On Zariski’s pairs of mth canonical discriminant curves

Vik.S. Kulikov ^{∗}

Abstract

In this note we give examples of Zariski’s pairs B_{1,m}, B_{2,m} (m ∈ ℕ and m ≥ 5) of plane cuspidal curves such that

(i) B_{i,m} is the discriminant curve of a generic morphism f_{i,m} : S_{i} → ℙ^{2}, i = 1, 2, (ii) S_{1} and S_{2} are homeomorphic surfaces of general type, (iii) f_{i,m} is given by linear three-dimensional subsystem of the mth canonical class of S_{i}.

Introduction

By definition, two plane curves B_{1}, B_{2} ⊂ ℙ^{2} over ℂ are called a Zariski pair if they have the same degree and homeomorphic tubular neighborhoods in ℙ^{2}, but the pairs (ℙ^{2}, B_{1}) and (ℙ^{2}, B_{2}) are not homeomorphic. A set of plane curves (B_{1}, ..., B_{k}) is called Zariski’s k-tuple if for each i ≠ j the pair (B_{i}, B_{j}) is a Zariski one.

Let B ⊂ ℙ^{2} be an irreducible plane curve with ordinary cusps and nodes, as its only singularities. Denote by 2d the degree of B, and let g be the genus of its desingularization, c = #{cusps of B}, and n = #{nodes of B}. The curve B is called the discriminant curve of a generic morphism if there exists a finite morphism f : S → ℙ^{2}, deg f ≥ 3, satisfying the following conditions:

(i) S is a non-singular irreducible projective surface;

(ii) f is unramified over ℙ^{2} \ B;

(iii) f^{∗}(B) = 2R + C, where R is irreducible and non-singular, and C is reduced; (iv) f_{|R} : R → B coincides with the normalization of B.

Such a morphism f is called generic.

Two generic morphisms (S_{1}, f_{1}), (S_{2}, f_{2}) with the same discriminant curve B are said to be equivalent if there exists an isomorphism φ : S_{1} → S_{2} such that f_{1} = f_{2} ∘ φ. The following assertion is known as Chisini’s Conjecture.

Conjecture 1 Let B be the discriminant curve of a generic morphism f : S → ℙ^{2} of degree deg f ≥ 5. Then f is uniquely determined by B.

∗Partly supported by RFFI (No. 96-01-00614) and INTAS (No. 96-0713).

In [5], Chisini’s conjecture was proved for discriminant curves of almost all generic morphisms. In particular, Chisini’s conjecture holds for the discriminant curves of the generic morphisms f_{m} : S → ℙ^{2} of the surfaces S of general type, where f_{m} is given by a linear threedimensional subsystem of the mth canonical class of S, m ∈ ℕ. The discriminant curve B of a generic morphism f_{m} : S → ℙ^{2}, given by a three-dimensional subsystem of the mth canonical class of S, will be called the mth canonical discriminant curve.

Let (ℙ^{2}, B_{1}) and (ℙ^{2}, B_{2}) be two diffeomorphic (resp. homeomorphic) pairs. One can show that if B_{1} is the discriminant curve of a generic morphism (S_{1}, f_{1}), then B_{2} is also the discriminant curve of some generic morphism (S_{2}, f_{2}). Moreover, if (S_{1}, f_{1}) is uniquely determined by B_{1}, then the same is true for (S_{2}, f_{2}) and S_{1} and S_{2} are diffeomorphic (resp. homeomorphic). It is a natural question to ask ([5]):

Question. Let S_{1}, S_{2} ⊂ ℙ^{r} be two homeomorphic (Case C)(resp. diffeomorphic (Case D)) surfaces of general type embedded by the mth canonical class, and let B_{i} be the mth canonical discriminant curve of a generic projection of S_{i} onto ℙ^{2}. Are pairs (ℙ^{2}, B_{1}) and (ℙ^{2}, B_{2}) homeomorphic (resp. diffeomorphic)?

In case C, the answer to this Question is negative, as the following theorem shows.

Theorem 1 For each positive integer k there exists a countable set of Zariski’s k-tuples (B_{1,m}, ..., B_{k,m}), m ∈ ℕ, m ≥ 5, of mth canonical discriminant curves B_{i,m} such that the corresponding generic morphisms f_{i,m} : S_{i} → ℙ^{2}, i = 1, ..., k, are the morphisms of pairwise homeomorphic surfaces of general type S_{i}.

The proof of this Theorem is based on technique developed in [5]. It also uses the examples constructed of homeomorphic (but not diffeomorphic) surfaces of general type with ample canonical class constructed by Catanese in [2] and [3].

The Question in the case D remains open.

This paper was written during my stay at the Max-Planck-Institut für Mathematik in Bonn. It is a pleasure to thank the Institut for its hospitality and financial support.

1 Proof of Theorem 1.

In [2] and [3], Catanese investigated smooth simple bidouble coverings φ : S → Q = ℙ^{1} × ℙ^{1} of type (a, b), (m, n); thus, φ is a finite (ℤ/2)^{2} Galois covering branched along two generic curves of respective bidegrees (2a, 2b), (2m, 2n). Let a, b, m, n be integers satisfying the following conditions:

\begin{matrix}a > 2n , n ≥ 3, & m > 2b , b ≥ 3 ; \\ a ≡ n (mod 2), & b ≡ m (mod 2). \end{matrix}

Put

u = (n + a − 2), v = (m + b − 2), w = (a − n), z = (m − b).  (1)

By [2], for a bidouble covering φ : S → Q = ℙ^{1} × ℙ^{1} of type (a, b), (m, n), S is a simply connected surface of general type whose canonical class is

K_{S} = φ^{∗}(O_{Q}(u, v))  (2)

and

K_{S}^{2} = 8uv, χ = χ(O_{S}) = \frac{3}{2}uv + (u + v) + 2 − \frac{1}{2}wz.  (3)

For u, v, w, z satisfying (1), the canonical class K_{S} is 2-divisible in H^{2}(S, ℤ) and the index

r = r(S) = max{s ∈ ℕ | (1/s)K_{S} ∈ H^{2}(S, ℤ)}

is equal to the greatest common divisor (u, v).

