Whitney theorem for complex polynomial mappings

We describe the topology of a general polynomial mapping $F=(f, g):X\to\Bbb C^2$, where $X$ is a complex plane or a complex sphere.


Introduction
Polynomial mappings F : C n → C n are the most classical objects in the complex analysis, yet their topology has not been studied up till now. To the best knowledge of the authors complex algebraic families of polynomials on affine varieties have not been investigated so far. Here we describe an idea of such study. For a smooth affine variety X k ⊂ C p we consider the family Ω X (d 1 , . . . , d m ) of polynomial mappings F : X → C m of degree bounded by (d 1 , . . . , d m ). In particular we prove a suitable version of Thom transversality theorem for this family, which is useful at least if d i ≥ k. We prove, that under this assumption a generic member of Ω X (d 1 , . . . , d m ) is transversal to the Thom-Boardman strata in J k (X, C m ).
Let us recall that in [10] the second author proved that if X, Y are smooth affine manifolds and Φ : M × X → Y is an algebraic family of polynomial mappings, such that the generic element of this family is proper, then two generic members of this family are topologically equivalent. In particular if X ⊂ C p is of dimension n and m ≥ n then any two generic members of the family Ω X (d 1 , . . . , d m ) are topologically equivalent. For example, if X is a smooth surface, then the numbers c X (d 1 , d 2 ) and d X (d 1 , d 2 ) of cusps and double folds of a generic member of the family Ω X (d 1 , d 2 ) are well-defined.
Our aim is to determine effectively the topology of such generic mappings. We consider in this paper the simplest case, when n = m = 2 and X = C 2 or X is the complex sphere S = {(x, y, z) ∈ C 3 : x 2 + y 2 + z 2 = 1}. In those cases we determine the topology of the set C(F ) of critical points of F and the topology of its discriminant ∆(F ). In particular we show that a generic polynomial mapping F ∈ Ω X (d 1 , d 2 ) has only cusps, folds and double folds as singularities and we compute the number c X (d 1 , d 2 ) of cusps and number d X (d 1 , d 2 ) of double folds of such generic polynomial mapping. Our ideas work well also in higher dimensions. This paper is the first step in a study of the topology of generic polynomial mappings F : C n → C n .
The problem of counting the number of cusps of a generic perturbation of a real planeto-plane singularity was considered by Fukuda and Ishikawa in [3]. They proved that the number modulo 2 of cusps of a generic perturbation F of a finitely determined map-germ F 0 : (R 2 , 0) → (R 2 , 0) is a topological invariant of F 0 . More recently, in [11] Krzyżanowska and Szafraniec gave an algorithm to compute the number of cusps for sufficiently generic fixed real polynomial mapping of the real plane.
Algebraic formulas to count the number of cusps and nodes of a generic perturbation of a finitely determined holomorphic map-germ F 0 : (C 2 , 0) → (C 2 , 0), were given by Gaffney and Mond in [4,5] (see also [16]). In this case any two generic perturbations F of F 0 defined on a sufficiently small neighborhood of 0 are topologically equivalent, so the number of cusps and nodes of F is an invariant of the map-germ F 0 .
Let us note that in some cases our result allows also to use local methods to study global mappings. Indeed, in the special case when gcd(d 1 , d 2 ) = 1 the numbers c(F ) and d(F ) can be computed by using local methods of Gaffney and Mond [5] and J. J. Nuno-Ballesteros, B. Orefice-Okamoto, J. N. Tomazella [14], or Ohmoto methods [15] based on Thom polynomials. Note that in this case the leading homogenous part F h of a generic mapping F = (f, g) is finitely determined. Moreover, we have a deformation F t (x) = (t d 1 f (t −1 (x)), t d 2 g(t −1 (x))). Now we can use the fact (which is first proved in our paper) that a generic (with respect to the Zariski topology) mapping F ∈ Ω X (d 1 , d 2 ) has only folds, cusps and double folds as singularities. Thus for the deformation F t ∈ Ω X (d 1 , d 2 ) of F all F t , t = 0 are generic mappings and all cusps and nodes of F t tend to 0 when t → 0. In this case our formulas for c(F ) and d(F ) coincide with formulas of Gaffney-Mond etc.
