Signature morphisms from the Cremona group over a non-closed field

DOI 10.4171/JEMS/983

Stéphane Lamy·Susanna Zimmermann

Signature morphisms from the Cremona group over a non-closed field

Received August 28, 2017

Abstract. We prove that the plane Cremona group over a perfect field with at least one Galois extension of degree 8 is a non-trivial amalgam, and that it admits a surjective morphism to a free product of groups of order 2.

Keywords. Cremona group, Sarkisov program, abelianization, non-closed fields

Introduction

TheCremona groupBir_k(P²)is the group of birational symmetries of the projective plane defined over a fieldk. Its elements are of the form

[x:y:z] 799K[f0(x, y, z):f1(x, y, z):f2(x, y, z)]

wheref₀, f₁, f₂ ∈ k[x, y, z] are homogeneous polynomials of equal degree with no common factor, and such that there exists an inverse of the same form. Equivalently, working in an affine chart one can define the Cremona group as the group of birational selfmaps of the affine plane, which is also (anti-)isomorphic to the group Aut_kk(x, y)of k-automorphisms of the fraction fieldk(x, y). The Cremona group contains the group of polynomial automorphisms of the affine plane over k. In particular, it is a rather huge group. It is neither finitely generated (see [Can17, Proposition 3.6]), nor finite- dimensional, even when working over a finite base field. It was recently shown that Birk(P²)is not a simple group, over any base fieldk[CL13,Lon16]. Then it is natural to ask for nice quotients of Birk(P²), for instance abelian ones. Over an algebraically closed field, it is known (see also §4.3) that the automorphism group Autk(P²)=PGL3(k)and the Jonquières group PGL2(k(T ))oPGL2(k)both embed in any quotient of the Cremona group. In particular in this situation any morphism from Birk(P²)to an abelian group, or to a finite group, is trivial. On the other hand, it was shown by the second author [Zim18]

S. Lamy, S. Zimmermann: Institut de Mathématiques de Toulouse UMR 5219, Université de Toulouse, 31062 Toulouse Cedex 9, France; current address: Laboratoire angevin de recherche en mathématiques (LAREMA), CNRS, Université d’Angers, 49045 Angers Cedex 1, France;

e-mail: [email protected], [email protected] Mathematics Subject Classification (2020):14E07, 14E30

that the situation is drastically different over the fieldRof real numbers. The real Cre- mona group admits an uncountable collection of morphisms toZ/2Z, and precisely we have the following result about the abelianization of Bir_R(P²):

BirR(P²)/[BirR(P²),BirR(P²)] 'M

(0,1]

Z/2Z. (†)

One consequence is that Bir_R(P²)is a nontrivial amalgamated product of two factors along their intersection [Zim17].

In this paper we explore a similar question, over any perfect base fieldkthat admits at least one Galois extension of degree 8. Observe that this condition corresponds to a large collection of fields, which includes the case of all number fields and finite fields.

The special role of degree 8 extensions is explained by their relation to Bertini invo- lutions. Indeed, given a point of degree 8 onP², that is, an orbit of cardinality 8 under the natural action of the absolute Galois group of the base fieldk, we can consider the sur- faceSobtained by blowing up this orbit. If the point is sufficiently general, the surfaceS is del Pezzo and admits another birational morphism toP², and the induced birational selfmap ofP²is an example of a Bertini involution. LetB ⊂Birk(P²)be a set of rep- resentatives of such Bertini involutions with a base point of degree 8, up to conjugacy by automorphisms. We prove that ifkadmits at least one Galois extension of degree 8, then the setBis quite large: it has at least the same cardinality ask, and in the case of a finite fieldF_qone can be more precise and give a lower bound for that cardinality which is polynomial inq(see §4.2).

These Bertini involutions are part of a system of elementary generators E for the group Bir_k(P²)which was found by Iskovskikh [Isk91]. The setE is always huge, be- cause for instance it contains all Jonquières maps (up to left-right composition with auto- morphisms).

Before stating our results we introduce a bit more notation. For each Bertini involution b ∈ B, we setG_b = hAutk(P²), bi. Moreover, we denote byG_e = hAutk(P²),ErBi the subgroup generated by automorphisms and all non-Bertini elementary generators.

Our first result gives an amalgamated product structure for Birk(P²), in terms of these subgroups.

Theorem A. Letkbe a perfect field admitting at least one Galois extension of degree8.

Consider the subgroupsG_ias defined above, fori∈B∪ {e}. ThenG_i∩G_j =Aut_k(P²) for alli6=j, and the Cremona group is the amalgamated product of theG_i along their common intersection:

Bir_k(P²)' ˚

Autk(P²)

G_i.

Moreover,Bir_k(P²)acts faithfully on the corresponding Bass–Serre tree.

It was shown by Cornulier [CL13, appendix] that the Cremona group over an algebraically closed field is not a non-trivial amalgam of two groups. In contrast, we deduce from the above theorem the following structure result, where we denote byGB = hAutk(P²),Bi the subgroup generated by allGb:

Corollary B. Letkbe a perfect field admitting at least one Galois extension of degree8.

ThenG_e∩G_B =Aut_k(P²), and

Bir_k(P²)'G_B∗_Aut

k(P²)G_e, andBirk(P²)acts faithfully on its Bass–Serre tree.

It turns out that each subgroupG_badmits a structure of free product, and this allows one to obtain a lot of morphisms from the Cremona group toZ/2Z:

Theorem C. Letkbe a perfect field with at least one Galois extension of degree8.

(1) For eachb ∈ Bwe haveG_b ' Aut_k(P²)∗Z/2Z, and we can write the Cremona group as a free product:

Bir_k(P²)'G_e∗

˚BZ/2Z .

(2) In particular, there is a surjective morphism Bir_k(P²)→˚

B

Z/2Z

whose kernel is the smallest normal subgroup containingG_e, and which sends each b∈Bto the corresponding generator on the right-hand side.

(3) In particular, the abelianization of the Cremona group overkcontains a subgroup isomorphic toL

BZ/2Z.

