aa r X i v : . [ m a t h . AG ] N ov RECENT PROGRESS IN THE GEOMETRY OF Q -ACYCLICSURFACES KAROL PALKA
Abstract.
We give a survey of results on the geometry of complexalgebraic Q -acyclic surfaces including some recent results. Contents
1. Introduction 12. Preliminaries 23. Balanced and standard completions 34. Basic properties 55. Case κ ( S ) = −∞ and moduli 66. Logarithmic Bogomolov-Miyaoka-Yau 87. Exceptional Q -homology planes 98. Non-logarithmic Q -homology planes 119. C ∗ -ruled Q -homology planes 1310. Contractible curves 1411. Smooth Q -homology planes of general type 1512. Smooth locus of general type 16References 171. Introduction
In this article we want to take the reader to the rich and yet not fullyexplored world of plane-like complex algebraic surfaces. We hope our surveywill give a general reader a taste of methods used and will serve as an updateon recent results for experts. All varieties considered are complex algebraic.The story begins with a surprising discovery of Ramanujam [Ram71] of acontractible affine surface non-isomorphic to C , one of the many nontrivial smooth homotopy planes , i.e. smooth contractible surfaces. Ramanujamdiscovered a first important characterization of C by showing that it isthe unique smooth homotopy plane which is simply connected at infinity.Since then affine algebraic geometers began the study of smooth (and more Mathematics Subject Classification.
Primary: 14R05; Secondary: 14J17, 14J26.
Key words and phrases.
Acyclic surface, homology plane, Q-homology plane.The author was supported by Polish Grant MNiSzW. generally normal) varieties with the same Betti numbers as C , so-called Q -homology planes . One of the motivations was the search for more practicalcharacterizations of the complex plane (the computation of the fundamentalgroup at infinity of an affine variety is usually very difficult). In the face ofthe topological simplicity the intriguing algebraic side of Q -acyclic surfacesis more clearly visible. Because of their homological similarity to the plane,smooth Q -homology planes, and especially smooth homotopy planes, playtoday an important role as a source of examples or counterexamples whenstudying working hypotheses as well as more challenging conjectures. Theyaccompany us when studying exotic structures on C n ’s (see [Za˘ı99]), theCancellation Conjecture (which was the motivation for Ramanujam andhas been proved in dimension two by Fujita and Miyanishi [Fuj79]), theJacobian Conjecture (see [Miy07, 5.2]) and others. Today, after almostforty years, mainly due to the tools of the theory of open surfaces, thescheme of the classification, and in most cases the classification itself, aresettled. Recently the author had the pleasure of adding his part to the story.Although we try to avoid notions which are not well-known to any well-versed algebraic geometer, we need to refer the reader for the basic notionsof the theory of open surfaces to [Miy01]. Since the section 4.3 loc. cit. is areview of smooth Q -homology planes we concentrate mainly on the singularcase, stating the updated results so to include the smooth case if possible(see also [Miy07]). Acknowledgements.
The author would like to thank professors P. Russelland S. Lu for the invitation to Montreal. He also thanks prof. P. Russellfor stimulating discussions. 2.
Preliminaries A normal pair ( X, D ) consists of a complete normal surface X and adivisor D with simple normal crossings (snc) contained in the smooth locusof X . A normal pair ( X, D ) is said to be smooth if X is smooth. If X is anormal surface then an embedding ι : X → X , where ( X, X \ X ) is a normalpair, is called a normal completion of X . A normal completion is a smoothcompletion if X is smooth. Two normal completions ι j : X → X j , j = 1 , f : X → X for which f ◦ ι = ι .A surface with isolated singularities is called logarithmic if all its singularpoints are of analytical type C /G for some finite subgroup G < GL (2 , C ).By an n -curve we mean a smooth rational curve with self-intersection n .An affine ruling (a P -ruling, a C ( n ∗ ) -ruling ) is a morphism from a surfaceonto a smooth curve with general fiber isomorphic to C (respectively to P , C with n points deleted). All these morphisms are called rational rulings. The m -dimensional Cancellation Conjecture: If X × C n ∼ = C n + m then X ∼ = C m . The Jacobian Conjecture: A polynomial map f : C n → C n with nowhere vanishingJacobian determinant | J ( f ) | has a polynomial inverse. ECENT PROGRESS IN THE GEOMETRY OF Q -ACYCLIC SURFACES 3 Definition 2.1.
Let p : X → B be a rational ruling of a normal surface X . A triple ( X, D, p ) is called a completion of p if and only if ( X, D ) is anormal completion of X and p : X → B is a P -ruling onto a smooth curve B ⊇ B extending p . Such a completion is minimal (we say also that ( X, D )is p -minimal) if it does not dominate any other completion of p .If an snc-divisor T (or rather its dual graph) is a chain and T = T + T + . . . + T n is its decomposition into irreducible components so that T i · T i +1 = 1for i = 1 , . . . , n −
1, we then write T = [ − T , . . . , − T n ]. As long as T is notconsidered as a twig attached to some other divisor containing T there is nopreferred choice of the tip ( T or T n ) of T , so in this case T = [ − T n , . . . , − T ]as well. If T is a twig of some fixed bigger divisor then by convention wealways choose the tip of T which is a tip of this bigger divisor as T .The Kodaira dimension of a divisor F on a smooth complete surface(hence projective by the theorem of Zariski) X is defined as κ ( F ) = sup n> dim Im(Φ | nF | ) ∈ {−∞ , , , } , where Φ | nF | : X P N is the mapping given by the linear system | nF | .Then the logarithmic Kodaira dimension of a smooth open surface X canbe defined by taking some smooth completion ( X, D ) of X and by putting κ ( X ) = κ ( K X + D ) , where K X is a canonical divisor on X . This is well-known to be indepen-dent of the smooth completion (see [Iit82] for the properties of κ ( F )). TheKodaira dimension of a singular surface is defined to be the Kodaira di-mension of any resolution. If R is an snc-divisor on a complete surface and Q ( R ) is the intersection matrix of R then we define the discriminant of R by d ( R ) = d ( − Q ( R )).3. Balanced and standard completions
One of the basic steps to take when dealing with an open surface is toconstruct a completion and boundary. Of course, these are not unique, butwe want to bring to the attention of the reader some normalizing conditions,which make them more unique and more useful in practice. We came to thisproblem when trying to distinguish (or to find an isomorphism) betweensome Q -acyclic surfaces. This type of analysis was done at least partiallyby many authors, with the most complete treatment in terms of weighteddual graphs in [Dai03] and [FKZ07]. Here we present the necessary resultsusing partially our own terminology. For simplicity we restrict ourselves todivisors whose dual graphs contain no loops (forests). Definition 3.1.
