Generic coverings of plane with A-D-E-singularities.

V.S. Kulikov ^{∗} and Vic.S. Kulikov ^{†}

Abstract

We investigate a presentation of an algebraic surface X with A-D-E-singularities as a generic covering f : X → ℙ^{2}, i.e. a finite morphism, having at most folds and pleats apart from singular points, isomorphic to a projection of a surface z^{2} = h(x, y) onto the plane x, y in neighbourhoods of singular points, and the branch curve B ⊂ ℙ^{2} of which has only nodes and ordinary cusps except singularities originated from the singularities of X. It is deemed that classics proved that a generic projection of a non-singular surface X ⊂ ℙ^{r} is of such form. In this paper this result is proved for an embedding of a surface X with A-D-E-singularities, which is a composition of the given one and a Veronese embedding. We generalize results of the paper [K] , in which a Chisini conjecture on the unique reconstruction of f by the curve B is investigated. For this fibre products of generic coverings are studied. The main inequality bounding the degree of a covering in the case of existence of two nonequivalent coverings with the branch curve B. This inequality is used for the proof of the Chisini conjecture for m-canonical coverings of surfaces of general type for m ≥ 5.

Introduction

Let S ⊂ ℙ^{r} be a non-singular projective surface, f : S → ℙ^{2} be its generic projection to the plane, B ⊂ ℙ^{2} be the branch curve, which we call the discriminant curve. It is deemed that classics proved (see [Z] , p.104) that (i) the map f is a finite covering, which has as singularities at most double points (folds), or singular points of cuspidal type (pleats); (ii) with this f^{∗}(B) = 2R+C, where the double curve R is non-singular and irreducible, and the curve C is reduced; (iii) the curve B is cuspidal, i.e. has at most nodes and ordinary cusps; over a node there lie two double points, and over a cusp – one point of cuspidal type; (iv) the restriction of f to R is of degree one. Any finite morphism f : S → ℙ^{2} is called a generic (or simple) covering, if it possesses the same properties as a generic projection. Two coverings of plane (S_{1}, f_{1}) and (S_{2}, f_{2}) are called equivalent, if there is a morphism φ : S_{1} → S_{2} such that f_{1} = f_{2} ∘ φ.

In this paper we consider a generalization of the notion of a generic covering to the case of surfaces with A-D-E-singularities. First of all we want to explain why we need such a generalization. A presentation of an algebraic variety as a finite covering of the projective space is one of the affective ways of studying projective varieties as well as their moduli. To compare we recall what such an approach gives in the case of curves. For a curve C of genus g a generic covering f : C → ℙ^{1} is such a covering that in every fibre there is at most one ramification point which is a double point (or a singular point of f of type A_{1} ). Let B ⊂ ℙ^{1} be the set of branch points, and d = deg B, i.e. d = ♯(B). Then according to the Hurwitz formula d = 2N + 2g − 2, where N = deg f. If N ≥ g + 1, then any curve of genus g can be presented as a simple covering of ℙ^{1} of degree N. The set of all simple coverings (up to equivalence) f : C → ℙ^{1} of degree N with d branch points is parametrized by a Hurwitz variety H = H^{N,d}. Let ℙ^{d} \ Δ (Δ – discriminant) be the projective space parametrizing the sets of d different points of ℙ^{1}, and let M_{g} be the moduli space of curves of genus g. There are two maps: a map h : H → ℙ^{d} \ Δ, sending f to the set of branch points B ⊂ ℙ^{1}, and a map μ : H → M_{g}, sending f to the class of curves isomorphic to C. Hurwitz introduced and investigated the variety H in 1891. He proved that the variety H is connected, and h is a finite unramified covering. In modern functorial language H was studied also by W.Fulton in 1969. The map μ is surjective (and has fibres of dimension N + (N − g + 1)). This gives one of the proofs ofirreducibility of the moduli space M_{g}.

∗Partly supported by RFFI (No. 99-01–01133) and INTAS-OPEN-97-2072.

†The work was done partially when the second author stayed at Max-Planck-Institut für Mathematik in Bonn.

In the case of surfaces we also can consider an analog of Hurwitz variety H of all generic coverings (up to equivalence) f : S → ℙ^{2} of degree N and with discriminant curve B of degree d with given number n of nodes and given number c of cusps. Let ℙ^{ν}, ν = \frac{d(d+3)}{2}, be a projective space parametrizing curves of degree d, and h : H → ℙ^{ν} be a map sending a covering f to its discriminant curve B. In [K] a Chisini conjecture is studied. It claims that if B is the discriminant curve of a generic covering f of degree N ≥ 5, then f is uniquely up to equivalence defined by the curve B. In other words, it means that the map h is injective (and, besides, N = deg f is determined by B). In [K] it is proved that the Chisini conjecture is true for almost all generic coverings. In particular, it is true for generic coverings defined by a multiple canonical class. A construction of the moduli space of surfaces of general type uses pluricanonical maps. As is known [BPV] , if S is a minimal surface of general type, then for m ≥ 5 the linear system |mK_{S}| blows down only (-2)-curves and gives a birational map of S to a surface X ⊂ ℙ^{r} (the canonical model) with at most A-D-E-singularities (in other terms, rational double points, Du Val singularities, simple singularities of Arnol’d and etc.). This requires a generalization of the notion of a generic covering to the case of surfaces with A-D-E-singularities.

In this paper we, firstly, generalize a classical result on singularities of generic projections of non-singular surfaces to the case of surfaces with A-D-E-singularities. We prove that if a surface X ⊂ ℙ^{r} has at most A-D-E-singularities, then (may be after a ”twist”) for a generic projection f : X → ℙ^{2} the discriminant curve B also has at most A-D-E-singularities. It follows from a slightly more general theorem.

Theorem 0.1 Let X ⊂ ℙ^{r} be a surface with at most isolated singularities of the form z^{2} = h(x, y) (= ”double planes”), X → ℙ^{2} be the restriction to X of a generic projection ℙ^{r}\L → ℙ^{2} from a generic linear subspace L of dimension r − 3. Then

(i) f is a finite covering;

(ii) at non-singular points of X the covering f has as singularities at most either double points (folds), or singular points of cuspidal type (pleats); in a neighbourhood of these points f is equivalent to a projection of a surface x = z^{2}, respectively y = z^{3} + xz, to the plane x, y;

(iii) in a neighbourhood of a point s ∈ Sing X the covering f is analytically equivalent to a projection of a surface z^{2} = h(x, y) to the plane x, y; from (ii) and (iii) it follows that the ramification divisor is reduced, i.e. f^{∗}(B) = 2R+C, where B = f(R), and R and C are reduced curves;

(iv) except singular points f(Sing X) the discriminant curve B is cuspidal;

(v) the restriction of f to R is of degree 1.

Actually, the main difficulty in the proof of this theorem lies in the classical case, when the surface X is non-singular. Unfortunately, authors do not know a complete (and mordern) proof of this theorem, and it seams that such a proof does not exist. Thus, the proof, even in the case of a non-singular surface, take interest. In this paper we prove a weakened version of Theorem 0.1, in which the initial embedding is ‘twisted’ by a Veronese embedding. This is quite enough for the purposes described above.

Thus, the curve B has, firstly, ‘the same’ singularities as the surface X (and as the curve R), which are locally defined by the equation h(x, y) = 0. These singularities on B we call s-singularities, in particular, s-nodes and s-cusps. Besides, there are nodes and cusps on B originated from singularities of the map f, which we call p-nodes and p-cusps. There are two double points of f over a p-node, at which f is defined locally as a projection of surfaces z_{1} = x^{2} and z_{2} = y^{2} to the plane x, y.

If S is a surface with A-D-E-singularities, then a covering f : S → ℙ^{2} is called generic, if it satisfies the properties of Theorem 0.1.

Secondly, we generalize the central result of [K] to the case of surfaces with A-D-E-singularities. It is proved there that if a generic covering f : S → ℙ^{2} of a non-singular surface S with discriminant curve B is of sufficiently big degree deg f = N, namely under condition

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

where 2d̄= deg B, g be the geometric genus of B, and c be the number of cusps, then B is the discriminant curve of a unique generic covering (the Chisini conjecture holds for B).

We can’t expect an analogous result in the case of singular surfaces, because for a curve B of even degree with at most A-D-E-singularities there always exists a double covering, which is generic. But if two generic coverings with given discriminant curve B are coverings of sufficiently big degree, then they are equivalent. More exactly, we prove the following theorem. Let there are two generic coverings f_{1} : X_{1} → ℙ^{2} and f_{2} : X_{2} → ℙ^{2} of surfaces with A-D-E-singularities and with the same discriminant curve B ⊂ ℙ^{2}. Let f_{i}^{∗}(B) = 2R_{i} + C_{i}, i = 1, 2. With respect to a pair of coverings f_{1} and f_{2} nodes and cusps of B are partitioned into four types: ss-, sp-, ps- and pp-nodes and cusps. For example, a sp-node b ∈ B is a node, which is a s-node for f_{1} and a p-node for f_{2}. The number of sp-nodes is denoted by n_{sp}. Then n = n_{ss} + n_{sp} + n_{ps} + n_{pp}. The analogous terminology is used for cusps.

Theorem 0.2 If f_{1} and f_{2} are nonequivalent generic coverings, then

deg f_{2} ≤ \frac{4(3d̄+ g_{1} − 1)}{2(3d̄+ g_{1} − 1) − ι_{1}} ,  (2)

where g_{1} = p_{a}(R_{1}) is the arithmetic genus of the curve R_{1}, and ι_{1} = 2n_{sp} + 2c_{sp} + c_{pp}.

We apply the main inequality (2) to the proof of the Chisini cojecture in the case of generic pluricanonical coverings. Let S be a minimal model of a surface of general type. According to a theorem of Bombieri [BPV] , if m ≥ 5, then the m-canonical map φ_{m} : S → ℙ^{p_{m}−1}, defined by the complete linear system numerically equivalent to |mK_{S}|, is a birational morphism, which blows down (-2)-curves on S. Then the canonical model X = φ_{m}(S) has at most A-D-E-singularities. A generic projection f : X → ℙ^{2} is called a generic m-canonical covering for S. We prove the following theorem.

Theorem 0.3 Let S_{1} and S_{2} be minimal models of surfaces of general type with the same (K_{S}^{2}) and χ(S), and let f_{1} : X_{1} → ℙ^{2}, f_{2} : X_{2} → ℙ^{2} be generic m-canonical coverings with the same discriminant curve. Then for m ≥ 5 the coverings f_{1} and f_{2} are equivalent.

Consider a subvariety H ⊂ Hilb × Gr, parametrizing m-canonical coverings. Here Hilb is a subscheme of the Hilbert scheme, parametrizing numerically m-canonical embeddings X ⊂ ℙ^{M} of surfaces with A-D-E-singularities and fixed (K_{S}^{2}) and χ(S), Gr is the Grassmann variety of projection centres from ℙ^{M} to ℙ^{2}, and H consists of pairs (X ⊂ ℙ^{M}, L) ) such, that a restriction to X of a projection with centre L is a generic covering. By theorem 0.1 there is a one-to-one correspondence between the set of irreducible (respectively, connected) components of Hilb and H. Let h : H → ℙ^{ν} be a map, taking a covering to its discriminant curve. Denote by D a variety of plane curves of degree d with A-D-E-singularities, among which the number of nodes ≥ n_{p}, and the number of cusps ≥ _{p}, where d, n_{p}, c_{p} are defined by invariants of S (see §6). By theorem 0.3 it follows (cf. [K] , §5) Corollary. The map, induced by h, from the set of irreducible (respectively, connected) components of the variety H to the set of irreducible (respectively, connected) components of the variety D is injective.

The proof of the main inequality (2) in [K] in the case of non-singular surfaces runs as follows. To compare two coverings f_{1} and f_{2}, a normalization X of the fibre product X_{1} ×_{ℙ^{2}} X_{2} is considered. Let g_{i} : X → X_{i}, i = 1, 2, be the corresponding mappings to the factors. The preimage g_{1}^{−1}(R_{1}) = R + C falls into two parts, where R is the curve mapped by g_{2} to R_{2}, and C is the curve mapped by g_{2} to C_{2}. If f_{1} and f_{2} are nonequivalent, then the surface X is irreducible, and if X_{i} are non-singular, then X is non-singular too. The main inequality is obtained by applying the Hodge index theorem to the pair of divisors R and C on X. We use the same idea also in the case of surfaces with A-D-E-singularities. For this we carry out the local analysis of the normalization of the fibre product X in the case of generic coverings of surfaces with A-D-E-singularities.

In §1 we generalize to the case of surfaces with A-D-E-singularities the theorem on generic projections. In §2 a local analysis of a normalization of the fibre product X is carried out.

In §3 we investigate the canonical cycle of an A-D-E-singularity, with the help of which we compute numerical invariants of a generic covering in §4. In §5 the main inequality (2) is proved. Finally, in §6 the Chisini conjecture for generic m-canonical coverings of surfaces of general type is proved.

1 Singularities of a generic projection of a surface with A-D-E-singalarities.

In this section we prove Theorem 0.1.

1.1. A generic projection to ℙ^{3}. Let X ⊂ ℙ^{r} be a surface of degree deg X = N with at most isolated hypersurface singularities x_{1}, …, x_{k}, i.e. such that the dimension of the tangent spaces dim T_{X,x_{i}} = 3. Denote by π_{L} : ℙ^{r} \ L → ℙ^{e−1} a projection from a linear subspace L of codimension e. It can be obtained as a composition of projections with centers at points. The Theorem 0.1 on projections of X to the plane (e = 3) is one of a series of theorems on generic projections for different e, beginning with projections from points (e = r) and finishing by projections to the line (e = 2), i.e. Lefschetz pencils.

A classical result is that, if r > 5 (= 2 dim X + 1), then the projection from a generic point gives an isomorphic embedding of X into ℙ^{r−1}. It follows that, if e ≥ 6, then the projection from a generic subspace L gives an isomorphic embedding of X into ℙ^{e−1}. In particular, by a generic projection the surface X is embedded into ℙ^{5}. When projecting to ℙ^{4}, e = 5, there appears isolated singularities on π_{L}(X), which is not difficult to describe. To prove Theorem 0.1 we are going to consider a generic projection of X into ℙ^{3}, e = 4, and to take advantage of the following theorem.

Theorem 1.1 If X ⊂ ℙ^{r} is a surface with at most isolated hypersurface singularities x_{i}, then the restriction of a projection π_{L} : ℙ^{r} \ L ⟶ ℙ^{3} with the centre in a generic subspace L ⊂ ℙ^{r} of codimension 4 gives a birational map of X onto a surface Y ⊂ ℙ^{3}, which is an isomorphism outside the double curve D ⊂ X not passing through the points x_{i}, and Y has, except the points π_{L}(x_{i}), at most ordinary singularities – the double curve Δ = π_{L}(D), on which there lie a finite number of ordinary triple points and a finite number of pinches. In neighbourhoods of these points in appropriate local analytic coordinates Y has normal forms as follows: uv = 0 for ordinary double points, uvw = 0 for ordinary triple points, u^{2} − vw^{2} = 0 for pinches (or ”Whitney umbrellas”).

The contemporary proof of this theorem one can find in the textbook [G-H]. The presence of singular points x_{i} do not add extra troubles: we need only to see to the centre of the projection L not to intersect the tangent spaces T_{X,x_{i}}, dim T_{X,x_{i}} = 3. A proof of this theorem one can find also in [M] .

We want to prove that for a generic point ξ ∈ ℙ^{3} the composition of projections π_{L} and π_{ξ} : ℙ^{3} \ ξ → ℙ^{2}, i.e. the projection ℙ^{r} \ π_{L}^{−1}(ξ) → ℙ^{2} with the centre π_{L}^{−1}(ξ), restricted to X, f = π_{ξ} ∘ π_{L}_{|_{X}} : X → ℙ^{2}, gives a covering satisfying the properties stated in Theorem 0.1 .