We see that even if no Bertini involutions were involved in [Zim18], we obtain a similar looking (even if less precise) result. In particular, theZ/2Zin the target group in (†) have nothing to do with the fact that the absolute Galois group ofRhas order 2, but rather with the fact that we are able to produce a natural set of generators for the Cremona group that contains involutions. In this sense, we like to think of the above morphisms Birk(P²)→ Z/2Zas some analogues of the classical signature morphism on the symmetric group. The huge collection of such morphisms corresponds to the existence of a system of generators with a lot of non-conjugate involutions. In this paper, we focus on Bertini involutions associated to a base point of degree 8 because they seem to be the easiest to handle technically. When the base field isR, a similar role was played by the so-called “standard quintic involutions”. It seems quite plausible that other “signature morphisms” exist on the Cremona group, associated to other type of involutions, such as the Geiser involution associated to a base point of degree 7. Also, we see no obvious obstruction why such morphisms could not exist in higher dimension, even over the field of complex numbers.

The strategy to prove the above results is to use the Sarkisov program. The Sarkisov program is a way to factorize a given birational map between Mori fiber spaces into elementary links. We recall that even if the starting map is a birational selfmap of a given varietyX(for instanceX=P²), the elementary links are not in general elements of the group Bir(X). In other words, even if one is primarily interested in the group Bir(X), the Sarkisov program naturally produces generators for the groupoid of birational maps Y 99K Z, whereY, Z can be any Mori fiber spaces birational to X. Nevertheless the

Sarkisov program turns out to be an efficient tool to produce some systems of generators for Bir_k(P²), and also to describe relations between them [Isk91, IKT93, Isk96]. The Sarkisov program was revisited recently in light of the progresses in the theory of the Minimal Model Program, and is now established in any dimension (overC) [HM13].

Moreover the relations between Sarkisov links were described by Kaloghiros [Kal13].

In Section2we encode Sarkisov links and relations between them in a square com- plexX on which the group Birk(P²)acts naturally. Then in Section3we give an account of the proof of the Sarkisov program in the simpler case of surfaces, but working over an arbitrary perfect field. This allows us to prove that the square complexXis connected and simply connected.

In Section4we recall the notion of elementary generators for the group Birk(P²), fol- lowing the work of Iskovskikh. Among the elementary generators we discuss in particular the Bertini involutions and prove their existence (and in fact abundance). We also discuss the Jonquières maps, and in any dimension we recall the following basic dichotomy: given a morphismϕ from the Cremona group to another groupH, either the subgroup gener- ated by the Jonquières maps lies in the kernel, orϕinduces an embedding of the subgroup intoH.

Finally, in Section5we use Bass–Serre theory to prove our results. The general idea is that the Bass–Serre trees of the various amalgams appearing in TheoremA, Corol- laryBand TheoremCare realized either as a quotient or as a subcomplex of the square complexX.

When one encounters a cube complex in geometric group theory, a natural question is whether this complex has non-positive curvature. It turns out that this is not the case for our square complexX, but we should mention thatX is essentially a subcomplex of an infinite-dimensional CAT(0)cube complex associated with the Cremona group that was constructed by Lonjou in her PhD thesis.

1. Birational maps between surfaces over an arbitrary field

In this section we review some results of the birational geometry of surfaces, with a focus on the case of an arbitrary perfect base field.

1.1. Factorization into blow-ups

Letk be a perfect field, andk^a an algebraic closure. All field extensions ofkthat we shall consider will be supposed to lie ink^a. By asurface(overk) we shall mean a smooth projective surface defined overk. We denote byS(k)the set ofk-rational points onS.

The Galois group Gal(k^a/k)acts onS×_SpeckSpeck^athrough the second factor. In par- ticular, Gal(k^a/k)acts on the setS(k^a)ofk^a-rational points. By apoint of degreed on S we mean an orbit p = {p1, . . . , p_d} ⊂ S(k^a)of cardinalityd under the action of Gal(k^a/k). Observe that the points inS(k)are exactly the fixed points for the action of Gal(k^a/k)onS(k^a), or in other words the points of degree 1. LetL/kbe a field extension such that thepi areL-rational points. We define the blow-up ofpto be the blow-up of thesedpoints, which is a morphismπ:S⁰→Sdefined overk, with exceptional divisor

E=C₁+ · · · +C_d, where theC_iare disjoint(−1)-curves defined overL, andE²= −d. We shall refer to this situation by saying thatEis anexceptional divisor of degreed.

We recall the following classical factorization results (see e.g. [Liu02, Theorems 9.2.2 and 9.2.7]).

Proposition 1.1. Letπ:S⁰ → S be a birational morphism between surfaces defined overk. Then π = π₁◦ · · · ◦π_n, where each π_i: S_i → S_i−1is the blow-up of a point of degreed_i ≥ 1onS_i−1, with exceptional divisorE_i onS_i satisfying E_i² = −d_i (in particularS=S₀, andS⁰=S_n).

Proposition 1.2. Letϕ:S 99KS⁰be a birational map between surfaces defined overk.

Then there exist a surfaceZ defined over k, and sequences of blow-upsπ: Z → S, π⁰:Z→S⁰of orbits of points underGal(k^a/k)such thatπ⁰=ϕ◦π.

We should mention that even if the Cremona group was explicitly defined in the intro- duction in terms of homogeneous polynomials, in practice we almost always think of an element of Bir_k(P²)as given by two sequences of blow-ups defined overk, as provided by Proposition1.2.

Remark 1.3. Over a non-perfect field k, there is no reason why the base points of a birational map should be defined over a separable closure ofk, and so we can no longer identify closed points with Galois orbits as we did in the statements of Propositions1.1 and1.2. As a simple example of this phenomenon, considerk=F₂(t ), and denote byt^1/2 the unique square root oft ink^a. Then k(t^1/2)/k is a non-separable extension. Now consider the birational involutionf ∈Bir_k(P²)given by

f: [x0:x1:x2] 799K[x0x2:x1x2:x₀²+t x₁²].

As a quadratic birational map,f admits three base points defined overk^a, which are p1= [0:0:1], p2= [t^1/2:1:0],

and a pointp₃infinitely near top₂. In particularp₂is not defined over a separable exten- sion ofk.

1.2. Negative maps, minimal and ample models, scaling

LetSbe a surface defined overk, andS^athe same surface overk^a. We define theNéron–

Severi spaceN¹(S^a)as the space of numerical classes ofR-divisors:

N¹(S^a):=Div(S^a)⊗R/≡.

The action of Gal(k^a/k) on N¹(S^a) factors through a finite group, and we denote by N¹(S) the subspace of invariant classes. Since we only consider surfaces with S(k)6= ∅andk^a[S^a]^∗ =(k^a)^∗,N¹(S)is also the space of classes of divisors defined overk(see [San81, Lemma 6.3(iii)]). The dimension of this finite-dimensionalR-vector space is called thePicard numberofSoverk, and denoted byρ(S).