However, in the general case these approaches do not work since any homogeneous mapping is not finitely determined if gcd(d 1 , d 2 ) = 1 (Gaffney-Mond, [5]). In particular in that case the local number of nodes can not be defined and the methods of Gaffney-Mond, Nuno-Ballesteros-Orefice-Okamoto-Tomazella and Ohmoto do not work. If gcd(d 1 , d 2 ) = 1 our formulas do not coincide with formulas of Gaffney-Mond, Nuno-Ballesteros-Orefice-Okamoto-Tomazella and Ohmoto. Hence in general even discrete global invariants can not be obtained by local methods or methods based on Thom polynomials. Now we will briefly describe the content of the paper. In Section 2 we state and prove general theorems. In Section 3 we describe the topology of the set of critical points of a generic mapping F ∈ Ω C 2 (d 1 , d 2 ). Moreover we compute the number c C 2 (d 1 , d 2 ) of cusps. In Section 4 we describe the topology of the discriminant ∆(F ) and compute the number d C 2 (d 1 , d 2 ) of nodes of ∆(F ). In Section 5 we describe the topology of the set of critical points of a generic mapping F ∈ Ω S (d 1 , d 2 ), and compute the number c S (d 1 , d 2 ), where S ⊂ C 3 is a complex sphere. In section 6 we describe the topology of the discriminant ∆(F ) and we compute the number d S (d 1 , d 2 ).
We conclude the paper with Section 7 where we introduce the notions of a generalized cusp and the index of a generalized cusp µ (see Definitions 7.1 and 7.3). We show that if F = (f, g) : X → C 2 is an arbitrary polynomial mapping with deg f ≤ d 1 , deg g ≤ d 2 and generalized cusps at points a 1 , . . . , a r then r i=1 µ a i ≤ c X (d 1 , d 2 ).

General polynomial mappings
Let Ω n (d 1 , . . . , d m ) denote the space of polynomial mappings F : C n → C m of multidegree bounded by d 1 , . . . , d m . Similarly if X ⊂ C p is a smooth affine variety we consider a family Ω X (d 1 , . . . , d m ) of polynomial mappings F : X → C m of multi-degree bounded by d 1 , . . . , d m (in the sense that F ∈ Ω X (d 1 , . . . , d m ) if there existsF ∈ Ω p (d 1 , . . . , d m ) such that F =F | X , note that for a generic F ∈ Ω p (d 1 , . . . , d m ) the multi-degrees of F and F | X coincide). Of course Ω X (d 1 , . . . , d m ) has the structure of the affine space.
By J q (C n , C m ) we denote the space of q-jets of polynomial mappings F = (f 1 , . . . , f n ) : C n → C m . We define it exactly as in [8]. However, if we fix coordinates in the domain and the target then we can identify J q (C n , C m ) with the space C n × C m × (C Nq ) m , where C Nq parameterizes coefficients of polynomials of n-variables and of degree bounded by q with zero constant term (which correspond to suitable Taylor polynomials). In further applications, in most cases, we treat the space J q (C n , C m ) in this simple way. In particular for a given polynomial mapping F : C n → C m we can define the mapping j q (F ) as If X n ⊂ C p is a smooth affine variety, then the space J q (X, C m ) has the structure of a smooth algebraic manifold and can be locally represented in the same simple way as above. Indeed, locally X is a complete intersection, i.e. for every point x ∈ X there is an open neighborhood U x of x such that U x = {g 1 = 0, . . . , g p−n = 0} (in some open set of C p ) and rank[ ∂g i ∂x j ] = p − n on U x . We can assume that the mapping (x 1 , . . . , x n , g 1 , . . . , g p−n ) is biholomorphic near x. In particular we have where C Nq parameterizes coefficients of polynomials of n-variables and of degree bounded by q with zero constant term (which correspond to suitable Taylor polynomials). In local coordinates we have a mapping We start with the following fact: Theorem 2.1. Let X n ⊂ C p be a smooth affine variety of dimension n. Let S 1 , . . . , S k be smooth algebraic submanifolds of J q (X, C m ). Let d 1 , . . . , d m be integers such that d i ≥ q for i = 1, . . . , m. Then there is a Zariski open dense subset V (S 1 , . . . , S k ) ⊂ Ω X (d 1 , . . . , d m ) such that for every F ∈ V (S 1 , . . . , S k ) we have Proof. First consider the case X = C n , p = n. For simplicity we can take m = 1 (the general case is analogous). Consider the mapping . Let a (0,...,−1,0,...,0) := x i (here −1 is on i th -position) be the coordinates in C n and a α for |α| ≥ 0 be the coordinates in We compute the matrix ∂Ψα ∂aα −1≤|α|≤q . It is easy to see that Hence Ψ is a submersion. Now assume that X is a general affine smooth variety. As above we can cover X by finite number of Zariski open subsets U i which have global local coordinates x i 1 , . . . , x in .