1.2. The disposition of lines with respect to a surface ℙ^{3}. To describe a projection π_{ξ} we need to investigate the disposition of lines l ⊂ ℙ^{3} with respect to the surface Y. A line l is called transversal to Y at a point y, if it is transversal to the tangent cone to Y at this point. It means that (l · Y)_{y} = 1, if y ∉ Sing Y; (l · Y)_{y} = 2, if y ∈ Δ \ Δ_{t} and (l · Y)_{y} = 3, if y ∈ Δ_{t}. We denote by Δ_{t} and Δ_{p} the set of triple points and the set of pinches. If l is not transversal to Y at a point y, we say that it is tangent to Y at this point. A line l is called a simple tangent to Y at y, if y ∉ Sing Y and (l · Y)_{y} = 2, or if y ∈ Δ \ (Δ_{t} ∪ Δ_{p}) and (l · Y)_{y} = 3, i.e. (l · Y_{i})_{y} = 2 for one of two branches Y_{i} at the point y. A line l is called stationary tangent, respectively simple stationary tangent to Y at y, if y ∉ Sing Y and (l · Y)_{y} ≥ 3, respectively = 3. A line l is called stationary tangent, respectively simple stationary tangent to Y, if l is transversal to Y at all points, except one, at which l is stationary tangent, respectively simple stationary tangent, and, besides the other points of intersection l ∩ Y are non-singular on Y. Finally, l is called simple bitangent, if l is transversal to Y at all points, except two of them, at which the contact is simple, the tangent planes at them are distinct, and, besides, l ∩ Sing Y = ∅. We want to prove that for a generic point ξ ∈ ℙ^{3} all lines l ∋ ξ are at most simple bitangents and simple stationary tangents with respect to Y.

To study the disposition of lines l ⊂ ℙ^{3} with respect to Y, we consider the Grassmann variety G = G(1, 3) and the flag variety 𝔽 = {(ξ, l) ∈ ℙ^{3} × G | ξ ∈ l}. There are two projections pr_{1} : 𝔽 → ℙ^{3} and pr_{2} : 𝔽 → G, which are ℙ^{2}- and ℙ^{1}-bundles respectively; dim 𝔽 = 5, and dim G = 4. In the sequal we consider points ξ ∈ ℙ^{3} as centres of projection π_{ξ} : ℙ^{3} \ ξ → ℙ^{2}. The fibre pr_{1}^{−1}(ξ) ≃ ℙ^{2} is mapped by the projection pr_{2} isomorphically onto ℙ_{ξ}^{2} ⊂ G. For ξ ∈ ℙ^{3} there is a section s_{ξ} : ℙ^{3} \ ξ → 𝔽 of the projection pr_{1}, y ⟼ (y, \overline{ξy}). Then π_{ξ} coincides with the restriction of the projection pr_{2} to s_{ξ}(ℙ^{3} \ ξ).

Firstly, we consider the case, when a surface Y is non-singular, and then we describe the necessary modifications and supplements in the case, when there is a double curve Δ and isolated singularities s_{i} on Y.

Consider a filtration of the variety 𝔽 by subvarieties

Z_{k} = {(ξ, l) ∈ 𝔽 | (l · Y)_{ξ} ≥ k}.

Then Z_{1} = pr_{1}^{−1}(Y), dim Z_{1} = 4. Over a generic point l ∈ G the map φ = pr_{2}_{|Z_{1}} : Z_{1} → G is an unramified covering of degree N. If there are no lines on Y, then φ is a finite covering, and Z_{2} is the ramification divisor of the covering.

Now consider restrictions of the projection pr_{1}. The variety Z_{2} is isomorphic to a projectivized tangent bundle, Z_{2} ≃ ℙ(Θ_{Y}), and Z_{2} → Y is a ℙ^{1}-fibre bundle, dim Z_{2} = 3. At a generic point y ∈ Y there are two asymptotic directions l_{1} and l_{2} in T_{Y,y}, for which (l_{1} · Y)_{y} and (l_{2} · Y)_{y} ≥ 3. Therefore, over a generic point the restriction of pr_{1} onto Z_{3}, ψ : Z_{3} → Y, is a two-sheeted covering, the branch curve of which P ⊂ Y is the parabolic curve consisting of points with coinciding asymptotic directions. Some fibres of the projection pr_{1} are exceptional curves of the map ψ. Their images on Y are points y, at which the restriction of the second differential of the local equation of Y onto the tangent plane T_{Y,y} vanishes. Such points y are called the planar points of the surface Y. The curve H = ψ(Z_{4}) ⊂ Y consists of points y, at which at least one of the numbers (l_{i} · Y)_{y} ≥ 4 (H is a curve, if the surface Y is not a quadric).

1.3. Absence of non simple stationary tangents. Consider a product Y × 𝔽 ⊂ Y × ℙ^{3} × G and projections pr′_{1} and pr′_{2} onto Y × ℙ^{3} and Y × G. We can consider the varieties Z_{k} as subvarieties in Y × G ⊂ ℙ^{3} × G. Consider a variety

I_{4} = {(y; ξ, l) ∈ Y × 𝔽 | (l · Y)_{y} ≥ 4} = (pr′_{2})^{−1}(Z_{4}).

The projection pr′_{2} = id_{Y} × pr_{2}, as well as pr_{2} : 𝔽 → G, is a ℙ^{1}-bundle. Therefore, dim I_{4} = 2 and dim Σ_{4} ≤ 2, where Σ_{4} = p_{2}(I_{4}), and p_{2} is a projection of Y × ℙ^{3} × G to ℙ^{3}. Then, if ξ ∈ ℙ^{3} \ Σ_{4}, we have that (l · Y)_{y} ≤ 3 for any line l ∋ ξ at any point y ∈ Y.

1.4. Absence of non simple bitangents. Consider a variety Σ_{2,3} ⊂ ℙ^{3}, made up of non simple bitangents, and show that Σ_{2,3} ≤ 2. Consider a product Y × Y × 𝔽 ⊂ Y × Y × ℙ^{3} × G and subvarieties I_{i,j}, which are closures of

I_{i,j}^{0} = {(y_{1}, y_{2}; ξ, l) ∈ Y × Y × 𝔽 : (Y · l)_{y_{1}} ≥ i, (Y · l)_{y_{2}} ≥ j, y_{1} ≠ y_{2}}.

Denote a projection of Y × Y × 𝔽 to Y × Y × G by pr′′_{2}, and let pr′′_{2}(I_{i,j}) = Ĩ_{i,j}. The projection pr′′_{2} and its restriction to I_{i,j}, I_{i,j} → Ĩ_{i,j} are ℙ^{1}-bundles.

Lemma 1.1

dim I_{2,3} ≤ 2.

Proof. Consider subvarieties

Y × Y × G ⊃ Ĩ_{1,1} ⊃ Ĩ_{2,1} ⊃ Ĩ_{2,2} ⊃ Ĩ_{2,3} ,

and let q_{1} be a projection onto the first factor. Obviously, Ĩ_{1,1} is an irreducible variety of dimension dim Ĩ_{1,1} = 4, birationally isomorphic to Y × Y. The projection q_{1} : Ĩ_{2,1} → Y is a fibration, fibers of which are curves q_{1}^{−1}(y) ≃ C_{y}, where

C_{y} = Y ∩ T_{Y,y}.

The curve C_{y} has a singularity at the point y, which is a node for a generic point y.

Furthermore, the ristriction of the projection to Ĩ_{2,2}, q_{1} : Ĩ_{2,2} → Y, is surjective, and its fibre over a point y ∈ Y is

q_{1}^{−1}(y) = {(y, y′, l) | l ⊂ T_{Y,y} and l is tangent to _{y} at y′}.

We want to prove that q_{1}(Ĩ_{2,3}) doesn’t coincide with Y, i.e. the embedding Y ⊂ ℙ^{3} possesses the following property (L_{1}) : there exists a point y ∈ Y such that all lines l ⊂ T_{Y,y}, passing through y, have at most simple contact with C_{y} \ {y}. We prove this below in 1.6 (Proposition 1.2) under the assumption that the embedding Y ⊂ ℙ^{3} is obtained by a projection of an embedding ”improved” by a Veronese embedding v_{k}, k ≥ 2.

Thus, dim q_{1}(Ĩ_{2,3}) ≤ 1. A generic fibre of the map q_{1} : Ĩ_{2,3} → Y is of dimension zero (it being one, Y is a ruled surface and we obtain a contradiction to the property (L_{1}) ), therefore, dim Ĩ_{2,3} ≤ 1 and, consequently, dim I_{2,3} ≤ 2.

Set Σ_{2,3} = p_{3}(I_{2,3}), where p_{3} is a projection of Y × Y × ℙ^{3} × G to ℙ^{3}. It follows from Lemma 1.1 that dim Σ_{2,3} ≤ 2. If ξ ∉ Σ_{2,3}, then any line l ∋ ξ, touching Y at two points y_{1} and y_{2}, has a simple contact at these points.

1.5. Absence of 3-tangents. Consider a product Y × Y × Y × 𝔽 ⊂ Y × Y × Y × ℙ^{3} × G and subvarieties I_{i,j,k}, which are closures of

I_{i,j,k}^{0} = {(y_{1}, y_{2}, y_{3}; ξ, l) ∈ Y × Y × Y × 𝔽 | (Y · l)_{y_{1}} ≥ i, (Y · l)_{y_{2}} ≥ j, (Y · l)_{y_{3}} ≥ k},

where y_{1} ≠ y_{2} ≠ y_{3} ≠ y_{1}. Denote a projection of Y × Y × Y × 𝔽 onto Y × Y × Y × G by pr_{2}^{(3)}, and let Ĩ_{i,j,k} = pr_{2}^{(3)}(I_{i,j,k}). As above, it is clear that dim Ĩ_{1,1,1} = 4, and pr_{1}^{(3)} being a ℙ^{1}-bundle, we have dim I_{1,1,1} = 5.

Lemma 1.2

dim I_{2,2,2} ≤ 2.

Proof. Again consider a projection of Y × Y × Y × G and of its sibvarieties

Y × Y × Y × G ⊃ Ĩ_{1,1,1} ⊃ Ĩ_{2,1,1} ⊃ Ĩ_{2,2,1} ⊃ Ĩ_{2,2,2},

to the first factor. Consider q_{1} : Ĩ_{2,2,2} → Y. For a point y ∈ Y we have q_{1}^{−1}(y) = {(y, y_{2}, y_{3}; l) | l ⊂ T_{Y,y}, l is tangent to _{y} at points y_{2} and y_{3} ∈ l}. Just as in Lemma 1.1 it is sufficient to prove that q_{1}(Ĩ_{2,2,2}) doesn’t coincide with Y. It means that there exists a point y ∈ Y, possessing the following property (L_{2}): none of the lines l ⊂ T_{Y,y}, passing through y, is not a bitangent, i.e. can’t touch C_{y} \ {y} at two different points. We prove this below in the following 1.6 (Proposition 1.2) under the assumption that the embedding Y ⊂ ℙ^{3} is obtained by a projection of an embedding ”improved” by a Veronese embedding v_{k}.

Set Σ_{2,2,2} = p_{4}(I_{2,2,2}), where p_{4} is a projection of Y × Y × Y × ℙ^{3} × G to ℙ^{3}. Then dim Σ_{2,2,2} ≤ 2 and if ξ ∉ Σ_{2,2,2}, then any line l ∋ ξ touches Y at most at two points.

1.6. Embeddings with a Lefschetz property. The properties (L_{1}) and (L_{2}) in the two previous subsections mean that there exists a point y ∈ Y, for which the projection π_{y} of the curve C_{y} \ {y} ⊂ T_{Y,y} ≃ ℙ^{2} from the point y is a Lefschetz pencil. Thus, to prove Lemmas 1.1 and 1.2 it is necessary to prove the existence of a point y ∈ Y possessing the following ”Lefschetz property” (L) with respect to the embedding into ℙ^{3}. We formulate it for a surface X embedded into a projective space of any dimension.

Let X ⊂ ℙ^{r} be an embedding into the projective space. We say, that a hyperplane L_{1} ⊂ ℙ^{r} possesses a property (L), if the curve X ∩ L_{1} has at most one node, i.e. L_{1} touches X at a unique point x, at which the curve X ∩ L_{1} has an ordinary quadratic singularity. In other words, the point [L_{1}] ∈ ℙ̌^{r}, corresponding to L_{1}, is a non-singular point of the dual variety X^{∨}. We say that a pair (L_{1}, L_{3}), where L_{3} ⊂ L_{1} is a linear subspace of dimension r − 3, possesses a property (L) , if : L_{1} possesses the property (L) , x ∈ L_{3}, and a projection of the curve X ∩ L_{1}− → ℙ^{1} from the centre L_{3} is a Lefschetz pencil, i.e. any fibre of this (rational) mapping contains one singular point, and this point is at most quadratic (is of multiplicity 2). We say that an embedding X ⊂ ℙ^{r} possesses a property (L) , if ∃x ∈ X, for which L_{1} = T_{X,x} possesses the property (L), and L_{1} can be added to a pair (L_{1}, L_{3}) with the property (L).

It is clear that, if a pair (L_{1}, L_{3}) possesses the property (L) and Y ⊂ ℙ^{3} is obtained from X by projection from a centre L_{4} ⊂ L_{3}, dim L_{4} = r − 4, then the embedding Y ⊂ ℙ^{3} possesses the property (L).

Proposition 1.1 If S ⊂ ℙ^{q} is an embedding of a non-singular surface, and X ⊂ ℙ^{r_{k}} is its composition with the Veronese embedding v_{k} defined by polynomials of degree k, then the embedding X ⊂ ℙ^{r_{k}} possesses the property (L).

Proof. Consider the hyperplane L_{1} corresponding to a point [L_{1}] ∈ X^{∨} \ Sing X^{∨}. Then the curve C = X ∩ L_{1} contains a unique singular point – a node x ∈ C. Let i : C → X be the embedding. Consider a projection π_{k,x} : ℙ^{r_{k}} \x → ℙ^{r_{k}−1} from the point x. To prove Proposition 1.1 it is enough to show that the image π_{k,x}(C) is a non-singular curve in ℙ^{r_{k}−1}. For then, if L′_{3}, dim L′_{3} = r_{k} − 3, is a centre of projection ℙ^{r_{k}−1} \ L′_{3} → ℙ^{1}, which is a Lefshetz pencil for π_{k,x}(C), then, obviously, the pair (L_{1}, L_{3}), where L_{3} = π_{k,x}^{−1}(L′_{3}) ∩ L_{1}, possesses the property (L).

Let I_{x} be the ideal sheaf of the point x on S, and O_{S}(1) be the sheaf of hyperplane sections. Under the identification v_{k} : S ≃ X, the map π_{k,x} is given by sections of H^{0}(S, O_{S}(k) ⊗ I_{x}). Let k = 2 and let σ : \overline{S} → S be a σ-process with centre at the point x. We can assume that \overline{S} is embedded into ℙ^{r_{2}−1}, where r_{2} − 1 = q(q + 3)/2, and the rational map σ^{−1} : S → \overline{S} is given by sections of H^{0}(S, O_{S}(2) ⊗ I_{x}), i.e. it coincides with π_{2,x}. Since the proper transform \overline{C} = σ^{−1}(C) ⊂ \overline{S} is a non-singular curve, we obtain Proposition 1.1 in the case k = 2. Besides, note that sections of i^{∗}(H^{0}(S, O_{S}(2) ⊗ I_{x})) give an embedding of \overline{C} into ℙ^{r_{2}−1}. Consequently, for k > 2 sections of i^{∗}(H^{0}(S, O_{S}(k) ⊗ I_{x})) also give an embedding of \overline{C} into ℙ^{r_{k}−1}, since there is a natural injection H^{0}(S, O_{S}(k − 2)) ⊗ H^{0}(S, O_{S}(2) ⊗ I_{x}) ⊂ H^{0}(S, O_{S}(k) ⊗ I_{x}), and sections of H^{0}(S, O_{S}(k − 2)) have no base points and fixed components. Therefore, sections of i^{∗}(H^{0}(S, O_{S}(k) ⊗ I_{x})) separate points and tangent directions on \overline{C}.