Remark 1.4. When working on a surfaceS, we can identify the spaceN¹(S)of divisors and the spaceN₁(S)of 1-cycles, and similarly the subspaces Eff(S)or NE(S)of effective divisors or 1-cycles. We shall use the notation that seems most natural in view of the extension of the results in higher dimension. For instance the Cone Theorem1.7is about 1-cycles, so there we use the notation NE(S).

Letπ:S⁰ → S be a birational morphism between surfaces defined overk, andD⁰ a Q-divisor onS⁰with push-forwardD =π∗(D⁰). By Proposition1.1, we can writeπ = π1◦ · · · ◦π_n, whereπ_i:S_i → Si−1is the blow-up of a point of degreed_i, withS =S0

andS⁰ = Sn. For anyi, we denote byEi the exceptional divisor ofπi, and byDi the push-forward ofD⁰onSi. We say thatπisD⁰-negativeifDi ·Ei <0 for alli. Observe also that onS⁰we can write

D⁰=π^∗D+X aiEi

for somea_i ∈Q, where theE_idenote strict transforms onS⁰.

Lemma 1.5. With the above notation, the morphism π is D⁰-negative if and only if a_i >0for alli.

Proof. OnS_i, we haveD_i =π_i^∗Di−1+a_iE_i, so that

0=π^∗Di−1·Ei =Di ·Ei−aiE_i².

Since E_i² = −d_i, where d_i ≥ 1 is the degree of the point blown up by π_i, we get a_i = −^Di·Ei

di , so thata_i andD_i·E_ihave opposite signs as expected. ut Ifπ:S⁰ →Sis aD⁰-negative birational morphism, andD=π∗(D⁰)is nef, we callSa D⁰-minimal modelofS⁰. Such a model, if it exists, is unique:

Lemma 1.6 ([Mat02, p. 94]). LetS1,S2be twoD⁰-minimal models ofS⁰. Then the in- duced mapS199KS2is an isomorphism.

We shall use the above setting for divisorsD⁰ of the formD⁰ = K_S⁰ +A, withAan ampleQ-divisor (orA=0). Observe that a(K_S⁰ +A)-negative birational morphism is also aK_S⁰-negative morphism, hence simply a sequence of inverse of blow-ups. There exist surfaces with infinitely many(−1)-curves: a classical example is given byP²blown up at (sufficiently general) nine points. This gives a countable collection of curvesC_iwith K_S⁰ ·C_i <0 andC_i²<0. However, after perturbingK_S⁰by adding anyQ-ample divisor, we get a finite collection:

Theorem 1.7 ([Rei97, Cone Theorem D.3.2]). LetSbe a surface defined overk^a. Then ifρ(S)≥3, allK_S-negative extremal rays of the coneNE(S)are of the formR_>0Cwith Ca(−1)-curve. Moreover, for any ampleQ-divisorAonS, there are only finitely many (−1)-curvesC_i such that(K_S+A)·C_i <0.

Comments on the proof. In the Cone Theorem, the main delicate point to check when working in arbitrary characteristic is vanishing. IfAis ample, by Serre duality we have H²(S, K_S+A)=H⁰(S,−A)^∗=0. So by Riemann–Roch on a surface we have, for the divisorD=K+A,

h⁰(S, D)≥h⁰(S, D)−h¹(S, D)= ¹

2D(D−KS)+χ (O_S). (1)

In fact when S is rational, we even haveH¹(S, K_S +A) = 0 (Kodaira vanishing in positive characteristic can only fail for surfaces of Kodaira dimension≥ 1, see [Ter99, Theorem 1.6]). But this extra information is not necessary in the argument given by Reid,

as inequality (1) is enough. ut

Remark 1.8. (1) Over an arbitrary perfect fieldk, we have an equivariant version of the Cone Theorem with respect to the action of Gal(k^a/k)[KM98, p. 48]. Essentially we only have to change “(−1)-curve” to “orbit of pairwise disjoint(−1)-curves under the action of Gal(k^a/k)”. By the Castelnuovo Contraction Theorem, we can contract such an orbit and obtain a new smooth projective surface. Thus by running the Minimal Model Program with respect to the canonical divisorK, or more generally with respect toK+A withAample, we stay in the category of smooth surfaces, and a posteriori this justifies that we restrict ourselves to this setting.

(2) In the case where the Picard numberρ(S)is equal to 2, NE(S)is a convex cone in a real 2-dimensional vector space, thus we have at most two extremal rays, which correspond either to the contraction of an exceptional divisor or to a Mori fibration. This case is particularly interesting in a relative setting, and is then often referred to as atwo rays game. More precisely, we start with a morphism π: S → Y from a surface S, with relative Picard number equal to 2. We assume that any curveC contracted byπ satisfiesK_S·C <0. Then there exist exactly two morphisms of relative Picard number 1, π_i:S→Y_i,i=1,2, such thatπ factors through each of theπ_i:

π1 S

~~ ^π2

π

Y1

Y2

~~Y

Theorem 1.9 (Base Point Free Theorem, see [Rei97, D.4.1]). LetS be a surface, and letDbe a nef divisor onSsuch thatD−εKSis ample for someε >0. Then the linear system|mD|is base point free for all sufficiently largem.

IfDis nef and of the formD =K+AwithAample, we denote byϕ_Dthe morphism fromS associated to the linear system|mD|form0. This morphism has connected fibers, and it contracts precisely the curvesCsuch that(K+A)·C =0. Now we extend the definition ofϕDto any pseudo-effective divisorD=K+A, by using the notion of scaling.

LetSbe a surface, andA,1ampleQ-divisors onS(we also admit the caseA=0).

TheK+Aminimal model program with scaling of 1is a sequence of birational mor- phismsπ_i:S_i−1→S_idefined iteratively as follows. We setS₀=S. IfS_i is constructed, we denote byA_iand1_i the direct images ofAand1onS_i, and we considert_i such that Di =KS_i +Ai +ti1i is nef but not ample onSi. Then the morphismπi+1fromSi is obtained by applying Theorem1.9toDi. Ifπi+1(Si)=Si+1is a surface, we repeat the construction. At some point,πi+1is a fibration to a curve or a point, in which case we have reached a Mori fiber space and the program stops. In particular, this process gives

a finite sequence of rational numbers

t0> t1>· · ·> t_n

such thatK_Si +A_i +t 1_i is ample onS_i for anyt > t_i. We shall say that a birational morphismπ:S→S⁰is(K+A)-negative with scaling of 1ifS⁰is one of theS_i in the above process.

