 1 ; z 0 : z 1 ]) = [αy 0 +βy 1 ; γy 0 +δy 1 : z 0 : z 1 ] for some α, β, γ, δ ∈ k with αδ-βγ 0, then P (y 0 , y 1 , z) satisfies P (y 0 , y 1 , z) = P (αy 0 + βy 1 , γy 0 + δy 1 , z).

(3) If a ≥ 1 and ϕ([y 0 : y 1 ; z 0 : z 1 ]) = [y 0 : y 1 + y 0 R(z 0 , z 1 ) : z 0 : z 1 ] for some R ∈ k[z 0 , z 1 ] a , then P is such that P (y 0 , y 1 , z) = P (y 0 , y 1 + y 0 R(z, 1), z).

Proof. Recall that the transition function is given by ν c,P ∈ Aut(F b × A 1 \ {0}), ν c,P : [x 0 : x 1 ; y 0 : y 1 ], z → [x 0 : x 1 z c + x 0 P (y 0 , y 1 , z); y 0 z a : y 1 ], 1 z and that the morphism π : Z b,c,P a → F a is given on the two charts by τ 0 , τ 1 : F b × A 1 → F a , which send ([x 0 : x 1 ; y 0 : y 1 ], z) onto, respectively, [y 0 : y 1 ; 1 : z] and [y 0 :

y 1 ; z : 1].
We construct a transition function ν c,P of the P 1 -bundle ϕπ : X → F a for some ϕ ∈ Aut(F a ) and show that it is equivalent to that of P (modulo the equivalence described in Lemma 3.3.6). To do this, we find corresponding birational maps θ 0 and θ 1 compatible with the following commutative diagram:

F b × A 1 θ 0 r r ν c,P G G τ 0 ' ' F b × A 1 . τ 1 Ó Ó θ 1 r r F b × A 1 ν c,P G G τ 0 ( ( F b × A 1 τ 1 Ó Ó F a ϕ q q F a
(1) When ϕ([y 0 : y 1 ; z 0 : z 1 ]) = [y 0 : y 1 ; αz 0 + βz 1 : γz 0 + δz 1 ] for some σ = on the two charts. To check that the action we gave on the moduli space is the right one, we only need to check it for generators of GL 2 .

(i) We first do the case where γ = 0 (upper-triangular matrices). The second chart of F a is preserved, and ϕ corresponds to

F b × A 1 -→ F b × A 1 ([x 0 : x 1 ; y 0 : y 1 ], z) → [x 0 : x 1 : δ a y 0 : y 1 ], αz+β δ
and to choose a transition function P (y 0 , y 1 , z) ∈ k[y 0 , y 1 , z] such that θ 0 = (ν c,P ) -1 θ 1 ν c,P is a local isomorphism at each point where z = 0. We compute that θ 0 ([x 0 : x 1 ; y 0 : y 1 ], z) is equal to

x 0 : βz + α δ c x 1 + x 0 z c R(y 0 , y 1 , z) : (βz + α) a y 0 : y 1 , δz βz + α ,
where R(y 0 , y 1 , z) = P (y 0 , y 1 , z) -P y 0 (βz + α) a , y 1 , δz βz+α . We then only need to choose the transition function P (y 0 , y 1 , z) ∈ k[y 0 , y 1 , z] so that θ 0 is a local isomorphism at each point where z = 0, which corresponds to saying that the valuation of z at R is at least c. We will observe that such a P exists in the equivalence class of P . Writing P (y 0 , y 1 , z) = b i 0 y i 0 y b-i 1 P i (z)z ai+1 , we get

P (y 0 βz + α) a , y 1 , δz βz+α = b i=0 y 0 i y 1 b-i δ ai+1 P i δz βz+α βz+α z ai+1 ,
and then we need to choose the P i such that δ ai+1 P i (δz/(βz+α)) βz+α -P i (z) has valuation at least c at z. Since

βz + α is invertible in k[z]/(z c-ai-1
), we find a unique solution in k[z]/(z c-ai-1 ), equal to α δ c-1 σ (P i ) by Lemma 3.4.3(2).

(ii) It remains to consider the case where σ = 0 1 1 0 . The two charts are exchanged here, so it suffices to choose ν c,P = (ν c,P ) -1 , which yields P (y 0 , y 1 , z) = -P (y 0 z a , y 1 , z -1 )z c , which is equivalent to

P y 0 z a , y 1 , 1 z z c = b i=0 (y 0 z a ) i y b-i 1 P i 1 z 1 z ai+1 z c = b i=0 y i 0 y b-i 1 P i 1 z z c-ai-2 z ai+1 Lemma 3.4.3(1) = b i=0 y i 0 y b-i 1 σ (P i ) z ai+1 .
(2) Now suppose a = 0 and that ϕ([y 0 : y 1 ; z 0 : z 1 ]) = [αy 0 + βy 1 ; γy 0 + δy 1 : z 0 : z 1 ]. Choosing θ 1 = θ 2 : ([x 0 : x 1 ; y 0 : y 1 ], z) → ([x 0 : x 1 : αy 0 +βy 1 : γy 0 +δy 1 ], z), we get ν c, P θ 1 = θ 2 ν c,P with P (y 0 , y 1 , z) = P (αy 0 + βy 1 , γy 0 + δy 1 , z).

(3) Now suppose a > 0 and ϕ([y 0 : y 1 ; z 0 :

z 1 ]) = [y 0 : y 1 + y 0 R(z 0 , z 1 ) : z 0 : z 1 ] for some R ∈ k[z 0 , z 1 ] a .
We then choose θ 1 : ([x 0 : x 1 ; y 0 : y 1 ], z) → ([x 0 : x 1 : y 0 : y 1 + y 0 R(z, 1)], z) and θ 2 : ([x 0 : x 1 ; y 0 : y 1 ], z) → ([x 0 : x 1 : y 0 : y 1 + y 0 R(1, z)], z) and get ν c, P θ 1 = θ 2 ν c,P with P (y 0 , y 1 , z) = P (y 0 , y 1 + y 0 R(z, 1), z). Corollary 3.4.6. Let a, b, c ∈ Z, with a ≥ 0, b ≥ 1, and c ≥ 2. We recall that GL 2 acts on k[z] ≤r (see Definition 3.4.1) and on k[y 0 , y 1 ] b when a = 0 (see Lemma 3.4.5(2)). We have the following equivariant isomorphisms: We now assume b = c -2 and define a non-degenerate bilinear form φ as follows:

(1) If a ≥ 1, then M b,c a P b i=0 y i 0 y b-i 1 • k[z] ≤c-2-ai as GL 2 -varieties. (2) If a = 0, then as GL 2 × GL 2 -varieties, M b,c 0 P k[y 0 , y 1 ] b ⊗ k[z] ≤c-2 = P Hom((k[y 0 , y 1 ] b ) * , k[z] ≤c-2 ) . Moreover, if b = c -2, then Hom GL 2 ((k[y 0 , y 1 ] b ) * , k[z] ≤b ) = {λI; λ ∈ k},
φ : k[y 0 , y 1 ] b × k[z] ≤b → k b j=0 c j y j 0 y b-j 1 , b i=0 d i z i → b l=0 c l d l .
The map φ is GL 2 -invariant. Indeed, by bilinearity, it suffices to check that for all g ∈ GL 2 , for all j = 0, . . . , b, and for all (u, v) ∈ k 2 , we have

φ        g • y j 0 y b-j 1 , g • b i=0 u i v b-i z i        = φ        y j 0 y b-j 1 , b i=0 u i v b-i z i        = u j v b-j .
This can be checked directly for g = 0 1 1 0 and g = α β 0 δ . Since these elements generate GL 2 , the GL 2invariance of φ follows.

Therefore, the GL 2 -representations k[y 0 , y 1 ] b and k[z] ≤b are dual to each other. Since they are both irreducible (a consequence of Remark 3.4.2), it follows from Schur's lemma that

Hom GL 2 ((k[y 0 , y 1 ] b ) * , k[z] ≤b ) = {λI; λ ∈ k}. Identifying k[z] ≤b with (k[y 0 , y 1 ] b ) * via
the map φ above, we see that {1, z, . . . , z b } is the dual basis of {y b 1 , y 0 y b-1 1 , . . . , y b 0 }. Hence the identity element corresponds to b i=0 y i 0 y b-i 1 z i as an element of k[y 0 , y 1 ] b ⊗ k[z] ≤b , and so it corresponds to the P 1 -bundle Z b,b+2,P a , with P as in the statement of the corollary.

Remark 3.4.7. In the proof of Corollary 3.4.6, we use the fact that the ground field is of characteristic zero to say that the GL 2 -representations k[y 0 , y 1 ] b and k[z] ≤b are irreducible (this is used to prove the last part of the statement). The statement of the corollary is true over any algebraically closed field of arbitrary characteristic, but a general proof is slightly more complicated since we can no longer invoke Schur's lemma to prove that Hom GL 2 ((k[y 0 , y 1 ] b ) * , k[z] ≤b ) = {λI; λ ∈ k}, and we have to do the calculation by hand (using Lemma 3.4.5).

The P 1 -bundles Ŝb

→ P 1 × P 1
We now study two families of non-decomposable P 1 -bundles X → F a , in § 3.5 and § 3.6, which will play an important role further on (see Proposition 3.7.4). Definition 3.5.1. For each integer b ≥ 1, we define Ŝb → F 0 = P 1 × P 1 to be the P 1 -bundle Z b,b+2,P 0 → F 0 , where P = b i=0 y i 0 y b-i 1 P i (z)z and P i (z) = z i . Remark 3.5.2. The P 1 -bundle Ŝb → F 0 naturally arises in Corollary 3.4.6(2), which explains why the image of Aut

• ( Ŝb ) → Aut • (F 0 ) contains the diagonal group H ∆ = {(g, g) | g ∈ PGL 2 } ⊂ PGL 2 × PGL 2 = Aut • (F 0 ).
Lemma 3.5.5 below makes this explicit and shows that Aut • ( Ŝb ) PGL 2 .

Remark 3.5.3. The P 1 -bundle Ŝb → F 0 has numerical invariants (0, b, b + 2) since it is equal to Z b,b+2,P 0 for some polynomial P (see Remark 3.3.5). Remark 3.5.4. We will prove in Lemma 4.2.4 that the P 1 -bundle Ŝb of Definition 3.5.1 coincides with the lift of the Schwarzenberger bundle S b → P 2 of Definition 1.2.7. Lemma 3.5.5. Let b ≥ 1 be an integer, and let us denote by π, π the P 1 -bundles π : Ŝb → P 1 × P 1 and π : F b+1,b+1 0 → P 1 × P 1 . Then, the following hold:

(1) For each i = 1, 2, denoting by pr i : P 1 × P 1 → P 1 the i-th projection, the morphism pr i π : Ŝb → P 1 is a F b -bundle. Denoting by S i ⊂ Ŝb the union of the (-b)-curves of the F b , the intersection C = S 1 ∩ S 2 is a curve isomorphic to the diagonal ∆ ⊂ P 1 × P 1 via π. It corresponds to the intersection of π -1 (∆) with the surface x 0 = 0 in both charts.

(2) We have Aut • ( Ŝb ) PGL 2 and a commutative diagram

X η C C t t Ŝb ψ G G B B F b+1,b+1 0 , s s P 1 × P 1
where all maps are PGL 2 -equivariant, the action of PGL 2 on P 1 × P 1 is the diagonal one, the action of PGL 2 on F b+1,b+1 0 is given by (3.2) [x 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ] → [x 0 : x 1 ; αy 0 + βy 1 : γy 0 + δy 1 ; αz 0 + βz 1 :

γz 0 + δz 1 ],
the morphism is the blow-up of the curve C, and the morphism η is the blow-up of the curve C ⊂ F b+1,b+1 0 given by

P 1 → F b+1,b+1 0 , [u : v] → [1 : 1 : u : v : u : v].
Moreover, every automorphism of the P 1 -bundle Ŝb → P 1 × P 1 is trivial.

(3) If char(k) does not divide b + 1, the curve C is the unique curve invariant by Aut • ( Ŝb ).

Proof. We write m = b + 1. The fact that (3.2) yields an action of PGL 2 on F m,m 0 follows from the fact that [x 0 : x 1 ; λy 0 : λy 1 ; λz 0 : λz 1 ] = [λ m x 0 : λ m x 1 ; y 0 : y 1 ; z 0 : z 1 ] = [x 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ] for each λ ∈ k * . The same argument shows that the map P 1 → F m,m 0 , [u : v] → [1 : 1 : u : v : u : v] is a well-defined closed embedding. The image C is sent to the diagonal ∆ ⊂ P 1 × P 1 , and the action on P 1 × P 1 (via the P 1 -bundle F m,m 0 → P 1 × P 1 ) is the diagonal action. We now construct a birational map ϕ : F m,m 0 Ŝm-1 = Ŝb . The two open subsets U 0 , U 1 ⊂ F m,m 0 where z 0 0 and z 1 0, respectively, are isomorphic to

F m × A 1 via F m × A 1 → F m,m 0 ι 0 : [x 0 : x 1 ; y 0 : y 1 ], z → [x 1 : x 0 ; y 0 : y 1 ; 1 : z], ι 1 : [x 0 : x 1 ; y 0 : y 1 ], z → [x 1 : x 0 ; y 0 : y 1 ; z : 1].
The transition function θ = (ι 1 ) -1 ι 0 ∈ Bir(F m × A 1 ) is then given by

[x 0 : x 1 ; y 0 : y 1 ], z → [x 0 : x 1 z m ; y 0 : y 1 ], 1 z ,
and the curve C ⊂ F m,m 0 yields a section of F m × A 1 → A 1 , given by A 1 → F m × A 1 , z → ([1 : 1; 1 : z], z
) and z → ([1 : 1; z : 1], z), respectively, on the two charts. We can then blow up C and contract the strict transform of π -1 (π (C )); we do it on the two charts via

F m × A 1 F m × A 1 ϕ 0 : [x 0 : x 1 ; y 0 : y 1 ], z → [x 0 (y 0 z -y 1 ) : x 1 -x 0 y m 0 ; y 0 : y 1 ], z , ϕ 1 : [x 0 : x 1 ; y 0 : y 1 ], z → [x 0 (y 0 -y 1 z) : x 1 -x 0 y m 1 ; y 0 : y 1 ], z . Computing the transition function θ = ϕ 1 θ (ϕ 0 ) -1 ∈ Bir(F b × A 1 ), we obtain [x 0 : x 1 ; y 0 : y 1 ], z → x 0 : x 1 z m+1 + x 0 z y m 0 z m -y m 1 y 0 z -y 1 ; y 0 : y 1 , 1 z ,
which is the transition function of the P 1 -bundle Ŝb → P 1 × P 1 (see Definition 3.5.1). Since C and π -1 (π (C )) = π -1 (∆) (where ∆ ⊂ P 1 × P 1 is the diagonal) are invariant by PGL 2 , the birational map ϕ is PGL 2 -equivariant, for some biregular action of PGL 2 on Ŝb , acting diagonally on P 1 × P 1 , and preserving the curve C ⊂ Ŝb being the image of the contracted surface π -1 (π (C )). Note that this curve is given by the intersection of x 0 = 0 with π -1 (∆) in both charts, where ∆ ⊂ P 1 × P 1 is the diagonal (this follows by replacing y 0 and y 1 by 1 and z in ϕ 0 and by z and 1 in ϕ 1 ). This yields the commutative diagram of (2), with ψ = ϕ -1 . To achieve the proof of (2), we only need to show that the homomorphism PGL 2 → Aut • ( Ŝb ) constructed by this map is surjective.

