Picard-Einstein Metrics and Class Fields Connected with Apollonius Cycle R-P. Holzapfel with Appendices by A. Pin˜eiro, N. Vladov November 16, 2013 Abstract WedefinePicard-EinsteinmetricsoncomplexalgebraicsurfacesasK¨ahler-Einsteinmetricswith negative constant sectional curvature pushed down from the unit ball via Picard modular groups allowing degenerations along cycles. We demonstrate how the tool of orbital heights, especially the Proportionality Theorem presented in [H98], works for detecting such orbital cycles on the projective plane. The simplest cycle we found on this way is supported by a quadric and three tangent lines (Apollonius configuration). We give a complete proof for the fact that it belongs to the congruence subgroup of level 1+i of the full Picard modular group of Gauß numbers together with precise octahedral- symmetric interpretation as moduli space of an explicit Shimura family of curves of genus 3. Proofs are based only on the Proportionality Theorem and classification results for hermitian lattices and algebraic surfaces. 1 11991MathematicsSubjectClassification: 11G15,11G18,11H56,11R11,14D05,14D22,14E20,14G35,14H10,14H30, 14J10,20C12,20H05,20H10,32M15 Key words: algebraic curves, moduli space, Shimura surface, Picard modular group, arithmetic group, Gauß lattice, K¨ahler-Einsteinmetric,negativeconstantcurvature,unitball SupportedbyDFG:HO1270/3-2and436BUL113/96/5 Contents 1 Introduction 2 2 The basic orbital surface: Plane with Apollonius cycle 4 3 Proportionality 9 4 Ball lattice conditions 14 5 The Gauss congruence ball lattice 21 6 Unimodular sublattices 25 7 Elements of finite order 30 8 The binary octehedron group and moduli of curves of Shimura equation type 37 9 Class fields corresponding to simple abelian CM threefolds of Q(i)- type (2,1) 43 10 Appendix 1 (by A. Pin˜eiro): The moduli space of hyperellitic genus 3 curves with Q(i)-multiplication 47 11 Appendix 2 (by N. Vladov): Determination of all proportional Apollonius cycles by MAPLE package “Picard” 51 1 1 Introduction ThemainpurposeofthisarticleistoshowthattheworldofcomplexalgebraicsurfacesisPicard-Einstein withauniversaldegenerationliftedfinitelyfromaquadricandthreetangentsonthecomplexprojective plane. The three tangent points are ”points at infinity” (cusp points) from the non-euclidean metric viewpoint. I found this projective complexified Apollonius configuration in connection with Fuchsian systems of partial differential equations in Sakurai-Yoshida [S-Y] (”mysterious phenomenon”, p. 1490; Figure 2, p. 1492). One calls a hermitian metric on a smooth complex surface ˚X Picard-Einstein (in a wide sense), if it is K¨ahler-Einstein with negative constant sectional curvature. If, moreover, ˚X is a Zariski open part of an algebraic surface X, then one says that X is Picard-Einstein (with Picard- Einstein metric) degenerating (at most) along X \ ˚X. The Bergman metric on the two-dimensional complex unit ball B is Picard-Einstein, see [BHH], Appendix B, for a short approach. For a ball lattice Γ⊂Aut Bthe(quasiprojective)quotientsurfaceX =X =B/Γ(alsoanycompactificationXˆ ofX)is hol Γ Picard-Einsteindegeneratingalongthebranchlocusofthecanonicalquotientmapp=p : B−→B/Γ Γ (and along the compactification cycle). The Picard-Einstein property lifts to each finite cover Yˆ of Xˆ degenerating (at most) along the preimages of branch loci of p and Yˆ −→ Xˆ. We call Yˆ Picard- Γ Einstein, if it is finitely lifted (that means via finite covering) from a ball quotient surface B/Γ such that the Baily-Borel compactification B(cid:100)/Γ of B/Γis the complex projectiveplane P2. If one finds a ball lattice with this property, then each complex projective surface is Picard-Einstein in the narrow sense because each such surface is a finite covering of P2, e.g. via general projections. The first proof for the fact that P2 is Picard-Einstein (degenerating along six lines) can be found in [H86]. There we used the Picard modular group of Eisenstein numbers. The main result of this paper is to show that P2 is Picard-Einstein degenerating along the Apollonius configuration de- scribed above, see theorem 5.1. The corresponding