AUTOMORPHISMS OF A1 -FIBERED AFFINE SURFACES JÉRÉMY BLANC AND ADRIEN DUBOULOZ Abstra t. We develop te hni s of birational geometry to study automorphisms of ane surfa es admitting many distin t rational brations, with a parti ular fo us on the intera tions between automorphisms and these brations. In parti ular, we asso iate to ea h surfa e S of this type a graph en oding equivalen e lasses of rational brations from whi h it is possible to de ide for instan e if the automorphism group of S is generated by automorphisms preserving these brations.

MSC 14R25, 14R20, 14R05, 14E05 Key words: ane surfa es; rational brations; automorphisms. Introdu tion

Motivated by the example of the ane plane A2 on whi h algebrai automorphisms a t transitively, it is a natural problem to determine whi h ane surfa es are homogeneous under the a tion of their automorphism group. It turned out that it is more interesting to onsider ane surfa es that are only almost homogeneous in the sense that the orbit of a general point has a nite omplement. Indeed, in his pioneer work, M.H. Gizatullin [10℄ obtained a geometri hara terization of su h surfa es in term of the stru ture of the boundary divisors in minimal proje tive ompletions of these. Namely, he established that up to nitely many ex eptional ases, su h surfa es are pre isely those whi h admit ompletions by so- alled zigzag s, that is, hains of proper nonsingular rational urves. The automorphism groups of su h surfa es have been studied later on by V.I. Danilov and M.H. Gizatullin [6, 7℄. Motivated again by the example of the ane plane due to J.-P. Serre [16℄, they established in parti ular that these automorphism groups an be realized as fundamental groups of graph of groups onstru ted from suitable famillies of proje tive ompletions. In prin iple, this des ription would allow to derive a more expli it presentation of these automorphism groups. This was done by V.I. Danilov and M.H. Gizatullin in the ase of surfa es admitting a ompletion by an irredu ible zigzag [7℄. But in general, the orresponding graphs of groups are innite and it be omes very di ult even to extra t any expli it des ription of potentially interesting subgroups. A noteworthy geometri feature of ane surfa es S ompletable by a zigzag is that they are rational, and admit A1 -brations π : S → A1 , that is, surje tive morphism with general bers isomorphi to the ane line. A tually, ex ept for the ase of A1 \ {0} × A1 , it turns out that every su h surfa e admits at least two brations of this type with distin t general bers (see e.g. [4℄). This motivates an alternative approa h onsisting of understanding the automorphisms of these surfa es in terms of their intera tions with A1 -brations. In parti ular, the following questions seem natural in this ontext: 1) Does the automorphism group Aut (S) of S a t transitively on the set of A1 -brations on S ? 2) Can Aut (S) be generated by automorphisms that ea h preserves an A1 -bration ? For instan e, both questions are answered armatively for the ane plane A2 , as onsequen es of the Abhyankar-Moh Theorem [1℄ and of the Jung-van der Kulk Theorem [11℄ giving the des ription of Aut A2 .

In this arti le, we develop a general method to address these questions, based on the study of birational relations between suitably hosen proje tive models. Namely, starting with an A1 -bered surfa e π : S → A1 , we onsider proje tive ompletions (X, B, π ¯ ) of S that we all 1-standard (see 1.0.2 below), following the notation of [6℄. Here X is a proje tive surfa e, B = X \ S is a boundary zigzag, and This resear h has been partially supported by FABER Grant 07-512-AA-010-S-179. 1

2

JÉRÉMY BLANC AND ADRIEN DUBOULOZ

π extends to a rational bration π ¯ : X → P1 . We introdu e two lasses of birational transformations ∼ ′ ′ ′ φ : (X, B, π ¯ ) 99K (X , B , π ¯ ) between su h ompletions that restri t to isomorphisms X \ B → X ′ \ B ′ . The rst ones, alled bered modi ations have the property that they are ompatible with the given rational brations on X and X ′ respe tively. The se ond ones, alled reversions as in [9℄, an be thought as the simplest possible birational transformations between su h ompletions that are not ompatible ¯ and π ¯ ′ . One of the main result of the arti le is the fa t that these with the rational brations π basi birational transformations are the building blo ks for general birational maps between 1-standard

ompletions preserving the omplement of the boundaries. More pre isely, we establish the following result (Theorem 3.0.2, proved in Se tion 3).

Theorem. Let φ : (X, B) 99K (X ′ , B ′ ) be a birational map between 1-standard pairs restri ting to an ∼ isomorphism X \ B → X ′ \ B ′ . If φ is not an isomorphism then it an be de omposed into a nite sequen e φ1

φ2

φn

φ = φn ◦ · · · ◦ φ1 : (X, B) = (X0 , B0 ) 99K (X1 , B1 ) 99K · · · 99K (Xn , Bn ) = (X ′ , B ′ )

of bered modi ations and reversions between 1-standard pairs (Xi , Bi ), i = 1, . . . , n. Furthermore, su h a fa torization of minimal length is unique up to omposition by isomorphisms between intermediate pairs. This leads in parti ular to a anoni al pro edure to fa tor an automorphism of an ane surfa e S

ompletable by a zigzag onsidered as a birational transformation of a xed 1-standard ompletion of S . Using this des ription, we asso iate to every su h surfa e S a onne ted graph FS with equivalen e

lasses of 1-standard ompletions (X, B) of S as verti es and with edges being given by reversions. This graph, whi h is in general smaller than the one onstru ted by V.I. Danilov and M. H. Gizatullin [6℄, en odes all the ne essary information to understand the intera tions between automorphisms of S and A1 -brations on it. For instan e, we establish that under mild assumptions on S , the automorphism group Aut (S) is generated by automorphisms of A1 -brations if and only if the asso iated graph FS is a tree. In general, we show that it is also possible to equip FS with an additional stru ture of a graph of groups having Aut (S) as its fundamental group. The arti le is organized as follows. Se tion 1 introdu es the basi denition on erning 1-standard pairs (X, B) and the existing rational brations on these. Se tion 2 ontains a detailed geometri study of bered modi ations and reversions. Se tion 3 is devoted to the proof of the main theorem above and se tion 4 presents the onstru tion and the interpretation of the graph FS together with its additional stru ture of graph of groups. Finally, in se tion 5, we apply our general ma hinery to the study of lassi al examples of ane surfa es ompletable by a zigzag. After explaining how to re over Jung's Theorem from our des ription, we onsider normal surfa es dened by an equation of the form uv = P (w) in A3 . In this ase we not only re over generators of their automorphism groups as obtained by M. Makar-Limanov [15℄ but we also show that they an be equipped an additional amalgamated produ t stru ture. As a byprodu t, we also re over D. Daigle transitivity Theorem [3℄ asserting that su h surfa es admit a unique equivalen e

lass of A1 -brations. We also prove that if the degree of P is at least 3, the group of automorphisms of the orresponding surfa e is not generated by automorphisms of A1 -brations. In the last subse tion, we give examples of an ane surfa es with the total inverse properties: in general, they admit innitely many equivalen e lasses of A1 -brations but the group is generated by automorphisms of A1 -brations. 1.

Preliminaries : Standard zigzags and asso iated rational fibrations

In what follows we x a eld k. All varieties o

uring in the sequel are impli itly assumed to be geometri ally integral and dened over k, and all morphisms between these are assumed to be dened over k.

Denition 1.0.1. A zigzag on a normal proje tive surfa e X is a onne ted SNC-divisor, supported in

the smooth lo us of X , with irredu ible omponents isomorphi to the proje tive line over k and whose dual graph is a hain.

AUTOMORPHISMS OF

If Supp (B) = a way that

Sr

i=0

A1 -FIBERED

AFFINE SURFACES

3

Bi then the irredu ible omponents Bi , i = 0, . . . , r, of B an be ordered in su h (

1 if |i − j| = 1, 0 if |i − j| > 1. A zigzag with su h an ordering on the set of its omponents is alled oriented and the sequen e (B0 )2 , . . . , (Br )2 is alled the type of B . The omponents B0 and Br are alled the boundaries of B . For an oriented zigzag B , the same zigzag with the reverse ordering is denoted by tB . An oriented sub-zigzag of an oriented zigzag is an SNC divisor B ′ with Supp (B ′ ) ⊂ Supp (B) whi h is a zigzag for the indu ed ordering. We say that an oriented zigzag B is omposed of sub-zigzags Z1 , . . . , Zs , and we write B = Z1⊲· · ·⊲Zs , if the Zi 's are oriented sub-zigzags of B whose union is B and the omponents of Zi pre ede those of Zj for i < j . Bi · Bj =

Denition 1.0.2. A zigzag B on a normal proje tive surfa e X is alled m-standard if it an be written

as B = F ⊲C ⊲E where F and C are smooth irredu ible rational urves with self-interse tions F 2 = 0 and C 2 = −m, m ∈ Z, and where E = E1 ⊲· · ·⊲Er is a (possibly empty) hain of irredu ible rational

urves with self-interse tions (Ei )2 ≤ −2 for every i = 1, . . . , r. An m-standard pair is a pair (X, B) onsisting of a normal rational proje tive surfa e X and an m-standard zigzag B . A birational map φ : (X, B) 99K (X ′ , B ′ ) between m-standard pairs is a birational ∼ map φ : X 99K X ′ whi h restri ts to an isomorphism X \ B → X ′ \ B ′ .

1.0.3. Sin e it is rational, the underlying proje tive surfa e of an m-standard pair (X, B = F ⊲C ⊲E)

omes equipped with a rational bration π ¯=π ¯|F | : X → P1 dened by the omplete linear system |F | (see e.g. [4℄). In the sequel, we will impli itly onsider m-standard pairs as equipped with this bration π ¯ . Re all that the generi ber of a rational bration π ¯ is isomorphi to the proje tive line over the fun tion eld of P1 , and that the total transform of the singular bers of π ¯ in a minimal resolution µ : Y → X of the singularities of X onsist of trees of nonsingular rational urves (see e.g., Lemma 1.4.1 p. 195 in [14℄ whi h remains valid over an arbitrary base eld). ¯ restri ts on the quasi-proje tive surfa e S = X \ B to an a faithfully at The rational bration π morphism π : S → A1 with generi ber isomorphi to the ane line over the fun tion eld of A1 . The general bers of π are isomorphi to ane lines and π has nitely many degenerate bers whose total transforms in a minimal resolution of singularities of S onsists of nonempty disjoint unions of trees of rational urves, with irredu ible omponents isomorphi to either ane or proje tive lines, possibly dened over nite algebrai extensions of k. In ontrast, the restri tion of π to the omplement of its degenerate bers has the stru ture of a trivial A1 -bundle. In what follows, su h morphisms will be simply refered to as A1 -brations or A1 -bered surfa es.

Denition 1.0.4. We say that two A1 -bered surfa es (S, π) and (S ′ , π′ ) are isomorphi if there exist

an isomorphism Ψ : S → S ′ and an automorphism ψ of A1 su h that π ′ ◦ Ψ = ψ ◦ π . On a surfa e S , two A1 -brations π, π ′ : S → A1 are said to be equivalent if (S, π) and (S, π ′ ) are isomorphi .

1.0.5. If B is moreover the support an ample divisor, then S is ane and π : S → A1 has a unique

π (E)) whi h onsists of a nonempty disjoint union of ane lines, again possidegenerate ber π −1 (¯ bily dened over nite algebrai extensions of k, when equipped with its redu ed s heme stru ture. Furthermore, if any, the singularities of S are all supported on the degenerate ber of π and admit a minimal resolution whose ex eptional set onsists of a hain of rational urves possibly dened over a nite algebrai extension of k (this follows from the same argument as in the proof of Lemma 1.4.4 in [14℄). In parti ular, if k is algebrai ally losed of hara teristi 0, then S has at worst Hirzebru h-Jung

y li quotient singularities.

1.0.6. Hereafter, we will mostly onsider 1-standard pairs (X, B). The simplest example (F1 , F ⊲C0 )

onsists of the Hirzebru h surfa e ρ : F1 = P (OP1 ⊕ OP1 (−1)) → P1 and the union of a ber F of ρ and the negative se tion C0 of ρ. More generally, we have the following des ription.

Lemma 1.0.7. Let (X, B = F ⊲C ⊲E, π¯ ) be a 1-standard pair and let µ : Y → X be the minimal resolution of the singularities of X . Then there exists a birational morphism η : Y → F1 , unique up

4

JÉRÉMY BLANC AND ADRIEN DUBOULOZ

to automorphisms of F1 , that restri ts to an isomorphism outside the degenerate bers of π¯ ◦ µ, and a

ommutative diagram ff Y YYYYYYYηY µ YYYYYY ffffff f f f f f , µ◦¯ π X XrfXXXXXX ff F1 XXXXX fffff f f f f X, 1 rff ρ π ¯ P .

Furthermore, if (X ′ , B ′ = F ′ ⊲C ′ ⊲E ′ , π¯ ′ ) is another 1-standard pair with asso iated morphism η : Y → F1 then (X, B, π ¯ ) and (X ′ , B ′ , π ¯ ′ ) are isomorphi if and only if there exists an automorphism of ′ −1 ′ −1 ′ F1 mapping isomorphi ally η µ∗ F onto η (µ )∗ F and sending isomorphi ally the base-points of η −1 (in luding innitely near ones) onto those of (η ′ )−1 . ′

Proof. Sin e B is supported in Xreg , its proper transform in Y oin ide with its total transform and is

again a 1-standard zigzag. We may therefore assume that X is smooth. Let us prove the rst assertion. By ontra ting su

essively all the (−1)- urves in the degenerate bers F1 , . . . , Fs of π ¯ , one obtains a birational morphism η : X → Fm onto a ertain Hirzebru h surfa e ρm : Fm → P1 , whi h maps C , F and the Fi 's onto a se tion and s + 1 distin t bers of ρm respe tively. Let r (η) = (η∗ C)2 ≥ −1. If r (η) = −1 then m = 1 and we are done. Otherwise, sin e C 2 = −1 in X , it follows that η ontra ts at least one of the irredu ible omponents of a degenerate ber, say ˜ m → Fm of p. F1 , onto the point p = η(F1 ) ∈ η(C). Therefore, η fa tors through the blow-up σ : F ˜ m → Fm±1 be the ontra tion of the stri t transform of the ber ρ−1 (ρm (p)), we obtain Letting τ : F m a new birational morphism η ′ = τ ◦ σ −1 ◦ η : X → Fm±1 satisfying r (η ′ ) = r (η) − 1. So the existen e of η : X → F1 follows by indu tion. Suppose that η ′ : X → F1 is another su h morphism. Then η ′ ◦ η −1 : F1 99K F1 is a birational map whi h does not blow-up any point of η(C) and does not ontra t any urve interse ting η(C). Sin e η(C) is a se tion and η ′ ◦ η −1 may be de omposed into elementary links between Hirzebrur h surfa es, η ′ ◦ η −1 is an isomorphism. The se ond assertion follows from the fa t that an isomorphism between 1-standard pairs (X, B, π ¯) and (X ′ , B ′ , π ¯ ′ ) indu es an isomorphism between B and B ′ whi h preserves the orientation, when e des ends to an automorphism of F1 . 2.

Two basi birational maps between

1-standard

pairs

2.1. Base-points and urves ontra ted. We will study isomorphisms between the omplements of the boundary as birational maps between 1-standard pairs; we an distinguish two dierent kind of su h maps, a

ording to the following result.

Lemma 2.1.1. Let φ : (X, B = F ⊲C ⊲E) 99K (X ′ , B ′ ) be a birational map between two 1-standard pairs, σ σ′ whi h is not an isomorphism, and let X ← Z → X ′ be a minimal resolution of φ. Then every urve ontra ted by φ and every base-point of φ is dened over k. Moreover, φ has a unique proper base-point q ∈ B , and one and exa tly one of the following o

ur: a) the stri t transform of C in Z is the unique (−1)- urve ontra ted by σ ′ , and q ∈ F \ C ; b) the stri t transform of F in Z is the unique (−1)- urve ontra ted by σ ′ , and q = F ∩ C . (Note that in both ases, it is possible that F and C are ontra ted by φ.) Proof. To any base-point of respe tively φ and φ−1 is asso iated a urve ontra ted by respe tively φ−1

and φ. Sin e any urve ontra ted by φ and φ−1 is ontained in the boundary, it is dened over k. This implies that all base-points also are dened over k. Ea h (−1)- urve in Z whi h is ontra ted by σ ′ is the proper transform of either C or F . Sin e 2 C = −1 and C · F = 1 in X , the two possibilities annot o

ur simultaneously, so φ−1 (and thus φ) has at most one proper base-point. If C is the (−1)- urve ontra ted by σ ′ , to avoid a positive self-interse tion for the urve F , there is one base-point on F \ C ( ase a). If F is the (−1)- urve

ontra ted by σ ′ there is one base-point on F ; either the base-point is F ∩ C ( ase b), or C be omes a non-negative urve, hen e the (0)- urve of B ′ , but this implies that only one urve is ontra ted by σ ′ , a ontradi tion. If no (−1)- urve is ontra ted by σ ′ , then σ ′ is an isomorphism and the dis ussion made above shows that so is σ .

AUTOMORPHISMS OF

A1 -FIBERED

AFFINE SURFACES

5

Remark 2.1.2. Be ause of this result, when dealing with birational maps between 1-standard pairs the fa t that k is not algebrai ally losed, and even its hara teristi is not relevant. There will only be some distin tion in the last se tion, where the onstru tion of the examples uses the birational morphism that blows-up points of F1 not ne essarily dened over k.