The second projection pr 2 : P 1 × P 1 satisfies that pr 2 π : Ŝb → P 1 is a F b -bundle, trivial on P 1 \ [0 : 1] and P 1 \ [1 : 0] with transition function θ. The union of the (-b)-curves of the F b is a surface S 2 ⊂ Ŝb , which corresponds to x 0 = 0 on both charts and is then sent by ψ onto the surface S 2 ⊂ F m,m 0 given by x 1 = 0.

The involution σ ∈ Aut(F m,m 0 ) given by σ : [x 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ] → [x 1 : x 0 ; z 0 : z 1 ; y 0 : y 1 ] commutes with PGL 2 and preserves C and π -1 (π (C)); hence σ = ψ -1 σ ψ = ϕσ ϕ -1 ∈ Aut( Ŝb ). Since σ and σ act on P 1 × P 1 by the exchange of the two factors, pr 1 π : Ŝb → P 1 is also an F b -bundle. The union of the (-b)-curves of the F b is a surface S 1 = σ (S 2 ) ⊂ Ŝb , which is then sent by ψ onto the surface S 1 ⊂ F m,m 0 given by x 0 = 0. The two surface S 1 , S 2 ⊂ F m,m 0 are disjoint and also disjoint from C . Their strict transforms on X are then again disjoint, and their images on Ŝb intersect only along C. This yields (1).

To prove (2), it remains to show that Aut • ( Ŝb ) PGL 2 and that every automorphism of the P 1bundle Ŝb → P 1 × P 1 is trivial. Every element of Aut • ( Ŝb ) permutes the fibres of the two F b -bundles pr 1 π, pr 2 π : Ŝb → P 1 , so S 1 and S 2 are both invariant, and the same holds for C = S 1 ∩ S 2 . Since π(C) = ∆ ⊂ P 1 × P 1 , the image of Aut • ( Ŝb ) → Aut • (P 1 × P 1 ) is the diagonal PGL 2 . It then suffices to see that every automorphism of the P 1 -bundle Ŝb → P 1 × P 1 is trivial. This amounts to showing that every automorphism of the P 1 -bundle F m,m 0 → P 1 × P 1 that fixes C is trivial. Indeed, every automorphism of the P 1 -bundle F m,m 0 → P 1 × P 1 preserves S 1 and S 2 , so is of the form [x 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ] → [λx 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ] for some λ ∈ k * . It preserves C if and only if λ = 1.

We finish the proof by proving (3). It follows from the construction that C = S 1 ∩ S 2 ⊂ Ŝb is invariant by Aut • ( Ŝb ). It remains to show that every curve ⊂ Ŝb invariant by Aut • ( Ŝb ) is equal to C when char(k) does not divide b + 1. The action of Aut • ( Ŝb ) on P 1 × P 1 being the diagonal action of PGL 2 , we find that π( ) is the diagonal ∆ and thus have to see that C is the only curve invariant by the action of Aut • ( Ŝb ) PGL 2 on the surface V = π -1 (∆), which is a P 1 -bundle V → ∆. To do this, it suffices to find an isomorphism V -→ P 1 × P 1 that sends C onto the diagonal. Indeed, the action of PGL 2 on P 1 × P 1 will then have to be the diagonal action, which only preserves the diagonal.

We restrict the transition function of Ŝb to V and get

[x 0 : x 1 ; 1 : z], z → [x 0 : x 1 z b+2 + x 0 (b + 1)z b+1 ; 1 : z], 1 z = [x 0 : x 1 z 2 + x 0 (b + 1)z; 1 z : 1], 1 z .
The curve C corresponds to x 0 = 0 on both charts. The isomorphism V -→ P 1 × P 1 can then be chosen on the two charts as

[x 0 : x 1 ; 1 : z], z → [x 1 : x 0 (b + 1) + x 1 z], [1 : z] , [x 0 : x 1 ; z : 1], z → [-(b + 1)x 0 + x 1 z : x 1 ], [z : 1] ,
which is an isomorphism since b + 1 0 (as we assumed that char(k) does not divide b + 1).

Umemura P 1 -bundles

In this section we introduce a new class of non-decomposable P 1 -bundles on Hirzebruch surfaces. To the best of the authors' knowledge, those appeared for the first time in the work of Umemura [Ume88, § 10], and that is the reason why we chose to call them Umemura bundles. Definition 3.6.1 (Umemura bundles). Let a, b ≥ 1 and c ≥ 2 be such that c = ak + 2 with 0 ≤ k ≤ b. We call Umemura P 1 -bundle the P 1 -bundle U b,c a → F a given by U b,c a = Z b,c,P a → F a with P = y k 0 y b-k 1 z c-1 . Remark 3.6.2. Recall (see Notation 3.3.4) that U b,c a = Z b,c,P a is obtained by the gluing of two copies of

F b × A 1 along F b × A 1 \ {0} by the automorphism ν ∈ Aut(F b × A 1 \ {0}), ν : ([x 0 : x 1 ; y 0 : y 1 ], z) → [x 0 : x 1 z c + x 0 y k 0 y b-k 1 z c-1 ; y 0 z a : y 1 ], 1 z = [x 0 : x 1 z c-ab + x 0 y k 0 y b-k 1 z c-ab-1 ; y 0 : y 1 z -a ], 1 z ,
and that U b,c a → F a sends ([x 0 : x 1 ; y 0 : y 1 ], z) ∈ F b × A 1 onto, respectively, [y 0 : y 1 ; 1 : z] ∈ F a and [y 0 : y 1 ; z : 1] ∈ F a on the two charts. It then has numerical invariants (a, b, c). Proof. The statement corresponds to showing that the element of M b,c a corresponding to U b,c a is fixed by the whole group Aut • (F a ). We only need to check it for generators of Aut • (F a ), using Lemma 3.4.5.

Recall that the polynomial P is given by P (y 0 , y 1 , z) = b i=0 y i 0 y b-i 1 P i (z)z ai+1 , with P i ∈ k[z] ≤c-2-ai (see Notation 3.3.4), where all P i are zero except one, namely P k , which is equal to 1. Lemma 3.4.5(1) shows that GL 2 fixes the class of U b,c a in M b,c a since P k is sent onto σ (P k ) ∈ k[z] ≤c-2-ak = k[z] ≤0 = k. Lemma 3.4.5(3) then shows that an automorphism [y 0 : y 1 ; z 0 : z 1 ]) = [y 0 : y 1 + y 0 R(z 0 , z 1 ) : z 0 : z 1 ] of F a sends P onto the polynomial P equivalent to the polynomial P (y 0 , y 1 , z) that satisfies y k 0 y b-k 1 z c-1 = P (y 0 , y 1 , z) = P (y 0 , y 1 + y 0 R(z, 1), z). The polynomial Pk is then equal to P k = 1, and all the polynomials Pi with i < k are zero and thus equal to P i . The other coefficients do not appear in the equivalence class since cai -2 < 0 for these. Remark 3.6.4. With the notation above, the group of automorphisms of P 1 -bundles of U b,c a identifies with the vector group b i=1 k[z 0 , z 1 ] ai-c whose action on U b,c a can be described on the first chart by the following biregular action:

F b × A 1 → F b × A 1 [x 0 : x 1 ; y 0 : y 1 ], z → [x 0 : x 1 + x 0 b i=1 y i 0 y b-i 1 q i (z, 1); y 0 : y 1 ], z , where Q = (q i ) i=1,...,b ∈ b i=1 k[z 0 , z 1 ] ai-c
. This can be computed for instance by using the transition function of U b,c a . The action on π -1 (s -a ), which corresponds to y 0 = 0 on both charts, is then trivial.

Remark 3.6.5. The group GL 2 acts on U b,c a by acting rationally on both charts via, respectively,

F b × A 1 → F b × A 1 ([x 0 : x 1 ; y 0 : y 1 ], z) → [x 0 : x 1 (βz+α) c αδ-βγ +x 0 β(βz+α) c-1 y k 0 y b-k 1 αδ-βγ ; y 0 (βz + α) a ; y 1 ], δz+γ βz+α , ([x 0 : x 1 ; y 0 : y 1 ], z) → [x 0 : x 1 (γz+δ) c αδ-βγ -x 0 γ(γz+δ) c-1 y k 0 y b-k 1 αδ-βγ
; y 0 (γz + δ) a ; y 1 ],

αz+β γz+δ as can directly be checked using the transition function.

Invariant fibres

The following result shows that one can reduce the study of P 1 -bundles X → F a to the case where the action of Aut • (X) on F a is transitive on the set on fibres of τ a : F a → P 1 . This is in particular the case when the action on F a yields a surjective group homomorphism Aut • (X) Aut • (F a ) (see Remark 2.4.5) and holds for decomposable P 1 -bundles (see Lemma 3.1.5), for the P 1 -bundles Ŝb → P 1 × P 1 (see Lemma 3.5.5), and for Umemura bundles (see Lemma 3.6.3). Lemma 3.7.1. Let π : X → F a be a P 1 -bundle. If there is a point p ∈ P 1 such that the surface τπ -1 (p) ⊂ X is invariant by Aut • (X) (where τ a : F a → P 1 is the standard P 1 -bundle), then there exist a decomposable bundle F b,c a → F a and a commutative diagram

X ψ G G π A A F b,c a F a ,
where ψ is a birational map satisfying ψ Aut

• (X)ψ -1 Aut • (F b,c a ).
Proof. The surface S = F a \ (τ a ) -1 (p) is isomorphic to P 1 × A 1 , and U = π -1 (S) is invariant by Aut • (X). Applying Proposition 3.2.2, one can perform finitely many Aut • (X)-equivariant birational maps and reduce to the case where τ a π : U → P 1 \ {b} is a trivial F b -bundle. We then find an integer c ≥ 0 and an inclusion U → F b,c a such that the action of Aut • (X) extends to a biregular action on F b,c a . This will yield a birational map as above, satisfying ψ Aut • (X)ψ -1 ⊂ Aut • (F b,c a ). Moreover, equality does not hold since Aut • (F b,c a ) acts transitively on the set of fibres of τ a : F a → P 1 (see Lemma 3.1.5 and Remark 2.4.5).

If b > 0, the action of every element g ∈ Aut • (X) on U corresponds to an automorphism of F b × A 1 of the form

[x 0 : x 1 ; y 0 : y 1 ], z → [x 0 : αx 1 + x 0 b i=0 y i 0 y b-i 1 p i (z); f 1 (y 0 , y 1 , z), f 2 (y 0 , y 1 , z)], az + b for some α ∈ k * and p i ∈ k[z],
where f 1 , f 2 ∈ k[y 0 , y 1 , z] correspond to the coordinates of the restriction of an automorphism of F a . The group Aut • (X) being an algebraic group, the degrees of the polynomials p i are bounded by an integer which does not depend on g ∈ Aut • (X). It then suffices to take an integer c > 0 big enough and to send ([x 0 : x 1 ; y 0 :

y 1 ], z) ∈ F b × A 1 to [x 0 : x 1 ; y 0 : y 1 ; z : 1] ∈ F b,c
a to be able to extend the action of all elements g ∈ Aut • (X) to F b,c a . (This can be checked for instance using the description of F b,c a provided in § 3.) If b = 0, the action of every element g ∈ Aut • (X) on U corresponds similarly to an automorphism of

F 0 × A 1 = P 1 × P 1 × A 1 of the form [x 0 : x 1 ; y 0 : y 1 ], z → [α(z)x 0 + β(z)x 1 : γ(z)x 0 + δ(z)x 1 ; f 1 (y 0 , y 1 , z), f 2 (y 0 , y 1 , z)], az + b , where α, β, γ, δ ∈ k[z] are such that α β γ δ ∈ PGL 2 (k[z]) (i.e. αδ -βγ ∈ k * ) and where f 1 , f 2 ∈ k[y 0 , y 1 , z]
correspond to the coordinates of the restriction of an automorphism of F a .

The morphism U → P 1 × A 1 , ([x 0 : x 1 ; y 0 : y 1 ], z) → ([x 0 : x 1 ], z) then yields an algebraic group homomorphism Aut • (X) → Aut(P 1 × A 1 ). The image H ⊂ Aut(P 1 × A 1 ) is then a connected algebraic subgroup of Bir(P 1 × A 1 ) that preserves the set of fibres P 1 × A 1 . There thus exist an integer c ≥ 0 and an inclusion P 1 × A 1 → F c which allows the image to extend. We then find an inclusion U → F b,c a which allows the action of Aut We now show that H is simple. Let N be a normal subgroup of H. As p i |H is onto, p i (N ) is a normal subgroup of PGL 2 . If p i (N ) = PGL 2 for some i, then N = H (as they have the same dimension). Otherwise, p 1 (N ) = p 2 (N ) = {1}, and so N = {1}. This achieves to prove that H is a simple group.

