Zariski-van Kampen method and transcendental lattices of certain singular surfaces
We present a method of Zariski-van Kampen type for the calculation of the transcendental lattice of a complex projective surface. As an application, we calculate the transcendental lattices of complex singular surfaces associated with an arithmetic Zariski pair of maximizing sextics of type that are defined over and are conjugate to each other by the action of .
2000 Mathematics Subject Classification:14J28, 14H50, 14H25
First we prepare some terminologies about lattices. Let be or , where is a prime integer or , is the ring of -adic integers for , and is the field of real numbers. An -lattice is a free -module of finite rank with a non-degenerate symmetric bilinear form
A -lattice is simply called a lattice. A lattice is called even if holds for any . Two lattices and are said to be in the same genus if the -lattices and are isomorphic for all (including ). Then the set of isomorphism classes of lattices are decomposed into a disjoint union of genera. Note that, if and are in the same genus and is even, then is also even, because being even is a -adic property. Let be a lattice. Then is canonically embedded into as a submodule of finite index, and extends to a symmetric bilinear form
Suppose that is even. We put
and define a quadratic form by
The pair is called the discriminant form of . By the following result of Nikulin (Corollary 1.9.4 in [MR525944]), each genus of even lattices is characterized by the signature and the discriminant form.
Two even lattices are in the same genus if and only if they have the same signature and their discriminant forms are isomorphic.
For a surface defined over a field , we denote by the Néron-Severi lattice of ; that is, is the lattice of numerical equivalence classes of divisors on with the intersection paring . Following the terminology of [MR0284440, §8] and [MR0441982], we say that a surface defined over a field of characteristic is singular if the rank of attains the possible maximum .
Let be a complex surface. Then the second Betti cohomology group is regarded as a unimodular lattice by the cup-product, which is even of signature . The Néron-Severi lattice is embedded into primitively, because we have . We denote by the orthogonal complement of in , and call the transcendental lattice of . Suppose that is singular in the sense above. Then is an even positive-definite lattice of rank . The Hodge decomposition induces a canonical orientation on . We denote by the oriented transcendental lattice of .
We denote by the symmetric matrix and put
on which acts by , where and . The set of isomorphism classes of even positive-definite lattices of rank is equal to
while the set of isomorphism classes of even positive-definite oriented lattices of rank is equal to
For a matrix , we denote by and the isomorphism classes represented by .
In [MR0441982], Shioda and Inose proved the following:
The map induces a bijection from the set of isomorphism classes of complex singular surfaces to the set .
The injectivity follows from the Torelli theorem by Piatetski-Shapiro and Shafarevich [MR0284440]. In the proof of the surjectivity, Shioda and Inose gave an explicit construction of the complex singular surface with a given oriented transcendental lattice, and they have proved the following:
Every complex singular surface is defined over a number field.
Let be a singular surface defined over a number field . We denote by the set of embeddings of into , and for , we denote by the complex singular surface . We define a map
by . Then we have the following theorem by Schütt [MR2346573], which is a generalization of a result that had been obtained in [tssK3].
Let be the genus of all such that is isomorphic to , and let be the pull-back of by the natural projection . Then the image of coincides with .
Therefore we obtain a surjective map
Remark that, by the classical theory of Gauss [MR837656], we can easily calculate the oriented genus from the finite quadratic form .
Let be a geometrically reduced and irreducible projective surface defined over a number field , and let be a desingularization of defined over a finite extension of . Suppose that is a singular surface. Then we can define a map
by the following:
The map factors as
where is the natural restriction map .
The purpose of this paper is to present a method to calculate the map from a defining equation of .
More generally, we consider the following problem. Let be a reduced irreducible complex projective surface. For a desingularization , we put
which is regarded as a lattice by the cup-product, and let be the sublattice of the cohomology classes of divisors on . We denote by
the orthogonal complement of in . Then we can easily see that the isomorphism class of the lattice does not depend on the choice of the desingularization , and hence we can define the transcendental lattice of to be . (See Lemma 3.1 of [Shioda_K3SPnew] or Proposition 2.1 of this paper.) We will give a method of Zariski-van Kampen type for the calculation of .
We apply our method to maximizing sextics. Following Persson [MR661198, MR805337], we say that a reduced projective plane curve of degree defined over a field of characteristic is a maximizing sextic if has only simple singularities and its total Milnor number attains the possible maximum , where is the algebraic closure of . The type of a maximizing sextic is the -type of the singular points of .
