Classification of singular -homology planes. I. Structure and singularities.
A -homology plane is a normal complex algebraic surface having trivial rational homology. We obtain a structure theorem for -homology planes with smooth locus of non-general type. We show that if a -homology plane contains a non-quotient singularity then it is a quotient of an affine cone over a projective curve by an action of a finite group respecting the set of lines through the vertex. In particular, it is contractible, has negative Kodaira dimension and only one singular point. We describe minimal normal completions of such planes.
Key words and phrases:Acyclic surface, homology plane, Q-homology plane
2000 Mathematics Subject Classification:Primary: 14R05; Secondary: 14J17, 14J26
We work with algebraic varieties defined over .
1. Main results
A -homology plane is a normal surface with Betti numbers of , i.e. with for . As for every open surface, one of its basic invariants is the logarithmic Kodaira dimension (see [Iit82]), which takes values in . Smooth -homology planes of non-general type, i.e. having the Kodaira dimension smaller than two, have been classified, see [Miy01, §3.4] for summary and for what is known in the case of general type. In this and in the forthcoming paper (see [Pal12]) we obtain a classification of singular -homology planes with smooth locus of non-general type. A lot of attention has been given to understand these surfaces in special cases (see [MS91, GM92, PS97, DR04, KR07]), let us mention explicitly at least the role of the contractible ones in proving the linearizability of -actions on (see [KR99]). To our knowledge, in the available literature on this subject it is always assumed that the planes are logarithmic, by what is meant that each singular point is analytically of type for some finite subgroup (singularity of ’quotient type’). This is a strong assumption, in particular it implies rationality of the surface ([GPS97]), and one of our goals was to avoid it.
Recall that a -homology plane is exceptional if and only if it has smooth locus of non-general type, which is neither - nor -ruled. There exist exactly three exceptional smooth -homology planes (see [Fuj82, 8.64]). In [Pal11] we described two singular ones. We obtain the following structure theorem, part (2) strengthening 1.4 loc. cit.
Singular -homology planes are affine and birationally ruled.
A singular -homology plane with smooth locus of non-general type satisfies one of the following.
It is logarithmic and - or -ruled.
It is either non-logarithmic or isomorphic to for some finite small non-cyclic subgroup . Its smooth locus has a -ruling which does not extend to a ruling of the -homology plane.
It is isomorphic to one of two exceptional singular -homology planes. These planes have Kodaira dimension zero and the Kodaira dimension of the smooth loci zero. They are quotients of smooth exceptional -homology planes and contain unique singular points, which are cyclic singularities of Dynkin type and respectively.
Logarithmic -homology planes which admit a - or a -ruling have been studied in [MS91], in particular singular fibers of the rulings have been described. In the forthcoming, second part of this paper we will give a detailed classification of these planes. As we see from the above, non-logarithmic -homology planes are of type (b). We obtain the following more detailed description.
A singular -homology plane which contains a non-quotient singularity (in particular each which is a non-rational surface) is isomorphic to a quotient of an affine cone over a smooth projective curve by an action of a finite group acting freely off the vertex of the cone and respecting the set of lines through the vertex. Moreover:
it is contractible and has a unique singular point,
its smooth locus has a unique -ruling and the ruling does not extend to a ruling of the plane,
it has negative Kodaira dimension and the Kodaira dimension of its smooth locus equals or ,
its minimal normal completion is unique up to isomorphism and the boundary is a rational tree with a unique branching component.
We show also that for -homology planes as above the singularity may be, but not need to be, rational in the sense of Artin (cf. 4.8). The following result is of independent interest (for the proof see 3.3).
If a singular -homology plane has smooth locus of general type then it has a unique singular point and the point is a quotient singularity. If a -homology plane has more than one singular point then either it is affine-ruled and the singularities are cyclic or it has exactly two singular points, both of Dynkin type .
Our methods rely on the theory of open algebraic surfaces, for which [Miy01] is a basic reference. We now give a more detailed overview of the paper. We denote a singular -homology plane by and its smooth locus by . From preliminaries in section 2 the result 2.6 is worth mentioning, as it gives a criterion for contractibility of divisors on complete surfaces. In section 3 we prove basic topological and geometric results, whose simpler versions for logarithmic -homology planes were known before, see [MS91]. In absence of restriction on the type of singularities arguments get more complicated. Once we prove that the Neron-Severi group of the smooth locus is torsion, we apply Fujita’s argument to show that the affiness of is a consequence of -acyclicity (cf. 3.2). It was proved in [PS97] that a logarithmic singular -homology plane is rational. We complete this result by showing in 3.4 that in general is birationally ruled.