Denition 2.1.3. If p ∈ X is the unique proper base-point of a birational map φ : (X, B) 99K (X ′ , B ′ ) (whi h indu es an isomorphism X \ B ∼ = X ′ \ B ′ ), we say that φ is entered at p and that p is the enter of φ. In subse tions 2.2 and 2.3, we review two basi lasses of birational transformations between 1standard pairs that will play a entral role in the sequel, and are the simplest examples of maps satisfying respe tively onditions a) and b) of Lemma 2.1.1. 2.2. Fibered Modi ations.

Denition 2.2.1. A a birational map φ : (X, B, π¯ ) 99K (X ′ , B ′ , π¯ ′ ) between (1)-standard pairs is bered if it restri ts to an isomorphism of A1 -bered quasi-proje tive surfa es ∼

S =X \B

φ

π ¯ |S

A1

∼

/ S′ = X ′ \ B′ ′

π¯ |S′ / A1 .

We say that φ is a bered modi ation if it is not an isomorphism.

Example 2.2.2. Let F1 = {((x : y : z), (s : t)) ⊂ P2 × P1 | yt = zs} be the Hirzebru h surfa e of index

1; the proje tion on the rst fa tor yields a birational morphism τ : F1 → P2 whi h is the blow-up of (1 : 0 : 0) ∈ P2 and the proje tion on the se ond fa tor yields a P1 -bundle ρ : F1 → P1 . Denote by C ⊂ F1 the ex eptional urve τ −1 ((1 : 0 : 0)) = (1 : 0 : 0) × P1 , and by F ⊂ F1 the ber ρ−1 ((0 : 1)). The map (x, y) 7→ ((x : y : 1), (y : 1)) yields an isomorphism A2 → F1 \ (C ∪ F ). Then every triangular automorphism Ψ of A2 of the form (x, y) 7→ (ax + b, cy + P (x)), where P ∈ k [x], preserves the A1 -bration prx = ρ|A2 : A2 → A1 and extends to a bered birational map φ : (F1 , F ⊲C, ρ) 99K (F1 , F ⊲C, ρ) of 1-standard pairs. The latter is a biregular automorphism if Ψ is ane and a bered modi ation otherwise.

More generally, we have the following des ription whi h says in essen e that every bered birational map between 1-standard pairs arises as the lift of a triangular automorphism of A2 as above.

Lemma 2.2.3. Let φ : (X, B = F ⊲C ⊲E, π¯ ) 99K (X ′ , B ′ = F ′ ⊲C ′ ⊲E ′ , π¯ ′ ) be a birational map between µ η µ′ η′ 1-standard pairs and let X ← Y → F1 and X ← Y ′ → F1 be the morphisms onstru ted in Lemma 1.0.7. Then the following are equivalent : ∼ a) φ restri ts to an isomorphism of A1 -bered surfa es (X \ B, π ¯ ) → (X ′ \ B ′ , π ¯ ′ ); ′ −1 ′ ′ 1 b) (µ ) ◦ φ ◦ µ : Y 99K Y is the lift via η and η of an isomorphism of A -bered ane surfa es ∼

A2 = F1 \ (η(F ) ∪ η(C))

Ψ

ρ|A2

A1

∼ ψ

/ A2 = F1 \ (η ′ (F ′ ) ∪ η ′ (C ′ )) ρ|A2 / A1

whi h maps isomorphi ally the base-points of η onto those of η′−1 . Furthermore, φ : (X, B) 99K(X ′ , B ′ ) is an isomorphism if and only if Ψ is ane. −1

Proof. One he ks that φ : (X, B, π¯ ) 99K (X ′ , B ′ , π¯ ′ ) restri ts to an isomorphism between the A1 -bered

−1 surfa es (X \ B, π ¯ ) and (X \ B, π ¯ ) if and only if its lift (µ′ ) ◦ φ ◦ µ restri ts to an isomorphism between 1 −1 ′ ′ −1 ′ ¯ ◦ µ) and (Y \ (µ )∗ B , π ¯ ′ ◦ µ′ ). We may thus assume that X and the A -bered surfa es (Y \ µ∗ B, π ′ X are smooth. σ′ σ Suppose that φ is not an isomorphism and let X ′ ← Z → X be a minimal resolution of φ where σ ′ and σ are sequen es of blow-ups with enters outside S ′ and S respe tively. Assume that φ satises (a), whi h implies that the rational brations π ¯ ′ and π ¯ lift to a same rational bration π ˜ : Z → P1 , and that the proper transforms of C ′ and C in Z oin ide with the unique se tion −1 ˜ ontained in the boundary D = (σ ′ ) (B ′ ) = σ −1 (B) in Z . Thus φ restri ts to a birational C˜ of π

6

JÉRÉMY BLANC AND ADRIEN DUBOULOZ

map C 99K C ′ . The only (−1)- urve of Z whi h is ontra ted by σ (respe tively σ ′ ) is therefore the proper transform of F (respe tively of F ′ ) in X . Consequently, φ restri ts to an isomorphism E → E ′ ; φ a tually restri ts to an isomorphism of A1 -bered surfa es (X \ (F ∪ C) , π ¯ ) ≃ (X ′ \ (F ′ ∪ C ′ ) , π ¯ ′ ). Conversely, su h isomorphisms extend to birational maps satisfying (a). Now the equivalen e follows from the one-to-one orresponden e between su h isomorphisms and ∼ those of the form Ψ : (F1 \ η∗ (F ∪ C) , ρ) → (F1 \ η∗′ (F ′ ∪ C ′ ) , ρ) whi h map isomorphi ally the base−1 ′ −1 points of η onto those of (η ) . The last assertion follows from the fa t that φ extends to an isomor∼ phism X → X ′ if and only if the orresponding automorphism Ψ of A2 extends to an automorphism of F1 (both onditions are equivalent to say that the proper transform of F is not ontra ted).

2.2.4. It follows from the above des ription (Lemmas 2.1.1 and 2.2.3) that a bered modi ation φ : (X, B = F ⊲C ⊲E, π ¯ ) 99K (X ′ , B ′ = F ′ ⊲C ′ ⊲E ′ , π ¯′ ) ∼

has a unique proper base-point q = F ∩ C . Letting Ψ : A2 → A2 , (x, y) 7→ (ax + b, cy + P (x)) be the ˜ of B in a minimal triangular automorphism asso iated with φ, one he ks that the total transform B ′ σ σ ′ ′ ˜ → (X , B ) of φ is a tree of rational urves with the following dual graph resolution (X, B) ← (Z, B) H′

F′

deg P − 2

−1

F

E = E′

− deg P

−2

deg P − 2

C = C′

−1

H where the two boxes represent hains of deg P − 2 (−2)- urves. Furthermore, the morphisms σ : Z → X and σ ′ : Z → X are given by the smooth ontra tions of the sub-trees H ∪ H ′ ∪ F ′ and H ∪ H ′ ∪ F onto the proper base-points q = F ∩ C and q ′ = F ′ ∩ C ′ of φ and φ−1 respe tively.

2.3. Zigzag Reversions.

Denition 2.3.1. A stri tly birational map φ : (X, B = F ⊲C ⊲E) 99K (X ′ , B ′ = F ′ ⊲C ′ ⊲E ′ ) between 1-standard pairs is alled a reversion if it admits a resolution of the form ˜ = t(C ⊲E)⊲H ⊲(C ′ ⊲E ′ )) (Z, B ZZZZσZ′ ZZ σdddd d ZZ, rdddd φ t X, tB = (C ⊲E)⊲F _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _/ (X ′ , B ′ = F ′ ⊲(C ′ ⊲F ′ ))

where H is a zigzag with boundaries F (left) and F ′ (right) and where σ : Z → X and σ ′ : Z → X ′ t ˜ onto their left and right are smooth ontra tions of the sub-zigzags H ⊲ (C ′ ⊲E ′ ) and (C ⊲E)⊲ H of B boundaries F and F ′ respe tively.

Example 2.3.2. Let n1 , n2 ≥ 3 be two integers, let Z be a normal rational proje tive surfa e and let ˜ ⊂ Z reg be a zigzag having the following dual graph B E2

−n2

E1

−n1

C

−1

F′

F

−2

n1−3

−3

n2−3

−2

C′

−1

E1′

−n2

E2′

−n1

H where the boxes represent hains of n1 −3 and n2 −3 (−2)- urves that there exist respe tively. One he ks ˜ → X, tB = E2 ⊲E1 ⊲C ⊲F and σ ′ : (Z, B) ˜ → (X ′ , B ′ = F ′ ⊲C ′ ⊲E ′ ⊲E ′ ) two birational morphisms σ : (Z, B) 1 2 ˜ starting with those of C ′

onsisting of a sequen e of smooth blow-downs of irredu ible omponents of B and C respe tively. By onstru tion, σ ′ ◦ σ −1 : (X, B) 99K (X ′ , B ′ ) is a reversion between 1-standard pairs of type (0, −1, −n1, −n2 ) and (0, −1, −n2 , −n1 ) respe tively.

The following lemma summarizes some of the main properties of reversions.

Lemma 2.3.3 (Properties of reversions). Let φ : (X, B = F ⊲C ⊲E) 99K (X ′ , B ′ = F ′ ⊲C ′ ⊲E ′ ) be a reversion and let ′ σ σ t ′ ′ ′ ′ ˜ t X, B ← Z, B = (C ⊲E)⊲H ⊲(C ⊲E ) → (X , B )

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be a minimal resolution of φ as in Denition 2.3.1 above. Then, the unique proper base-point of φ (respe tively φ−1 ) belongs to F \ C (respe tively F ′ \ C ′ ); moreover the following hold : ˜ onsists of a unique smooth rational urve (F = H = F ′ ) if and only if E a) The sub-zigzag H of B

onsists of a possibly empty hain of (−2)- urves. b) Otherwise, (if E ontains at least an irredu ible omponent with self-interse tion ≤ −3) then H

an be written as H = F ⊲ H˜ ⊲F ′ and the morphisms σ : Z → X and σ′ : Z → X ′ are the ontra tions ˜ to a point of X and X ′ respe tively. ˜ ⊲F ′ ⊲(C ′ ⊲E ′ ) and t(C ⊲E)⊲F ⊲ H of the sub- hains H c) If E is not empty then the birational morphisms σ : Z → X and σ ′ : Z → X ′ fa tor into unique sequen es of smooth blow-downs starting with the ontra tions of the (−1)- urve C ′ and C respe tively and ending with the ontra tions of the right boundaries of E ′ and E . Proof. Sin e C is not ae ted by σ, it has self-interse tion −1 in Z ; sin e C is ontra ted by σ′ , the

unique proper base-point of φ belongs to F \C (Lemma 2.1.1). Ex hanging C and C ′ yields the analogue result for φ−1 . Sin e the ex eptional lo us a smooth ontra tion annot ontain two (−1)- urves whi h interse t, it follows that F or F ′ has self-interse tion −1 in Z if and only if H = F = F ′ and E = E ′ = ∅; this is a degenerate ase of a). We may now assume that F 2 , (F ′ )2 ≤ −2 in Z . Let us prove that σ ′ does not fa tor through the

ontra tion η ′ : Z → X˜ ′ of H ′ ⊲C ′ ⊲E ′ , for some stri t subzigzag H ′ ⊂ (H \ F ). Suppose the ontrary. Sin e σ ontra ts a onne ted urve whi h is (H \ F )⊲C ′ ⊲E ′ the same holds for η . Then η(H \ F ) is a ontra tible onne ted urve, ontaining a unique (−1)- urve whi h is its right boundary. This implies (sin e F is the right boundary of H and F 2 ≤ −2 in Z ) that F has self-interse tion ≤ −1 in X ′ , a ontradi tion. This observation proves the following two results: (i) if C ′ ⊲E ′ is ontra tible whi h is equivalent to say that ea h omponent of E ′ has self-interse tion −2 then H = F ; the onverse being obvious we obtain assertion a) and b). (ii) if E ′ is not empty the last urve ontra ted by σ ′ is the right boundary of E ′ . The same argument for σ a hieves to prove c).

2.3.4 (Des ription of reversions between 0-standard pairs by means of elementary links). A reversion between 0-standard pairs was introdu ed in [9℄. Given some pair with a zigzag of type (..., n1 , 0, n2 , ...), the blow-up of the point on the (0)- urve whi h also belongs to the next omponent, followed by the ontra tion of the proper transform of the (0)- urve yields to a pair with a zigzag of type (..., n1 + 1, 0, n2 − 1, ...). Starting from a 0-standard pair (X, B) of type (−nr , ..., −n1 , 0, 0), one an then onstru t a birational map ϕ1 : (X, B) 99K (X1 , B1 ) to a pair with a zigzag of type (−nr , ..., −n2 , 0, 0, −n1 ). Repeating this pro ess yields birational maps ϕ1 , ..., ϕr , and a reversion φ = ϕr ◦ ... ◦ ϕ1 : (X, B) 99K (Xr , Br ), where Br has type (0, 0, −nr , ..., −n1 ). ϕ2

ϕ1

−nr

−n2 −n1 ϕr

0

0

0

0

−nr

−nr

−n2

−n2

0

0

ϕ3

−n1 −nr

0

0

−n2

−n1

−n1

The onstru tion also de omposes the reversion into birational maps ϕi , where ea h ϕi preserves the A1∗ -bration on the open part that is given by the (0)- urve involved. However, the disadvantage of the de omposition is that (ϕi )−1 and ϕi+1 have the same proper base-point, whi h is the interse tion of the two (0)- urves of Xi+1 .†

2.3.5 (Des ription of reversions between 1-standard pairs by means of elementary links). On 1-standard pairs, the analogue of onstru tion 2.3.4 is possible. We start with a pair (X, B) of type (−nr , ..., −n2 , −n1 , −1, 0). We hoose a point p ∈ X that belongs to the (0)- urve of B but not to its (−1)- urve. The ontra tion of the (−1)- urve of B followed by the blow-up of p yields a birational map θ0 : (X, B) 99K (X0 , B0 ) to a pair with a zigzag of type (−nr , ..., −n2 , −n1 + 1, 0, −1). As before, we an † Note also that the same problem holds when dealing with reversion and bered modi ation on 0-standard pairs, whi h have the same proper base-point. There is thus no analogue of Lemma 2.1.1 for 0-standard pairs.

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produ e a birational map ϕ1 : (X0 , B0 ) 99K (X1′ , B1′ ), where B1′ is of type (−nr , ..., −n2 , −1, 0, −n1 + 1). The blow-down of the (−1)- urve followed by the blow-up of the point of interse tion of the (0)- urve with the urve immediately after it yields a birational map θ1 : (X1′ , B1′ ) 99K (X1 , B1 ) where B1 is a zigzag of type (−nr , ..., −n2 + 1, 0, −1, −n1). Repeating this pro ess yields birational maps θ0 , ϕ1 , θ1 , ..., ϕr , θr des ribed by the following gure. ϕ2

ϕ1

−nr

−nr

−n1

−1

θ0

−n2

θ2

−1

0 −n2+1 −n1

−nr

0

−n2 −n1+1 ϕr

−1

0

−1 −nr

0

−nr+1 −n2

−n2

−n1

θr

−1

0

0

−1

θ1 −n1+1 −nr −n2+1

−nr

−n2

0

−1

−n1

−n1

Then, the omposition φ = θr ϕr · · · θ1 ϕ0 θ0 is a birational map φ : (X, B) 99K (Xr , Br ) between two 1-standard pairs.

Lemma 2.3.6. The map φ dened in §2.3.5 is a reversion between the two 1-standard pairs (X, B) and (X ′ , B ′ ). Proof. Sin e φ is a birational map of pairs, it has one proper base-point only (Lemma 2.1.1), whi h is p: the unique proper base-point of θ0 . Denote by σ : Z → X the blow-up of the base-points of φ, so that σ ′ = φσ is a morphism. If q is a base-point of φ, distin t from p, then q is innitely near to p and orresponds to a base-point of some ϕi or some θi ; so q belongs to exa tly two omponents of the ˜ of B in Z is a zigzag, equal to B ⊲ H , for total transform of B . Consequently, the total transform B some zigzag H (here B ⊂ Z is the stri t transform of B ⊂ X ). Doing the same for φ−1 shows that the resolution given by σ and σ ′ satises the properties of Denition 2.3.1.

