On a Chisini Conjecture

Vik.S. Kulikov ^{∗}

Abstract

Chisini’s conjecture asserts that for a cuspidal curve B ⊂ ℙ^{2} a generic morphism f of a smooth projective surface onto ℙ^{2} of degree ≥ 5, branched along B, is unique up to isomorphism. We prove that if deg f is greater than the value of some function depending on the degree, genus, and number of cusps of B, then the Chisini conjecture holds for B. This inequality holds for many different generic morphisms. In particular, it holds for a generic morphism given by a linear subsystem of the mth canonical class for almost all surfaces with ample canonical class.

Introduction

Let B ⊂ ℙ^{2} be an irreducible plane curve over ℂ with ordinary cusps and nodes, as the 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}. We shall call B 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.

We shall call such f a generic morphism.

Note that if S ⊂ ℙ^{r}, f is the restriction to S of a generic projection of ℙ^{r} onto ℙ^{2}, and B is the branch curve of f, then (S, f) is a generic morphism and B is its discriminant curve.

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. In the sequel, ”f is unique” means ”f is unique up to equivalence”.

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, for B, the generic morphism f is unique.

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

If B ⊂ ℙ^{2} is the dual curve of a smooth cubic, then B is the discriminant curve of four generic non-equivalent morphisms ([6], [5]). Three of them have degree four, and the last one has degree three. This is the only known (up to now) example of a discriminant curve, for which there exist several non-equivalent generic morphisms.

In general case, as it follows from [5], the number of non-equivalent generic morphisms with a given discriminant curve B is less than or equal to 2^{2g+c−1}.

B. Moishezon proved the Chisini Conjecture for the discriminant curves of generic projections of smooth hypersurfaces in ℙ^{3}. His proof is based on the presentation of the fundamental group of the complement in ℙ^{2} of the discriminant curve of projection, obtained by him in [13]. A short review of other results relating to the Chisini Conjecture, and of some attempts to prove it can be found in [5].

The main result of this paper is

Theorem 1 Let B be the discriminant curve of a generic morphism f : S → ℙ^{2} of deg f = N. If

N > \frac{4(3d + g − 1)}{2(3d + g − 1) − c}.  (1)

Then, for B, the generic morphism f is unique and thus, the Chisini Conjecture holds for B.

Theorem 1 shows that if the degree of a generic morphism with given discriminant curve B is sufficiently large, then this generic morphism is unique for B. Almost all generic morphisms interesting from algebraic geometric point of view satisfy this condition. More precisely, let E = f^{∗}(ℙ^{1}) be the preimage of a generic line ℙ^{1} ⊂ ℙ^{2}. In order to obtain the following theorems, which are consequences of the main result, we check inequality (1) for morphisms given by three-dimensional subsystems of different linear systems |E| on surfaces of different types.

Theorem 2 Let S be a surface of general type with ample canonical bundle K_{S}, f : S → ℙ^{2} a generic morphism such that E ≡ mK_{S}, m ∈ ℕ (≡ means numerical equivalence). Then, for the discriminant curve B of f, the generic morphism f is unique except, possibly, for:

1) m = 1, K_{S}^{2} = 5, p_{a} = 1, deg B = 20, g = 51, c = 108, n = 12;

2) m = 1, K_{S}^{2} = 5, p_{a} = 2, deg B = 20, g = 51, c = 96, n = 24;

3) m = 1, K_{S}^{2} = 6, p_{a} = 1, deg B = 24, g = 61, c = 132, n = 60, where p_{a} = χ(O_{S}) is the arithmetic genus of S.

In all the exceptional cases, if for B there exist non-equivalent generic morphisms, then these morphisms have degree ≤ 6.

Theorem 3 Let S be a Del Pezzo surface and f : S → ℙ^{2} be a generic morphism such that E ∈ | − mK_{S}|, m ∈ ℕ. Then, for the discriminant curve B of f, the generic morphism f is unique.

Theorem 4 Let f : S → ℙ^{2} be any generic morphism of S = ℙ^{1} × ℙ^{1}. Then, for the discriminant curve B of f, the generic morphism f is unique.

Theorem 5 Let S be a K3 surface and f : S → ℙ^{2} any generic morphism. Then, for the discriminant curve B of f, the generic morphism f is unique.

Theorem 6 Let S be an Enriques surface and f : S → ℙ^{2} any generic morphism. Then, for the discriminant curve B of f, the generic morphism f is unique except, possibly, for deg f = 4. In the exceptional case, deg B = 12, g = 19, c = 36, n = 0 and if such a morphism exists, then:

1) for B, there exist at least two non-equivalent generic morphisms,

2) any generic morphism f′ with such a discriminant curve B has deg f′ ≤ 4. In particular, Chisini’s Conjecture holds for the discriminant curves of the generic morphisms of Enriques surfaces.

Theorem 7 Let S be an abelian surface and f : S → ℙ^{2} any generic morphism. Then, for the discriminant curve B of f, the generic morphism f is unique except, possibly, for deg f = 6. In the exceptional case, deg B = 18, g = 28, c = 72, n = 36 and if, for B, there exists a generic morphism f′ which is not equivalent to f, then deg f′ ≤ 6.

Theorem 8 Let S ⊂ ℙ^{N} be a complete intersection and f : S → ℙ^{2} the restriction of a generic projection. Then, for the discriminant curve B of f, the generic morphism f is unique.

Theorem 9 Let S be a projective non-singular surface and L an ample divisor on S, f : S → ℙ^{2} a generic morphism given by a three-dimensional subsystem {E} ⊂ |mL|, m ∈ ℚ, and B its discriminant curve. Then there exists a constant m_{0} (depending on L^{2}, (K_{S}, L), K_{S}^{2}, p_{a}) such that, for B, the generic morphism f is unique if m ≥ m_{0}. In particular, if L = K_{S}, then one can take m_{0} = 2.

Theorem 10 The Chisini Conjecture holds for the dual curve B of a nodal plane curve except, possibly, for:

1) deg B = 30, g = 10, c = 72, n = 324;

2) deg B = 20, g = 6, c = 45, n = 120;

3) deg B = 18, g = 5, c = 39, n = 92;

4) deg B = 16, g = 4, c = 33, n = 68.

In all the exceptional cases, if, for B, there exist non-equivalent generic morphisms, then these morphisms have degree ≤ 6.

Theorem 11 The Chisini Conjecture holds for a curve B of genus g ≤ 3.

Theorem 12 The Chisini Conjecture holds for B satisfying the inequality

d > 3(g − 1).

Unfortunately, in general case we have no a satisfactory description, purely in terms of algebraic geometry, of the set of discriminant curves with given degree, genus, and the number of cusps. But it is possible to give such a description in some particular cases. For example, in [23], Zariski showed that a sextic with 6 ordinary cusps is the discriminant curve of some generic morphism if and only if these 6 cusps lie on a conic, and he proved that there exist sextics with 6 cusps which do not lie on a conic. We also give, in terms of algebraic geometry, a description of discriminant curves of morphisms given by three-dimensional subsystems of the mth canonical class for m ≥ 21 (see section 4.2 below). On the other hand, the set of the discriminant curves can be completely described in terms of the fundamental group of their complement in ℙ^{2} (see Proposition 1 below).

In section 1, we recall some well-known facts on generic morphisms and their discriminant curves. Section 2 is devoted to the proof of Theorem 1. In section 3, we check inequality (1) in different cases in order to prove Theorems 2 - 12. Section 4 contains a more detailed investigation of the case when a generic morphism is given by a linear subsystem of the mth canonical class. In section 5, we briefly discuss the question about the number of irreducible components of moduli space of discriminant curves with given degree, genus, and the number of cusps, and we apply Theorem 2 to find new examples of Zariski’s pairs.

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 Auxiliary results.

1.1. Let B be a discriminant curve of a generic morphism f : S → ℙ^{2}, deg f = N, E = f^{∗}(ℙ^{1}). We have (E^{2})_{S} = N.

Lemma 1 deg B = 2d is even and, consequently, d ∈ ℕ.

Proof. By Hurwitz’s formula,

2g(E) − 2 = −2N + deg B.

Hence deg B is even.

Since g(E) ≥ 0, we have the following inequality

deg f ≤ d + 1.  (2)

Now we recall some inequalities contracting the genus, degree, and the number of cusps of B and following from Plücker’s formulas and Nori’s result [15]. For the dual curve B^{∗} of B, put δ = deg B^{∗}, γ = #{cusps of B^{∗}} and ν = #{nodes of B^{∗}}. From Plücker’s formulas

\begin{aligned}δ & = & 2d(2d − 1) − 2n − 3c; \\ 2d & = & δ(δ − 1) − 2ν − 3γ; \\ 2g & = & (2d − 1)(2d − 2) − 2n − 2c; \\ 2g & = & (δ − 1)(δ − 2) − 2ν − 2γ \end{aligned}

it follows that

\begin{aligned}δ & = & 4d − c + 2g − 2 ; \\ γ & = & 6d − 2c + 6g − 6. \end{aligned}

Since δ ≥ 0 and γ ≥ 0, we have Lemma 2

\begin{aligned}c & ≤ & 4d + 2g − 2 ; \\ c & ≤ & 3d + 3g − 3 . \end{aligned}

Corollary 1 c < 2(3d + g − 1). Lemma 3 3d + g − 1 ≤ 2c. Proof. If B is a discriminant curve of a generic morphism, then π_{1}(ℙ^{2} \ B) is not abelian. Hence by [15], B^{2} − 6c ≤ 2n. For a plane cuspidal curve B of deg B = 2d we have g + c + n = (2d − 1)(d − 1). Hence B^{2} − 2n = 4d^{2} − 2n = 2c + 6d + 2g − 2 ≤ 6c. Lemma 4

R^{2} = 2d^{2} − c − n =  (3)
= 3d + g − 1  (4)

Proof. We have K_{S} = −3E + R. Hence (4) follows from adjunction formula:

\frac{(K_{S} + R, R)}{2} = \frac{(−3E + 2R, R)}{2} = −3d + R^{2} = g − 1.