Now assume thatDis a pseudo-effective Q-divisor onS of the formD =K+1, with1ample. We run theK minimal model program with scaling of1, and we look where the coefficientt =1 corresponding toDfits into the sequencet₀> t₁>· · ·> t_n. More precisely, we setj =max{i;t_i ≥1}and we denoteϕ_D =π_j ◦ · · · ◦π₁. Observe that the morphismϕ_D is birational if and only ifj ≤ n−1, and in any caseϕ_D(D)is ample. We say thatϕD is theample modelofD.

Remark 1.10. (1) The above construction only depends on the numerical class ofD, which would not be the case for more generalD(that is, not of the formK+1with1 ample) [KKL16, Example 4.8].

(2) The morphismϕD coincides with the morphism fromS to Proj(L

H⁰(Z, mD)) [KKL16, Remark 2.4]. In particular, if we writeD=K+1₁+1₂with1₁, 1₂ample, and run theK+1₁minimal model program with scaling of1₂, we will get the same morphismϕ_D, but possibly by another sequence of contractions.

2. A square complex associated to the Cremona group

In this section we construct a square complex that encodes Sarkisov links and relations between them. First we introduce the key notion of rankrfibration.

2.1. Rank r fibrations

If not stated otherwise, all varieties and morphisms are defined overk. LetSbe a surface, andr ≥ 1 an integer. We say thatS is a rank r fibration if there exists a surjective morphismπ:S →Bwith connected fibers, whereBis a point or a smooth curve, with relative Picard number equal to r, and such that the anticanonical divisor −K_S is π- ample. The last condition means that for any curveCcontracted to a point byπ, we have K_S·C <0. Observe that the condition on the Picard number translates asρ(S)=rifB is a point, andρ(S)=r+1 ifBis isomorphic toP¹. IfSis a rankrfibration, we will writeS/B if we want to emphasize the basis of the fibration, andS^r when we want to emphasize the rank. An isomorphism between two fibrationsS/BandS⁰/B⁰(necessarily of the same rankr) is an isomorphismS →^∼ S⁰such that there exists an isomorphism on the bases (necessarily uniquely defined) that makes the following diagram commute:

S ^∼ //

π

S⁰

π⁰

B ^∼ //B⁰

As the following examples make it clear, there are sometimes several choices for a structure of rankr fibration on a given surface, that may even correspond to distinct ranks.

Example 2.1. (1)P²with the morphismP²→pt, or the Hirzebruch surfaceFnwith the morphismFn→P¹, are rank 1 fibrations.

(2) F1 with the morphismF1 → pt is a rank 2 fibration. Idem forF0 → pt. The blow-upS² → Fn → P¹of a Hirzebruch surface along a point of degreed, such that each point of the orbit is in a distinct fiber, is a rank 2 fibration overP¹.

(3) The blow-up of two distinct points on P², or of two points ofFn not lying on the same fiber, give examples of rank 3 fibrations, with morphisms to the point or toP¹ respectively.

Remark 2.2. Observe that the definition of a rankrfibration puts together several well- known notions. IfB is a point, thenS is a del Pezzo surface of Picard rankr(over the base fieldk). IfBis a curve, thenSis a conic bundle of relative Picard rankr: a general fiber is isomorphic toP¹, and (overk^a) any singular fiber is the union of two(−1)-curves secant at one point. Note also that rank 1 fibrations are exactly the usual 2-dimensional Mori fiber spaces.

We will be only interested in rational surfaces, and we define amarking on a rankr fibrationS/B to be a choice of a birational mapϕ: S 99K P². Observe that ifS is ra- tional andB is a curve, thenBis isomorphic toP¹. We say that two marked fibrations ϕ:S/B99KP²andϕ⁰:S⁰/B⁰99KP²areequivalentifϕ⁰⁻¹◦ϕ:S/B →S⁰/B⁰is an iso- morphism of fibrations. We denote by(S/B, ϕ)an equivalence class under this relation.

The Cremona group Bir_k(P²)acts on the set of equivalence classes of marked fibrations by post-composition:

f ·(S/B, ϕ):=(S/B, f ◦ϕ).

IfS⁰/B⁰ andS/Bare marked fibrations of respective ranksr⁰ > r ≥ 1, we say that S⁰/B⁰factorizesthroughS/Bif the birational mapS⁰ →Sinduced by the markings is a morphism, and moreover there exists a (uniquely defined) morphismB → B⁰such that the following diagram commutes:

S^{0 π}

0 //

!!

B⁰

S ^π //B

<<

(2)

In fact, ifB⁰ =pt the last condition is empty, and ifB⁰ 'P¹it means thatS⁰ →Sis a morphism of fibrations over a common basisP¹.

2.2. Square complex

We define a 2-dimensional complexX as follows. Vertices are equivalence classes of marked rankr fibrations, with 3 ≥ r ≥ 1. There is an oriented edge from(S⁰/B⁰, ϕ⁰)

to(S/B, ϕ)ifS⁰/B⁰factorizes throughS/B. Ifr⁰ > rare the respective ranks ofS⁰/B⁰ andS/B, we say that the edge hastyper⁰, r. For each triplet of pairwise linked vertices (S⁰⁰³/B⁰⁰, ϕ⁰⁰), (S⁰²/B⁰, ϕ⁰),(S¹/B, ϕ), we glue a triangle. In this way we obtain a 2- dimensional simplicial complexX on which the Cremona group acts.

Lemma 2.3. For each edge of type3,1fromS⁰⁰/B⁰⁰toS/B, there exist exactly two tri- angles that admit this edge as a side.

Proof. In short, the proof is a two rays game (see Remark1.8). By assumptionS⁰⁰/B⁰⁰ factorizes throughS/B, so by settingY = S (ifB ' B⁰) orY = B (ifB ' P¹ and B⁰=pt), we obtain via diagram (2) a morphismπ:S⁰⁰→Ywith relative Picard number ρ(S⁰⁰/Y )equal to 2. We have exactly two extremal rays in the cone NE(S⁰⁰/Y ), and since ρ(S⁰⁰) = 3 or 4, both correspond to divisorial contractions. Denote byS⁰⁰ → S⁰ and S⁰⁰→ ˜S⁰these two contractions. Then the two expected triangles areS⁰⁰/B⁰⁰,S⁰/B⁰,S/B

andS⁰⁰/B⁰⁰,S˜⁰/B⁰,S/B. ut

In view of the lemma, by gluing all the pairs of triangles along edges of type 3,1, and keeping only edges of types 3,2 and 2,1, we obtain a square complex that we still de- noteX. Letvertices of typeP²be the vertices in the orbit of the vertex(P²/pt,id)under the action of Birk(P²). When drawing subcomplexes ofX we will often drop part of the information which is clear by context, about the markings, the equivalence classes and/or the fibration. For instanceS/Bmust be understood as(S/B, ϕ)for an implicit marking ϕ, and(P², ϕ)as(P²/pt, ϕ).

