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Adrien DUBOULOZ

On the cancellation problem for algebraic tori Tome 66, n^o6 (2016), p. 2621-2640.

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ON THE CANCELLATION PROBLEM FOR ALGEBRAIC TORI

by Adrien DUBOULOZ (*)

Dedicated to Wlodek Danielewski.

Abstract. — We address a variant of Zariski Cancellation Problem, asking whether two varieties which become isomorphic after taking their product with an algebraic torus are isomorphic themselves. Such cancellation property is easily checked for curves, is known to hold for smooth varieties of log-general type by virtue of a result of Iitaka-Fujita and more generally for nonA¹∗-uniruled varieties.

We show in contrast that for smooth affine factorialA¹∗-ruled varieties, cancellation fails in any dimension bigger than or equal to two.

Résumé. — Nous considérons une variante du Problème de Simplification de Zariski pour les tores algébriques: deux variétés algébriques dont les produits car- tésiens avec un même tore algébrique sont isomorphes sont-elles isomorphes ? Un argument élémentaire montre que les courbes algébriques possèdent cette propriété de simplification. Un résultat très général de simplification du à Iitaka et Fujita implique qu’il en est de même pour les variétés de type log-général ou non A¹∗- réglées. Dans cet article, nous construisons en toute dimension supérieure ou égale à deux des couples de variétés factoriellesA¹∗-réglées ne possèdant pas la propriété de simplification par des tores.

Since the late seventies, the Cancellation Problem is usually understood in its geometric form as the question whether two algebraic varietiesX and Y with isomorphic cylindersX×A¹andY×A¹are isomorphic themselves.

This problem is intimately related to the geometry of rational curves on X and Y: in particular, if X or Y are smooth quasiprojective and not A¹-uniruled, in the sense that they do not admit any dominant generically finite morphism from a variety of the formZ×A¹, then every isomorphism Φ : X ×A¹ →^∼ Y ×A¹ descends to an isomorphism between X and Y, a property which is sometimes called strong cancellation. Over an alge- braically closed field of characteristic zero, the non A¹-uniruledness of a

Keywords:cancellation problem, algebraic tori, principal bundles.

Math. classification:14R05, 14L30.

(*) This project was partialy funded by ANR Grant “BirPol” ANR-11-JS01-004-01.

smooth quasi-projective varietyX is guaranteed in particular by the exis- tence of nonzero pluri-forms with logarithmic poles at infinity on suitable projective completions ofX. The existence of nonzero such pluri-forms of highest degree can in turn be read off from the non-negativity of a nu- merical invariant of X, called its (logarithmic) Kodaira dimension κ(X), introduced by S. Iitaka [12] as the analogue of the usual notion of Ko- daira dimension for complete varieties. In this setting, it was established by S. Iitaka et T. Fujita [13] that strong cancellation does hold for a large class of smooth varieties, namely wheneverX or Y has non-negative Ko- daira dimension. This general result implies in particular that cancellation holds for smooth affine curves, due to the fact that the affine lineA¹is the only such curve with negative Kodaira dimension.

All these assumptions turned out to be essential, as shown by a famous unpublished example due to W. Danielewski [3] of a pair of non-isomorphic smooth complex A¹-ruled affine surfaces with isomorphic cylinders. The techniques introduced by W. Danielewski have been sources of progress on the Cancellation Problem during the last decade but, except for the case of the affine planeA² which was solved earlier affirmatively by M. Miyanishi and T. Sugie [17], the question whether cancellation holds for the complex affine spaceAⁿ remains one of the most challenging and widely open prob- lem in affine algebraic geometry. In contrast, the same question in positive characteristic was recently settled in the negative by N. Gupta [9], who checked using algebraic methods developed by A. Crachiola and L. Makar- Limanov that a three-dimensional candidate constructed by T. Asanuma [1]

is indeed a counter-example.

In this article, we consider another natural cancellation problem in which A¹ is replaced by the punctured affine line A¹∗ ' Spec(C[x^±1]) or, more generally, by an algebraic torus Tⁿ = Spec(C[x^±1₁ , . . . , x^±1_n ]), n >1. The question is thus whether two, say smooth quasi-projective, varietiesX and Y such that X×Tⁿ is isomorphic to Y ×Tⁿ are isomorphic themselves.

When restricted to affine algebraic varieties, this cancellation problem is of course equivalent to the more algebraic question of the uniqueness of the coefficient ring in a Laurent polynomial ring. In contrast with the usual Cancellation Problem, this version seems to have received much less atten- tion, one possible reason being that the analogue in this context of the Can- cellation Problem forAⁿ, namely the question whether an affine varietyX such thatX×Tⁿis isomorphic toT^n+mis itself isomorphic to the torusTⁿ, admits an elementary positive answer derived from the knowledge of the structure of the automorphism groups of algebraic tori: indeed, the action of

the torusTⁿ= Spec(C[M⁰]) by translations on the second factor ofX×Tⁿ corresponds to a grading of the algebra of the torusT^n+m= Spec(C[M]) by the lattice M⁰ of characters of Tⁿ, induced by a surjective homomor- phismσ:M →M⁰ from the lattice of charactersM ofT^n+m. The kernel ofσis a sub-latticeM⁰⁰of rankmofM for which we have isomorphisms of geometric quotientsX'X×Tⁿ/Tⁿ 'T^n+m/Tⁿ'T^m= Spec(C[M⁰⁰]).

