COMPLETIONS OF C SURFACES

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Niveau: Supérieur, Doctorat, Bac+8
COMPLETIONS OF C?-SURFACES HUBERT FLENNER, SHULIM KALIMAN AND MIKHAIL ZAIDENBERG Prepublication de l'Institut Fourier no 684 (2005) www-fourier.ujf-grenoble.fr/prepublications.html Dedicated to Masayoshi Miyanishi Abstract. Following an approach of Dolgachev, Pinkham and Demazure, we classified in [FlZa1] normal affine surfaces with hyperbolic C?-actions in terms of pairs of Q-divisors (D+, D?) on a smooth affine curve. In the present paper we show how to obtain from this description a natural equivariant completion of these C?-surfaces. Using elementary transformations we deduce also natural completions for which the boundary divisor is a standard graph in the sense of [FKZ] and show in certain cases their uniqueness. This description is especially precise in the case of normal affine surfaces completable by a zigzag i.e., by a linear chain of smooth rational curves. As an application we classify all zigzags that appear as boundaries of smooth or normal C?-surfaces. Keywords: C?-action, C+-action, affine surface. Resume. Dans une publication recente [H. Flenner, M. Zaidenberg, Normal affine surfaces with C?-actions. Osaka J. Math. 40, 2003, 981–1009] nous avons classifie, en suivant une approche due a Dolgachev, Pinkham et Demazure, les surfaces affines normales V sur C ad- mettant une action C? hyperbolique, en termes de couples de Q-diviseurs (D+, D?) sur une courbe affine lisse.

zigzag can

gizatullin surface

deduisons egalement de completions naturelles

equivariant completions

c?-actions

?˜2 - · ·

?n ?n

c?-surface

Sujets

Russell

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Publié par	profil-zyak-2012
Nombre de lectures	33

Extrait

COMPLETIONS OF

C∗S-RUAFECS

HUBERT FLENNER, SHULIM KALIMAN AND MIKHAIL ZAIDENBERG

Pr´epublicationdel’InstitutFourierno684 (2005) www-fourier.ujf-grenoble.fr/prepublications.html

Dedicated to Masayoshi Miyanishi

Abstract.Following an approach of Dolgachev, Pinkham and Demazure, we classiﬁed in [FlZa1] normal aﬃne surfaces with hyperbolicC∗-actions in terms of pairs ofQsrosiiv-d (D+, D− the present paper we show how to obtain from In) on a smooth aﬃne curve. this description a natural equivariant completion of theseC∗ elementary-surfaces. Using transformations we deduce also natural completions for which the boundary divisor is a standard graph in the sense of [FKZ] and show in certain cases their uniqueness. This description is especially precise in the case of normal aﬃne surfaces completable by a zigzag i.e., by a linear chain of smooth rational curves. As an application we classify all zigzags that appear as boundaries of smooth or normalC∗-surfaces. Keywords:C∗noit,ac-C+-action, aﬃne surface.

R´esu´,gbnredeai.Z,Mernnle.FH[etnece´rnoitacDsnnupebuilNormal aﬃne surfaces me.a withC∗-actionsnsuivantusnseiﬁ´e,evanocsal00]9onsu,903–181.4th200,akasaM.JO. approchedue`aDolgachev,PinkhametDemazure,lessurfacesaﬃnesnormalesVsurCad-mettant une actionC∗hyperbolique, en termes de couples deQ-diviseurs (D+, D−) sur une courbeaﬃnelisse.Icinousmontronscommentonpeutobtenir,a`partirdecettedescription, unecomple´tionnaturelle´equivarianted’unetellesurface.Enutilisantdestransformations e´l´ementaires,nousend´eduisonse´galementdecompl´etionsnaturellespourlesquellesledi-viseur au bord a un graphe dual standard au sens de [H. Flenner, S. Kaliman, M. Zaidenberg, Birational transformations of weighted graphs. math.AG/0511063], et nous montrons qu’elles sontuniquesdanscertainscas.Cettedescriptionestspe´cialementpre´cisedanslecasdes surfacesaﬃnesnormalespouvanteˆtrecompl´ete´esparunzigzag,c’esta`dire,parunechaˆıne line´airedecourbesrationnelleslisses.Commeapplication,nousclassiﬁonstousleszigzags qui apparaissent en tant que bords des surfacesC∗lisses ou normales.

Mots-cle´s: actionC∗, actionC+, surfaces aﬃnes.

Mathematics Subject Classiﬁcation (2000): 14R05, 14R20, 14J50.

HUBERT FLENNER, SHULIM KALIMAN AND MIKHAIL ZAIDENBERG

Contents

1. Introduction 2. Equivariant completions of aﬃneG-surfaces 2.1. Equivariant completions 2.2. Standard and semistandard completions 2.3. Uniqueness of standard completions 3. Equivariant completions ofC∗-surfaces 3.1. Generalities 3.2. Equivariant completions of hyperbolicC∗-surfaces 3.3. Equivariant resolution of singularities 3.4. Parabolic and ellipticC∗-surfaces 4. Boundary zigzags of GizatullinC∗-surfaces 4.1. Smooth Gizatullin surfaces 4.2. Toric Gizatullin surfaces 4.3. Smooth GizatullinC∗-surfaces 5. Extended graphs of GizatullinC∗-surfaces 5.1. Extended graphs 5.2. Extended graphs on GizatullinC∗-surfaces 5.3. Danilov-GizatullinC∗-surfaces References