We say that a linear subspace L_{4} of dimension r − 4 possesses a property (L) with respect to an embedding X ⊂ ℙ^{r}, if the projection π_{L_{4}} to ℙ^{3} from the centre L_{4} maps X onto a surface Y = π_{L_{4}}(X) with ordinary singularities.

As is known (see [G-H] ), there is an open subset U in the Grassmannian G_{4} = Gr(r − 4, r), points of which correspond to linear subspaces with the property (L).

Proposition 1.2 If an embedding X ⊂ ℙ^{r} possesses the property (L), then there exists a linear subspace L_{4} with the property (L), which can be added to a flag L_{1} ⊃ L_{3} ⊃ L_{4} such that the pair (L_{1}, L_{3}) possesses the property (L). In other words, there exists a projection to ℙ^{3}, for which the embedding Y ⊂ ℙ^{3}, where Y is the image of X, possesses the property (L).

Proof. Let G_{1} = ℙ̌^{r} be the dual space to ℙ^{r}, G_{3} = G(r − 3, r) be the Grassmann variety of linear subspaces L_{3} of dimension r − 3, and 𝔽 = 𝔽_{1,3,4} ⊂ ℙ̌^{r} × G_{3} × G_{4} be the variety of flags L_{1} ⊃ L_{3} ⊃ L_{4}. Let X^{∨} be the dual variety, W ⊂ X × X^{∨} ⊂ ℙ^{r} × ℙ̌^{r} be a closed subvariety W = {(x, L_{1}) : L_{1} ⊃ T_{X,x}}. Then the projection of W → X^{∨} is an isomorphism over X_{0}^{∨} = X^{∨} \ Sing X^{∨}, W_{0} ≃ X_{0}^{∨}.

Denote by Z ⊂ X × 𝔽 a closed subvariety

Z = {(x; L_{1} ⊃ L_{3} ⊃ L_{4}) | (x, L_{1}) ∈ W, L_{3} ∋ x} ,

and by Z_{0} ⊂ Z an open subset: (x, L_{1}) ∈ W_{0}. Then Z is an irreducible variety. Consider a projection Z_{0} → W_{0}. The fibres are not empty by the previous proposition, and each of the fibres contains an open set of points z, for which the pair (L_{1}, L_{3}) possesses the property (L) (because the centres of projections for Lefschetz pencils form an open set). Therefore, Z contains an open set Z_{L}, for points of which the pair (L_{1}, L_{3}) possesses the Lefschetz property.

Obviously, the map Z → G_{4} is surjective. Therefore, p_{4}^{−1}(U), where p_{4} is a projection of Z to G_{4}, is a non empty Zariski open set. Then Z_{L}∩p_{4}^{−1}(U) is not empty, and if (x; L_{1} ⊃ L_{3} ⊃ L_{4}) is a point of this set, then L_{4} possesses the desired property.

1.7. Projecting in a neighbourhood of a generic point of the double curve Δ. Now let Y ⊂ ℙ^{3} has ordinary singularities along the double curve Δ and isolated singularities s_{i} ∈ Y \ Δ, i = 1, …, k, which are double planes. Under the incidence varieties, defined in the previous subsections, we mean the closures of the corresponding varieties, initially defined for an open surface Y \ Sing Y.

Consider Y × 𝔽. In addition to notations in 1.3, let q_{1} and q_{2} the projections of Y × G to Y and G. Consider the intersection Ã = (Δ × G) ∩ Z_{3}. Then the restriction of the projection Y × G → Y to Ã, Ã → Δ, is a covering of degree 4 over a generic point : at a point y ∈ Δ there are two asymptotic directions for each of two branches of Y at y. Therefore, Ã is a curve. Set A = (pr′_{2})^{−1}(Ã). It is a ruled surface. Set Σ_{Δ} = p_{2}(A). Then, if ξ ∉ Σ_{Δ}, we have that for a generic point y ∈ Δ the lines l ∋ ξ have at most simple contact with branches of Y.

Denote by Σ_{0} the union of planes in ℙ^{3} composing the tangent cones at the rest points of Δ, including triple points and pinches, and also at singular points s_{i} ∈ Y.

1.8. Projecting in a neighbourhood of a triple point. If ξ ∉ Σ_{0}, then in a neighbourhood of a point y ∈ Σ_{t} all lines l ∋ ξ are transversal to each of the three branches of Y at y, and therefore, locally these branches are mapped isomorphically onto ℙ^{2}.

1.9. Projecting in a neighbourhood of a pinch. In a neighbourhood of a pinch y ∈ Y there are coordinates, by which Y is locally defined by an equation u^{2} = vw^{2}. The double curve Δ ⊂ Y is a line u = w = 0, and the tangent cone C_{Y,y} to Y at y has an equation u = 0. In a neighbourhood of a pinch a normalization ℂ^{2} → Y is defined by formulae u = tw, v = t^{2}, w = w. Since X is non-singular and π_{L} is a finite map, we can assume that the projection π_{L} is the normalization. If a point ξ does not belong to the tangent cone C_{Y,y}, then the projection π_{ξ} locally is a map of gedrr 2. A projection f : X → ℙ^{2} a neighbourhood of the preimage of a pinch is a two-sheeted covering of non-singular varieties, and, hence, locally is defined by equations v = t^{2}, w = w. Thus, the curve R̄ ⊂ Y goes through the pinch, and pinches are projected to non-singular points of the discriminant curve B.

1.10. Normal forms of a generic projection at points of the ramification curve.

Lemma 1.3 ([A]) Let (X, 0) ⊂ (ℂ^{3}, 0) be a non-singular surface, and (ℂ^{3}, 0) → (ℂ^{2}, 0) be a smooth morphism, the restriction of which f : X → ℂ^{2} is a finite covering of degree μ. Then one can choose local coordinates x, y in ℂ^{2} and x, y, z in ℂ^{3} such, that X is defined by an equation

y = z^{μ} + λ_{1}(x)z^{μ−2} + … + λ_{μ−2}(x)z,

and f is a projection along z axis.

Proof. This is Lemma 1 in Arnol’d paper [A]. It is obtained, if we consider the covering f as a 2-paremeter family of 0-dimensional hypersurface singularities of multiplicity μ, and, consequently, f is induced by the miniversal deformation of the singularity of type A_{μ−1}.

We proved that at a generic point P of the ramification curve a projection f : X → ℙ^{2} is of degree μ = 2, and at isolated points it is of degree μ = 3. By Lemma 1.3 for μ = 2 we obtain that at a generic point of the ramification curve a generic projection is equivalent to a projection of the surface X : x = z^{2} to the x, y-plane, i.e. it is a fold. For μ = 3 we obtain

Corollary 1.1 For a non-singular surface X a finite covering X → ℂ^{2} of degree 3 locally is a projection to the x, y-plane of one of the surfaces

y = z^{3} + x^{k}z, k = 1, 2, …, or y = z^{3} (k = ∞).

In the case k ≠ ∞ the ramification curve C is reduced and has an equation 3z^{2} +x^{k} = 0 in local coordinates x, z on X. The curve C is non-singular only for k = 1. The discriminant curve B has an equation 4x^{3k} + 27y^{2} = 0, i.e. B is a cusp. It is an ordinary cusp only for k = 1.

Proof. By lemma 1.3 we have that X is defined by an equation y = z^{3} + λ_{1}(x)z. We obtain the stated normal form of the covering f, where k is the order of vanishing of λ_{1}(x) at the point x = 0. The ramification curve C is defined by equation J = 0, where J = 3z^{2} + x^{k} is the Jacobian of the covering f. The discriminant curve B is defined by 0th Fitting ideal F_{0} (f_{∗}O_{C}) of the sheaf f_{∗}O_{C}. To obtain an equation of B — the generator of the Fitting ideal, we need to take a finite presentation f_{∗}O_{X} →^{J} f_{∗}O_{X} → f_{∗}O_{C} → 0 of the sheaf f_{∗}O_{C}, where (f_{∗}O_{X})_{0} = ℂ{x, z} = ℂ{x, y} · 1 ⊕ ℂ{x, y}z ⊕ ℂ{x, y}z^{2}, and to compute a determinant of the ℂ{x, y}-linear map J, which is a multiplication by the Jacobian J.

Now we show that for a generic projection the discriminant curve B has at most ordinary nodes and cusps. Let b ∈ B be a point corresponding to a bitagent l under projecting π_{ξ} : ℙ^{3}\ξ → ℙ^{2} from a point ξ. Let l touches Y at points P_{1} and P_{2}, to which correspond branches B_{1} and B_{2} of the discriminant curve B at a point b. We have to show that for a generic projection the point b is a node, i.e. the branches B_{1} and B_{2} have different tangents. Determine where does the centres ξ of ”bad” projections lie. Let a line λ ⊂ ℙ^{2} is a common tangent to branches B_{1} and B_{2} at a point b. Then the plane π_{ξ}^{−1}(λ) is bitangent – it touches the surface Y at points P_{1} and P_{2}. Consider the dual surface Y^{∨} ⊂ ℙ̌^{3}. Then the point [π_{ξ}^{−1}(λ)] ∈ Sing Y^{∨} = γ^{∨}. Set γ = τ^{−1}(γ^{∨}), where τ : Y → Y^{∨} is the Gauss map. Let Σ_{u} ⊂ ℙ^{3} be a ruled surface composed by lines P_{1}P_{2}, where P_{1}, P_{2} ∈ γ, τ(P_{1}) = τ(P_{2}) = [π_{ξ}^{−1}(λ)]. Then, if ξ ∉ Σ_{u}, then at points b, corresponding to bitangents l, the curve B has at most nodes.

Now let b ∈ B be a point corresponding to a stationary tangent l at a point P ∈ Y. As was noted above, in a neighbourhood of P the projection π_{ξ} is equivalent to a projection of a surface y = z^{3} + x^{k}z to the x, y-plane. We have to show that for a generic projection the exponent k = 1. The fact is that, if k > 1, then the point P is a planar point of the surface Y. Excepting the centres of projection lying in tangent planes to Y at planar points, we obtain that in a neighbourhood of a point with μ = 3 the projection f is equivalent to a projection of a surface y = z^{3} + xz to the x, y-plane, i.e. it is a pleat.

1.11. Projecting in a neighbourhood of an isolated double plane singularity.

Lemma 1.4 If (X, 0) ⊂ (ℂ^{3}, 0) is an (isolated) double plane singularity z^{2} = h(x, y) , π : X ⟶ ℂ^{2} be a projection from any point p ∈ ℂ^{3}, not lying in the tangent cone z = 0, then the ramification curve of π is reduced, and the discriminant curve B ⊂ ℂ^{2} is locally analytically isomorphic to the singularity h(x, y) = 0.

Proof. The singularity (X, 0) is of multiplicity 2. Therefore, π is a covering of degree 2, and, consequently, is locally a projection of a double plane w^{2} = g(u, v) to the (u, v)-plane. Thus, the germs of singularities h(x, y) and g(u, v) are stably isomorphic, and hence isomorphic ([AGV] , vol.1).

2 Local structure of fibre products.

2.1. Local structure of a generic covering. Let f : X → ℙ^{2} be a generic covering of the plane by a surface X with A-D-E-singularities, and let B ⊂ ℙ^{2} be the discriminant curve, f^{∗}(B) = 2R + C. Singular points o ∈ Sing X will be called s-points of the surface X (from the word singularity). In a neighbourhood of a s-point o the covering f is isomorphic to the projection to x, y-plane of a surface z^{2} = h(x, y), where h(x, y) has one of the A-D-E-singularities. Singular points o on X correspond to singular points of the same type as o on B. With respect to f non-singular points of X are partitioned into r-points (from the word regular), at which the morphism f is étale, and p-points (from the words singularity of projection) – they are points of the ramification curve R. A p-point is either a fold (or a singular p-point of type A_{1}), in a neighbourhood of which f : (x, z) ⟼ (x, y), y = z^{2}, or a pleat o ∈ R ∩ C (or a singular p-point of type A_{2}), in a neighbourhood of which f : (x, z) ⟼ (x, y), y = z^{3} − 3xz (more details about this see in section 2.4 below).

The singular points of B ‘originated’ from singular points Sing X we call s-points . There are additional singular points of type A_{1} (nodes) and of type A_{2} (cusps), which we call p-nodes and p-cusps. Over a generic point b ∈ B there lie one fold and N − 2 r-points; over p-node there lie two folds and N − 4 r-points; over a p-cusp there lie one pleat and N − 3 r-points; over a s-node or a s-cusp, as over any s-point b ∈ B, there lie one singular point of X and N − 2 r-points.

2.2. Types of points on the fibre product. With respect to a pair of generic coverings f_{1} : X_{1} → ℙ^{2} and f_{2} : X_{2} → ℙ^{2} with the same discriminant curve B ⊂ ℙ^{2} nodes and cusps on B are partitioned by this time into 4 types: ss- , sp- , ps- and pp-nodes and cusps. For example, a ps-node it is a node b ∈ B, such that there are two folds on X_{1} over b, and on X_{2} over b there is a singular point of type A_{1}. The analogous terminology is used for the classification of points x̄ = (x_{1}, x_{2}) on the fibre product X^{×} = X_{1} ×_{ℙ^{2}} X_{2} : we say about rs-points, ss-points, sp-points, etc. For example, we say that x̄ is a ps-point of type A_{2}, if x_{1} ∈ X_{1} is a p-point of type A_{2}, and x_{2} ∈ Sing X_{2} is a singular s-point of type A_{2}.

In this section we describe the structure of a normalization ν : X = (X^{×})^{(ν)} → X^{×} of the fibre product X^{×}. Denote by g_{1}, g_{2} and f the morphisms of X to X_{1}, X_{2} and ℙ^{2}. Since the normalization is defined locally, we can replace ℙ^{2} by a neighbourhood of the point 0 ∈ ℂ^{2} and to assume that X_{1} and X_{2} are neighbourhoods of points x_{1} ∈ X_{1} and x_{2} ∈ X_{2}. We pass on to an item-by-item examination of all possible types of points x̄ = (x_{1}, x_{2}) ∈ X^{×}. We do it up to the permutation of factors X_{1} and X_{2}.

At first we consider quite trivial cases. 2.2.1. If x̄ is a r∗-point (where ∗ = r, s, p), then X^{×} at the point x̄ is locally the same as X_{2} at the point x_{2}, and f^{×} : X^{×} → ℂ^{2} is locally the same as f_{2} : X_{2} → ℂ^{2}. 2.2.2. If x̄ is a 2 × 2-point, i.e. x_{1} and x_{2} are points of ‘double planes’, z_{1}^{2} = h(x, y) , z_{2}^{2} = h(x, y), then X^{×} = X_{1} ×_{ℂ^{2}} X_{2} in a neighbourhood of the point x̄ = (x_{1}, x_{2}) is a surface in ℂ^{4} ∋ (x, y, z_{1}, z_{2}), defined by equations z_{1}^{2} = h(x, y), z_{2}^{2} = h(x, y). We obtain that z_{1}^{2} = z_{2}^{2}

and hence X^{×} = X1^{×} ∪ X2^{×}, where X1^{×} : z_{1}^{2} = h(x, y), z_{2} = z_{1} , X2^{×} : z_{2}^{2} = h(x, y), z_{1} = −z_{2}.