If is of non-general type and admits no - and no -rulings then the plane is called exceptional. By general structure theorems for open surfaces (cf. [Miy01, 2.2.1, 2.5.1, 2.6.1]) exceptional -homology planes have . Under the assumption that singularities are topologically rational we have proved in [Pal11] that there are exactly two such surfaces up to isomorphism. Here we show that the mentioned assumption can be omitted.
We begin section 4 by proving that if is non-logarithmic then it is of very special kind, namely there is a unique -ruling of and it does not extend to a ruling of . This implies (cf. 4.1). We analyze the -ruling and classify all non-logarithmic -homology planes in 4.5. To reconstruct them we use the contractibility criterion 2.6. We analyze the Kodaira dimension of the smooth locus and the singularity in 4.7 and 4.8. Finally we argue that a non-logarithmic admits a -action with a unique fixed point, which leads to an isomorphism with a quotient of an affine cone as in 1.2.
Acknowledgements. The paper contains results obtained during the graduate studies of the author at the University of Warsaw and their improvements obtained during his stay at the Polish Academy of Sciences and McGill University. The author thanks his thesis advisor prof. M. Koras for numerous discussions and for reading preliminary versions of the paper. He also thanks prof. P. Russell for useful comments.
- 1 Main results
- 2 Preliminaries
- 3 Topology and Singularities
- 4 Non-logarithmic
2.1. Divisors and pairs
We mostly follow the notational conventions and terminology of [Miy01], [Fuj82] and also of [Pal11, §1, §2]. Let be an snc-divisor on a smooth complete surface (hence projective by the theorem of Zariski) with distinct irreducible components . We write for a reduced divisor with the same support as and denote the branching number of by . A component is branching if . If contains a branching component then it is branched. The determinant of , where is the intersection matrix of , is denoted by , by definition. Considering as a topological subspace of a complex surface with its Euclidean topology it is easy to check that if is connected then there is a homotopy equivalence
where is a geometric realization of a dual graph of . In particular, .
If is a chain (i.e. it is reduced and its dual graph is linear) then writing it as a sum of irreducible components we always assume that for . If is a chain and some tip (a component with ), say , is fixed to be the first one then we distinguish between and . We write in case is a rational chain. If is a rational chain with for each we say that is admissible. Let be some fixed reduced snc-divisor which is not an admissible chain. A rational chain not containing branching components of and containing one of its tips is a twig of . In this situation we always assume that the tip of is the first component of . For any admissible (ordered) chain we define
Now and are defined as the sums of respective numbers computed for all maximal admissible twigs of . (A an admissible twig of is maximal if it is not contained in another admissible twig of with a bigger number of components.)
An snc-pair consists of a complete surface and a reduced snc-divisor contained in the smooth part of . We write for in this case. The pair is a normal pair (smooth pair) if is normal (resp. smooth). If is a normal surface then an embedding , where is a normal pair, is called a normal completion of . If is smooth then is smooth and is called a smooth completion of . We often identify with by and neglect in the notation. A morphism of two completions , is a morphism , such that .
Let be a birational morphism of normal pairs. We put , i.e. is the reduced total transform of . Assume . If is a blowup then we call it subdivisional (sprouting) for if its center belongs to two (one) components of . In general we say that is subdivisional for (and for ) if for any component of we have .
The exceptional locus of a birational morphism between two surfaces , denoted by , is defined as the locus of points in for which is not a local isomorphism. The canonical divisor of a complete surface is denoted by and the numerical equivalence of divisors by . For a divisor the arithmetic genus of is .
A -curve is a smooth rational curve with self-intersection . A divisor is snc-minimal if all its -curves are branching.
A birational morphism of surfaces is a connected modification if it is proper, is a smooth point on and contains a unique -curve. In case is a morphism of pairs , such that and , then we call it a connected modification over .
Note that since for a connected modification the exceptional locus contains a unique -curve, the modification can be decomposed into a sequence of blowdowns , such that for the center of belongs to the exceptional divisor of . A sequence of blowdowns (and its reversing sequence of blowups) whose composition is a connected modification will be called a connected sequence of blowdowns (blowups).
Let and be -divisors on a smooth complete surface, such that is negative definite and for each irreducible component of . Denote the integral part of a -divisor by .
If is effective then is effective.
If and is a -divisor then .
(i) We can assume that and are -divisors and is effective and nonzero. Write , where are distinct irreducible components of . Choose , such that the sum is the smallest possible among divisors , such that is effective. If for some then
by the assumptions. Hence contains some , a contradiction with the definition of . Thus is effective.