Let be a maximizing sextic defined over a number field . The double covering branching exactly along is defined over . Let be the minimal resolution defined over a finite extension of . Then is a singular surface defined over . We denote by the oriented genus . By Proposition 1.5, we have a surjective map
As an illustration of our Zariski-van Kampen method, we calculate for a reducible maximizing sextic of type defined over by the homogeneous equation
This equation was discovered by means of Roczen’s result [MR1175728] (see §5).
We can calculate by the method of Yang [MR1387816], and obtain
Let be the embeddings of into given by . We have two surjective maps from to . Remark that, since the two complex maximizing sextics and cannot be distinguished by any algebraic methods, we have to employ some transcendental method to determine which surjective map is the map . By the method described in §3 of this paper, we obtain the following:
We have shown in [AZP] and [nonhomeo] that, for a complex maximizing sextic , the transcendental lattice of the double covering branching exactly along is a topological invariant of . Thus the curves and form an arithmetic Zariski pair. (See [AZP] for the definition.) The proof of Proposition 1.6 illustrates very explicitly how the action of the Galois group of over affects the topology of the embedding of into .
The first example of arithmetic Zariski pairs was discovered by Artal, Carmona and Cogolludo [MR2247887] in degree by means of the braid monodromy. It will be an interesting problem to investigate the relation between the braid monodromy of a maximizing sextic and our lattice invariant .
In the study of Zariski pairs of complex plane curves, the topological fundamental groups of the complements (or its variations like the Alexander polynomials) have been used to distinguish the topological types. (See, for example, [MR1257321], [MR1167373] or [MR1421396] for the oldest example of Zariski pairs of -cuspidal sextics [MR1506719, MR1507244].) We can calculate the fundamental groups and of our example in terms of generators and relations by the classical Zariski-van Kampen theorem. (See, for example, [MR1341806, MR1952329].) It will be an interesting problem to determine whether these two groups are isomorphic or not. Note that, by the theory of algebraic fundamental groups, their profinite completions are isomorphic.
The plan of this paper is as follows. In §2, we prove Proposition 1.5. In §3, we present the Zariski-van Kampen method for the calculation of the transcendental lattice in full generality. In §4, we apply this method to the complex maximizing sextics and prove Proposition 1.6. In §5, we explain how we have obtained the equation (1.1) of .
Thanks are due to the referee for his/her comments and suggestions on the first version of this paper.
2. The map
Proof of Theorem 1.4.
It is easy to see that the image of is contained in . (See Theorem 2 in [tssK3] or Proposition 3.5 in [nonhomeo].) In [tssK3] and [MR2346573], using Shioda-Inose construction, we constructed a singular surface defined over a number field such that (and hence ) holds and that the image of coincides with . (See also §4 of [nonhomeo].) We choose an arbitrary . Then there exists such that . Since and are isomorphic over by Theorem 1.2, there exists a number field containing both and such that we have an isomorphism
over . Consider the commutative diagram
where and are the natural surjective restriction maps. The surjectivity of then follows from the surjectivity of .
Proof of Proposition 1.5.
Let and be two desingularizations of a reduced irreducible complex projective surface . Then . If and are singular surfaces, then .
Using a desingularization of , we obtain a complex smooth projective surface with birational morphisms and . Since the transcendental lattice of a complex smooth projective surface is invariant under a blowing-up, and any birational morphism between smooth projective surfaces factors into a composite of blowing-ups, we have and .
3. Zariski-van Kampen method for transcendental lattices
For a -module , we denote by
the maximal torsion-free quotient of . If we have a bilinear form , then it induces a canonical bilinear form .
Let be a reduced irreducible complex projective surface. Our goal is to calculate . Let be a desingularization. We choose a reduced curve on with the following properties:
the classes of irreducible components of the total transform of span over , and
the desingularization induces an isomorphism .
and consider the free -module
with the intersection paring
Then is a free -module, and the intersection paring induces a non-degenerate symmetric bilinear form
The transcendental lattice is isomorphic to the lattice .
By the condition (D2), we can regard as a Zariski open subset of . Consider the homomorphism
induced by the inclusion . Under the isomorphism of lattices
induced by the Poincaré duality, the image of is contained in by the condition (D1) on . Using the argument in the proof of Theorem 2.6 of [AZP] or Theorem 2.1 of [nonhomeo], we see that the homomorphism
is surjective. Note that we have
for any , where is the cup-product on . Since is non-degenerate, we conclude that .