Without such precise information on automorphism groups, the question for general varietiesX andY is more complicated. Of course, appropriate conditions on the structure of invertible functions onX andY can be im- posed to guarantee that cancellation holds (see [6] for a detailed discussion of this point of view): this is the case for instance when either X or Y does not have non constant such functions. Indeed, given an isomorphism Φ : X ×A¹∗ ∼

→ Y ×A¹∗, the restriction to every closed fiber of the first projection pr₁:X×A¹∗→X of the composition of Φ with the second pro- jection pr₂ :Y ×A¹∗ →A¹∗ induces an invertible function on X, implying that Φ descends to an isomorphism betweenXandY as soon as every such function onX is constant.

But from a geometric point of view, it seems that the cancellation prop- erty forA¹∗is again related to the nature of affine rational curves contained in the varietiesXandY, more specifically to the geometry of images of the punctured affine lineA¹∗on them. It is natural to expect that strong cancel- lation holds for varieties which are not dominantly covered by images ofA¹∗, but this property is harder to characterize in terms of numerical invariants.

In particular, in every dimension>2, there exist smoothA¹∗-uniruled affine varietiesX of any Kodaira dimension κ(X) ∈ {−∞,0,1, . . . ,dimX−1}. In contrast, a smooth complex affine varietyX of log-general type, i.e., of maximal Kodaira dimensionκ(X) = dimX, is not A¹∗-uniruled, and an- other general result of I. Itaka and T. Fujita [13] does indeed confirm that strong cancellation holds for products of algebraic tori with smooth affine varieties of log-general type. Combined with the fact thatA¹_∗is the unique smooth affine curve of Kodaira dimension 0, this is enough for instance to conclude that cancellation holds for smooth affine curves.

Our main result, which can be summarized as follows, shows that sim- ilarly as in the case of the usual Cancellation Problem for A¹, these as- sumptions are essential:

Theorem. — For everyd>3and everyκ∈ {2, . . . , d−1}, there exists non isomorphic smooth factorial affineA¹∗-uniruled varieties X and Y of dimensiondand Kodaira dimensionκwith isomorphicA¹∗-cylindersX×A¹∗

and Y ×A¹∗. Furthermore, there exists non isomorphic smooth factorial affine surfaces of Kodaira dimension1with isomorphicA¹_∗-cylinders.

In dimension d>3 and Kodaira dimensionκ=d−1, these families are obtained in the form of total spaces of suitable Zariski locally trivialA¹∗- bundles over smooth affine varieties of log-general type. The construction guarantees the isomorphy between the correspondingA¹_∗-cylinders thanks to a fiber product argument reminiscent to the famous Danielewski fiber product trick in the case of the usual Cancellation Problem. The two- dimensional counter-examples are produced along the same lines, at the cost of replacing the base varieties of theA¹∗-bundles involved in the con- struction by appropriate orbifold curves. The article is organized as fol- lows: in the first section, we establish a variant of the Iitaka-Fujita strong cancellation Theorem for Zariski locally trivial Tⁿ-bundles over smooth affine varieties of log-general type. This criterion is applied in the second section to deduce the existence of families of Zariski locally trivial A¹∗- bundles over smooth affine varieties of log-general type with non isomor- phic total spaces but isomorphic A¹∗-cylinders. The two-dimensional case is treated in a separate sub-section. The last section contains a generaliza- tion of some of these constructions to the cancellation problem for higher dimensional toriTⁿ over varieties of dimension at least three, leading in particular to the existence of counter-examples of arbitrary Kodaira di- mensionκ∈ {2, . . . , d−1}, and a complete discussion of the cancellation problem forA¹∗ in the special case of smooth factorial affine surfaces.

1. A criterion for cancellation

1.1. Recollection on locally trivial Tⁿ-bundles

In what follows, we denote by Tⁿ the spectrum of the Laurent poly- nomial algebra C[t^±1₁ , . . . , t^±1_n ] in n variables. We use the notation Tⁿ to indicate that we consider Tⁿ as an algebraic group: the direct product Gⁿm ofn-copies of the multiplicative groupGm. The automorphism group Aut(Tⁿ) ofTⁿis isomorphic to the semi-direct productTⁿoGLn(Z), where Tⁿ acts on Tⁿ by translations and where GLn(Z) acts by (aij)i,j=1,...,n· (t1, . . . , tn) = (Qn

i=1t^a_i¹ⁱ, . . . ,Qn i=1t^a_iⁿⁱ).

Definition 1.1. — A Zariski locally trivial Tⁿ-bundle over a scheme X, is anX-schemep:P →Xfor which every point ofX has a Zariski open neighbourhoodU ⊂X such thatp⁻¹(U)'U×Tⁿ as schemes overU.