As dim(H) = 3 and H is of rank at most 2, the classification of simple root systems yields that H is isomorphic to SL 2 or PGL 2 ; see [START_REF] Humphreys | Linear algebraic groups[END_REF]§ 32 and Appendix]. Since p i |H : H → PGL 2 is a bijective morphism, H cannot be isomorphic to SL 2 (otherwise ±I 2 would be sent to I 2 ), and so H PGL 2 . In this case, H is conjugate to the diagonal embedding of PGL 2 in PGL 2 × PGL 2 . Indeed, this follows from the fact that a bijective morphism of algebraic groups is an isomorphism in characteristic zero (e.g. by Zariski's main theorem), together with the fact that all automorphisms of PGL 2 are inner [Dem65, Theorem 1. for n ∈ Z. Proposition 3.7.4. Let a ≥ 0, and let π : X → F a be a P 1 -bundle. Then, there exist b, c ∈ Z such that one of the following holds:

(1) X is isomorphic to a decomposable P 1 -bundle F b,c a → F a (Definition 3.1.1); (2) X is isomorphic to an Umemura P 1 -bundle U b,c a → F a (Definition 3.6.1);

(3) we have a = 0, and (X, π) is square isomorphic to the P 1 -bundle Ŝb → F 0 of Definition 3.5.1; or (4) there exist a P 1 -bundle τ : F a → P 1 and a closed point p ∈ P 1 such that (τπ) -1 (p) is invariant by Aut • (X).

In cases (3) and (4), there exist a decomposable bundle F b,c a → F a and a commutative diagram

X ψ G G π A A F b,c a F a ,
where

ψ is a birational map satisfying ψ Aut • (X)ψ -1 Aut • (F b,c a ).
Proof. Lemma 3.7.1 and Lemma 3.5.5 (2) give the existence of the birational map ψ : X F b,c a in cases (3) and (4), with ψ Aut • (X)ψ -1 ⊂ Aut • (F b,c a ). Moreover, we cannot have equality since the action of Aut • (F b,c a ) → Aut(F a ) is surjective (see Lemma 3.1.5). It remains to show that we can reduce to the four cases above.

We denote by

H ⊂ Aut • (F a ) the image of Aut • (X) by the natural homomorphism Aut • (X) → Aut • (F a ) (see Lemma 2.1.1).
If the preimage of one fibre of τ a : F a → P 1 is invariant by Aut • (X), we get case (4). We can in particular assume that all fibres of τ a π : X → P 1 are isomorphic to F b for the same b ≥ 0 (no jumping fibre; see Proposition 3.2.2). Hence, X → P 1 has numerical invariants (a, b, c) for some c ∈ Z, which is positive if b = 0 (see Proposition 3.3.1 and Definition 1.4.1). We can moreover assume that X → F a is not a decomposable bundle since otherwise we obtain case (1). This implies that b ≥ 1 and c ≥ 2 and that X → F a is isomorphic to Z b,c,P a → F a , where P (y 0 , y 1 , z) = b i=0 y 0 i y 1 b-i P i (z)z ai+1 and P i (z) ∈ k[z] ≤c-2-ai for i = 0, . . . , b (see Corollary 3.3.7).

For each h ∈ Aut • (F a ), we have h ∈ H if and only if there exists an ĥ ∈ Aut • (X) such that the following diagram commutes:

(3.3) X ĥ G G π X π F a h G G F a .
This is thus equivalent to asking that the class of X → F a in M b,c a (see Corollary 3.3.8) is fixed by the action of h.

We now consider two cases, depending on whether a ≥ 1 or a = 0. Case a ≥ 1. We denote by i 0 the smallest integer such that P i 0 0. For each element σ ∈ PGL 2 , there exists an element of H ⊆ Aut(F a ) whose action on P 1 , via τ, corresponds to σ in PGL 2 . We can write this element as ϕ 1 ϕ 2 , where ϕ 1 , ϕ 2 ∈ Aut(F a ) are given by ϕ 1 : [y 0 : y 1 ; z 0 : z 1 ] → [y 0 : y 1 ; αz 0 + βz 1 : γz 0 + δz 1 ],

ϕ 2 : [y 0 : y 1 ; z 0 : z 1 ] → [y 0 : y 1 + y 0 R(z 0 , z 1 ) : z 0 : z 1 ],
where σ = α β γ δ ∈ GL 2 represents the class σ ∈ PGL 2 and R ∈ k[z 0 , z 1 ] a . Lemma 3.4.5 describes the action of ϕ 1 and ϕ 2 on the class [π]. The element ϕ 2 does not change the polynomial P i 0 (see Lemma 3.4.5(3)), so σ (P i 0 ) has to be equal to a multiple of P i 0 , for the action of GL 2 on k[z] ≤c-ai 0 given in Definition 3.4.1 (see Lemma 3.4.5(1)). This implies that the class of P i 0 in P(k[z] ≤c-2-ai 0 ) is fixed by the corresponding action of PGL 2 . This happens if and only if

(k[z] ≤c-2-ai 0 ) SL 2 is non-zero. As k[z] ≤c-2-ai 0 is an irreducible SL 2 -representation (see Remark 3.4.2), (k[z] ≤c-2-ai 0 ) SL 2 is non-zero if and only if k[z] ≤c-2-
ai 0 is the trivial representation; that is, c -2 = ai 0 and P i 0 is a constant polynomial. Since i 0 is the smallest integer such that P i 0 0, we have P i = 0 for all i < i 0 . Moreover, P i = 0 for i > i 0 since c -2ai < 0. This implies that P (y 0 , y 1 , z) = λy i 0 0 y b-i 0 1 z ai 0 +1 , and so that X → F a is an Umemura bundle (see Definition 3.6.1). Case a = 0. Let a = 0 and π : X → P 1 × P 1 be a P 1 -bundle. If H = PGL 2 × PGL 2 = Aut • (F 0 ), then the moduli space M b,c 0 must contain a fixed point for the natural H-action, described in Corollary 3.4.6(2). This cannot happen since the SL 2 × SL 2 -representation k[y 0 , y 1 ] b ⊗ k[z] ≤c-2 is irreducible, and non-trivial when b ≥ 1 (as we assumed above). Now assume that H PGL 2 × PGL 2 is a proper subgroup. If one of the two projections H → PGL 2 is not onto, then one fibre of a projection P 1 × P 1 → P 1 is invariant, and we get case (4). We then assume that H surjects onto PGL 2 via both projections. By Lemma 3.7.2, H is conjugate to

H ∆ := {(h, h) | h ∈ PGL 2 } in PGL 2 × PGL 2 . If H is conjugate to H ∆ ,
((k[y 0 , y 1 ] b ) * , k[z] ≤c-2 ) = 0 when b c -2. On the other hand, if b = c -2, then Hom SL 2 ((k[y 0 , y 1 ] b ) * , k[z] ≤c-2 ) = {λId; λ ∈ k},
and so M b,b+2 0 has a unique fixed point corresponding to the identity; the latter is given by P (y 0 , y 1 , z) = b i=0 y i 0 y b-i 1 z i+1 (see Corollary 3.4.6(2)) and yields case (3).

Remark 3.7.5. In the proof of Proposition 3.7.4, the assumption that the base field k is of characteristic zero is required. Indeed, the results from the representation theory of SL 2 that we use in the proof are not valid in positive characteristic. In positive characteristic, there are actually more P 1 -bundles to consider.

P 1 -bundles over P 2

The results in § 4.1 are valid over an algebraically closed field k of arbitrary characteristic, but in § 4.2 we need to assume char(k) 2 (due to the fact that we work with a quadratic form and need 2 to be invertible). In § 4.3, Lemma 4.3.1 and Proposition 4.3.4 both rely on Proposition 3.7.4, and so are valid only in characteristic zero, while Lemma 4.3.3 holds in characteristic different from 2.

Decomposable bundles over P 2

In this section, we give an explicit description of the decomposable P 1 -bundles over P 2 , similar to the one provided in § 3.1 for the P 1 -bundles over the Hirzebruch surfaces. We also give global coordinates on decomposable P 1 -bundles over P 2 . Definition 4.1.1. Let b ∈ Z. Define P b to be the quotient of (A 2 \ {0}) × (A 3 \ {0}) by the action of (G m ) 2 given by

(G m ) 2 × (A 2 \ {0}) × (A 3 \ {0}) → (A 2 \ {0}) × (A 3 \ {0}) (µ, ρ), (y 0 , y 1 ; z 0 , z 1 , z 2 ) → (µρ -b y 0 , µy 1 ; ρz 0 , ρz 1 , ρz 2 ).
The class of (y 0 , y 1 , z 0 , z 1 , z 2 ) will be written [y 0 : y 1 ; z 0 : z 1 : z 2 ]. The projection

P b → P 2 , [y 0 : y 1 ; z 0 : z 1 : z 2 ] → [z 0 : z 1 : z 2 ]
identifies P b with b) as a P 1 -bundle over P 2 . As before, we get an isomorphism of P 1 -bundles P b P -b by exchanging y 0 with y 1 and will then often assume b ≥ 0 in the following. Proof. The existence of ρ is given by Lemma 2.1.1. The fact that it is surjective can be seen by making GL 3 act naturally on P b , do nothing on y 0 , y 1 , and act naturally on z 0 , z 1 , z 2 .

P O P 2 (b) ⊕ O P 2 = P O P 2 ⊕ O P 2 (-
Remark 4.1.3. Assume b ≥ 1. The group of automorphisms of P 1 -bundle of P b identifies with the (connected) group

1 0 p λ ∈ GL 2 (k[z 0 , z 1 , z 2 ]) λ ∈ k * and p ∈ k[z 0 , z 1 , z 2 ] b
whose action on P b is as follows:

(λ, p) • [y 0 : y 1 ; z 0 : z 1 : z 2 ] = [y 0 : λy 1 + y 0 p(z 0 , z 1 , z 2 ); z 0 : z 1 : z 2 ].
This can be seen directly from the global description of P b in Definition 4.1.1, and by using trivialisations on open subsets isomorphic to A 2 .

Schwarzenberger P 1 -bundles over P 2

In this subsection, we study the Schwarzenberger P 1 -bundles S b → P 2 , with b ≥ -1, given by

P(κ * O P 1 ×P 1 (-b -1, 0)) → P 2 (
U 0 = {[X : Y : Z]|X 0} A 2 , U 1 = {[X : Y : Z]|Z 0} A 2 ,
the restriction of S b on P 2 \ {[0 : 1 : 0]} is obtained by gluing P 1 × U 0 and P 1 × U 1 along P 1 × (U 0 ∩ U 1 ) via the isomorphism given by θ :

P 1 × U 0 P 1 × U 1 x 0 x 1 , [1 : u : v] → α 11 (u, v) α 12 (u, v) α 21 (u, v) α 22 (u, v) x 0 x 1 , 1 v : u v : 1 , where α ij (u, v) ∈ k[u, v] are the polynomials satisfying α 11 (s + t, st) α 12 (s + t, st) α 21 (s + t, st) α 22 (s + t, st) =                1 0 0 -st if b = -1, 1 s -t s b -t b st(s b-1 -t b-1 ) s b+1 -t b+1 st(s b -t b ) if b ≥ 0. Proof. Recall that S b = P(κ * O P 1 ×P 1 (-m, 0))
, where m = b + 1 and κ is given by

κ : P 1 × P 1 → P 2 , [y 0 : y 1 ], [z 0 : z 1 ] → [y 0 z 0 : y 0 z 1 + y 1 z 0 : y 1 z 1 ]
(see Definition 1.2.7). The preimages of U 0 , U 1 ⊂ P 2 by κ are then two open subsets T 0 = κ -1 (U 0 ) and T 1 = κ -1 (U 1 ) of P 1 × P 1 isomorphic to A 2 using the standard coordinates s 0 =

y 1 y 0 , t 0 = z 1 z 0 , s 1 = y 0 y 1 , t 1 = z 0 z 1
in k(P 1 × P 1 ):

T 0 = κ -1 (U 0 ) = [y 0 : y 1 ], [z 0 : z 1 ] ∈ P 1 × P 1 | y 0 z 0 0 = Spec k[s 0 , t 0 ] , T 1 = κ -1 (U 1 ) = [y 0 : y 1 ], [z 0 : z 1 ] ∈ P 1 × P 1 | y 1 z 1 0 = Spec k[s 1 , t 1 ] .
The line bundle O P 1 ×P 1 (-m, 0) is trivial on T 0 and T 1 , so O P 1 ×P 1 (-m, 0) is the gluing of two copies of A 3 , via the transition function

(A 1 \ {0} × A 1 \ {0}) × A 1 -→ (A 1 \ {0} × A 1 \ {0}) × A 1 (s 0 , t 0 , a 0 ) → 1 s 0 , 1 t 0 , a 0 s m 0 , which corresponds to the identifications a 1 = a 0 s m 0 , s 1 = 1 s 0 , t 1 = 1 t 0
. This transition function implies that a section on T 0 ∩ T 1 correspond on the first chart to an element f (s 0 , t 0 ) ∈ k[s 0 ±1 , t 0 ±1 ] and on the second chart to s -m 1 f ( 1

s 1 , 1 t 1 ) ∈ k[s 1 ±1 , t 1 ±1 ].
To compute the transition on U 0 and U 1 , we take standard coordinates

u 0 = Y X , v 0 = Z X , u 1 = Y Z , v 1 = X Z on U 0 = Spec(k[u 0 , v 0 ]) and U 1 = Spec(k[u 1 , v 1 ]
). We then observe that for i = 1, 2, the morphisms

κ | T i : T i → U i corresponds to the injective algebra morphism k[U i ] = k[u i , v i ] → k[T i ] = k[s i , t i ]
that sends u i and v i onto s i + t i and s i t i , respectively.