We obtain (3) if instead of g we substitute g = (2d − 1)(d − 1) − c − n. Lemma 5

N ≤ \frac{4d^{2}}{3d + g − 1},  (5)

and (5) is the equality if and only if either E ≡ mK_{S} for some m ∈ ℚ^{∗}, or K_{S} ≡ 0. Proof. By Hodge’s Index Theorem,

| \begin{matrix}E^{2} & (E, R) \\ (E, R) & R^{2} \end{matrix} | = | \begin{matrix}N & 2d \\ 2d & 3d + g − 1 \end{matrix} | ≤ 0,

i.e.

N ≤ \frac{4d^{2}}{3d + g − 1} ,

and we have the equality if and only if E and R are linearly dependent in NS(S) ⊗ ℚ, where NS(S) is the Neron-Severi group of S. In the last case, since K_{S} = −3E + R, either E ≡ mK_{S} for some m, or K_{S} ≡ 0. Lemma 6

\begin{aligned}K_{S}^{2} & = & 9N + 2d^{2} − 12d − c − n = \\ & = & 9N − 9d + g − 1. \end{aligned}

Proof. K_{S}^{2} = (−3E + R, −3E + R) = 9E^{2} − 6(E, R) + R^{2} = 9N − 12d + R^{2}.

Lemma 7 The topological Euler characteristic

\begin{aligned}e(S) & = & 3N + 4d^{2} − 6d − 3c − 2n = \\ & = & 3N + 2(g − 1) − c. \end{aligned}

Proof. Using a generic pencil of lines in ℙ^{2} and its preimage {E_{t}} in S, we get a formula for e(S):

\begin{aligned}e(S) + N & = & δ + 2(2 − 2g(E)) = 2d(2d − 1) − 3c − 2n − 2(K_{S} + E, E) = \\ & = & 2d(2d − 1) − 3c − 2n − 2(−2E + R, E) = 4N + 4d^{2} − 6d − 3c − 2n. \end{aligned}

From Noether’s formula K_{S} + e(S) = 12p_{a} it follows

Lemma 8 The Euler characteristic of O_{S}

\begin{aligned}p_{a} = 1 − q + p_{g} & = & N + \frac{d(d − 3)}{2} − \frac{c}{3} − \frac{n}{4} = \\ & = & N + \frac{3g − 3 − 9d − c}{12}. \end{aligned}

Corollary 2 ([14]).

\begin{aligned}c & ≡ & 0 (mod 3) ; \\ n & ≡ & 0 (mod 4) . \end{aligned}

Lemma 9 The divisor R is ample on S.

Proof. It suffices to show that (R, Γ) > 0 for any irreducible curve Γ. If (R, Γ) ≤ 0, then by Hodge’s Index Theorem, Γ^{2} < 0 and (R, Γ) = 0, since by Lemma 4, R^{2} > 0 and R is irreducible. If Γ^{2} < 0 and Γ is irreducible, then (K_{S}, Γ) ≥ −1, i.e. −3(E, Γ) + (R, Γ) ≥ −1, which contradicts (R, Γ) = 0. 1.2. Let us fix p ∈ ℙ^{2} \ B and denote by π_{1} = π_{1}(ℙ^{2} \ B, p) the fundamental group of the complement of B. Choose any point x ∈ B\Sing B and consider a line Π = ℙ^{1} ⊂ ℙ^{2} intersecting B transversely at x. Let γ ⊂ Π be a circle of small radius with center at x. If we choose an orientation on ℙ^{2}, then it defines an orientation on γ. Let Γ be a loop consisting of a path L in ℙ^{2} \ B joining the point p with a point q ∈ γ, the circuit in positive direction along γ beginning and ending at q, and a return to p along the path L in the opposite direction. Such loops Γ (and the corresponding elements in π_{1}) will be called geometric generators. It is well-known that π_{1} is generated by geometric generators, and any two geometric generators are conjugated in π_{1}, since B is irreducible.

For each singular point s_{i} of B we choose a small neighborhood U_{i} ⊂ ℙ^{2} such that B ∩ U_{i} is defined (in local coordinates in U_{i}) by equation y^{2} = x^{3} if s_{i} is a cusp, and y^{2} = x^{2} if s_{i} is a node. Let p_{i} be a point in U_{i} \ B. It is well-known that if s_{i} is a cusp, then π_{1}(U_{i} \ B, p_{i}) is isomorphic to the braid group Br_{3} of 3-string braids and generated by two geometric generators (say a and b) satisfying the following relation

aba = bab.

If s_{i} is a node, then π_{1}(U_{i} \B, p_{i}) is isomorphic to ℤ⊕ℤ generated by two commuting geometric generators.

Let us choose smooth paths γ_{i} in ℙ^{2}\B joining p_{i} and p. This choice defines homomorphisms ψ_{i} : π_{1}(U_{i} \ B, p_{i}) → π_{1}. Denote the image ψ_{i}(π_{1}(U_{i} \ B, p_{i})) by G_{i} if s_{i} is a cusp, and Γ_{i} if s_{i} is a node.

A generic morphism of degree N determines a homomorphism φ : π_{1} → 𝔖_{N}, where 𝔖_{N} is the symmetric group. This homomorphism φ is determined uniquely up to inner automorphism of 𝔖_{N}.

Proposition 1 The set of the non-equivalent generic morphisms of degree N possessing the same discriminant curve B is in one to one correspondence with the set of the epimorphisms φ : π_{1}(ℙ^{2} \ B) → 𝔖_{N} (up to inner automorphisms of 𝔖_{N}) satisfying the following conditions:

(i) for a geometric generator γ the image φ(γ) is a transposition in 𝔖_{N};

(ii) for each cusp s_{i} the image φ(G_{i}) is isomorphic to 𝔖_{3} generated by two transpositions; (iii) for each node s_{i} the image φ(Γ_{i}) is isomorphic to 𝔖_{2} ×𝔖_{2} generated by two commuting

transpositions.

Proof. It is well-known that each homomorphism φ : π_{1} → 𝔖_{N} defines a finite morphism f : S → ℙ^{2} of degree N, unramified over the complement of B, and such that S is a normal surface, and vice versa.

Condition (i) is equivalent to one that f^{∗}(B) = 2R + C, where R is irreducible and C is reduced (cf. [10] and [11]).

Conditions (ii) and (iii) are equivalent to one that S and R are non-singular at the points in f^{−1}(s_{i}) (cf. [5]).

Since S is irreducible and deg f = N, φ(π_{1}) must act transitively on the set \overline{N} = \{1, ..., N\}, and since π_{1} is generated by the geometric generators, φ(π_{1}) is generated by some subset of transpositions. It is easy to check that subgroup of 𝔖_{N}, generated by some subset of transpositions acting on \overline{N} transitively, must coincide with 𝔖_{N}. Therefore φ must be an epimorphism.

Remark 1 By Proposition 1, if we allow N = 2, then a two-sheeted covering f : S → ℙ^{2}, branched along a non-singular curve B ⊂ ℙ^{2}, can be also considered as a generic morphism.

Remark 2 If B ⊂ ℙ^{2} is the dual curve of a smooth cubic, then by Proposition 1, using the presentation of π_{1}(ℙ^{2} \ B) obtained in [22], it is easy to show that B is the discriminant curve of exactly four generic non-equivalent morphisms.

1.3. Let s_{j} ∈ B be a cusp and (x, y) local coordinates around s_{j} such that B ∩ U_{j} is given by y^{2} = x^{3}. Choose a neighborhood V_{j} ⊂ S of p_{j} ∈ R, f(p_{j}) = s_{j}, such that f = f_{|V_{j}} : V_{j} → U_{j} is a three-sheeted covering ramified along R ∩ V_{j}. It is well known that f : V_{j} → U_{j} = U is unique up to equivalence and, in particular, f is equivalent to the standard covering \overline{f} : \overline{V} → U given by the normalized equation of third degree:

\{ \begin{matrix}\overline{V} = \{(w, x, y) | (x, y) ∈ U, w^{3} − 3xw + 2y = 0 \}; \\ \overline{f}(w, x, y) = (x, y) . \end{matrix}

\overline{V} is non-singular and (x, w) are local coordinates in \overline{V}. The ramification divisor \overline{R} = \{(x, w) ∈ \overline{V} | x−w^{2} = 0 \} of \overline{f} is smooth and \overline{f}^{−1}(B) = 2\overline{R}+\overline{C}, where \overline{C} = \{(x, w) ∈ \overline{V} | 4x−w^{2} = 0 \}. Note that \overline{R} is tangent to \overline{C} at the origin o = (0, 0), and the intersection multiplicity of \overline{R} and \overline{C} at o is equal to 2.

It was mentioned above that π_{1}(U \ B) ≃ Br_{3} =< a, b | aba = bab >. Then \overline{f} : \overline{V} → U corresponds to the homomorphism \overline{φ} : π_{1}(U \B) → 𝔖_{3} given by \overline{φ}(a) = (1, 2) and \overline{φ}(b) = (2, 3).