Example 2.4. LetSbe the surface obtained by blowing upP² in two distinct pointsa andbof degree 1. Denote byF1,a/P¹a,F1,b/P¹_bthe two intermediate Hirzebruch surfaces with their fibrations toP¹. Finally, denote byF0the surface obtained by contracting the strict transform onSof the line throughaandb. All these surfaces fit into the subcomplex ofXpictured in Figure1, where the dotted arrows are the edges of type 3,1 that we need to remove from the simplicial complex in order to get a square complex.

S/pt

S/P¹a S/P¹_b

F0/P¹a F0/P¹_b F0/pt

F1,a/pt F1,b/pt

F1,a/P¹a F1,b/P¹_b P²/pt

yy %%

yy %%

eeyy 99 %%

99 ee

ee 99oo //

OO

Fig. 1

Example 2.5. Consider the blow-ups of three points a, b, c on P². These give three squares around the corresponding vertex of typeP²(see Figure2). In particular, the square complexXis not CAT(0), as mentioned in the introduction.

S_a,b/pt

S_b,c/pt S_a,c/pt

F1,a/pt F1,b/pt F1,c/pt

P²/pt

ff 88

88 ff OO&& xx

88 ff

Fig. 2

2.3. Sarkisov links and elementary relations

In this section we show that the complexX encodes the notion of Sarkisov links, and of elementary relations between them.

First we rephrase the usual notion of Sarkisov links between 2-dimensional Mori fiber spaces. Let(S/B, ϕ),(S⁰/B⁰, ϕ⁰)be two marked rank 1 fibrations. We say that the induced birational mapS99KS⁰is aSarkisov linkif there exists a marked rank 2 fibrationS⁰⁰/B⁰⁰ that factorizes through bothS/B andS⁰/B⁰. Equivalently, the vertices corresponding to S/BandS⁰/B⁰are at distance 2 in the complexX, with middle vertexS⁰⁰/B⁰⁰:

S⁰⁰/B⁰⁰

ww ((

S/B S⁰/B⁰

This definition is in fact equivalent to the usual definition of a link of type I, II, III or IV fromS/BtoS⁰/B⁰(see [Kal13, Definition 2.14] for the definition in arbitrary dimension).

Below we recall these definitions in the context of surfaces, in terms of commutative diagrams where each morphism has relative Picard number 1 (such a diagram corresponds to a “two rays game”), and we give some examples. Note that these diagrams are not part of the complexX: in each case, the corresponding subcomplex ofX is just a path of two edges, as described above.

• Type I:Bis a point,B⁰'P¹, andS⁰→Sis the blow-up of a point of degreed≥1 such that we have a diagram

S⁰

S

P¹

~~pt

Then we takeS⁰⁰/B⁰⁰:=S⁰/pt.

Examples are given by the blow-up of a point of degree 1, or a general point of de- gree 4, onS =P². The fibrationS⁰/P¹corresponds respectively to the lines through the point of degree 1, or to the conics through the point of degree 4.

• Type II:B=B⁰, and there exist two blow-upsS⁰⁰→SandS⁰⁰→S⁰that fit into a diagram of the form

S⁰⁰



S

S⁰

~~B

Then we takeS⁰⁰/B⁰⁰:=S⁰⁰/B.

An example is given by blowing up a point of degree 2 on S = P², and then by contracting the transform of the unique line through this point. The resulting surfaceS⁰is a del Pezzo surface of degree 8, which has rank 1 overk, but has rank 2 overk^a(being isomorphic to P¹_ka ×P¹_ka). Other examples, important for this paper, are provided by blowing up a point of degree 8 onP²: see §4.2.

• Type III: a situation symmetric to a link of type I.

• Type IV:(S, ϕ)and(S⁰, ϕ⁰)are equal as marked surfaces, but the fibrations toB andB⁰are distinct. In this situationB andB⁰must be isomorphic toP¹, and we have a diagram

S



B

B⁰

~~pt

Then we takeS⁰⁰/B⁰⁰:=S/pt.

For rational surfaces, a type IV link always corresponds to the two rulings onF0 = P¹×P¹, that is,S/B = F0/P¹ is one of the rulings,S⁰/B⁰ = F0/P¹ the other one, andS⁰⁰/B⁰⁰ =F0/pt. See [Isk96, Theorem 2.6 (iv)] for other examples in the context of non-rational surfaces.

Apath of Sarkisov linksis a finite sequence of marked rank 1 fibrations (S₀/B₀, ϕ₀), . . . , (S_n/B_n, ϕ_n)

such that for all 0≤i≤n−1, the induced mapg_i: S_i/B_i 99KS_i+1/B_i+1is a Sarkisov link.

Proposition 2.6. Let (S⁰/B, ϕ) be a marked rank3 fibration. Then there exist finitely many squares inX withS⁰as a corner, and the union of these squares is a subcomplex ofX homeomorphic to a disk with center corresponding toS⁰.

Proof. Sinceρ(S⁰)=3 or 4, we can factorize the fibrationS⁰/Binto S⁰→S→Y →B,

whereS⁰→Sis a divisorial contraction, andY is either a surface or a rational curve. By playing the two rays game onS⁰/Y (see Remark1.8), we obtain another surfaceS˜and a divisorial contractionS⁰→ ˜Sthat fits into a commutative diagram

S

''S⁰

66''

Y //B S˜

77

IfY is a surface, we obtain the following square inX: S⁰/B

{{ ##

S/B

##

S/B˜

{{Y /B

On the other hand, ifY 'P¹is a curve (and soBis a point), then we obtain the following two squares inX:

S⁰/pt

ww ''

S/pt

S/pt˜

S⁰/P¹

ww ''

S/P¹ S/˜ P¹

Now in both cases we consider the two rays game onS/B: this produces˜ Y˜, which is either a surface or a curve, and which fits into a diagram

S

''S⁰

66''

Y ''˜

S

77&&

B Y˜

77

Then by considering the two rays game onS⁰/Y˜, we produce one or two new squares inX that are adjacent to the previous ones. After finitely many such steps, the process must stop and produce the expected disk, because by Theorem1.7there are only finitely many divisorial contractions that we can use to factorS⁰/B. ut Remark 2.7. If B ' P¹, then at each step Y is a surface, and in this case the disk produced by the proof consists of exactly four squares inX.