The space of sections

κ * O P 1 ×P 1 (-m, 0)(U i ) = O P 1 ×P 1 (-m, 0)(T i ) k[T i ] = k[s i , t i ] is a free k[U i ]
-module of rank 2 generated by 1 and ξ i = s it i . This basis being chosen, it determines a transition function, which is of the form

U 0 × A 2 U 1 × A 2 u 0 , v 0 , b 0 b 1 → u 0 v 0 , 1 v 0 , α 11 (u 0 , v 0 ) α 12 (u 0 , v 0 ) α 21 (u 0 , v 0 ) α 22 (u 0 , v 0 ) • b 0 b 1 for some α ij ∈ k[u 0 , v 0 ±1 ].
We then take a section on U 0 ∩ U 1 , which is given on the first chart by

f 0 (u 0 , v 0 ), f 1 (u 0 , v 0 ) ∈ k[u 0 , v 0 ±1
] and on the second chart by

g 0 (u 1 , v 1 ) = α 11 u 1 v 1 , 1 v 1 f 0 u 1 v 1 , 1 v 1 + α 12 u 1 v 1 , 1 v 1 f 1 u 1 v 1 , 1 v 1 ∈ k[u 1 , v 1 ±1 ], g 1 (u 1 , v 1 ) = α 21 u 1 v 1 , 1 v 1 f 0 u 1 v 1 , 1 v 1 + α 22 u 1 v 1 , 1 v 1 f 1 u 1 v 1 , 1 v 1 ∈ k[u 1 , v 1 ±1 ].
The corresponding section on T 0 ∩ T 1 is then given on the two charts by

f (s 0 , t 0 ) = f 0 (s 0 + t 0 , s 0 t 0 ) + (s 0 -t 0 )f 1 (s 0 + t 0 , s 0 t 0 ) ∈ k[s ±1 0 , t ±1 0 ], g(s 1 , t 1 ) = s -m 1 f 1 s 1 , 1 t 1 = g 0 (s 1 + t 1 , s 1 t 1 ) + (s 1 -t 1 )g 1 (s 1 + t 1 , s 1 t 1 ) ∈ k[s ±1 1 , t ±1 1 ]
. We then use the equalities

2g 0 (s 1 + t 1 , s 1 t 1 ) = g(s 1 , t 1 ) + g(t 1 , s 1 ) ∈ k s ±1 1 , t ±1 1 , 2g 1 (s 1 + t 1 , s 1 t 1 ) = g(s 1 ,t 1 )-g(t 1 ,s 1 ) s 1 -t 1 ∈ k[s ±1 1 , t ±1 1 ], g 0 u 0 v 1 , 1 v 1 = α 11 (u 0 , v 1 )f 0 (u 0 , v 1 ) + α 12 (u 0 , v 1 )f 1 (u 0 , v 1 ) ∈ k[u 0 , v 1 ±1 ], g 1 u 0 v 1 , 1 v 1 = α 21 (u 0 , v 1 )f 0 (u 0 , v 1 ) + α 22 (u 0 , v 1 )f 1 (u 0 , v 1 ) ∈ k[u 0 , v 1 ±1 ]
to compute the α ij :

2g 0 s 0 +t 0 s 0 , 1 s 0 t 0 = g 1 s 0 , 1 t 0 + g 1 t 0 , 1 s 0 = s m 0 f (s 0 , t 0 ) + t m 0 f (t 0 , s 0 ), = (s m 0 + t m 0 )f 0 (s 0 + t 0 , s 0 t 0 ) + (s m 0 -t m 0 )(s 0 -t 0 )f 1 (s 0 + t 0 , s 0 t 0 ), 2g 1 s 0 +t 0 s 0 t 0 , 1 s 0 t 0 = -g 1 s 0 , 1 t 0 -g 1 t 0 , 1 s 0 s 0 t 0 s 0 -t 0 = -s m 0 f (s 0 , t 0 ) -t m 0 f (t 0 , s 0 ) s 0 t 0 s 0 -t 0 , = -s 0 t 0 s m 0 -t m 0 s 0 -t 0 f 0 (s 0 + t 0 , s 0 t 0 ) -s 0 t 0 (s m 0 + t m 0 )f 1 (s 0 + t 0 , s 0 t 0 ) and get α 11 (s 0 + t 0 , s 0 t 0 ) α 12 (s 0 + t 0 , s 0 t 0 ) α 21 (s 0 + t 0 , s 0 t 0 ) α 22 (s 0 + t 0 , s 0 t 0 ) = 1 2       s m 0 + t m 0 (s m 0 -t m 0 )(s 0 -t 0 ) -s 0 t 0 s m 0 -t m 0 s 0 -t 0 -s 0 t 0 (s m 0 + t m 0 )       . If m = 0,
s m + t m (s m -t m )(s -t) -st s m -t m s-t -st(s m + t m ) 2 s + t 0 -1 = 4st t -s s m-1 -t m-1 st(s m-2 -t m-2 ) s m -t m st(s m-1 -t m-1
) .

by

P 1 × T 0 P 1 × T 1 x 0 x 1 , [1 : s], [1 : t] → 1 s -t s b -t b st(s b-1 -t b-1 ) s b+1 -t b+1 st(s b -t b ) • x 0 x 1 , 1 s : 1 , 1 t : 1
(see Lemma 4.2.1). We then define two open embeddings ι i :

P 1 × T i → F b × A 1 , i = 0, 1: P 1 × T 0 ι 0 → F b × A 1 [x 0 : x 1 ], ([1 : s], [1 : t]) → [-x 0 -tx 1 : x 1 ; 1 : s], t , P 1 × T 1 ι 1 → F b × A 1 [x 0 : x 1 ], ([s : 1], [t : 1]) → [-x 0 + tx 1 : x 1 ; s : 1], t and compute ι 1 θ(ι 0 ) -1 ∈ Bir(F b × A 1 ): F b × A 1 ι 1 θ(ι 0 ) -1 F b × A 1 [x 0 : x 1 ; 1 : y], z → y b z x 0 : y b+1 -z b+1 y-z x 0 + z b+1 x 1 ; 1 y : 1 , 1 z = [x 0 : x 1 z b+2 + x 0 z b i=0 y i z b-i ; 1 : y], 1 z This then yields ν b,P ι 0 = ι 1 θ, where ν b,P ∈ Aut(F b × A 1 \ {0}) is given by ν b,P : [x 0 : x 1 ; y 0 : y 1 ], z → [x 0 : x 1 z b+2 + x 0 P (y 0 , y 1 , z); y 0 : y 1 ], 1 z , with P (y 0 , y 1 , z) = b i=0 y i 0 y b-i 1 P i (z)z and P i (z) = z i ∈ k[z] ≤b for i = 0, . . . , b.
This shows that the P 1 -bundle is isomorphic to Ŝb → P 1 × P 1 over T 0 ∪ T 1 (see Definition 3.5.1) and thus over the whole P 1 × P 1 by Lemma 2.3.1. This achieves the proof of (1).

(2) By Lemma 3.5.5(3), there is a unique curve C ⊂ Ŝb invariant by Aut • ( Ŝb ) which corresponds to the intersection of π -1 (∆) with the surface x 0 = 0 in both charts P 1 × A 1 . It then corresponds to the curve given in P 1 × T 0 and P 1 × T 1 by

[x 0 : x 1 ], ([1 : t], [1 : t]) ∈ P 1 × T 0 | x 0 + tx 1 = 0 , [x 0 : x 1 ], ([t : 1], [t : 1]) ∈ P 1 × T 1 | x 0 -tx 1 = 0 .
Sending this curve into S b yields a bijection C → D, where D ⊂ S b is given locally as above.

Lemma 4.2.5. Let π : S b → P 2 be the b-th Schwarzenberger P 1 -bundle, with b ≥ 1, and let Γ ⊂ P 2 be the conic which is the ramification locus of κ : P 1 × P 1 → P 2 (see Definition 1.2.7).

(1)

If L ⊂ P 2 is a line, the P 1 -bundle π -1 (L) π → L P 1 is isomorphic to F b → P 1 if L is a tangent line to Γ ; F 0 → P 1 if L is a not a tangent line to Γ and b is even; or F 1 → P 1 if L is a not a tangent line to Γ and b is odd.
(2) The action of Aut • (S b ) on P 2 yields a group isomorphism

Aut • (S b ) -→ Aut(P 2 , Γ ) = {g ∈ Aut(P 2 ) | g(Γ ) = Γ } PGL 2 if b ≥ 2; or Aut(P 2 ) PGL 3 if b = 1.
Moreover, every automorphism of the P 1 -bundle S b → P 2 is trivial.

Proof. Lemma 4.2.4 implies that π : Ŝb → P 1 × P 1 is isomorphic to S b × P 2 (P 1 × P 1 ) → P 1 × P 1 . Lemma 3.5.5(2) then yields Aut • ( Ŝb ) PGL 2 , with an action on P 1 × P 1 which is the diagonal one, and thus corresponds, via κ : P 1 × P 1 → P 2 , to Aut(P 2 , Γ ) PGL 2 . This implies, in particular, that the image of the group homomorphism Aut • (S b ) → Aut(P 2 ) (given by Lemma 2.1.1) contains Aut(P 2 , Γ ) PGL 2 .

Taking a line L ⊂ P 2 tangent to Γ , the preimage κ -1 (L) ⊂ P 1 × P 1 consists of two fibres f 1 , f 2 of the two projections, exchanged by the involution associated to the double cover κ. In particular, π -1 (L) is isomorphic to π-1 (f 1 ) π-1 (f 2 ) F b since Ŝb has numerical invariants (0, b, b + 2).

To finish the proof of (1), we take a line L ⊂ P 2 not tangent to Γ and show that π -1 (L) is isomorphic to F 0 or F 1 , depending on whether b is even or odd. Using the action of Aut • (S b ), we can choose the line of equation Y = 0. Using the notation of Proposition 4.2.1, the restriction of S b → P 2 on P 2 \ {[0 : 1 : 0]} is given by gluing P 1 × U 0 and P 1 × U 1 with θ :

P 1 × U 0 P 1 × U 1 x 0 x 1 , [1 : u : v] → α 11 (u, v) α 12 (u, v) α 21 (u, v) α 22 (u, v) x 0 x 1 , 1 v : u v : 1 , where a ij (u, v) ∈ k[u, v] are the polynomials satisfying α 11 (s + t, st) α 12 (s + t, st) α 21 (s + t, st) α 22 (s + t, st) = 1 s -t s b -t b st(s b-1 -t b-1 ) s b+1 -t b+1 st(s b -t b ) .
We then observe that s+t divides s n -t n s-t if n is even (by replacing t with -s), and that s n -t n s-t ≡ s n-1 ≡ ±(st)

n-1 2 (mod s + t) if n is odd. This yields α 11 (0, v) α 12 (0, v) α 21 (0, v) α 22 (0, v) =                        0 ±v b 2 ±v b 2 0       if b is even,       ±v b-1 2 0 0 ±v b+1 2       if b
is odd and achieves the proof of (1).

To prove (2), we study the group homomorphism ρ : Aut(S b ) → Aut(P 2 ). As we observed, the image contains Aut(P 2 , Γ ) PGL 2 . It is then equal to Aut(P 2 , Γ ) if b ≥ 2 (this follows from (1)). If b = 1, then ρ is surjective since S 1 P(T P 2 ) (see Corollary 4.2.2 and Remark 4.2.3). To finish the proof of (2), it remains to show that every automorphism α of the P 1 -bundle S b → P 2 is trivial. The lift of α yields an element α ∈ Aut( Ŝb ) which acts trivially on P 1 × P 1 and thus is trivial by Lemma 3.5.5(2). Remark 4.2.6. Let Γ ⊂ P 2 be the smooth conic of Definition 1.2.7. Then the natural embedding of Aut(P 2 , Γ ) PGL 2 in Aut(P 2 ) = PGL 3 is the one induced from the injective group homomorphism

(4.1) GL 2 (k) → GL 3 (k), a b c d → 1 ad -bc          a 2 ab b 2 2ac ad + bc 2bd c 2 cd d 2          .
Also, PGL 2 acts on P 2 with two orbits, which are Γ and its complement. Indeed, the morphism κ of Definition 1.2.7 is PGL 2 -equivariant, where PGL 2 acts diagonally on P 1 × P 1 , and as P 1 × P 1 is the union of two orbits (namely, the diagonal and its complement), the result follows.

Remark 4.2.7. Let m = b +1 ≥ 2. By Lemma 4.2.5, the group PGL 2 acts on S b , and the (2 : 1)-cover Ŝb → S b is PGL 2 -equivariant. Hence, by Remark 3.5.7, S b has an open PGL 2 -orbit, say PGL 2 /F, where F is a finite subgroup scheme that fits into an exact sequence 0 → H → F → Z/2Z → 0 with H = α 0 0 α -1 ; α m = 1 . Actually, using (4.1), an explicit computation of the stabiliser of a general point in the P 1 -fibre over [0 : 1 : 0] ∈ P 2 yields that F is conjugate to H, 0 1 -1 0 ; see [Ume88, Lemma 4.5] for details.

Remark 4.2.8. Writing X = P(T P 2 ), Lemma 4.2.5 shows that the action of Aut • (X) on P 2 yields an isomorphism Aut • (X) -→ PGL 3 . This classical observation can also be seen as follows. Let B be a Borel subgroup of G = PGL 3 , and let P be a maximal parabolic subgroup of G containing B.

Reduction to P 1 -bundles of the four families

We first show that the Schwarzenberger and decomposable P 1 -bundles over P 2 are those which have large automorphism groups.

Lemma 4.3.1. Let π : X → P 2 be a P 1 -bundle, let Γ ⊂ P 2 be the conic of equation Y 2 = 4XZ, and let H ⊂ Aut(P 2 ) be the image of Aut • (X). Then, the following hold:

( It then remains to assume Aut(P 2 , Γ ) ⊂ H and deduce from it that π : X → P 2 is decomposable or isomorphic to S b → P 2 for some b ≥ 1. We use the double cover κ : P 1 × P 1 → P 2 ramified over Γ given in Definition 1.2.7 and define a P 1 -bundle π : X = X × P 2 (P 1 × P 1 ) → P 1 × P 1 . By construction, the image of Aut • ( X) → Aut • (P 1 × P 1 ) contains the group PGL 2 embedded diagonally, namely the subgroup of Aut • (P 1 × P 1 ) preserving the diagonal ∆, the branch locus of κ. This implies that π : X → P 1 × P 1 is either a decomposable P 1 -bundle or square isomorphic to Ŝb for some b ≥ 1 (see Proposition 3.7.4). In the latter case, the isomorphism of P 1 × P 1 that comes in the square has to preserve the diagonal and lifts to Aut • ( X) (see Lemma 3.5.5(2)), so π : X → P 1 × P 1 is in fact isomorphic to Ŝb . Denoting by σ ∈ Aut( X) the involution induced by the automorphism of P 1 × P 1 exchanging the two factors, we find X = X/σ .