A rational chain D = [ a , . . . , a n ] is balanced if a , . . . , a n ∈{ , , , . . . } or if D = [1]. A reduced snc-forest D is balanced if all rationalchains contained in D which do not contain branching components of D arebalanced. A normal pair ( X, D ) is balanced if D is balanced. KAROL PALKA
The word balanced stands here for the property that on one hand wedo not allow non-branching ( − b -curves with positive b . The following operation isresponsible for non-uniqueness of balanced completions of a given surface. Definition 3.2.
Let (
X, D ) be a normal pair. Let L be a 0-curve which isa non-branching component of D . Make a blowup of a point c ∈ L , suchthat c ∈ L ∩ ( D − L ) if L · ( D − L ) = 2 and contract the proper transform of L . The resulting pair ( X ′ , D ′ ), where D ′ is the reduced direct image of thetotal transform of D is called an elementary transform of ( X, D ). The point c ∈ L is the center of the transformation. A composition of elementarytransformations of D and its subsequent elementary transforms is called aflow inside D .For example, taking T = [0 , , a , . . . , a n ] one can obtain by a flow exactlythe chains of type [0 , b, a , . . . , a n ], [ a , . . . , a k − , a k − b, , b, a k +1 , . . . , a n ] or[ a , . . . , a n , b,
0] where 1 ≤ k ≤ n and b ∈ Z . In particular, it is easy to seewhen two rational chains differ by a flow. The following result is the mainproperty of balanced completions, it follows from [FKZ07, 3.36]. Proposition 3.3.
Any normal surface which admits a normal completionwith a forest as a boundary has a balanced completion. Two such completionsdiffer by a flow inside the boundary. In particular, all balanced boundariesof a given surface are isomorphic as curves.
Of course, two balanced boundaries of a given surface are in general non-isomorphic as weighted curves (weights are here the self-intersections of theircomponents). One introduces normalizing conditions to deal with this. Fol-lowing [FKZ07] the components of the divisor which remain after subtractingall non-rational and all branching components of an snc-divisor D will becalled the segments of D . An snc-divisor is standard if and only if any ofits connected components is either [1] or has all segments of type [0], [0 , , , , , a , . . . , a n ] or [ a , . . . , a n ] with a , . . . , a n ≥
2. It follows fromthe example above and from the Hodge index theorem that each balancedchain can be carried to a standard form by a flow.
Example 3.4.
The affine plane C has a balanced completion ( F , T ), where F is the second Hirzebruch surface and T = [0 , C is of type [0 , a ] for some a ≥
2, it is standardif and only if a = 0. Note that the isomorphism type of the boundary canchange when we admit non-balanced completions, for example the boundaryof C embedded in P is [ − T = [0 , , a , . . . , a n ] with some a , . . . , a n ≥ ECENT PROGRESS IN THE GEOMETRY OF Q -ACYCLIC SURFACES 5 the second is the reversion of the first one, T rev = [0 , , a n , . . . , a ]. Givenone weighted dual graph of some standard boundary of a given surface eachother can be easily described, which gives a refined method of distinguishingbetween many open surfaces.4. Basic properties
Let R be a ring. An R -homology plane is a normal surface X with H ∗ ( X, R ) ∼ = R . (This is a bit nonstandard, as usually R -homology planesare defined as smooth by definition). We say that X is a homotopy plane if π i ( X ) ∼ = 0 for i >
0. A Z -homology plane with trivial π is a homotopyplane by the theorem of Hurewicz. By a theorem of Whitehead homotopyplanes are contractible. Example 4.1.
Let
G < GL (2 , C ) be a finite subgroup without pseudore-flections. Then C /G is a singular logarithmic homotopy plane for whichthe smooth locus has negative Kodaira dimension. Indeed, the linear con-traction of C to 0 ∈ C descends to a contraction of the quotient and thequotient morphism C − { } → C − { } /G is ´etale, so κ ( C − { } /G ) = κ ( C − { } ) = −∞ .We now fix the notation for the rest of the paper. Let ǫ : S → S ′ bea minimal snc-resolution of a singular Q -homology plane S ′ . Denote theexceptional divisor of ǫ by b E . Let ( S, D ) be a smooth completion of S andlet S be the smooth locus of S ′ . Since S ′ is normal, its singular locusconsists of a finite number of points. For topological spaces A ⊆ X we write H i ( X, A ) for H i ( X, A, Q ) and b i ( X, A ) for dim H i ( X, A ).Logarithmic Q -homology planes are known to be affine by an argumentof Fujita (cf. [Fuj82, 2.4]). They are also known to be rational due toGurjar-Pradeep-Shastri [PS97], [GPS97], [GP99]. On the other hand, thefollowing example shows that non-logarithmic Q -homology planes can benon-rational. Example 4.2.
Let C ⊆ P n be a projectively normal embedding of a smoothprojective curve. Then the affine cone over C is normal and contractible.It has a standard cylinder resolution, for which the exceptional divisor isisomorphic to C . In case C is not rational, the cone is non-rational andnon-logarithmic.In general we have the following result (cf. [Pal09, 3.2, 3.4]). Theorem 4.3.
Every singular Q -homology plane is affine and birationallyequivalent to a product B × C for some curve B . As for the affiness we note that the above mentioned argument of Fujitaworks if one assumes only the inclusion ι : D ∪ b E → S induces an isomorphism H ( ι ) : H ( D ∪ b E ) → H ( S ). One can prove that this condition is alwayssatisfied. If b E is a rational forest this can be seen as follows. First notethat H i ( S, D ∪ b E ) ∼ = H − i ( S ) by the Lefschetz duality. One can prove that KAROL PALKA b ( S ) = b ( S ) = b ( b E ), so if b E is a rational forest then b ( S, D ∪ b E ) = b ( S, D ∪ b E ) = 0 and H ( ι ) is an isomorphism. For b ( b E ) = 0 the argumentis more complicated. For the rest we refer to loc. cit. Corollary 4.4. If S ′ is a singular Q -homology plane then its boundary isconnected and the homology groups H i ( S ′ , Z ) vanish for i ≥ .Proof. Since S ′ is affine, by [Kar77] its boundary is connected and S ′ ishomotopy equivalent to a CW-complex of real dimension at most two, hence H ( S ′ , Z ) is torsionfree and H ( S ′ , Z ) = H ( S ′ , Z ) = 0. Since b ( S ′ ) = 0, weget H ( S ′ , Z ) = 0. (cid:3) The rationality of smooth Q -homology planes has strong consequences forvector bundles. Theorem 4.5.