More precisely, with the notation of the proof, the morphismsS/Y andS/Y˜ corre- spond to the blow-ups of pointsp andp, and˜ S⁰ is the blow-up of both. Observe that

p,p˜can have arbitrary degrees, but they are not on the same fiber ofY /P¹because other- wiseS⁰/P¹would not be a rank 3 fibration (the anticanonical divisor would not be ample because of the presence of(−2)-curves). LetE_p, Ep˜ be the corresponding exceptional divisors inS⁰, andF_p, Fp˜the strict transforms of the fibers throughpandp˜respectively.

Then the four squares correspond to the four possible choices of contraction of two divi- sors fromS⁰overP¹: eitherE_porF_p, and independently eitherEp˜orFp˜.

On the other hand, ifB is a point the number of squares might vary: for instance in Example2.4we saw five squares.

In the situation of Proposition2.6, by going around the boundary of the disk we obtain a path of Sarkisov links whose composition is the identity (or strictly speaking, an automor- phism). We say that this path is anelementary relationbetween Sarkisov links, coming fromS⁰³/B. More generally, any composition of Sarkisov links that corresponds to a loop in the complexX is called arelationbetween Sarkisov links.

3. Relations in the Sarkisov program in dimension 2

In this section we prove that the complexX is connected and simply connected, which will be the key in proving TheoremA. These connectedness results will follow from the Sarkisov program, and more precisely from the study of relations in the Sarkisov program, which we can state as follows:

Theorem 3.1 (Sarkisov program).

(1) Any birational mapf: S 99KS⁰between rank1fibrations is a composition of Sar- kisov links(and automorphisms).

(2) Any relation between Sarkisov links is generated by elementary relations.

In arbitrary dimension overC, these results correspond to [HM13, Theorem 1.1] and [Kal13, Theorem 1.3]. We give an account of the proof of these results in the simpler case of surfaces, but working over an arbitrary perfect base field.

3.1. Polyhedral decomposition

Let{(S₁/B₁, ϕ₁), . . . , (S_m/B_m, ϕ_m)}be a finite collection of marked rank 1 fibrations. In the context of Theorem3.1, we will takem = 2,S1 = S,S2 = S⁰ when proving(1), or the entire collection of rank 1 fibrations visited by a relation of Sarkisov links when proving(2).

By repetitively applying Proposition1.2, we produce a marked surfaceZdominating all the S_i, that is, such that all induced birational mapsf_i: Z → S_i are morphisms.

We pick a sufficiently small ampleQ-divisorAonZsuch that for alli,fi is(K +A)- negative, and−KS_i −fi∗(A)is relatively ample overBi. The point of choosing such an ample divisorAis to ensure that there exist only finitely many(K+A)-negative birational morphisms fromZ(up to post-composition with an isomorphism). Indeed, this follows

from Theorem1.7, which says that at each step there are only finitely many possible divisorial contractions. If there are only finitely many birational morphisms fromZ(for instance ifZis a del Pezzo surface), we also admit the choiceA=0.

For eachi=1, . . . , m, applying the following two steps we construct an (effective) ample divisor1_i onZsuch thatf_i:Z→S_iis(K+A)-negative with scaling of1_i, and more preciselyf_i will be a(K+A+1_i)-ample model ofZ:

(1) If B_i ' P¹, pick a large multipleG_i of the fiber ofS_i/B_i such that −K_Si − f_i∗(A)+G_i is ample onS_i. IfB_i = {pt}, then−K_Si −f_i∗(A)is already ample, so we just setG_i =0. In both casesG_i is a nef divisor onS_i.

(2) Now pick an effectiveQ-divisorP_ionS_i, equivalent to the ample divisor−K_Si− f_i∗(A)+G_i, and set1_i = A+f_i^∗(P_i), which is ample as the sum of ample and nef divisors. One checks that

f_i∗(K+A+1_i)=K_Si+f_i∗(A)+f_i∗(A)−K_Si −f_i∗(A)+G_i =f_i∗(A)+G_i, which is an ample divisor onS_i as expected.

We can assume that the family{1i}^m

i=1generatesN¹(Z)(throw in more ample di- visors if necessary). We choose some rationalsri > 0 such that eachK+A+ri1i is ample. We say that aQ-divisor1is asubconvex combinationof theri1i if

1=

m

X

i=1

t_ir_i1_i witht_i ≥0,X t_i ≤1.

This family{1_i}^m

i=1of ample divisors being fixed, let{g_j}^s

j=1be the finite collection of (K+A+1)-negative birational morphisms (up to isomorphism on the range), for all choices of a subconvex combination1as above. Observe that this collection is indeed finite because of Theorem1.7. Note also that the initial f_i,i = 1, . . . , m, are part of this collection by construction, and so is the identity id: Z → Z (corresponding for instance to taking one of theti equal to 1, and all the other equal to 0, because then D =K +A+ri1i is ample by assumption, and the corresponding embeddingϕD is equivalent to the identity morphism).

For any family{D_i}_i∈I ∈N¹(Z)of divisors in the vector spaceN¹(Z), we denote by Conv^◦(D_i; i∈I )the cone over the convex hull of theD_i. Now we intersect the cone

Conv^◦(K+A, K+A+r₁1₁, . . . , K+A+r_m1_m) with the pseudo-effective cone Eff(Z)to get a coneC^◦. Explicitly,

C^◦= n

D=λ

K+A+

m

X

i=1

tiri1i

; λ, ti≥0,X

ti≤1, Dpseudo-effectiveo

⊆N¹(Z).

We denote byCthe intersection ofC^◦with an affine hyperplane defined byK+A:

C:= {D∈C^◦;(K+A)·D= −1}.

Recall from Proposition1.1that eachg_j:Z →S_jis a finite sequence of contractions of exceptional divisors, that is, orbits under Gal(k^a/k)of pairwise disjoint(−1)-curves.