We first assume that X → P 1 × P 1 is a decomposable P 1 -bundle, say F m 1 ,m 2 0

. For i = 1, 2, the fibres of pr i • π : F m 1 ,m 2 0 → P 1 are the Hirzebruch surfaces F |m i | , where pr i : P 1 × P 1 → P 1 is the i-th projection. As σ exchanges the fibres, we get m 1 = ±m 2 . We now prove that up to conjugation by an isomorphism of P 1 -bundles, we have one of the following situations for the action of σ on X:

σ : F m,-m 0 → F m,-m 0 [x 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ] → [x 0 : x 1 ; z 0 : z 1 ; y 0 : y 1 ], m ≥ 0; σ : F m,-m 0 → F m,-m 0 [x 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ] → [x 0 : -x 1 ; z 0 : z 1 ; y 0 : y 1 ], m ≥ 0; or σ : F m,m 0 → F m,m 0 [x 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ] → [x 1 : x 0 ; z 0 : z 1 ; y 0 : y 1 ], m ≥ 0.
If m 1 = m 2 = 0, then F 0,0 0 is isomorphic to (P 1 ) 3 , so σ is of the form [x 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ] → [ι(x 0 : x 1 ); z 0 : z 1 ; y 0 : y 1 ] for some element ι ∈ PGL 2 , which is either the identity or of order 2. Applying conjugation, we get one of the above cases (the last two cases being conjugate).

We then assume m 1 = m 2 = m > 0. The union of -m-curves in the fibres of pr i • π : F m,m 0 → P 1 is then given by x 0 = 0 and x 1 = 0. This implies that σ is of the form [x 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ] → [x 1 : ξx 0 ; z 0 : z 1 ; y 0 : y 1 ] for some ξ ∈ k * . Conjugating by an automorphism [x 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ] → [µx 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ], we can assume ξ = 1. This gives the third case.

The remaining cases are when m 1 = -m 2 = m > 0. The union of -m-curves in the fibres of pr i • π is then given by x 0 = 0, for i = 1, 2. This implies that σ is of the form [x 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ] → [x 0 : µx 1 + x 0 P (y 0 , y 1 , z 0 , z 1 ); z 0 : z 1 ; y 0 : y 1 ], where µ ∈ k * and P is of bidegree m, m in y 0 , y 1 , z 0 , z 1 . As σ is an involution, we get µ = ±1 and µP (y 0 , y 1 , z 0 , z 1 ) + P (z 0 , z 1 , y 0 , y 1 ) = 0. In both cases, conjugating by the automorphism [x 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ] → [x 0 : x 1 ± 1 2 x 0 P (y 0 , y 1 , z 0 , z 1 ); y 0 : y 1 ; z 0 : z 1 ], we can assume that P = 0 and obtain one of the first two cases.

We now study the three cases and show that only the first case can appear, in which case X is a decomposable P 1 -bundle over P 2 . In the first case, we obtain that X = F m,-m 0 /σ = P(O P 2 (-m) ⊕ O P 2 ) is a decomposable P 1 -bundle over P 2 . In the second case, we consider U := [1 : x; 1 : y; 1 : z] A 3 , which is a σ -stable affine open subset of F m,-m 0 . The corresponding action on A 3 is (x, y, z) → (-x, z, y), which is conjugate to (x, y, z) → (-x, -y, z). In the third case, we consider V := [x -1 : x + 1; 1 : y + z; 1 : -y + z] A 3 , which is a σ -stable affine open subset of F m,m 0 . The action is again given by (x, y, z) → (-x, -y, z). In both cases, the quotient is singular since the invariant algebra is k

[x 2 , xy, y 2 , z] = k[X 1 , X 2 , X 3 , X 4 ]/(X 1 X 3 -(X 2 ) 2 ), which is impossible since X is smooth.
It remains to study the case where X → P 1 × P 1 is isomorphic to Ŝb → P 1 × P 1 . By Lemma 4.2.4, X is equal to the pull-back of S b → P 2 , so there is an involution σ ∈ Aut( X), acting on P 1 × P 1 as the exchange of the two factors τ ∈ Aut(P 1 × P 1 ), such that X/σ = S b . Since every automorphism of the P 1 -bundle Ŝb → P 1 × P 1 is trivial (see Lemma 3.5.5(2)), σ = σ is the unique lift of τ, which shows that π : X → P 2 is isomorphic to S b → P 2 . Remark 4.3.2. If k = C, then Lemma 4.3.1(1) is a particular case of a classical result due to Van de Ven [START_REF] Van De Ven | On uniform vector bundles[END_REF]Theorem], and Lemma 4.3.1(2) follows from a result due to Vallès [START_REF] Vallès | Fibrés de Schwarzenberger et coniques de droites sauteuses[END_REF]Theorem].

The following result is well known to specialists. Here we give a very short proof relying on recent results by Laurent.

Lemma 4.3.3. If G Aut(P 2 ) = PGL 3 is a proper connected algebraic subgroup that does not fix any point of P 2 , then one of the following holds:

(1) G = Aut(P 2 , Γ ) PGL 2 , where Γ ⊂ P 2 is a smooth conic; or (2) G is a subgroup of Aut(P 2 , L) GL 2 k 2 , where L ⊂ P 2 is a line, equal either to Aut(P 2 , L) or to SL 2 k 2 ⊂ Aut(P 2 , L).

Moreover, in both cases, G acts on P 2 with two orbits: the smooth conic Γ in case (1) or the line L in case (2), and its open complement.

Proof. If G is solvable, then it acts on P 2 with a fixed point by the Borel fixed-point theorem (see [START_REF] Humphreys | Linear algebraic groups[END_REF]§ 21.2,Theorem]). If G is semisimple, then by [Lau18, Proposition 3.4.1], the only possibility is case (1). If G is not solvable nor semisimple, then by [Lau18, Proposition 3.4.4], the only possibility is case (2). The last sentence of the statement can be checked with a direct computation.

Proposition 4.3.4. Let π : X → P 2 be a P 1 -bundle. If X is not decomposable or isomorphic to a Schwarzenberger bundle S b → P 2 , for some b ≥ 1, then there exists a p ∈ P 2 such that the fibre

f = π -1 (p) is Aut • (X)-invariant. The blow-up η : X → X of f yields a commutative diagram with Aut • (X)-equivariant maps X η G G π X π F 1 τ G G P 2 ,
where X → F 1 is a P 1 -bundle over F 1 and τ is the blow-up of p.

Proof. Let H ⊂ PGL 3 be the image of the natural group homomorphism Aut • (X) → Aut(P 2 ). The assumption implies that H does not contain any group Aut(P 2 , Γ ) for some smooth conic Γ (see Lemma 4.3.1).

If H fixes a point p ∈ P 2 , we blow up simultaneously p in P 2 and f = π -1 (p) in X. Computing explicitly these blow-ups in a trivializing local chart containing p 0 , we see that X → F 1 is again a P 1 -bundle. Moreover, since p and f are Aut • (X)-stable in P 2 and X, respectively, the group Aut • (X) acts on X.

It remains to show that the case where no point of P 2 is fixed by H is impossible. By Lemma 4.3.3, this case can only happen if H = Aut(P 2 , L) GL 2 (k 2 , +) or H SL 2 (k 2 , +) Aut(P 2 , L), where L ⊂ P 2 is a line. We take a point p ∈ P 2 \ L and denote by G 0 and H 0 the subgroups of G = Aut • (X) and H stabilizing p (note that H 0 GL 2 or H 0 SL 2 ). Blowing up the point p and its fibre yields a P 1 -bundle X → F 1 equipped with a G 0 -action. As the group H 0 acts transitively on P 2 \ (L ∪ {p}) A 2 \ {0}, it acts transitively on the exceptional divisor of the blow-up in F 1 , and thus H 0 acts transitively on P 1 via the structure morphism F 1 → P 1 . Therefore, by Proposition 3.7.4, the P 1 -bundle X → F 1 is a decomposable or an Umemura P 1 -bundle. In both cases, the natural group homomorphism Ĝ = Aut • ( X) → Aut(F 1 ) is onto (see Lemmas 3.1.5 and 3.6.3), and so H 0 acts transitively on the complement of the exceptional section of F 1 (see Remark 2.4.4). Moreover, by Lemma 2.3.1, the group Ĝ identifies with a subgroup of G; in particular, G must act transitively on P 2 \ {p}, contradicting the equality H = Aut(P 2 , L).

The classification

In this section, we first reduce to four families of P 1 -bundles, in Proposition 5.1.1, which uses Propositions 3.7.4 and 4.3.4 and is thus valid only in characteristic zero. We then study equivariant square birational maps between the four families, in § § 5.2-5.7; the content of these subsections is valid over any algebraically closed field of characteristic different from 2. We then apply the results obtained in § § 5.1-5.7 to get the main result in § 5.8, valid only in characteristic zero.

Reduction to the four families

As a first step towards Theorem A, we prove the following.

Proposition 5.1.1. Let π : X → S be a P 1 -bundle over a smooth projective rational surface S. There exists an Aut • (X)-equivariant square birational map (X, π) (X , π ) such that (X , π ) is isomorphic to one of the following:

(a) a decomposable P 1 -bundle F b,c a -→ F a for some a, b ≥ 0, c ∈ Z; (b) a decomposable P 1 -bundle P b -→ P 2 for some b ≥ 0; (c) an Umemura P 1 -bundle U b,c a -→ F a for some a, b ≥ 1, c ≥ 2; (d) a Schwarzenberger P 1 -bundle S b -→ P 2 for some b ≥ 1.

Proof. Using the descent lemma (Lemma 2.3.2), we can assume S = P 2 or S = F a for some a ≥ 0.

In the case where S = F a , we apply Proposition 3.7.4 to reduce to the case of decomposable or Umemura bundles.

In the case where S = P 2 , we apply Proposition 4.3.4 to reduce to the case of decomposable Schwarzenberger bundles, or to the case of F 1 , already studied.

First links

Remark 5.2.1. We now study equivariant square birational maps from a P 1 -bundle X → S to another P 1 -bundle X → S , both in the above families. The action being equivariant, we get a birational map η : S S which conjugates the image H ⊂ Aut • (S) of Aut • (X) to a subgroup of Aut • (S ). Hence, η is a sequence of blow-ups of points fixed by H followed by a sequence of contractions of curves invariant by H. The nature of the pair (H, S) given above implies that no point of S is fixed and that the only (-1)-curve invariant by H is the exceptional curve of F 1 . We then only need to consider this case and study the P 1 -bundle over P 2 obtained (which is given by the descent lemma (Lemma 2.3.2)) and square isomorphisms/birational maps of P 1 -bundles (doing nothing on S).

We then observe that no decomposable P 1 -bundle over F 1 has a maximal group of automorphisms. base locus of ϕ is then given, in these coordinates, by u = 0 and x 0 = 0, corresponding to a curve ⊂ X such that π yields an isomorphism π| : → K. The blow-up of this curve followed by the contraction of the strict transform of the surface π -1 (K) is locally given by the matrix R. As det(M ) = λu n-1 , we proceed by induction and obtain the result. (1) The curve l 00 given by x 0 = y 0 = 0 is invariant if and only if ab > 0 or ac < 0.

Links between decomposable bundles over Hirzebruch surfaces

(2) The curve l 10 given by x 1 = y 0 = 0 is invariant if and only if ac > 0.

(3) No irreducible curve ⊂ F b,c a with l 00 and l 10 is invariant.

Proof. By Lemma 3.1.5, the morphism π : F b,c a → F a yields a surjective group homomorphism Aut

• (F b,c a ) Aut • (F a ). Hence, if ⊂ F b,c
a is a curve invariant by Aut • (F b,c a ), then π( ) = s -a and a > 0 (see Remarks 2.4.3 and 2.4.4). Moreover, G m acts on F b,c a via [x 0 : x 1 ; y 0 : y 1 ; z 0 , z 1 ] → [x 0 : tx 1 ; y 0 : y 1 ; z 0 : z 1 ], so any point of should satisfy x 0 = 0 or x 1 = 0 (otherwise we get a whole fibre of a point of s -a which is contained in ); hence has to be equal to l 00 or l 10 . This yields (3).

We can now assume a > 0 and show (1) and (2) by proving when l 00 and l 10 are invariant. The surface π -1 (s -a ) F c being invariant, the curve l 00 is invariant when c < 0 and l 10 is invariant when c > 0. Moreover, the fibres of the F b -bundle F b,c a → P 1 (given in Remark 3.1.4) are exchanged by Aut • (F b,c a ). If b > 0, the surface S -b given by x 0 = 0 is the union of the negative sections and is then invariant, so l 00 = S -b ∩π -1 (s -a ) is invariant.

It remains to show that l 10 and l 00 are not invariant in the other cases. If c ≤ 0, the group G a acts on F b,c a via (t, [x 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ]) → ([x 0 : x 1 + tx 0 y b 1 z -c 1 ; y 0 : y 1 ; z 0 : z 1 ]), so l 10 is not invariant. If b = 0 and c ≥ 0, then G a acts on F b,c a via (t, [x 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ]) → ([x 0 + tx 1 z c 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ]), so l 00 is not invariant. Lemma 5.4.2.

(1) For all a, b, c ∈ Z with a, b ≥ 0, the blow-up of the curve l 00 ⊂ F b,c a given by x 0 = y 0 = 0 followed by the contraction of the strict transform of the surface π -1 (s -a ) onto l 10 ⊂ F b+1,c+a a given by x 1 = y 0 = 0 yields a birational map ϕ :

F b,c a F b+1,c+a a [x 0 : x 1 ; y 0 : y 1 ; z 0 : z 1 ] → [x 0 : x 1 y 0 ; y 0 : y 1 ; z 0 : z 1 ] .