2 4 4 5 8 11 11 12 15 18 19 19 20 21 24 24 25 29 34

1.rtnIcudoonti An irreducible normal aﬃne surfaceX= SpecAendowed with an eﬀectiveC∗-action will be called aC∗-surface the. Inelliptic casethe action possesses an attractive or repulsive ﬁxed point and in theparabolic casean attractive or repulsive curve consisting of ﬁxed points. A simple and convenient description for these surfaces, based on the fact that theC∗-action corresponds to a grading of the coordinate ringAofX, was elaborated by Dolgachev, Pinkham and Demazure, so it was called in [FlZa1, I] aneseitatno-DrpPD. Namely, in the elliptic case our surface is represented as X= SpecAwithA=MH0(COC(bkDc))∙uk k≥0

whereuis an indeterminate,Dis an ampleQ-divisor on a smooth projective curveCand bkDc curve Thedenotes the integral part.C= ProjAis then the orbit space of theC∗-action on the complement of its unique ﬁxed point inX in the parabolic case. Likewise, X= SpecA0[D] withA0[D] =MH0(CO(bkDc))∙uk k≥0 where nowDis aQ-divisor on a smooth aﬃne curveC= SpecA0, which again is the orbit space of ourC∗-action on the complement of its ﬁxed point set inX. All otherC∗-surfacesXarechpyreobil . Theirﬁxed points are all isolated, attractive in one and repulsive in the other direction. Any such surface is isomorphic to

SpecA0[D+ D−] withA0[D+ D−] :=A0[D+]⊕A0A0[D−] whereD±is a pair ofQ-divisors on a normal aﬃne curveC= SpecA0withD++D−≤0 [FlZa1, I]. In this paper we are mainly interested in an explicit description of the completions of such C∗contained in section 3, where we describe a canonical of the main results is -surfaces. One

COMPLETIONS OFC∗-SURFACES 3 equivariant completion of a hyperbolicC∗-surface in terms of the divisorsD±, see for instance Corollary 3.18 for the dual graph of its boundary divisor. We also treat in brief the case of elliptic and parabolic surfaces, see Section 3.4. In [FKZ], Corollary 3.36 we have shown that any normal aﬃne surfaceVadmits a comple-tion for which the dual graph of the boundary is standard (see 2.8). Given a DPD presentation of aC∗-surfaceV, the results of Section 3 provide an explicit equivariant standard completion ¯ VstofVSection 2 we investigate the question as to when such equivariant generally, in . More standard completions can be found for actions of an arbitrary algebraic groupG. We show that this is indeed possible for normal aﬃneG-surfacesVexcept for

P2\QP1×P1\Δ Vd,1 whereQis a non-singular quadric inP2, Δ is the diagonal inP1×P1andVd,1,d≥1, are the Veronese surfaces, see Theorem 2.9. Moreover, equivariant standard completions always exist ifG Weis a torus. also deduce their uniqueness in certain cases, see Theorem 2.13. In this paper we study mostlyC∗-actions on Gizatullin surfaces. By aGizatullin surfacewe mean a normal aﬃne surface completable by azigzagthat is, a simple normal crossing divisor Dwith rational components and a linear dual graph ΓD. These surfaces are remarkable by a variety of reasons. By a theorem of Gizatullin [Gi, Theorems 2 and 3] (see also [Be, BML], and [Du1] for the non-smooth case), the automorphism group Aut(X) of a normal aﬃne surfaceXhas an open orbit with a ﬁnite complement inXif and only if eitherX∼=C∗×C∗ orX automorphism groups of Gizatullin surfaces were further Theis a Gizatullin surface. studied in [DaGi]. Like in the case ofX=A2C, such a group has a natural structure of an amalgamated free product. These surfaces can also be characterized by the Makar-Limanov invariant: a normal aﬃne surfaceX= SpecAdiﬀerent fromA1C×C∗is Gizatullin if and only if its Makar-Limanov invariant is trivial that is, ML(X) :=Tker∂=Cwhere the intersection is taken over all locally nilpotent derivations ofA. Among the hyperbolicC∗-surfacesX= SpecA0[D+ D−] the Gizatullin ones are characterized by the property that each of the fractional parts{D±}= D±− bD±cis either zero or supported at one point{p±}, see [FlZa1, II]. In Theorem 4.4(a) we show that an arbitrary ample zigzag can be realized as a boundary divisor of a GizatullinC∗ However, not every such zigzag-surface and even a toric one. appears as the boundary divisor of asmoothC∗ precisely we give in 4.4-4.6 a-surface. More numerical criterion as to when a zigzag can be the boundary divisor of a smooth Gizatullin C∗criterion we can exhibit many smooth Gizatullin surfaces which do this -surface. Using not admit anyC∗-action, see Corollary 4.8. We note that everyQ-acyclic Gizatullin surface1 is aC∗-surface [Du2 latter class was studied e.g., in [DaiRu, MaMi, II.5.10]. The1, Du2]. Finally, in 5.13 we investigateC∗-actions on Danilov-Gizatullin surfaces, by which we mean complements Σn\Sof an ample sectionSin a Hirzebruch surface Σn a theorem of. By Danilov-Gizatullin [DaGi], the isomorphism class of such a surfaceVk+1depends only on the self-intersection numberS2=k+ 1> n particular it does not depend on. Innand is stable under deformations ofSinside Σn. According to Peter Russell2, given any naturalkthere are exactlykpairwise non-conjugatedC∗-actions onVk+1 give another proof of this result. We using our DPD-presentations. In a forthcoming paper we will show that a Gizatullin surface which possesses at least 2 non-conjugatedC∗-actions is isomorphic to a Danilov-Gizatullin surface.

1That isHi(X,Q) = 0∀i >0. 2 are grateful to WeAn oral communication.Peter Russell for generously sharing results from unpublished notes [CNR].