group Γ(1 + i) is the congruence sublattice of √ Γ := SU(diag(1,1,−1),O), O = Z+Zi, i = −1, belonging to the ideal O(1+i). This is a Picard modular group of Gauß numbers. Therearesomepaperswhichcamealreadyneartothisresult. FirstIhavetomentionMatsumoto’s article [Mat]. There is proved that P1×P1 is the compactified ball quotient surface by a subgroup of (cid:179) (cid:180) 0−i0 Γ(1+i) of index 2 but with Γ = SU( i 0 0 ,O). His proof is based on Mostow-Deligne’s theorem 0 0 1 [D-M] conversing multivalued solutions (hypergeometric functions) of a special Picard-Fuchs system by a (in [D-M] explicitly unknown) monodromy group acting on the ball. In the recent monograph of Yoshida[Y97]itappearsintermsofadmissiblesequences, seeCh. VI,Table1, cased=4, 2+2+2+2 (∞,∞,∞). (cid:100) Alreadyin[Ho83]weclassifiedpreciselythesurfaceB/Γ. Theproofisreproducedin[H98],addition- ally with explicit description of the branch locus of p . The ramification locus (on B) has been found Γ before by Shvartsman [Sv1], [Sv2] via classification of some hermitian O-lattices. He calculated the (cid:100) Euler number of B/Γ. The rationality of this surface has been proved before by Shimura [Sm64] after his celebrated general interpretations of arithmetic quotient varieties in [Sm63], which are called now ”Shimura varieties”. Since Shvartsmann’s classification of Γ-elliptic points is not avalaible in publica- tions,wefillthatgapinsections6,7classifyingpreciselytheindefiniteunimodularrank−2sublattices of the Gauß lattice Λ=O3 endowed with our diagonal hermitian metric of signature (2,1). Very useful is Hashimoto’s paper [Has] for this purpose. The most natural way for finding a configuration (reduced cycle Z) on an orbiface (two-dimensional orbifold), which could be the degenerate locus of a Picard-Einstein metrics has been described in [H98]. Beside of quotient singularities we allow also cusp singularities on the surface. The irreducible com- ponents of the configuration (points and irreducible curves) are endowed with natural numbers or ∞ (weights) in an admissible manner. Then one gets an orbital cycle. The surface X together with the orbital cycle Z is called an orbital surface. The orbital surface germs around points are irreducible components of the orbital cycle are called orbital points or orbital curves, respectively. Points or curves withweight∞arecalledcusp points orcusp curves,respectively. TheyformasubcycleZ ofZ whose ∞ support is denoted by X . The finitely weighted points are quotient points. For details we refer to ∞ [H98], where we corresponded rational numbers to our orbital objects called orbital hights. The orbital surface hights (global hights H) generalize volumes of Γ- fundamental domains on B of arbitrary ball latticesΓ. Theorbitalcurvehights(localhightsh)dothesameforthecomplexunitdiscDandD-lattice groups. EulerformandsignatureformdefineonthiswaytwodifferentorbitalhightsH , H andh , h e τ e τ called Euler or signature hights, respectively. A finite uniformization Y −→ X of an orbital surface 2 X = (X,Z) is a finite Galois covering Y −→ X such that Y is smooth (outside cusp points) and the weights of the components of Z coincide with corresponding ramification indices. A ball uniformization of X is a (locally finite) infinite Galois covering (quotient map by a ball lattice) B −→ X := X \X f ∞ again with wights equal to corresponding ramification indices. We announce the following Theorem 1.1 For an orbital surface X=(X,Z) the following conditions are equivalent: (i) X has a ball uniformization (ii) The proportionality conditions (Prop 2) H (X)=3H (X)>0 e τ (Prop 1) h (C)=2h (C)<0 for all orbital curves C⊂Z e τ are satisfied, and there exists a finite uniformization Y of X, which is of general type. The direction (i) ⇒ (ii) has been proved in [H98], see Proportionality Theorem IV.9.2. Notice that our h is 3 times h of [H98]. The other direction follows from the degree homogenity of the global heights τ τ and a well-known theorem of R.Kobayashi-Miyaoka-Yau applied to Y. Namely, it is easy to see that the (Prop 2)-condition lifts to the logarithmic Chern number condition c¯2 =3c¯ for Y. 1 2 (cid:164) In section 3 we use the explicit orbital hight machine for detecting suitable wights for points and curves on the Apollonius configuration on P2 such that the corresponding orbital surface satisfies the proportionality conditions. This has been done for demonstrating and understanding a general ap- proach to detect Picard-Einstein metrics on surfaces. Any orbital configuration (X,Z) defines a system Dioph(X,Z) of diophantine equations. It comes out from a system of a quadratic and some linear equations with rational coefficients closely related with (Prop 2) or (Prop 1), respectively, for which we have to determine inverse of natural numbers as solutions (the inverse of the weights we look for). There are at most finitely many solutions, see [H98], IV.10. Inthenextsectionwetransformthedetectedweightstosevenproperties(i),...,(vii)ofauniformizing ball lattice Γ(cid:48) we look for using the Proportionality Theorem via the system Dioph(X,Z) again, this time in converse direction: We know the weights but the data (Chern numbers, selfintersections) of X, Z are unknown. With the eight postulated properties we are able to determine these data and to classifysurfaceandcurvestogetB(cid:91)/Γ(cid:48) =P2 andtheApolloniusconfigurationback. Inthesections5,6,7 we prove that the congruence lattice Γ(1+i) has all the eight properties. In section 5 we prove that the structure of the factor group Γ/Γ(2) is isomorphic to the binary octahedron group 2O. An essential point is to decide which of two possible unitary codes in F8 is 2 defined by the intermediate factor group Γ(1 + i)/Γ(2). This is done by a non-elementary tool of algebraic topology (Armstrong’s Theorem, see Theorem 8.2). Its application is well-prepared by the sectionsbefore. KnowingthecodewefindanintermediateballlatticeΓ(2)⊂Γ ⊂Γ(1+i)withquotient 2 surface P1×P1 and factor group Γ /Γ isomorphic to the binary dieder group 2S of order 12. Together 2 3 with the appendix we prove that P1×P1 is the moduli space of the obviously 2S -symmetric family of 3 (double distinguished) curves C : Y3 = (X −1)(X +1)(X −b )2(X −b )2(X −b )2. The projective b 1 2 3 plane appears as moduli space of the (distinguished) curves via the map C (cid:55)→Pb=(b :b :b ). b 1 2 3 In order to connect the family with octahedral-symmetric Picard modular forms it is important to know the surface B(cid:92)/Γ(2) because van Geemen [vGm] found a structure result for the ring of Γ(2)- modularformsintermsofthetaconstantsandleftopentheproblemofprecisesurfaceclassification. For theta constants of Matsusaka’s Γ(cid:48)-level we refer also to [Mat]. Until now we know and announce that 2 B(cid:92)/Γ(2) is a smooth rational surface with six cusp points. The curve part of the corresponding orbital cyclecontainspreciselytensmoothrationalcurvesofwight2andselfintersection−1ontheblowingup model of the six cusp points. Nearby should be also congruence subgroups whose quotient surfaces are modelsofE×E,E ellipticcurvewithcomplexQ(i)-multiplicationoroftheKummersurfaceE×E/(cid:104)ι(cid:105), ιtheinvolutionsendingP to−P ontheabeliansurfaceE×E. Bothsurfacestogetherwithballquotient presentationsareimportant. ThefirstoneshouldrecognizeHirzebruch’sabeliancoversofE×E defined in [Hir] as Picard modular surfaces as it was done for Eisenstein numbers in [Ho86]. The second one could join Hilbert’s 12-th problem for our special Shimura surface(s) with the 3-dimensional congruent number problem, see Narumiya-Shiga [N-S]. 3 Next we turn our attention to a conjecture of Kobayashi [Kob] about complements of hypersurfaces in Pn to be (Kobayashi-) hyperbolic, if the degree of the sum of hypersurfaces is high enough. We refer to Dethloff- Schumacher-Wong [DSM] and the references given there for more details, restrict ourselves ton=2andaskforcurveconfigurationsZ onP2 suchthatP2\supp(Z)isPicard-Einsteindegenerated precisely along Z. This is a much stronger problem. For the complements of three quadrics or of a quadric and three lines the hyperbolicity is proved in general, but not individual. The orbital hight machine should be used to present more Picard-Einstein models. Is degree 5, as for the