We denote by{C_i}_i∈I the finite collection of classes inN¹(Z)obtained as pull-back of such exceptional divisors contracted byg_j, for allj =1, . . . , s. We introduce a notation for the hyperplane and half-spaces defined byC_i inN¹(Z):

C_i^⊥= {D∈N¹(Z);D·Ci =0}, C^≥_i = {D∈N¹(Z);D·C_i ≥0}, and similarly forC_i^>,C_i^≤,C_i^<.

LetD∈Cbe a big divisor. Then we get a partitionI =I⁺∪I⁻such that D∈ \

i∈I⁺

C_i^>∩ \

i∈I⁻

C_i^≤.

There existsj such that the morphismϕDassociated withD(see discussion after Theo- rem1.9) coincides withgj :Z → Sj. The classesCi,i ∈ I⁻, correspond to the curves contracted byϕD=gj.

Lemma 3.2. The coneC^◦ ⊂ N¹(Z)is rational polyhedral and convex, hence so is the affine sectionC.

Proof. First we prove that nef(Z)∩C^◦=\

i∈I

C_i^≥∩Conv^◦(K+A, K+A+r111, . . . , K+A+rm1m),

from which it follows that nef(Z)∩C^◦is a rational polyhedral convex cone.

IfD=K+A+1∈C^◦is not nef, then by construction ofϕ_Dthere existsi∈Isuch that the exceptional divisorC_iis contracted byϕ_D, andD·C_i <0. On the other hand, if D ∈C^◦is nef, then for any irreducible componentC in the support of one of theC_i we haveD·C≥0, henceD·C_i ≥0 for alli.

Now we show that C^◦is a rational polyhedral convex cone. Let D ∈ C^◦ be a big divisor, andϕ_D: Z → S the associated birational morphism. Up to reordering theC_i, we can assume thatC₁, . . . , C_r are the classes contracted byϕ_D, whereris the relative Picard number ofZoverS. Then we can write

D=ϕ_D^∗(ϕD∗(D))+

r

X

i=1

aiCi,

whereϕ^∗_D(ϕD∗(D))∈nef(Z)∩C^◦, and theai are positive by Lemma1.5. Together with K+A+r_i1_i ∈nef(Z)∩C^◦this gives

C^◦⊆Conv^◦(nef(Z)∩C^◦, K+A)∩Conv^◦(nef(Z)∩C^◦, C_i; i∈I ).

This inclusion is in fact an equality because the first cone on the right-hand side is just Conv^◦(K +A, K +A+r111, . . . , K +A+rm1m)and the second cone consists of

pseudo-effective divisors, so the right-hand side is contained inC^◦. It follows thatC^◦is

rational polyhedral and convex, as expected. ut

We set

A_j := \

i∈I⁺

C_i^>∩ \

i∈I⁻

C_i^≤∩C.

In particular,A_j is a rational polyhedral subset ofC, and is equal to the set of divisors D∈ Csuch thatϕ_D =g_j. Observe that the chamberAof ample divisors inCis one of theA_j, associated tog_j =id, and to the partitionI⁺ =I,I⁻= ∅. Clearly theA_j form a partition of the interior ofC.

We have just re-proved [KKL16, Theorem 4.2] (which is stated in arbitrary dimension, but overC):

Theorem 3.3. The interior of the coneCadmits a finite partition into polyhedral cham- bers,Int(C)=S

jA_j.

For further reference we sum up the above discussion:

Set-Up 3.4. • We start with a finite collection {(S1/B1, ϕ1), . . . , (Sm/Bm, ϕm)} of marked rank 1 fibrations.

• We pickZa common resolution with Picard numberρ(Z)≥4.

• We chooseAan ampleQ-divisor onZsuch that each mapZ→S_iis(K+A)-negative with scaling of an ample divisor1_i. We may add some ample divisors1_i so that the family{1_i}^m

i=1generatesN¹(Z). If there are finitely many birational morphisms from Z, we allowA=0.

• We construct a convex cone C^◦ in N¹(Z), by considering the union of all seg- ments [1, K +A] ∩Eff(Z), for all convex combinations 1 of the ample divisors K+A+r_i1_i,i=1, . . . , m, and by taking the cone over these. In practice, we work withC, the section ofC^◦by the affine hyperplane corresponding to classesDsuch that (K+A)·D= −1.

• Each class D in the interior of C corresponds to a (K +A)-birational morphism ϕD:Z→SD.

• Conversely, given a(K+A)-negative birational morphismgj:Z→Sj, the divisorsD inCsuch thatgj =ϕDform a polyhedral chamberA_j with non-empty interior.

Remark 3.5. In higher dimension several complications arise that we avoided in the above discussion. In particular in dimension ≥ 3 it is not true anymore that eachA_j spansN¹(Z), because of the appearance of small contractions.

We should also mention that in dimension 2, the decomposition of Theorem3.3can be phrased in terms of Zariski decompositions (see [BKS04]). Namely, in each chamberA_j the support of the negative part of the Zariski decomposition is constant, and corresponds to the support of the classesCi withi∈I⁻.

Finally, as already noticed in Remark1.10, since we only consider adjoint divisors of the formK+1with1ample, we can work directly in the Néron–Severi spaceN¹(Z) instead of choosing a subspace of the space of Weil divisors, as in [HM13,Kal13].

Example 3.6. In [Kal13, Figures 1 and 6] the above construction is illustrated by the case ofP²blown up at two or three distinct points. Here we consider the case ofP²blown up at two points (of degree 1), with one infinitely near the other. More precisely, letZbe the surface obtained fromP²by blowing up a pointp=L∩L⁰of intersection of two lines, producing an exceptional divisor E, and thenp⁰ = E∩L⁰, producing an exceptional divisor E⁰ (see Figure 3, where the numbers in brackets denote self-intersection, and where we use the same notation for a curve and its strict transforms).

L[+1] L⁰[+1]

H

L⁰[0]

L[0]

H H

[−1]E

E[−2] L⁰[−1]

L[0] p

p⁰ E⁰[−1]

Fig. 3. P²blown up atpandp⁰.

On Z, the curvesE,E⁰ andL⁰are the only irreducible divisors with negative self- intersection, and they generate the pseudo-effective cone Eff(S). We also denote byH the class of a generic line fromP², and as usualKis the canonical divisor. OnZwe have

H=L⁰+E+2E⁰, −K=3L⁰+2E+4E⁰. The classesC_iof contracted curves are

C1=L⁰, C2=E⁰, C3=E+E⁰.