Isomorphy classes of Zariski locally trivial Tⁿ-bundles over X are in one-to-one correspondence with elements of the Čech cohomology group Hˇ¹(X,Aut(Tⁿ)X), where Aut(Tⁿ)X denotes the sheaf of groups on the category of schemes overX equipped with the Zariski topology which as- sociates to everyX-scheme S the group Aut(Tⁿ)(S) = AutS(S×Tⁿ) of S-automorphisms ofS×Tⁿ. Letting GLn(Z)Xbe the locally constant sheaf GLn(Z) onX, we derive from the short exact sequence

0→TⁿX =Gⁿm,X →Aut(Tⁿ)X →GLn(Z)X→0

of sheaves of groups over X for the Zariski topology the following long exact sequence of pointed sets in non abelian Čech cohomology

0→Hˇ⁰(X,TⁿX) //Hˇ⁰(X,Aut(Tⁿ)X) //Hˇ⁰(X,GLn(Z)X)

// ˇH¹(X,TⁿX) // ˇH¹(X,Aut(Tⁿ)X) // ˇH¹(X,GLn(Z)X), (see e.g. [7, Proposition 3.4.1]). IfX is irreducible, then ˇH¹(X,GLn(Z)X) = 0 and so, every Zariski locally trivialTⁿ-bundle overX can be equipped with the additional structure of a principal homogeneousTⁿ-bundle. More- over, two principal homogeneous Tⁿ-bundles have isomorphic underlying Tⁿ-bundles if and only if their isomorphy classes in ˇH¹(X,TⁿX)'H¹(X,TⁿX) belong to the same orbit of the natural action of ˇH⁰(X,GLn(Z)X) ' GLn(Z) which, for every (aij)i,j=1,...n∈GLn(Z), sends the isomorphy class of theTⁿ-bundlep:P →X with actionTⁿ×P →P, ((t1,· · ·, tn), p)7→

(t1, . . . , tn)·pto the isomorphy class ofp:P →X equipped with the ac- tion ((t1, . . . tn), p)7→((aij)i,j=1,...,n·(t1, . . . , tn))·p. In other words, for an irreducibleX, isomorphy classes of Zariski locally trivialTⁿ-bundles over X are in one-to-one correspondence with elements ofH¹(X,TⁿX)/GLn(Z).

1.2. Cancellation for Tⁿ-bundles over varieties of log-general type

Recall that the(logarithmic) Kodaira dimensionκ(X) of a smooth com- plex algebraic variety X is the Iitaka dimension of the invertible sheaf ω_X/C(logB) = (det Ω¹_X/C)⊗ OX(B) on a smooth complete model X of X with reduced SNC boundary divisor B =X \X. Soκ(X) is equal to tr.deg_C(L

m>0H⁰(X, ω_X/C(logB)^⊗m))−1 ifH⁰(X, ω_X/C(logB)^⊗m)6= 0 for sufficiently large m, and, by convention to −∞ otherwise. The so- defined element of{−∞} ∪ {0, . . . ,dim_CX} is independent of the choice

of a smooth complete model (X, B) [12] and coincides with the usual Ko- daira dimension in the case whereX is complete. A smooth varietyX such thatκ(X) = dim_CX is said to be of log-general type.

The following Proposition is a variant for Zariski locally trivial Tⁿ- bundles of the Iitaka-Fujita’s strong Cancellation Theorem [13, Theorem 3]

for products of varieties of log-general type with affine varieties of Kodaira dimension equal to 0, such as algebraic toriTⁿ.

Proposition 1.2. — LetX and Y be smooth algebraic varieties and let p : P → X and q : Q → Y be Zariski locally trivial Tⁿ-bundles. If eitherX orY is of log-general type then for every isomorphism of abstract algebraic varietiesΦ :P →^∼ Qbetween the total spaces of P andQ, there exists an isomorphismϕ:X →^∼ Y such that the diagram

P →^Φ Q

p↓ ↓q

X →^ϕ Y

commutes.

Proof. — The proof is very similar to that of [13, Theorem 1]. We may assume without loss of generality thatY is of log-general type. It enough to show thatq◦Φ is constant on the fibers of p. Indeed, if so, then there exists a unique set-theoretic map ϕ : X → Y such that ϕ◦p = q◦Φ, and the fact thatphas local sections in the Zariski topology implies that ϕis actually a morphism. Furthermore, since Φ is an isomorphism,ϕwill be bijective whence an isomorphism by virtue of Zariski Main Theorem [8, 8.12.6]. Now sincep: P → X is Zariski locally trivial andκ(Tⁿ) = 0, it follows from [12] that for every prime Weil divisor D on X, the Kodaira dimension κ(p⁻¹(Dreg)) of the inverse image of the regular part of D is at most equal to dimD. This implies in turn that the restriction ofq◦Φ to p⁻¹(D) cannot be dominant since otherwise we would have κ(Y) 6 κ(p⁻¹(Dreg))<dimX = dimY, in contradiction with the assumption that κ(Y) = dimY. So there exists a prime Weil divisor D⁰ on Y such that the image of p⁻¹(D) by Φ is contained in q⁻¹(D⁰), whence is equal to it since they are both irreducible of the same dimension. Now given any closed pointx∈ X, we can find a finite collection of prime Weil divisors D₁, . . . , Dnsuch thatD₁∩· · ·Dn={x}. LettingD_i⁰be a collection of prime Weil divisors onY such that Φ(p⁻¹(Di)) =q⁻¹(D_i⁰) for everyi= 1, . . . , n,

we have

q⁻¹( \

i=1,...,n

D⁰_i) = \

i=1,...,n

q⁻¹(D⁰i) = \

i=1,...,n

Φ(p⁻¹(Di)) 'Φ( \

i=1,...,n

p⁻¹(Di)) 'Φ({x} ×Tⁿ)'Tⁿ.