Proposition 3.1 shows that, in order to obtain , it is enough to calculate and . Enlarging if necessary, we have a surjective morphism
onto a Zariski open subset of an affine line such that its general fiber is a connected Riemann surface. By the condition (D1) on , the general fiber of is non-compact. Let be a smooth irreducible projective surface containing as a Zariski open subset such that extends to a morphism
Let and be the irreducible components of the boundary , where are the vertical components (that is, is a point), and are the horizontal components (that is, ). Since the general fiber of is non-compact, we have at least one horizontal component. We put
Adding to some fibers of and making smaller if necessary, we can assume the following:
the surjective morphism has only ordinary critical points,
is étale over , and
, where .
Note that has no critical points on by the condition (2). We denote by the critical values of , and put
By the assumptions, is locally trivial (in the category of topological spaces and continuous maps) over with the fiber being a connected Riemann surface of genus with punctured points, where is the degree of . We then choose a base point , and put
For each , we choose a loop
that is sufficiently smooth and injective in the sense that holds only when or , and that defines the same element in as a simple loop (a lasso) around in . For each critical value , we choose a sufficiently smooth and injective path
such that , and for . We choose these loops and paths in such a way that any two of them intersect only at . Then, by a suitable self-homeomorphism of , the objects , , , and on are mapped as in Figure 3.1.
In particular, the union of and is a strong deformation retract of . Note that is locally trivial over .
Let be an oriented one-dimensional sphere. We fix a system of oriented simple closed curves
on in such a way that their union is a strong deformation retract of . In particular, we have
where is the homology class of . For each and , let
be an embedding such that the diagram
commutes and that
is equal to . We put
where stands for the monodromy along , and denote the homology class of by
Let be the topological space obtained from by contracting to a point ; that is, is a cone over with the vertex . Let be the natural projection. Let be a critical value of , and let be the critical points of over . For each critical point , we choose a thimble
along the path corresponding to the ordinary node of . Namely, the thimble is an embedding such that
commutes, and that . (See [MR592569] for thimbles and vanishing cycles.) Then the simple closed curve
on represents the vanishing cycle for the critical point along . We denote its homology class by
We can assume that and are disjoint if .
There are two choices of the orientation of the thimble (and hence of the vanishing cycle ).
Then the union
is homotopically equivalent to . Since the -dimensional CW-complex is a strong deformation retract of , the homology group is equal to the kernel of the homomorphism
The intersection pairing on is calculated by perturbing the system of loops and paths with the base point to a system with the base point . We make the perturbation in such a way that the following hold.
There exists a small open disk containing both and such that
where are small positive real numbers.
If intersects or , then their intersection points are contained in , and the intersections are transverse.
If intersects or with , then their intersection points are contained in , and the intersections are transverse.
Any intersection point of and is either the common end-point , or a transversal intersection point contained in .
We then perturb the topological -chains over and over to topological -chains over and over , respectively. Let be one of or , and let be the loop or the path over which locates. Let be one of or over the loop or the path . We can make the perturbation in such a way that and intersect transversely at each intersection points. Suppose that
Then are contained in the union of fibers except for the case where and for some and . If and , then and also intersect at the critical point transversely with the local intersection number . (See Lemma 4.1 of [MR1433320].) For each , let and be the -cycles on the open Riemann surface given by
We denote by the local intersection number of the -chains and on at , which is or by the assumption on the perturbation. We also denote by the intersection number of and on the Riemann surface . Then the intersection number of and is equal to
The number is calculated as follows. Let
be the intersection pairing on , which is anti-symmetric. If , then the -cycle on can be deformed to the vanishing cycle on along the path in . We put
Suppose that . Then we have , where and are small positive real numbers. If the number such that is contained in , then can be deformed to the -cycle on along the path in . If , then can be deformed to the -cycle on along the path in . We define by
We define from in the same way. Since is topologically trivial over , we have
The formulae (3.2) and (3.3) give the intersection number of topological -chains and . Even though the number depends on the choice of the perturbation, it gives the symmetric intersection paring on . Thus we obtain and .
4. Maximizing sextics of type
Recall from Introduction that (resp. ) is the set of isomorphism classes of even positive-definite lattices (resp. oriented lattices) of rank .
Let be the map of forgetting orientation. We say that is real if consists of a single element, and that