We then have ϕ Aut

• (F b,c a )ϕ -1 ⊂ Aut • (F b+1,c+a a
) if and only if ab > 0 or ac < 0, and we have ϕ

-1 Aut • (F b+1,c+a a )ϕ ⊂ Aut • (F b,c a ) if and only if a(c + a) > 0. (2) For all a, b, c ∈ Z with a, b ≥ 0, every Aut • (F b,c a )-equivariant birational map of P 1 -bundles F b,c
a X is a composition of birational maps as in (1) (and of their inverses) and of isomorphisms of P 1 -bundles.

Proof. As we can check in local coordinates, the birational map F b F b+1 given by [x 0 : x 1 ; y 0 : y 1 ] → [x 0 : x 1 y 0 ; y 0 : y 1 ] is the composition of the blow-up of the point [0 : 1; 0 : 1], followed by the contraction of the strict transform of y 0 = 0 onto the point [1 : 0; 0 : 1]. Doing this within a family yields ϕ. We then have ϕ Aut

• (F b,c a )ϕ -1 ⊂ Aut • (F b+1,c+a a ) if and only if Aut • (F b,c a ) preserves l 00 and ϕ -1 Aut • (F b+1,c+a a )ϕ ⊂ Aut • (F b,c a ) if and only if Aut • (F b+1,c+a a
) preserves l 10 . Hence, (1) follows from Lemma 5.4.1 for b ≥ 0. It remains to prove (2). By Lemma 5.3.1, we only need to consider elementary links F b,c a X, obtained by blowing up an invariant curve ⊂ F b,c a , followed by contracting π -1 (π( )). The only curves in F b,c a that are invariant are l 00 and l 10 (see Lemma 5.4.1), and the links associated to these are given in (1): if we start with l 00 , then it is equal to ϕ as in (1); if we start with l 10 , then it is equal to ϕ -1 as in (1) when b ≥ 1 and is the composition of the isomorphism F 0,c a F 0,-c a exchanging x 0 and x 1 with a link ϕ as in (1) if b = 0.

We recall that the notions of stiff and superstiff P 1 -bundle, used in the next result and later, were defined in the introduction (Definition 1.2.3).

Corollary 5.4.3. Let a, b ≥ 0 and c ∈ Z be such that c ≤ 0 when b = 0. Then, Aut • (F b,c a ) is maximal if and only if a 1 and one of the following holds:

(1) a = 0; i.e. F b,c a is a decomposable P 1 -bundle over

F 0 = P 1 × P 1 ; (2) b = c = 0; i.e. F b,c a is isomorphic to F 0,0 a F a × P 1 ; (3) -a < c < ab.
Moreover, F b,c a is superstiff in cases (1) and (2) and is not stiff in case (3). More precisely, denoting by r the smallest integer such that cra ≤ 0, we find r ≤ b and get an infinite sequence of elementary links

F b-r,c-ra a • • • F b,c a F b+1,c+a a • • • F b+n,c+an a . . . which conjugates Aut • (F b,c a ) to Aut • (F b+s,c+as a
) for each integer s ≥ -r. This gives all Aut • (F b,c a )-equivariant square birational maps from F b,c a to another P 1 -bundle.

Proof. If a = 1, then Aut • (F b,c a ) is not maximal, by Lemma 5.2.2. We can thus assume a 1. In this case, as we have a surjective group homomorphism Aut • (F b,c a ) Aut • (F a ), every Aut • (F b,c a )-equivariant square birational map starting from F b,c a is in fact the composition of an element of Aut • (F b,c a ) with an Aut • (F b,c a )equivariant birational map of P 1 -bundles. These are compositions of links given in Lemma 5.4.2(1) and isomorphisms of P 1 -bundles, as explained in Lemma 5.4.2(2).

If a = 0 or b = c = 0, then every Aut • (F b,c a )-equivariant birational map of P 1 -bundles starting from F b,c a is in fact an isomorphism of P 1 -bundles (see Lemma 5.4.2). This shows that F b,c a is superstiff and thus that Aut • (F b,c a ) is maximal in these cases, corresponding to (1) and (2). We then assume a ≥ 2 and suppose (b, c) (0, 0). ) by Lemma 5.4.1(1). We can now assume -a < c < ab. Denoting by r smallest integer such that cra ≤ 0, we find r ≤ b and get an infinite sequence of elementary links

Let us show that

F b-r,c-ra a • • • F b,c a F b+1,c+a a • • • F b+n,c+an a . . . which conjugates Aut • (F b,c a ) to Aut • (F b+s,c+as a
) for each integer s ≥ -r (see Lemma 5.4.2). Every Aut • (F b,c a )equivariant birational map of P 1 -bundles F b,c a X is a composition of birational maps such as these and of isomorphisms of P 1 -bundles since l 10 is not invariant by Aut • (F b-r,c-ra a ) (see Lemma 5.4.1(2)) as cra ≤ 0 and br ≥ 0.

Links between Umemura bundles over Hirzebruch surfaces

We can similarly treat the case of Umemura P 1 -bundles. Recall that such bundles U b,c a → F a are defined by positive integers a, b ≥ 1 and c ≥ 2 such that c = ak + 2 with 0 ≤ k ≤ b (see Definition 3.6.1 and Remark 3.6.2 for more details).

We start with the case where a = 1 and study the P 1 -bundle V b → P 2 obtained from U b,2 1 → F 1 as follows.

Lemma 5.5.1. Let η : F 1 → P 2 , [y 0 : y 1 ; z 0 : z 1 ] → [y 0 z 0 : y 1 : y 0 z 1 ] be the blow-up of [0 : 1 : 0], which induces an isomorphism between Û = η -1 (U ) and U = P 2 \ {[0 : 1 : 0]}.

(1) For each integer b ≥ 1, there exist a P 1 -bundle π : V b → P 2 , unique up to isomorphism of P 1 -bundles, and a birational morphism ψ : U b,2 1 → V b such that the following hold: (i ) ψ is a square birational map over η; (ii ) ψ induces an isomorphism π-1 ( Û ) -→ π -1 (U ); (iii ) ψ is the blow-up of the smooth rational curve π -1 ([0 : 1

: 0]) ⊂ V b . (2) If L ⊂ P 2 is a line, the P 1 -bundle π -1 (L) π → L P 1 is isomorphic to F b → P 1 if L is a line through [0 : 1 : 0]; F |b-2| → P 1 if L is a line not passing through [0 : 1 : 0]. (3) The image in Aut(P 2 ) of Aut • (V b ) is equal to Aut(P 2 ) if b = 1 and to Aut(P 2 , [0 : 1 : 0]) if b ≥ 2. (4) We have ψ Aut • (U b,2 1 )ψ -1 ⊂ Aut • (V b ), with equality if b ≥ 2.
Proof. The existence of a unique birational map ψ : U b,2 1 V b satisfying (1)(i )-(ii ) follows from the descent lemma (Lemma 2.3.2). We now prove that ψ also satisfies (iii ), which implies that ψ is a birational morphism. To do this, we denote by W ⊂ F 1 the open subset given by y 1 0 and show that π : U b,2 1 → F 1 is trivial over W . As W contains the exceptional curve s -1 ⊂ F 1 of η (given by y 0 = 0), this will show that one can contract π-1 (s -1 ) P 1 × P 1 and obtain a birational morphism having the desired properties. To show the triviality of π over W , we take the transition function of U b,2 1 , given by ν

∈ Aut(F b × A 1 \ {0}) as follows ν : [x 0 : x 1 ; y 0 : y 1 ], z → [x 0 : x 1 z 2 + x 0 y b 1 z; y 0 z : y 1 ], 1 z . The intersection of W with each chart is isomorphic to P 1 × A 2 , via the inclusion P 1 × A 2 → F b × A 1
given by ([x 0 : x 1 ], y, z) → [x 0 : x 1 ; y : 1], z), and the transition function becomes ([x 0 : x 1 ], y, z) → ([x 0 : x 1 z 2 + x 0 z], yz, z -1 ), which yields a trivial P 1 -bundle since 0 1 -1 z -1 • 1 0 z z 2 • 1 -z 0 1 = z 0 0 z . This achieves the proof of (1).

(2)-( 3)-(4) The existence and unicity of ψ and V b being proven, we then observe that ψ Aut • (U b,2 1 )ψ -1 ⊂ Aut • (V b ) also follows from the descent lemma (see Lemma 2.3.2). This shows in particular that the subgroup H b ⊂ Aut(P 2 ), being the image of Aut • (V b ), contains the group Aut(P 2 , [0 : 1 : 0]) GL 2 k 2 .

We now take the open subsets U 0 , U 1 ⊂ P 2 given by U

0 = {[X : Y : Z]|X 0} A 2 and U 1 = {[X : Y : Z]|Z 0} A 2 and observe that Ûi = η -1 (U i ) U i for i = 0, 1. Hence, π -1 (U i )
π-1 ( Ûi ), and the transition function is computed as follows: a point [1 : u : v] ∈ U 0 ∩ U 1 corresponds to the point [1 : u; 1 : v] ∈ F 1 , and thus its preimage to ([x 0 :

x 1 ; 1 : u], v) ∈ F b × A 1 on the first chart is sent onto ([x 0 : x 1 v 2 + x 0 u b v; v : u], v -1 ) = ([x 0 : x 1 v 2-b + x 0 u b v 1-b ; 1 : v -1 ], v -1
) on the second chart. The transition function is then given by

P 1 × U 0 P 1 × U 1 [x 0 : x 1 ], [1 : u : v] → [x 0 : x 1 v 2-b + x 0 u b v 1-b ], [ 1 v : u v : 1] . For b = 1,
we find the transition function of S 1 P(T P 2 ) (see Corollary 4.2.2), which yields V b S 1 and thus H 1 = Aut(P 2 ) (see Remark 4.2.3 or Lemma 4.2.5(2)). In particular, ψ Aut • (U b,2 1 )ψ -1 Aut • (V b ) in this case. Assertions (3) and (4) are then proven for b = 1.

We now prove (2) (for each b ≥ 1). We observe that if L passes through [0 : 1 : 0], its strict transform on F 1 is a fibre f of the P 1 -bundle F 1 → P 1 , so π-1 (f ) F b . Since π -1 (L) π-1 (f ) (because ψ is a blow-up of a curve), we obtain π -1 (L) F b . We now take a line L not passing through [0 : 1 : 0], use the action of Aut • (V b ) to restrict to the case where L is the line given by Y = 0, and set u = 0 in the transition function above to get π -1 (L) F |b-2| . Assertion (3) for b ≥ 2 is now given as follows: [0 : 1 : 0] has to be fixed by H b because of (2), and Aut(P 2 , [0 : 1 : 0]) ⊂ H b was already proven.

It remains to show (4), and then to prove that every element g ∈ Aut • (V b ) belongs to ψ -1 gψ ∈ Aut • (U b,2 1 ) when b ≥ 2. Assertion (3) implies that g preserves the curve π -1 ([0 : 1 : 0]), so this follows from (1)(iii ). (1) The curve l 00 given by x 0 = y 0 = 0 on both charts is invariant.

(2) The curve l 10 given by x 1 = y 0 = 0 on both charts is invariant if and only if k > 0 (i.e. when c > 2).

(3) These are the two curves (respectively, the only curve) of U b,c a invariant by Aut • (U b,c a ) if c > 2 (respectively, when c = 2).

Proof. Since a ≥ 1, the curve s -a ⊂ F a is invariant by Aut • (F a ); hence the surface π -1 (s -a ) is given by y 0 = 0 on both charts. The fibres of the F b -bundle U b,c a → P 1 are exchanged by Aut • (U b,c a ). Since b ≥ 1, the surface S -b given by x 0 = 0 is the union of the negative sections and is then invariant. This yields (1).

Recall that the transition function of U b,c a is given by ν : [x 0 : x 1 ; y 0 : y 1 ], z → [x 0 : x 1 z c + x 0 y k 0 y b-k 1 z c-1 ; y 0 z a : y 1 ], 1 z (see Remark 3.6.2). If k > 0, the surface π -1 (s -a ), corresponding to y 0 = 0, is isomorphic to F c , and the curve l 10 given by x 1 = y 0 = 0 on both charts corresponds to the curve s -c ⊂ F c (with c > 0) and is thus invariant.

It remains to see that l 10 is not invariant if k = 0 and that no curve distinct from l 10 or l 00 can be invariant.

(3) Let ⊂ U b,c a be an invariant curve. As the morphism π : U b,c a → F a yields a surjective group homomorphism Aut • (U b,c a ) Aut • (F a ) (see Lemma 3.6.3), and because a ≥ 1, we have π( ) = s -a . We then use the fact that has to be invariant by the GL 2 -action given explicitly in Remark 3.6.5. We consider the action of the upper-triangular group by taking γ = 0 and obtain that the image of ([x 0 : x 1 ; 0 : 1], 0), on the first chart, is equal to [x 0 : x 1 α c-1 δ ; 0 : 1], 0 if k > 0 and to [x 0 : x 1 α c-1 δ -

x 0 α c-2 β δ

; 0 : 1], 0 if k = 0. We then find that either ([0 : 1; 0 : 1], 0) ∈ or ([1 : 0; 0 : 1], 0) ∈ and k > 0. In the first case, we get = l 00 since l 00 is an orbit. In the second case, l 10 is an orbit and = l 10 . This achieves the proof.

Lemma 5.5.3.

(1) For each Umemura P 1 -bundle U b,c a → F a , the blow-up of the curve l 00 ⊂ U b,c a followed by the contraction of the strict transform of the surface π -1 (s -a ) onto the curve 10 yields a birational map ϕ : U b,c a U b+1,c+a a satisfying ϕ Aut • (U b,c a )ϕ -1 = Aut • (U b+1,c+a a ).

(2) For each a ≥ 1, we have a birational map ϕ : U 1,a+2 a F 0,2 a such that ϕ(Aut • (U 1,a+2 a ))ϕ -1 Aut • (F 0,2 a ).

Proof.