Proposition 2.3.7 (Uni ity of reversions). For every 1-standard pair (X, B = F ⊲C ⊲E) and every point q ∈ F \ C , there exists a 1-standard pair (X ′ , B ′ ) and a reversion φ : (X, B) 99K (X ′ , B ′ ), unique up to an isomorphism at the target, having q as a unique proper base-point. Furthermore, if B is of type (0, −1, −n1, . . . , −nr ) then B ′ is of type (0, −1, −nr , . . . , −n1 ). Proof. The existen e follows from §2.3.5 and Lemma 2.3.6 (it was also des ribed in the proof of [4, Proposition 2.10℄). It remains to prove uni ity. For i = 1, 2, let φi : (X, B = F ⊲C ⊲E) 99K (Xi , Bi = Fi ⊲Ci ⊲Ei ) be a σi′ σ ˜i ) → (Xi , Bi ) su h that reversion entered at q ∈ F \ C , admitting a minimal resolution (X, B) ←i (Zi , B t ˜ ˆ ˆ Bi = ( Ei ⊲Ci )⊲Hi ⊲(C⊲E). Denoting by η : (X, B) → (X, B) the blow-up of the ommon base-points of σ1 and σ2 , we have a ommutative diagram (W, D) OOOνO2 o o o ' wo ˜2 ) ˜1 ) (Z ,B (Z1 , B τ2 o 2 ?? ?? OOτO1 o wo ?? σ′ ?? ' σ1′ ?? 2 ? ˆ B) ˆ ??(X, ?? σ σ 2 1 ? ? ? η (X1 , B1 ) o_ _ _ _ _ _ (X, B) _ _ _ _ _ _/ (X2 , B2 ), ν1

φ1

φ2

ν2 ˜2 ) is the minimal resolution of ˜1 ) ← (W, D) → (Z2 , B where τi are birational morphisms, where (Z1 , B −1 (τ2 ) ◦ τ1 , and where ea h map is an isomorphism on the open part. We prove now that φ2 ◦ (φ1 )−1 is an isomorphism. We rst prove that either τ1 or τ2 is an isomorphism. Suppose the ontrary; then, for i = 1, 2 the ˆ (be ause so is σ −1 , Lemma 2.3.3). Re all that map (τi )−1 has a unique proper base-point pi ∈ X i ˆ orresponding to the ex eptional divisor of η . ˆ = A⊲F ⊲C ⊲E for some non-empty subzigzag A ⊂ B B ˆ, ˆ , belonging to the same omponent D ⊂ B Furthermore p1 , p2 are two distin t singular points of B whi h is the unique (−1)- urve of A. Assume that p1 belongs to the omponent of A whi h pre edes ˜2 = (tE2 ⊲C2 )⊲H2 ⊲(C ⊲E) belongs to a omponent whi h D, whi h implies that the point τ2−1 (p1 ) ∈ B ν1

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pre edes C2 ⊂ τ2−1 (p2 ) and thus belongs to E2 . Sin e ν2 blows-up τ2−1 (p1 ) and sends C1 onto this point, the map φ2 ◦ (φ1 )−1 sends the urve C1 on a point of E2 ⊂ B2 whi h is a proper base-point of φ1 ◦ (φ2 )−1 , ontrary to Lemma 2.1.1. We may now suppose that τ2 is an isomorphism, and onsider that it is the identity. We prove that ˆ = Z2 . Sin e σ1 so is τ1 . Suppose on the ontrary that (τ1 )−1 has a unique proper base-point p1 ∈ X

ontra ts a hain whi h ontains only one (−1)- urve, the point p1 belongs to the unique (−1)- urve of Z2 ontra ted by η = σ2 , i.e. p1 ∈ C2 . Sin e φ2 (φ1 )−1 ontra ts C1 on σ2′ (p1 ), the point σ2′ (p1 ) is a proper base-point of φ1 (φ2 )−1 and onsequently belongs to F2 , so p1 = F2 ∩ C2 ∈ Z2 . Hen e the ˜1 = (tE1 ⊲C1 )⊲H1 ⊲(C ⊲E). The fa t that φ1 is a reversion stri t transform of tE2 ⊲C2 pre edes C1 in B ˜1 is ontra ted by σ ′ , hen e φ1 (φ2 )−1 does not ontra t implies that no omponent at the left of C1 in B 1 any urve of B2 ex ept perhaps F2 ; this implies that φ1 (φ2 )−1 is a bered modi ation. But φ2 (φ1 )−1

annot be a bered modi ation, as it ontra ts C1 . ˜2 ) = B ˜1 . This means that The ontradi tion shows that τ1 is an isomorphism, hen e τ1 (τ2 )−1 (B neither φ1 (φ2 )−1 nor φ2 (φ1 )−1 ontra ts any urve, hen e both maps are isomorphisms. If the type of the subzigzag E of B = F ⊲ C ⊲ E is not a palindrome, then the omposition of two reversions annot be a reversion. However, the following shows that this may o

ur.

Lemma 2.3.8 (Composition of two reversions). For i = 1, 2, let φi : (X, B) 99K (Xi , Bi ) be a reversion, and assume that every irredu ible urve of B has self-interse tion ≥ −2. If the proper base-points of φ1 and φ2 are distin t (respe tively equal) the map φ2 ◦ (φ1 )−1 is a reversion (respe tively an isomorphism). Proof. Denote by r ≥ 0 the number of omponents of E (ea h one is a (-2)- urve). For i = 1, 2, ′

σi σ ˜i ) → (Xi , Bi = Fi ⊲ Ci ⊲ Ei ) be a minimal resolution of φi , su h that let (X, B = F ⊲ C ⊲ E) ←i (Zi , B ˜i = (tEi ⊲Ci )⊲Hi ⊲(C⊲E). Observe that Hi is the proper transform of F and Fi by respe tively (σi )−1 B and (σi′ )−1 , and that Ei is a hain of r (-2)- urves. We therefore have a ommutative diagram

˜i ) ′ (Zi , B OOσOOi o woo ' (X, B) _ _ _ _ _ _/ (Xi , Bi ) φi OOO o O wooνo′ νi ' ′ (Wi , Bi ), i σi o

where νi and νi′ ontra t the urves E ⊲C and Ei ⊲Ci respe tively. Sin e ν1 and ν2 ontra t the same

urves, we may assume that ν1 = ν2 = ν and (W1 , B1′ ) = (W2 , B2′ ) = (W, B ′ ). This yields the following

ommutative diagram: ˜2 ) ′ ˜1 ) (Z , B (Z1 , B σ2 o 2 OOσOO2 OOσOO1 o o o o o wo ' ' wo (X1 , B1 ) o_WW _ _ _ _ _ (X, B) _ _ _ _ _ _g/ (X2 , B2 ), WWWφW1W φ2 ggg WWWWW ν sggggggνg′g + ν1′ ′ 2 (W, B ) σ1′

where the proper base-point of (νi′ )−1 is equal to the image by ν of the proper base-point of φi (and (σi )−1 ). Consequently, if these two base-points are equal then φ2 ◦(φ1 )−1 = (ν2′ )−1 ◦ν1′ is an isomorphism, and otherwise it is a reversion.

Remark 2.3.9. By denition, a reversion φ : (X, B, π¯ ) 99K (X ′ , B ′ , π¯ ′ ) restri ts to an isomorphism ∼

φ : S = X \ B → S ′ = X ′ \ B ′ of quasi-proje tive surfa es, whi h, in ontrast with the ase of bered ¯ |S ) and (S ′ , π ¯ ′ |S ). Indeed, modi ations, is never an isomorphism of A1 -bered surfa es between (S, π 1 ′ ′ 1 it is easily seen that the rational brations π ¯ : X → P and π ¯ : X → P lift to rational brations with ˜ of φ. This implies that the indu ed A1 -brations distin t general bers on the minimal resolution (Z, B) π ¯ |S and π ¯ ′ ◦ φ |S on S have distin t general bers.

2.4. Summary on the base-points and urves ontra ted. Re all that the enter of a birational map (X, B) 99K (X ′ , B ′ ) is its unique proper base-point.

10

JÉRÉMY BLANC AND ADRIEN DUBOULOZ

Lemma 2.4.1. Let φ : (X, B = F ⊲C ⊲E) 99K (X ′ , B ′ = F ′ ⊲C ′ ⊲E ′ ) be a birational map. a) If φ is a bered modi ation, it is entered at p = F ∩ C , and F ′ is the only irredu ible omponent of B ′ ontra ted by φ−1 . b) If φ is a reversion, it is entered at a point p ∈ F \ C , and φ−1 ontra ts the urves C ′ and E ′ on p, and also ontra ts F ′ on p if and only if some irredu ible omponent of E ′ has self-interse tion ≤ −3. Proof. Follows respe tively from 2.2.4 and Lemma 2.3.3. 3.

Fa torization of birational maps between

1-standard

pairs

This se tion is devoted to the proof of the following result.

Theorem 3.0.2. Let φ : (X, B) 99K (X ′ , B ′ ) be a birational map between 1-standard pairs restri ting ∼ to an isomorphism X \ B → X ′ \ B ′ . If φ is not an isomorphism then it an be de omposed into a nite sequen e φ2

φ1

φn

φ = φn ◦ · · · ◦ φ1 : (X, B) = (X0 , B0 ) 99K (X1 , B1 ) 99K · · · 99K (Xn , Bn ) = (X ′ , B ′ )

of bered modi ations and reversions between 1-standard pairs (Xi , Bi ), i = 1, . . . , n. Furthermore, su h a fa torization of minimal length is unique, whi h means that if φ′1

φ′n

φ = φ′n ◦ · · · ◦ φ′1 : (X, B) = (X0′ , B0′ ) 99K · · · 99K (Xn′ , Bn′ ) = (X ′ , B ′ )

is another fa torization, then there exist isomorphisms of pairs αi : (Xi , Bi ) → (Xi′ , Bi′ ) for i = 1, ..., n su h that αi ◦ φi = φ′i ◦ αi−1 for i = 2, ..., n.

3.0.3. Let us ompare Theorem 3.0.2 with the existing results in the literature. Sin e φ restri ts to an isomorphism between X \ B and X ′ \ B ′ , we know that it an be fa tored into a sequen e of smooth blow-ups and ontra tion with enters on the su

essive boundaries. A rened des ription of su h fa torizations, based on a areful study of base-points of the birational maps under onsideration, was obtained by V. Danilov and M. Gizatullin [6℄. Namely, they established that one an always nd a fa torization as above with the additional property that the boundaries of all intermediate pairs onsist of a ertain type of zigzags alled standard in lo . it. Moreover, su h a fa torization of minimal length is unique up to omposition by automorphisms of the intermediate proje tive surfa es preserving the boundaries. In general, the intermediate pairs whi h arise in a Danilov-Gizatullin fa torization ψ1

ψ2

ψs

φ : (X, B) = (X0 , B0 ) 99K (X1 , B1 ) 99K · · · 99K (Xs , Bs ) = (X ′ , B ′ )

of φ of minimal length are not all 1-standard. However, there is an obvious way to on atenate these , B maps into a sequen e of birational maps φ : X between all su

essive , B 99K X α j+1 α α α j j j+1 j+1 1-standard pairs Xαj , Bαj among the pairs (Xi , Bi ) o

urring in the fa torization. Theorem 3.0.2 would follow provided that we show that the birational maps obtained by this pro edure are either reversions or bered modi ations. This is the ase, and the uniqueness properties a tually imply that a minimal fa torization as in Theorem 3.0.2 oin ides with a one obtained from a minimal DanilovGizatullin fa torization by the above pro edure. But a proof of this fa t would require to redo a areful analysis of the base-points of the birational maps φ : (X, B) 99K (X ′ , B ′ ) under onsideration. So we nd it simpler and more enlightening to give a omplete and self- ontained proof. We pro eed in two steps. First we show in 3.1 below that every birational map φ : (X, B) 99K (X ′ , B ′ ) ∼ between 1-standard pairs restri ting to an isomorphism φ : X \ B → X ′ \ B ′ an be de omposed in an essentially unique sequen e of elementary birational maps between a ertain lass of pairs whi h stri tly ontains the 1-standard ones. Then we he k in 3.2 that these elementary birational maps an be on atenated into sequen es of reversions and bered modi ations between the 1-standard pairs o

urring in the fa torization.

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3.1. Elementary birational links between almost standard pairs. Here we onstru t an enlargement of the lass of 1-standard pairs onsisting of pairs (X, B) with a boundary zigzag B of a more general type. We show that within this lass every birational map ∼ φ : (X, B) 99K (X ′ , B ′ ) between 1-standard pairs restri ting to an isomorphism φ : X \ B → X ′ \ B ′

an be de omposed into an essentially unique sequen e of suitable elementary birational links onsisting of either smooth blow-ups or ontra tions. Some of the results of this subse tion are losely related to those of in [9℄.

Denition 3.1.1. A pair (X, B) onsisting of a normal rational proje tive surfa e X and a nonempty zigzag B = B1 ⊲· · ·⊲Bs (ea h Bi being irredu ible) supported in the regular part of X is alled almost standard if the following hold : a) There exists a unique irredu ible omponent Bm of B with non-negative self-interse tion, alled the positive urve of B ; b) There exists at most one irredu ible omponent Bl of B with self-interse tion (Bl )2 = −1. Furthermore, if it exists it is alled the (−1)- urve of B and Bm · Bl = 1, i.e., l = m ± 1. Denition 3.1.2. Let (X, B) be an almost standard pair and let Bm be the positive urve of B . A birational map φ : (X, B) 99K (X ′ , B ′ ) between almost standard pairs is alled an elementary link if it onsists of one of the following four operations : I) The ontra tion of the (−1)- urve of B if it exists, II) If B ontains a (−1)- urve Bm±1 , the blow-up of the interse tion point p of Bm with Bm±1 , immediately followed by the ontra tion of the stri t transform of Bm when (Bm )2 = 0 in X . Bm Bm±1

a

−1

Bm

a−1

Bm±1

−1

−2

a=0

Bm±1

−2

0

III) If B ontains no (−1)- urve and if Bm is not a boundary of B , the blow-up of one of the two points p = Bm−1 ∩ Bm or p = Bm ∩ Bm+1 , immediately followed by the ontra tion of the stri t transform of Bm when (Bm )2 = 0 in X . Bm±1 Bm Bm∓1

−b

a

−c

Bm±1 Bm

−b

a−1

−1

Bm∓1

−c−1

a=0

Bm±1

−b+1

0

Bm±1

−c−1

IV) If B ontains no (−1)- urve and if Bm is a boundary of B , the blow-up of an arbitrary point p ∈ Bm , immediately followed by the ontra tion of the stri t transform of Bm when (Bm )2 = 0 in X . As before, the elementary links of type II), III) and IV) are said to be entered at p.

Proposition 3.1.3. Let φ : (X, B) 99K (X ′ , B ′ ) be a birational map between almost standard pairs ∼ restri ting to an isomorphism X \ B → X ′ \ B ′ . Then φ is either an isomorphism or it an be fa tored into a nite sequen e ϕ1

ϕ2

ϕr

φ = ϕr ◦ · · · ◦ ϕ1 : (X, B) = (X1 , B1 ) 99K (X2 , B2 ) 99K · · · 99K (Xr , Br ) = (X ′ , B ′ )

of elementary links between almost standard pairs. Proof. We pro eed by indu tion on the total number of base-points s φ, φ−1 of φ and φ−1 . If

σ s φ, φ−1 = 0 then φ is an isomorphism. We assume thus that s φ, φ−1 > 0, and let (X, B) ← ′ σ ˜ → (Z, B) (X ′ , B ′ ) be the minimal resolution of φ, where the birational morphisms σ and σ ′ onsist of blow-ups of the su

essive base points of φ and φ−1 respe tively. The map σ ′ ontra ts at most two (−1)- urves of B , namely the proper transforms of the positive urve Bm and of the unique possible (−1)- urve Bl of B if it exists. If the proper transforms of Bm and Bl in Z are both (−1)- urves then Bm · Bl = 1 in Z ne essarily and so, they annot be both ex eptional for σ . Therefore φ has at most one proper base-point. In turn, this implies that φ and all its su

essive lifts to the intermediate pairs o

urring in the de omposition of σ into a sequen e of smooth blow-ups have a unique proper base-point. A similar des ription holds for φ−1 . If the proper base-point of φ (respe tively φ−1 ) orresponds to the proper transform in Z of the unique possible (−1)- urve of B ′ (respe tively of B ) we fa tor φ by the ontra tion of this (−1)- urve;

12

JÉRÉMY BLANC AND ADRIEN DUBOULOZ

this de reases the total number of base-points of φ (respe tively φ−1 ). We may thus assume that the unique proper base-points p and q of φ and φ−1 respe tively orrespond to the positive urve of B ′ and B respe tively (if φ ontra ts the positive urve, it has to have a base-point, so either both have base-points or φ is an isomorphism, a ase eliminated before). We de rease s φ, φ−1 by performing an elementary link with enter at p ∈ Bm . The existen e of su h a link is lear if Bm is a boundary of B and if there is no (−1)- urve in B . Otherwise, we distinguish two ases : a) If B ontains a (−1)- urve Bl , then Bm interse ts it and p = Bm ∩Bl ne essarily. Indeed, otherwise, after the ontra tion of the proper transform of Bm by σ ′ , Bl would be a urve with non-negative selfinterse tion and not ontra ted by φ−1 , whi h ontradi ts our assumptions. b) If Bm interse ts two irredu ible omponents Bm−1 and Bm+1 of B with self-interse tion ≤ −2, then p = Bm−1 ∩Bm or p = Bm ∩Bm+1 . Indeed, otherwise, after the ontra tion of the proper transform of Bm by σ ′ , the total transform of B would not B an SNC divisor, whi h is absurd. We on lude that in any of these two ases, φ an be fa tored through an elementary link with enter at p, of type II) in ase a) and of type III) in ase b). This ompletes the proof.

Example 3.1.4. (Fa torization of bered modi ations). Let φ : (X, B = F ⊲C ⊲E) 99K (X ′ , B ′ )

be a bered modi ation between 1-standard pairs lifting a triangular automorphism φ : (x, y) 7→ (ax + b, cy + P (x)) of A2 , where d = deg P ≥ 2 as in Lemma 2.2.3. It follows from the des ription of the resolution of su h a birational map given in 2.2.4 that φ fa tors into a sequen e of d − 1 elementary links of type IV) with enters at the interse tion of the positive urve with the proper transform of C , followed by a sequen e of d − 1 elementary link of type IV) with enters at points outside the proper transform of C . One easily he ks that this fa torization is obtained by the lift (des ribed in Lemma 2.2.3) of the fa torization of the orresponding birational map φ : (F1 , F1 ⊲C0 ) 99K (F1 , F1 ⊲C0 ), whi h onsists of a sequen e of d − 1 elementary transformations σi : (Fi , Fi ⊲C0 ) 99K (Fi+1 , Fi+1 ⊲C0 ) with enter at Fi ∩ C0 , i = 1, . . . , d − 1, followed by a sequen e of d − 1 elementary transformations ′ ′ σi′ : Fi+1 , Fi+1 ⊲C0 99K (Fi , Fi′ ⊲C0 ) with enter at a point of Fi+1 \ C0 , i = d − 1, . . . , 1.