Put \overline{W} = \{(w_{1}, w_{2}, w_{3}) | w_{1} + w_{2} + w_{3} = 0 \}. We have morphisms f̃ : \overline{W} → U and g : \overline{W} → \overline{V} given by

\begin{aligned}x & = & −\frac{1}{3}(w_{1}w_{2} + w_{1}w_{3} + w_{2}w_{3}), \\ y & = & −\frac{1}{2}w_{1}w_{2}w_{3}, \\ w & = & w_{1} \end{aligned}

such that f̃ = \overline{f} ∘ g, deg f̃ = 6 and deg g = 2. The morphism g is a two-sheeted covering branched along \overline{C}, g^{∗}(\overline{C}) = 2C̃_{2}, where C̃_{2} is given in coordinates (w_{1}, w_{2}) by w_{1} + 2w_{2} = 0; and g^{∗}(\overline{R}) = R̃ + C̃_{1}, where R̃ and C̃_{1} are given by w_{1} = w_{2} and 2w_{1} + w_{2} = 0 respectively.

Note that f̃ corresponds to the homomorphism φ̃: π_{1}(U \ B) → 𝔖_{6} = 𝔖(𝔖_{3}) defined by \overline{φ}.

2 Proof of Theorem 1.

2.1. Assume that there exist two non-equivalent generic morphisms (S_{1}, f_{1}) and (S_{2}, f_{2}) with the same discriminant curve B, deg f_{1} = N_{1} and deg f_{2} = N_{2}. Put f_{1}^{∗}(B) = 2R_{1} + C_{1} and f_{2}^{∗}(B) = 2R_{2} + C_{2}. Let us consider

S_{1} ×_{ℙ^{2}} S_{2} = { (x, y) ∈ S_{1} × S_{2} | f_{1}(x) = f_{2}(y) }

and let X = { \atop S_{1} ×_{ℙ^{2}} S_{2}} be the normalization of S_{1} ×_{ℙ^{2}} S_{2}. Denote by g_{1} : X → S_{1}, g_{2} : X → S_{2}, and f_{1,2} : X → ℙ^{2} the corresponding natural morphisms. We have deg g_{1} = N_{2}, deg g_{2} = N_{1}, and deg f_{1,2} = N_{1}N_{2}.

Proposition 2 If (S_{1}, f_{1}) and (S_{2}, f_{2}) are non-equivalent, then X is irreducible.

Proof. The morphism f_{1,2} corresponds to the homomorphism

φ_{1,2} = φ_{1} × φ_{2} : π_{1} → 𝔖_{N_{1}} × 𝔖_{N_{2}} ⊂ 𝔖_{N_{1}N_{2}},

where φ_{1} : π_{1} → 𝔖_{N_{1}} (resp. φ_{2}) is an epimorphism corresponding to f_{1} (resp. f_{2}). Put G = φ_{1,2}(π_{1}). The group G, as a subgroup of 𝔖_{N_{1}} × 𝔖_{N_{2}}, acts on \overline{N_{1}} × \overline{N_{2}}. Without loss of generality we can assume that for some geometric generator γ the image φ_{1,2}(γ) = ((1, 2), (1, 2)) is a product of transpositions (1, 2) ∈ 𝔖_{N_{i}}. Let p_{i} : G → 𝔖_{N_{i}} be the restriction to G of the projection pr_{i} : 𝔖_{N_{1}} × 𝔖_{N_{2}} → 𝔖_{N_{i}}.

Lemma 10 Let G be a subgroup of 𝔖_{N_{1}} × 𝔖_{N_{2}}, N_{i} > 2, such that p_{i} : G → 𝔖_{N_{i}} is an epimorphism for i = 1, 2, and such that ((1, 2), (1, 2)) ∈ G. Let St_{(1,1)} ⊂ G be the stabilizer of (1, 1) ∈ \overline{N_{1}} × \overline{N_{2}}. Then the index of St_{(1,1)} in G

(G : St_{(1,1)}) = N_{1}N_{2}

except for the case when N_{1} = N_{2} = N and G = Δ ⊂ 𝔖_{N} × 𝔖_{N} (up to inner automorphism of one of factors), where Δ is the diagonal subgroup.

Proof. The inclusion \overline{N_{i} − 1} ≃ \{2, ... , N_{i}\} ⊂ \{1, 2, ... , N_{i}\} defines the embedding 𝔖_{N_{i}−1} ⊂ 𝔖_{N_{i}}. Then

St_{(1,1)} = G ∩ (𝔖_{N_{1}−1} × 𝔖_{N_{2}−1}).

Put

H_{1} × {e_{2}} = G ∩ (𝔖_{N_{1}} × {e_{2}})  and  {e_{1}} × H_{2} = G ∩ ({e_{1}} × 𝔖_{N_{2}}),

where e_{i} is the unit element of 𝔖_{N_{i}}. We note that ker p_{2} = H_{1} × {e_{2}} and ker p_{1} = {e_{1}} × H_{2}. Therefore H_{1} × {e_{2}} and {e_{1}} × H_{2} are normal subgroups of G. Since p_{i} : G → 𝔖_{N_{i}} is an epimorphism for i = 1, 2, H_{i} is a normal subgroup of 𝔖_{N_{i}}.

It is well known that if H is a normal subgroup of 𝔖_{N}, then either H = 𝔖_{N}, or H = 𝔄_{N} is the alternating group, or H = {e}, and if N = 4, then there exist one more possibility: H is the Klein four group

K_{4} = {e, (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)}.

Consider all possible cases.

Case I : H_{1} = 𝔖_{N_{1}}. Since p_{2} is an epimorphism, G = 𝔖_{N_{1}} × 𝔖_{N_{2}}. Therefore | G |= N_{1}!N_{2}! and | St_{(1,1)} |= (N_{1} − 1)!(N_{2} − 1)!. Hence (G : St_{(1,1)}) = N_{1}N_{2}.

Case II : H_{1} = 𝔄_{N_{1}}. Since p_{2} is an epimorphism and ker p_{2} = H_{1} × {e},

| G |= \frac{N_{1}!N_{2}!}{2}.

Similarly, if we consider p_{1}, then we obtain | ker p_{1}| = N_{2}!/2, hence H_{2} = 𝔄_{N_{2}}. Therefore (σ_{1}, σ_{2}) ∈ G if and only if σ_{1} and σ_{2} have the same sign. Hence

| St_{(1,1)} |= \frac{(N_{1} − 1)!(N_{2} − 1)!}{2}

and (G : St_{(1,1)}) = N_{1}N_{2}.

Case III : H_{1} = {e_{1}}. Therefore p_{2} is an isomorphism and there exist two possibilities: either H_{2} = {e_{2}} and p_{1} is also an isomorphism or H_{2} ≠ {e_{2}}. If p_{1} and p_{2} are isomorphisms, then N_{1} = N_{2} = N and G = Δ ⊂ 𝔖_{N} × 𝔖_{N} up to automorphism of one of factors, and since ((1, 2), (1, 2)) ∈ G, this automorphism must be inner. If H_{2} ≠ {e_{2}}, then p_{1} ∘p_{2}^{−1} : 𝔖_{N_{2}} → 𝔖_{N_{1}} is an epimorphism (not isomorphism). Since N_{i} > 2, 𝔖_{N_{2}} must coincide with 𝔖_{4}, 𝔖_{N_{1}} = 𝔖_{3} and H_{2} = K_{4}. The rest of the proof of this case will be left to the reader.

Case IV : N_{1} = 4 and H_{1} = K_{4}. The case H_{2} = 𝔖_{N_{2}} is impossible. In fact, if we consider p_{1}, then we obtain | G |= 4!N_{2}!. On the other hand, if to consider p_{2}, then | G |= 4N_{2}!, a contradiction.

The case H_{2} = 𝔄_{N_{2}} is also impossible. In fact, if we consider p_{1}, then | G |= 4!N_{2}!/2. On the other hand, if to consider p_{2}, then | G |= 4N_{2}!, a contradiction.

The case H_{2} = {e_{2}} coincides (up to indexing) with one in Case III.

The case N_{2} = 4 and H_{2} = K_{4} will be left to the reader.

To complete the proof of Proposition 2 we note that deg f_{1,2} = N_{1}N_{2} and there exists an irreducible component X_{(1,1)} of X such that deg f_{|X_{(1,1)}} = (G : St_{(1,1)}). Hence, by Lemma 9, deg f_{|X_{(1,1)}} = N_{1}N_{2} and, consequently, X is irreducible always except the case when N_{1} = N_{2} = N and G ≃ Δ ⊂ 𝔖_{N} × 𝔖_{N}. But the exceptional case corresponds to one when (S_{1}, f_{1}) and (S_{2}, f_{2}) are equivalent.

Proposition 3 X is non-singular.

Proof. We need to check the smoothness of X only at z ∈ f_{1,2}^{−1}(B) such that p_{1} = g_{1}(z) ∈ R_{1} ⊂ S_{1} and p_{2} = g_{2}(z) ∈ R_{2} ⊂ S_{2}. Put f_{1,2}(z) = s and choose a small neighborhood V_{1} ⊂ S_{1} of p_{1} (resp. V_{2} ⊂ S_{2} of p_{2}) and a small neighborhood U ⊂ ℙ^{2} of s such that f_{i}(V_{i}) = U and, in the chosen neighborhoods, there exist local holomorphic coordinates for which equations defining f_{i} have the simplest form.