Apollonius configuration, the minimal possible degree ? In the mean time N.Vladov writes a detecting algorithm onMAPLEproducingfirstexperimentalresults: Thedeclarationofallthethreetangentpointsascusp pointsleadsonlytoourseriesofwightsofcyclecomponentssuchthatthecorrespondingorbitalsurface satisfiestheproportionalityconditions. Buttherearealsoweightsolutionsforthecasesthatonly2, 1or no of these three points are declared to be of cusp type (the other of quotient type), see Theorem 11.1. It seems to be quite possible that a good part of the 27 cases of the PDM (Picard-Deligne-Mostow) list ofwightedlinesonP1×P1 orP2,seee.g. [BHH],p. 201,canbeliftedfromtheApolloniusconfiguration. We finish the introduction with the following problem: Consider the class F of all smooth compact complexalgebraicsurfacesfinitelycoveringP2withbranchlocusontheAppoloniusconfiguration. Finite curve coverings of P1 branched over three points only are characterized as curves defined over number fields by a famous Theorem of Belyi [Bel] (for proof see also [Se89]). Find a similar characterization for the class F of surfaces ! Belyi’s curves are also characterized as compactified quotients curves by subgroups of Sl (Z) acting on the upper half plane H, see Shabat, Voevodsky [S-V], also [Se89], app. of 2 5.4. Which of the surfaces of F are ball quotients by (a group commensurable with) a Picard modular group of Gauß numbers ? 2 The basic orbital surface: Plane with Apollonius cycle We consider an orbital surface (1) X(cid:98) =(Xˆ;C(cid:98) +C(cid:98) +C(cid:98) +C(cid:98) +P +P +P +K +K +K ) 0 1 2 3 1 2 3 1 2 3 with smooth compact complex algebraic surface Xˆ supporting the orbital cycle (2) Z(X(cid:98))=C(cid:98) +C(cid:98) +C(cid:98) +C(cid:98) +P +P +P +K +K +K , 0 1 2 3 1 2 3 1 2 3 which consists of four orbital curves C(cid:98) , j =0,1,2,3, on X(cid:98) with weight 4, three (finite) orbital abelian j points P , j =1,2,3, of type C2/Z ×Z where Z ×Z ⊂Gl (C) denotes the abelian group generated j 4 4 4 4 2 by 2 opposite reflections of order 4, and K ,K ,K are precisely the orbital points at infinity. For the 1 2 3 surface Xˆ and the reduced cycle (3) Z(Xˆ)=Cˆ +Cˆ +Cˆ +Cˆ +P +P +P +K +K +K 0 1 2 3 1 2 3 1 2 3 we claim the following conditions: (i) The surface Xˆ is the projective plane P2 (ii) a) Cˆ is a quadric on P2; 0 b) Cˆ ,Cˆ ,Cˆ are projective lines on P2; 1 2 3 c) P ,P ,P are the three different intersection points of these lines; 1 2 3 d) Cˆ is the tangent line of Cˆ at K , j =1,2,3; j 0 j e) The configuration divisor Cˆ +Cˆ +Cˆ +Cˆ is symmetric. This means that there is an 0 1 2 3 effective action of the symmetric group S on P2 preserving Cˆ +Cˆ +Cˆ +Cˆ . 3 0 1 2 3 Definition 2.1 If these conditions are satisfied we call Cˆ +Cˆ +Cˆ +Cˆ a plane Apollonius configu- 0 1 2 3 ration or Apollonius configuration on P2, the cyle Z(Xˆ) a reduced plane Apollonius cycle and each effective cycle with this support a plane Apollonius cycle. 4 Thepropertiesa),b),c),d)meanthattheApolloniusconfigurationonP2 consistsofaplanequadricand three different tangent lines of it. We will see below that e) is automatically satisfied with a unique S - 3 action. The following graphic describes the corresponding configuration together with three additional lines L joining P and K . For the rest of this section we work on Xˆ =P2 and omit the hats over C . j j j j Moreover, we assume that all quadrics are non-degenerate, if the opposite is not stated. .........(cid:173).............P(cid:173)............(cid:173).......(cid:173).......(cid:173).............(cid:173)............(cid:173)......(cid:173)......K..........................................................(cid:173).....................................................................2.................................(cid:173)...............................................................................(cid:173)................................................................(cid:173).......................................................(cid:173).............C.........................L.....................(cid:173).............2...................2............................