Example 3.1.5. (Fa torization of reversions). Let φ : (X, B = F ⊲C ⊲E) 99K (X ′ , B ′ ) be a reversion

between 1-standard pairs, where B is of type (0, −1, −n1, ..., −nr ). A

ording to Lemma 2.3.6 and Proposition 2.3.7, φ may be de omposed as φ = θr ϕr ...ϕ0 θ0 , where ϕi : (Xi−1 , Bi−1 ) 99K (Xi′ , Bi′ ) and θi : (Xi′ , Bi′ ) 99K (Xi , Bi ) are dened in §2.3.4 (note that (X, B) = (X0′ , B0′ )).

ϕ2

ϕ1

−nr

−nr

−n1

−1

θ0

−n2

θ2

−1

0 −n2+1 −n1

−nr

0

−n2 −n1+1 ϕr

−1

0

−1 −nr

0

−nr+1 −n2

−n2

−n1

θr

−1

0

0

−1

θ1 −n1+1 −nr −n2+1

−nr

−n2

0

−1

−n1

−n1

If ni ≥ 3, then (Xi−1 , Bi−1 ) and (Xi′ , Bi′ ) are almost-standard pairs and ϕi : (Xi−1 , Bi−1 ) 99K (Xi′ , Bi′ ) is the omposition of a link of type II and ni − 3 links of type III. If ni = 2, then ϕi is an isomorphism between two pairs whi h are not almost-standard (there are two (−1)- urves in the boundary). Let ni , ni + 1, ..., ni + m be a sequen e of multipli ities equal to 2 (with m ≥ 0), su h that either ni−1 respe tively ni+m+1 does not exist (i = 1 or i + m = r) or is stri tly bigger than 2. Then, the map θi+m ϕi+m · · · θi ϕi θi−1 is the omposition of m + 2 links of type I and m + 2 links of type III or IV. The remaining maps to de ompose are the θi whi h are between two almost standard pairs. Then θi is the omposition of a link of type I and a link of type III (respe tively IV), if i > 0 (respe tively if i = 0). 3.2. Con atenating elementary links into birational maps between 1-standard pairs.

3.2.1. Given a birational map φ :(X, B) 99K (X ′ , B ′ ) between 1-standard pairs, restri ting to an iso∼

morphism X \ B → X ′ \ B ′ , it follows from Proposition 3.1.3 that there exists a fa torization ϕ1

ϕ2

ϕr

φ = ϕr ◦ · · · ◦ ϕ1 : (X, B) = (X1 , B1 ) 99K (X2 , B2 ) 99K · · · 99K (Xr , Br ) = (X ′ , B ′ )

AUTOMORPHISMS OF

A1 -FIBERED

AFFINE SURFACES

13

of φ into a nite sequen e of elementary links between almost standard pairs. By on atenating these elementary links into birational maps φj = ϕαj+1−1 ◦ · · · ϕαj +1 ◦ ϕαj : Xαj , Bαj 99K Xαj+1 , Bαj+1 between all su

essive 1-standard pairs Xαj , Bαj among the pairs (Xi , Bi ) o

urring in the fa torization φ = ϕr ◦ · · ·◦ ϕ1 , we obtain a new fa torization of φ = φn ◦ · · ·◦ φ1 into a nite sequen e of birational maps between 1-standard pairs. The following lemma gives the rst part of the proof of Theorem 3.0.2. Lemma 3.2.2. The birational maps φj : Xαj , Bαj 99K Xαj+1 , Bαj+1 dened above are either rever-

sions or bered modi ations.

Proof. There is only two possible elementary links starting with a 1-standard pair (X, B = F ⊲C ⊲E),

namely the ontra tion of the (−1)- urve C , or the blow-up of the point F ∩C followed by the ontra tion of the proper transform of F . It is enough to show that ea h possibility gives rise to a birational map whi h is reversion in the rst ase and a bered modi ation on the se ond one. a) If ϕ1 is the ontra tion of the (−1)- urve C then one he ks easily that the only possible subsequen e of elementary links o

urring in the de omposition of φ before we rea h the rst 1-standard pair (Xα1 , Bα1 ) oin ides with the one des ribed in Example 3.1.5 above; indeed at ea h step there are only two possible links, one being the inverse of the last link produ ed. This shows that if ϕ1 is the

ontra tion of the (−1)- urve C , then φ1 : (X, B) 99K (Xα1 , Bα1 ) is a reversion. b) If ϕ1 : (X, B) 99K (X1 , B1 ) is the blow-up of the point F ∩ C followed by the ontra tion of the proper transform of F , then the proper transform of C has self-interse tion −2, and interse ts the (0)- urve Fi produ ed, whi h is the boundary of Bi , for i = 1. Until the self-interse tion of (the proper transform of) C be omes −1 again, the elementary links ϕi+1 : (Xi , Bi ) 99K (Xi+1 , Bi+1 ), i = 1, . . . , d − 1 onsist ne essarily of a sequen e of the blow-up of a point of the (0)- urve Fi of Bi having self-interse tion 0 followed by the ontra tion of the proper transform of this urve. Consequently, the map φ1 does not ontra t the urve C , whi h is a se tion of the bration on (X, B), and thus φ1 : (Xα1 , Bα1 ) 99K (Xα2 , Bα2 ) is a bered modi ation. As a onsequen e of the des riptions, we re over [6, Corollary 2℄ :

Corollary 3.2.3. If (X, B) and (X ′ , B ′ ) are two 1-standard pairs of type (0, −1, −n1 , ..., −nr ) and (0, −1, −n′1, ..., −n′s ) su h that X \ B ∼ = X ′ \ B ′ , then r = s and either ni = n′i for ea h i or ni = n′r+1−i for ea h i. Proof. Denote by φ : X 99K X ′ the birational map obtained by extension of the isomorphism. Lemma 3.2.2 yields a de omposition of φ into bered modi ations and reversions; the bered modi ations do not

hange the type of the zigzag and the reversions reverse the order of the ni . Now that the existen e of the fa torization of Theorem 3.0.2 is proved, it remains to dedu e the uni ity. It is a onsequen e of the following lemma, whi h ompletes the proof of the theorem.

Lemma 3.2.4. Let φ : (X, B = F ⊲C ⊲E) 99K (X ′ , B ′ ) be a stri tly birational map between 1-standard ∼ pairs restri ting to an isomorphism X \ B → X ′ \ B ′ and let φ2

φ1

φn

φ = φn ◦ · · · ◦ φ1 : (X, B) = (X0 , B0 ) 99K (X1 , B1 ) 99K · · · 99K (Xn , Bn ) = (X ′ , B ′ )

be a de omposition of φ, for n ≥ 1, satisfying that φi is either a reversion or a bered modi ation. Then, the following are equivalent: (1) the de omposition above is minimal (i.e. there does not exist another su h de omposition with less than n fa tors); (2) for any i < n, the enters of (φi )−1 and φi+1 are distin t, and if φi and φi+1 are reversions then E ontains at least one urve of self-interse tion ≤ −3. Furthermore, if the onditions are satised, the following hold: (a) the map φ is not an isomorphism, and the enters of φ and φ1 (respe tively of φ−1 and (φn )−1 ) are equal; φ′1

φ′n

(b) if φ = φ′n ◦ · · · ◦ φ′1 : (X, B) = (X0′ , B0′ ) 99K · · · 99K (Xn′ , Bn′ ) = (X ′ , B ′ ) is another fa torization, there exist isomorphisms αi : (Xi , Bi ) → (Xi′ , Bi′ ) for i = 1, ..., n su h that αi ◦ φi = φ′i ◦ αi−1 for i = 2, ..., n.

14

JÉRÉMY BLANC AND ADRIEN DUBOULOZ

Proof. For any i, we write (Xi , Bi = Fi ⊲Ci ⊲Ei ), and re all that the type of Ei is equal to the type of

E or of tE (Corollary 3.2.3). We now prove the impli ation (1) ⇒ (2), or in fa t its ontraposition. First assume that (φi )−1 and φi+1 have the same proper base-point p ∈ Bi . A

ording to Lemma 2.4.1, either p = Fi ∩ Ci and both (φi )−1 and φi+1 are bered modi ations, or p ∈ Fi \ Ci and (φi )−1 and φi+1 are reversions. In the rst

ase, φi+1 ◦ φi is a bered modi ation and in the se ond this map is an isomorphism; the de omposition is thus not minimal. Assume now that φi and φi+1 are reversions and E (and thus Ei ) is a hain of (−2)- urves. Lemma 2.3.8 shows that φi+1 ◦ φi is either a reversion or an isomorphism, and on e again the de omposition is not minimal. We now prove (2) ⇒ (a). Sin e (2) is symmetri , it su es to assume (2) and to prove by indu tion on n that φ is not an isomorphism and that the enter of φ−1 and (φn )−1 are equal. If n = 1, this is obvious. If n > 1, the map ψ = φn−1 ◦ ... ◦ φ1 ontra ts some urve on the enter p ∈ Bn−1 of (φn−1 )−1 , by indu tion hypothesis. Then p is not a base-point of φn , and φn ontra ts a urve Γ ⊂ Bn−1 that

ontains p. Consequently, the map φ = φn ◦ ψ ontra ts a urve on φn (Γ) = φn (p). This point is furthermore the enter of (φn )−1 (Lemma 2.4.1). Assume now that two de ompositions φ = φn ◦ ... ◦ φ1 = φ′m ◦ ... ◦ φ′1 satisfying (2) exist. Then, the identity map fa tors as φ′m ◦ ... ◦ φ′1 ◦ (φ1 )−1 ◦ ... ◦ (φn )−1 . Sin e it is an isomorphism, ondition (2) is not satised for this de omposition. Three possibilities o

ur; in ea h one we prove that φ′1 ◦ (φ1 )−1 is an isomorphism. i) both φ′1 and (φ1 )−1 are bered modi ations. In this ase, φ′1 ◦(φ1 )−1 is either a bered modi ation or an isomorphism; the rst ase is not possible as it yields a de omposition of the identity satisfying (2). ii) both φ′1 and (φ1 )−1 are reversions with the same enter. Proposition 2.3.7 shows that φ′1 ◦ (φ1 )−1 is an isomorphism. iii) both φ′1 and (φ1 )−1 are reversions with distin t enters, and ea h irredu ible urve of E has selfinterse tion ≥ −2. This ase is not possible, as it implies that φ′1 ◦ (φ1 )−1 is a reversion (Lemma 2.3.8) and yields a de omposition of the identity satisfying (2). Denote by α1 the isomorphism φ′1 ◦ (φ1 )−1 and repla e it in the de omposition of the identity written above. Writing ψ2′ = φ′2 ◦ α1 , whi h is again a reversion or a bered reversion, we nd as before that ψ2′ ◦ (φ2 )−1 is an isomorphism, that we denote by α2 . By indu tion, we dene ψr′ = φ′r ◦ αr−1 and obtain an isomorphism αr = ψr′ ◦ (φr )−1 for r = 2, ..., m + 1. The last relation obtained is αm = id, whi h shows that m = n. Choosing α0 = id we nd that αi ◦ φ1 = φ′i ◦ αi−1 for i = 1, ..., n. This proves the two remaining impli ations needed, whi h are (2) ⇒ (1) and (2) ⇒ (b).

4.

Graphs asso iated to pairs and fibrations

In this se tion, we asso iate a graph to every normal quasi-proje tive surfa e S admitting a ompletion by a 1-standard pair. The graph ree ts the A1 -brations on S and the links between these.

Denition 4.0.5. To every normal quasi-proje tive surfa e S we asso iate the oriented graph FS , dened in the following way: a) A vertex of FS is an equivalen e lass of 1-standard pairs (X, B) su h that X \ B ∼ = S , where two 1-standard pairs (X1 , B1 , π1 ), (X2 , B2 , π2 ) dene the same vertex if and only if the A1 -bered surfa es (X1 \ B1 , π1 ) and (X2 \ B2 , π2 ) are isomorphi . b) Any arrow of FS is an equivalen e lass of reversions. If φ : (X, B) 99K (X ′ , B ′ ) is a reversion, then the lass [φ] of φ is an arrow starting from the lass [(X, B)] of (X, B) and ending at the lass [(X ′ , B ′ )] of (X ′ , B ′ ). To follow the notation of [16℄, we write o([φ]) = [(X, B)] and t([φ]) = [(X ′ , B ′ )] for respe tively the origin and target. Two reversions φ1 : (X1 , B1 ) 99K (X1′ , B1′ ) and φ2 : (X2 , B2 ) 99K (X2′ , B2′ ) dene the same arrow if and only if there exist two isomorphisms θ : (X1 , B1 ) → (X2 , B2 ) and θ′ : (X1′ , B1′ ) → (X2′ , B2′ ), su h that φ2 ◦ θ = θ′ ◦ φ1 . Remark 4.0.6. Note that, as in [16, 2.1℄, this graph is oriented, and that any arrow a admits an inverse

arrow a ¯, whi h is the lass of θ−1 for any θ su h that a = [θ]. However, ontrary to the denition of [16℄, here it is possible that a = a ¯. The fa torization theorem yields the following basi properties for the graph FS .

AUTOMORPHISMS OF

A1 -FIBERED

AFFINE SURFACES

15

Proposition 4.0.7. Let S be a normal quasi-proje tive surfa e with a non-empty graph FS . Then, the following hold. a) The graph FS is onne ted. b) There is a natural bije tion between the set of verti es of FS and the set of equivalen e lasses of A1 -brations on S (see Denition 1.0.4). c) Assume that (X, B) is a 1-standard pair with X \ B ∼ = S and that B ontains at least one urve of self interse tion ≤ −3. Then, the graph FS is a tree if and only if Aut(S) is generated by automorphisms of A1 -brations on S . Moreover, we have a natural exa t sequen e 1 → H → Aut(S) → Π1 (FS ) → 1,

where H is the (normal) subgroup of Aut(S) generated by all automorphisms of A1 -brations and Π1 (FS ) is the fundamental group of the graph FS . Proof. The onne tedness is a dire t onsequen e of Theorem 3.0.2. In the sequel, we x a vertex v of

FS and a 1-standard pair (Xv , Bv ) with Xv \ Bv = S and v = [(X, B)]. If α : S → A1 is a A1 -bration, then there exists a 1-standard pair (X, B, π) and an isomorphism (S, α) → (X \ B, π) of A1 -bered surfa es. The isomorphism lass of (S, α) gives the one of (X \ B, π), whi h is equal to the vertex [(X, B)]. This yields b). Given any birational map φ : (Xv , Bv ) 99K (Xv , Bv ), we use Theorem 3.0.2 to write φ = θn+1 rn · · · θ2 r1 θ1 , where n ≥ 0, ea h ri is a reversion and ea h θi is a bered birational map between 1-standard pairs (whi h may be the identity). We asso iate to φ the element [rn ][rn−1 ] · · · [r2 ][r1 ] of the fundamental group Π1 (FS , v). Observe that be ause B ontains at least a urve of self-interse tion ≤ −3, the element of Π1 (FS , v) does not depend of the hoi e of the de omposition (Lemma 3.2.4) and the map dened is a surje tive homomorphism ν : Aut(S) → Π1 (FS , v). Given an A1 -bration β : S → A1 , let ψ : (Xv , Bv ) 99K (X, B, π) be a birational map of pairs su h that ψ restri ts to an isomorphism (S, β) → (X \ B, π). Then, the group Aut(S, β) is equal to ψ −1 Aut(X \ B, π)ψ . By onstru tion, this group is ontained in the kernel of ν . Take an element φ = θn+1 rn · · · θ2 r1 θ1 as before, and assume that ν(φ) = 1. We prove by indu tion on the number of reversions in the de omposition (here n) that φ belongs to H . If n = 0, φ is a bered birational map of (Xv , Bv ). Otherwise, [ri+1 ][ri ] vanishes in Π1 (FS ), for some i, whi h means that ri+1 = γ ◦ (ri )−1 ◦ δ for ertain isomorphisms of 1-standard pairs γ and δ . Writing φ = ϕ′ ri+1 θi+1 ri θi ϕ, we have φ = ϕ′ γ(ri )−1 δθi+1 ri θi ϕ = ϕ′ γθi ϕ(ri θi ϕ)−1 δθi+1 (ri θi ϕ). Sin e (ri θi ϕ)−1 δθi+1 (ri θi ϕ) ∈ H , we may on lude by applying indu tion hypothesis to ϕ′ γθi ϕ.

Then, we give to the graph FS a natural stru ture of graph of groups. Before doing it in Denition 4.0.9, we re all the notion of graph of groups, following [16, 4.4℄.

Denition 4.0.8. Let G be a graph. • A graph of groups stru ture on G is given by the hoi e of

a) a group Gv , for any vertex v of G ; b) a group Ga and an inje tive morphism ρa : Ga → Gt(a) , for any arrow a of G ; ¯ = x for any x ∈ Ga . c) an anti-isomorphism ¯: Ga → Ga¯ , for any arrow a, su h that x • A path in the graph of groups is a sequen e gn an−1 gn−1 · · · a2 g2 a1 g1 , where ai is an arrow from vi to vi+1 and gi ∈ Gvi . The path starts at v1 and ends at vn , and is losed if and only if v1 = vn . • The fundamental group of the graphs of groups at the vertex v onsists of losed paths starting and ¯ and a¯ ending at v , modulo the relations ρa (h) · a = a · ρa¯ (h) a = 1 for any arrow a and any h ∈ Ga .