Let Ω n (d 1 , . . . , d m )(x i 1 , . . . , x in ) ⊂ Ω X (d 1 , . . . , d m ) denote the set of polynomial mappings, which depend only on variables x i 1 , . . . , x in . Note that we have where mappings in W have parameters independent from parameters in Ω n (d 1 , . . . , d m ) (x i 1 , . . . , x in ). If we restrict our attention to the set U i we have where W i denotes the set of holomorphic mappings of variables x i 1 , . . . , x in , which are defined on U i and which have parameters independent from parameters in Ω n (d 1 , . . . , d m ) (x i 1 , . . . , x in ). Now we can prove as above that Ψ : By the transversality theorem with a parameter the set of polynomials On the other hand this set is constructible in Ω X (d 1 , . . . , d m ).

We conclude that there is a Zariski open dense subset
By Theorem 2.1 the subset of one-generic mappings contains a Zariski open dense subset of Ω X (d 1 , . . . , d n ). We have he following result: Theorem 2.3. Let X be a smooth Stein manifold of dimension n. Let F : X → C n be a proper holomorphic one-generic mapping. Let C(F ) denote the set of critical points of F . Then there is an open and dense subset U ⊂ C(F ) such that for every a ∈ U the germ F a : (X a , a) → (C n , F (a)) is holomorphically equivalent to a fold.
Proof. Let ∆ = F (C(F )) be the discriminant of F . Take U = C(F ) \ F −1 (Sing(∆)). The set U is a Zariski open dense subset of C(F ). Take a point a ∈ U and consider the germ F a : (X a , a) → (C n , F (a)). By the choice of the point a the germ of the discriminant of F a is smooth. Hence by [9], Corollary 1.11, the germ F a is biholomorphically equivalent to a k-fold: has to be a submersion at 0. This is possible only for k = 2.
In the sequel we use the Thom-Boardman singularities (see [1], [12]) which give a stratification in the jest space J k (X, C m ). The strata are smooth and locally Zariski closed ( [1], [12]). In fact we will use here mainly singularities of type S i , S i,j (see [1], [12] where they are denoted as Σ i , Σ i,j ).
We have the following general result: Theorem 2.4. Let X k ⊂ C n be a smooth algebraic variety. Assume that d i ≥ k, i = 1, . . . , m. Then there is a Zariski open subset U ⊂ Ω X (d 1 , . . . , d m ) such that for every F ∈ U the mapping F is transversal to the Thom-Boardman strata in J k (X, C m ).
Proof. Note that J k (X, C m ) is an algebraic variety and Thom-Boardman strata are smooth algebraic subvarieties. Now we can apply Theorem 2.1.
Remark 2.5. If there is an index i such that d i < k, then the mapping Ψ is not a submersion. However we can omit this problem as follows: Consider the mapping Ψ : then we can proceed as above. We show that it is indeed the case at least when X = a plane or X = a sphere. However we believe that it is a general principle. We go back to this problem in the future.

Plane mappings
Here we will study the set Ω 2 (d 1 , d 2 ). Let us denote coordinates in J 1 (C 2 , C 2 ) by (x, y, f, g, f x , f y , g x , g y ).