Let S ′ be a smooth Q -homology plane. Then any vector bun-dle over S ′ is a sum of a line bundle and a trivial vector bundle. Moreover, Pic S ′ ∼ = H ( X, Z ) , so if S ′ is a Z -homology plane then all vector bundlesover S ′ are trivial.Proof. The first part of the theorem is a result of Murthy [Mur69] validfor smooth affine surfaces birational to a product of a curve with an line.Since S ′ is rational, we have Pic S ′ ∼ = H ( S ′ , Z ). The groups H ( S ′ , Z )and H ( S ′ , Z ) are finite, so by the universal coefficient theorem H ( S ′ , Z ) ∼ = H ( S ′ , Z ). (cid:3) We note that the question if a vector bundle over any smooth contractiblethreefold is necessarily trivial is open.5.
Case κ ( S ) = −∞ and moduli Let us start with recalling some known results in case S has negativeKodaira dimension. This is a strong assumption, which in particular forces S ′ to be logarithmic (and hence rational). This can be seen as follows.Suppose S is affine-ruled. The affine ruling extends to a P -ruling ofsome smooth completion ( S, D + b E ) of S , where b E is the exceptional di-visor of some (not necessarily minimal) snc-resolution of singularities of S ′ .The unique section of this extension contained in the boundary is in factcontained in D . Indeed, if it is contained in b E , then D is vertical for thisruling (i.e. all its components intersect trivially with fibers), so since thehomology classes of components of D + b E generate H ( S ), the intersectionform on S is semi-negative definite, contradicting the Hodge index theorem.Thus the affine ruling of S extends to an affine ruling of S ′ , so S ′ has atmost cyclic singularities by [Miy81, Theorem 1]. There is no bound on thenumber of singularities, H ( S ′ , Z ) can be any finite abelian group. The ex-tension has a unique fiber contained in the boundary (which can be assumedto be smooth) and each singular fiber contains a unique component not con-tained in D ∪ b E (see [MS91a, § §
2] for more details). Note that from 1.2
ECENT PROGRESS IN THE GEOMETRY OF Q -ACYCLIC SURFACES 7 loc. cit. if follows that C is the only smooth Z -homology plane of negativeKodaira dimension (see [Fuj79] for the first proof of this) and this charac-terization implies the positive solution of the two-dimensional CancellationConjecture.Now if S is not affine-ruled (this can happen only if S ′ is singular) then itfollows from an important structure theorem by Miyanishi-Tsunoda [MT84]that it contains an open subset U with a very special C ∗ -ruling called a Platonic fibration . In fact one shows that S = U (cf. [KR07, 3.1]), whichimplies that S ′ ∼ = C /G for a finite small non-cyclic subgroup G < GL (2 , C ).It is well known that there are only finitely many fake projective planes ,i.e. smooth projective surfaces with Betti numbers of P (cf. [BHPVdV04,V.1]). This is clearly not the case for Q -homology planes (the name fakeaffine planes is not used). Moreover, there are arbitrarily high dimensionalnontrivial families of Q -homology planes with a common weighted bound-ary. The following example of a family of singular Q -homology planes is amodification of a similar example in [FZ94, 4.16]. We sketch the arguments. Figure 1.
Singular fibers in the example 5.1
Example 5.1.
Let p : F → B ∼ = P be the P -ruling of the first Hirze-bruch surface and let N be a positive integer. Choose N + 3 distinct points x ∞ , x , x , . . . , x N +2 on the negative section. Blow up successively over each x i so to produce singular fibers with reductions F ∞ = [0], F = [3 , , , F = [2 , , F ′ i = [2 , ,
2] for i = 2 , . . . , N + 2 lying respectively over x ∞ , x , . . . , x N +2 . Denote the resulting surface by V and the proper trans-form of the negative section by D h . Note there are chains b E = [2 ,
2] and b E = [2] contained in F and F respectively, where we can assume that b E , b E do not intersect D h . For each i ∈ { , . . . , N } choose a point y i on the( − D i of F ′ i and blow up once. For i ∈ { , . . . , N } denote the unique( − F i by C i . Let S y , where y = ( y , . . . , y N +2 ) be the resultingsurface and put D = F ∞ + D h +( F − C − b E )+( F − C − b E )+ P Ni =3 ( F i − C i ).One checks that the surface S ′ y obtained by the contraction of b E and b E on S − D is a singular Q -homology plane. Clearly, the above family S ′ y is N -dimensional. Now if S ′ y ∼ = S ′ z then the isomorphism lifts to S − D andthen, since F ∞ is the only 0-curve in D , extends to S y − F ∞ ∼ = S z − F ∞ by3.3, which in turn descends to an automorphism of U = F − F ∞ − D h ∼ = C fixing fibers. However, if x, y are respectively the horizontal and verticalcoordinate on U , each automorphism of U fixing fibers can be written as( x, y ) → ( x, λy + P ( x )) for some P [ x ] ∈ C [ x ] and its lifting to V acts by λ on D i in some coordinates on D i (the multiplicity of D i in the fiber is 2).Thus if we consider an ( N − y ∈ D then λ = 1, so the mentioned action on each D i is trivial, hence y = z anddifferent members of this subfamily are non-isomorphic.6. Logarithmic Bogomolov-Miyaoka-Yau
In studies on singular Q -homology planes we use often the logarithmicversion of the Bogomolov-Miyaoka-Yau inequality proved by Kobayashi (cf.[Kob90]). Usually this inequality is stated for surfaces of general type interms of the so-called strongly minimal model . This is not necessary andin fact the inequality works for surfaces of non-negative Kodaira dimension.For example, the following lemma has been proved in [Pal10, 2.5] as aneasy corollary from an inequality of Bogomolov-Miyaoka-Yau type provedby Langer (cf. [Lan03]), which in particular generalizes both the inequalityof Kobayashi and another inequality of Miyaoka [Miy84]. For the notion ofthe Zariski decomposition and of the bark Bk D of an effective snc-divisor D see [Miy01, § D = D − Bk D . Lemma 6.1.
Let ( X, D ) be a smooth pair with κ ( K X + D ) ≥ . Then:(i) χ ( X − D ) + 14 (( K X + D ) − ) ≥ ( K X + D ) . (ii) For each connected component of D , which is a connected componentof Bk D (hence contractible to a quotient singularity) denote by G P thelocal fundamental group of the respective singular point P . Then χ ( X − D ) + X P | G P | ≥
13 ( K X + D ) . We now illustrate the usefulness of the second inequality in our context.