Note that ρa (g) is written g a in [16℄; furthermore, the two groups Ga and Ga¯ are said to be equal, whi h yields the same stru ture as our denition, but is less onvenient for the following denition.

Denition 4.0.9. Let S be a normal quasi-proje tive surfa e and let FS its asso iated graph. Then, a graph of groups stru ture on FS is given by the hoi e of a) for any vertex v of FS , a xed 1-standard pair (Xv , Bv , πv ) in the lass v . The group Gv is then equal to Aut(Xv \ Bv , πv ); ra b) for any arrow a of FS , a reversion ra in the lass of a, whi h is (Xa , Ba , πa ) 99K (Xa′ , Ba′ , πa′ ), and ′ ′ ′ also an isomorphism µa : (Xa \ Ba , πa ) → (Xt(a) \ Bt(a) , πt(a) ). The group Ga is then equal to {(φ, φ′ ) ∈ Aut(Xa , Ba ) × Aut(Xa′ , Ba′ ) | ra ◦ φ = φ′ ◦ ra },

16

JÉRÉMY BLANC AND ADRIEN DUBOULOZ

and ρa : Ga → Gt(a) is given by ρa ((φ, φ′ )) = µa ◦ φ′ ◦ (µa )−1 . We ask further that ra¯ = (ra )−1 , and ¯: Ga → Ga¯ is the map (φ, φ′ ) 7→ (φ′ , φ).

4.0.10. In most ases (in parti ular when the subzigzag E of B = F ⊲C ⊲E is not a palindrome) a 6= a¯

for any arrow a of FS , and it is lear that a graph of groups stru ture exists on FS . If a = a ¯ for a

ertain a, then we may hoose that (Xa , Ba , πa ) = (Xa′ , Ba′ , πa′ ) and we have (ra )−1 = λ ◦ ra ◦ µ for some elements λ, µ ∈ Aut(Xa , Ba ). Repla ing ra by µra we may hoose that µ = 1. Consequently, (ra )2 = ra¯ ra ∈ Aut(Xa , Ba ). But, it is not lear that this one an always be hosen to be the identity. However, we will see that this property is satised for all the ases that we deal with in the sequel.

Theorem 4.0.11. Let (X, B) be a 1-standard pair su h that at least one omponent of B has selfinterse tion ≤ −3, and let S = X \ B . If FS admits a stru ture of graph of groups, then the fundamental group of the graph of groups obtained is naturally isomorphi to Aut(S). Proof. Let us x a graph of groups stru ture for FS , as in Denition 4.0.9. We will work with g -sequen es s = gn an gn−1 · · · a1 g0 , where n ≥ 1, gi ∈ Gvi for i = 0, ..., n and

ai is an arrow for i = 1, ..., n, satisfying o(ai ) = vi−1 , t(ai ) = vi . We write t(s) = vn and o(s) = v1 . There is a natural way of on atening g -sequen es s1 , s2 to s2 s1 satisyfying t(s1 ) = o(s2 ), by multiplying the last term of s1 with the rst of s2 . Then, to any g -sequen e s, we an asso iate a birational map ψs : (Xo(s) , Bo(s) ) 99K (Xt(s) , Bt(s) ), by saying that ψa = µa ◦ra ◦(µa¯ )−1 : (Xo(a) , Bo(a) ) 99K (Xt(a) , Bt(a) ) for any arrow a of FS , that ψg = g : (Xv , Bv ) 99K (Xv , Bv ) for any g ∈ Gv , and that ψss′ = ψs ◦ ψs′ , for any g -sequen es s, s′ with t(s) = o(s′ ). Let us prove now that for any two verti es v, v ′ and any birational map φ : (Xv , Bv ) 99K (Xv′ , Bv′ ) there exists a g -sequen e s su h that φ = ψs . We de ompose φ into a minimal sequen e of bered modi ations and reversions, using Theorem 3.0.2, and pro eed by indu tion on the number of reversions that o

ur in the de omposition. If there is no reversion, φ is a bered birational map, thus v = v ′ and φ ∈ Gv . Otherwise, φ = φ′ ◦ θ2 ϕθ1 , where θ1 and θ2 are bered birational maps (whi h may be isomorphisms), ϕ is a reversion and the de omposition of φ′ involves less reversions than the one of φ. Up to isomorphisms, whi h hange the maps θ1 and θ2 , we may assume that ϕ = ra for some arrow a starting from v . Sin e ra is a birational map starting from (Xa , Ba ) and both θ1 and (µa¯ )−1 are bered modi ation or isomorphisms (Xv , Bv ) 99K (Xa , Ba ), the map µa¯ θ1 belongs to Gv . We write φ = φ′ θ2 (µa )−1 ψa (µa¯ θ1 ) and use indu tion hypothesis on the map φ′ θ2 (µa )−1 : (Xt(a) , Bt(a) ) 99K (Xv , Bv ) to on lude. Let us x a vertex w of FS , write S = Xw \ Bw , and denote by Λ the group of g -sequen es s su h that t(s) = o(s) = w. The map s 7→ ψs yields a surje tive homomorphism Ψ : Λ → Aut(Xw \ Bw ) = Aut(S). The fundamental group of the graph of groups at w is the quotient of the group Λ by the relations ¯ , and a¯ ρa (h) · a = a · ρa¯ (h) a = 1 for any arrow a and any h ∈ Ga . To prove the theorem we prove that these relations generates the kernel of Ψ. Let a be an arrow. Then, ψa = µa ◦ ra ◦ (µa¯ )−1 and ψa¯ = µa¯ ◦ ra¯ ◦ (µa )−1 . Sin e ra¯ = (ra )−1 , we have ψa¯a = id. Re all that Ga = {(φ, φ′ ) ∈ Aut(Xa , Ba ) × Aut(Xa′ , Ba′ ) | ra ◦ φ = φ′ ◦ ra }. If h = (φ, φ′ ) ∈ Ga , ¯ = µa¯ ◦ φ ◦ (µa¯ )−1 ∈ Go(a) by denition. Consequently, then ρa (h) = µa ◦ φ′ ◦ (µa )−1 ∈ Gt(a) and ρa¯ (h) ′ −1 are equal. This the birational maps ψρa (h)a = µa ◦ φ ◦ ra ◦ (µa¯ )−1 and ψaρa¯ (h) ¯ = µa ◦ ra ◦ φ ◦ (µa ¯) shows that ea h relation of the fundamental group is satised in Aut(S). Let s = gn an gn−1 · · · a1 g0 ∈ Λ as above, and suppose that ψs = id. We prove by indu tion on n that s is trivial in the fundamental group. If n = 0, then s = g0 ∈ Gv , and ψg0 = id means that g0 = 1. Assume now that n > 0. We x ϕ0 = (µa1 )−1 ◦ g0 , ϕi = (µai+1 )−1 ◦ gi ◦ µai for 1 ≤ 1 ≤ n − 1 and ϕn = gn ◦ µan . Sin e ψai = µai ◦ rai ◦ (µai )−1 for ea h i, ψs de omposes as ψs = ϕn ran · · · ϕ1 ra1 ϕ0 , where ea h rai is a reversion and ea h ϕi is a bered birational map. Be ause ψs is the identity, there are simpli ations in this de omposition, whi h means (by Theorem 3.0.2, and be ause the boundary ontains at least a urve of self-interse tion ≤ −3) that for some j ∈ {1, ..., n − 1} the map ϕj is an isomorphism of 1-standard pairs whi h sends the proper base-point of (raj )−1 on the one of raj+1 . Consequently, (raj )−1 and raj+1 ϕj are two reversions entred at the same point, so raj+1 ϕj = θ(raj )−1 for some isomorphism of pairs θ. This means that aj = [(raj )−1 ] = [raj+1 ] = aj+1 , when e (raj )−1 = raj = raj+1 . Moreover, (ϕj , θ) ∈ Gaj+1 = Gaj , so the element h = (θ, ϕj ) belongs to Gaj = Gaj+1 . Thus we have ¯ ϕj = (µaj )−1 ◦ gj ◦ µaj , whi h means that ρaj (h) = µaj ◦ ϕj ◦ (µaj )−1 = gj . Sin e aj ρaj (h)aj = ρaj (h)

AUTOMORPHISMS OF

A1 -FIBERED

AFFINE SURFACES

17

¯ ∈ Gt(a ) in the de omposition of s, and in the fundamental group, we may repla e aj+1 gj aj by ρa¯j (h) j+1 redu e its length. By indu tion, we nd that s is trivial in the fundamental group.

5.

Expli it examples of affine surfa es

In this se tion, we apply the tools used before (espe ially Lemma 1.0.7 and Theorem 3.0.2) to des ribe examples of ane surfa es. 5.1. Expli it form - notation. A

ording to Lemma 1.0.7 the resolution of singularities of any 1standard pair may be obtained by some blow-up of points on a ber of F1 . We embedd F1 into P2 × P1 as F1 = {((x : y : z), (s : t)) ⊂ P2 × P1 | yt = zs}; the proje tion on the rst fa tor yields the birational morphism τ : F1 → P2 whi h is the blow-up of (1 : 0 : 0) ∈ P2 and the proje tion on the se ond fa tor yields a P1 -bundle ρ : F1 → P1 . We denote by F, L ⊂ P2 the lines with equations z = 0 and y = 0 respe tively. We also all F, L ⊂ F1 their proper transforms on F1 , and denote by C ⊂ F1 the ex eptional urve τ −1 ((1 : 0 : 0)) = (1 : 0 : 0) × P1 . The ane line L \ C ⊂ F1 and its image L \ (1 : 0 : 0) ⊂ P2 will be alled L0 . In the sequel, we asso iate to any 1-standard pair (X, B, π), its minimal resolution of singularities µ : (Y, B, π ◦ µ) → (X, B, π) and a birational morphism η : Y → F1 whi h is the blow-up of a nite number of points. Ea h of these points belongs as proper or innitely near point to the ane line L0 = L \ C ⊂ P1 , is dened over k but not ne essarily over k; however, the set of all points blown-up ˆ ⊂ Y ontained in ˆ , for some (possibly redu ible) urve E by η is dened over k. We have D = F ⊲C ⊲ E η −1 (L0 ), and where π ◦ µ = ρ ◦ η , as in the diagram of Lemma 1.0.7. We x embeddings of A2 = Spec (k [x, y]) into F1 and P2 as (x, y) 7→ ((x : y : 1), (y : 1)) and (x, y) 7→ ∼ ∼ (x : y : 1), whi h give natural isomorphisms A2 → F1 \(F ∪C) and A2 → P2 \F . The restri tion to the line y = 0 yields a anoni al isomorphism A1 → L0 whi h sends α ∈ k on (α : 0 : 1) ∈ L0 ⊂ P2 . The group of ane automorphisms of A2 whi h is the group of automorphisms that extend to automorphisms of P2 is denoted by Aff and the group of triangular or de Jonquières automorphisms automorphisms of the bered surfa e (A2 , ρ|A2 ) is denoted by Jon. Expli itly, we have Aff = {(x, y) 7→ (a1 x + a2 y + a3 , b1 x + b2 y + b3 ) | ai , bi ∈ k, a1 b2 6= a2 b1 )} , Jon = {(x, y) 7→ (ax + P (y), by + c) | a, b ∈ k∗ , c ∈ k, P ∈ k[y])} .

Two 1-standard pairs are isomorphi (respe tively indu e isomorphi ane bered surfa es) if and only their orresponding set of points blown-up are equivalent after the a tion of some element of Aff ∩ Jon (respe tively of Jon); this follows from Lemma 2.2.3 and is explained more pre isely in Lemma 5.2.1 below. 5.2. Links between 1-standard pairs isomorphisms of brations. Here we des ribe the links between 1-standard pairs obtained from isomorphisms of ane bered-surfa es. In general it is possible that for two non-isomorphi 1-standard pairs (X, B, π) and (X ′ , B ′ , π ′ ), the ane A1 -bered surfa es (X \ B, π) and (X ′ \ B ′ , π ′ ) are isomorphi ; the following simple result des ribes the situation.

Lemma 5.2.1. For i = 1, 2, let (Xi , Bi , πi ) be a 1-standard pair, with a minimal resolution of singularities µi : (Yi , Bi , πi ◦ µi ) → (Xi , Bi , πi ) and let ηi : Yi → F1 be a birational morphism as in §5.1 above. Then, the following relations are equivalent: (1) the A1 -bered surfa e (X1 \ B1 , π1 ) and (X2 \ B2 , π2 ) (respe tively the pairs (X1 , B1 , π1 ) and (X2 , B2 , π2 )) are isomorphi ; (2) there exists an element of Jon (respe tively of Jon ∩ Aff ) whi h sends the points blown-up by η1 onto those blown-up by η2 and sends the urves ontra ted by µ1 onto those ontra ted by µ2 . Proof. Follows dire tly from Lemma 2.2.3.

Re all that ea h point of the ex eptional urve obtained by blowing-up a point p on a surfa e is in the rst neighbourhood of p, and that if q is in the m-th neighbourhood of p, then any point in the rst neighbourhood of q is in the (m + 1)-th neighbourhood of p; by onvention, a point p is in its 0-th neighbourhoud.

18

JÉRÉMY BLANC AND ADRIEN DUBOULOZ

Corollary 5.2.2. Let (X, B, π) be a 1-standard pair with X \ B ane. The group of automorphisms of the ane bered-spa e (X \ B, π) ontains a subgroup isomorphi to the polynomial group (k[x], +), a ting algebrai ally on X and trivially on the base of the bration. Proof. Consider as before a minimal resolution of singularities µ : (Y, B, π ◦ µ) → (X, B, π) and let η :

Y → F1 be a birational morphism as in 5.1. Denote by m the maximal height of the points blown-up by η (so that ea h point blown-up belongs to the i-th neighbourhood of a point of F1 , for i ≤ m). Then, the algebrai subgroup H (of innite dimension) of Jon equal to (x, y) 7→ (x + y m+1 · P (y), y) | P ∈ k[y]) a ts trivially on the line y = 0 and on the set of points that belong to the i-th neighbourhood of this line, for i ≤ m. Consequently, H xes any point blown-up by η and then preserves any irredu ible

urve ontra ted by µ; a

ording to Lemma 2.2.3, µη −1 onjugates H to a group of automorphims of the ane bered-surfa e (X \ B, π).

5.3. The simplest ase: 1-standard pairs of type (0, −1) / the ane plane. The simplest 1standard zigzag is of type (0, −1). Assuming that the omplement of the boundary is ane implies that it is isomorphi to A2 , as the following simple result shows:

Lemma 5.3.1. Let (X, B, π) be a 1-standard pair of type (0, −1), and assume that S = X \ B is ane, then (X, B) ∼ = (F1 , F ⊲C), and thus X \ B ∼ = A2 . µ

η

Proof. Let (X, B, π) ← (Y, B, πµ) be a minimal resolution of the singularities of X and let Y → F1 be

the morphism dened in 5.1. Suppose that some point p ∈ L0 ⊂ F1 is blown-up by η ; then η −1 (p) is a

onne ted tree of smooth rational urves of negative interse tion, and at least one irredu ible urve in η −1 (p) is a (−1)- urve, whi h is not ontra ted by µ by minimality. The image of this (−1)- urve by µ is thus a proje tive urve, whi h does not interse t B be ause no singular point of X belongs to B . This ontradi ts the fa t that S is ane, so η and onsequently µ is an isomorphism. As a dire t onsequen e of our approa h, we nd the following well-known results for this simple ∼

ase. Re all the notation of 5.1 for the natural isomorphism A2 → F1 \ (F ∪ C), su h that ρ : F1 → P1 2 restri ts on A to the proje tion on the rst fa tor.

Proposition 5.3.2. Let (X, B, π) = (F1 , F⊲C, ρ), and let S = A2 = X \B , with the natural isomorphism (x, y) 7→ ((x : y : 1), (y : 1)) (as in §5.1), su h that π extends the map π : S → A1 whi h is (x, y) 7→ x. Then: (1) the automorphism Ψ1 : (x, y) 7→ (y, x) of S extends to a reversion Ψ1 : (X, B) 99K (X, B); (2) letting J = Aut(S, π), A =< Aut(X, B), Ψ1 >, the group Aut(S) is the free produ t of A and J amalgamated over their interse tion: Aut(S) = A ⋆A∩J J.

(3) the following equalities o

ur: Aff = Jon =

A = J =

{(x, y) 7→ (a1 x + a2 y + a3 , b1 x + b2 y + b3 ) | ai , bi ∈ k, a1 b2 6= a2 b1 )} , {(x, y) 7→ (ax + P (y), by + c) | a, b ∈ k∗ , c ∈ k, P ∈ k[y])} ;

(4) there exist innitely many A1 -brations on S , but only one up to automorphisms of S ; (5) the group Aut(S) is generated by automorphisms of A1 -brations.

Remark 5.3.3. Assertion (2) is the famous Jung's theorem, proved from many dierent manners sin e

the original proof [11℄ of Jung. We refer to [12℄ and [8℄ for the proofs whi h are the losest to our approa h.

Proof. Any reversion or bered modi ation that starts from (X, B) gives a 1-standard pair with a zigzag of type (−1, 0) (Corollary 3.2.3), and thus whi h is isomorphi to (X, B) (Lemma 5.3.1). This implies assertion (4). Observe that a reversion onsists of the ontra tion of the (−1)- urve of F1 , followed by the blow-up of a point of F ; it is therefore the lift of an automorphism of P2 , whi h sends the point (1 : 0 : 0) on another point of F (the line z = 0), whi h yields a new bration. Sin e the map Ψ1 : (x : y : z) 7→ (y : x : z) of P2 is an example of su h map, assertion (1) is lear.