(ii) Let denote the fractional part of a -divisor , i.e. . Let be some effective divisor, such that . Then as -divisors. Since is effective by (i), the coefficient of each irreducible component of is bounded below by the coefficient of the same component in . Since is a -divisor and the coefficients of components in are fractional and positive, is effective. Moreover, being a -divisor is equal to , so the rational function giving the equivalence of and gives an equivalence of and . ∎
2.2. Singularities and contractible divisors
Let be the reduced exceptional divisor of the (unique) minimal good (i.e. such that is an snc-divisor) resolution of a singular point on a normal surface . Then is connected and is negative definite. Recall that a point is of quotient type if there exists an analytical neighborhood of and a small (i.e. not containing any pseudo-reflections) finite subgroup of , such that is analytically isomorphic to for some ball around . Then . Note that by a result of Tsunoda ([Tsu83]) for normal surfaces quotient singularities are the same as log-terminal singularities. For a singular point of quotient type it is known ([Bri68]) that is cyclic if and only if is an admissible chain and that is non-cyclic if and only if it is non-abelian if and only if is an admissible fork (rational snc-minimal fork with three twigs and with negative definite intersection matrix, cf. [Miy01, 2.3.5]), in each case . In case is a fork, we will say that is of type if the maximal twigs of have equal to . Quotient singularities are rational, as the first direct image of the structure sheaf of their resolutions vanishes. It follows from [Art66, 1] that for a rational singularity is a rational tree, hence rational singularities are topologically rational, which by definition means that . This notion is a bit stronger than the quasirationality in the sense of Abhyankar (cf. [Abh79]), for which only the rationality of components of is required.
Let be given by . Then the blowup of in has an exceptional line contained in the singular locus, hence is not normal. Since the blowup of a normal surface with rational singularity remains normal by [Lip69, 8.1], is not a rational singularity. On the other hand, it is topologically rational.
More generally, let be a Pham-Brieskorn surface given by the equation , where . This surface is contractible (note it has a -action with the singularity as the unique fixed point) and it is known that is a topologically rational singularity if and only if one of is coprime with two others or are integers coprime in pairs. (In [Ore95] and [FZ03, 0.1] the above is stated as a condition for quasirationality, but in both cases the graph of the resolution is a tree by looking at the proof or by using [OW71]). On the other hand, the rationality of is by [FZ03, 2.21] equivalent to each of the following conditions:
is of quotient type,
We have the following corollary from the Nakai criterion.
Let and be effective snc-divisors on a smooth complete surface having disjoint supports. If for every irreducible curve on either or then for sufficiently large and sufficiently divisible one has:
has no base points,
is birational and contracts exactly the curves in ,
is normal, projective and isomorphic to
Let be a smooth completion of a smooth surface and let be the Neron-Severi group of consisting of numerical equivalence classes of divisors. The Neron-Severi group of is defined as the cokernel of the natural map , where is a free abelian group generated by irreducible components of . We denote by .
The above definition does not depend on a smooth completion of (cf. [Fuj82, 1.19]). Contrary to the case when is complete, in general can have torsion.
Let and be effective snc-divisors on a smooth complete surface having disjoint supports. Assume that is connected, is negative definite and . Then there exists a normal affine surface and a morphism contracting connected components of , such that is an isomorphism.
Smooth complete surface is projective by the theorem of Zariski. Since , there exists a divisor with and , which is numerically equivalent to an ample divisor on . Then is ample, because ampleness is a numerical property by Nakai’s criterion. To use 2.4 we need to show that there exists an effective divisor , such that and for all irreducible curves . To deal with curves we use Fujita’s argument ([Fuj82, 2.4]). Let consist of all effective divisors , such that and for any prime component of . Writing , where , are effective and have no common component, we see that is nonempty because . Suppose is an element of with maximal number of components. For an irreducible curve satisfying one would get for , hence by the connectedness of .
Suppose an irreducible curve satisfies . Since , we have . We can choose some reduced divisor , such that irreducible components of give a basis of . Let us write , where , the divisors are effective and have no common component. For each we have , so , hence because . We have
so . Thus the divisor is nonzero, effective and numerically trivial, a contradiction. Let for as in lemma 2.4. Then contracts connected components of . We have also , where is a very ample divisor on , which implies that is affine. ∎
Note that any divisor with negative definite intersection matrix can be contracted in the analytical category by the theorem of Grauert (cf. [Gra62]). However, in general it is a more subtle problem if this can be done in the algebraic category (see [Art66] for results concerning rational singularities).