(cid:173)....................................................(cid:173).................................................(cid:173).................................(cid:74)....(cid:74)............K......(cid:173)..............................(cid:74)..................................................................................................................................................................................................................................................................................(cid:173)....(cid:173)......L.............(cid:74)....P......................................3................(cid:74)..3......................................C...............(cid:74).....................................L...0...........(cid:74)....C........................................1.........(cid:74)........1........................................................(cid:74)...............................................................(cid:74)............................................................(cid:74)...............................................................K.................(cid:74).................................................................C........................................................................(cid:74)1.............(cid:74).......(cid:74)......(cid:74).............(cid:74).............(cid:74)PP......(cid:74)..2.....(cid:74)...................... (cid:173) 1 3 3 2 (cid:74) Figure 1. Remark 2.2 The three quadruples {P ,K ,K ,K }, j = 1,2,3, are in general position. This means j 1 2 3 that each subtriple spans P2. Namely, the different points K ,K ,K cannot ly on one line L because 1 2 3 (L·C ) = 2. For the same reason, for example, the (tangent) line through P , K cannot contain K 0 1 2 1 or K . By symmetry the argument is complete. 3 Especially, we can choose the S -symmetric 3 Normalized Model 2.3 C :(X+Y −Z)2−4XY =X2+Y2+Z2−2XY −2XZ−2YZ =0; 0 C :X = 0 C :Y = 0 C :Z = 0, 1 2 3 P = (1:0:0) P = (0:1:0) P = (0:0:1); 1 2 3 K = (0:1:1) K = (1:0:1) K = (1:1:0); 1 2 3 L :Y = Z L :X = Z L :X = Y . 1 2 3 For finding the (unique) quadratic equation we refer to the end of this section (Lemma 2.10). Proposition 2.4 Up to PGl -equivalence the Apollonius configuration is unique-ly determined. All 3 Apollonius configurations are S -symmetric. 3 Proof. Let D ⊂ P2 be another quadric and D ,D ,D three tangents touching D in M ,M ,M , 0 1 2 3 0 1 2 3 respectively. The intersection point of D , D is denoted by Q for {i,j,k} = {1,2,3}. Let π be the i j k correspondence P (cid:55)→ Q , K (cid:55)→ M , j = 1,2,3. By the main theorem of (elementary) projective ge- 1 1 j j ometry, this map extends uniquely to a projective transformation Π : P2 −→ P2, because the points P ,K ,K ,K (andtheirimages)areingeneralpositionbytheaboveremark. ΠsendstheC -tangents 1 1 2 3 0 C , C (through P and K , K , respectively) to the D -tangents D , D . A quadric is uniquely de- 2 3 1 2 3 0 2 3 termined by two given tangent lines and a point on it different from the touching points of the two tangents. Namely, the algebraic family of all plane quadrics is 5-dimensional. Going through three given points and two given tangent lines at two of them yield five linear conditions for the five (affine) parametersforthequadrics. Viaprojectivetransformationthiscanbealsocheckednowmoreexplicitly by example: Take D = X-axis, D = Y-axis in C2 ⊂ P2 , P = (0,0), K = (1,0), K = (0,1). It is 2 3 1 2 3 an easy calculation to see that the only quadric with tangents D , D at K or K , respectively, going 2 3 2 3 5 through K := (2,1) is the circle (X −1)2 +(Y −1)2 = 1 with center (1,1). Turning back to the 1 general situation we see that Π sends the quadric C to the quadric D . But then the tangent line C 0 0 1 at K ∈C is sent to the tangent D at M ∈D . 