Corollary 6.2.
Let S be the smooth locus of a singular Q -homology plane S ′ and let b E , . . . , b E q be the connected components of the exceptional divisorof the snc-minimal resolution.(i) If κ ( S ) = 2 then S ′ is logarithmic and q = 1 .(ii) If κ ( S ) = 0 or then either q = 1 , or q = 2 and b E = b E = [2] .Proof. Let ( S m , D m ) be the almost minimal model of ( S, D + b E ). Since S ′ is affine, S m − D m is isomorphic to an open subset of S satisfying χ ( S m − ECENT PROGRESS IN THE GEOMETRY OF Q -ACYCLIC SURFACES 9 D m ) ≤ χ ( S ) = 1 − q . Let Q be the set of singular points which havebeen created by contracting connected components of D as in 6.1(ii). Since Q ≤ q , the above inequality gives13 (( K S m + D m ) + ) ≤ χ ( S m − D m ) + X P ∈ Q | G P | ≤ − q + Q ≤ − q . Now if κ ( S ) = 2 then (( K S m + D m ) + ) > q = 1 and0 < P P ∈ Q | G P | , so there is a unique singular point on S ′ and it is of quotienttype. If q > κ ( S ) = 0 , K S m + D m ) + ) = 0 and we get q = 2and 1 = 1 / | G P | + 1 / | G P | , so | G P | = | G P | = 2. (cid:3) Exceptional Q -homology planes The structure theorem for Q -homology planes with smooth locus of non-general type is based on general structure theorems for open surfaces. Definition 7.1. A Q -homology plane for which the smooth locus is neitherof non-general type, nor C - or C ∗ -ruled is exceptional .Now the mentioned structure theorems lead to the fact that for excep-tional S ′ one has κ ( S ) = 0. It was proved by Fujita (cf. [Fuj82, 8.64])that each exceptional smooth Q -homology plane is up to isomorphism oneof three surfaces called Y { , , } , Y { , , } and Y { , , } (the Fujita’s sur-faces of type H [ k, − k ] with k ≥ C ∗ -ruled). The snc-minimal boundaryof Y { a, b, c } is a rational fork (a tree with one branching component) and itsthree maximal twigs have discriminants equal to a, b, c respectively. A sin-gular exceptional Q -homology plane having a boundary with this propertywill be denoted by SY { a, b, c } . Together with the description of exceptionalsingular Q -homology planes in [Pal10] we have the following theorem (to beprecise one still needs to prove that smooth Y { a, b, c } ’s are not C ∗ -ruled,but this can be done as in loc. cit.). Theorem 7.2. If S ′ is a Q -homology plane with smooth locus S of non-general type then S is affine-ruled or C ∗ -ruled or S ′ is up to isomor-phism one of five exceptional Q -homology planes having smooth locus ofKodaira dimension zero: smooth Y { , , } , Y { , , } , Y { , , } and sin-gular SY { , , } , SY { , , } . For the last two surfaces κ ( S ′ ) = 0 and thesingular locus consists of a unique point of Dynkin type A and A , respec-tively. None of the exceptional surfaces is a Z -homology plane. The maximaltwigs of their snc-minimal boundaries consist of ( − Q -homology planes. Namely, Y { , , } and Y { , , } have automorphismgroups Z and Z respectively (the automorphism group of Y { , , } istrivial) and the actions have unique fixed points. The quotients are two non-isomorphic Q -homology planes with smooth loci of Kodaira dimension zero. Suppose, say, the quotient S ′ = Y { , , } / Aut Y { , , } is not exceptional.Then its smooth locus S is C ∗ -ruled, so the Stein factorization of the pull-back of this C ∗ -ruling gives a C ∗ -ruling of the complement of the fixed pointof Y { , , } . Since Y { , , } is exceptional, the closures of the fibers meetin the fixed point, hence the closures of the fibers of the C ∗ -ruling of S meet in the singular point of S ′ . Thus the last C ∗ -ruling does not extend toa ruling of S ′ . Since the singularity is cyclic, it follows from 5.4 [Pal09] that κ ( S ) = −∞ , a contradiction. (One can also get a contradiction with thefact that Y { , , } does not contain infinitely many contractible curves, cf.10.1).We recall here the construction of SY { , , } , mainly because of its beau-tiful connection with classical geometry. A projective configuration of type ( a c , b d ) is an arrangement of b lines in a projective space and a points onthese lines, such that each point belongs to c lines and each line contains d points. Clearly, ac = bd for such a configuration. Figure 2.
Singular Y { } , dual Hesse configuration. Example 7.3.
Up to a projective automorphism there exists a unique pro-jective configuration H of type (12 , ) (the uniqueness is easy to show us-ing information on the automorphism group of the configuration which wegive below). The dual configuration H ∗ is the famous Hesse configuration(9 , ) of flexes of an elliptic curve. It is known that Aut ( H ) ∼ = Aut ( H ∗ )has order 216 and is isomorphic to the group of special affine transformationsof F , i.e. F ⋊ SL (2 , F ) (cf. [Dol04, § Q , Q , Q of H none two of which lie on a common line of H and choose some line U incident to Q . By taking the dual choice in H we check using linear algebrathat Aut ( H ) acts transitively on the set of such choices and the stabilizerΓ < Aut ( H ) has order three. For i = 1 , T ,i , T ,i , T ,i be the lines inci-dent to Q i and let E , E , U be the lines incident to Q . Blow up once in eachof eight points not incident to U . Let S be the resulting complete surface.We denote divisors and their proper transforms by the same letters. Let B be ECENT PROGRESS IN THE GEOMETRY OF Q -ACYCLIC SURFACES 11 the exceptional curve over Q , put D = B + T , + T , + T , + T , + T , + T , and b E = E + E . Clearly, all components of D − B + b E are ( − D and b E are disjoint and D is a fork. Put S ′ = ( S − D ) / b E . One can showthat Aut ( S ′ ) ∼ = Γ.To see that S ′ is Q -acyclic note that b ( S ) = 0 and b ( S ) = b ( D ∪ b E ) = 9.Now since d ( D + b E ) = 0, the natural morphism H ( D ∪ b E ) → H ( S ) is anisomorphism. The homology exact sequence of the pair ( S, D ) and Lefschetzduality give b ( S ) = b ( S ) = b ( S ) = 0 and b ( S ) = b E ( S = S \ D ). Weknow from the above that H ( b E ) → H ( S ) is a monomorphism, so thehomology exact sequence of the pair ( S, b E ) gives that S ′ is Q -acyclic.We check easily that in both cases K S + D = K S + D + b E intersectstrivially with all components of D + b E , hence κ ( S ) = κ ( S ′ ) = 0. See [Pal10, §
5] for an explicit realization of H and for a proof that the constructed Q -homology plane does not admit a C ∗ -ruling.8. Non-logarithmic Q -homology planes Let us start with a generalization of example 4.2.