2.3. Minimal models
Let us give a brief sketch of the notion of minimality for open surfaces for unfamiliar readers. By the Castelnuovo criterion a smooth projective surface is minimal if and only if there is no irreducible curve on for which and , which is equivalent to being a -curve. Similarly, we can say that a smooth pair is relatively minimal if and only if there is no irreducible curve on for which and . In case this implies that is a -curve intersecting in at most one point and transversally. However, if then the conditions are equivalent to and
and hence to being a smooth rational curve with negative self-intersection and branching number . Contraction of such an immediately leads out of the category of smooth pairs, as in particular any tip of any admissible maximal twig of would have to be contracted. Thus one repeats the definition of a relatively minimal pair for pairs consisting of a normal projective surface and reduced Weil divisor (cf. [Miy01, 2.4.3]). Then a relatively minimal model of a given pair (which can be singular and not unique) is obtained by successive contractions of curves satisfying the above conditions. To go back to the smooth category one can translate the conditions for to be relatively minimal in terms of the properties of its minimal resolution. This leads to the notion of an almost minimal pair, which we recall now for the convenience of the reader (cf. [Miy01, 2.3.11]).
First, for any smooth pair we define the bark of . For non-connected bark is a sum of barks of its connected components, so we will assume is connected. If is an snc-minimal resolution of a quotient singularity (i.e. is an admissible chain or an admissible fork) then we define as a unique -divisor with , such that
In other case let be all the maximal admissible twigs of . (If and is snc-minimal then all rational maximal twigs of are admissible, cf. [Fuj82, 6.13]). In this case we define as a unique -divisor with , such that
The definition implies that is an effective -divisor with negative definite intersection matrix and its components can be contracted to quotient singular points. In fact all components of in its irreducible decomposition have coefficients at smaller than , unless is an admissible chain or fork consisting of -curves. Thus if is not such a -chain or a -fork then is an effective divisor with .
We now say that a smooth pair is almost minimal if for each curve on either or but the intersection matrix of is not negative definite. Consequently, an almost minimal model of a given pair can be obtained by successive contractions of curves for which and is negative definite. These are the non-branching -curves in and -curves for which and is negative definite. Minimalization does not change the logarithmic Kodaira dimension. One shows that is almost minimal if and only if after taking the contraction of connected components of to singular points the pair is relatively minimal. Moreover, if is almost minimal and then and are the numerically effective (nef) and negative definite parts of the Zariski decomposition of . The reader can find a more detailed description of the process of minimalization in loc. cit. We will need in particular the following fact.
Let be a smooth pair which is not almost minimal, but for which is snc-minimal. Let be a curve witnessing the non-almost-minimality, i.e. is a -curve not contained in , such that and is negative definite. Then meets transversally, in at most two points, each connected component of at most once. Moreover, if meets in two points then one of the connected components is an admissible chain and both points of intersection belong to .
2.4. Rational rulings
By a rational ruling of a normal surface we mean a surjective morphism of this surface onto a smooth curve, for which a general fiber is a rational curve. If its general fiber is isomorphic to it is called a -ruling.
If is a rational ruling of a normal surface then by a completion of we mean a triple , where is a normal completion of and is an extension of to a -ruling with being a smooth completion of . We say that is a minimal completion of if is -minimal, i.e. if does not dominate any other completion of .
Note that if and are as above then is -minimal if and only if each non-branching -curve contained in is horizontal.
For any rational ruling as above there is a completion . Let be a general fiber of . We call a -ruling (a -ruling) if (if ). Any fiber of a -ruling has vanishing arithmetic genus and self-intersection. The following well-known lemma shows that these conditions are also sufficient.
Let be a connected snc-divisor on a smooth complete surface . If then there exists a -ruling and a point for which .
The proof given in [BHPVdV04, V.4.3] after minor modifications works with the above assumptions. ∎
For an irreducible vertical curve we denote its multiplicity in the fiber containing it by (or if is fixed).
If is a singular fiber of a -ruling then, since , it is a rational tree with components of negative self-intersection. Its structure is well-known (see [Fuj82, §4]). First of all, since (by the adjunction formula ), it contains a -curve and if the -curve is unique then its multiplicity is bigger than one. In fact each -curve of intersects at most two other components of , so its contraction leads to a snc-fiber with a smaller number of components and, by induction, to a smooth fiber. In the process of contraction the total number of -curves in a fiber decreases, unless .