So the intersection of the D_i⁰, i = 1, . . . , n, consists of a unique closed point y ∈ Y for which we have by construction Φ(p⁻¹(x)) = q⁻¹(y), as

desired.

Remark 1.3. — The proof above shows in fact that the conclusion of the proposition holds under the more geometric hypothesis that eitherX or Y is not A¹_∗-uniruled, i.e., does not admit any dominant generically finite morphism from a variety of the form Z ×A¹∗. In particular strong cancellation holds for products of algebraic toriTⁿ with nonA¹∗-uniruled varieties.

Corollary 1.4. — Two smooth curvesC andC⁰ admit Zariski locally trivial Tⁿ-bundles p : P → C and p⁰ : P⁰ → C⁰ with isomorphic total spacesP andP⁰ if and only if they are isomorphic.

Proof. — If eitherCorC⁰is of (log-)general type, then the assertion fol- lows from Proposition 1.2. Note further thatCis affine if and only if so isP. Indeed,p:P→Cis an affine morphism and conversely, ifP is affine, then viewingP as a principal homogeneousTⁿ -bundle with geometric quotient P//Tⁿ ' C, the affineness of C follows from the fact that the algebraic quotient morphismP →P//T= Spec(Γ(P,O^P)^T) is a categorical quotient in the category of algebraic varieties, so thatC 'Spec(Γ(P,O^P)^T). Thus CandC⁰are simultaneously affine or projective. In the first case,CandC⁰ are isomorphic to either the affine lineA¹ or the punctured affine line A¹∗

which both have a trivial Picard group. SoP andP⁰are trivialTⁿ-bundles and the isomorphy of C and C⁰ follows by comparing invertible function on P and P⁰. In the second case, if either C or C⁰ has genus 1, say C⁰, then, being rational, the image of a fiber ofp:P →C by an isomorphism Φ :P →^∼ P⁰ must be contained in a fiber of p⁰ :P⁰→C⁰, and we conclude similarly as in the proof of the previous proposition that Φ descends to an

isomorphism betweenCandC⁰.

The automorphism group Aut(X) of a scheme X acts on the set of iso- morphy classes of principal homogeneous Tⁿ-bundles over X via the lin- ear representationη : Aut (X) → GL(H¹(X,TⁿX)), where for an element

ψ∈Aut (X), η(ψ) maps the isomorphy class of a principal homogeneous Tⁿ-bundlep:P →X to the one of theTⁿ-bundle pr₂:P×^p,X,ψX →X. This action commutes with action of GLn(Z) introduced in page 2625 above, and Proposition 1.2 implies the following characterization:

Corollary 1.5. — Over a smooth varietyXof log-general type, the set H¹(X,TⁿX)/(Aut(X)×GLn(Z))parametrizes isomorphy classes as abstract varieties of total spaces of Zariski locally trivialTⁿ-bundlesp:P →X.

2. Non-Cancellation for the 1-dimensional torus Candidates for non-cancellation of the 1-dimensional torus T = A¹∗ = Spec(C[t^±1]) can be constructed along the following lines: given say a smooth quasi-projective variety X and a pair of non-isomorphic princi- pal homogeneous G^m-bundles p : P → X and q : Q→ X whose classes generate the same subgroup ofH¹(X,Gm), the fiber productW =P×^XQ is a principal homogeneousT²-bundle overX, which inherits the structure of a principalG^m-bundle over P and Qsimultaneously, via the first and the second projection respectively. The pull-back p^∗P = P ×^X P → P ofP to itself having a section, it is a trivial G^m-bundle, which implies in turn that the subgroup ofH¹(X,G^m) generated by the class ofP belongs to the kernel of p^∗ :H¹(X,G^m)→H¹(P,G^m). For the same reason, the subgroup of H¹(X,Gm) generated by the class of Q belongs to the ker- nel ofq^∗ :H¹(X,G^m)→H¹(Q,G^m). The assumption that the classes of P and Q generate the same subgroup ofH¹(X,G^m) guarantees that the classes of pr₁:W 'p^∗Q→P and pr₂:W 'q^∗P→QinH¹(P,G^m) and H¹(Q,G^m) respectively are both trivial, and so, we obtain isomorphisms P×G^m'W 'Q×G^m of locally trivialT²-bundles overX.

Then we are left with finding appropriate choices of X and classes in H¹(X,Gm) which guarantee that the total spaces of the corresponding principal homogeneous G^m-bundles p : P → X and q : Q → X are not isomorphic as abstract algebraic varieties.

2.1. Non-cancellation for smooth factorial affine varieties of dimension>3

A direct application of the above strategy leads to families of smooth factorial affine varieties of any dimension>3 for which cancellation fails:

Proposition 2.1. — LetX be the complement of a smooth hypersur- faceD of degree ` in P<sup>κ</sup>,κ>2, such that`>κ+ 2 and |(Z/`Z)<sup>∗</sup>|>3, and letp: P → X and q : Q→ X be the G<sup>m</sup>-bundles corresponding to the line bundles OP<sup>κ</sup>(1)|<sup>X</sup> and OP<sup>κ</sup>(r) |<sup>X</sup>, for some r∈ Z such thatr ∈ (Z/`Z)^∗\ {1, `−1}under the isomorphismH<sup>1</sup>(X,G<sup>m</sup>)'Pic(X)'Z/`Z. ThenP andQare not isomorphic as abstract algebraic varieties butP×A¹∗

andQ×A¹∗ are isomorphic as schemes overX.