Example 8.1.
Let U be an affine cone over a projectively normal curveand let G be a finite group acting on it so that the vertex of the cone is theunique fixed point, the action is free on its complement and recpects linesof the cone. Then the quotient S ′ = U/G is a normal contractible surface.What is surprising is that all non-logarithmic Q -homology planes arise inthe above way. Namely, we have the following theorem (see [Pal09, 5.8]). Theorem 8.2.
Every singular Q -homology plane containing a non-quotientsingularity is a quotient of an affine cone over a smooth projective curveby an action of a finite group which is free off the vertex of the cone andrespects the set of lines through the vertex. In particular, it is contractible,has negative Kodaira dimension, has a unique singular point and its smoothlocus is C ∗ -ruled.Proof. (sketch) By 6.2 and 7.2 the smooth locus of a non-logarithmic Q -homology plane S ′ is C - or C ∗ -ruled. By the results of section 5, κ ( S ) ≥ S is C ∗ -ruled. One can show (cf. 3.6 loc. cit.) that if thisruling extends to a C ∗ -ruling of S ′ then S ′ is necessarily logarithmic, so wecan further assume that this is not the case. This means that there is a P -ruling p : S → P of some completion ( S, D + b E ) of S as before, suchthat for a general fiber f we have f · D = f · b E = 1. We can assume that thecompletion is p -minimal. Then each singular fiber of p is a so-called columnarfiber , which means that it is a rational snc-chain of discriminant zero, itscomponents have negative self-intersections, it contains a unique ( − C i which is also a unique S -component of the fiber (i.e. a component notcontained in D ∪ b E ) and it is intersected by the horizontal components of D and b E in tips. It follows that D and b E contain unique branching components D h and b E h . Write the singular fibers as D i + C i + E i , where D i ⊆ D and E i ⊆ b E (see Fig. 3). Contracting the singular fibers we get a P -bundleover a complete curve, so all non-logarithmic singular Q -homology planescan be reconstructed starting from such a bundle by producing columnarsingular fibers, taking out D and contracting b E . The mentioned bundleadmits a usual C ∗ -action fixing pointwise the images of D h and b E h . Thisaction induces a C ∗ -action on S ′ with the singular point as the unique fixedpoint. Thus by [Pin77, 1.1] S ′ is a quotient of an affine cone over a smoothprojective curve by an action of a finite group. In fact without knowingthis global description the contractibility follows also from [Fuj82, 5.9, 4.19]and the Whitehead theorem, as one can show that π ( b E h ) → π ( S ) is anisomorphism and π ( S ) → π ( S ′ ) is an epimorphism. Since S = S − D isaffine-ruled, we get κ ( S ′ ) = −∞ . (cid:3) Figure 3.
Construction of non-logarithmic Q -homology planes.We see that one can obtain any non-logarithmic Q -homology plane bycreating its snc-minimal completion S of the resolution as in the proof aboveby starting with a non-trivial P -bundle over some smooth complete curve B and by blowing up some number n of columnar fibers. Then exactly one oftwo forks, say b E , separated by vertical ( − b E h ∼ = B ∼ = P . Although b E is a rational tree, thesingularity does not have to be a rational singularity. Indeed, using Artin’scriterion one can show that if b E h + n ≤ S ′ has a rational and if b E h + n ≥ b E h + n = 1is more subtle (cf. [Pin77, 5.8]). ECENT PROGRESS IN THE GEOMETRY OF Q -ACYCLIC SURFACES 13 C ∗ -ruled Q -homology planes By the results described in previous sections the classification of Q -homo-logy planes of non-general type reduces now to the classification of loga-rithmic (and hence rational) Q -homology planes which are C ∗ -ruled, or inother words, for which the smooth locus is C ∗ -ruled and the ruling extendsto a C ∗ -ruling of S ′ . Note that the existence of a C ∗ -ruling implies that κ ( S ) = 2 by the ’easy addition theorem’ (cf. [Miy01, 2.1.5]). This casewas analyzed in [MS91a], where one can find a description of singular fibersand the computation of κ ( S ) and H ( S ′ , Z ) in terms of these fibers. Wenote here that in [Pal09, 6.8] we have redone some incorrect computationsof κ ( S ) from loc. cit. (identified then with the Kodaira dimension of S ′ )and we have computed κ ( S ′ ) too. We discuss two issues here.First, it is practically useful to know when a C ( n ∗ ) -ruled surface is a Q -homology plane. To formulate a criterion we need to recall the definition ofsome numbers characterizing rational rulings. Having a fixed P -ruling of asmooth complete surface X and a reduced divisor T we defineΣ X − T = X F * T ( σ ( F ) − , where σ ( F ) is the number of ( X − T )-components (i.e. irreducible compo-nents not contained in T ) of a fiber F (cf. [Fuj82, 4.16]). T h is the horizontalpart of T , which consists of components of T intersecting nontrivially with ageneral fiber. If T h = 0 then T is vertical . The numbers h and ν are definedrespectively as T h and as the number of fibers contained in T . Lemma 9.1.
Let ( S, T ) be a smooth pair and let p : S → P be a P -ruling.Assume the following conditions are satisfied:(i) there exists a unique connected component D of T which is not vertical,(ii) D is a rational tree,(iii) Σ S − T = h + ν − ,(iv) d ( D ) = 0 .Then the surface S ′ defined as the image of S − D after contraction ofconnected components of T − D to points is a rational Q -homology plane and p induces a rational ruling of S ′ . Conversely, if p ′ : S ′ → B is a rationalruling of a rational Q -homology plane S ′ (not necessarily singular) then anycompletion ( S, T, p ) of the restriction of p ′ to the smooth locus of S ′ has theabove properties. The conditions (iii)-(iv) are equivalent to the fact that H ( D ∪ b E ) → H ( S ) is an isomorphism, similar criteria were used by many authors. Whatis important, in case of a C ∗ -ruling the most problematic condition (iv) canbe replaced by an easier and more geometric condition (cf. [Pal09, 6.1]).Second, there is a question about the uniqueness of a C ∗ -ruling of S ′ . Letus assume S ′ is singular. In case κ ( S ) = 1 it is easy to prove that there isa unique C ∗ -ruling of S ′ and it is induced by the C ∗ -ruling of S given by the linear system of some multiple of the logarithmic canonical divisor of S . In case κ ( S ) = 0 generically there are two C ∗ -rulings of S ′ , but theremay be zero (exceptional Q -homology planes), one or three as well (cf. 6.12loc. cit.). We will see that this information is important for example whencomputing the number of contractible curves on S ′ .As for the completions of S ′ , a non-affine-ruled S ′ has a unique balancedcompletion unless it admits an untwisted C ∗ -ruling with base C (untwistedmeans that for some completion of the surface the ruling extends to a P -ruling for which the horizontal part of the boundary consists of two ir-reducible components). In the latter case there are exactly two balancedcompletions (cf. 6.11 loc. cit.).10. Contractible curves
It is well known that the logarithmic Bogomolov-Miyaoka-Yau inequalityimposes restrictions on the number of contractible curves on S ′ . Accordingto the author’s knowledge this number is known except the cases when S ′ is non-logarithmic or when S ′ is singular and κ ( S ) = 0 (see [Za˘ı87], [Za˘ı91][GM92], [GP95]). In the non-logarithmic case it is easily seen to be infinityby 8.2 and, as we sketch below, in the last case it can be deduced from theknowledge on the number (and types) of C ∗ -rulings of S [Pal09, 6.12]. Thefinal result is as follows. Theorem 10.1.