Let s ∈ B be a non-singular point of B or a node. Then f_{i} : V_{i} → U is given by

\begin{aligned}u_{i,1}^{2} & = & v_{1} ; \\ u_{i,2} & = & v_{2} , \end{aligned}

where v_{1} = 0 is an equation of B ∩ U (or one of the branches of B if s is a node). Therefore V_{1} ×_{U} V_{2} in V_{1} × V_{2} is given by equations

\begin{aligned}u_{1,1}^{2} & = & u_{2,1}^{2} ; \\ u_{1,2} & = & u_{2,2} , \end{aligned}

or, equivalently,

\begin{aligned}u_{1,1} & = & ±u_{2,1} ; \\ u_{1,2} & = & u_{2,2} . \end{aligned}

Hence V_{1} ×_{U} V_{2} consists of two irreducible non-singular components one of which corresponds to the sign + and the other does to −. Therefore the normalization { \atop V_{1} ×_{U} V_{2}} of V_{1} ×_{U} V_{2} is the disjoint union of two non-singular surfaces.

Let s = s_{j} ∈ B be a cusp and (x, y) local coordinates around s chosen in section 1.3. Let V_{1} ⊂ S_{1} (resp. for S_{2}) be a neighborhood of p_{1} = p_{1,j} such that f_{1} = f_{1|V_{1}} : V_{1} → U is a threesheeted covering ramified along R_{1} ∩ V_{1}. Put Y = V_{1} ×_{U} V_{2} and let Ỹ be the normalization of Y. Denote by g_{i} : Ỹ → V_{i} and f_{1,2} : Ỹ → U the corresponding natural morphisms. Since (V_{1}, f_{1}) and (V_{2}, f_{2}) are equivalent, f_{1,2} corresponds to the homomorphism

φ_{1,2} = φ_{1} × φ_{2} : π_{1}(U \ B) → Δ ⊂ 𝔖_{3} × 𝔖_{3} ⊂ 𝔖_{9},

thus, φ_{1,2}(π_{1}(U \ B)) acts on \overline{9} ≃ \overline{3} × \overline{3}. It is easy to check that there are two orbits of the action of φ_{1,2}(π_{1}(U \ B)): one of them is the orbit of (1, 1), and the other one is the orbit of (1, 2). Therefore Ỹ is the disjoint union of Ỹ_{(1,1)} and Ỹ_{(1,2)}. It is easily seen (cf. Lemma 1.6 in [5]) that (Ỹ_{(1,1)}, f_{1,2}) is isomorphic to (\overline{V} , \overline{f}) (in notation of 1.3) and g_{i} : Ỹ_{(1,1)} → V_{i} is an isomorphism for i = 1, 2; (Ỹ_{(1,2)}, f_{1,2}) is isomorphic to (\overline{W} , f̃), and each (Ỹ_{(1,2)}, g_{i}) is isomorphic to (\overline{W} , g). Hence X is non-singular.

Remark 3 If (S_{1}, f_{1}) and (S_{2}, f_{2}) with the same discriminant curve are equivalent, then 1) X is non-singular;

2) X is the disjoint union of two irreducible components: X = X_{(1,1)} ⨆ X_{(1,2)}, such that g_{i|X_{(1,1)}} : X_{(1,1)} → S_{i} is an isomorphism, i = 1, 2, and deg g_{i|X_{(1,2)}} = N −1, where N = N_{1} = N_{2}.

2.2. Let R̃ ⊂ X be a curve g_{1}^{−1}(R_{1})∩g_{2}^{−1}(R_{2}), C̃ = g_{1}^{−1}(C_{1})∩g_{2}^{−1}(C_{2}), C̃_{1} = g_{1}^{−1}(R_{1})∩g_{2}^{−1}(C_{2}), and C̃_{2} = g_{1}^{−1}(C_{1}) ∩ g_{2}^{−1}(R_{2}).

Proposition 4

\begin{aligned}R̃^{2} & = & 2(3d + g − 1) − c , \\ C̃_{1}^{2} & = & (N_{2} − 2)(3d + g − 1) − c , \\ C̃_{2}^{2} & = & (N_{1} − 2)(3d + g − 1) − c , \\ (R̃, C̃_{i}) & = & c for i = 1, 2. \end{aligned}

Proof. It follows from the local considerations in the proof of Proposition 3 that deg g_{1|R̃} = 2 and g_{1|R̃} is étale.

One can check that R̃ and C̃_{i} are intersected only at points over the cusps of B. Consider one of cusps of B, say s = s_{j}, and p_{i} = p_{i,j} ∈ R_{i}∩f^{−1}(s). In notation of the proof of Proposition 3, let U be a neighborhood of s. It is easily seen that one of the branches of R̃ ∩ Ỹ lies in Ỹ_{(1,1)} and the other one does in Ỹ_{(1,2)}. Since (Ỹ_{(1,1)}, f_{1,2}) is isomorphic to (V_{i}, f_{i}), the intersection Ỹ_{(1,1)} ∩ C̃_{i} = ∅ for i = 1, 2. Consider Ỹ_{(1,2)} ∩ R̃ and Ỹ_{(1,2)} ∩ C̃_{i}. Since each (Ỹ_{(1,2)}, g_{i}) is equivalent to (\overline{W} , g), where (\overline{W} , g) was defined in section 1.3, we can identify (Ỹ_{(1,2)}, g_{1}) with (\overline{W} , g). Then Ỹ_{(1,2)} ∩ R̃, Ỹ_{(1,2)} ∩ C̃_{1}, and Ỹ_{(1,2)} ∩ C̃_{2} can be identified, respectively, with R̃, C̃_{1}, and C̃_{2} ⊂ \overline{W}. Since in the neighborhood \overline{W} the intersection multiplicity of R̃ and C̃_{i} is equal to 1, hence (R̃, C̃_{i}) = c.

To calculate R̃^{2}, consider again the local case. Let g : \overline{W} → V, g(0) = 0, be a two-sheeted covering given in local analytic coordinates by

\begin{aligned}w_{1}^{2} & = & v_{1} ; \\ w_{2} & = & v_{2} , \end{aligned}

Denote by C ⊂ V a curve given by v_{1} = 0 and R given by v_{2}^{2} = v_{1}. Then C is the branch curve, g^{∗}(C) = 2C̃_{2}, and g^{∗}(R) = R̃ + C̃_{1}, where C̃_{2} is given by w_{1} = 0, and R̃ and C̃_{1} are given by w_{2} = ±w_{1}. Let σ : Ṽ → V be the composition of two σ-processes with centers at points and such that σ^{−1}(R + C) = R + C + L_{1} + L_{2} is a divisor with normal crossings, where L_{1} is the exceptional divisor of the first σ-process, L_{2} is the exceptional divisor of the second σ-process, and, for simplicity of notation, we again denote by R and C, respectively, the strict preimages of R ⊂ V and C ⊂ V. Since we performed two σ-processes with centers at points lying in R, R^{2} is decreased by 2 (if R is considered as a complete curve). We can perform two σ-processes σ̃ : W̃ → \overline{W} (the first one with center at the origin and the second one with center at the intersection point of the strict preimage of the curve {w_{2} = 0} and the exceptional divisor of the first σ-process). It is easy to check that we again obtain a morphism \overline{g} : W̃ → Ṽ. Since we performed only one σ-process with center at a point lying in R̃, R̃^{2} is decreased by 1. Besides, \overline{g}_{|R̃} : R̃ → R is an isomorphism (locally), and \overline{g} is non-ramified at each point lying in R̃.

The considerations described above allow us to calculate R̃^{2}. Indeed, performing at each point p_{1,j} ∈ R_{1} two σ-processes as above, R_{1}^{2} is decreased by 2c. Performing at each point of g_{1}^{−1}(p_{1,j})∩R̃ two σ-processes either as above or if a neighborhood of the point in consideration is isomorphic to Ỹ_{(1,1)} we perform σ-processes as in V_{1} as, in view of the fact that g_{1} : Ỹ_{(1,1)} → V_{1} is an isomorphism. After performing all these σ-processes, R̃^{2} is decreased by 3c, and we can find a neighborhood Ṽ_{1} of the strict preimage of R_{1} and a neighborhood W̃ of the strict preimage of R̃ such that the restriction g̃_{1|W̃} : W̃ → Ṽ of the obtained morphism g̃_{1} to W̃ is a non-ramified two-sheeted covering. Hence

R̃^{2} − 3c = 2(R_{1}^{2} − 2c).

Thus, by Lemma 4, R̃^{2} = 2(3d + g − 1) − c.

Since deg g_{1} = N_{2},

N_{2}R_{1}^{2} = (R̃ + C̃_{1}, R̃ + C̃_{1}) = R̃^{2} + 2(R̃, C̃_{1}) + C̃_{1}^{2}.

Hence

C̃_{1}^{2} = (N_{2} − 2)(3d + g − 1) − c.

Proposition 4 is proved. 2.3. To complete the proof of Theorem 1, we apply Hodge’s Index Theorem. Since by Corollary 1, R̃^{2} > 0,

| \begin{matrix}R̃^{2} & (R̃, C̃_{i}) \\ (C̃_{i}, R̃) & C̃_{i}^{2} \end{matrix} | = | \begin{matrix}2(3d + g − 1) − c & c \\ c & (N_{j} − 2)(3d + g − 1) − c \end{matrix} | ≤ 0,

i.e.

2(N_{j} − 2)(3d + g − 1)^{2} − N_{j}(3d + g − 1)c ≤ 0 .

Hence

N_{j}[2(3d + g − 1) − c] ≤ 4(3d + g − 1) .

Thus, if there exist two non-equivalent generic morphisms (S_{1}, f_{1}) and (S_{2}, f_{2}), then their degrees

N_{j} ≤ \frac{4(3d + g − 1)}{2(3d + g − 1) − c}.