Since K_{S} is 2-divisible, the intersection form on H_{2}(S, ℤ) is even. Therefore, by Freedman’s theorem ([4]) two such bidouble coverings S_{1} and S_{2} of Q are homeomorphic if and only if K_{S_{1}}^{2} = K_{S_{2}}^{2} and χ(O_{S_{1}}) = χ(O_{S_{2}}). In [2] and [3], it has been proved that for each integer k there exists at least one k-tuple S_{1}, ..., S_{k} of bidouble coverings of Q of respective types (a_{i}, b_{i}), (m_{i}, n_{i}) satisfying the conditions described above and such that

(i) S_{i} and S_{j} are homeomorphic for each 1 ≤ i, j ≤ k.

(ii) r(S_{i}) ≠ r(S_{j}) for i ≠ j (and therefore S_{i} and S_{j} are not diffeomorphic). We shall call such k-tuple of bidouble coverings of Q a Catanese k-tuple.

Example. Two bidouble coverings S_{1} of type (16, 22), (52, 4) and S_{2} of type (28, 10), (28, 10) form a Catanese pair with

K_{S}^{2} = K_{S_{i}}^{2} = 10368, χ = χ(O_{S_{i}}) = 1456

and

r(S_{1}) = 18, r(S_{2}) = 36.

Consider the m-canonical embedding Φ_{|mK_{S_{i}}|} : S_{i} → ℙ^{r}, m ≥ 5 ([1]), and let f_{m,i} : S_{i} → ℙ^{2} be the restriction to Φ_{|mK_{S_{i}}|}(S_{i}) of a generic projection pr : ℙ^{r} → ℙ^{2}. Denote by R_{m,i} ⊂ S_{i} the ramification curve of the generic projection f_{m,i}, and by B_{m,i} ⊂ ℙ^{2} the discriminant curve of f_{m,i}.

Proposition 1 Let f_{i} : S_{i} → ℙ^{2}, i = 1, 2, be two generic morphisms with respective discriminant curves B_{i}. Assume that there exist a homeomorphism of pairs ψ : (ℙ^{2}, B_{1}) → (ℙ^{2}, B_{2}). Then there exists a commutative diagram

\xymatrix{S_{1} \ar[d]_{f_{1}} \ar[r]^{Ψ} & S_{2} \ar[d]^{f_{2}} \\ ℙ^{2} \ar[r]_{ψ} & ℙ^{2}}

such that (i) Ψ is also a homeomorphism; (ii) Ψ(R_{1}) = R_{2}, where R_{i} is the branch curve of f_{i}.

Proof coincides with the proof of proposition 8 in [5].

Let a generic morphism f_{m} : S → ℙ^{2} be given by a three-dimensional linear subsystem of the mth canonical class on S. Then, by the adjunction formula, the ramification curve R_{m} ⊂ S of f_{m} is equivalent to (3m + 1)K_{S}. Therefore, if we have two generic morphisms f_{i,m} : S_{i} → ℙ^{2} given by three-dimensional linear subsystems of the mth canonical classes and such that the corresponding pairs (ℙ^{2}, B_{1,m}) and (ℙ^{2}, B_{2,m}) are homeomorphic, then by proposition 1 there exists a homeomorphism Ψ : S_{1} → S_{2} such that Ψ(R_{1,m}) = R_{2,m}. In this case, we should have r(S_{1}) = r(S_{2}), since Ψ induces an isomorphism Ψ_{∗} : H_{2}(S_{1}, ℤ) → H_{2}(S_{2}, ℤ) such that Ψ_{∗}(R_{1,m}) = R_{2,m}. But it is impossible, since, by definition, r(S_{1}) ≠ r(S_{2}) for the Catanese pair S_{1}, S_{2}. Theorem 1 is proved.

Example. (Continuation). Applying Theorem 1 for m-canonical discriminant curves of generic morphisms of surfaces S_{1} and S_{2} in the example considered above, we obtain Zariski’s pairs (B_{1,m}, B_{2,m}), m ≥ 5, of plane cuspidal curves. Applying the computation in the proof of theorem 3 in [5], one can see that the m-canonical discriminant curve B_{i,m} has degree

deg B_{i,m} = 10368m(3m + 1),

genus

g = 5184(3m + 2)(3m + 1) + 1,

and

c = 10368(12m^{2} + 9m) − 13632

ordinary cusps.

References

[1] E. Bombieri, Canonical models of surfaces of general type // Publ. Math. IHES, 42 (1973), 171-219.

[2] F. Catanese, On the moduli spaces of surfaces of general type // J. Diff. Geom, 19, (1984), 483-515.

[3] F. Catanese, Connected components of moduli space // J. Diff. Geom, 24, (1986), 395-399.

[4] M.H. Freedman, The topology of four dimensional manifolds // J. Diff. Geom, 17, (1982), 357-453.

[5] Vik.S. Kulikov, On a Chisini Conjecture // Preprint MPI 98-28.

Steklov Mathematical Institute

victor@olya.ips.ras.ru