1 0 1 1 0 If a configuration is symmetric, then each projective transform of it is, by conjugation of the S - 3 action. Since 2.3 is the symmetric we are through. (cid:164) Corollary 2.5 The action of the symmetric group S on P2 preserving the configuration C +C + 3 0 1 C +C is unique. It is determined by extending permutations of points π : K (cid:55)→ K , P (cid:55)→ P , 2 3 i π(i) i π(i) i=1,2,3, π ∈S , to Π∈AutP2 = PGl (C). Especially for the normalized model 2.3 the group S acts 3 3 3 by permutation of canonical projective coordinates (x:y :z) on P2. Proof. ThegeneralstatementisaspecialcaseconsideredintheproofofProposition2.4settingD =C 0 0 and taking for D ,D ,D an arbitrary permutation of C ,C ,C . For the normalized model the action 1 2 3 1 2 3 is obvious. (cid:164) Remark 2.6 The lines L ,L ,L defined in (1.4) have a common point. 1 2 3 Proof. This can be checked now on any special model. The normalized model 2.3 yields (1 : 1 : 1) as intersection point of the three lines. (cid:164) Lemma 2.7 Each projective representation G ⊂ AutP2 of a finite group can be lifted to a linear representation G˜ ⊂ Gl (C). For given d ∈ N there is a unique central lift (group extension) G˜ ⊂ 3 + d Gl (C) of G with the group Z ⊂ C* of 3d-th unit roots as (central) kernel. It consists of all lifts of 3 3d elements of G with determinant in Z . d Each finite lift G˜ ⊂Gl (C) of G is a subgroup of G˜ for a suitable d∈N . The special lift G˜ ⊂Sl (C) 3 3d + 1 3 has kernel Z over G. 3 Proof. For each g ∈G we can find a lift g˜∈Gl (C) because of the exact sequence 3 1−→C* −→Gl (C)−→PGl (C)−→1 3 3 of group homomorphism. The coset C*g˜ consists of all lifts of g. We can choose a special lift g˜ with determinant in Z . Then the subset Z g˜coincides with the set of all lifts of g with determinant in Z . d 3d d Bysuchchoiceg˜foreachg ∈GweobtainthegroupG˜ :={Z g˜;g ∈G}⊂Gl (C)togetherwithexact d 3d 3 sequences 1−→Z −→G˜ −→G−→1 3d d det 1 −→ SG˜ −→ G˜ −→ Z −→ 1 d d d with central kernels. Now it is clear that each finite representative lift of G to Gl (C) is contained in 3 one of the G˜ because the corresponding determinant group must be finite. d Write both exact sequences together in one diagram and complete it to a diagram with three exact rows and three exact columns. Then one gets an exact sequence 1−→Z −→SG˜ −→G−→1. 3 d Obviously, SG˜ does not depend on d. It coincides with G˜ . So the last sequence is nothing else but d 1 (4) 1−→Z −→G˜ −→G−→1. 3 1 (cid:164) 6 Denote for an arbitrary group H by CF(H) the set of conjugation classes of finite subgroups of H. Obviously we get for all n∈N by C*- factorization a surjective map + (5) CF(Gl (C))(cid:179)CF(PGl (C)). n n Let CF (Gl (C)) be the subset of CF(Gl (C)) consisting of all complete lifts G˜ with determinant d of d 3 3 d finite subgroups G of PGl (C). For n=3 we get a bijective restriction of (5) to 3 (6) CF (Gl (C))←→CF(PGl (C)). d 3 3 For d = 1 one gets especially a bijection (7) CF(Sl (C))=CF (Gl (C))←→CF(PGl (C)). 3 1 3 3 Let P ⊂ Gl (C) be the subgroup of permutation matrices. We multiply the elements of P by their 3 determinants to get a subgroup P of Sl (Z) isomorphic to S . We call it the group of unimodular 1 3 3 permutation matrices. Corollary 2.8 Let G ⊂ PGl (C) be a finite group isomorphic to S . Then G can be lifted uniquely to 3 3 a subgroup of Sl (C) conjugated to the group P of unimodular permutation matrices. There exists a 3 1 projective coordinate system on P2 such that G acts by permutation of coordinates. Proof. The second statement follows from the first because projective conjugations correspond to projective coordinate changes, and P acts obviously by permutation of canonical coordinates. Take 1 the lift G˜ ⊂ Sl (C). It splits into Z ×Σ by (6) and Σ projects isomorphically to G ∼= S . Since 1 3 3 3 3 3 Σ is not abelian, the representation Σ ⊂ Gl (C) is not diagonizable that means it doesn’t split 3 3 3 into three characters. The only irreducible representations of S are the characters 1, sgn (signature 3 of permutations) and the faithful rank-2- representation δ realized as the dieder group of a regular triangle. Therefore there are, up to conjugation, only two faithful rank-3 representation, namely 1+δ and δ+sgn, where only the latter has determinant +1. Therefore it is equivalent to the representation P ⊂ Sl (C). This means that the groups Σ and P are Gl (C)-conjugated, hence Sl (C)-conjugated. 