3 Uniqueness of generic morphisms for certain types of discriminant curves.

Let us write inequality (1) in the form

N[2(3d + g − 1) − c] − 4(3d + g − 1) > 0  (6)

3.1. Proof of Theorem 2. Put k = K_{S}^{2}, then E^{2} = m^{2}k and hence

N = deg f = m^{2}k.  (7)

We have K_{S} = f^{∗}(K_{ℙ^{2}}) + R = −3E + R, hence R ≡ (3m + 1)K_{S}. Therefore deg B = (E, R) = m(3m + 1)k, i.e.

d = \frac{m(3m + 1)k}{2} .  (8)

By adjunction formula, 2(g − 1) = (K_{S} + R, R). Therefore

g − 1 = \frac{(3m + 2)(3m + 1)k}{2} .  (9)

From (8) and (9) it follows that

3d + g − 1 = (3m + 1)^{2}k.  (10)

By Lemma 8,

c = (12m^{2} + 9m + 3)k − 12p_{a}.  (11)

Let us substitute (7), (10), and (11) into (6). We have

\begin{aligned}m^{2}k[(6m^{2} + 3m − 1)k + 12p_{a}] − 4(3m + 1)^{2}k & = \\ m^{2}k[(6m^{2} + 3m − 1)k + 12p_{a} − 4(3 + \frac{1}{m})^{2}] & > & 0. \end{aligned}

If m ≥ 2, then

[(6m^{2} + 3m − 1)k + 12p_{a} − 4(3 + \frac{1}{m})^{2}] ≥ 29k + 12p_{a} − 49 > 0,

and, since k ≥ 1 and p_{a} ≥ 1, the last inequality does not hold only for k = 1 and p_{a} = 1. In the exceptional case if m ≥ 3, then

[(6m^{2} + 3m − 1)k + 12p_{a} − 4(3 + \frac{1}{m})^{2}] > 62 + 12 − 45 > 0.

The case m = 2, k = p_{a} = 1 is impossible. Indeed, in this case by (8) - (11), we have deg B = 14, g = 29, and c = 57, and the number of nodes n must be non-negative. But by adjunction formula,

n = \frac{1}{2}(deg B − 1)(deg B − 2) − g − c = 13 · 6 − 29 − 57 < 0,

a contradiction. Therefore inequality (1) holds always for m ≥ 2.

If m = 1, then

[(6m^{2} + 3m − 1)k + 12p_{a} − 4(3 + \frac{1}{m})^{2}] = 8k + 12p_{a} − 64 > 0,

and, consequently, inequality (1) is equivalent to

2k + 3p_{a} > 16.

Since k = E^{2} = N ≥ 3, the last inequality does not hold only for 1) k = 3, p_{a} ≤ 3;

2) k = 4, p_{a} ≤ 2;

3) k = 5, p_{a} ≤ 2;

4) k = 6, p_{a} = 1.

The cases 1) and 2) are impossible. Indeed, by (8) - (11), deg B = 4k, g = 10k + 1, and c = 12(2k − p_{a}), and the number of nodes n must be non-negative. But by adjunction formula,

n = \frac{1}{2}(deg B−1)(deg B−2)−g−c = (4k−1)(2k−1)−10k−1−24k+12p_{a} = 4(2k^{2}−10k+3p_{a}) < 0

for k = 3, p_{a} ≤ 3 and k = 4, p_{a} ≤ 2.

The assertion on the degree of generic morphisms in the exceptional cases follows from Lemma 5. 3.2. Proof of Theorem 3. Put k = K_{S}^{2}, then E^{2} = m^{2}k and hence

N = deg f = m^{2}k.  (12)

We have K_{S} = f^{∗}(K_{ℙ^{2}}) + R = −3E + R, hence R ∈ |(−3m + 1)K_{S}|. Therefore deg B = (E, R) = m(3m − 1)k, i.e.

d = \frac{m(3m − 1)k}{2} .  (13)

By adjunction formula, 2(g − 1) = (K_{S} + R, R). Therefore

g − 1 = \frac{(3m − 2)(3m − 1)k}{2} .  (14)

It follows from (13) and (14) that

3d + g − 1 = (3m − 1)^{2}k.  (15)

By Lemma 8,

c = (12m^{2} − 9m + 3)k − 12.  (16)

Let us substitute (12), (15), and (16) into (6). We have

m^{2}k[2(3m − 1)^{2}k − (12m^{2} − 9m + 3)k + 12] − 4(3m − 1)^{2}k =
m^{2}k[(6m^{2} − 3m − 1)k + 12 − 4(3 − \frac{1}{m})^{2}]  =
m^{2}k[(6(m − \frac{1}{4})^{2} − \frac{11}{8})k + 12 − 4(3 − \frac{1}{m})^{2}]  > 0.

If m ≥ 3, then inequality (1) holds, since

[(6m^{2} − 3m − 1)k + 12 − 4(3 − \frac{1}{m})^{2}] > 26k + 12 − 36 > 0.

If m = 2, then inequality (1) holds, since

[(6m^{2} − 3m − 1)k + 12 − 4(3 − \frac{1}{m})^{2}] = 17k + 12 − 25 > 0.

If m = 1, then inequality (1) also holds, since in this case k ≥ 3, hence

[(6m^{2} − 3m − 1)k + 12 − 4(3 − \frac{1}{m})^{2}] = 2k + 12 − 16 > 0.

3.3. Proof of Theorem 4. Let f^{−1}(ℙ^{1}) = E ∈ |aL_{1} + bL_{2}|, where L_{1} and L_{2} are the natural generators of P ic S. Without loss of generality, we can assume that a ≥ b > 0. Then

N = deg f = 2ab.  (17)

We have K_{S} = −2L_{1}−2L_{2} and K_{S} = f^{∗}(K_{ℙ^{2}})+R = −3E+R, hence R ∈ |(3a−2)L_{1}+(3b−2)L_{2}|. Therefore deg B = (E, R) = a(3b − 2) + b(3a − 2), i.e.

d = 3ab − a − b.  (18)

By adjunction formula, 2(g − 1) = (K_{S} + R, R). Therefore

g − 1 = 9(ab − a − b) + 8.  (19)

It follows from (18) and (19) that

3d + g − 1 = 18ab − 12a − 12b + 8.  (20)

By Lemma 7,

c = 24ab − 18a − 18b + 12.  (21)

Let us substitute (17), (20), and (21) into (6). We have

\begin{aligned}2ab[12ab − 6a − 6b + 4] − 4(18ab − 12a − 12b + 8) & = \\ 4ab[3a(b − 1) + 3b(a − 1) + \frac{12}{b} + \frac{12}{a} − 16 − \frac{8}{ab}] & > & 0 . \end{aligned}

If a ≥ b ≥ 2, then inequality (1) holds, since

3a(b − 1) + 3b(a − 1) + \frac{12}{b} + \frac{12}{a} − 16 − \frac{8}{ab} ≥ 3a + 6(a − 1) + \frac{8}{a} + \frac{12}{b} − 16 = 9a + \frac{8}{a} + \frac{12}{b} − 22 > 0.

If a > b = 1, then inequality (1) also holds, since

3a(b − 1) + 3b(a − 1) + \frac{12}{b} + \frac{12}{a} − 16 − \frac{8}{ab} = 3(a − 1) + 12 + \frac{4}{a} − 16 = 3a + \frac{4}{a} − 7 > 0.

If a = b = 1, then f is a two-sheeted covering of ℙ^{2} branched along smooth conic. 3.4. Proof of Theorems 5-7. Let

deg f = N = E^{2} = 2k  (22)

(E^{2} is even, since 2K_{S} is trivial). We have K_{S} = f^{∗}(K_{ℙ^{2}}) + R = −3E + R, hence R ≡ 3E. Therefore deg B = 6k, i.e.

d = 3k.  (23)

By adjunction formula, 2(g − 1) = R^{2}. Therefore

g − 1 = 9k.  (24)

It follows from (23) and (24) that

3d + g − 1 = 18k.  (25)

By Lemma 8,

c = 24k − 12p_{a}.  (26)

Let us substitute (22), (25), and (26) into (6). We have

\begin{aligned}2k[12k + 12p_{a}] − 72k & = \\ 24k[k + p_{a} − 3] & > & 0 . \end{aligned}

and k ≥ 2, since N > 2. If S is a K3 surface, then inequality (1) holds, since p_{a} = 2.

If S is an Enriques surface, then p_{a} = 1 and inequality (1) also holds except, possibly, for k = 2. For abelian varieties (p_{a} = 0) inequality (1) holds except, possibly, for k = 2 and k = 3.

For abelian surfaces the case k = 2 is impossible, since such a curve B does not exist. Indeed, in this case it follows from (23), (24), and (26) that deg B = 12, g = 19, c = 48. Then

n = \frac{1}{2}(deg B − 1)(deg B − 2) − g − c = 55 − 19 − 48 < 0,

a contradiction.

For Enriques surface in the exceptional case we have deg f = 4. Consider the epimorphism φ : π_{1} = π_{1}(ℙ^{2} \ B) → 𝔖_{4} corresponding to f. It follows from (23), (24), and (26) that deg B = 12, g = 19, and c = 36, hence n = 0. Therefore, by Proposition 1, the epimorphism φ′ : π_{1} → 𝔖_{3}, which is the composition of φ and the natural epimorphism 𝔖_{4} → 𝔖_{3} = 𝔖_{4}/K_{4}, where K_{4} is the Klein four group, gives rise to a generic morphism f′ : S′ → ℙ^{2} of degree 3 with the same discriminant curve B.