For a mapping F = (f, g) ∈ Ω 2 (d 1 , d 2 ), we have which justifies our notation. The set S 1 is given by the equation φ(x, y, f, g, f x , f y , g x , g y ) = f x g y −f y g x = 0. Since S 1 describes elements of rank one it is easy to see that it is a smooth (non-closed) subvariety of J 1 (C 2 , C 2 ). Now we would like to describe the set S 1,1 effectively. We restrict our attention only to sufficiently general jets. In the space J 2 (C 2 , C 2 ) we introduce coordinates (x, y, f, g, f x , f y , g x , g y , f xx , f yy , f xy , g xx , g yy , g xy ).
A generic mapping F satisfies rank d a F ≥ 1 for every a (because codim S 2 = 4). We can assume that F = (f, g) and ∇ a f = 0. The critical set of F is exactly the set S 1 (F ) and it has a reduced equation ∂f ∂x (x, y) ∂g ∂y (x, y) − ∂f ∂y (x, y) ∂g ∂x (x, y) = 0, which by simplicity we write as f x g y − f y g x = 0. In particular the tangent line to S 1 (f ) is given as Consequently the condition for [F a ] ∈ S 1,1 is: Let us note that the last equation contains terms g xx f 2 y and g yy f 2 x hence for ∇f = 0 these two equations form a complete intersection. In general, if we omit the assumption ∇f = 0 the set S 1,1 is given in J 2 (C 2 , C 2 ) by three equations: As above by symmetry the set S 1,1 is smooth and locally is given as a complete intersection of either L 1 , L 2 or L 1 , L 3 .
Remark 3.1. These formulas give a description of S 1,1 also in the case of a general affine surface X, however, it might be only locally in the Zariski topology of J 2 (X, C 2 ). Definition 3.2. Let F : (C 2 , a) → (C 2 , F (a)) be a holomorphic mapping. We say, that F has a fold at a if F is biholomorphically equivalent to the mapping (C 2 , 0) ∋ (x, y) → (x, y 2 ) ∈ (C 2 , 0). Moreover, we say that F has a simple cusp at a if F is biholomorphically equivalent to the mapping (C 2 , 0) ∋ (x, y) → (x, y 3 + xy) ∈ (C 2 , 0).
A direct consequence of Theorem 2.4 is: [17]) Let X ⊂ C n be a smooth algebraic surface. Assume that d 1 , d 2 > 1. Then there is a Zariski open subset U ⊂ Ω X (d 1 , d 2 ) such that for every F ∈ U the mapping F has only folds and simple cusps as singularities.
Now we compute the number of cusps of a generic polynomial mapping F ∈ Ω 2 (d 1 , d 2 ). To do this we need a series of lemmas: (1) ∂f ∂x = 0 ⋔ ∂f ∂y = 0 , Proof. The case d 1 = 1 is trivial so assume d 1 > 1. Let us note that the set S ⊂ J 1 (C 2 , C 2 ) given by {f x = f y = 0} is smooth. Hence (1) follows from Theorem 2.1. To prove (2) it is enough to assume that f ∈ H d , where H d denotes the set of homogenous polynomials of two variables of degree d. Let Ψ : , ∂f ∂y (x, y)) ∈ C 2 is transversal to the point (0, 0). In particular Ψ −1 f (0, 0) is either zero-dimensional or the empty set. Since f is a homogenous polynomial the first possibility is excluded. This means that ∂f Lemma 3.5. Let L ∞ denote the line at infinity of C 2 . There is a non-empty open subset V ⊂ Ω 2 (d 1 , d 2 ) such that for all F = (f, g) ∈ V :  (1) it is sufficient to show that X has dimension strictly smaller than the dimension of H d 1 ,d 2 .
Note that all fibers of the projection X → P 1 are isomorphic to X 0 . Thus dim(X) = dim(X 0 ) + dim(P 1 ) and to prove (1) it is sufficient to show that X 0 has codimension at least 2 in Y .
Let p = (q, F ) ∈ Y and let a i and b i be the parameters in H d 1 ,d 2 giving respectively the coefficients of f at x d 1 −i y i and of g at To conclude the proof of (1) we will show that the codimension of {a 0 b 0 = 0} ∩ X 0 in Y is at least 2 and ∇J and ∇J 1,1 are linearly independent outside {a 0 b 0 = 0} ∩ X 0 and thus the variety X 0 has codimension 2 in Y .
has codimension at least 2 and we may assume in further calculations that a 0 (F ) = 0 and similarly b 0 (F ) = 0.