Let l be the number of contractible curves on a Q -homologyplane S ′ . Let S be the smooth locus of S ′ . Then:(i) if κ ( S ) = 2 then l = 0 ,(ii) if S ′ is exceptional (hence κ ( S ′ ) = κ ( S ) = 0 ) then l = 0 ,(iii) if S ′ is non-logarithmic (hence κ ( S ′ ) = −∞ , κ ( S ) = 0 , ) then l = ∞ ,(iv) if κ ( S ) = −∞ then l = ∞ ,(v) in other cases ( S ′ is C ∗ -ruled and κ ( S ) = 0 , ), l = 1 or .Proof. (sketch) If κ ( S ) = −∞ then S ′ is affine-ruled or isomorphic to C /G for some G < GL (2 , C ), so l = ∞ . If S ′ is non-logarithmic then by 8.2 l = ∞ .We can therefore assume that κ ( S ) ≥ S ′ is logarithmic. Suppose S ′ contains a contractible curve L . It follows from the logarithmic Bogomolov-Miyaoka-Yau inequality that κ ( S − L ) ≤ S ′ isrational, P ic ( S ) = Coker(Pic( D + b E ) → Pic S ) is torsion, so the class of L in Pic( S ) is torsion. Then there is a morphism f : S − L → C ∗ and takingits Stein factorization one gets a C ∗ -ruling of S − L , which (as κ ( S ) ≥ C ∗ -ruling of S . Since S ′ is logarithmic, each C ∗ -ruling of S extends in turn to a C ∗ -ruling of S ′ . Therefore, any contractible curve on S ′ is vertical for some C ∗ -ruling of S ′ . In particular, if l > S ′ , and hence S , is necessarily C ∗ -ruled, hence S ′ cannot be exceptional. The analysis offibers (Suzuki’s formula) leads to the corollary that there exist one or twocontractible vertical curves for a given C ∗ -ruling. Now if κ ( S ) = 1 then thisruling is unique (given by a multiple of the logarithmic canonical divisor of ECENT PROGRESS IN THE GEOMETRY OF Q -ACYCLIC SURFACES 15 S ), hence l = 1 ,
2. Consider now the case κ ( S ) = 0. Here the problem ismore difficult, as there may be more C ∗ -rulings of S ′ . In case S ′ is smoothit was shown in [GP95] that l = 1. In case S ′ is singular we have computedthe number and types of possible C ∗ -rulings of S in [Pal09, 6.12] (one cando the same if S ′ is smooth in a similar way). Since this number is finite wesee that l is finite and nonzero. Looking more closely at the proof one canshow that l ≤ (cid:3) Example 10.2.
Let π : S → C be the restriction of the projection ( x, y, z ) → ( x, y ) to the surface S = { ( x, y, z ) ∈ C : z n = f ( x, y ) } , where n isa positive natural number. Then π is a branched cover with the curve C = { ( x, y ) ∈ C : f ( x, y ) = 0 } as the branch locus. Suppose S is asmooth Q -homology plane. Then C is smooth and we have 1 − χ ( C ) =1 − χ ( π − ( C )) = 1 − χ ( S ) + χ ( S − π − ( C )) = nχ ( C − C ) = n (1 − χ ( C )),hence χ ( C ) = 1. If U is any smooth affine curve then it is non-complete, soits Euler characteristic is smaller than the Euler characteristic of the smoothcompletion U , hence χ ( U ) ≤ − g ( U ), where g is the genus. In case U isirreducible it follows that χ ( U ) ≥ U ∼ = C or U ∼ = C ∗ . In particular,some component of C ∼ = π − ( C ) is an affine line, so by 10.1 S is not of gen-eral type. In fact either κ ( S ) = −∞ or S is C ∗ -ruled. Smooth Q -homologyplanes of this kind have been classified in [Mah09].11. Smooth Q -homology planes of general type Let S ′ be a smooth Q -homology plane of general type, i.e. κ ( S ′ ) = 2. Weknow already that S ′ is rational. By 10.1 S ′ contains no contractible curves,which implies that its snc-minimal completion ( S, D ) is almost minimal. Wecan assume that this completion is balanced. Since S ′ is neither C - nor C ∗ -ruled, D contains no non-branching 0-curves, so any flow inside D is trivial.It follows that the balanced completion of S ′ is unique. As for now there isno classification, but there are some partial results.The first example of such a surface which was shown to be non-isomorphicto C is the famous example of Ramanujam ([Ram71]). We know nowinfinitely many examples of this kind. One method of construction of Q -homology planes of general type is to use C ( n ∗ ) -rulings with n ≥
2. For n = 2 this was done in [MS91b]. Another method was used by tom Dieckand Petrie. Definition 11.1.
Let (
S, D ) be a completion of S ′ for which there existsa birational morphism f : ( S, D ) → ( P , f ∗ D ). Then f ∗ D is called a planedivisor of S ′ . If f ∗ D is a sum of lines then it is a linear plane divisor andwe say that S ′ comes from the line arrangement f ∗ D .In [tD90a] it was noticed that an inequality of Hirzebruch bounds thenumber of types of possible linear plane divisors for smooth Q -homologyplanes, a list of these divisors has been given. In [tDP93] a general algo-rithm for recovering smooth Q -homology planes (in fact countable series of them) starting from a given rational divisor on a minimal rational completesurface is described. The conjecture that all smooth Q -homology planes ofgeneral type have linear plane divisors is not true by an example of tomDieck (cf. [tD90b]). Tom Dieck’s smooth Z -homology plane has a nontrivialautomorphism group (it is necessarily finite, as S ′ is of general type), so isalso a counterexample to the earlier conjecture of Petrie [Pet89]. Example 11.2.