The situation when has a unique -curve, say , is of special interest. In this case is produced by a connected sequence of blowups from a smooth -curve. Let be the branching components of written in order in which they are created, put . It is convenient to write as , where the divisor is a reduced chain consisting of all components of created not later than . We call the i-th branch of . The proof of the following result is straightforward.
Let be a singular fiber of a -ruling of a smooth complete surface. If contains a unique -curve then:
and there are exactly two components of with multiplicity one. They are tips of the fiber and belong to the first branch,
if then either or is a tip of and then or is a -fork of type ,
if is branched then the connected component of not containing curves of multiplicity one is a chain (possibly empty).
Recall that having a fixed -ruling of a smooth surface and a divisor we define
where is the number of -components of a fiber (cf. [Fuj82, 4.16]). The horizontal part of is a divisor without vertical components, such that is vertical. The numbers and are defined respectively as and as the number of fibers contained in . We will denote a general fiber by .
With the notation as above the following equation is satisfied (cf. loc. cit. and [Pal11, 2.2] for a short proof):
Let be a completion of a -ruling of a normal surface . We say that the original ruling is twisted if is a 2-section. If consists of two sections we say that is untwisted. A singular fiber of is columnar if and only if it is a chain not containing singular points of and which can be written as
with a unique -curve , such that meets exactly in and , in each once and transversally. The chains and are called adjoint chains.
By abuse of language we call twisted or untwisted depending on the type of . Twisted and untwisted -rulings are called respectively gyoza (a Chinese dumpling) and sandwich in [Fuj82].
3. Topology and Singularities
Let be a singular -homology plane. Let be a good resolution and a smooth completion of . Denote the singular points of by and the smooth locus by . We put and assume that is snc-minimal. Define as the boundary of the closure of , where is a tubular neighborhood of . The construction of can be found in [Mum61]. We may assume that is a disjoint sum of closed oriented 3-manifolds. We write for and for .
Let us mention that the results we obtain below are generalizations of similar results obtained in the logarithmic case by Miyanishi and Sugie. However, restriction to quotient singularities is a strong assumption, which makes the considerations easier, even if at the end we prove that not so many non-logarithmic -homology planes do exist.
Let , , and be the inclusion maps. One has:
for some finite group of order ,
and are isomorphisms for positive ,
is connected, is an isomorphism and ,
is an isomorphism,
is an epimorphism, it is an isomorphism if ,
if then .
(i) By [Mum61] there is an exact sequence
where is a finite group of order and is induced by the composition of embedding of into the closure of with retraction onto . Since is free abelian, it follows that .
(ii) Let . We look at the reduced homology exact sequence of the pair . The pairs and are ’good CW-pairs’ (see [Hat02, Thm 2.13]), so for we have
and then induced by is an isomorphism for . Now
so and is also an isomorphism. Since are epimorphisms, the Mayer-Vietories sequence for splits into exact sequences:
Since is injective, is injective by exactness, so it is an isomorphism.
(iii)-(iv) By (ii) , so the homology exact sequence of the pair yields , hence by the Lefschetz duality (see [Dol80, 7.2]), which implies the connectedness of . The components of are numerically independent because , hence they are independent in . This implies that the inclusion induces a monomorphism on . By (ii) we can write the exact sequence of the pair as:
Now is a monomorphism, so by the Lefschetz and Poincare duality
On the other hand , so is an isomorphism.
Since is an isomorphism, the homology exact sequence of the pair yields an exact sequence:
We have by (ii) and by (i), so is an epimorphism. We need to prove that and . Note that and
so . This implies that if and only if .
so is a monomorphism. We can therefore assume that is not a rational forest, in particular is not logarithmic. Note that since is an epimorphism, is affine by 2.6, so we can use 3.3(i) below to infer that . Suppose , then is -ruled (cf. [Kaw79, 2.3]). Since modifications over do not change and , we can assume that this ruling extends to . The divisor is not vertical, otherwise would be semi-negative definite, which contradicts the Hodge index theorem. On the other hand, is not vertical because is not a rational forest, so each of and contains a unique section. Then , so we are done. We can now assume . Suppose , then . Put , where is a connected component of with . Let be an almost minimal model of with and being the direct images of and . Each curve contracted in the process of minimalization intersects the image of , because is affine. Note that not all components of are contracted in this process, as is not negative definite by the Hodge index theorem. Moreover, by 2.7 such a curve cannot intersect a connected component of with nontrivial . Thus the divisors and are disjoint, so
Since by 2.2(ii)
we have , so , which contradicts