AUTOMORPHISMS OF

A1 -FIBERED

AFFINE SURFACES

19

The group Aut(X, B) is the lift of the group of automorphisms of P2 that x (1 : 0 : 0) and leave F invariant; sin e Ψ1 ex hanges the two points (1 : 0 : 0), (0 : 1 : 0) ∈ F , the group A is equal to Aut(P2 , F ) = Aff . The equality Jon = Aut(S, π) being obvious, (3) follows dire tly. Let us prove now that A and J generate the group Aut(S). Any element φ ∈ Aut(S) extends to a birational map φ : (X, B) 99K (X, B), whi h belongs either to Aut(X, B) ⊂ A or fa torises as φn ◦ ... ◦ φ1 , where ea h φi is a reversion or a bered modi ation (Theorem 3.0.2). Sin e ea h pair whi h o

urs in this de omposition is isomorphi to (X, B), we may assume that φi ∈ Aut(S). The bered modi ations belong to J and the reversions are equal to αΨ1 β , for some α, β ∈ Aut(X, B) (follows from Proposition 2.3.7 and from the transitivity of the a tion of Aut(X, B) on L), this yields the equality Aut(S) =< A, J >. Sin e J ontains Aut(X, B) we also have Aut(S) =< Ψ1 , J >. Note that elements of A are produ ts of reversions, and then are either reversions or elements of Aut(X, B) (Lemma 2.3.8). To prove the amalgamated produ t stru ture, we take an element g = gn ◦ · · · ◦ g1 ◦ g0 ∈ Aut(S), where n ≥ 1 and the gi belong alternatively to A \ J or to J \ A, and prove that g is not the identity. Sin e both A and J ontain Aut(X, B), elements of A \ J are reversions and elements of J \ A are bered modi ations. The fa t that g is not trivial and furthermore is not an automorphism follows from Theorem 3.0.2, or more pre isely of Lemma 3.2.4. Assertion (2) is now lear. It remains to prove assertion (5). Sin e Ψ1 orresponds to (x, y) 7→ (y, x), it preserves the A1 -bration (x, y) 7→ x + y . The equality Aut(S) =< Ψ1 , J > yields the assertion. 5.4. 1-standard pairs of type (0, −1, −n) / surfa es with equation uv = P (w) in A3 . In (5.4.2) below, we onstru t a 1-standard pair of type (0, −1, −n) assso iated to any polynomial P ∈ k[w] of degree n. Then, Lemma 5.4.3 shows that any su h pair is obtained by this way. Lemma 5.4.4 provides an isomorphism of the surfa e with the hypersurfa e of A3 with equation uv = P (w).

5.4.1. Surfa es dened by an equation of the form uv = P (w) have been intensively studied during

the last de ade, with a parti ular fo us on the lassi ation of additive group a tions on them. In parti ular, L. Makar-Limanov [15℄ determined by areful algebrai analysis of the oordinate ring a set of generators of their automorphism group. Every surfa e with equation uv = P (w) admits at least two A1 -bration over A1 indu ed respe tively by the restri tions of the proje tions pru and prv . The latter obviously dier by the omposition of the involution of the surfa e whi h ex hanges u and v . In [3℄, D. Daigle used similar algebrai methods as L. Makar-Limanov to show every A1 -bration over A1 on these is of the form pru ◦ φ, where φ is an automorphism of the surfa e. Here we re over these results as

orollaries of the des ription of birational maps between 1-standard pairs asso iated with these surfa es. It follows from a general des ription due to V. I. Danilov and M.H. Gizatullin [7℄ (see also S. Lamy [13℄ for a self- ontained proof) that the automorphism group of smooth ane quadri with equation uv = w2 − 1 admits the stru ture of an amalgamated produ t analogous to the one of the automorphism group of the plane. In Theorem 5.4.5, we show that this holds more generaly for every surfa e with equation uv = P (w). We keep the notation of 5.1. Re all that L0 = L \ C ⊂ A2 ⊂ F1 is identied with A1 via the in lusion α 7→ (α, 0) of A1 into A2 ⊂ F1 .

5.4.2. To any polynomial P ∈ k[x] of degree n ≥ 2, we asso iate a birational morphism ηP : Y → F1

whi h is the blow-up of n points. For ea h root α ∈ k of P of multipli ity r, the point α ∈ F0 (k) ⊂ F1 is blown-up by ηP , and for i = 1, ..., r − 1, the point in the i-th neighbourhood of α that belongs to the proper transform of F is also blown-up. It follows from the denition of ηP that it is dened over k. In this onstru tion, any irredu ible urve of Y ontra ted by η has self-interse tion −1 or −2; the urves of self-interse tion −1 interse t E and the others do not interse t E ; furthermore L2 = −n in Y . The ontra tion of every irredu ible urve ontra ted by ηP whi h has self-interse tion −2 gives rise to a birational morphism µP : Y → X to a 1-standard pair (X, B = F ⊲ C ⊲ L) with a zigzag of type (0, −1, −n). The following gure des ribes the situation. In the sequel, all the gures will represent all

urves and their interse tions over k. Moreover, the birational morphism ηP ◦ (µP )−1 : X → F1 is lo ally given by the blow-up of the ideal (P (x), y) in A2 .

20

JÉRÉMY BLANC AND ADRIEN DUBOULOZ

r1 − 1

rl − 1

−1

F

C

0

−1

L

0

ηP

F

C

E

0

−1

−n

−1

.. .

µP

F

C

E

0

−1

−n

η

.. .

µ

P P (F1 , F ⊲ C ⊲ L) ← (Y, B) → (X, B), where Pl r1 , ..., rl are the multipli ities of the roots of P , the degree of P is n = i=1 ri , and where a blo k with label t onsists of a zigzag of t (−2)- urves.

Figure 5.1.

The des ription of the morphisms

Lemma 5.4.3 (Isomorphism lasses of surfa es of type (0, −1, −n)). Let (X, B = F ⊲ C ⊲ E, π) be a 1-standard pair, su h that B is of type (0, −1, −n) (n ≥ 2), with a minimal resolution of singularities µ : (Y, B, π ◦ µ) → (X, B, π) and let η : Y → F1 be a birational morphism as in §5.1 above. Assuming that X \ B is ane, the following hold: (1) the morphisms η , µ are equal to the morphisms ηP , µP dened in 5.4.2, for some polynomial P of degree n; (2) any 1-standard pair (X ′ , B ′ , π ′ ) su h that (X \ B, π) ∼ = (X ′ \ B ′ , π ′ ) is isomorphi to (X, B, π). (3) The isomorphism lasses of 1-standard pairs (X, B) of type (0, −1, −n) with X \ B ane orrespond to polynomials in k[X] of degree n up to ane automorphisms of the line.

Proof. Sin e S = X \ B is ane, only one ber of π is singular and ea h singularity of X is solved by a hain of rational urves of Y (see 1.0.3). Note that E ⊂ X, Y is the proper transform of L ⊂ F1 , and has self-interse tion −n in X and Y . Denote by f ⊂ Y the unique singular ber of π ◦ µ; then f

ontains E , whi h is in the boundary B = F ⊲C ⊲E of Y , and f \ E is ontained in the ane part S . Denote by Γ a onne ted omponent of f \ E . Then, Γ ontains one irredu ible urve Γ0 not ontra ted by µ whi h interse t E , and a (possibly empty) set of onne ted hains of smooth rational urves, ea h of self-interse tion ≤ −2, ontra ted by µ. Sin e Γ is ontra ted by η , it ontains a (−1)- urve, whi h is ne essarily Γ0 , and therefore Γ \ Γ0 is a hain of smooth rational urves of self-interse tion −2. This shows that ea h point blown-up by η belongs as a proper or innitely near point to L. Denote by a1 , ..., al ∈ L0 (k) ⊂ A2 ⊂ F1 the proper base-points of η −1 . For i = 1, ..., l, denote by ri ∈ NPthe number of omponents of η −1 (ai ) ⊂ Y . Sin e F 2 = 0 in F1 and F 2 = E 2 = −n in Y , we r have i=1 ri = n. Then, η and µ orrespond to the morphisms ηP and µP dened in 5.4.2, for the Q polynomial P = li=1 (x − ai )ri ∈ k[x]. This gives the rst assertion. Let us prove the remaining assertions. Denote by B(ηP−1 ) the set of points blown-up by ηP , whi h belong to L0 ⊂ F1 as proper or innitely near points. Let α ∈ Jon. A

ording to Lemma 5.2.1, to prove (2) it su es to show that there exists β ∈ Aff ∩ Jon su h that β −1 α xes ea h point of B(ηP−1 ). The map α restri ts to an automorphism of the ane line L0 = L \ C ⊂ F1 , whi h extends to an element β ∈ Aff ∩ Jon = Aut(F1 , F ⊲ C). Then, β −1 α a ts trivially on L ⊂ F1 and onsequently xes ai for i = 1, ..., r; it also xes ea h point of B(ηP−1 ), sin e these points belong to the proper transform of L. This yields (2). Assertion (3) follows dire tly from Lemma 5.2.1.

Lemma 5.4.4 (Reversions between pairs of type (0, −1, −n)). Let P, P ′ ∈ k[x] be two polynomials of degree n ≥ 2, and let µP

(X, B = F ⊲C ⊲E, π) ←− µP ′ ′ (X , B ′ = F ′ ⊲C ′ ⊲E ′ , π ′ ) ←−

ηP

(Y, B, πµP ) −→ (F1 , F ⊲C ⊲L) ηP ′ ′ ′ ′ ′ (F1 , F ⊲C ⊲L) (Y , B , π µP ) −→

be the orresponding onstru tion made in 5.4.2. Suppose that there exists a reversion φ : (X, B) 99K (X ′ , B ′ ) entred at p ∈ F \ C , with φ−1 entred at p′ ∈ F ′ \ C ′ . Then, the following hold: (1) Let a1 , ..., al ∈ k and a′1 , ..., a′l′ ∈ k be the roots of P and P ′ respe tively. For i = 1, ..., l, let ri ∈ N be the multipli ity of ai , whi h is the number of omponents of ηP−1 (ai ) ⊂ Y ; we denote by Ai the omponent of self-interse tion −1 and by Ai the union of the ri − 1 omponents of self-interse tion −2. We also denote by Di the stri t transform by (τ ηP )−1 of the line of P2 passing through p and ai . Doing the same with primes for P ′ , we get on Y and Y ′ the following dual graphs of urves:

AUTOMORPHISMS OF

A1

−1 A

A1

AFFINE SURFACES

D1

0

D1′

D

Dt′

r1 −1

.. . Al

A1 -FIBERED

0

−1l rl −1 0l E

C

−n

0

F

−1

A′1

r1′ −1

A′1

.. −1 . A′r ′

rt′−1

At

−1

′

′

C

0

−1

F

0

21

E′

−n

(2) The numbers l and l′ are equal, and after renumbering ri = ri′ for ea h i and there exists an automorphism of L0 whi h sends ai on a′i for ea h i. Moreover (X, B) is isomorphi to (X ′ , B ′ ). (3) Let ψ = (µP ′ )−1 φµP : Y 99K Y ′ be the lift of the reversion. Then, ψ restri ts to an isomorphism from respe tively Ai , Di and Ai to Di′ , A′i and A′i . And ψ de omposes as in the following diagram A1

−1 A

−1l E

−n

A1 r1−1

.. . Al

rl−1

C

−1

A′1

A1

0

−1

A′l

A

−1l

0

F θ0 E 0

−n+1

A1 r1−1

.. . Al

rl−1

A′1

−1

A′l

0

r1−1

ϕ1

A

−1l

−1

F

A1

A1

−1

Ep

E p′

−1

−1

.. . Al

rl−1

F

′

0

A′1

−1

A1

A′l

0l

0

A

−1

E

′

−n+1

θ1

A1 r1−1

A′1

.. −1 . Al ′

rl−1

′

′

C

0

−1

F

Al

−1

E′

−n

where θ0 , ϕ1 and θ1 orrespond to the maps des ribed in §2.3.5, and Ep and Ep′ orrespond to the ex eptional urve ontra ted on p and p′ respe tively, whi h are the proper transforms of respe tively E ′ and E . (4) We have the following ommutative diagram of birational maps ψ a a a ` ` ` _ __ _ ^ ^ ^ ] ] ] \ \ \ [ [ b b b c c [ Z d ϕ Z * U [ d c 1 UUUU \ ^ θ_1 ` b iiciZ-1 Y ′ Y dUU[UUU\ ^ θ_0 ` b iiciii1 U UUUU UUUU i ii UU* tiiiiiiEp UU* tiiiiiiC ′ C E ′ p ′

{Ai } η P {Ai }

{Ai } {Ai }

{Ai } {A′i }

′ ηP ′ {Ai′ } {Ai }

{Ai } {A′i } d _ Z + {Ai } {A′i } F1 F F F1 SSS ψ S 1 1 1 S k k S k k S SSτSS k k τ SSSS kkk SSS kkkkkEkk C ) 2 u_ _ _p _ _ _ _ _ _ _ _ _ _ _Ep_′ _ S_S)/ 2 ukkkk C ′ P P Ψ1

where ea h straight line is a birational morphism whi h ontra ts the urves (or proper transform of

urves) written above the arrow, where ψ1 is a birational map whi h preserves the ruling of F1 , and where Ψ1 is given by Ψ1 : (x : y : z) 99K (xyz n−2 : G(x, z) : yz n−1 ),

up to automorphisms of (Y, B) and (Y ′ , B ′ ). (5) letting χ : P2 99K (P1 )3 be the rational map χ : (x : y : z) 99K (y : z), (G(x, z) : yz n−1 ), (x : z) ,

the map χ ◦ τ ηP (µP )−1 restri ts to an embedding of X \ B to the hypersurfa e of given by

A3 = (u : 1), (v : 1), (w : 1) ∈ (P1 )3 , u, v, w ∈ k

(u, v, w) ∈ A3 | uv = P (w) .

The restri tions of the three anoni al proje tions A3 → A1 give respe tively the A1 -bration π, the A1 -bration π ◦ (Ψ1 |X\B ) obtained by means of the reversion ψ0 , and the A1∗ -bration given by the pen il of lines of P2 passing through the point where the reversion Ψ1 is entered. Proof. Ea h urve Di has self-interse tion 0 in Y , and interse ts B transversally and only at p. Moreover, it interse ts also the urve orresponding to the blow-up of ai ; this latter urve is Ai if ri = 1 and the

omponent at the right side of Ai otherwise. Sin e Di does not interse t any other urve Aj or in Aj , we obtain (1). De ompose ψ into ψ = θ1 ϕ1 θ0 , as in §2.3.5, and ompute the diagram of (3) above for the Ai , Di , Ai . This shows that Di is sent by ψ on a urve of self-interse tion −1, interse ting B ′ only at E ′ , and transversally. In onsequen e, any Di is sent on a urve A′j . Moreover, if ri > 0 the singular point of

22

JÉRÉMY BLANC AND ADRIEN DUBOULOZ

Pl P l′ µP (Di ) is sent by φ onto the singular point of µP (A′j ). Sin e t=1 rt = t=1 rt′ = n, we obtain the equality l = l′ and get that after renumbering Di is sent on A′i and Ai is sent on A′i , when e ri = ri′ . Assertion (3) is now proved. The proje tion by ηP (p) ∈ P2 on the line L = ηP (E) indu es an isomorphism Ep → E whi h sends Ep ∩ Di onto ai , and sends Ep ∩ F onto E ∩ C . Using the diagram of (3), the map θ1 ϕ1 restri ts to an isomorphism Ep → E ′ whi h sends Ep ∩ Di = onto E ′ ∩ A′i and sends Ep ∩ F onto E ′ ∩ C ′ . Combining the two isomorphisms, and sin e E ′ is the proper transform of L by ηP ′ , we obtain (2). In the de omposition of ψ given in (3), θ0 de omposes as the ontra tion of C , followed by the blowup of p. Moreover, ea h of these two steps do not hange the self-interse tion of any of the omponents of {Ai }li=1 , {Ai }li=1 , ontra ted by ηP , whi h are thus still ontra tible in the surfa es obtained from Y by ontra ting C and blowing-up p. Doing the same with (θ1 )−1 , we obtain the diagram of (4). Sin e any urve ontra ted by the map φ1 : F1 99K F1 is a bre of the ruling, φ1 preserves the ruling. The lift of the group of automorphisms of P2 of the form (x : y : z) → (x + λz : y : z), λ ∈ k gives a group of automorphisms of (Y, B) or (X, B) whi h a ts transitively on the k-points of F . Thus, we may assume, up to automorphisms of (Y, B) and (Y ′ , B ′ ), that p = p′ = (0 : 1 : 0). It remains to observe that Ψ1

an be given in this ase by the map Ψ0 : (x : y : z) 99K (xyz n−2 : G(x, z) : yz n−1 ). The map Ψ0 learly preserves the lines passing through (0 : 1 : 0), and this point is a base-point of Ψ0 of multipli ity n − 1. One an moreover he k that it has 2n − 2 other base-points dened as follows. a) The base-points whi h orresponds to the n − 2 base-points of ϕ1 , all innitely near of p = (0 : 1 : 0) and lying on L, b) the n points blown-up by ηP , whi h are {(ai : 0 : 1)}li=1 and points innitely near, all on F . Thus, Ψ1 and Ψ0 have the same base-points, and (4) is now proved. Letting ζ = τ ηP (µP )−1 : X 99K P2 , we prove now that χ ◦ ζ restri ts to an embedding of S = X \ B into A3 . The rst oordinate of (P1 )3 orresponds to the proje tion of P2 by (1 : 0 : 0) and then restri ts exa tly to π : S → A1 ; the se ond oordinate is obtained by means of the reversion, it restri ts to the A1 -bration (πΨ1 )|X\B ; the last one orresponds to the proje tion of P2 by (0 : 1 : 0) and restri ts to a A1∗ -bration on S . This last map separates the points of the dierent regular bers of π and separates the omponents of the redu ed ber. Sin e ea h of these omponents is a se tion of the A1 -bration (πΨ1 )|X\B , the map (χ ◦ ζ)|X\B is an embedding of S into A3 ⊂ (P1 )3 . Taking oordinates (u : 1), (v : 1), (w : 1) on A3 ⊂ (P1 )3 , we dedu e from the expli it form of χ that the image is the surfa e with equation uv = P (w). The reversion Ψ1 omputed before orresponds to the automorphism (u, v, w) 7→ (v, u, w) of the surfa e, and the bration π is the proje tion on the rst fa tor.