Proof. — The Picard group ofX is isomorphic to the groupµ`'Z/`Z of`-th roots of unity, generated by the restriction of OP<sup>κ</sup>(1) to X. Since ris relatively prime with`, OP^κ(r)|^X is also a generator of Pic(X). This guarantees thatP×A¹_∗ is isomorphic to Q×A¹_∗ by virtue of the previous discussion. Since`>κ+ 2, the divisorK_P^κ+D is linearly equivalent to a positive multiple of a hyperplane section, and soX is of log-general type.

We can therefore apply Proposition 1.2 to deduce that for every isomor- phism of abstract algebraic varieties Φ :P →^∼ Q, there exists an automor- phismϕofX such thatP is isomorphic toϕ^∗Qas a Zariski locally trivial A¹∗-bundle over X. In view of Corollary 1.5, this means equivalently that as aG^m-bundle overX,ϕ^∗Qis isomorphic to either P or its inverseP⁻¹ in H¹(X,G^m). Since the choice of r guarantees that the G^m-bundle Q is isomorphic neither toP nor toP⁻¹, the conclusion follows from the obser- vation that the natural action of Aut(X) onH¹(X,G^m) is the trivial one.

Indeed, through the open inclusionX ,→P^κ, we may consider an automor- phismϕofX as a birational self-map ofP^κ restricting to an isomorphism outsideD. If ϕis not biregular on the whole P^κ, thenD would be an ex- ceptional divisor of ϕ⁻¹, in particular, D would be birationally ruled, in contradiction with the ampleness of its canonical divisorKDguaranteed by the condition`>κ+ 2. So every automorphism ofX is the restriction of a linear automorphism of the ambient spaceP^κ, and since PGL(κ+ 1) acts trivially on Pic(P^κ), it follows that Aut(X) acts trivially on Pic(X).

Example 2.2. — In the setting of Proposition 2.1 above, an isomor- phismP ×A¹∗ ∼

→Q×A¹∗ can be constructed “explicitly” as follows. The complement X ⊂ P^κ = Proj(C[x0, . . . , xκ]) of a smooth hypersurface D defined by an equation F(x0, . . . , xκ) = 0 for some homogeneous polyno- mial of degree`can be identified with the quotient of the smooth factorial affine variety ˜X ⊂A<sup>κ+1</sup>with equationF(x0, . . . , xr) = 1 by the free action of the group µ` = Spec(C[ε]/(ε<sup>`</sup>−1)) of `-th roots of unity defined by ε·(x0, . . . , xκ) = (εx0, . . . , εxκ). TheGm-bundles overX corresponding to the line bundlesOP^κ(r)|^X,r∈Z, then coincide with the quotients of the trivialA¹∗-bundles ˜X×A¹∗= ˜X×Spec(C[t^±1]) by the respectiveµ`-actions

ε·(x0, . . . , xκ, t) = (εx0, . . . , εxκ, ε^rt), r ∈Z. Now let q: Q →X be the Gm-bundle corresponding toOP^κ(r) |^X for some r ∈ {2, . . . , `−2} rela- tively prime with `, and leta, b ∈ Zbe such that ar−b`= 1. Then the following isomorphism

˜Φ : ˜X×T²= ˜X×Spec(C[t^±1₁ , u^±1₁ ])→^∼ X˜ ×T²= ˜X×Spec(C[t^±1₂ , u^±1₂ ]) (t1, u₁)7→(t2, u₂) = (t^r₁u^a₁, t^b`₁u₂)

of schemes over ˜X is equivariant for the actions, say µ`,1 and µ`,r, of µ`

defined respectively byε·(x0, . . . , xκ, t1, u1) = (εx0, . . . , εxκ, εt1, u1) and ε·(x0, . . . , xκ, t₂, u₂) = (εx0, . . . , εxκ, ε^rt₂, u₂), hence descends to an iso- morphism

Φ : ( ˜X×T²)/µ`,1'P×A<sup>1</sup><sub>∗</sub>→<sup>∼</sup> Q×A<sup>1</sup><sub>∗</sub>'( ˜X×T<sup>2</sup>)/µ`,r

of schemes overX'X/µ˜ `.

2.2. Non-cancellation for smooth factorial affine surfaces Since the Picard group of a smooth affine curve C of log-general type is either trivial ifCis rational or of positive dimension otherwise, there is no direct way to adapt the previous construction using principalGm-bundles over algebraic curves to produce 2-dimensional candidate counter-examples for cancellation byA¹_∗. Instead, we will use locally trivialA¹_∗-bundles over certain orbifold curves ˜C which arise from suitably chosen A¹∗-fibrations π:S→C on smooth affine surfacesS.

