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COMPLETIONS OF
CS-RUAFECS
HUBERT FLENNER, SHULIM KALIMAN AND MIKHAIL ZAIDENBERG
Pr´epublicationdelInstitutFourierno684 (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:Cnoit,ac-C+-action, affine surface.
R´esu´,gbnredeai.Z,Mernnle.FH[etnece´rnoitacDsnnupebuilNormal affine surfaces me.a withC-actionsnsuivantusnsei´e,evanocsal00]9onsu,903181.4th200,akasaM.JO. approchedue`aDolgachev,PinkhametDemazure,lessurfacesanesnormalesVsurCad-mettant une actionChyperbolique, en termes de couples deQ-diviseurs (D+, D) sur une courbeanelisse.Icinousmontronscommentonpeutobtenir,a`partirdecettedescription, unecomple´tionnaturelle´equivariantedunetellesurface.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 surfacesanesnormalespouvanteˆtrecompl´ete´esparunzigzag,cesta`dire,parunechaˆıne line´airedecourbesrationnelleslisses.Commeapplication,nousclassionstousleszigzags qui apparaissent en tant que bords des surfacesClisses ou normales.
Mots-cle´s: actionC, actionC+, surfaces affines.
Mathematics Subject Classification (2000): 14R05, 14R20, 14J50.
2
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))ukk0
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))ukk0 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++D0 [FlZa1, I]. In this paper we are mainly interested in an explicit description of the completions of such Ccontained 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,1whereQis a non-singular quadric inP2, Δ is the diagonal inP1×P1andVd,1,d1, 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×CorX 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×Cis Gizatullin if and only if its Makar-Limanov invariant is trivial that is, ML(X) :=Tker=Cwhere 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 Ccriterion 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) = 0i >0. 2 are grateful to  WeAn oral communication.Peter Russell for generously sharing results from unpublished notes [CNR].
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