The assertion on the degree of generic morphisms in the exceptional cases follows from Lemma 5. 3.5. Proof of Theorem 8. Let S = X(m_{1}, ..., m_{k}) ⊂ ℙ^{k+2} be a complete intersection of multidegree (m_{1}, ..., m_{k}), m_{i} > 1. Then for a generic projection onto ℙ^{2}

deg f = N = ∏_{i=1}^{k} m_{i} .  (27)

By adjunction formula, K_{S} = (m_{1} + ... + m_{k} − k − 3)E. Since K_{S} = f^{∗}(K_{ℙ^{2}}) + R = −3E + R, we have R = (m_{1} + ... + m_{k} − k)E. Therefore deg B = (m_{1} + ... + m_{k} − k)E^{2}, i.e.

d = \frac{1}{2}(∑_{i=1}^{k}(m_{i} − 1)) ∏_{i=1}^{k} m_{i} .  (28)

By adjunction formula, 2(g − 1) = R^{2} + (R, K_{S}). Therefore

g − 1 = \frac{1}{2}(∑_{i=1}^{k}(m_{i} − 1))(2 ∑_{i=1}^{k}(m_{i} − 1) − 3) ∏_{i=1}^{k} m_{i}.  (29)

It follows from (28) and (29) that

3d + g − 1 = (∑_{i=1}^{k}(m_{i} − 1))^{2} ∏_{i=1}^{k} m_{i}.  (30)

By Lemma 7,

c = 3N + 2(g − 1) − e(S) = 3 ∏_{i=1}^{k} m_{i} + (∑_{i=1}^{k}(m_{i} − 1))(2 ∑_{i=1}^{k}(m_{i} − 1) − 3) ∏_{i=1}^{k} m_{i} − e(S). (31)

Let us substitute (27), (30), and (31) into (6). We have

∏_{i=1}^{k} m_{i}[3 ∏_{i=1}^{k} m_{i}(∑_{i=1}^{k}(m_{i} − 1)) + e(S) − 4(∑_{i=1}^{k}(m_{i} − 1))^{2}] > 0 .

If k ≥ 2 , then

[3 ∏_{i=1}^{k} m_{i}(∑_{i=1}^{k}(m_{i} − 1)) + e(S) − 4(∑_{i=1}^{k}(m_{i} − 1))^{2}] > (∑_{i=1}^{k}(m_{i} − 1))[3 ∏_{i=1}^{k} m_{i} − 4(∑_{i=1}^{k}(m_{i} − 1))] > 0

and, obviously, the last inequality is true.

If k = 1 , then e(S) = m_{1}^{3} − 4m_{1}^{2} + 6m_{1}, hence

3 ∏_{i=1}^{k} m_{i}(∑_{i=1}^{k}(m_{i} − 1)) + e(S) − 4(∑_{i=1}^{k}(m_{i} − 1))^{2} =
3m_{1}(m_{1} − 1) + m_{1}^{3} − 4m_{1}^{2} + 6m_{1} − 4(m_{1} − 1)^{2} =
m_{1}^{3} − 5m_{1}^{2} + 11m_{1} − 4  =
(m_{1} − 2)^{3} + (m_{1} − 2)^{2} + 3(m_{1} − 2) + 6  ≥ 6 > 0

for m ≥ 2, i.e. inequality (1) also holds in the case k = 1. 3.6. Proof of Theorem 9. Put k = K_{S}^{2}, L^{2} = a, and (K_{S}, L) = b. Note that a > 0. If E = mL, then

N = deg f = m^{2}a.  (32)

We have K_{S} = f^{∗}(K_{ℙ^{2}}) + R = −3E + R, i.e. R = K_{S} + 3mL. Therefore

\begin{aligned}R^{2} & = & 9m^{2}a + 6mb + k, \\ (K_{S}, R) & = & 3mb + k. \end{aligned}

By adjunction formula,

2(g − 1) = (K_{S} + R, R) = 9m^{2} + 9mb + 2k  (33)

and by Lemma 4,

3d + g − 1 = 9m^{2}a + 6mb + k.  (34)

By (32), (33), and Lemma 7,

c = 12m^{2}a + 9mb + 2k − e(S).  (35)

Let us substitute (32), (34), and (35) into (6). We have

\begin{aligned}m^{2}a[6m^{2}a + 3mb + e(S)] − 4(9m^{2}a + 6mb + k) & = \\ m^{2}a[6m^{2}a + 3mb + e(S) − 36a − \frac{24b}{ma} − \frac{4k}{m^{2}a}] & > & 0, \end{aligned}

Now it is clear that there exist a constant m_{0} such that the last inequality holds for m ≥ m_{0}.

The proof in the case L = K_{S} coincides with one of Theorem 2. 3.7. Proof of Theorem 10. Let B^{∗} ⊂ ℙ^{2∗} be a nodal curve of genus g, deg B^{∗} = δ, and B the dual curve of B^{∗}. Then (cf. [5]) B is the discriminant curve of some generic morphism f of degree δ. In fact, let S be the normalization of

X = { ((x_{1}, x_{2}, x_{3}), (y_{1}, y_{2}, y_{3})) ∈ ℙ^{2} × ℙ^{2∗} | ∑_{i} x_{i}y_{i} = 0 , (y_{1}, y_{2}, y_{3})) ∈ B^{∗} }

and f be induced by the projection pr_{1} : P^{2} × ℙ^{2∗} → P^{2}. It is clear that (x_{1}, x_{2}, x_{3}) ∉ B iff ∑_{i} x_{i}y_{i} = 0 is not tangent to B^{∗}, i.e. f^{−1}((x_{1}, x_{2}, x_{3})) has exactly δ points and

N = deg f = δ,  (36)

hence B is the branch curve of f and it is clear that f is a generic morphism.

It follows from Plücker’s formulas that

d = δ + (g − 1)  (37)

and

c = 3δ + 6(g − 1).  (38)

Let us substitute (36) - (38) into inequality (1) :

δ > \frac{4(3δ + 4(g − 1))}{3δ + 2(g − 1)} = 8 − \frac{12δ}{3δ + 2(g − 1)}.

Thus, inequality (1) holds for δ ≥ 8.

Consider the cases δ ≤ 7.

The case δ = 7. Inequality (1) does not hold iff

\frac{12δ}{3δ + 2(g − 1)} ≤ 1 ⟹ 9δ ≤ 2(g − 1) ⟹ g ≥ 33.

On the other hand,

g ≤ \frac{(δ − 1)(δ − 2)}{2} = \frac{6 · 5}{2} < 33.

Thus, in the case δ = 7 inequality (1) is true.

The case δ = 6. Inequality (1) does not hold iff

\frac{12δ}{3δ + 2(g − 1)} ≤ 2 ⟹ 3δ ≤ 2(g − 1) ⟹ g ≥ 10.

On the other hand,

g ≤ \frac{(δ − 1)(δ − 2)}{2} = \frac{5 · 4}{2} = 10.

Thus, in the case δ = 6 inequality (1) is not true iff ν = 0, i.e., possibly, there exist two non-equivalent generic morphisms only when B possesses the following invariants: deg B = 30, g = 10, c = 72, n = 324. For such B, if there exist another (non-equivalent to f) generic morphism f_{1} of deg f_{1} = N_{1}, then

N_{1} ≤ \frac{4(3d + (g − 1))}{6d + 2(g − 1) − c} = \frac{4(3 · 15 + (10 − 1))}{6 · 15 + 2(10 − 1) − 72} = 6.

The case δ = 5. Inequality (1) does not hold iff

\frac{12δ}{3δ + 2(g − 1)} ≤ 3 ⟹ δ ≤ 2(g − 1) ⟹ g ≥ 4.

On the other hand,

g ≤ \frac{(δ − 1)(δ − 2)}{2} = \frac{5 · 4}{2} = 6.

Thus, in the case δ = 5 inequality (1) is not true iff either ν = 0, or ν = 1, or ν = 2, i.e., possibly, there exist two non-equivalent generic morphisms only in the following three cases:

0) deg B = 20, g = 6, c = 45

1) deg B = 18, g = 5, c = 39.

2) deg B = 16, g = 4, c = 33. In all cases 0) - 2) if there exist two non-equivalent generic morphisms f and f_{1} with the same discriminant curve B, then the computation, similar to one described above, gives rise to deg f_{1} = N_{1} ≤ 5.

The case δ = 4. We have g ≤ 3. If g = 3, then deg B = 12 and c = 24. It is easy to check that if there exist two non-equivalent generic morphisms f and f_{1} with such discriminant curve B, then deg f_{1} = N_{1} ≤ 5. Let us show that such a curve can not be the discriminant curve of a generic morphism f_{1} : S_{1} → ℙ^{2} of degree 5. In fact, in this case by Lemmas 6 - 8, K_{S_{1}}^{2} = −7, e(S_{1}) = −5, and p_{a} = −1. Hence S_{1} is a ruled surface over a curve C of genus g = 2. Let \overline{S}_{1} be a relatively minimal model of S_{1}. Then e(\overline{S}_{1}) = −4, hence e(S_{1}) ≥ −4, a contradiction.

The case g ≤ 2 will be considered in the proof of Theorem 11. 3.8. Proof of Theorem 11. In the next subsection we shall prove Theorem 12. By that theorem, the Chisini Conjecture holds for a curve B of g ≤ 1, and if in the case g = 2 there exist two non-equivalent generic morphisms f_{1} and f_{2}, then d ≤ 3.

Consider the case g = 2 and d = 3. Then the inequality opposite to (1) takes the following form

N_{i} ≤ \frac{40}{20 − c}.

Therefore, if N_{i} ≥ 5, then

5 ≤ \frac{2c}{20 − c} ⟺ 12 ≤ c ≤ 19.

On the other hand,

c = (2d − 1)(d − 1) − g − n = 8 − n ≤ 8,

a contradiction.

A curve B with invariants g = 2 and deg B = 4 (i.e. d = 2) can not be a discriminant curve. Indeed, in this case either c = 1, or n = 1, which contradicts Corollary 2.