LetSbe a smooth affine surface equipped with a flat fibrationπ:S→C over a smooth affine rational curve C whose fibers, closed or not, are all isomorphic toA¹∗over the corresponding residue fields when equipped with their reduced structure ⁽¹⁾. It follows from the description of degenerate fibers ofA¹∗-fibrations given in [16, Theorem 1.7.3] thatSadmits a relative completion into aP¹-fibered surfaceπ:S→C obtained from a trivialP¹- bundle pr₁:C×P¹→C with a fixed pair of disjoint sectionsH₀ andH_∞ by performing finitely many sequences of blow-ups of the following type:

the first step consists of the blow-up of a closed pointci∈H₀,i= 1, . . . , s, with exceptional divisor E1,i followed by the blow-up of the intersection point of E_1,i with the proper transform of the fiber Fi = pr⁻¹₁ (pr₁(ci)), the next steps consist of the blow-up of an intersection point of the last exceptional divisor produced with the proper transform of the union ofFi (1)In particular,πis an untwistedA¹∗-fibration in the sense of [16, §1.7 p. 201].

and the previous ones, in such a way that the total transform ofFiis a chain of proper rational curves with the last exceptional divisors produced, say Ei,ni, as the unique irreducible component with self-intersection −1. The projection pr₁ :C×P¹→C lifts on the so-constructed surfaceS to aP¹- fibrationπ:S→CandSisomorphic to the complement of the union of the proper transforms ofH0andH_∞and of the divisorsFi∪Ei,1∪· · ·∪Ei,ni−1, i= 1, . . . , s. The restriction ofπtoS is indeed an A¹∗-fibrationπ:S →C withs degenerate fibersπ⁻¹(pr₁(ci)) isomorphic to Ei,ni∩S 'A¹∗ when equipped with their reduced structure and whose respective multiplicities depend on the sequences of blow-ups performed.

It follows in particular from this construction that S admits a proper action of the multiplicative groupG^m which lifts the one on (C×P¹)\ (H0∪H_∞) ' C ×A¹_∗ by translations on the second factor. The local descriptions given in [5] can then be re-interpreted for our purpose as the fact that theA¹∗-fibrationπ:S→Cfactors through an étale locally trivial A¹∗-bundle ˜π : S → C˜ over an orbifold curve δ : ˜C → C obtained from Cby replacing the finitely many pointsc₁, . . . , cs over the which the fiber π⁻¹(ci) is multiple, say of multiplicitymi >1, by suitable orbifold points

˜

ci depending only on the multiplicity mi. More precisely, ˜C is a smooth separated Deligne-Mumford stack of dimension 1, of finite type overCand with trivial generic stabilizer, which, Zariski locally aroundδ⁻¹(ci) looks like the quotient stack [ ˜Uci/Z^mi], where ˜Uci → Uci is a Galois cover of ordermi of a Zariski open neighborhoodUci ofci, totally ramified overci

and étale elsewhere [2].

Example 2.3. — Let G^m act on A²∗ = Spec(C[x, y])\ {(0,0)} by t · (x, y) = (t²x, t⁵y). The quotient P(2,5) = A²∗/G^m is isomorphic to P¹ and the quotient morphism q : A²∗ → P¹ = A²∗/G^m is an A¹∗-fibration with two degenerate fibers q⁻¹([0 : 1]) and q⁻¹([1 : 0]) of multiplicities 5 and 2 respectively, corresponding to the orbits of the points (0,1) and (1,0). In contrast, the quotient stack [A²∗/G^m] is the Deligne-Mumford curve P[2,5] obtained from P¹ by replacing the points [0 : 1] and [1 : 0]

by “stacky points” with respective Zariski open neighborhoods isomorphic to the quotients [A¹/Z5] and [A¹/Z2] for the actions of Z5 and Z2 on A¹ = Spec(C[z]) given by z 7→ exp(2iπ/5)z and z 7→ −z. The quotient morphismq:A²∗ →P¹ factors through the canonical morphism ˜q:A²∗ → P[2,5] which is an étale local trivialA¹_∗-bundle, and the induced morphism δ:P[2,5]→P¹ =A²∗/G^m is an isomorphism over the complement of the points [0 : 1] and [1 : 0].

In the next paragraphs, we construct two smooth affine surfaces S1 and S2 with anA¹_∗-fibrationπi :Si→A¹, i= 1,2, factoring through a locally trivialA¹∗-bundle ˜π:Si →A¹[2,5] over the affine Deligne-Mumford curve A¹[2,5] obtained from the one P[2,5] of Example 2.3 above by removing a general scheme-like point.

The first oneS1is equal to the complement in the projective planeP²= Proj(C[x, y, z]) of the union of the cuspidal curve D₁ =

x⁵−y²z³= 0 and the line Lz ={z = 0}. Equivalently, S1 is the complement in A² = Spec(C[x, y]) =P²\Lz of the curveD1∩A²={x⁵−y²= 0}. This curve being an orbit with trivial isotropy of theG^m-actiont·(x, y) = (t²x, t⁵y) on A², the composition of the inclusionS1,→A²_∗with the canonical morphism q: A²∗ →P[2,5] = [A²∗/G^m] defines a locally trivial A¹∗-bundle ˜π₁ : S₁ → P[2,5]\q(D1∩A²∗)'A¹[2,5]. The rational pencilξ1: P²99K P¹ induced by π₁ = δ◦π˜₁ : S₁ → A¹ coincide with that generated by the pairwise linearly equivalent divisors D1, 5Lx and 3Lz+ 2Ly, where Lx, Ly and Lz denote the lines {x = 0}, {y = 0} and {z = 0} in P² respectively.