The surfaces X1^{×} and X2^{×} meet along a curve z_{1} = z_{2} = 0, h(x, y) = 0. We obtain that a normalization X = X̃^{×} = X1^{×} ⊔ X2^{×} locally consists of two disjoint components X1^{×} and X2^{×} isomorphic to X_{1} and X_{2}.

In particular, we obtain a description of the normalization in a neighbourhood of a pp-point (x_{1}, x_{2}) ∈ X^{×} lying over a non-singular point of B, B : x = 0,

Fig. 1

g_{1}^{∗}(R_{1}) = R′ + R′′, g_{2}^{∗}(R_{2}) = R′ + R′′.

Every ss-point is a 2×2-point. Thus, in a neighbourhood of a ss-point the normalization has the same local structure as in the case of a pp-point above: X locally consists of two disjoint components isomorphic to X_{1} and X_{2}.

It remains to ezamine less trivial cases when x̄ is a pp- or sp-point of type A_{1} or A_{2}. This is done in the following two subsections.

2.3. On fibre product of double planes. 2.3.1. The ordinary quadratic singularity – the singularity of type A_{1} on a surface X_{0} : z^{2} = xy can be considered as a 2-sheeted covering of the plane f_{0} : X_{0} → ℂ^{2} branched along a node B : xy = 0. As is known, the singularity X_{0} itself can be considered as a quotient singularity under the action of cyclic group ℤ_{2} = {±1}, X_{0} = X/ℤ_{2}, where X = ℂ^{2} ∋ (z_{1}, z_{2}), and a generator of ℤ_{2} acts by the rule: z_{1} ⟼ −z_{1}, z_{2} ⟼ −z_{2}. The factorization morphism g_{0} : X → X_{0} is defined by formulae

x = z_{1}^{2}, y = z_{2}^{2}, z = z_{1}z_{2}.

We obtain a 4-sheeted covering f = f_{0} ∘ g_{0} : X → ℂ^{2}, which can be considered as a factorization under the action of the group G = ℤ_{2} × ℤ_{2} on X. Then the factorization g_{0} corresponds to a subgroup of order two ℤ_{2} = G_{0} = {(1, 1), (−1, −1)}, imbedded diagonally into G. In G there are two more subgroups of order two: G_{1} = {1} × ℤ_{2} and G_{2} = ℤ_{2} × {1}. Considering X_{1} = ℂ_{2}/G_{1} ≃ ℂ_{2} and X_{2} = ℂ_{2}/G_{2} ≃ ℂ_{2}, we obtain two more decompositions of f and a commutative diagram

[figure: (x, y) ∈ ℂ^{2}; X = ℂ^{2} ∋ (z_{1}, z_{2}); (z_{1}, y) ∈ ℂ^{2} = X_{1}; X_{2} = ℂ^{2} ∋ (x, z_{2}) ,; g_{0}; f_{0}; f_{1}; f_{2}; g_{1}; X_{0}; g_{2}; (∗_{2})]

where g_{1} : y = z_{2}^{2}, f_{1} : x = z_{1}^{2}, g_{2} : x = z_{1}^{2}, f_{2} : y = z_{2}^{2}. Denote by B_{1} : x = 0, B_{2} : y = 0 the branches of B : xy = 0, and by R′ : z_{1} = 0, R′′ : z_{2} = 0 the branches of their proper transform z_{1}z_{2} = 0 on X.

The diagram (∗_{2}) shows that we can consider X as a normalization in three cases: 2.3.2. X is a normalization in a neighbourhood of a ps-point of type A_{1}, x̄ ∈ X_{1}^{×} = X_{1}×_{ℂ^{2}}X_{0},

Fig. 2

f_{1}^{∗}(B_{1}) = 2R_{1}, f_{1}^{∗}(B_{2}) = C_{1} ; g_{1}^{∗}(R_{1}) = R′, g_{1}^{∗}(C_{1}) = 2R′′ ;
f_{0}^{∗}(B) = 2R_{2} = 2(R′_{2} + R′′_{2}) ; g_{0}^{∗}(R′_{2}) = R′ ; g_{0}^{∗}(R′′_{2}) = R′′ .

(g_{0} is unramified outside the point 0 ∈ X_{0} ). 2.3.3. X is a normalization in a neighbourhood of a sp-point of type A_{1}, x̄ ∈ X_{2}^{×} = X_{0}×_{ℂ^{2}}X_{2} (the case symmetric to 2.3.2.)

2.3.4. X is a normalization in a neighbourhood of a pp-point of A_{1}, x̄ ∈ X^{×} = X_{1}×_{ℂ^{2}}X_{2} , which is not a 2 × 2-point, f_{1} : x = z_{1}^{2} , f_{2} : y = z_{2}^{2}.

Fig. 3

f_{1}^{∗}(B) = 2R_{1} + C_{1}, g_{1}^{∗}(R_{1}) = R′, g_{1}^{∗}(C_{1}) = 2R′′,
f_{2}^{∗}(B) = 2R_{2} + C_{2}, g_{2}^{∗}(R_{2}) = R′′, g_{2}^{∗}(C_{2}) = 2R′.

Using 2.3.2-2.3.4, now we can describe a normalization X over a neighbourhood of a node b ∈ B. 2.3.5 Over a neighbourhood of a ps-node b ∈ B (as well as a sp-node) a normalization of X^{×} in a neighbourhood of a ps-point looks like as

Fig. 4

g_{1}^{∗}(R′_{1}) = R^{′,′} , g_{1}^{∗}(C′_{1}) = 2R^{′,′′} , g_{1}^{∗}(R′′_{1}) = R^{′′,′} , g_{1}^{∗}(C′′_{1}) = 2R^{′′,′′} ,
g_{2}^{∗}(R′_{2}) = R^{′,′} + R^{′′,′′} , g_{2}^{∗}(R′_{2}) = R^{′,′′} + R^{′′,′}

On Fig. 4 the normalization in neighbourhoods of pr-, rs- and rr-points of X^{×} is not pictured. 2.3.6. Over a neighbourhood of a pp-node b ∈ B a normalization of X^{×} in a neighbourhood of a pp-point looks like as:

Fig. 5

g_{1}^{∗}(R′_{1}) = R^{′,′} + R^{′,′′} + C^{′,′} , g_{1}^{∗}(R′′_{1}) = R^{′′,′} + R^{′′,′′} + C^{′′,′} ,
g_{2}(R^{′,′}) = R′_{2} , g_{2}(R^{′,′′}) = R′_{2} , g_{2}(R^{′′,′}) = R′′_{2} , g_{2}(R^{′′,′′}) = R′′_{2} ,
g_{2}(C^{′,′}) = C′′_{2} , g_{2}(C^{′′,′}) = C′_{2} , (g_{1}(C^{′,′′}) = C′_{1} , g_{1}(C^{′′,′′}) = C′′_{1}) .

2.4. On coverings of ℂ^{2} unbranched outside a cusp B : y^{2} = x^{3}. To describe a normalization of the fibre product in a neighbourhood of a sp- and pp-point of type A_{2} in a natural context, we begin with reminding of a small topic from singularity theory. 2.4.1 The singularity of cuspidal type of a map (pleat) and the miniversal deformation of a singularity of type A_{2}. A cusp (B, 0) ⊂ (C^{2}, 0) is defined by a germ of function x^{3} − y^{2} stable equivalent to a germ of function x^{3}. It is a simple singularity of type A_{2}. It is interesting that a cusp (a singularity of type A_{2} ) appears also on the discriminant in the base of the miniversal deformation of the same singularity of type A_{2}.

As is known, the miniversal unfolding of the function t = z^{3} is

ℂ × ℂ^{2} → ℂ × ℂ^{2} , (z, a_{2}, a_{3}) ⟼ (z^{3} + a_{2}z + a_{3}, a_{2}, a_{3}) .

The restriction of this map over {0}×ℂ^{2} gives a miniversal deformation F of a zero-dimensional singularity z^{3} = 0, ℂ^{3} ⊃ X ⟶^{F} ℂ^{2}. Here X is a surface z^{3} + a_{2}z + a_{3} = 0, and F is induced by projection onto (a_{2}, a_{3}). The surface X is a graph of function −a_{3} = z^{3} + a_{2}z ; z and a_{2} are local coordinates on X,

[figure: (a_{2}, a_{3}) ∈ ℂ^{2}; (a_{2}, z) ∈ ℂ^{2}; ⟶^{∼}; , G : \{ \begin{matrix}a_{2} & = & a_{2} \\ −a_{3} & = & z^{3} + a_{2}z. \end{matrix}; F; G; X ⊂ ℂ^{3}]

We obtain a 3-sheeted covering G : ℂ^{2} → ℂ^{2}, the ramification curve of which R is defined by the equation 3z^{2} + a_{2} = 0, and the discriminant (branch) curve B = G(R) is defined by equation

4a_{2}^{3} + 27a_{3}^{2} = 0.

To bring the equation of B to the form y^{2} = x^{3}, we make a substitution

a_{2} = −3x , a_{3} = 2y ,

and denote ℂ^{2} ≃ X by X_{3}, and G by f_{3}.

Fig. 6

We obtain a 3-sheeted covering f_{3} : X_{3} → ℂ^{2},

f_{3} : x = x, y = −\frac{1}{2}(z^{3} − 3xz).

Then x^{3} − y^{2} = x^{3} − \frac{1}{4}(z^{3} − 3xz)^{2} = (x − z^{2})^{2}(x − \frac{1}{4}z^{2}) and, consequently,

f_{3}^{∗}(B) = 2R + C,

where R : x = z^{2} is the ramification curve, and C : x = \frac{1}{4}z^{2}. Note that C and R are tangent of order two, (C · R) = 2.

By Lemma 1.3 the singular point of the covering f_{3} is uniquely characterized as a singular point of a 3-sheeted covering f : X → ℂ^{2} by a non-singular surface X, the discriminant curve of which is an ordinary cusp. 2.4.2 The Viète map f_{6}. We produce a well known regular covering of ℂ^{2} with group S_{3} branched along a cusp B : y^{2} = x^{3}, which appears to be a normalization of the fibre product in a neighbourhood of a sp-point of type A_{2}. This covering naturally appears in singularity theory.

Consider a quotient of the space ℂ^{3} under the action of permutation group S_{3}. We get the Viète map

v : ℂ^{3} → ℂ^{3} , (z_{1}, z_{2}, z_{3}) ⟼ (a_{1}, a_{2}, a_{3}),

where (z − z_{1})(z − z_{2})(z − z_{3}) = z^{3} + a_{1}z^{2} + a_{2}z + a_{3} , i.e.

a_{1} = −(z_{1} + z_{2} + z_{3}), a_{2} = z_{1}z_{2} + z_{2}z_{3} + z_{3}z_{1}, a_{3} = −z_{1}z_{2}z_{3}.

The map v is a map of degree 6 unramified outside Δ = ∪_{i≠j}{z_{i} = z_{j}}, and v(Δ) = D is defined by the discriminant of a polynomial of degree three.

The action of S_{3} on ℂ^{3} is reducible: ℂ^{3} is a direct sum ℂ^{3} = ℂ ⊕ ℂ^{2} of invariant subspaces – of the line ℂ = {z_{1} = z_{2} = z_{3}} and of the plane ℂ^{2} = {z_{1} + z_{2} + z_{3} = 0}. Consider the restriction of v to this plane ℂ^{2},

(z_{1}, z_{2}, z_{3}) ∈ ℂ^{3} ⊃ {z_{1} + z_{2} + z_{3} = 0} = ℂ^{2} ⟶ ℂ^{2} = {a_{1} = 0} ⊂ ℂ^{3} ∋ (a_{1}, a_{2}, a_{3}) .

Set ℂ^{2} ∩ Δ = L , ℂ^{2} ∩ D = B. Then L consists of three lines

L = L_{1} + L_{2} + L_{3}, where L_{i} : z_{j} = z_{k}, z_{1} + z_{2} + z_{3} = 0 , {i, j, k} = {1, 2, 3} ,

and the curve B : 4a_{2}^{3} + 27a_{3}^{2} = 0 is defined by the discriminant of the polynomial z^{3} + a_{2}z + a_{3}. Since π_{1}(ℂ^{2} \ L) = π_{1}(ℂ^{3} \ Δ) , π_{1}(ℂ^{2} \ B) = π_{1}(ℂ^{3} \ D) = Br_{3}, we obtain a covering v : ℂ^{2} → ℂ^{2} of degree 6 unbranched apart from B. Denote this map by f_{6}. In coordinates x, y, where a_{2} = −3x, a_{3} = 2y, this map

ℂ^{2} = {z_{1} + z_{2} + z_{3} = 0} = X_{6} ⟶^{f_{6}} ℂ^{2} ∋ (x, y)

is defined by formulae

f_{6} : x = −\frac{1}{3}(z_{1}z_{2} + z_{2}z_{3} + z_{3}z_{1}) , y = −\frac{1}{2}z_{1}z_{2}z_{3},

the discriminant B has equation y^{2} = x^{3}, and f^{∗}(B) = 2L = 2L_{1} + 2L_{2} + 2L_{3} (it is easy to see that x^{3} − y^{2} = \frac{1}{4·27}(z_{2} − z_{1})^{2}(z_{3} − z_{2})^{2}(z_{1} − z_{3})^{2} under condition z_{1} + z_{2} + z_{3} = 0 ).

Consider a two-sheeted covering unbranched outside B

(x, y, w) ∈ ℂ^{3} ⊃ X_{2} ⟶^{f_{2}} ℂ^{2} ∋ (x, y) ,

where X_{2} is defined by equation w^{2} = x^{3} −y^{2}, and f_{2} is induced by projection. Such a structure has a generic covering f : X → ℙ^{2} in a neighbourhood of a s-point of type A_{2}.

Lemma 2.1 ([C]) If f : (X, 0) → (ℂ^{2}, 0) is a finite covering by a normal irreducible surface X, unbranched outside an ordinary cusp B ⊂ ℂ^{2}, and the ramification curve of which is reduced, i.e. f^{∗}(B) = 2R + C (R and C reduced curves), then f is equivalent to one of the coverings f_{2}, f_{3} and f_{6}.

The proof is obtained by means of studying the possible monodromy homomorphisms ρ : π_{1} → S_{N}, where π_{1} = π_{1}(ℂ^{2} \ B) = Br is the fundamental group of a cusp, and N = deg f.

We obtain one more characterization of the covering f_{3} as a finite covering f : (X, 0) → (ℂ^{2}, 0) by a normal irreducible surface, unbranched outside a cusp B, and with a reduced and non-singular ramification curve R. 2.4.3 Description of a normalization of the fibre product in a neighbourhood of a sp-point of type A_{2}. The map f_{6} factors through the maps f_{2} and f_{3}, and we have a commutative diagram

[figure: (x, y) ∈ ℂ^{2} ⊃ B : y^{2} = x^{3}; X_{6} = {z_{1} + z_{2} + z_{3} = 0}; {w^{2} = x^{3} − y^{2}} = X_{2}; X_{3} = {z^{3} − 3xz^{2} + 2y = 0} ,; f_{2}; f_{3}; g_{3}; f_{6}; g_{2}; (∗_{3})]

where g_{2} and g_{3} are defined by formulae: x and y are defined by the same formulae as f_{6}, and z = z_{1} for g_{2}, and w = \frac{1}{6\sqrt{3}}(z_{2} − z_{1})(z_{3} − z_{2})(z_{1} − z_{3}) for g_{3}. It is easy to see that g_{3} is a factorization under the action of a cyclic group ℤ_{3} = A_{3} ⊂ S_{3}, X_{2} = X_{6}/A_{3}, and g_{2} is a factorization under the action of a cyclic group of order two ℤ_{2} ≃ S_{2} = {(1), (2, 3)} ⊂ S_{3}.