Theorem 5.4.5. Let (X, B = F ⊲ C ⊲ E, π) be a 1-standard pair, su h that B is of type (0, −1, −n) (n ≥ 2) and su h that the surfa e S = X \ B is ane. Then, there exists an isomorphism of bered-surfa es from (S, π) to the hypersurfa e of A3 given by (u, v, w) ∈ A3 | uv = P (w) ,

for some polynomial P of degree n, equipped by the u-bration; and any su h surfa e is obtained in this way. Furthermore, the following assertions hold: (1) the isomorphism lass of the surfa e is given by the polynomial P , up to a multiple and up to an automorphism of A1 = Spec (k [w]); (2) there exist innitely many brations on S , but only one up to an automorphism of S ; (3) the graph FS is :•; (4) if n ≥ 3, the group Aut(S) is not generated by the automorphisms of A1 -bration; (5) if n = 2, the group Aut(S) is generated by the automorphisms of A1 -bration; (6) the involution (u, v, w) 7→ (v, u, w) on S orresponds to a reversion Ψ1 : (X, B) 99K (X, B); (7) the group Aut(S) is the free produ t of A =< Aut(X, B), Ψ1 > and J = Aut(S, π), amalgamated over their interse tion A ∩ J = Aut(X, B): Aut(S) = A ⋆A∩J J;

(a) if n = 2, the ontra tion of C ⊲ E gives a birational morphism of pairs (X, B) → (Z, D), whi h onjugates A to the group Aut(Z, D). Moreover if P has two distin t roots in k, Z is a smooth quadri in P3 , and D is an hyperplane se tion; if one adds that the two roots are dened over k, then Z is isomorphi to P1 × P1 and D be omes a diagonal. If P has only

AUTOMORPHISMS OF

A1 -FIBERED

AFFINE SURFACES

23

one multiple root (ne essarily dened over k), Z is the weighted plane P(1, 1, 2) obtained by

ontra ting the (−2)- urve of F2 and D is the image of a se tion of self-interse tion 2. (b) if n > 3, we denote by Aut(X, B, Ψ1 ) the subgroup of Aut(X, B) whi h xes the unique proper base-point of Ψ1 ; then A is the free produ t of Aut(X, B) and A0 =< Aut(X, B, Ψ1 ), Ψ1 >, amalgamated over their interse tion A0 ∩ Aut(X, B) = Aut(X, B, Ψ1 ), and we also have another amalgamated produ t stru ture for Aut(S): Aut(S) = A0 ⋆A0 ∩J J.

(8) denoting by H, I, T, T0 , Sp the following subgroups of automorphisms of S : {(u, v, w) 7→ (au, a−1 v, w) | a ∈ k∗ }; {(u, v, w) 7→ (v, u, w), (u, v, w) 7→ (u, v, w)}; {(u, v, w) 7→ (u, u−1 · (P (w + uq(u)) − P (w)), w + uq(u)) | q ∈ k[u]}; {(u, v, w) 7→ (u, u−1 · (P (w + au) − P (w)), w + au) | a ∈ k}; {(u, v, w) 7→ (u, cv, aw + b) | a, c ∈ k∗ , b ∈ k, P (aw + b) = cP (w)}; k∗ if p has only one root, then, H ∼ = k and Sp ∼ = k∗ , I ∼ = Z/2Z, T ∼ = k[u], T ⊃ T0 ∼ = Z/mZ otherwise (m = 1 in general). H I T T0 Sp

= = = = =

Furthermore, the following o

ur:

J =< H, Sp, T >∼ = k[u] ⋊ (k∗ × Sp) is the group of automorphisms of (S, π). Aut(X, B) =< H, Sp, T0 >∼ = k ⋊ (k∗ × Sp); Aut(X, B, Ψ1 ) =< H, Sp >∼ = (k∗ × Sp); ∗ ∼ A0 =< H, Sp, I >= (k ⋊ Z/2Z) × Sp; k ⋊ (k∗ × Sp) if n = 2, (e) A =< H, Sp, I, T0 >∼ = Aut(X, B) ⋆Aut(X.B)∩A0 A0 otherwise.

(a) (b) ( ) (d)

µ

η

P Proof. A

ording to Lemma 5.4.3, there exist two morphisms (X, B = F ⊲ C ⊲ E, π) ←P (Y, B, πµP ) →

(F1 , F ⊲ C ⊲ L) as in 5.4.2, for some polynomial P of degree n. The isomorphism between S and the surfa e {(u, v, w) ∈ A3 | uv = P (w)} follows from Lemma 5.4.4. The isomorphism lass of the pair (X, B) is determined by the points blown-up by ηP , up to an a tion of Aff ∩ Jon (Lemma 5.4.4), and

onsequently by the polynomial P up to multiple and to an automorphism of L0 = A1 (follows from the des ription of ηP and µP , made in 5.4.2). Sin e any reversion or bered modi ation that starts from (X, B) yields an isomorphi 1-standard pair (Lemma 5.4.4), any 1-standard pair (X ′ , B ′ ) su h that X \ B ∼ = X ′ \ B ′ is isomorphi to (X, B) (Theorem 3.0.2). This implies with the dis ussion made above the assertions (1) and (2); it also shows that the graph FS ontains only one vertex; we prove now that it ontains only one arrow. The group of automorphism of P2 that x ea h point of L, and preserve the line F lift to a subgroup of Aut(X, B) whi h a ts transitively on F \ C . Consequently, if φ : (X, B) 99K (X ′ , B ′ ) is a reversion, there exists α ∈ Aut(X, B) su h that φα is a reversion entered at the same point as Ψ1 . Proposition 2.3.7 implies that Ψ1 = βφα, for some isomorphism β : (X ′ , B ′ ) → (X, B). This yields assertions (3) and thus (4) (using Proposition 4.0.7). Let us prove assertion (5). Assume that n = 2, and let α be an element of Aut(X, B) whi h does not x the proper base-point of Ψ1 . The reversions Ψ−1 1 = Ψ1 and Ψ1 α have thus distin t base-points, so Ψ1 αΨ1 is a reversion (Lemma 2.3.8), equal to βΨ1 γ , for some β, γ ∈ Aut(X, B). Consequently −1 Ψ1 = (α−1 Ψ1 )(βΨ1 γ) = α−1 (Ψ1 βΨ−1 1 )γ ; sin e Ψ1 βΨ1 preserves the bration Ψ1 π , the reversion Ψ1 1 is generated by automorphisms of A -brations. The equality Aut(S) =< Aut(X, B), Ψ1 , J > yields assertion (5). Assertion (6) follows from Lemma 5.4.4. It remains to prove the main assertions, i.e. (7) and (8). Let us write I =< Ψ1 > and J = Aut(S, π) (automorphisms of S whi h preserve the bration π ). We prove now that Aut(X, B), I, J generate Aut(S). Any element g ∈ Aut(S) extends to a birational map g : (X, B) 99K (X, B); either g belongs to Aut(X, B) or it may be written using Theorem 3.0.2) as gn g2 g1 g = gn ◦ · · · ◦ g1 : (X, B) = (X0 , B0 ) 99K (X1 , B1 ) 99K · · · 99K (Xn , Bn ) = (X, B) where gi is a reversion or a bered modi ation. We proved previously that ea h (Xi , Bi ) is isomorphi to (X, B), we may thus assume, by hanging the gi , that (Xi , Bi ) = (X, B). Consequently, gi may be viewed as an element of Aut(S). If it is a bered modi ation, it belongs to J . Otherwise, it is a

24

JÉRÉMY BLANC AND ADRIEN DUBOULOZ

reversion; sin e Aut(X, B) a ts transitively on F \ E , gi = αΨ1 β , for some α, β ∈ Aut(X, B). This a hieves the proof of the equality Aut(S) =< Aut(X, B), I, J >. Writing A =< Aut(X, B), I >, the group Aut(S) is generated by A and J . Let us prove that it is an amalgamated free produ t. Let g = an ◦ jn ◦ ... ◦ a1 ◦ j1 , where ea h ai ∈ A \ J and ji ∈ J \ A. Then, ai is a produ t of reversions whi h is not an isomorphism, and ji is a bered modi ation. Theorem 3.0.2 (or more pre isely Lemma 3.2.4) implies that g does not belong to Aut(X, B) and then is not the identity. This shows that Aut(S) = A ⋆A∩J J . Assume that n = 2. Then, C ⊲ E is a zigzag of type (−1, −2); the ontra tion of this zigzag gives rise to birational morphism of pairs ν : (X, B) → (Z, D) for some proje tive surfa e Z , and some urve D. Furthermore, ν indu es an isomorphism F → D. Let us des ribe the pair (Z, D), using the maps µP ηP ν τ (Z, D) ← (X, B) ← (Y, B) → (F1 , F ⊲C⊲L) → (P2 , F ⊲L). If P has two distin t roots in k, then ηP is the blow-up of two distin t points of L0 = L\C ⊂ F1 and µP is an isomorphism. Sin e both ν and τ ontra t the same urve C , the birational map (P2 , F⊲L) 99K (Z, D) onsists of the blow-up of two distin t points of F \ L, followed by the ontra tion of F . This implies that Z is isomorphi to a smooth quadri in P3 and that D (whi h is the image of a line F ⊂ P2 ) is an hyperplane se tion of Z . Moreover, if the two roots of P are dened over k, Z is isomorphi to P1 × P1 and D is a diagonal (i.e. a urve of bidegree (1, 1)). If P has one root of multipli ity two, then ηP is the blow-up of a point p1 ∈ L0 = L \ C ⊂ F1 , followed by the blow-up of the point p2 in the rst neighborhood of p1 , whi h belongs to the proper transform of the line L. Furthermore, µP onsists of the ontra tion of the ex eptional urve Ep1 of p1 (whi h is a (−2)- urve) on the unique singular point of X . On e again, both ν and τ ontra t the same

urve C . The map (P2 , F ⊲L) 99K (Z, D) is therefore the omposition of the blow-up of p1 , p2 and the

ontra tion of the two urves Ep1 and L. The blow-up of p1 goes to a surfa e isomorphi to F1 , where the ex eptional se tion is E1 and where C be omes a se tion of self-interse tion 1. Then, the blow-up of p2 followed by the ontra tion of F is an elementary link F1 99K F2 ; the urves E1 and F be ome se tions of self-interse tion −2 and 2 respe tively. The ontra tion of E2 gives the birational morphism F2 → P(1, 1, 2) = Z . Now that Z is des ribed in ea h ase, let us prove that Aut(Z, D) = νAν −1 . Sin e ea h of the three urves F , C , E is preserved by any automorphism of (X, B), the group ν −1 Aut(X, B)ν is ontained in Aut(Z, D), and orresponds in fa t to the subgroup of elements of Aut(Z, D) whi h x the point ν(C ∪E). Note that νΨ1 ν −1 is an automorphism of (Z, D) whi h sends the point ν(C ∪E) ∈ D onto the point ν(p) ∈ D, where p is the base-point of Ψ1 . Sin e the a tion of Aut(Z, D) on D yields a surje tive morphism ρ : Aut(Z, D) → Aut(D) ∼ = PGL(2, k), and be ause ν −1 Aut(X, B)ν ontains the kernel of ρ and its image by ρ is a maximal group, then ν −1 Aut(X, B)ν and ν −1 Ψ1 ν generate Aut(Z, D). This shows 7(a). Assume now that n ≥ 3, let Aut(X, B, Ψ1 ) be the group of automorphisms of (X, B) whi h x the proper base-point p of Ψ1 , and let A0 =< Aut(X, B, Ψ1 ), Ψ1 >. Then, learly A0 and Aut(X, B) (respe tively J ) generate A (respe tively Aut(S)). Let us prove that we have an amalgamated free produ t in both ases. Let g = an ◦jn ◦...◦a1 ◦j1 , where ea h ai ∈ A0 \Aut(X, B) and ji ∈ Aut(X, B)\A0 . Then, ai is a reversion entered at p and ji is an automorphism of (X, B) whi h moves p. Consequently, the de omposition g = (an jn ) ◦ ... ◦ (a1 j1 ) has no simpli ation and is minimal (Lemma 3.2.4), so g is not trivial. Assume now that ea h ai belongs to A0 \ J and ea h ji belongs to J \ A0 . On e again, ea h ai is a reversion entered at p, and now ji is either an automorphism whi h moves p or a bered modi ation. We may group the ji whi h belongs to Aut(X, B) with ai and obtain a de omposition of g of minimal length (applying on e again Lemma 3.2.4), so g is not trivial. This yields 7(b). It remains to prove the expli it forms of (8). Let ψ = ηP ◦ (µP )−1 : (X, B) 99K (F1 , F ⊲C), and re all that ψ restri ts to a birational morphism S = X \B → A2 = F1 \(F ∪C). A

ording to Lemma 2.2.3, J = ψ −1 J ′ ψ , where J ′ is the group of eleements of Jon = {(x, y) 7→ (ax + P (y), by + c) | a, b ∈ k∗ , c ∈ k, P ∈ k[y])} whi h preserve the points blown-up by ψ −1 (or ηP−1 ); furthermore, Aut(X, B) = ψ −1 (J ′ ∩ Aff )ψ . The proper base-points of η −1 are the points (xi , 0) where P (xi ) = 0. Furthermore, the other base-points lying on the transform of the line L (whi h orresponds to y = 0), J ′ is the subgroup of elements of Jon whi h preserve the set of points of the form (xi , 0) with P (xi ) = 0. This means that J ′ is generated by H′ Sp′ T′

= = =

{(x, y) 7→ (x, ay) | a ∈ k∗ }, {(x, y) 7→ (ax + b, y) | a ∈ k∗ , b ∈ k, P (ax + b) is a multiple of P (x)}, and {(x, y) 7→ (x + yQ(y), y) | Q ∈ k[y]}.

AUTOMORPHISMS OF

A1 -FIBERED

AFFINE SURFACES

25

The lift of these groups give respe tively H, Sp, T , whi h generate J . Note that J ′ ∩ Aff is generated by H ′ , Sp′ , and T0′ = T ′ ∩ Aff . The lift of these groups give H, Sp, T0 , whi h generate Aut(X, B). The proper base-point of Ψ1 orresponds to (0 : 1 : 0) ∈ P2 (see Lemma 5.4.4), whi h orresponds in A2 to the pen il of lines of the form ax + b = 0. The group Aut(X, B, Ψ1 ) is thus the lift of < H ′ , Sp′ >. The remaining parts of (8) follow dire tly. 5.5. 1-standard pairs with a zigzag of type (0, −1, −2, −3) or (0, −1, −3, −2). The surfa es with a zigzag of type (0, −1, −n1, −n2 ) are the most simple immediately after the surfa es des ribed in the previous se tion. All these surfa es an give new examples of ane surfa es with unexpe ted properties. We give here the spe ial ase where the surfa e is smooth, the zigzag is of type (0, −1, −2, −3) or (0, −1, −3, −2), and where ea h omponent of the degenerate bre is k-rational. Properties distin t from the previous surfa es already show up in this simple example (Proposition 5.5.4). The general ase will be treated in a forth oming arti le. Firstly, we des ribe a family of 1-standard pairs (5.5.1), and then prove that these are the only examples (Lemma 5.5.2). We give the links between these maps by studying the possible reversions (Lemma 5.5.3), and then use this result to des ribe the properties of the A1 -brations and of the automorphism group (Proposition 5.5.4).