A minimal resolution ˜ξ₁ : ˜S₁ → P¹ of ξ₁ is depicted in Figure 2.1. A relatively minimal SNC completion (S1, B1) of S1, with boundary B1 = D₁∪H_∞,1∪H_0,1∪E_∞,3 ∪E_∞,1∪E_0,1∪E_0,2∪E_∞,2∪E_0,3, on which π1 extends to a P¹-fibration π1 : S1 → P¹ is then obtained from ˜S1 by contracting the proper transform ofLz.

The second surface S₂ is obtained as follows. In the Hirzebruch surface ρ : F3 = P(OP¹ ⊕ OP¹(−3)) → P¹ with exceptional section C0 of self- intersection−3, we choose a sectionC ofρin the linear system|C₀+ 4F|, whereF denotes a general fiber ofρ, and a sectionC1in the linear system

|C₀+ 3F|intersectingCwith multiplicity 4 in a unique pointp₀. The fact that such pairs of sections exists follows for instance from [4, Lemma 3.2].

Letξ2:F399KP¹be the pencil generated by the linearly equivalent divisors C₀+5Cand 6C1+2F0whereF₀=ρ⁻¹(ρ(p0)). LetD₂be a general member of ξ and let S2 ⊂ F3 be the complement of C0 ∪C1∪D2. A minimal resolution ˜ξ₂ : ˜S₂ → P¹ of ξ₂ : F3 99K P¹ is depicted in Figure 2.2. A relatively minimal SNC completion (S2, B2) of S2 with boundary B2 = D₂∪H_∞,2∪H_0,2∪E_∞,1 ∪E_0,5∪C₀∪S9

i=6E_0,i, on which π₂ extends to a morphism π2 : S2 → P¹ is then obtained from ˜S2 by contracting successively the proper transforms ofC₁,E_0,4,E_0,3,E_0,2E_0,1. The proper transform ofC in S₂ is a unique (−1)-curve in the fiberπ⁻¹₂ (π2(C)), and the image ofC0 ⊂ π⁻¹₂ (π2(C)) by the successive contractions of C,E0,6, E_0,7 and E_0,9 is a smooth rational curve of self-intersection 0. Similarly, F0 is a unique (−1)-curve in π⁻¹₂ (π2(C)) and the image of E_∞,1 by the

P²

∞

0 Ly

Lx

Lz D1

S˜1

D1 0 H∞,1=E∞,4

−1 E∞,3

−2

E∞,1

−3

Lx

−1

E0,1

−2

E0,2

−3

H0,1=E0,4

−1

E∞,2

−3

Lz

−1

Ly

−2

E0,3

−2

P¹ ξ˜1

Figure 2.1. Minimal resolution of the pencil ξ₁ : P² P¹. The exceptional divisors E0,i and E_∞,i, i = 1, . . . ,4, over the respective proper base points 0 = [0 : 0 : 1] and ∞ = [0 : 1 : 0] of ξ₁ are numbered according to the order of their extraction.

general member ofξ and letS2⊂F3be the complement ofC0∪C1∪D2. A minimal resolution ˜ξ2: ˜S2→P¹ of ξ2 :F3 P¹ is depicted in Figure 2.2. A relatively minimal SNC completion (S2, B₂) of S₂ with boundary B2=D2∪H_∞,2∪H0,2∪E_∞,1∪E0,5∪C0∪9

i=6E0,i, on whichπ2extends to a morphism π₂ : S₂ → P¹ is then obtained from ˜S₂ by contracting successively the proper transforms ofC1,E0,4,E0,3,E0,2E0,1. The proper transform of C in S₂ is a unique (−1)-curve in the fiberπ⁻¹₂ (π2(C)), and the image of C0 ⊂ π⁻¹₂ (π2(C)) by the successive contractions of C,E0,6, E0,7 and E0,9 is a smooth rational curve of self-intersection 0. Similarly, F0 is a unique (−1)-curve in π⁻¹₂ (π2(C)) and the image of E_∞,1 by the contractions ofF0andE0,5is a smooth rational curve of self-intersection 0.

Counting the number of points blown-up fromF3and of curves contracted, we conclude that the smooth projective surface obtained fromS2by making all these contractions has Picard rank 2, hence is a Hirzebruch surface, in which the images of C0 and E_∞,1 are fibers of a P¹-bundle structure having the images of H_∞,2 and H0,2 as disjoint sections. It follows that π₂ : S₂ → P¹ is a P¹-fibration which has π⁻¹₂ (π2(C)) and π⁻¹₂ (π2(F0)) has unique reducible fibers. By construction,π2 restricts on S2 to anA¹_∗- fibration π₂ : S₂ → A¹ = P¹\ξ(D2) with two degenerate fibers : one of multiplicity 5 supported byC∩S2A¹∗, and one of multiplicity 2 supported

Figure 2.1. Minimal resolution of the pencilξ1:P²99KP¹. The excep- tional divisors E_0,i and E_∞,i, i= 1, . . . ,4, over the respective proper base points 0 = [0 : 0 : 1] and ∞ = [0 : 1 : 0] of ξ1 are numbered according to the order of their extraction.

contractions ofF0andE0,5is a smooth rational curve of self-intersection 0.