By the property of universallity of fibre products we have a morphism X_{6} → X_{2}×_{ℂ^{2}}X_{3}. The fibre product X_{2}×_{ℂ^{2}}X_{3} is irreducible, since each its component Z is mapped onto X_{2} and X_{3}, and, therefore, the degree of Z → ℂ^{2} have to be divided by 2 and 3, i.e. have to be equal to 6. Thus, X_{6} is a normalization of X_{2}×_{ℂ^{2}}X_{3}, and the diagram (∗_{3}) describes a normalization of the fibre product in a neighbourhood of a sp-point of type A_{2}.

The diagram (∗_{3}) can be visually-schematic presented as follows

Fig. 7

Direct computations show that x − z^{2} = \frac{1}{3}(z_{2} − z_{1})(z_{1} − z_{3}) , and x − \frac{1}{4}z^{2} = \frac{1}{12}(z_{3} − z_{2})^{2}, i.e.

g_{2}^{∗}(R) = L_{2} + L_{3} , g_{2}^{∗}(C) = 2L_{1}.

And, besides, g_{3}^{∗}(R_{1}) = L_{1} + L_{2} + L_{3}. 2.4.4 Description of a normalization of the fibre product in a neighbourhood of a pp-point of type A_{2}. Let x_{1} ∈ X_{1} and x_{2} ∈ X_{2} be p-points of type A_{2} for f_{1} and f_{2}, f_{1}^{∗}(B) = 2R_{1} + C_{1} , f_{2}^{∗}(B) = 2R_{2} + C_{2}. In this case the 3-sheeted coverings f_{1} and f_{2} are the same (equivalent), and the monodromy homomorphisms φ_{1}, φ_{2} : π_{1} = π_{1}(ℂ^{2} \ B, y_{0}) → S_{3} are epimorphic. The fibre (f^{×})^{−1}(y_{0}) of the 9-sheeted covering f^{×} : X^{×} = X_{1}×_{ℂ^{2}}X_{2} → ℂ^{2} consists of pairs f_{1}^{−1}(y_{0}) × f_{2}^{−1}(y_{0}) = {(i, j), 1 ≤ i, j ≤ 3}, and the monodromy homomorphism is (equivalent to) a diagonal homomorphism φ : π_{1} → S_{3} × S_{3} ⊂ S_{9}. Since φ_{i} are epimorphic, the fibre (f^{×})^{−1}(y_{0}) consists of two orbits w.r.t. the action of π_{1} — the orbit of the point (1, 1), which consists of 3 elements, and the orbit of the point (1, 2), which consists of 6 elements. From this and from Lemma 2.1 it follows that in a neighbourhood of the x̄ = (x_{1}, x_{2}) a normalization X of the product X^{×} consists of two disjoint components X = X_{3} ∐ X_{6}, and on X_{3} the morphism f : X → ℂ^{2} coincides with f_{3}, the morphisms g_{1} and g_{2} are isomorphisms, and on X_{6} the morphism f = f_{6}, the morphisms g_{1} and g_{2} are the same as g_{2} in the diagram (∗_{3})

There are 4 curves on X^{×}: C_{1}×_{B}C_{2}, R_{1}×_{B}C_{2}, C_{1}×_{B}R_{2}, R_{1}×_{B}R_{2}, preimages of which on the normalization X are C_{3}, L_{2}, L_{1}, and L_{3}, R_{3}. Under such a numeration of the lines L_{i} we have

g_{1}^{∗}(R_{1}) = R_{3} + L_{2} + L_{3} , g_{1}^{∗}(C_{1}) = C_{3} + 2L_{1},
g_{2}^{∗}(R_{2}) = R_{3} + L_{1} + L_{3} , g_{2}^{∗}(C_{2}) = C_{3} + 2L_{2}.

2.4.5 A lift of the diagram (∗_{3}). Consider the diagram (∗_{3}). For computation of intersection numbers in §5 we need to resolve the singular point of type A_{2} on the surface X_{2}, and to ‘disjoint’ the curves L_{2} and L_{3} on X. A resolution of the singular point of type A_{2}, as of any ‘double plane’, can be obtained, if we firstly take an imbedded resolution σ : ℂ̄^{2} → ℂ^{2} of the branch curve B ⊂ ℂ^{2}, and then take a normalization of X_{2}×_{ℂ^{2}} ℂ̄^{2}.

Actually we’ll make more – we lift the whole of the diagram (∗_{3}) on ℂ̄^{2}.

1) The singular point of B is resolved by one σ-process σ_{1}. It is enough for the resolving of the singular point on X_{2}, but to resolve the total transform of B up to a divisor with normal crossings, one need two more σ-processes. We picture the resolution process schematically by ‘drawing’ the total transform of the curve B. Denote by E_{i} the curve glued in under the i-th σ-process, and also its proper transform under subsequent σ-processes.

Fig. 9

Along each curve we indicate two numbers: the negative is the self-intersection number, the positive is its multiplicity in the total transform of the curve B.

2) Denote by σ : ℂ̄^{2} → ℂ^{2} the composition σ_{3} ∘ σ_{2} ∘ σ_{1}. We add on the diagram (∗_{3}) over ℂ̄^{2} and obtain a diagram as follows, on which all morphisms on the right face are finite coverings.

\xymatrix{ & X_{6} \ar[dl]_{g_{3}} \ar@{-->}[ddr]^{g_{2}} &  & \overline{X}_{6} \ar[ll]_{σ_{6}} \ar[dl]_{_{\overline{g}_{3}}} \ar[ddr]^{_{\overline{g}_{2}}} \\ X_{2} \ar[ddr]_{f_{2}} &  & _{\overline{X}_{2}} \ar[ll]^{σ_{2}} \ar[ddr]^{_{\overline{f}_{2}}} \\  &  & X_{3} \ar@{-->}[dl]_{f_{3}} &  & _{\overline{X}_{3}} \ar@{-->}[ll]_{σ_{3}} \ar[dl]^{_{\overline{f}_{3}}} \\  & ℂ^{2} &  & _{\overline{ℂ}_{2}} \ar[ll]_{σ}}  (∗)

The right square of the diagram (∗) is obtained as a fibre product (∗_{3})×_{ℂ^{2}} ℂ̄^{2}, i.e. X̄_{i} are normalizations of X_{i}×_{ℂ^{2}} ℂ̄^{2}, and morphisms are induced by morphisms of the diagram (∗_{3}) and projections. We describe how one can construct the diagram (∗) not uniformly as a normalization of the lift, but step-by-step. To facilitate the following of the description we begin with the final picture. We draw the right square of the diagram (∗) by replacing the varieties at its vertices by the total transforms of the curve B

Fig. 10 The rule of notation is as follows. The exceptional curves E_{1}, E_{2}, E_{3} on ℂ̄^{2} are already denoted. Under double indexing E_{i,j} the first index indicates the variety X_{i}, where E_{i,j} lies, and the second index indicates to what curve E_{j} the curve E_{i,j} is mapped on ℂ̄^{2}.

3) We begin a description of the diagram (∗) with X̄_{6} (‘from the top’). To disjoint the lines L_{i}, we make σ-process with centre at the point 0 ∈ X_{6} = ℂ^{2} = {z_{1} + z_{2} + z_{3} = 0} ⊂ ℂ^{3}. By this the curve E_{6,3} = ℙ^{1} = {t_{1} + t_{2} + t_{3} = 0} ⊂ ℙ^{2} ∋ (t_{1} : t_{2} : t_{3}) is glued, and we obtain a variety X′_{6}. The action of S_{3} on X_{6} is extended to X′_{6} and, in particular, to ℙ^{1}. On ℙ^{1} there are 8 exceptional points forming exceptional orbits:

p_{1} = E_{6,3} ∩ L_{1} = (−2 : 1 : 1), p_{2} = E_{6,3} ∩ L_{2} = (1 : −2 : 1), p_{3} = E_{6,3} ∩ L_{3} = (1 : 1 : −2);

P_{1} = (0 : 1 : −1), P_{2} = (1 : 0 : −1), P_{3} = (1 : −1 : 0);
Q_{1} = (1 : ζ : ζ^{2}), Q_{2} = (1 : ζ̄: ζ̄^{2}),

where ζ = ^{3} \sqrt{1} is a primitive root, and ζ̄ = ζ^{2}. Denote by ξ = (123) a generator of the cyclic group of order three ℤ_{3} = A_{3} = {(1), (123), (132)} ⊂ S_{3}, and by ε = (23) a generator of the cyclic group of order two ℤ_{2} = S_{2} = {(1), (23)} ⊂ S_{3}. Then

ξ(p_{1}) = p_{2}, ξ(p_{2}) = p_{3}, ξ(p_{3}) = p_{1}; ξ(P_{1}) = P_{2}, ξ(P_{2}) = P_{3}, ξ(P_{3}) = P_{1};
ξ(Q_{1}) = (ζ^{2} : 1 : ζ) = (1 : ζ : ζ^{2}) = Q_{1}, ξ(Q_{2}) = (ζ : 1 : ζ^{2}) = (1 : ζ^{2} : ζ) = Q_{2};
ε(p_{1}) = p_{1}, ε(p_{2}) = p_{3}, ε(p_{3}) = p_{2}; ε(P_{1}) = P_{1}, ε(P_{2}) = P_{3}, ε(P_{3}) = P_{2}; ε(Q_{1}) = Q_{2}.

If we take a quotient X′_{6} under the action of Z_{3} = A_{3}, then the stationary points Q_{1} and Q_{2} give two quotient singularities on X′_{2} = X′_{6}/A_{3}, resolving of which X′′_{2} → X′_{2} glues the curves E′_{2,1} and E′′_{2,1} with (E′_{2,1}^{2}) = −3, (E′′_{2,1}^{2}) = −3. To lift X′_{6} → X′_{2} onto X′′_{2}, we have to blow up the points Q_{1} and Q_{2} , X′′_{6} → X′_{6}, and by this we obtain X′′_{6} → X′′_{2},

Fig. 11

4) The map f_{2} is a factorization under the cyclic group ℤ_{2} = S_{2}. The action extends to X′′_{2}. The stationary point on E_{2,3} – the image of the point P_{1} on E_{6,3} gives a singular point of type A_{2} on X′′_{2}/ℤ_{2} (= C̄^{2}′). A resolution of this point glues a (-2)-curve E_{2}, and we obtain ℂ̄^{2}. To lift X′′_{2} → X′′_{2}/ℤ_{2} onto the resolution ℂ̄^{2}, we have to blow up a point on X′′_{2}. By this a (-1)-curve E_{2,2} is glued, and we obtain X̄_{2}. To obtain ḡ_{3} : X̄_{6} → X̄_{2}, we have to perform 3 σ-processes with centres at points P_{1}, P_{2}, P_{3} on X′′_{6}, by which three lines E′_{6,2}, E′′_{6,2} and E′′′_{6,2} are glued. We obtain the left side ḡ_{3} and f̄_{2} of the right square of diagram (∗), pictured on Fig.

10. Note thay the map ḡ_{3} is ramified along the curves E′_{6,1} and E′′_{6,1}, and the map f̄_{2} is ramified along the curves E_{2,2} and R_{1}.

We can blow down the (-1)-curve E_{2,2} on X̄_{2}, and then to blow down the (-1)-curve E_{2,3}. By this we obtain a minimal resolution of the singular point of type A_{2} on X_{2},

Fig. 12

5) The map ḡ_{2} is a factorization under the group Z_{2} = S_{2} = {(1), (23)}. We obtain the surface X̄_{3} = X̄_{6}/S_{2}, ḡ_{2} : X̄_{6} → X̄_{3}. The map ḡ_{2} is ramified along the curves E′_{6,2} and L_{2}, which are mapped onto E′_{3,2} and C correspondingly. The diagram is completed by the map f̄_{3} : X̄_{3} → ℂ̄^{2}. The surface X̄_{3} is obtained from X_{3} = ℂ^{2}, if we at first blow up the point of tangency of curves C and R gluing E′_{3,2}; then we blow up the point of intersection of C and R gluing E_{3,3}; finally, we blow up two more points on E_{3,3}:

Fig. 13

3 The canonical cycle of a Du Val singularity

We intend to apply Hodge index theorem to obtain the basic inequality for generic coverings of ℙ^{2} by surfaces with A-D-E-singularities. For this we need intersection theory and, therefore, a resolution of singularities of X. In this section we examine the local situation and find out how the resolution affects the canonical class and the ramification curve.

3.1. Definition of canonical cycle. Let (X, x) be a 2-dimensional A-D-E-singularity. Let π : X̄ → X be a minimal resolution, L = π^{−1}(x) be the exceptional curve. As is known, the canonical class KX̄ is trivial in a neighbourhood of L, that is we can choose a divisor in KX̄ with a support not intersecting L. In other words, there is a differetial form ω on X̄, which has neither poles nor zeroes in a neighbourhood of L. Such a form can be obtained, for example, as follows. As is known, (X, x) is a quotient singularity, X = ℂ^{2}/G, where G ⊂ SL(2, ℂ) . The form du ∧ dv on ℂ^{2} ∋ (u, v) is invariant w.r.t. G and it defines a form on X (φ^{∗}(ω) = du ∧ dv, where φ : ℂ^{2} → X). Hence, the divisor (ω) = ∑ k_{i}L_{i}. Since L_{i} are (-2)-curves, (L_{i} · (ω)) = 0, and we obtain (ω) = 0.

On the other hand, (X, x) can be considered as a double plane, that is as a 2-sheeted covering X ⟶^{f} Y of the plane Y = ℂ^{2} (locally). Let z^{2} = h(x, y) be an equation of (X, x), B : h(x, y) = 0 be the discriminant curve, f^{−1}(B) = R, defined by the equation z = 0, be the ramification curve. We can consider the differential form ω = f^{∗}(dx ∧ dy) lifted from Y. Then on X̄ the divisor (ω) = (z) = R̄ + Z, where R̄ ⊂ X̄ is the proper transform of R , Z = ∑ γ_{i}L_{i} is a cycle on L = π^{−1}(x). We shall say that Z is the canonical cycle of a 2-dimensional A-D-E-singularity. Thus, −Z is a cycle on the exceptional curve L, which is equivalent to the ramification curve R̄ in a neighbourhood of L. Let us calculate the canonical cycle for all A-D-E-singularities.

3.2. On resolution of double planes. As for any double plane, a resolution of an A-D-E-singularity can be obtained by means of a resolution of the discriminant curve B ⊂ Y = ℂ^{2}, B : h(x, y) = 0. Let σ : Ȳ → Y be a composition of σ-processes, such that the total transform of B is a divisor with normal crossings. Let σ^{∗}(B) = B̄ + ∑_{i=1}^{r} α_{i}l_{i}, where B̄ is the proper transform of B, l_{i} ≃ P^{1}, i = 1, …, r, are the exceptional curves, as well as their proper transforms, glued by σ-processes. Let X̄ be the normalization of Ȳ ×_{Y} X, and f̄ and π be induced by projections,

[figure: π^{−1}(x) = L = L_{1} ∪ … ∪ L_{r} ⊂ X̄; X ⊃ R ∋ x, R : z = 0; π; σ^{∗}(B) = \overline{B} + ∑_{i=1}^{r} α_{i}l_{i} ⊂ Ȳ; Y ⊃ B.; f; f̄; σ; (♯)]

Set f̄^{−1}(l_{i}) = L_{i}. The curve L_{i} is either irreducible or consists of two components L_{i} = L′_{i} + L′′_{i}, where L′_{i} ≃ ℙ^{1} , L′′_{i} ≃ ℙ^{1}. The mapping f̄ is a 2-sheeted covering branched along the curve B̄ + ∑_{α_{i}−odd} l_{i}. To be more graphic we denote the curves l_{i}, for which α_{i} are odd, also by l̄_{i}, and L_{i} – respectively by L̄_{i}. The surface X̄ has singularities of type A_{1} over nodes of the branch curve B̄ + ∑ l̄_{i}. If this curve is non-singular, that is, a disconnected union of components (one can reach this by performing one additional σ-processes for each node), then X̄ is non-singular and is a resolution of the singularity (X, x). Let R̄ be the proper transform of R w.r.t. π (= the proper transform of B̄ w.r.t. f̄). We have f̄^{∗}(l̄_{i}) = 2L̄_{i}, if α_{i} is odd, and f̄^{∗}(l_{i}) = L_{i}, if α_{i} is even. We have

((σ ∘ f̄)^{∗}h(x, y)) = (z^{2}) = 2R̄ + ∑_{α_{i}−odd} 2α_{i}L̄_{i} + ∑_{α_{i}−even} α_{i}L_{i}

and, consequently, (z) = R̄ + Z, where

Z = ∑_{α_{i}−odd} α_{i}L̄_{i} + ∑_{α_{i}−even} \frac{1}{2}α_{i}L_{i} .