1 3 3 1 3 3 Since each Sl -lift of G must be contained in G˜ (via Lemma 2.7) we see that Σ is the only possibility 3 1 3 of isomorphic lifting. (cid:164) Remark 2.9 Let O be an integral subdomain of the field C not containing primitive 3-rd unit roots. A finitegroupG⊂PGl (C)hasatmostoneunimodularliftG˜ ⊂Sl (O). Ifitexists, itmustbeisomorphic 3 3 to G. Especially, S has the unique representation P ⊂Sl (O). 3 1 3 Proof. The first statement is true because (cid:110)(cid:179) (cid:180) (cid:111) ω 0 0 Z3 = 0 ω 0 ; ωa3−rdunitroot 0 0 ω intersects Sl (O) trivially and because of the exact sequence (6). For the second one has only to lift 3 the representation of S permuting canonical coordinates and to apply the uniqueness statement of 3 Corollary 2.8. (cid:164) Lemma 2.10 For three projective lines C ,C ,C on P2 intersecting each other in different points and 1 2 3 for a given subgroup Σ ∼= S of PGl permuting them there is precisely one quadric C with tangents 3 3 3 0 C ,C ,C . For the canonical coordinate axes X =0, Y =0, Z =0 of P2 the corresponding quadric (see 1 2 3 2.3, normalized model) has equation X2+Y2+Z2−2XY −2XZ−2YZ =(X+Y −Z)2−4XY =0. Proof. Assume that we find an Apollonius configuration C +C +C +C extending the given lines. 0 1 2 3 BecauseofPGl -equivalenceofsuchconfigurations(Proposition2.4)wecanassumethattheselinesare 3 the coordinate axis C :X =0 , C :Y =0 , C :Z =0 1 2 3 7 with intersection points P =(1:0:0) , P =(0:1:0) , P =(0:0:1). 1 2 3 Moreover, the action of S on P2 can assumed to be the most natural one permuting coordinates 3 (Corollary 2.5) with unique lift S ⊂Sl (C) represented by permutation matrices (Corollary 2.8). Let 3 3 aX2+bY2+cZ2+2dXY +2eXZ+2fYZ =0. be the homogeneous equation of an arbitrary plane quadric. All S -invariant quadrics must satisfy 3 simultaneously three equations aY2+bX2+cZ2+2dXY +2eYZ+2fXZ = 0 aZ2+bY2+cX2+2dYZ+2eXZ+2fXY = 0 aX2+bZ2+cY2+2dXZ+2eXY +2fYZ = 0, which have to be the same up to a factor. It follows that a=b=c=1 (without loss of generality) and d=e=f. Therefore the S -invariant quadrics form a 1-parameter family 3 X2+Y2+Z2+2dXY +2dXZ+2dYZ =0. The curve C has contact of order 2 with C : X = 0 at a point K = (0 : 1 : t). Substituting the 0 1 1 coordinates of K in the quadratic equation, this means that the equation 1 0=1+t2+2dt=(t+d)2+(1−d2) must have a unique solution for t. Therefore d = −t = 1 or d = −t = −1. By symmetry we conclude that there are precisely two S -invariant quadrics with tangent C , namely 3 1 X2+Y2+Z2−2XY −2XZ−2YZ =(X+Y −Z)2−4XY =0 and X2+Y2+Z2+2XY +2XZ+2YZ =(X+Y +Z)2 =0. But the latter one is degenerated. (cid:164) For later use we need the more general Definition 2.11 An Apollonius cycle on a smoooth compact complex algebraic surface Y is a cycle (8) Z=v L +v L +v L +v L +P +P +P +K +K +K , 0 0 1 1 2 2 3 3 1 2 3 1 2 3 where the v ’s are positive integers, P ,P ,P ,K ,K ,K are points on Y, and the L ’s are smooth i 1 2 3 1 2 3 i complete algebraic curves on Y with the following intersection behaviour: L ·L =2K forj =1,2,3; L ·L =P for{i,j,k}={1,2,3}. 0 j j i j k ThesupportingreducedcurveL +L +L +L iscalledanApolloniusconfigurationonY.TheApollonius 0 1 3 3 cycle (configuration) is called symmetric, iff there is an algebraic S -action on Y, which preserves the 3 cycle Z permuting effectively its components Cˆ , P , K , j =1,2,3, respectively. j j j Remark 2.12 Obviously, v = v = v holds in the symmetric case. If Y is the projective plane, then 1 2 3 the Apollonius configuration is automatically of the (symmetric) plane Apollonius cycle consisting of a quadric and three tangent lines as defined in 2.1 . Namely, from (L ·L )=1 for 0<j <k ≤3 and Bezout’s theorem it follows that the L ’s are smooth j k j curves of degree 1 on P2. Therefore L ,L ,L are projective lines. Since (L ·L )=2 we conclude with 1 2 3 0 j the same argument that L is a quadric touched by L at K . For symmetry we refer to Proposition 0 j j 2.4. 