Consider the case g = 3. By Theorem 12, if there exist two non-equivalent generic morphisms f_{1} and f_{2} such that N_{1} ≥ 5, then d ≤ 6. By Lemmas 2 and 3,

\frac{3}{2}d + 1 ≤ c ≤ 3d + 6,  (39)

and the inequality opposite to (1) takes the following form

5 ≤ \frac{12d + 8}{6d + 4 − c} ⟺ \frac{18d + 12}{5} ≤ c ≤ 6d + 3.  (40)

If d = 6, then by (39), 10 ≤ c ≤ 24. On the other hand, by (40), c ≥ 24 and, consequently, c = 24 and n = 28. Thus, we obtain the case considered already in the proof of Theorem 10.

If d = 5, then by (39), 9 ≤ c ≤ 21. On the other hand, by (40), c ≥ 21 and, consequently, c = 21 and n = 12. By Plücker’s formula, δ = 10 · 9 − 2 · 12 − 3 · 21 = 3, contrary to g = 3.

If d = 4, then by (39), 7 ≤ c ≤ 18. On the other hand, by (40), c ≥ 17 and, consequently, by Corollary 2, c = 18 and n = 0. By Plücker’s formula, δ = 8 · 7 − 3 · 21 ≤ 0, a contradiction.

If d = 3, then by (39), 6 ≤ c ≤ 15. On the other hand, by (40), c ≥ 14 and, consequently, by Corollary 2, c = 15, which contradicts the inequality g = (2d − 1)(d − 1) − c − n ≥ 0.

If d = 2, i.e. deg B = 4. Hence B is non-singular. Thus, B can not be a discriminant curve. 3.9. Proof of Theorem 12. Consider again the inequality opposite to (1)

N_{i} ≤ \frac{4(3d + (g − 1))}{6d + 2(g − 1) − c}.

Thus, if there exist two non-equivalent generic morphisms such that one of them has degree N_{i} ≥ 5, then

\frac{4(3d + (g − 1))}{6d + 2(g − 1) − c} ≥ 5  ⟹
5c ≥ 18d + 6(g − 1) ⟹ (by Lemma 2)
15d + 15(g − 1) ≥ 18d + 6(g − 1) ⟺
3(g − 1) ≥ d.

4 Canonical discriminant curves.

4.1. A curve B is said to be a(n m-)canonical discriminant curve if B is the discriminant curve of a generic morphism f : S → ℙ^{2} given by a linear subsystem {E} ⊆ |mK_{S}|, m ∈ ℕ.

Let B ⊂ ℙ^{2} be a curve of even degree 2d with ordinary cusps and nodes as the only singularities and ν̃ : R → B the normalization. Put 𝔢 = ν̃^{−1}(ℙ^{1} ∩ B), 𝔠 = 2 ∑′ ν̃^{−1}(s_{i}), and 𝔫 = ∑′′ ν̃^{−1}(s_{i}), where we denote by ∑′ (resp. by ∑′′) summation over all cusps (resp. nodes) s_{i} ∈ B.

Proposition 5 Let B and R be as above. If B is a canonical discriminant curve, then

(i) \frac{2d}{g − 3d − 1} := m ∈ ℕ ,

(ii) \frac{(g − 3d − 1)^{2}}{3d + g − 1} := k ∈ ℕ ,

(iii) \frac{4d^{2}}{3d + g − 1} := N ∈ ℕ ,

(iv) N + \frac{3g − 3 − 9d − c}{12} := p_{a} ∈ ℕ ,

(v) There exists a divisor 𝔨 ∈ P ic R such that

K_{R} = (3m + 2)𝔨  and  𝔢 = m𝔨.

Besides,

dim H^{0}(R, O_{R}(r𝔨)) = \frac{r(r − 1)}{2}k + p_{a}

for r = 2, ... , 3m.

(vi) 𝔠 + 𝔫 = [(2d − 6)m − 2]𝔨 ,

Proof. By (8) and (9),

\{ \begin{matrix}2g − 2 & = & (9m^{2} + 9m + 2)k \\ 6d & = & (9m^{2} + 3m)k \end{matrix} ⟹

\begin{aligned}g − 1 − 3d & = & (3m + 1)k & ⟹ \\ m & = & \frac{2d}{g − 3d − 1}. \end{aligned}

Now (ii) follows from (8), (iii) follows from (7), and (iv) does from (11)

The element 𝔨 ∈ P ic R is the restriction of K_{S} to R. Since K_{S} is ample,

dim H^{1}(S, O_{S}(−rK_{S})) = 0

for r > 0. Therefore from the exact sequence

0 ⟶ O_{S}((r − 3m − 1)K_{S}) ⟶ O_{S}(rK_{S}) ⟶ O_{R}(rK_{S}) ⟶ 0

it follows that H^{0}(S, O_{S}(rK_{S})) is isomorphic to H^{0}(R, O_{R}(rK_{S})) for 1 ≤ r ≤ 3m. By Riemann-Roch formula, we have

dim H^{0}(S, O_{S}(rK_{S})) = \frac{r(r − 1)}{2}K_{S}^{2} + p_{a}

for r > 1.

To prove (vi), let us blow up all singular points of B. Denote the composition of these c+n σ-processes by σ : ℙ̃^{2} → ℙ^{2} and let L_{i} = σ^{−1}(s_{i}) be the exceptional curve over s_{i} ∈ Sing B. Then σ^{∗}(B) = R + 2 ∑ L_{i}, where the strict preimage R of B is a non-singular curve, since all singularities of B are ordinary cusps and nodes. We have

\begin{aligned}Kℙ̃^{2} & = & −3σ^{∗}(ℙ^{1}) + ∑ L_{i} , \\ R & = & deg Bσ^{∗}(ℙ^{1}) − 2 ∑ L_{i} . \end{aligned}

Hence by adjunction formula,

O_{R}(K_{R}) = O_{R}(Kℙ̃^{2} + R) = O_{R}((deg B − 3)σ^{∗}(ℙ^{1}) − ∑ L_{i}).

But O_{R}(𝔠 + 𝔫) = O_{R}(∑ L_{i}). Therefore

O_{R}(𝔠 + 𝔫) = O_{R}(((2d − 6)m − 2)𝔨),

since O_{R}(σ^{∗}(ℙ^{1})) = O_{R}(𝔢) = O_{R}(m𝔨) and O_{R}(K_{R}) = O_{R}((3m + 2)𝔨) .

Proposition 6 Let B satisfy conditions (i) - (vi) of Proposition 5. If B is a discriminant curve of a generic morphism f : S → ℙ^{2} of deg f = N, then S is a surface of general type with ample canonical bundle and f is given by a three-dimensional linear subsystem of |mK_{S} + α|, where α ∈ P ic S is an element of order 2. Moreover, if m is even, then α = 0.

Proof. Let f be given by linear subsystem of |E|. We have

E^{2} = N = \frac{4d^{2}}{3d + g − 1},

R^{2} = 3d + g − 1 and (E, R) = 2d. Then

| \begin{matrix}E^{2} & (E, R) \\ (R, E) & R^{2} \end{matrix} | = | \begin{matrix}\frac{4d^{2}}{3d + g − 1} & 2d \\ 2d & 3d + g − 1 \end{matrix} | = 0.

Therefore by Hodge’s Index Theorem,

R ≡ \frac{3d + g − 1}{2d}E ≡ \frac{3m + 1}{m}E.

Since K_{S} = −3E + R, we have

\begin{aligned}E & ≡ & mK_{S}, \\ R & ≡ & (3m + 1)K_{S}, \end{aligned}

and if E = mK_{S} + α, where α ≡ 0, and O_{R}(K_{S}) = O_{R}(𝔨 + β), where deg β = 0, then

O_{R}(E) = O_{R}(mβ + α + m𝔨) = O_{R}(m𝔨)

and

O_{R}(R) = O_{R}(K_{S} + 3E) = O_{R}((3m + 1)β + 3α + (3m + 1)𝔨).

Hence O_{R}(K_{R}) = O_{R}((3m + 2)β + 3α + (3m + 2)𝔨). Therefore

\begin{aligned}O_{R}(mβ) & = & O_{R}(−α), \\ O_{R}((3m + 2)β) & = & O_{R}(−3α). \end{aligned}

Hence 2β = 0 and O_{R}(2α) = O_{R}, and if m is even, then O_{R}(α) = O_{R}.

Proposition 6 will be proved once we prove

Lemma 11 (cf. Appendix to Chapter V in [23]) Let S be a smooth projective surface and i : R ↪ S a smooth irreducible curve. If O_{S}(R) is ample, then i^{∗} : P ic^{0} S → P ic R is injective, where P ic^{0} S ⊂ P ic S is the subgroup of numerically equivalent to zero classes of divisors.

Proof. In the commutative diagram

[figure: 0 = H^{1}(S, O_{S}(−R)); P ic^{0} S; 0; H^{1}(S, ℤ); H^{1}(S, O_{S}); j^{∗}; H^{1}(S, O_{S}^{∗}); i^{∗}; H^{2}(S, ℤ); 0; H^{1}(R, ℤ); H^{1}(R, O_{R}); H^{1}(R, O_{R}^{∗}); H^{2}(R, ℤ)]

with exact rows, the morphism j^{∗} is embedding. Therefore if for α ∈ P ic^{0} S the image i^{∗}(α) = 0, then rα = 0 for some r ∈ ℕ, since T ors H^{2}(S, ℤ) is a finite abelian group and H^{1}(S, ℤ) → H^{1}(R, ℤ) is embedding. An element α ∈ P ic S of finite order r defines a non-ramified abelian covering φ : S_{r} → S of deg φ = r. If α ≠ 0 and i^{∗}(α) = 0, then φ^{−1}(R) is the disjoint union of r irreducible curves, which contradicts the ampleness of R. 4.2. In notation of section 4.1, consider the natural homomorphism

μ : Sim^{2}H^{0}(R, O_{R}(𝔢)) ⟶ H^{0}(R, O_{R}(2𝔢)).