Let us compute the cycle Z for each type of A-D-E-singularities (despite of abundance of papers concerning Du Val singularities, the authors do not know any of them, where the cycle Z is written out; so we have to perform these computations).

3.3. Computation of the canonical cycle. Consider the minimal resolution of each type of A-D-E-singularities described above. The following lemma contains the results of computations of σ^{∗}(B), of the exceptional curve π^{−1}(x) = L and of the canonical cycle Z.

Lemma 3.1 Below we picture schematically the total transform σ^{∗}(B) = B̄ + ∑_{i=1}^{r} α_{i}l_{i} (near each curve l_{i} a positive number α_{i} and a negative number (l_{i}^{2}) are shown), and over it we picture the curve π^{−1}(R), consisting of R̄ and (-2)-curves, and besides we write down the canonical cycle Z:

1) The singularity A_{2k−1} : y^{2} = x^{2k}, k ≥ 1,

Z = L_{1} + 2L_{2} + … + kL_{k};

2) The singularity A_{2k} : y^{2} = x^{2k+1}, k ≥ 1,

Z = L_{1} + 2L_{2} + … + kL_{k};

3) The singularity D_{2k+2} : x(y^{2} + x^{2k}), k ≥ 1,

Z = 3L_{1} + 5L_{2} + … + (2k + 1)L_{k} + 2L_{k+1} + 4L_{k+2} + … + 2kL_{2k} + (k + 1)L_{2k+1} + (k + 1)L_{2k+2} ; 4) The singularity D_{2k+3} : x(y^{2} + x^{2k+1}), k ≥ 1,

Z = 3L_{1} + 5L_{2} + … + (2k + 1)L_{k} + 2L_{k+1} + 4L_{k+2} + … + 2kL_{2k} + (2k + 2)L_{2k+2} + (k + 1)L_{2k+1} ;

L_{2k+1} = L′_{2k+1} + L′′_{2k+1} ;

5) The singularity E_{6} : x^{3} + y^{4} ,

Z = 3L_{1} + 2L_{2} + 4L_{3} + 6L_{4} ; 6) The singularity E_{7} : x(x^{2} + y^{3}) ,

Z = 3L_{1} + 5L_{2} + 9L_{3} + 6L_{4} + 5L_{5} + 8L_{6} + 3L_{7} ;

7) The singularity E_{8} : x^{3} + y^{5} ,

Z = 3L_{1} + 5L_{2} + 9L_{3} + 15L_{4} + 10L_{5} + 8L_{6} + 12L_{7} + 6L_{8} .

3.4. Defect of a singularity. Define a defect δ of a A-D-E-singularity by the formula

δ = \frac{1}{2}(R̄ · Z) .

Corollary 3.1 For different types of A-D-E-singularities the defect equals

δ = \{ \begin{matrix}[\frac{n+1}{2}] & for type A_{n}; \\ [\frac{n}{2}] + 1 & for type D_{n}; \\ [\frac{n+1}{2}] & for types E_{n}, n = 6, 7, 8 . \end{matrix}

In particular, for the type A_{1} (nodes) and A_{2} (cusps) the defect δ = 1.

Actually one can show that defect δ is the δ-invariant (genus) of the one-dimensional A-D-E-singularity.

4 Numerical invariants of a generic covering

Now we consider a global situation. Let X be a surface with A-D-E-singularities,

Sing X = ∑_{k≥1} a_{k}A_{k} + ∑_{k≥4} d_{k}D_{k} + ∑_{k=6,7,8} e_{k}E_{k} ,

that means that X has a_{k} singularities of type A_{k}, d_{k} – of type D_{k} and e_{k} – of type E_{k}. Let f : X → P^{2} be a generic covering of degree N, and B ⊂ ℙ^{2} be the discriminant curve. Let deg B = d and let B has n nodes and c cusps, n_{s} = a_{1} and c_{s} = a_{2} of which originates from Sing X, and n_{p} and c_{p} are p-nodes and p-cusps. Let R ⊂ X be the ramification curve, f^{∗}(B) = 2R + C, and L ⊂ X be the preimage of a generic line l ⊂ ℙ^{2}. Let π : S → X be the minimal resolution of X, and f̄ = f ∘ π : S → ℙ^{2}. Denote by R̄ and L̄ the proper transforms of R and L on S. Then R̄ is a normalization of the curve R ≃ B, and L̄ ≃ L.

4.1. The canonical class K_{S} and the canonical cycle Z. Let

Z = ∑_{x∈SingX} Z_{x}

be the canonical cycle of S, where Z_{x} are the canonical cycles of singularities x ∈ Sing X. It follows from 3.2 that

K_{S} = (f ∘ π)^{∗}K_{ℙ^{2}} + R̄ + Z = −3L̄ + R̄ + Z .  (4.1)

Besides, the singularities of X being Gorenstein, the divisor R is locally principal, and

π^{∗}(R) = R̄ + Z .  (4.2)

4.2. The intersection numbers.

Lemma 4.1 The intersection numbers of L̄, R̄ and Z on S are equal

\binom{}{L̄^{2}} = N , L̄ · R̄ = d , L̄ · Z = 0 , R̄ · Z = 2δ_{X} , (Z^{2}) = −2δ_{X} ,  (4.3)

where

δ_{X} = ∑_{x∈SingX} δ_{x} = ∑ a_{k} [\frac{k + 1}{2}] + ∑ d_{k} ([\frac{k}{2}] + 1) + ∑ e_{k} [\frac{k + 1}{2}]  (4.4)

is the defect of the surface X.

Proof. Obviously, we have \binom{}{L̄^{2}} = deg f = N, and L̄ · R̄ = deg B = d. By 3.4 we have R̄ · Z = 2δ_{X}. The divisor Z being exceptional, we have π(Z) = Sing X, dim π(Z) = 0, and L̄ = π^{∗}(L), R̄ + Z = π^{∗}(R), and therefore, L̄ · Z = 0, and (R̄ + Z) · Z = 0, and, consequently, (Z^{2}) = −(R̄ · Z).

It remains to compute \binom{}{R̄^{2}}.

4.3. The evenness of degree deg B = d = 2d̄. The restriction of f̄ to L̄ , L̄ → l ≃ ℙ^{1}, is a covering of degree N, with ramification indices 2 at the points of intersection of L̄ and R̄. We have L̄ · R̄ = d, and from Hurwitz formula we obtain 2g(L̄) − 2 = −2N + d. It follows that deg B = d is even. Let d = 2d̄. Besides, since

g(L̄) = \frac{1}{2}d + 1 − N ≥ 0 ,

we obtain a bound for the degree of covering,

N ≤ d̄+ 1 .

4.4. The self-intersection number \binom{}{R̄^{2}} and the arithmetical genus of the curve R. Denote by δ the defect of the curve B,

δ = δ_{B} = ∑_{s∈SingB} δ_{s} = n + c + δ_{0},  (4.5)

where

δ_{0} = ∑_{x∈SingB, x not A_{1}andA_{2}} δ_{x}.  (4.6)

The numbers δ and δ_{0} are the extremal values of defects δ_{X} of surfaces X with given discriminant curve B : δ_{0} corresponds to a surface X, all nodes and cusps of which are p-nodes and p-cusps, n = n_{p}, c = c_{p}, and δ corresponds to a surface X (for example, a 2-sheeted covering of ℙ^{2}), all nodes and cusps of which are s-nodes and s-cusps, n = n_{s}, c = c_{s}.

At first we express the geometric genus of B, g = g(B) = g(R̄), in terms of the defect δ. For this we consider a surface X, which is a 2-sheeted covering of ℙ^{2} with the discriminant curve B. In this case (Z^{2}) = −(R̄ · Z) = −2δ, and f^{∗}(B) = 2R and, consequently, d · L̄ ∼ 2R̄ + 2Z,

because B ∼ d · l. From (4.1) and the adjunction formula g(R̄) = \frac{(R̄, R̄ + KX̄)}{2} + 1 we obtain

g = \frac{(d − 1)(d − 2)}{2} − δ .  (4.7)

If it is known that the defect δ coincides with the δ-invariant of a one-dimensional singularity, then this formula coincides with the well known formula for the geometric genus g(R) =^{df} g(R̄) of a singular curve R, g(R) = p_{a}(R) − ∑_{x∈SingR} δ_{x}.

We return to a generic covering X of degree N, n = n_{s} + n_{p}, c = c_{s} + c_{p}. Then

δ_{X} = n_{s} + c_{s} + δ_{0} = δ − n_{p} − c_{p}.  (4.8)

Lemma 4.2 The self-intersection number of the proper transform of the ramification curve R̄ ⊂ S is equal

\binom{}{R̄^{2}} = 3d̄+ g − 1 − δ_{X} ,  (4.9)

and

(R̄ + Z)^{2} = 3d̄+ g − 1 + δ_{X} = 3d̄+ p_{a}(R) − 1 ,  (4.10)

where

p_{a}(R) = g + δ_{X} = \frac{(d − 1)(d − 2)}{2} − n_{p} − c_{p}  (4.11)

is the arithmetical genus of R.

Proof. From (4.1) and the adjunction formula 2g(R̄)−2 = (R̄, R̄+K_{S}) = (R̄, −3L̄+2R̄+Z) we obtain \binom{}{R̄^{2}} = \frac{3}{2}(R̄ · L̄) + g − 1 − \frac{1}{2}(R̄ · Z). Applying formulae (4.3), we obtain the proof.

From formulae (4.1), (4.3) and (4.9) we obtain a corollary.

Corollary 4.1

(K_{S}^{2}) = 9N − 9d̄+ p_{a}(R) − 1,  (4.12)

or, substituting p_{a}(R) from (4.11),

(K_{S}^{2}) = 9N + \frac{1}{2}d(d − 12) − n_{p} − c_{p}.  (4.12′)

4.5. A bound for the covering degree.

Lemma 4.3 For a generic covering of degree N with discriminant curve of degree d = 2d̄ and genus g, we have

N ≤ \frac{4d̄^{2}}{3d̄+ g − 1 + δ_{X}} ,  (4.13)

where δ_{X} is the defect of singularities of X, and moreover, the equality holds if and only if L̄ ≡ mK_{S} for some m ∈ ℚ^{∗}, or mK_{S} ≡ 0.

Proof. Applying Hodge index theorem to divisors L̄ and π^{∗}(R) = R̄ + Z on S, we obtain

| \begin{matrix}L̄^{2} & (L̄, R̄ + Z) \\ (L̄, R̄ + Z) & (R̄ + Z)^{2} \end{matrix} | = | \begin{matrix}N & d \\ d & 3d̄+ g − 1 + δ_{X} \end{matrix} | ≤ 0 ,

and it is the desired inequality. The equality holds only if L̄ and R̄ + Z are linear dependent in the Néron-Severi group NS(X̄) ⊗ Q. Since K_{S} = −3L̄ + R̄ + Z, we obtain the assertion about possible equality.

4.6. The topological Euler characteristic e(S).

Lemma 4.4 The topological Euler characteristic of a surface S, obtained by the minimal resolution of singularities of X, is connected with the defect δ_{X} and invariants of a generic covering f by a formula

e(S) = 3N + 2g − 2 + 2δ_{X} − c_{p},  (4.14)

where N = deg f, and c_{p} is the number of p-cusps on B (or the number of pleats of f).

Proof is obtained in the same way as in the case of a non-singular surface X ([K], §1 Lemma 7), considering a generic pencil of lines on ℙ^{2} and the corresponding hyperplane sections on S, and lifting the morphism f̄ : S → ℙ^{2} to a morphism of fiberings of curves over ℙ^{1}. One can obtain a proof by direct computations. At first we find e(X) = 3N−e(B)−n_{p}−c_{p} by considering the finite covering f : X → ℙ^{2}, the stratification ℙ^{2} = (ℙ^{2} \ B) ∪ (B \ Sing B) ∪ Sing B, and applying the additivity property of Euler characteristic, and then we find e(S).

From Noether’s formula (K_{S}^{2}) + e(S) = 12p_{a} and formulae (4.12) and (4.14) we have 12p_{a} = 12N − 9d̄+ 3p_{a}(R) − 3 − c_{p}. Substituting p_{a}(R) from (4.11), we obtain a corollary.

Corollary 4.2 The Euler characteristic of the structure sheaf O_{S} equals

p_{a} = 1 − q − p_{g} = N + \frac{d̄(d̄− 3)}{2} − \frac{n_{p}}{4} − \frac{c_{p}}{3} .  (4.15)

Thus, as in the case of a non-singular surface X, we obtain

Corollary 4.3

n_{p} ≡ 0 (mod 4) , c_{p} ≡ 0 (mod 3).

5 Proof of the main inequality.

5.1. A fiber product of two generic coverings. Let a curve B be a common discriminant curve for two generic coverings f_{1} : X_{1} → ℙ^{2} and f_{2} : X_{2} → ℙ^{2} of degrees deg f_{1} = N_{1} and degf_{2} = N_{2}. Let

Sing B = nA_{1} + cA_{2} + ∑_{k>2} a_{k}A_{k} + ∑_{k≥4} d_{k}D_{k} + ∑_{k=6,7,8} e_{k}E_{k} .

With respect to a pair of coverings f_{1} and f_{2} nodes and cusps of B are subdivided into four types,

n = n_{ss} + n_{sp} + n_{ps} + n_{pp} , c = c_{ss} + c_{sp} + c_{ps} + c_{pp},  (5.1)

where n_{♭♯} and c_{♭♯} are numbers of ♭♯-nodes and ♭♯-cusps of . In particular, n_{ss} + n_{sp} = a_{1} is the number of singularities of type A_{1}, and _{ss} +_{sp} = a_{2} is the number of singularities of type A_{2} on the surface X_{1}.

Consider a normalization X of the fiber product X^{×} = X_{1}×_{P^{2}}X_{2} and the corresponding commutative diagram

[figure: (x, y) ∈ ℂ^{2}; X = ℂ^{2} ∋ (z_{1}, z_{2}); (z_{1}, y) ∈ ℂ^{2} = X_{1}; X_{2} = ℂ^{2} ∋ (x, z_{2}) ,; g_{0}; f_{0}; f_{1}; f_{2}; g_{1}; X_{0}; g_{2}; (∗_{2})]

The surface X is a N_{1}N_{2}-sheeted covering of ℙ^{2} and it has at most A-D-E-singularities, which lie over Sing B. Lemma. If coverings f_{1} and f_{2} are non equivalent, then the surface X is irreducible.

Proof is word for word the same as in the case of generic coverings of non-singular surfaces ([K] Proposition 2).