8 3 Proportionality Turnbacknowtothemoreprecisenotationsof2.1notassuminginthissectionthesymmetrycondition (ii) e). We blow up each of the the points K twice such that the proper transforms of Cˆ for i = j i 1,2,3 on the resulting surface X˜ do not intersect the proper transform of Cˆ . The exceptional divisor 0 E(X˜ −→ Xˆ) on X˜ consists of three connected components. Each of them is a pair of transversally crossing smooth rational curves with selfintersection -1 or -2, respectively. Then we contract the three -2-curves to get a surface X(cid:48) with three quotient singularities of type C2/±(10) lying on exceptional 01 curves E ,E ,E ⊂X(cid:48). On this way we get an orbital birational morphism X(cid:48) −→Xˆ being isomorphic 1 2 3 outside X(cid:48) = E +E +E and Xˆ = K +K +K . The proper transforms of the Cˆ are denoted ∞ 1 2 3 ∞ 1 2 3 j by C(cid:48), j =0,1,2,3, respectively. On this way we get a complete orbital surface j X(cid:48) =(X(cid:48);C(cid:48) +C(cid:48) +C(cid:48) +C(cid:48) +P +P +P +E +E +E ) 0 1 2 3 1 2 3 1 2 3 called the canonical locally abelian model of Xˆ. The finite part supported by X =X(cid:48) =X(cid:48)\X(cid:48) is the f ∞ open orbital surface X=(X;C +C +C +C +P +P +P ) 0 1 2 3 1 2 3 with supporting non-compact curves C = C(cid:48) = C(cid:48)\X(cid:48) . The orbital cycle Z(X(cid:48)) is described in the j jf j ∞ following picture (cid:84) (cid:183) (cid:84)(cid:183) P (cid:183)(cid:84) 3 (cid:183) (cid:84) (cid:183) (cid:84) (cid:183) (cid:84) (cid:183) (cid:84) (cid:183) (cid:84) (cid:183) (cid:84) (cid:183) (cid:84) (cid:183)(cid:183)C(cid:183)E(cid:72)(cid:48)2(cid:183)2(cid:183)(cid:72)(cid:183)(cid:114)(cid:183)(cid:72)(cid:183)(cid:72)(cid:183)(cid:72)........................................................................................................................................(cid:72)....................(cid:72)..............................................................................................................................................................C................................(cid:48)0...................................................................................................................(cid:169)................(cid:169)..........................................S..................................(cid:169)...............1.......................................................................................................(cid:169)(cid:84)(cid:84)(cid:169)T(cid:84)1(cid:169)(cid:84)R(cid:114)(cid:84)1(cid:169)(cid:84)C(cid:84)(cid:48)1E(cid:84)1(cid:84) (cid:183) (cid:84) (cid:183) (cid:84) (cid:183)P C(cid:48) P (cid:84) (cid:183) 1 (cid:114) 3 2 (cid:84) E 3 Figure 2. • singularity of type <2,1> The open orbital curves can be written as C =4C ,C =(4C ;P +P ),C =(4C ;P +P ),C =(4C ;P +P ) 0 0 1 1 2 3 2 2 1 3 3 3 1 2 The corresponding atomic graphs of the four orbital curves look like 4 C0 :.....<........4...............................................2..................,......................................1..-................<1....>..............................................................2........................(cid:102)............,...................................................1....................................>.................................................-..................4..................................................................................................................1.........(cid:118)..............................-..........................2........................................................................................................................4..........(cid:102)....................................-........................................1.......................................................<........................................2.............................,..........1>C1,C2,...4...........C........<.............................................3..2............(cid:102).....,.........:.......1...............>.................................................-..................4..................................................................................................................1.........(cid:118)..............................-..........................1.......................................................4.....(cid:102).................................................4......................... 9