5.5.1. We dene here four families of 1-standard pairs of surfa es (X, B) of type (0, −1, −2, −3) or

(0, −1, −3, −2), where X \ B is ane and smooth. The map η : X → F1 (as in 1.0.7) is des ribed here by its set of base-points whi h are in ea h ase four points belonging, as proper or innitely near points, to L0 = L \ C ⊂ A2 ⊂ F1 . Re all that A2 is viewed in F1 via the embedding (x, y) → ((x : y : 1), (y : 1)), and that L0 is the line of equation y = 0 in A2 (see 5.1). I: Redu ed ase of type (0, −1, −2, −3): there is only one surfa e here, alled (X1 , B1 ). The map η1 : X1 → F1 is the blow-up of (0, 0), (1, 0) ∈ L0 , and of the two points in the rst neighbourhood of (0, 0) ∈ L0 orresponding to the two dire tions x = 0 and x = y . E1 ⊂ X1 is the proper transform of L and E2 ⊂ X1 is the urve obtained by blowing-up (0, 0). The following gure des ribes the morphism η1 (F1 , F ⊲C ⊲L) ← (X1 , B1 ). −1 −1−1 F

C

L

0

−1

0

η1

F

C

0

−1

E1

−2

E2

−3

II: Redu ed ase of type (0, −1, −3, −2): there is a family here, parametrised by a parameter a ∈ k\{0, 1}. The pair is alled (X2,a , B2,a ). The map η2,a : X2,a → F1 is the blow-up of (0, 0), (1, 0), (a, 0) ∈ L0 , and of the point in the rst neighbourhood of (0, 0) ∈ L0 orresponding to the two dire tion x = 0. E1 ⊂ X2,a is the proper transform of L and E2 ⊂ X2,a is the urve obtained by blowing-up (0, 0). The η2,a following gure des ribes the morphism (F1 , F ⊲C ⊲L) ← (X2,a , B2,a ). −1−1 −1 F

C

L

0

−1

0

η2,a F 0

C

−1

E1

−3

E2

−2

III: Non-redu ed ase of type (0, −1, −2, −3): there is a family here, parametrised by a parameter a ∈ k \ {0, 1}. The pair is alled (X3,a , B3,a ). The map η3,a : X3,a → F1 is the blow-up of p0 = (0, 0) ∈ L0 , of the point p1 in the rst neighbourhood of p0 orresponding to the two dire tion x = 0, and of two points in the neighbourhood of p1 . In oordinates, (u, v) 7→ (u, u2 v) is the blow-up of p0 and p1 , and the last two points orrespond to (u, v) = (0, 1) and (u, v) = (0, a). E1 ⊂ X3,a is the proper transform of L and E2 ⊂ X3,a is the urve obtained by blowing-up p1 (in the above oordinates η3,a it orresponds to u = 0). The following gure des ribes the morphism (F1 , F ⊲C ⊲L) ← (X3,a , B3,a ). −1−1 −2 F

C

L

0

−1

0

η3,a F 0

C

−1

E1

−2

E2

−3

IV: Non-redu ed ase of type (0, −1, −3, −2): there only one pair here, alled (X4 , B4 ). The map η4 : X4 → F1 is the blow-up of (1, 0) ∈ L0 , of p0 = (0, 0) ∈ L0 , of the point p1 in the rst neighbourhood of p0 orresponding to the two dire tion x = 0, and of one more point in the neighbourhood of p1 .

26

JÉRÉMY BLANC AND ADRIEN DUBOULOZ

In oordinates, (u, v) 7→ (u, u2 v) is the blow-up of p0 and p1 , and the last point orresponds here to (u, v) = (0, 1). E1 ⊂ X4 is the proper transform of L and E2 ⊂ X4 is the urve obtained by blowing-up p1 (in the above oordinates it orresponds to u = 0). The following gure des ribes the morphism η4 (F1 , F ⊲C ⊲L) ← (X4 , B4 ). −1 −1 −2 F

C

L

0

−1

0

η4

F

C

0

−1

E1

−2

E2

−3

Lemma 5.5.2 (Isomorphism lasses of surfa es of type (0, −1, −2, −3)). Let (X, B = F ⊲C ⊲E, π) be a 1-standard pair, su h that B is of type (0, −1, −2, −3) or (0, −1, −3, −2), su h that X \ B is smooth and ane and let η : X → F1 be a birational morphism as in §5.1 above. Assuming that any omponent in the singular bre of π is k-rational, the following hold: (1) there exist an automorphism α of (F1 , F ⊲C ⊲L) su h that αη is equal to one of the morphisms η1 , η2,a , η3,a , η4 dened in §5.5.1 above. In parti ular, (X, B) is isomorphi to one of the pairs given in 5.5.1; (2) any 1-standard pair (X ′ , B ′ , π ′ ) su h that (X \ B, π) ∼ = (X ′ \ B ′ , π ′ ) is isomorphi to (X, B, π). Proof. Sin e S = X\B is ane and smooth, only one ber of π is singular and any irredu ible omponent

of this bre tou hes B or belongs to B . The self-interse tions of the omponents in the boundary being given, η : X → F1 is the blow-up of exa tly four points. One he ks that all possibilities are given in the four ases des ribed in 5.5.1. The se ond assertion an be he ked dire tly, using the des ription of the base-points and applying Lemma 5.2.1.

Lemma 5.5.3. Let (X, B) and (X ′ , B ′ ) be pairs des ribed in 5.5.1, given with maps η : X → F1 and η ′ : X ′ → F1 . Suppose that there exists a reversion ψ : (X, B) 99K (X ′ , B ′ ), entred at p ∈ X , whose inverse is entred at p′ ∈ X ′ , where p and p′ orrespond respe tively via η and η′ to ((λ : 1 : 0), (1 : 0)), ((λ′ : 1 : 0), (1 : 0)) ∈ F1 for some λ, λ′ ∈ k. Then, up to automorphisms of the pairs (X, B), (X ′ , B ′ ), one of the following situations o

urs for ψ or its inverse, and every su h situation an be realised: (1) (X, B) = (X1 , B1 ), λ ∈ k \ {0, 1}, (X ′ , B ′ ) = (X2,1−1/λ , B2,1−1/λ ), λ′ ∈ k∗ . (2) (X, B) = (X1 , B1 ), λ ∈ {0, 1}, (X ′ , B ′ ) = (X4 , B4 ), λ′ ∈ k. (3) (X, B) = (X2,a , B2,a ), λ = 0, (X ′ , B ′ ) = (X3,a , B3,a ), λ′ ∈ k. Proof. Let us x some notation. We denote by ǫp : Xp → X the blow-up of p, and by Ep ⊂ Xp the

ex eptional urve produ ed, and write B = F ⊲C ⊲E1 ⊲E2 , B ′ = F ′ ⊲C ′ ⊲E1′ ⊲E2′ , and π ′ : X ′ → P1 the bration asso iated to F ′ . We also denote by R′ the set of omponents of the singular bre of π ′ whi h interse t B ′ . There are 1, 2 or 3 elements in R′ , depending in whi h family the pair (X ′ , B ′ ) is. We will

ompute the number of elements of R′ and their self-interse tion using the information on (X, B) and λ to know in whi h of the four families the pair (X ′ , B ′ ) is. Denote by T the set of urves of X whi h are sent by ψ on urves of R′ . It follows from the de omposition of ψ given in 2.3.5 (or from its resolution given in 2.3.2) that ψ fa tors through ǫp and that ψp = ψ ◦ (ǫp )−1 restri ts to an isomorphism from Ep \F to E2′ \E1′ . Moreover, if a urve of R′ has self-interse tion −r, the orresponding urve in T has self-interse tion −r + 1, and it interse ts the boundary B transversally and only at p. Sin e r ∈ {−1, −2}, the urve of T is the proper transform by (τ η)−1 of a line passing through τ η(p) = (λ : 1 : 0) and through one or two points blown-up by η . We des ribe now the set of urves in T for ea h family and ea h λ. (I) If (X, B) = (X1 , B1 ), the line of equation z = x − λy passes through (λ : 1 : 0) and through the point (1 : 0 : 1) blown-up by η1 . Hen e, its transform on X gives an element of T of self-interse tion 0, and thus an element of R′ of self-interse tion −1. / {0, 1}, there is no other element of T , hen e (X ′ , B ′ ) is equal to (X2,a , B2,a ) for some (Ia) If λ ∈ a ∈ k \ {0, 1}. (Ib) If λ ∈ {0, 1}, the line x = λy passes through the point (0 : 0 : 1), whi h is blown-up by η1 , and by one of the two points in its neighbourhood whi h are also blown-up η1 . In this ase, the transform of the line is an element of self-interse tion −1 of T , and gives and element of R′ of self-interse tion −2. In onsequen e, (X ′ , B ′ ) = (X4 , B4 ).

AUTOMORPHISMS OF

A1 -FIBERED

AFFINE SURFACES

27

(II) If (X, B) = (X2,a , B2,a ) for some a ∈ k \ {0, 1}, the lines of equation z = x − λy and az = x − λy pass through (λ : 1 : 0) and respe tively through (1 : 0 : 1) and (a : 0 : 1). Hen e, their transforms give two elements of T of self-interse tion 0 on X , and so two elements of R′ of self-interse tion −1. Moreover, the omposition of the proje tion from (λ : 1 : 0) to the line L ⊂ P2 (of equation y = 0) with ǫp gives rise to an isomorphism Ep \F → L\F , whi h indu es with ψ ◦ (ηp )−1 an isomorphism L\F → E2′ \E1′ . Call D0 ⊂ X the proper transform of the line of P2 of equation x = λy , whi h passes through (λ : 1 : 0) and the point (0 : 0 : 1) blown-up by η2,a . (IIa) If λ 6= 0, D0 has self-interse tion 0 in X and interse ts E2 . It does not belong to T , so (X ′ , B ′ ) is equal to (X1 , B1 ). Moreover, ψ(D0 ) also has self-interse tion 0, and interse ts B ′ into two points, whi h are p′ = ψ(E2 ) and (ψ ◦ (ηp )−1 )(D0 ∩ Ep ) ∈ E2′ . In onsequen e, ψ(D0 ) is the lift by (η1 )−1 of the line of equation x = λ′ y . The isomorphism L\F → E2′ \E1′ sends (1 : 0 : 1) and (a : 0 : 1) onto the dire tions of the lines x = y and x = 0, ea h of these two urves passing through (0 : 0 : 1) = η1 (E2′ ). Moreover, the point (0 : 0 : 1) ∈ L is sent onto the dire tion of the line x = λ′ y . Up to an ex hange of the two dire tions x = y and x = 0 (whi h is indu ed by an automorphism of (X1 , B1 )), we obtain an automorphism of A1 whi h sends respe tively 1, a, 0 onto 0, 1, λ′ . This automorphism is x 7→ (x − 1)/(a − 1), so λ′ = 1/(1 − a). (IIb) If λ = 0, the urve D0 belongs to T , sin e the line of equation x = λy passes through the point (0 : 0 : 1), and by the point in its neighbourhood whi h is also blown-up η2,a . In this ase, (D0 )2 = −1 and D0 orresponds to an element of R′ of self-interse tion −2. In onsequen e, (X ′ , B ′ ) = (X3,b , B3,b ), for some b ∈ k\{0, 1}. Let us prove that b = a. The isomorphism Ep → E2′ sends respe tively the dire tion of z = x − λy , az = x − λy and 0 = x − λy onto the points orresponding to the three urves of R′ . Taking the oordinates (u, v) as in the denition of family III, the three points orrespond respe tively to 1, b, 0. We get an automorphism of A1 whi h sends respe tively 1, a, 0 onto 1, b, 0. In onsequen e, b = a. (III) If (X, B) = (X3,a , B3,a ) for some a ∈ k \ {0, 1}, the line of equation x = λy passes through (λ : 1 : 0) and through (0 : 0 : 1), blown-up by η3,a . Its transform is the unique element of T , of self-interse tion 0 on X . Hen e, (X ′ , B ′ ) = (X2,b , B2,b ) for some b ∈ k \ {0, 1}. Moreover, b = a sin e this reversion is the inverse of the one des ribed in (IIa). (IV) If (X, B) = (X4 , B4 ), the line of equation z = x − λy passes through (λ : 1 : 0) and (1 : 0 : 1), blown-up by η4 . Its transform is the unique element of T , and has self-interse tion 0 on X . Hen e, (X ′ , B ′ ) = (X1 , B1 ). By the above list, there are three possible ases for ψ or its inverse, whi h are I → II , I → IV and II → III . It remains to study ea h ase and to give the values of the parameters asso iated to the surfa es or to the points. I → II . It follows from (Ia) and (IIa) that (λ, λ′ ) ∈ k\{0, 1} × k∗ and that ea h ouple of this form is possible. We prove now that here the parameter of the surfa e (X2,a , B2,a ) is a = 1 − 1/λ (whi h proves in parti ular that any element of family II an be obtained by a reversion on (X1 , B1 )). This link being the inverse of the one des ribed in (IIa), the equality λ = 1/(1 − a), follows from the equality

omputed above. I → IV . The equality λ = 0 follows from (Ib). Moreover, (IV) shows that λ′ an take all possible values in k. II → III . It follows from (IIb) that λ = 0, and that the parameters of ea h pair are the same. Moreover, (III) shows that λ′ an take all possible values in k.

Proposition 5.5.4. All pairs des ribed in 5.5.1 give the same ane surfa e S , up to isomorphism. Moreover, the graph FS asso iated is the following: (X4 , B4 ) o

(X2,a , B2,a ) o g3 gs gggg .. / (X1 , B1 ) kWWWWWW . W+ (X2,b , B2,b ) o

/ (X3,a , B3,a )

.. . / (X3,b , B3,b )

where the a, b orrespond to all values in k \ {0, 1}, up to equivalen e a ∼ a−1 . There are innitely many equivalen e lasses of A1 -brations on S if and only if k is innite. Furthermore, Aut(S) is generated by the automorphisms of A1 -bration.

28

JÉRÉMY BLANC AND ADRIEN DUBOULOZ

Remark 5.5.5. The stru ture of Aut(S) an be des ribed by this method; it is an amalgamated produ t of the group of automorphisms of A1 -brations.

Proof. A

ording to Lemma 5.5.3, we may obtain (X4 , B4 ) and any surfa e of type (X2,a , B2,a ) by

applying a reversion on (X1 , B1 ). Applying a reversion on (X2,a , B2,a ), we get either (X1 , B1 ) or (X3,a , B3,a ). Due to the des riptions of the families, (X2,a , B2,a ) is isomorphi to (X2,b , B2,b ) if and only if there exists an element of Aff whi h sends the points blown-up by η2,a onto points blown-up by η2,b . This amounts to ask for the existen e of an automorphism of the ane line L0 ⊂ A2 whi h xes (0, 0) and sends {(1, 0), (a, 0)} onto {(1, 0), (b, 0)}, and is thus equivalent to say that a = b±1 . The ase of family III is similar. Moreover, two pairs are isomorphi if and only if they indu e the same ane bred surfa es (Lemma 5.5.2). This gives the fa t that all ane surfa es provided by the four families are isomorphi and also the des ription of the graph FS . We obtain the last assertion by applying Proposition 4.0.7.

Remark 5.5.6. In fa t, taking a, b ∈ k∗ , c ∈ k, a 6= b, the following equations in A4 = Spec(k[w, x, y, z]) dene a smooth ane surfa e Sa,b,c , already studied in [5℄ (see also [2℄). xz yw xw

y(y − a)(y − b) z(z − c) (y − a)(y − b)(z − c)

= = =

The proje tion on the x-fa tor indu es a A1 -bration whi h an be ompa tied by a pair of family II ([5℄). In fa t, we an he k that the surfa e is (X2,b/a , B2,b/a ). The proje tion on the w-fa tor also gives an A1 -bration, and one an observe that this one belongs to family I if c 6= 0 and to family IV otherwise. Proposition 5.5.4 gives information on this ane surfa e and also shows that the isomorphism

lass does not depend of the parameters (a, b, c) ∈ (k∗ )2 × k. Referen es

Embeddings of the line in the plane Ane surfa es with . On lo ally nilpotent derivations of Completions of normal ane surfa es with a trivial Makar-Limanov invariant Embeddings of Danielewski surfa es in ane spa es. Automorphisms of ane surfa es. I. Automorphisms of ane surfa es. II. A new geometri proof of Jung's theorem on fa torisation of automorphisms of . Birational transformations of weighted graphs Quasihomogeneous ane surfa es Über ganze birationale Transformationen der Ebene Une preuve géométrique du théorème de Jung Sur la stru ture du groupe d'automorphismes de ertaines surfa es anes Open Algebrai Surfa es On groups of automorphisms of a lass of surfa es Arbres, amalgames,

1. S. S. Abhyankar, T. T. Moh, , J. Reine Angew. Math. 276 (1975), 148166. 2. T. Bandman, L. Makar-Limanov, AK(S) = C Mi higan Math. J. 49 (2001), no. 3, 567582. k[X1 , X2 , Y ]/(φ(Y ) − X1 X2 ), J. Pure Appl. Algebra 181 (2003), 3. D. Daigle, no. 2-3, 181-208. 4. A. Dubouloz, , Mi higan Math. J. 52 (2004), no. 2, 289-308. 5. A. Dubouloz, Comment. Math. Helv. 81 (2006), no. 1, 4973. 6. V. I. Danilov, M. H. Gizatullin, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 3, 523-565. 7. V. I. Danilov, M. H. Gizatullin, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 1, 54-103. C2 8. J. Fernández de Bobadilla, Pro . Amer. Math. So . 133 (2005), no. 1, 1519. 9. H. Flenner, S. Kaliman, M. Zaidenberg, , Ane and algebrai geometry, 107147. Osaka Univ. Press, Osaka 2007. 10. M. H. Gizatullin, , Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 10471071. 11. H.W.E. Jung, . J. reine angew. Math. 184 (1942), 161-174. 12. S. Lamy, , Enseign. Math. (2) 48 (2002), no. 3-4, 291-315. 13. S. Lamy, , Publ. Mat. 49 (2005), no. 1, 3-20. 14. M. Miyanishi, , CRM Monogr. Ser., 12, Amer. Math. So ., Providen e, RI, 2001. 15. L. Makar-Limanov, , Israel J. Math. 69 (1990), no. 2, 250-256. 16. J.-P. Serre, SL2 . Astérisque, No. 46. So iété Mathématique de Fran e, Paris, 1977. Jérémy Blan , Université de Genève, Se tion de mathématiques, 2-4 rue du Lièvre Case postale 64, 1211 Genève 4, Suisse

E-mail address : Jeremy.Blan unige. h Adrien Dubouloz, Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9 avenue Alain

Savary - BP 47870, 21078 Dijon edex, Fran e

E-mail address : Adrien.Duboulozu-bourgogne.fr