Counting the number of points blown-up fromF3and of curves contracted, we conclude that the smooth projective surface obtained fromS2by making all these contractions has Picard rank 2, hence is a Hirzebruch surface, in which the images of C0 and E_∞,1 are fibers of a P¹-bundle structure having the images of H∞,2 and H0,2 as disjoint sections. It follows that π₂ : S₂ → P¹ is a P¹-fibration which has π⁻¹₂ (π2(C)) and π⁻¹₂ (π2(F0)) has unique reducible fibers. By construction,π2 restricts onS2 to anA¹_∗- fibration π₂ : S₂ → A¹ = P¹ \ ξ(D2) with two degenerate fibers: one of multiplicity 5 supported by C∩S2 ' A¹∗, and one of multiplicity 2 supported byF₀∩S₂'A¹∗. So by virtue of page 2631,π₂factors through a locally trivialA¹∗-bundle ˜π2:S2→A¹[2,5].

Proposition 2.4. — The surfacesS1andS2are smooth affine rational and factorial. They are non isomorphic butS₁×A¹∗is isomorphic toS₂×A¹∗. Proof. — Since S1 is a principal open subset of A², it is smooth affine rational and factorial. The smoothness and the rationality of S₂ are also clear. SinceD₂belongs to the linear system 6C0+20F, it is ample by virtue of [11, Theorem 2.17]. This implies in turn thatC0+C1+D2is the support of an ample divisor, whence thatS₂is affine. Since the divisor class group of F3is generated byC0andF, the identityF ∼7C1−C0−D2in the divisor

2634 Adrien DUBOULOZ

byF₀∩S₂A¹∗. So by virtue of§2.2.2,π₂factors through a locally trivial A¹_∗-bundle ˜π2:S2→A¹[2,5].

F3 C₀

C1 F0 C D2

•p₀

S˜₂ H∞,2=E∞,2

−1 E∞,1−2

F0 −2

E0,1

−2

E0,2

−2

E_0,3

−2 E0,4 −2

C1

−1 E_0,5

−6 H0,2=E0,10

−1

C0

−5

−1 C

E0,6

−2

E0,7

−2

E0,8

−2

E_0,9

−2 D2 0

P¹ ξ˜₂

Figure 2.2. Minimal resolution of the pencil ξ₂ : F3 P¹. The exceptional divisorsE_∞,1, E_∞,2 and E_0,i, i= 1, . . . ,10, over the res- pective proper base points∞=F0∩C0and 0 =p0ofξ2are numbered according to the order of their extraction.

Proposition 2.4. — The surfacesS1andS2are smooth affine rational and factorial. They are non isomorphic butS₁×A¹∗is isomorphic toS₂×A¹∗. Démonstration. —SinceS₁is a principal open subset ofA², it is smooth affine rational and factorial. The smoothness and the rationality ofS2 are also clear. SinceD₂belongs to the linear system 6C0+ 20F, it is ample by virtue of [11, Theorem 2.17]. This implies in turn thatC0+C1+D2 is the support of an ample divisor, whence thatS₂is affine. Since the divisor class group ofF3is generated byC0andF, the identityF∼7C1−C0−D2in the divisor class group of F3 implies that every Weil divisor on S2 is linearly equivalent to one supported on the boundary F3\S₂ = C₀∪C₁∪D₂. So S2 is factorial. By construction, S1 and S2 both have the structure of locally trivial A¹∗-bundles ˜πi : Si → A¹[2,5]. The fiber product W = S1×A¹[2,5]S2 thus inherits via the first and the second projection respec- tively the structure of an étale locally trivial A¹∗-bundle over S₁ and S₂. Since H_ét¹(Si,G^m) H¹(Si,G^m) by virtue of Hilbert’s Theorem 90 and

ANNALES DE L’INSTITUT FOURIER

Figure 2.2. Minimal resolution of the pencilξ2:F399KP¹. The excep- tional divisors E∞,1,E∞,2andE0,i,i= 1, . . . ,10, over the respective proper base points ∞ = F₀ ∩C₀ and 0 = p₀ of ξ₂ are numbered according to the order of their extraction.

class group ofF3implies that every Weil divisor onS2is linearly equivalent to one supported on the boundaryF3\S₂=C₀∪C₁∪D₂. SoS₂is factorial.

By construction,S1 and S2 both have the structure of locally trivial A¹∗- bundles ˜πi : Si → A¹[2,5]. The fiber product W = S₁ ×A¹[2,5] S₂ thus inherits via the first and the second projection respectively the structure of an étale locally trivialA¹_∗-bundle overS₁ andS₂. SinceH_ét¹(Si,Gm)' H¹(Si,G^m) by virtue of Hilbert’s Theorem 90 andSi is factorial, it follows thatW is simultaneously isomorphic to the trivialA¹_∗-bundlesS1×A¹_∗ and S₂×A¹∗. It remains to check thatS₁andS₂are not isomorphic. Suppose on the contrary that there exists an isomorphismϕ:S1 ∼

→S2and consider its natural extension as a birational mapϕ: S₁ 99K S₂ between the smooth SNC completions (S1, B1) and (S2, B2) of S1 and S2 constructed above.

Thenϕ must be a biregular isomorphism. Indeed, if eitherϕor ϕ⁻¹, say ϕ, is not regular then we can consider a minimal resolutionS₁←^σ X →^σ0 S₂ of it. By definition of the minimal resolution, there is no (−1)-curve in the unionB of the total transforms of B1 andB2 byσandσ⁰respectively which is exceptional for σ and σ⁰ simultaneously, and σ⁰ consists of the contraction of a sequence of successive (−1)-curves supported on B. The