(1) For each a ≥ 1, b ≥ 0, and 0 ≤ k ≤ b, we write c = ak + 2, denote by ν b,c a ∈ Aut(F b × A 1 \ {0}) the birational map

F b × A 1 F b × A 1 ν b,c
a : [x 0 : x 1 ; y 0 : y 1 ], z → [x 0 : x 1 z c + x 0 y k 0 y b-k 1 z c-1 ; y 0 z a : y 1 ], 1 z , and observe that ν b,c a is the transition function of U b,c a if b ≥ 1 (see Remark 3.6.2). We then denote by ϕ b the birational map

F b × A 1
F b+1 × A 1 ϕ b : ([x 0 : x 1 ; y 0 : y 1 ], z) → ([x 0 : x 1 y 0 ; y 0 : y 1 ], z) and observe that ϕ b ν b,c a ϕ -1 b = ν b+1,c+a a . If b ≥ 1, the blow-up of l 00 ⊂ U b,c a , followed by the contraction of the strict transform of the surface π -1 (s -a ), yields a birational map given in the two charts by ϕ b . This then corresponds to a birational map U b,c a U b+1,c+a a , which is the blow-up of the curve l 00 ⊂ U b,c a followed by the contraction of the strict transform of the surface π -1 (s -a ). Since l 00 and π -1 (s -a ) are invariant by Aut • (U b,c a ) (see Lemma 5.5.2), we get ϕ Aut • (U b,c a )ϕ -1 ⊂ Aut • (U b+1,c+a a

). We then observe that ϕ -1 is the blow-up of l 10 ⊂ U b+1,c+a a followed by the contraction of the strict transform of the surface π -1 (s -a ). As l 10 is invariant by Aut • (U b+1,c+a a ), we obtain ϕ Aut • (U b,c a )ϕ -1 = Aut • (U b+1,c+a a

). This achieves the proof of (1). We now consider the above construction in the case b = 0 (which yields k = 0 and c = 2). The transition function ν b+1,c+a a still corresponds to the transition function of U b+1,c+a a = U 1,a+2 a , but the transition function ν b,c a corresponds to a transition function on a P 1 -bundle over F a with numerical invariants (a, b, c), which is therefore decomposable and isomorphic to F b,c a = F 0,2 a (see Proposition 3.3.1(2)). The maps (ϕ b ) -1 on both charts then yield an elementary link ϕ : U 1,a+2 a F 0,2 a centred at l 10 , which is then Aut • (U 1,a+2 a )equivariant. We moreover have ϕ(Aut • (U 1,a+2 a ))ϕ -1 Aut • (F 0,2 a ) since Aut • (F 0,2 a ) contains a torus of dimension 3 (see Remark 3.1.3), which is not the case for Aut • (U 1,a+2 a ) (as follows from Remark 3.6.4). This then achieves the proof of (2).

Corollary 5.5.4. Let U b,c a be an Umemura bundle. Then, Aut • (U b,c a ) is maximal if and only if one of the following holds:

(1) a ≥ 2 and cab < 2;

(2) a = 1 and cab < 1.

In this case, U b,c a is not stiff. More precisely, denoting by k the integer such that c = ak + 2, we find 0 ≤ k < b and get a sequence of birational maps )ψ -1 = Aut • (V b-k ) (see Lemma 5.5.1). It remains to show that we cannot get any further link. At the level of surfaces, the only Aut(F 1 )-equivariant birational maps F 1 S, where S is a smooth projective surface, are isomorphisms or blow-ups F 1 → P 2 of a point of P 2 . We then only need to show that there is no square birational map V b-k X, to a π-bundle X → P 2 , which is not a square isomorphism. We can reduce to the case of birational of P 1 -bundles (doing nothing on P 2 ) and then use Lemma 5.3.1, and only need to observe that no curve of P 2 is invariant by the action of Aut • (V b-k ), which acts as Aut(P 2 , [0 : 1 : 0]). The result then follows from Lemma 5.3.1. Proof. Let ρ : Aut • (S b ) → Aut(P 2 ) be the group homomorphism induced by π. If b = 1, then ρ is surjective (see Lemma 4.2.5(2)), so there is no curve of S b which is invariant. Now suppose b ≥ 2, in which case ρ yields an isomorphism Aut • (S b ) -→ Aut(P 2 , Γ ) PGL 2 (again by Lemma 4.2.5(2)). Every curve of S b is then contained in the invariant surface X = π -1 (Γ ) ⊂ S b . To understand the action of Aut • (S b ) PGL 2 on X, we use the corresponding action on the P 1 -bundle π : Ŝb = S b × P 2 (P 1 × P 1 ) → P 1 × P 1 , obtained by Lemma 4.2.4. The pull-back of X on Ŝb is the surface X = π(∆) ⊂ Ŝb (where ∆ ⊂ P 1 × P 1 is the diagonal), isomorphic to X via a PGL 2 -equivariant isomorphism. By Lemma 4.2.4, there is a unique curve in X invariant by Aut • ( Ŝb ) PGL 2 , which is sent onto the curve of X given locally as above. which correspond locally to the blow-up of D, followed by the contraction of the strict transform of π -1 (Γ ), in the two charts (see Lemma 5.6.1 for the equation of D). We then check that ϕ 1 θ = θϕ 0 , where θ : P 1 × U 0 P 1 × U 1 is the transition function of S b given in Lemma 4.2.1. This follows from the equality

U b-k,2 a ϕ -k U b-k+1,2+a a . . . ϕ -1 U b,c a ϕ 0 • • • ϕ n-1 U b+n,c+na

Links between Schwarzenberger bundles

-s -t 2 -2st s + t s b -t b st(s b-1 -t b-1 ) s b+1 -t b+1 st(s b -t b ) = s b + t b st(s b-1 + t b-1 ) s b+1 + t b+1 st(s b + t b ) = s b -t b st(s b-1 -t b-1 ) s b+1 -t b+1 st(s b -t b ) s + t 2st -2 -s -t
and then yields a birational map of P 1 -bundles ϕ : S b S b , given by the blow-up of D, followed by the contraction of the strict transform of π -1 (Γ ) onto D.

By Lemma 5.3.1, every birational map of P 1 -bundles S b → X which is not an isomorphism is a composition of ϕ with an isomorphism of P 1 -bundles. Proof. If b = 1, the P 1 -bundle has no invariant curves (see Remark 4.2.3) (it is actually a homogeneous variety), and we conclude by Lemma 5.3.1. If b ≥ 2, we apply Lemma 5.6.2.

Theorem B .

 . The P 1 -bundles of Theorem A are superstiff only in the cases: (a) F b,c a with a = 0 or b = c = 0; (b) P b for b ≥ 0; and (c) S 1 P(T P 2 ).

  see Definition 3.1.1) and that P b = P(O P 2 (b) ⊕ O P 2 ) = P(O P 2 ⊕ O P 2 (-b)) → P 2 (see Definition 4.1.1).

  we keep the equation AC = BD, do not change the degree of det(B) = det(C) or the set Z, and can then assume A(λ) = y m 0 0 y ñ . Writing B(λ) = β 11 β 12 β 21 β 22 yields C(λ) = A(λ) -1 B(λ)D = β 11 y m-m β 12 y n- m β 21 y m-ñ β 22 y n-ñ = β 11 y - β 12 y -b- β 21 y b+ β 22 y .

  Replacing B and C by B and C does not change the equation AC = BD, since D commutes with ∆, and decreases the degree of det(B). A similar argument works if the second column of B is zero.

  and reduce to case (a). (d) If = 0 and b > 0, then β 12 = 0 and β 22 ∈ k. If β 22 = 0, we do as above. If β 22 0, we get

Notation 3.3. 4 .

 4 Let a, b, c ∈ Z, with a, b ≥ 0, and c ≤ 0 if b = 0. For each

  respectively, [y 0 : y 1 ; 1 : z] ∈ F a and [y 0 : y 1 ; z : 1] ∈ F a on the two charts.Remark 3.3.5. Proposition 3.3.1 shows that every P 1 -bundle over F a with no jumping fibre is isomorphic to Z b,c,P a → F a for some b, c ∈ Z with b ≥ 0, and c ≤ 0 if b = 0, and someP ∈ k[y 0 , y 1 , z, z -1 ] b ,and that it has numerical invariants (a, b, c) (see Definition 1.4.1). Moreover, Z b,c,0 a → F a is isomorphic to F b,c a (see Remark 3.1.4). Lemma 3.3.6. Let π : Z b,c,P a → F a and π : Z b ,c ,P a → F a be two P 1 -bundles as in Notation 3.3.4, with b ≥ 1. Then the following are equivalent: (1) The P 1 -bundles π : Z b,c,P a → F a and π : Z b ,c ,P a → F a are isomorphic. (2) We have b = b and c = c, and there exist λ

Lemma 3.4. 3 .

 3 Let r ≥ 0, and let P ∈ k[z] ≤r .

  GL 2 , the action of ϕ on F a corresponds to [y 0 : y 1 ; 1 : z] → [y 0 : y 1 ; α + βz : γ + δz] = (α + βz) a y 0 : y 1 ; 1 : γ+δz α+βz , [y 0 : y 1 ; z : 1] → [y 0 : y 1 ; αz + β : γz + δ] = (γz + δ) a y 0 : y 1 ; αz+β γz+δ : 1

[y 0 :

 0 y 1 ; 1 : z] → [y 0 : y 1 ; α + βz : δz] = (α + βz) a y 0 : y 1 ; 1 : δz α+βz , [y 0 : y 1 ; z : 1] → [y 0 : y 1 ; αz + β : δ] = δ a y 0 : y 1 ; αz+β δ : 1 on the two charts. It then suffices to choose θ 1 :

  and the identity element corresponds to Z b,b+2,P a , with P = b i=0 y i 0 y b-i 1 P i (z)z and P i (z) = z i for i = 1, . . . , b. Proof. Assertion (1) and the first part of (2) follow from Corollary 3.3.8 and Lemma 3.4.5.

  Remark 3.5.6. When char(k) divides b + 1, there are actually two curves invariant by Aut • ( Ŝb ) PGL 2 . Remark 3.5.7. Let m = b + 1. An element g = α β γ δ ∈ PGL 2 sends p 0 = [1 : 1; 1 : 0; 0 : 1] ∈ F m,m 0 to [1 : 1; α : γ; β : δ]. Therefore, g • p 0 = p 0 if and only if β = γ = 0 and α m = δ m = 1, and so H := Stab p 0 PGL 2 = α 0 0 α -1 ; α m = 1 . As dim(PGL 2 /H) = 3, we see that PGL 2 /H is a dense open orbit for the diagonal action of PGL 2 on F m,m 0 . As Ŝb is PGL 2 -equivariantly birational to F m,m 0 , the same holds for Ŝb .

  Lemma 3.6.3. Let a, b ≥ 1 and c ≥ 2 be such that c = ak + 2 with 0 ≤ k ≤ b. The morphism π : U b,c a → F a yields a surjective group homomorphism ρ : Aut • (U b,c a ) Aut • (F a ).

  3 and § 3.6]. Remark 3.7.3. In characteristic p > 0, the above result is false: we get infinitely many embeddings of PGL 2 into PGL 2 × PGL 2 with pairwise distinct images, up to conjugation, given by a b c d → a b c d , a p n b p n c p n d p n

  then the moduli space M b,c 0 contains a fixed point for the natural H ∆action. By Corollary 3.4.6(2), M b,c 0 P(Hom((k[y 0 , y 1 ] b ) * , k[z] ≤c-2 ) as an H ∆ -variety, where we identify H ∆ = PGL 2 . Hence M b,c 0 contains a fixed point if and only if Hom SL 2 ((k[y 0 , y 1 ] b ) * , k[z] ≤c-2 ) {0}. As the SL 2 -representations (k[y 0 , y 1 ] b ) * and k[z] ≤c-2 are irreducible and of dimension b + 1 and c -1, respectively, it follows from Schur's lemma that Hom SL 2

Lemma 4.1. 2 .

 2 For each b ∈ Z, the morphism π : P b → P 2 yields a surjective group homomorphism ρ : Aut • (P b ) Aut(P 2 ) = PGL 3 .

  we get simply the matrix defined above. If m ≥ 1, we change the transition function, by observing that 1 s+t st 0 2

Lemma 5.4. 1 .

 1 Let a, b, c ∈ Z with a, b ≥ 0. The curves of F b,c a invariant by Aut • (F b,c a ) are given as follows:

F.

  Aut • (F b,c a ) is not maximal when c ≥ ab. Note that b > 0 in this case, by assumption (we suppose c < 0 when b = 0). Lemma 5.4.2 yields an Aut • (F b,c a )-equivariant birational map ψ : F b,c a F 0,c-ab a given as the composition of the links F b,c a By construction, ψ Aut • (F b,c a )ψ -1 ⊂ Aut • (F 0,c-ab a ) preserves the curve l 00 , which then implies that ψ Aut • (F b,c a )ψ -1 Aut • (F 0,c-ab a ) by Lemma 5.4.1(1). We now prove that Aut • (F b,c a ) is not maximal when c ≤ -a. In this case, we use the Aut • (F b,c a )-equivariant link ϕ : F b,c a F b+1,c+a a given by Lemma 5.4.2, which is made such that ϕ Aut • (F b,c a )ϕ -1 ⊂ Aut • (F b+1,c+a a ) preserves the curve l 10 and thus yields ϕ Aut • (F b,c a )ϕ -1 Aut • (F b+1,c+a a

Lemma 5.5. 2 .