We set

g_{1}^{−1}(R_{1}) = R + C,

where R is a part, which is mapped by g_{2} onto R_{2}, and C is a part, which is mapped g_{2} onto C_{2}. We are interested in the intersection number of R and C after a resolution of singularities of X in a neighbourhood of the curve R + C.

Consider a restriction R + C → R_{1} of the covering g_{1} over the curve R_{1}. As follows from 2.2.1 and 2.2.2, it is an étale covering of degree N_{2} over a generic point x_{1} ∈ R_{1}, where R → R_{1} is a 2-sheeted, and C → R_{1} is a (N_{2} − 2)-sheeted covering. The same picture is over a point x_{1} ∈ R_{1}, which is a s-point of X_{1}, lying over a ss-point of B.

Denote by π̃ : S → X a minimal resolution of singularities of X, and denote by R̃ and˜ the proper transforms of R and C on S. Our goal is to calculate the intersection numbers (R̃^{2}), (R̃ · C̃) and (C̃^{2}), and also the analogous intersection numbers for divisors π̃^{−1}(R) = R̃ + Z_{R} and π̃^{−1}() =˜ + Z, where Z_{R} and Z_{C} are the sums of canonical cycles corresponding to singular points x ∈ Sing X and lying on R and C respectively.

5.2. The structure of a fibre product over a neighbourhood of a singular point of the discriminant curve. Let U ⊂ ℙ^{2} be a sufficiently small neighbourhood (in complex topology) of a point b ∈ Sing B. The preimage f_{1}^{−1}(U) is a disjoint union of two parts, f_{1}^{−1}(U) = V_{1} ⊔ V′_{1}, where V_{1} is a part containing the ramification curve R_{1}, and V′_{1} is a part not containing R_{1} and étale mapped to U. Analogously f_{2}^{−1}(U) = V_{2} ⊔ V′_{2}. Then f^{−1}(U) is a disjoint union of four open sets – of normalizations of fibre products W = \overline{V_{1}×_{U}V_{2}}, W′ = \overline{V_{1}×_{U}V′_{2}}, \overline{V′_{1}×_{U}V_{2}} and \overline{V′_{1}×_{U}V′_{2}}. And only W and W′ meet the curve g_{1}^{−1}(R_{1}). The open sets W ⊂ X were studied in detail in §2. The surface X in the neighbourhood W is non-singular except the case of ss-points b. The open set W′ consists of N_{2} − k components (k = 2, 3 or 4 depending on the type of the singular point b), which are mapped isomorphically onto V_{1}. And W′ does not meet R, and W′ ∩ C consists of N_{2} − k components isomorphic to V_{1} ∩ R_{1}.

It follows from the investigation of the local structure of X in §2 that X and the curves R and C are of the following form over neighbourhoods of singular points b ∈ Sing B of different types.

1) Over a ss-point b the neighbourhood W has 2 , and W′ has N_{2} − 2 components, which are mapped isomorphically onto V_{1} by the map g_{1}. Correspondingly R ∩ W consists of two, and C ∩ W′ consists of (N_{2} − 2) components isomorphic to R_{1} ∩ V_{1}.

2) Over a sp-point b ∈ B of type A_{1} the neighbourhood W′ consists of (N_{2} − 4) components isomorphic to V_{1} and having a singular point of type A_{1}. Correspondingly C consists of N_{2} − 4 nodal curves. The neighbourhood W consists of two components: see Fig. 4, where

R = R^{′,′} + R^{′′,′} , and C = R^{′,′′} + R^{′′,′′}

(it ought to change places of the left and right parts of Fig. 4, g_{1} stands for g_{2}, and g_{2} – for g_{1} ). We see that in the neighbourhood W the curves R and C are non-singular and meet transversally in two points.

3) Over a ps-point b ∈ B of type A_{1} the neighbourhood V_{1} ⊂ X_{1} consists of two components and on each of them the map f_{1} has a fold. The neighbourhood W′ consists of disjoint union of (N_{2} − 2) pieces isomorphic to V_{1}. The neighbourhood W consists of two components: see Fig. 4, on which

R = R^{′,′} + R^{′′,′} , and C = ∅.

We see that on W the curve R is non-singular and does not meet C.

4) Over a pp-point b ∈ B of type A_{1} the neighbourhood V_{1} ⊂ X_{1} consists of two irreducible components and on each of them f_{1} has a fold. The neighbourhood W′ is non-singular and consists of N_{2} − 4 components isomorphic to V_{1}. The neighbourhood W is represented on Fig. 5, on which

R = R^{′,′} + R^{′,′′} + R^{′′,′} + R^{′′,′′}, C = C^{′,′} + C^{′′,′}.

We see that the curves R and C are non-singular and do not meet.

5) Over a sp-point b ∈ B of type A_{2} the neighbourhood V_{1} has a singular point of type A_{2}, and W′ consists of (N_{2} − 3) components isomorphic to V_{1}. The neighbourhood W is pictured on Fig. 7, on which

R = L_{2} + L_{3} , C = L_{1}.

We see that R has a double point, C is non-singular and intersect transversally each of the branches of R at the intersection point, and, consequently, (R · C) = 2.

6) Over a ps-point b ∈ B of type A_{2} the neighbourhood V_{1} is non-singular, and W′ consists of (N_{2} − 2) components isomorphic to V_{1}. The neighbourhood W is pictured on Fig. 7 (on which it ought to change places of the left and right parts, g_{1} stands for g_{2}, and g_{2} – for g_{3} ), where

R = L_{2} + L_{3} , C = ∅ .

We see that R has a double point and does not meet C.

7) Over a pp-point b ∈ B of type A_{2} the neighbourhood W′ consists of N_{2} − 3 components isomorphic to V_{1}. The neighbourhood W is pictured on Fig. 8, on which

R = R_{3} + L_{3} , C = L_{2} .

We see that R is non-singular and meets with C transversally at one point.

From the obtained local description it follows that the surface X is non-singular at the points of intersection of R and C, and the intersection is not void only over the points b ∈ B of types: over sp-points of type A_{1}, where (R·C) = 2, over sp-points of type A_{2}, where (R·C) = 2, and over pp-pointe of type A_{2}, where (R · C) = 1. Therefore,

(R̃ · C̃) = 2n_{sp} + 2c_{sp} + c_{pp}.  (5.3)

5.3. A lift of the fibre product to a resolution of the discriminant curve. To compute intersection numbers on S we consider firstly an auxiliary surface X̄, which is not a minimal resolution of X, and then we ‘descend’ to S. Let σ : ℙ̄^{2} → ℙ^{2} be a composition of σ-processes resolving the curve B and needed to obtain a minimal resolution of a double plane singularities, lying over B (see §3), and, besides, let σ includes two additional σ-processes as in 2.4.5 for each cusp, which is not a ss-cusp. Consider a lift of the diagram (∗_{1}) to ℙ̄^{2}, namely consider the diagram

\xymatrix{ &  & S \ar[dl]_{π̃} \\  & X \ar[dl]_{g_{1}} \ar@{-->}[ddr]^{g_{2}} &  & \overline{X} \ar[ll]_{π} \ar[ul]_{\overline{π}} \ar[dl]_{_{\overline{g}_{1}}} \ar[ddr]^{_{\overline{g}_{2}}} \\ X_{1} \ar[ddr]_{f_{1}} &  & _{\overline{X}_{1}} \ar[ll]^{π_{1}} \ar[ddr]^{_{\overline{f}_{1}}} \\  &  & X_{2} \ar@{-->}[dl]_{f_{2}} &  & _{\overline{X}_{2}} \ar@{-->}[ll]_{π_{2}} \ar[dl]^{_{\overline{f}_{2}}} \\  & ℙ^{2} &  & _{\overline{ℙ}_{2}} \ar[ll]_{σ}}  (5.4)

in which X̄_{i} and X̄ are normalizations of X_{i} ×_{ℙ^{2}} ℙ̄^{2} and X ×_{ℙ^{2}} ℙ̄^{2}. Then morphisms ‘on the right wall’ of diagram (5.4) are finite coverings. The surface X̄ is non-singular, and π̄ : X̄ → S blows down the ‘superfluous’ exceptional curves of the first kind. Let R̄_{1} be the proper transform of R_{1} on X̄_{1}, and R̄ and C̄ (respectively R̃ and C̃) be the proper transforms of R and C on X̄ (respectively on S). Then ḡ_{1}^{∗}(R̄_{1}) = R̄ + C̄, and R̄ → R̄_{1} and C̄ → R̄_{1} are finite coverings of degree 2 and N_{2} − 2 respectively, and R̄ and C̄ are disjoint. Therefore,

\binom{}{R̄^{2}} = 2 \binom{}{R̄_{1}^{2}} , \binom{}{C̄^{2}} = (N_{2} − 2) \binom{}{R̄_{1}^{2}} , R̄ · C̄ = 0.  (5.5)

Actually from 3) and 4) one can see that over ps- and pp-nodes b in a neighbourhood of R+C the surface X is non-singular, the curves R and C are non-singular and disjoint. Therefore, one can suppose that ps- and pp-nodes on B are not blown up (and on the surface S there remain singular points, which lie over these nodes).

5.4. Computation of intersection numbers. First we find (R̄_{1}^{2}). Recall that by (4.9) we have on the minimal resolution X̃_{1} of the surface X_{1}

(R̃_{1}^{2}) = 3d̄+ g − 1 − δ_{1},  (5.6)

where δ_{1} = δ_{X_{1}} = n_{s} + c_{s} + δ_{0}, and n_{s} = n_{ss} + n_{sp} and _{s} = _{ss} +_{sp} are the numbers of singular points of type A_{1} and A_{2} on the surface X_{1}.

Let π_{1} = π̃_{1} ∘ π̄_{1}, where π̃_{1} : X̃_{1} → X_{1} is a minimal resolution, and π̄_{1} : X̄_{1} → X̃_{1} is the blowing down of the ”superfluous” exceptional curves. The surface X̄_{1} differs from the surface X̃_{1} only over the cusps of B, which are not ss-cusps. Let Ū = σ^{−1}(U), and V̄_{1} = π_{1}^{−1}(V_{1}), Ṽ_{1} = π̃_{1}^{−1}(V_{1}) be neighbourhoods of X̄_{1} and X̃_{1} lying over Ū and containing the proper transform of R_{1}. Analogously V̄′_{1} and Ṽ′_{1}.

For a sp-cusp b ∈ B the blowing down V̄_{1} ⟶^{π̄_{1}} Ṽ_{1} is represented on Fig. 12. We see that π̄ includes one σ-process with a centre R_{1}. For ps- and pp-cusps b ∈ B the blowing down V̄_{1} ⟶^{π̄_{1}} Ṽ_{1} is represented on Fig. 13 (where R stands for R_{1}, and C stands for C_{1}). We see that it needs two σ-process with centres on R_{1} to disjoint C_{1} and R_{1}. Therefore,

(R̄_{1}^{2}) = (R̃_{1}^{2}) − c_{sp} − 2c_{ps} − 2c_{pp}.  (5.7)

Now we examine how the intersection numbers (R̄^{2}) and (C̄^{2}) change under the blowing down π̄. For a neighbourhood U ⊂ ℙ^{2} of a point b ∈ Sing B set W̄ = π^{−1}(W), W̄′ = π^{−1}(W′), W̃ = π̃^{−1}(W), W̃′ = π̃^{−1}(W′). Then ḡ_{1}^{−1}(V̄_{1}) = W̄ ⊔ W̄′. We examine one after another the blowing down π̄ : X̄ → S in neighbourhoods W̄ ⊔ W̄′ ⊂ X̄ separately for different types of singular points b ∈ Sing B (the numbering of cases corresponds to the numbering of cases in 5.2).

2) For a sp-point b of type A_{1} the neighbourhood W̄′ is a disjoint union of (N_{2} −4) open sets isomorphic to V̄_{1} – to the minimal resolution of singular points of type A_{1}. The neighbourhood W is represented on Fig. 4 , and π̄ : W̄ → W is a blowing up of two points R^{′,′} ∩ R^{′,′′} and R^{′′,′} ∩ R^{′′,′′}. Therefore, the blowing down π̄ : W̄ → W̃ ≃ W increases (R̄^{2}) and (C̄^{2}) on 2 for one point b and, consequently, on 2n_{sp} for all points of this type.

5) For a sp-point b of type A_{2} the neighbourhood W̄ is represented on the upper part of Fig. 10 . It is obtained from the neighbourhood W, pictured on Fig. 7 , by blowing up the point of intersection of lines L_{1}, L_{2} and L_{3}, and then by blowing up 5 points on the glued line E_{6,3} and not lying on the proper transform of these lines. The blowing down π̄ : W̄ → W̃ ≃ W is the converse procedure, i.e. the blowing down of five exceptional curves of the first kind, and then blowing down the curve E_{6,3}. In this case R = L_{2} + L_{3}, and C = L_{1}. Since (R^{2}) = (L_{2}^{2}) + 2(L_{2} · L_{3}) + (L_{3}^{2}) and (L_{2}^{2}), (L_{3}^{2}) are diminished on 1, and L_{2} and L_{3} are no longer intersected after the σ-process with the centre at the point L_{2} ∩ L_{3}, the blowing down π̄ increases (R̄^{2}) on 4 for one point b and on 4c_{sp} for all points of this type.

The neighbourhood W̄′ consists of (N_{2} − 3) components isomorphic to V̄_{1}, for each of which π̄ is represented on Fig. 12 . As above for (R̄_{1}^{2}), we see that the blowing down π̄ increases (C̄^{2}) on (N_{2} − 3) + 1 (taking account of the neibourhood W̄ ) for one point b and on (N_{2} − 2)c_{sp} for all points of this type.

6) For a ps-point b of type A_{2} the neighbourhood W̄ and the blowing down π̄ : W̄ → W̃ ≃ W are the same as in 5) , but in this case R = L_{2} + L_{3}, and C ∩ W = ∅. Therefore, as in 5) we obtain that the blowing down π̄ increases (R̄^{2}) on 4c_{ps}.

The neighbourhood W̄′ consists of (N_{2} − 2) components isomorphic to V̄_{1}, for each of which π̄ is represented on Fig. 13 . As above for (R̄_{1}^{2}), we see that the blowing down π̄ increases (C̄^{2}) on 2(N_{2} − 2) for one point b and on 2(N_{2} − 2)c_{ps} for all points of this type.

7) For a pp-point b of type A_{2} the neighbourhood W̄ consists of two components: one is the same as in 5) and the other is the same as V̄′_{1} and represented on the left side of Fig. 13 . Since in the neibourhood W, represented on Fig. 8, R = R_{3} + L_{3}, and C = L_{2}, we obtain that the blowing down π̄ : W̄ → W̃ ≃ W increases (R̄^{2}) on 1 + 2 = 3 for one point b and on 3c_{pp} for all points of this type. Besides, (C̄^{2}) is increased on c_{pp}.

The neighbourhood W̄′ consists of (N_{2}−3) components isomorphic to V̄′_{1}, and is represented on Fig. 13 (on which C stands for R ). Therefore, taking account of the neighbourhood W̄, the blowing down π̄ increases (C̄^{2}) on 2(N_{2} − 3)c_{pp} + c_{pp} = (2N_{2} − 5)c_{pp}.