Let C ⊆ P be an irreducible curve. Recall that a singularpoint p ∈ C is a cusp if C is locally irreducible at p . Since C ֒ → P inducesa monomorphism on H ( · , Q ), by the Lefschetz duality and by the longexact sequence of the pair ( P , C ) the Betti numbers of S C = P \ C are b ( S C ) = b ( C ) and b i ( S C ) = 0 for i = 1 and i >
2. Assume that C isrational and cuspidal. Then S C is a smooth Q -homology plane (this is infact equivalent). By [Wak78] if C has two cusps then κ ( S C ) ≥ S C is of general type. The literature onplane cuspidal curves is rich, see for example [FZ96], [FdBLMHN07] andreferences there.We now list some conjectural properties of smooth Q -homology planes. Conjecture.
Let S ′ be a smooth Q -homology plane of general type and let( S, D ) be its minimal smooth completion.(A) S ′ has a plane divisor consisting of lines and conics.(B) S ′ admits a C (3 ∗ ) -ruling.(C) ( K S + D ) = −
2, or equivalently K S · ( K S + D ) = 0.(D) S ′ is rigid and has unobstructed deformations.(E) The set of all possible Eisenbud-Neumann diagrams for smooth Q -homology planes is finite.In [tD92] conjectures (A)-(C) have been stated and verified for all knownsmooth Z -homology planes. Sugie [Sug99] analyzed C ( n ∗ ) -rulings on smooth Q -homology planes and classified possible singular fibers for n = 2. Flennerand Zaidenberg [FZ94, 6.12] have shown that part (D), which implies (C),holds for smooth Q -homology planes of general type having linear planedivisors. See [Za˘ı05] for (D), (E) and related conjectures.Recently, the following result was proved [GKMR10]. Theorem 11.3.
Let S ′ be a smooth homotopy plane of general type. Then Aut S ′ is cyclic, its action on S ′ has a unique fixed point and is free off thispoint. The situation for Aut S ′ of a Q -homology plane is not known.12. Smooth locus of general type
Let S ′ be a singular Q -homology plane with smooth locus S of generaltype, i.e. κ ( S ) = 2. By 6.2 S ′ has a unique singular point and this point isof quotient type. This means that b E is either a chain if this point is a cyclic ECENT PROGRESS IN THE GEOMETRY OF Q -ACYCLIC SURFACES 17 singularity or a negative definite rational fork if not. Since S ′ is logarithmic,it is rational. By 10.1 it contains no contractible curves, which implies thatthe snc-minimal completion ( S, D + b E ) of S is almost minimal. Arguingas above we see that in fact this snc-minimal completion is unique and doesnot contain non-branching 0-curves. Again, there is no classification, butthere are some partial results.Note that due to the existence of the transfer homomorphism for branchedcoverings (cf. [Smi83]) the quotient of a smooth Q -homology plane of generaltype by its automorphism group is a Q -homology plane (with smooth locusof general type).A priori there is no restriction on the Kodaira dimension of S ′ . However,refining the methods of Koras-Russell [KR07] M. Koras and the author haveobtained the following theorem (this is also a part of the thesis of the authorwritten under the supervision of M. Koras, cf. [PK10]). Theorem 12.1.
Singular Q -homology planes with smooth locus of generaltype have non-negative Kodaira dimension. The cases κ ( S ′ ) = 0 , κ ( S ′ ) = 2 have not beenanalyzed systematically as for now. Also the number of known examples ismuch smaller than in the case of smooth Q -homology planes. For example,similarly as in the smooth case, one could ask if S has a completion ( S, D + b E ) admitting a birational morphism f : S → P with f ∗ ( D + b E ) a sum oflines and conics. Even the case when f ∗ ( D + b E ) is a sum of lines has notbeen studied.The known examples of singular Z -homology planes of general type thelocal fundamental group of the singular point (which has order equal to d ( b E )) is cyclic of order not bigger than six. As for now the following resultin this direction has been proven ([GKMR10]): Theorem 12.2.
A singular Z -homology plane with smooth locus of generaltype has a cyclic quotient singularity. References [BHPVdV04] Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven,
Compact complex surfaces , second ed., Ergebnisse der Mathematik undihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics[Results in Mathematics and Related Areas. 3rd Series. A Series of ModernSurveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004.[Dai03] Daniel Daigle,
Classification of weighted graphs up to blowing-up andblowing-down , arXiv:math/0305029 (2003).[Dol04] Igor V. Dolgachev,
Abstract configurations in algebraic geometry ,The Fano Conference, Univ. Torino, Turin, 2004, pp. 423–462,(arXiv:math/0304258).[FdBLMHN07] J. Fern´andez de Bobadilla, I. Luengo, A. Melle-Hern´andez, andA. N´emethi,
On rational cuspidal plane curves, open surfaces and lo-cal singularities , Singularity theory, World Sci. Publ., Hackensack, NJ,arXiv:math/0604421, 2007, pp. 411–442. [FKZ07] Hubert Flenner, Shulim Kaliman, and Mikhail Zaidenberg,
Birationaltransformations of weighted graphs , Affine algebraic geometry, OsakaUniv. Press, Osaka, 2007, pp. 107–147.[Fuj79] Takao Fujita,
On Zariski problem , Proc. Japan Acad. Ser. A Math. Sci. (1979), no. 3, 106–110.[Fuj82] , On the topology of noncomplete algebraic surfaces , J. Fac. Sci.Univ. Tokyo Sect. IA Math. (1982), no. 3, 503–566.[FZ94] Hubert Flenner and Mikhail Zaidenberg, Q -acyclic surfaces and theirdeformations , Classification of algebraic varieties (L’Aquila, 1992), Con-temp. Math., vol. 162, Amer. Math. Soc., Providence, RI, 1994, pp. 143–208.[FZ96] , On a class of rational cuspidal plane curves , Manuscripta Math. (1996), no. 4, 439–459.[GKMR10] R. V. Gurjar, M. Koras, M. Miyanishi, and P. Russell, A homology planeof general type can have at most a cyclic quotient singularity , preprint, 62pages, 2010.[GM92] R. V. Gurjar and M. Miyanishi,