 2 Let a, b ≥ 1 and c ≥ 2 be such that c = ak + 2 with 0 ≤ k ≤ b. The curves of U b,c a invariant by Aut • (U b,c a ) are given as follows:

  a ϕ n • • • such that ϕ n Aut • (U b+n,2+na a )ϕ -1 n = Aut • (U b+n+1,2+(n+1)a a) for each n ≥ -k. If a ≥ 2, this gives all Aut • (U b,c a )equivariant square birational maps from U b,c a to another P 1 -bundle. If a = 1, we add the birational morphismU b-k,2 a = U b-k,2 1 → V b-k,1 of Lemma 5.5.1.Proof. We denote as usual by k the integer satisfying 0 ≤ k ≤ b such that c = ak + 2 and find thatcab < 2 ⇔ k < b. If cab ≥ 2, then k = b, so c = ab + 2. We construct an Aut • (U b,c a )-equivariant birational map of P 1 -bundles U b,c a U 1,a+2 a which is the composition of birational maps U b,c a = U b,ab+2 a U b-1,a(b-1)+2 a • • • U 1,a+2 a (see Lemma 5.5.3(1)). Since Aut • (U 1,a+2 a) is not maximal (see Lemma 5.5.3(2)), neither is Aut • (U b,c a ). We now assume cab < 2, which means k < b. Lemma 5.5.3(1) yields a sequence of birational mapsU bsuch that ϕ n Aut • (U b+n,2+na a )ϕ -1 n = Aut • (U b+n+1,2+(n+1)a a ) for each n ≥ -k. If a ≥ 2, every Aut • (U b,c a )-equivariant birational map of P 1 -bundles U b,ca X is a composition of birational maps as these and of isomorphisms of P 1 -bundles since l 10 is not invariant by Aut • (U b-k,2 a ) (see Lemma 5.5.2). We then get the result in this case (a ≥ 2).It remains to do the case where a = 1, which yields c = k + 2 and cab= kb + 2. If cab = 1, then bk = 1, which implies that Aut • (U b-k,2 a) is not maximal (see Lemma 5.5.1) and thus neither is Aut • (U b,c a ). If cab < 1, then bk > 2. We thus get a birational morphism ψ : U b-k,2 a → V b-k which satisfies ψ Aut • (U b-k,2 1

Lemma 5.6. 1 .

 1 Let b ≥ 1, and let π : S b → P 2 be the b-th Schwarzenberger P 1 -bundle. If b = 1, no curve of S b is invariant by Aut • (S b ). If b ≥ 2, there is a unique curve invariant by Aut • (S b ), which is given on the two charts of Lemma 4.2.1 by[x 0 : x 1 ], ([1 : 2t : t 2 ]) ∈ P 1 × U 0 | x 0 + tx 1 = 0 , [x 0 : x 1 ], ([t 2 : 2t : 1]) ∈ P 1 × U 1 | x 0tx 1 = 0 .

Lemma 5.6. 2 .

 2 Let b ≥ 2, and let π : S b → P 2 be the b-th Schwarzenberger P 1 -bundle. There is a birational involution ϕ :S b S b such that ϕ Aut • (S b )ϕ -1 = Aut • (S b ). Moreover, every Aut • (S b )-equivariant birational map of P 1 -bundle S bX is either an isomorphism of P 1 -bundles or a composition of ϕ with an isomorphism of P 1 -bundles.Proof. By Lemma 5.6.1, there is a unique curve D ⊂ S b which is invariant by Aut • (S b ) and satisfies π(D) = Γ . We consider the birational involutionsϕ 0 : P 1 × U 0 P 1 × U 0 [x 0 : x 1 ], [1 : u : v] → [-ux 0 -2vx 1 : 2x 0 + ux 1 ], [1 : u : v] , ϕ 1 : P 1 × U 1 P 1 × U 1 [x 0 : x 1 ], [v : u : 1] → [ux 0 -2vx 1 : 2x 0vx 1 ], [v : u : 1]

  Corollary 5.6.3. Let b ≥ 1. Then, Aut • (S b ) is maximal and S b is stiff. It is moreover superstiff if and only if b = 1.

  PGL 3 and is thus equal to Aut • (S 1 ), as S 1 → P 2 , P(T P 2 ) → P 2 , and PGL 3 /B → PGL 3 /P are isomorphic, by Remark 4.2.8.

( J5) is Aut • (PGL 2 /D 2n ), with n ≥ 4, and is thus equal to Aut •

  Let H be a maximal proper connected subgroup of PGL 2 × PGL 2 . Then H is B × PGL 2 or PGL 2 ×B, where B ⊂ PGL 2 is a Borel subgroup (conjugate to the group of upper-triangular matrices), or H is isomorphic to PGL 2 . In the latter case, H is conjugate to the diagonal embedding of PGL 2 in PGL 2 × PGL 2 . Proof. Let p 1 and p 2 be the two natural projections PGL 2 × PGL 2 → PGL 2 . If p i (H) PGL 2 , then p i (H) is contained in a Borel subgroup B of PGL 2 (e.g. by [Hum75, § 30.4, Theorem (a)]), and so H is B × PGL 2 or PGL 2 ×B since H is maximal. We now assume p i (H) = PGL 2 for i = 1, 2. Let K be the kernel of p 1 |H : H → PGL 2 . As K is a normal subgroup of H and p 2 |H is onto, p 2 (K) is a normal subgroup of PGL 2 . As H is a proper subgroup of PGL 2 × PGL 2 , we must have p 2 (K) = {1}, and so K = {1} and p 1 |H is a bijective morphism of algebraic groups H → PGL 2 . In particular, dim(H) = 3 by [Hum75, § 4.1, Theorem].

	Lemma 3.7.2.

• (X) to extend.

  see Definition 1.2.7). As we will observe, only the cases b ≥ 1 are interesting since S -1 and S 0 are decomposable (see Corollary 4.2.2).

Lemma 4.2.1. Denoting by U 0 , U 1 ⊂ P 2 the two open subsets

  Then the structure morphism X → P 2 identifies with the projection G/B → G/P (see e.g. [IP99, Example 2.1.8]). Therefore, by [Dem77, Theorem 1], we have Aut • (X) = Aut • (G/B) = G.

  1) H = Aut(P 2 ) if and only if π : X → P 2 is a decomposable bundle or is isomorphic to the projectivised tangent bundle. (2) PGL 2 Aut(P 2 , Γ ) ⊂ H if and only if π : X → P 2 is a decomposable bundle or is isomorphic to a Schwarzenberger bundle S b → P 2 for some b ≥ 1. Proof. If π : X → P 2 is a decomposable bundle, then H = Aut(P 2 ) (see Lemma 4.1.2). The same holds if π : X → P 2 is isomorphic to S 1 P(T P 2 ) (see Corollary 4.2.2 and Lemma 4.2.5). If π : X → P 2 is isomorphic to S b for some b ≥ 2, then H = Aut(P 2 , Γ ) (see Lemma 4.2.5(2)).

Automorphisms of P 1 -bundles over rational surfaces

J. Blanc, A. Fanelli and R. Terpereau

J. Blanc, A. Fanelli and R. Terpereau

The authors acknowledge support by the Swiss National Science Foundation Grant "Algebraic subgroups of the Cremona groups" 200021_159921. The second-named author is currently supported by the DFG grant "Gromov-Witten Theorie, Geometrie und Darstellungen" (PE 2165/1-2). The third-named author is supported by the ANR Project FIBALGA ANR-18-CE40-0003-01 and is grateful to the Max-Planck-Institut für Mathematik of Bonn for the warm hospitality and support provided during part of the preparation of this work. The IMB receives support from the EIPHI Graduate School (contract ANR-17-EURE-0002).

We then recover the following result already observed by Schwarzenberger [START_REF] Schwarzenberger | Vector bundles on the projective plane[END_REF]Proposition 7].

Corollary 4.2.2. We have the following isomorphisms of P 1 -bundles: S -1 P 1 = P(O P 2 (1) ⊕ O P 2 ), S 0 P 0 = P 1 × P 2 , and S 1 P(T P 2 ).

Proof. Let us note that to obtain the transition function of the tangent bundle T P 2 , it suffices to differentiate the map corresponding to the change of coordinates between two affine charts of P 2 . It follows that the transition function of the projectivised tangent bundle over P 2 \ [0 : 1 : 0] is

1 . Applying Proposition 4.2.1, the transition functions for b = -1, 0, 1 correspond, respectively, to the matrices

1 0 , and 1 0 u v , which gives the result.

Remark 4.2.3. One can also see S 1 P(T P 2 ) as

with π : S 1 → P 2 the projection onto the first factor. We again find the same transition function, by trivialising the P 1 -bundle over U 0 and U 1 via

Seeing the equation with vectors, as t x • y = 0, the group PGL 3 then acts via (x, y) → (Ax, t A -1 y).

We will see in Lemma 4.2.5(2) that Aut • (S 1 ) PGL 3 , so the above action yields all elements of Aut • (S 1 ) (but not of Aut(S 1 ), as we also have (x, y) → (y, x)). Lemma 4.2.4. For each b ≥ 1, the following hold:

(1) The P 1 -bundle Ŝb → P 1 × P 1 is isomorphic to

obtained by pulling back the Schwarzenberger bundle S b → P 2 via the double cover κ : P 1 × P 1 → P 2 . In particular, Ŝb is isomorphic to

(2) The only curve invariant by Aut • ( Ŝb ) is the curve C ⊂ Ŝb , being the preimage of the curve D ⊂ S b given on the two charts of Lemma 4.2.1 by

Proof.

(1) We take as usual coordinates ([y 0 : y 1 ], [z 0 : z 1 ]) on P 1 × P 1 and denote by T 0 , T 1 ⊂ P 1 × P 1 the open subsets given by, respectively, y 0 z 0 0 and y 1 z 1 0. The restriction of π :

). Proof. The birational morphism η : F 1 → P 2 , [y 0 : y 1 ; z 0 : z 1 ] → [y 0 z 0 : y 1 : y 0 z 1 ] is the blow-up of [0 : 1 : 0], and by the construction of ϕ, we find that this is a square birational map above τ. We then write U = P 2 \ {[0 : 1 : 0]} and Û = τ -1 (U ) ⊂ F 1 and observe that ϕ induces an isomorphism π-1 ( Û ) -→ π -1 (U ). By Lemma 2.3.2, ϕ is the unique square birational map above τ having this property (up to composition by an isomorphism of P 1 -bundles at the target) and is Aut

) acts transitively on P 2 (see Lemma 4.1.2), but every element of ϕ Aut • (F b,c 1 )ϕ -1 acts on P 2 by fixing [0 : 1 : 0].

Reduction of birational maps to elementary links

We show here that every birational map of P 1 -bundles between the four types in Proposition 5.1.1 is a sequence of elementary links.

Lemma 5.3.1. Let G be a connected algebraic group acting on a P 1 -bundle X → S, let H ⊂ Aut • (S) be the image of G under this action, and assume either that no curve of S is invariant by H or that (H, S) is one of the following pairs:

If π : X → S is a P 1 -bundle and ϕ : X X is a G-equivariant birational map of P 1 -bundles (as in Definition 1.2.1) which is not an isomorphism, then we have a sequence of P 1 -bundles π i : X i → S, i = 0, . . . , n, with π 0 = π and π n = π , and we can write ϕ = ϕ n • • • • • ϕ 1 , where ϕ i : X i X i+1 is the blow-up of an irreducible curve i ⊂ X i , followed by the contraction of the strict transform of π i -1 (π i ( )), and where

yields an isomorphism between i and either s -a ⊂ F a with a ≥ 1, ∆ ⊂ P 1 × P 1 , or Γ ⊂ P 2 . Moreover, taking n minimal, the sequence ϕ 1 , . . . , ϕ n is unique up to isomorphisms of P 1 -bundles.

Proof. Taking an open subset U ⊂ S isomorphic to A 2 , the P 1 -bundle X is trivial, so corresponds to P 1 ×A 2 . On this chart, the birational map ϕ is of the form

with no common factor; the polynomials a, b, c, d are unique, up to multiplication by an element of k * . The zero locus of the determinant P = adbc thus corresponds exactly to the subset of U over which ϕ is not an isomorphism.

Denoting by K ⊂ S the subset over which ϕ is not an isomorphism, we find that K is a union of closed irreducible curves. The map ϕ being G-equivariant, K is invariant by H. The assumption made on (H, S) implies that either K = ∅, in which case ϕ is an isomorphism, or (K, S) is one of the three cases (s -a , F a ), (∆, P 1 × P 1 ), or (Γ , P 2 ).

In the three cases, we can choose, for each point p ∈ K, an open set U ⊂ S and an isomorphism U -→ A 2 which sends K ∩ A 2 onto the line u = 0. Writing ϕ with a, b, c, d ∈ k[u, v] as above, we find that adbc = λu n for some integer n ≥ 1 and λ ∈ k * . As u does not divide all polynomials a, b, c, d, the matrix

Rigidity for decomposable bundles over P 2

Lemma 5.7.1. Let b ≥ 0, and let π : P b → P 2 be a decomposable P 1 -bundle.

(1) P b does not contain any Aut • (S b )-invariant curve.

(2) Aut • (P b ) is maximal and P b is superstiff.

Proof. Assertion (1) follows from the fact that Aut • (P b ) surjects onto Aut(P 2 ) (see Lemma 4.1.2).

(2) Let ϕ : P b X be an Aut • (P b )-square birational map of P 1 -bundles, over η : P 2 X (where X is a smooth projective rational surface). We want to show that ϕ is a square isomorphism. The action of Aut • (P b ) on P 2 being transitive, the birational map η is an isomorphism, so we can assume X = P 2 , and that η is the identity. We then apply Lemma 5.3.1 to get that ϕ is an isomorphism.

Last step

We have all the ingredients needed to prove the main theorems of this paper stated in the introduction.

Proof of Theorems A and B. These are simply a consequence of Proposition 5.1.1, together with Corollaries 5.4.3, 5.5.4, 5.6.3 and Lemma 5.7.1.

Remark 5.8.1. Our original motivation was to study the maximal connected algebraic subgroups of the Cremona group Bir(P 3 ). These are studied in the article [START_REF] Blanc | Connected Algebraic Groups Acting on three-dimensional Mori Fibrations[END_REF]. Most of the families appearing in our classification in fact give maximal connected algebraic subgroups of the Cremona group, even if some sporadic cases (like S 2 and P 1 ) disappear as they are conjugate to bigger subgroups, with a birational map which does not preserve any P 1 -bundle structure.

Remark 5.8.2. If we assume the characteristic of the algebraically closed field k to be positive, the strategy to prove Theorems A and B should be analogous, but some of the results that we use in characteristic zero are no longer valid in positive characteristic; see e.g. Remarks 3.7.3 and 3.7.5. As a consequence, it seems that new P 1 -bundles X → S analogous to Umemura bundles and Schwarzenberger bundles, and such that Aut • (X) is maximal, could show up in the classification. The positive characteristic case will be studied by the authors in a future work.