Summing all modifications of (R̄^{2}) and (C̄^{2}), we obtain

(R̃^{2}) = (R̄^{2}) + 2n_{sp} + 4c_{sp} + 4c_{ps} + 3c_{pp},  (5.8)
(C̃^{2}) = (C̄^{2}) + 2n_{sp} + (N_{2} − 2)c_{sp} + 2(N_{2} − 2)c_{ps} + (2N_{2} − 5)c_{pp}.  (5.9)

Applying (5.5) and substituting (R̄_{1}^{2}) from (5.7), we obtain

(R̃^{2}) = 2((R̃_{1}^{2}) − c_{sp} − 2c_{ps} − 2c_{pp}) + 2n_{sp} + 4c_{sp} + 4c_{ps} + 3c_{pp} =
= 2(R̃_{1}^{2}) + 2n_{sp} + 2c_{sp} − c_{pp},  (5.10)

(C̃^{2}) = (N_{2} − 2)((R̃_{1}^{2}) − c_{sp} − 2c_{ps} − 2c_{pp}) + 2n_{sp} + (N_{2} − 2)c_{sp} + 2(N_{2} − 2)c_{ps} + (2N_{2} − 5)c_{pp} =

= (N_{2} − 2)(R̃_{1}^{2}) + 2n_{sp} − c_{pp}.  (5.11)

5.5. Computation of intersection numbers (continuation). Now we find (R̃+Z_{R})^{2}, (C̃+Z_{C})^{2} and (R̃ + Z_{R}) · (C̃ + Z_{C}), where the divisor Z_{R}, respectively Z_{C}, equals to ∑ Z_{x}, where Z_{x} is the canonical cycle of a point x ∈ Sing X, and the summation runs over x ∈ R, respectively x ∈ C. The analogous sums ∑ δ_{x} we denote by δ_{R} and δ_{C} respectively. By (4.2) we have

(R̃ · Z_{R}) = −(Z_{R}^{2}) = 2δ_{R} , (C̃ · Z_{C}) = −(Z_{C}^{2}) = 2δ_{C} .

Obviously,

(R̃ + Z_{R}) · (C̃ + Z_{C}) = R̃ · C̃ ,  (5.12)

and

(R̃ + Z_{R})^{2} = (R̃^{2}) + 2(R̃ · Z_{R}) + (Z_{R}^{2}) = (R̃^{2}) + 2δ_{R}.  (5.13)

Analogously, (R̃_{C} + Z_{C})^{2} = (C̃^{2}) + 2δ_{C}.

It remains to determine how many singular points x ∈ Sing X lie on R, respectively, on C. From 5.2 it follows that over each ss-point on R there lie 2, and on C there lie (N_{2} − 2) singular points. There are no other singular points on R. There are singular points on C of the following type: over a sp-point of type A_{1} there are (N_{2} − 4) singular points of type A_{1}, over a sp-point of type A_{2} there are (N_{2} − 3) singular points of type A_{2}. We obtain

δ_{R} = 2(δ_{0} + n_{ss} + c_{ss}) = 2(δ_{1} − n_{sp} − c_{sp}),  (5.14)
δ_{C} = (N_{2} − 2)(δ_{0} + n_{ss} + c_{ss}) + (N_{2} − 4)n_{sp} + (N_{2} − 3)c_{sp} =
= (N_{2} − 2)δ_{1} − 2n_{sp} − c_{sp}.

Substituting (R̃^{2}) from (5.10) and δ_{R} from (5.14) to (5.13), we obtain

(R̃ + Z_{R})^{2} = 2(R̃_{1}^{2}) + 2n_{sp} + 2c_{sp} − c_{pp} + 4(δ_{1} − n_{sp} − c_{sp}) =
= 2((R̃_{1}^{2}) + 2δ_{1}) − 2n_{sp} − 2c_{sp} − c_{pp}.

Analolously we find

(R̃_{C} + Z_{C})^{2} = (N_{2} − 2)(R̃_{1}^{2}) + 2n_{sp} − c_{pp} + 2(N_{2} − 2)δ_{1} − 4n_{sp} − 2c_{sp} =
= (N_{2} − 2)((R̃_{1}^{2}) + 2δ_{1}) − 2n_{sp} − 2c_{sp} − c_{pp}.

Set

2n_{sp} + 2c_{sp} + c_{pp} = ι_{1},  (5.15)

and let

g_{1} = p_{a}(R_{1}) = g + δ_{1}  (5.16)

be the arithmetic genus of the curve R_{1}. Since by (5.6) (R̃_{1}^{2})+2δ_{1} = 3d̄+g−1+δ_{1} = 3d̄+g_{1} −1, finally we obtain:

(R̃+Z_{R})^{2} = 2(3d̄+g_{1} −1)−ι_{1}, (C̃ +Z_{C})^{2} = (N_{2} −2)(3d̄+g_{1} −1)−ι_{1}, (R̃+Z_{R})·(C̃ +Z_{C}) = ι_{1}.

(5.17)

5.7. The self-intersection number of the divisor R̃ + Z_{R} is positive.

Lemma 5.1

(R̃ + Z_{R})^{2} > 0.  (5.18)

Proof. Recall that 2d̄= d = deg B, and δ_{1} = δ_{0} + n_{sp} + c_{sp}. Therefore,

\begin{aligned}(R̃ + Z_{R})^{2} & = & 2(3d̄+ g − 1 + δ_{1}) − 2n_{sp} − 2c_{sp} − c_{pp} = \\ & = & d + (2d + 2g − 2) + 2δ_{0} − c_{pp}. \end{aligned}  (5.19)

Now we apply the Hurwitz formula for a generic projection φ : B → ℙ^{1} of the curve B from a point P ∈ ℙ^{2} onto the line ℙ^{1}, more precisely for the covering φ̄ : B̄ → ℙ^{1}, where φ̄ = φ ∘ n, and n : B̄ → B is a normalization of the curve B. Obviously, the covering φ̄ is ramified at the following points. Firstly, φ̄ has a ramification of the second order at points b̄ ∈ B̄, which correspond to non-singular points b ∈ B, for which the line \overline{P b} is tangent to B. The number of such points is d̂= deg B̂, where B̂ is a curve dual to B. Secondly, φ̄ has a ramification of order m_{k} at points b̄, which correspond to the branches B_{k} of the curve B at the singular points b. Here m_{k} is the multiplicity (order) of the corresponding branch. Denote by

ν = ∑_{k}(m_{k} − 1),  (5.20)

where the summation runs over all branches of the curve (at singular points). The covering φ̄ is of degree d = deg B. By the Hurwitz formula we obtain

2g − 2 = −2d + d̂+ ν.  (5.21)

Remark 5.1 Actually we derived one of the Plücker formulae

d̂= 2d + (2g − 2) − ν

for a plane curve with singularities.

Obviously, the number ν for A-D-E-singularities is equal:

ν(A_{2k−1}) = ν(D_{2k+2}) = 0, ν(A_{2k}) = ν(D_{2k+3}) = ν(E_{7}) = 1, ν(E_{6}) = ν(E_{8}) = 2.

Therefore, for the curve B the number ν = ν(B) is equal:

ν = c + ν′, where ν′ = ∑_{k>1} a_{2k} + ∑ d_{2k+3} + 2e_{6} + e_{7} + 2e_{8} .  (5.22)

Returning to the proof of the inequality, we obtain from (5.19), (5.21) and (5.22)

(R̃ + Z_{R})^{2} = d + (d̂+ ν) + 2δ_{0} − c_{pp} = d + d̂+ 2δ_{0} + ν′ + (c − c_{pp}) > 0.  (5.23)

5.8. Conclusion of the main inequality. Applying the Hodge index theorem to divisors R̃ + Z_{R} and C̃ + Z_{C} on the surface S, we obtain

| \begin{matrix}2(3d̄+ g_{1} − 1) − ι_{1} & ι_{1} \\ ι_{1} & (N_{2} − 2)(3d̄+ g_{1} − 1) − ι_{1} \end{matrix} | ≤ 0.

Therefore,

2(N_{2} − 2)(3d̄+ g_{1} − 1)^{2} − N_{2}(3d̄+ g_{1} − 1)ι_{1} ≤ 0

or

N_{2}[2(3d̄+ g_{1} − 1) − ι_{1}] ≤ 4(3d̄+ g_{1} − 1) .  (5.24)

Thus, if there are two nonequivalent generic coverings f_{1} and f_{2}, then

N_{2} ≤ \frac{4(3d̄+ g_{1} − 1)}{2(3d̄+ g_{1} − 1) − ι_{1}} .  (5.25)

6 Proof of the Chisini conjecture for pluricanonical embeddings of surfaces of general type.

6.1. The numerical invariants in the case of a m-canonical embedding. Let S be a minimal model of a surface of general type with numerical invariants (K_{S}^{2}) = k and e(S) = e. Let X be a canonical model of the surface S, and π : S → X be the blowing down of (-2)-curves. Let f : X → ℙ^{2} be a generic m-canonical covering, i.e. a generic projection onto ℙ^{2} of X = φ_{m}(S), where φ_{m} is a m-canonical map, φ_{m} : S → ℙ^{p_{m}−1}, defined by the complete linear system |mK_{S}|, p_{m} = \frac{1}{2}m(m − 1)k + χ(S). As is well known [BPV] , by a theorem of Bombieri φ_{m}(S) ≃ X for m ≥ 5, and φ_{m} gives the blowing down π.

Let B ⊂ ℙ^{2} be the discriminant curve. We concerve the notations of §4. Then

L̄ = mK_{S}, K_{S} · Z = 0, R̄ = (3m + 1)K_{S} − Z.  (6.1)

By formulae (4.3), we obtain

N = m^{2}k, d = m(3m + 1)k.  (6.2)

By formulae (4.10), we find

3d̄+ p_{a}(R) − 1 = (3m + 1)^{2}k ,  (6.3)

and

p_{a}(R) − 1 = \frac{1}{2}(3m + 1)(3m + 2)k .  (6.4)

6.2. Invariants of a surface and of the discriminant curve define invariants of the covering. Now let S_{1} and S_{2} be two surfaces of general type with numerical invariants k and e. Let f_{i} : X_{i} → ℙ^{2}, i = 1, 2, be their m_{i}-canonical coverings having the same discriminant curve B ⊂ P^{2}. Then by the second formula of (6.2) it follows that m_{1} = m_{2} = m. Then also deg f_{1} = deg f_{2} = N. We show that the other numerical invariants of f_{1} and f_{2} are the same.

By formula (6.4) it follows that p_{a}(R_{1}) = p_{a}(R_{2}), and since p_{a}(R) = g + δ_{X}, we have δ_{X_{1}} = δ_{X_{2}}.

By formulae (4.14) and (4.11) it follows that the number of p-cusps c_{p} and the number of p-nodes n_{p} for both coverings are the same. Then n_{p} = n_{pp} + n_{ps} = n_{pp} + n_{sp}, c_{p} = c_{pp} + c_{ps} = c_{pp} + c_{sp} and, consequently, n_{ps} = n_{sp} and c_{ps} = c_{sp}.

6.3. The main inequality in the case of surfaces of general type. To prove that m-canonical projections f_{1} and f_{2} are equivalent, by (5.24) it is sufficient to show that an inequality

N (2(3d̄+ p_{a}(R) − 1) − ι) > 4(3d̄+ p_{a}(R) − 1)

holds (here R stands for R_{1}), or

(N − 2)(3d + 2p_{a}(R) − 2) − N · ι > 0,  (6.6)

where

ι = 2n_{sp} + 2c_{sp} + c_{pp} = 2n_{sp} + c_{sp} + c_{p}.  (6.7)

Let us obtain an estimate for the number ι. We can express c_{p} by formulae (4.14)

c_{p} = 3N + 2p_{a}(R) − 2 − e.  (6.8)

To estimate 2n_{sp} +c_{sp} we use the Hirzebruch-Miyaoka inequality ([BPV] ,p.215): if the minimal surface of general type S contains s disjoint (-2)-curves, then

s ≤ \frac{2}{9} (3e(S) − (K_{S}^{2})) .  (6.9)

Since we can take one (-2)-curve for each of the singular points of types A_{1} and A_{2} on X, we have

n_{s} + c_{s} ≤ \frac{2}{9} (3e − k) .  (6.10)

Remark 6.1 Instead of the Hirzebruch-Miyaoka inequality we can use the estimate 2n_{sp}+c_{sp} ≤ 2(h^{1,1} − 1) = 2(e − 2 + 4q − 2p_{g} − 1) and the inequalities p_{g} ≥ q, p_{g} ≤ \frac{1}{2}(K_{S}^{2}) + 2 (the Noether’s inequality).

By (6.7), (6.8) and (6.10), we obtain an estimate

\begin{aligned}ι & ≤ \frac{4}{9}(3e − k) + 3N + 2p_{a}(R) − 2 − e = \\ & = \frac{1}{3}e − \frac{4}{9}k + 3N + 2p_{a}(R) − 2. \end{aligned}

Applying the Noether’s inequality ([BPV] ,p.211),

e ≤ 5k + 36,  (6.11)

we obtain

ι ≤ \frac{11}{9}k + 12 + 3N + 2p_{a}(R) − 2.  (6.12)

Combining (6.12) and (6.6), we obtain a corollary.

Lemma 6.1 If the inequality

3N(d − N) − 6d − 4(p_{a}(R) − 1) − (\frac{11}{9}k + 12) N > 0  (6.13)

holds, then a generic m-canonical projection of a surface of general type S with given k and e is unique.

6.4. Proof of Theorem 0.3 . Express the inequality (6.13) in terms of m. Substitute N and d from (6.2) and p_{a}(R) − 1 from (6.4) to (6.13). We obtain

3m^{3}(2m + 1)k^{2} − 6m(3m + 1)k − 2(3m + 1)(3m + 2)k − (\frac{11}{9}k + 12)km^{2} > 0,

i.e.

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

Dividing by m^{2}, we obtain

3m(2m + 1)k − (\frac{11}{9}k + 12) − 4 (3 + \frac{1}{m})^{2} > 0,

or, dividing by k,

3m(2m + 1) > \frac{11}{9} + \frac{1}{k} (12 + 4(3 + \frac{1}{m})^{2}) .  (6.14)

The right side of inequality decreases, when k and m increase. This inequality holds for all k ∈ N, if it holds for k = 1. For k = 1 and m = 3 the right side equals \frac{11}{9} + 12 + 4 · (\frac{10}{3})^{2} = \frac{173}{3} < 9 · 7 = 63. Thus, the inequality (6.14), and, consequently, the inequality (6.6), holds for m ≥ 3 and for all k. This completes the proof of Theorem 0.3 .

We can mention in addition that for m = 2 the inequality (6.14) holds, if k > 2, and for m = 1 it holds, if k > 9.

References

[A] Arnol’d, V.I.: Indices of singular points of 1-forms on a manifold with a boundary, convolution of invariants of groups generated by reflections, and singular projections of smooth surfaces. – Usp. Math. Nauk., 34, No. 2 (1979), 3–38. (Engl. translation in Russ. Math. Surv., 34, No. 2 (1979), 1–42.)

[AGV] Arnol’d, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Sigularities of differentiable Maps, Vol. I, II. – Birkhäuser, (1985).

[BPV] Barth W., Peters C., Van de Ven A.: Compact complex surfaces.– Springer (1984).

[C] Catanese F.: On a Problem of Chisini. – Duke Math. J., 53, No. 1 (1986), 33-42.

[G-H] Griffiths, Ph., Harris, J.: Principles of algebraic geometry.– John Wiley & Sons, New York (1978).

[K] Kulikov, Vic.S. On a Chisini Conjecture. – Izvestiya: Mathematics, V.63, No.6 (1999).

[M] Moishezon B.: Complex Surfaces and Connected Sums of Complex Projective Planes. – LNM 603, Springer (1977).

[Z] Zariski,O.: Algebraic surfaces.– Berlin.Verlag von Julius Springer (1935) (Springer-Verlag (1971)).

V.S. Kulikov  Vic. S. Kulikov
Chair of Mathematics,  Department of algebra,
Moscow State University of Printing,  Steklov Mathematical Institute,

    E-mail: valentin@masha.ips.ras.ru  E-mail: victor@olya.ips.ras.ru