Affine lines on logarithmic Q -homologyplanes , Math. Ann. (1992), no. 3, 463–482.[GP95] R. V. Gurjar and A. J. Parameswaran, Affine lines on Q -homology planes ,J. Math. Kyoto Univ. (1995), no. 1, 63–77.[GP99] R. V. Gurjar and C. R. Pradeep, Q -homology planes are rational. III ,Osaka J. Math. (1999), no. 2, 259–335.[GPS97] R. V. Gurjar, C. R. Pradeep, and Anant. R. Shastri, On rationality oflogarithmic Q -homology planes. II , Osaka J. Math. (1997), no. 3, 725–743.[Gra62] Hans Grauert, ¨Uber Modifikationen und exzeptionelle analytische Mengen ,Math. Ann. (1962), 331–368.[Iit82] Shigeru Iitaka, Algebraic geometry , Graduate Texts in Mathematics,vol. 76, Springer-Verlag, New York, 1982, An introduction to birationalgeometry of algebraic varieties, North-Holland Mathematical Library, 24.[Kar77] K. K. Karˇcjauskas,
A generalized Lefschetz theorem , Funkcional. Anal. iPriloˇzen. (1977), no. 4, 80–81.[Kob90] Ryoichi Kobayashi, Uniformization of complex surfaces , K¨ahler metricand moduli spaces, Adv. Stud. Pure Math., vol. 18, Academic Press,Boston, MA, 1990, pp. 313–394.[KR07] Mariusz Koras and Peter Russell,
Contractible affine surfaces with quo-tient singularities , Transform. Groups (2007), no. 2, 293–340.[Lan03] Adrian Langer, Logarithmic orbifold Euler numbers of surfaces with ap-plications , Proc. London Math. Soc. (3) (2003), no. 2, 358–396.[Mah09] Alok Maharana, Q -homology planes as cyclic covers of A , J. Math. Soc.Japan (2009), no. 2, 393–425.[Miy81] Masayoshi Miyanishi, Singularities of normal affine surfaces containingcylinderlike open sets , J. Algebra (1981), no. 2, 268–275.[Miy84] Yoichi Miyaoka, The maximal number of quotient singularities on surfaceswith given numerical invariants , Math. Ann. (1984), no. 2, 159–171.[Miy01] Masayoshi Miyanishi,
Open algebraic surfaces , CRM Monograph Series,vol. 12, American Mathematical Society, Providence, RI, 2001.[Miy07] ,
Recent developments in affine algebraic geometry: from the per-sonal viewpoints of the author , Affine algebraic geometry, Osaka Univ.Press, Osaka, 2007, pp. 307–378.[MS91a] M. Miyanishi and T. Sugie,
Homology planes with quotient singularities ,J. Math. Kyoto Univ. (1991), no. 3, 755–788. ECENT PROGRESS IN THE GEOMETRY OF Q -ACYCLIC SURFACES 19 [MS91b] Masayoshi Miyanishi and Tohru Sugie, Q -homology planes with C ∗∗ -fibrations , Osaka J. Math. (1991), no. 1, 1–26.[MT84] Masayoshi Miyanishi and Shuichiro Tsunoda, Noncomplete algebraic sur-faces with logarithmic Kodaira dimension −∞ and with nonconnectedboundaries at infinity , Japan. J. Math. (N.S.) (1984), no. 2, 195–242.[Mur69] M. Pavaman Murthy, Vector bundles over affine surfaces birationallyequivalent to a ruled surface , Ann. of Math. (2) (1969), 242–253.[Pal09] Karol Palka, On the classification of singular Q -acyclic surfaces ,arXiv:0806.3110 (2009).[Pal10] , Exceptional singular Q -homology planes , arXiv:0909.0772, to ap-pear in Annales Inst. Fourier (2010).[Pet89] Ted Petrie, Algebraic automorphisms of smooth affine surfaces , Invent.Math. (1989), no. 2, 355–378.[Pin77] H. Pinkham, Normal surface singularities with C ∗ action , Math. Ann. (1977), no. 2, 183–193.[PK10] Karol Palka and Mariusz Koras, Singular Q -homology planes of negativekodaira dimension have smooth locus of non-general type , arXiv:1001.2256(2010).[PS97] C. R. Pradeep and Anant R. Shastri, On rationality of logarithmic Q -homology planes. I , Osaka J. Math. (1997), no. 2, 429–456.[Ram71] C. P. Ramanujam, A topological characterisation of the affine plane as analgebraic variety , Ann. of Math. (2) (1971), 69–88.[Smi83] Larry Smith, Transfer and ramified coverings , Math. Proc. CambridgePhilos. Soc. (1983), no. 3, 485–493.[Sug99] Toru Sugie, Singular fibers of homology planes with pencils of rationalcurves , Mem. Fac. Educ. Shiga Univ. III Nat. Sci. (1999), no. 49, 29–40(2000).[tD90a] Tammo tom Dieck,
Linear plane divisors of homology planes , J. Fac. Sci.Univ. Tokyo Sect. IA Math. (1990), no. 1, 33–69.[tD90b] , Symmetric homology planes , Math. Ann. (1990), no. 1-3,143–152.[tD92] ,
Optimal rational curves and homotopy planes , Bol. Soc. Mat.Mexicana (2) (1992), no. 1-2, 115–138, Papers in honor of Jos´e Adem(Spanish).[tDP93] Tammo tom Dieck and Ted Petrie, Homology planes and algebraic curves ,Osaka J. Math. (1993), no. 4, 855–886.[Wak78] Isao Wakabayashi, On the logarithmic Kodaira dimension of the comple-ment of a curve in P , Proc. Japan Acad. Ser. A Math. Sci. (1978),no. 6, 157–162.[Za˘ı87] M. G. Za˘ıdenberg, Isotrivial families of curves on affine surfaces, and thecharacterization of the affine plane , Izv. Akad. Nauk SSSR Ser. Mat. (1987), no. 3, 534–567, 688.[Za˘ı91] , Additions and corrections to the paper: “Isotrivial families ofcurves on affine surfaces, and the characterization of the affine plane”[Izv. Akad. Nauk SSSR Ser. Mat. (1987), no. 3, 534–567] , Izv. Akad.Nauk SSSR Ser. Mat. (1991), no. 2, 444–446.[Za˘ı99] M. Za˘ıdenberg, Exotic algebraic structures on affine spaces , Algebra iAnaliz (1999), no. 5, 3–73, arXiv:9801075.[Za˘ı05] Mikhail Za˘ıdenberg, Selected problems , arXiv:0501457 (2005).
Karol Palka: Department of Mathematics, McGill University, Montreal,QC, Canada
E-mail address ::