In this simple example, it turns out that any class D in the pseudo-effective cone Eff(Z)corresponds to a birational morphismϕ_D. There are six possibilities for this mor- phismϕ_D, and Figure4shows the chamber decomposition for the whole Eff(Z).

• •

•

•

•

•

• A^s_E,L0 A^s_E

H E

L

E⁰ E+E⁰

L⁰

−K•

Fig. 4

However, Set-Up3.4only guarantees the chamber decomposition for classes in C, which is pictured on Figure 5 (where we work with the choice A = 0). Observe in particular that the chambersA^s_E andA^s_E,L0 in Figure4correspond to singular surfaces, namely the blow-down ofE, or the blow-down of E, L⁰. Equivalently, a wall between chambers in Figure4does not always correspond to the contraction of a(−1)-curve. In contrast, the chambersA₁, . . . ,A₄ in Figure5 correspond to smooth surfaces, and the chambers are delimited by the hyperplanesC_i^⊥.

C₁=L⁰ C₂=E⁰

H

−K

L

K+r₂1₂

K+r₁1₁ C₁^⊥

C^⊥₃ C₂^⊥ A₄

A₃

A₂ A₁

C

Fig. 5. The partition Int(C)=S₄

i=1A_i(in gray).

The following two propositions correspond to [HM13, Theorem 3.3] in the case of sur- faces.

Proposition 3.7. Assume Set-Up3.4, letA_jbe one of the polyhedral chambers given by Theorem3.3, andg_j:Z→S_j the associated birational morphism. IfD∈AĎ_jrA_j with associated morphismsϕ_D:Z → Y, then there exists a unique morphismf: S_j → Y such thatϕ_D =f ◦g_j. Moreover, the morphismϕ_Donly depends on the faceF^r ⊂AĎ_j such thatDis in the relative interior ofF^r.

Proof. SetD₀ =D ∈ AĎ_j rA_j, and pickD₁ ∈ A_j. Then for allt ∈ (0,1], we have D_t :=t D₁+(1−t )D₀∈A_j. The curves contracted byg_j =ϕ_Dt are the curvesCsuch thatD_t ·C ≤0 fort ∈ (0,1], and so also fort =0. Hence NE(ϕ_D)⊂NE(g_j), and the Rigidity Lemma below gives the expected morphismf:S_j →Y.

Finally, the curves contracted byf are the curvesConSj such that(gj)∗D·C =0, and these conditions onDdefine the minimal faceF^r containingD. ut Lemma 3.8 (Rigidity Lemma, see [Deb01, Proposition 1.14]). LetX, Y, Y⁰ be projec- tive varieties and letπ: X → Y,π⁰:X → Y⁰ be morphisms with connected fibers. If NE(π )⊂ NE(π⁰), then there is a unique morphism f:Y → Y⁰with connected fibers such thatπ⁰=f ◦π.

Proposition 3.9. Assume Set-Up3.4. Letj, kbe two indices such thatAĎ_j∩A_k 6= ∅, and letg_j,k:S_j → S_k be the morphism(given by Proposition3.7)such thatg_k =g_j,k◦g_j. Then the relative Picard number ofgj,k:Sj →Skis equal to the codimension ofAĎj∩AĎk

inAĎ_j.

Proof. By definition there exist two partitions of the set of indices, I = I_j⁺∪I_j⁻ = I_k⁺∪I_k⁻, such that

Aj = \

i∈I_j⁺

C_i^>∩ \

i∈I_j⁻

C_i^≤∩C, Ak= \

i∈I_k⁺

C_i^>∩ \

i∈I_k⁻

C_i^≤∩C.

The conditionAĎ_j∩A_k 6= ∅means thatI_j⁻(I_k⁻. Let{i₁, . . . , i_t} =I_j⁺∩I_k⁻. Then AĎ_j∩AĎ_k = \

i∈I_k⁺

C_i^≥∩ \

i∈{i1,...,it}

C_i^⊥∩ \

i∈I_j⁻

C_i^≤∩C.

The morphismg_j,kcorresponds to the contraction of the classesC_i1, . . . , C_it, andt is by construction the codimension ofAĎj∩AĎkinAĎj. ut

3.2. Boundary ofC

Assuming Set-Up3.4, we define∂⁺Cas the intersection ofC with the boundary of the pseudo-effective cone inN¹(Z), or equivalently as the classes inC that are not big. If ρ=ρ(Z)is the Picard number ofZ, then by assumptionC^◦is a cone of full dimensionρ, hence the affine sectionC is homeomorphic to a ballB^ρ−1and∂⁺C is homeomorphic either to a ballB^ρ−2, or to a sphereS^ρ−2, depending on whether the pseudo-effective cone is contained inC or not. For instance the sphere situation arises ifZ is del Pezzo, A=0, and the initial collection of rank 1 fibrationsS_i/B_i is the full (finite) collection of such fibrations dominated byZ.

Now we put a structure of polyhedral complex on∂⁺C. We consider the set of cham- bersA_j such thatAĎ_j∩∂⁺Ccontains a codimension 1 faceWof the closed polytopeAĎ_j. We call such a faceWawindowofA_j. The collection of suchWdefines a unique struc- ture of polyhedral complex on∂⁺Csuch that theWare the maximal faces of the complex.

More generally, we denote byF^r a (closed) codimensionr face in∂⁺C. Here the codi- mension is taken relative to the ambient spaceN¹(Z), in particular windows correspond to codimension 1 faces. We say thatF^r is aninnerface if it intersects the relative interior of∂⁺C. Equivalently,F^r is inner if it can be written as the intersection ofrwindows.

For a windowWof the chamberAj,Din the relative interior ofW, and sufficiently smallε >0, the divisorD⁰:=D−ε(K+A)is inAj, and the imagesS=ϕ_D⁰(Z)and B =ϕD(Z)correspond to a Mori fibrationS/Bthat depends only onW(and not on the particular choice ofDorε).

We see that the codimension 1 faces in∂⁺Care in bijection with the rank 1 fibrations S¹/Bdominated by a(K+A)-negative map fromZ, and such thatS¹/Bis(−K−A)- ample. More generally, we have:

Proposition 3.10. (1) Let F^r be an inner codimension r face in the polyhedral com- plex∂⁺C. Then there exists a rankrfibrationS^r/Bsuch that

• the induced mapZ→S^r is equal togj for somej ∈ {1, . . . , s}, in particular this is a(K+A)-negative birational morphism;

• the chamberA_jassociated withgj satisfiesAĎ_j ⊇F^r.