The kernel ker μ generates the homogeneous ideal I in the homogeneous coordinate ring ℜ = ⊕Sym^{r}H^{0}(R, O_{R}(𝔢)) of the projective space ℙ = ℙH^{0}(R, O_{R}(𝔢)). Put S_{I} = P roj ℜ/I. The normalization ν̃: R → B determines a 3-dimensional subspace

L = ν̃^{∗}(H^{0}(B, O_{B}(ℙ^{1} ∩ B))) ⊆ H^{0}(R, O_{R}(𝔢)).

The subspace L defines a projection pr : ℙ → ℙ^{2} with base locus ℙL ⊂ ℙ. Let f_{I} : S_{I} → ℙ^{2} be the restriction of pr to S_{I}.

Proposition 7 Let B satisfy conditions (i) - (vi) of Proposition 5. Assume that m is even and ≥ 21. Then the curve B is an m-canonical discriminant curve if and only if S_{I} is a non-singular surface of deg S_{I} = N and f_{I} is a generic morphism with discriminant curve B.

Proof. Let A be a very ample divisor on S and D a numerically effective one. By [7], if the embedding of S into ℙ is given by |K_{S} +4A+D|, then the homogeneous ideal I(S) is generated by quadrics.

By [3], A = 5K_{S} is very ample (we assume that K_{S} is ample). Therefore, if S is embedded by |mK_{S}|, m ≥ 21, then I(S) is generated by quadrics. On the other hand, R ∈ |(3m + 1)K_{S}| and R ⊂ S ⊂ ℙ is embedded by |m𝔨|. As it was mentioned in the proof of Proposition 5, the restriction map

H^{0}(S, O_{S}(rmK_{S})) ⟶ H^{0}(R, O_{R}(rmK_{S})) = H^{0}(R, O_{R}(r𝔢))

is an isomorphism for r = 1, 2. Hence the set of quadrics containing S coincides with the one containing R, and Proposition 7 follows from Propositions 5 and 6.

Remark 4 Proposition 7 holds also for odd m ≥ 21 if in view of Proposition 6 we slightly change the definition of m-canonical discriminant curves. Moreover, Proposition 7 is a particular case of more general assertion.

Proposition 7′ Let B satisfy conditions (i) - (iv) of Proposition 5. Assume that m ≥ 21. Then the curve B is the discriminant curve of morphism given by three-dimensional subsystem of |L|, where L is numerically equivalent to the mth canonical class, if and only if S_{I} is a non-singular surface of deg S_{I} = N and f_{I} is a generic morphism with discriminant curve B.

5 On Zariski’s pairs

5.1. The set of plane curves of degree 2d is naturally parameterized by the points in ℙ^{d(2d+3)}. The subset of plane irreducible curves of degree 2d and genus g with c ordinary cusps and some nodes, as the only singularities, corresponds to a quasi-projective subvariety M(2d, g, c) ⊂ P^{d(2d+3)} ([21]). One can show that if two non-singular points of the same irreducible component of M_{red}(2d, g, c) correspond to curves B_{1} and B_{2}, then the pairs (ℙ^{2}, B_{1}) and (ℙ^{2}, B_{2}) are diffeomorphic. In particular, in this case the fundamental groups π_{1}(ℙ^{2} \ B_{1}) and π_{1}(ℙ^{2} \ B_{1}) are isomorphic.

The following Proposition is a simple consequence of Proposition 1 and local considerations in 1.2 and 1.3.

Proposition 8 Let (ℙ^{2}, B_{1}) and (ℙ^{2}, B_{2}) be two diffeomorphic (resp. homeomorphic) pairs. 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 unique, then the same is true for (S_{2}, f_{2}) and S_{1} and S_{2} are diffeomorphic (resp. homeomorphic).

Conversely, for S ⊂ ℙ^{r} a projection f : S → ℙ^{2} is defined by a point in Grassmannian Gr_{r+1,r−2} (the base locus of the projection). It is well-known that the set of generic projections is in one to one correspondence with some Zariski’s open subset U_{S} of Gr_{r+1,r−2}. A continuous variation of a point in U_{S} gives rise to a continuous family of generic projections of S whose branch curves belong to the same continuous family of plane cuspidal curves. Therefore discriminant curves of two generic projections of S ⊂ ℙ^{r} belong to the same irreducible component of M(2d, g, c).

Moreover, if two surfaces S_{1} and S_{2} of general type with the same K_{S}^{2} = k and p_{a} = p are embedded by the mth canonical class into the same projective space ℙ^{r} and belong to the same irreducible component of coarse moduli space M_{S}(k, p) of surfaces with given invariants ([8]), then there exist generic projections f_{1} of S_{1} and f_{2} of S_{2} belonging to the same continuous family of generic projections. Therefore, discriminant curves of two generic projections of S_{1} and S_{2}, belonging to the same irreducible component of a moduli space M_{S}(k, p), belong to the same irreducible component of M(2d, g, c) (cf. [21]). By Theorem 2 and Propositions 5 and 6, for a surface of general type with ample canonical class the triple of integers (m, k, p) is uniquely determined by the invariants (d, g, c) of mth canonical discriminant curve, and vice versa. Hence by Proposition 1 and Theorem 2,

i(2d(k, p), g(k, p), c(k, p)) ≥ i(k, p),

where i(2d, g, c) (resp. i(k, p)) is the number of irreducible components of M(2d, g, c) (resp. M_{S}(k, p)).

In [4], F. Catanese showed that for each positive integer h there exist integers k, p such that M_{S}(k, p) has at least h irreducible components. Hence for each positive integer h there exist integers d, g, c such that M(2d, g, c) has at least h irreducible components. Note that the lower bound estimates for i(k, p), obtained in [4] and [12], hold also for i(2d, g, c), where d, g, and c are the invariants of the corresponding mth canonical discriminant curves.

5.2. Recently, there were published several articles (see, for example, [1], [16], [19], [20]) devoted to so called Zariski’s pairs. By Artal-Bartolo’s definition, two plane curves C_{1}, C_{2} ⊂ ℙ^{2} are called a Zariski pair if they have the same degree and homeomorphic tubular neighborhoods in ℙ^{2}, but the pairs (ℙ^{2}, C_{1}) and (ℙ^{2}, C_{2}) are not homeomorphic.

The first example of such pairs was obtained by O. Zariski and it is just two curves of degree 6 with 6 cusps mentioned in the Introduction.

In view of Theorem 2, applying Proposition 8, it is easy to prove the existence of a lot of Zariski pairs. To this end, we can find pairs of non-homeomorphic minimal surfaces of general type with ample canonical class and the same K_{S}^{2} and p_{a}, and consider the corresponding mth canonical discriminant curves. For example, Proposition 9 For each integer m ≥ 5, there is at least one Zariski’s triple (not only a pair) of

plane cuspidal curves B_{m,i}, i = 3, 4, 5, of degree m(3m+1) and genus g = \frac{1}{2}(3m+1)(3m+2)+1 with c = 3(4m^{2} + 3m − 3) cusps.

Proof. There exist ([9], [18], [2]) five non-homeomorphic surfaces S_{i} (i = 1, ... , 5 ) of general type with p_{g} = 0, p_{a} = 1, and K_{S_{i}}^{2} = 1. They are distinguished by TorsH_{1}(S_{i}, ℤ) = ℤ/iℤ. At least for i ≥ 3 there exists a surface S_{i} with ample canonical class possessing such invariants. Let ϕ_{m,i} : S_{i} ↪ ℙ^{\frac{1}{2} m(m−1)} be the mth canonical embedding, m ≥ 5 ([3]), and f_{m,i} : S_{i} → ℙ^{2} a generic projection with discriminant curve B_{m,i}. Applying the computation in the proof of Theorem 2, the discriminant curve B_{m,i} has

deg B_{m,i} = m(3m + 1), g = \frac{1}{2}(3m + 1)(3m + 2) + 1, c = 3(4m^{2} + 3m − 3).

By Proposition 8, since S_{i} and S_{j} are not homeomorphic for i ≠ j, the pairs (ℙ^{2}, B_{m,i}) and (ℙ^{2}, B_{m,j}) also are not homeomorphic.

The easiest way to construct two non-homeomorphic surfaces S_{1} and S_{2} of general type, for which general morphisms, given by linear subsystems of the mth canonical class, give rise to Zariski’s pair B_{1} and B_{2}, is to construct two surfaces with different irregularities q_{1} and q_{2}, where q_{i} = dim H^{1}(S_{i}, O_{S_{i}}), and the same K_{S}^{2} and p_{a}. For example, let A be an abelian surface and C ⊂ A a non-singular curve such that the class of C in P ic A is divisible by 2. Consider the two-sheeted covering φ : S → A branched along C. It is easy to check that S is a surface of general type with ample canonical bundle and irregularity q ≥ 2. If C^{2} = 8p, then the simplest computation gives rise to K_{S}^{2} = 4p and p_{a} = p. On the other hand, in [17], U. Persson proved that for any positive integers x, y satisfying

2x − 6 ≤ y ≤ 8(x − cx^{2/3}),  (41)

where c = \frac{9}{\sqrt[3]{12}}, there exists a simply connected minimal surface of general type with K^{2} = y

and p_{a} = x. It is easily seen that x = p and y = 4p satisfy (41) if p ≥ 486.

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

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Steklov Mathematical Institute victor@olya.ips.ras.ru