Proof. Let us consider two subsets in J 1 (C 2 , C 2 ): such that for every F ∈ V 1 the mapping j 1 (F ) is transversal to R 1 and R 2 . Since these subsets have codimension three, we see that the image of j 1 (F ) is disjoint with R 1 and R 2 .
Proof. Let us consider the (non-closed) subvariety S ⊂ J 2 (2) given by equations: It is easy to check that S is a smooth complete intersection and it has codimension three. The set of generic mappings F which are transversal to S contains a Zariski open dense subset Proof. There is a Zariski open subset V 3 which contains only generic mappings which satisfy hypotheses of all lemmas above. We can also assume that the curves ∂f ∂x = 0 and ∂f ∂y = 0 intersect transversally. We have to show that the curves J(f, g) and J 1,1 (f, g) intersect transversally at every point a ∈ J(f, g) ∩ J 1,1 (f, g). If ∇f = 0 then it follows from transversality of the mapping F to the set S 1,1 . Hence we can assume ∂f ∂x (a) = 0 and ∂f ∂y (a) = 0 . By Lemma 3.6 we have ∂g ∂x (a) = 0 and ∂g ∂y (a) = 0. Let us denote: Now we are in a position to prove: such that for every mapping F ∈ U the mapping F has only two-folds and cusps as singularities and the number of cusps is equal to then the set C(F ) of critical points of F is a smooth connected curve, which is topologically equivalent to a sphere with g = (d 1 +d 2 −3)(d 1 +d 2 −4) 2 handles and d 1 + d 2 − 2 points removed.
Proof. If d 1 = d 2 = 1 then the theorem is obvious. Hence we can assume that d 1 > 1. Assume first that also d 2 > 1. Note that every point a of the intersection of curves J(f, g) and J 1,1 (f, g) with ∇ a f = 0 is a cusp. Moreover for a generic mapping F points with ∇ a f = 0 are not cusps (Lemma 3.8). By Bezout Theorem we have that in J(f, g)∩J 1,1 (f, g) there are exactly (d 1 − 1) 2 points with ∇f = 0 and that the number of cusps of a generic mapping is equal to If d 2 = 1 then we can replace the space J 2 (C 2 , C 2 ) by its subspace Y given by equations g xx = g xy = g yy = 0. Note that varieties S 1 , S 1,1 are transversal to Y. Moreover the mapping Ψ : Ω 2 (d, 1) × C 2 → Y is a submersion and we can proceed as above. We leave the details to the reader.
Finally by Lemma 3.5 we have that C(F ) = S 1 (F ) is a smooth affine curve which is transversal to the line at infinity. This means that C(F ) is also smooth at infinity, hence it is a smooth projective curve of degree d = d 1 + d 2 − 2. Thus by the Riemmann-Roch Theorem the curve C(F ) has genus g = (d−1)(d−2) 2 . This means in particular that C(F ) is homeomorphic to a sphere with g = (d−1)(d−2) 2 handles. Moreover, by the Bezout Theorem it has precisely d points at infinity.

The discriminant
Here we analyze the discriminant of a generic mapping from Ω(d 1 , d 2 ). Let us recall that the discriminant of the mapping F : . Moreover if the fiber over F is finite then F |C(F ) is injective outside a finite set given by the fiber.
Note that all fibers of the projection X → C 2 × C 2 are isomorphic to X 0 . Thus dim(X) = dim(X 0 ) + 4 and to prove (1) it is sufficient to show that X 0 has codimension at least 4 in Y .
Let (p, q, F ) ∈ Y and let a ij and b ij be the parameters in Ω 2 (d 1 , d 2 ) giving respectively the coefficients of f and g at (k−j)! a ij (F ) and similarly for g and b ij . The condition F (p) = F (q) yields the equations ∂(a 01 ,b 01 ,a 10 ,a 11 ) is triangular and its determinant is equal to b 01 (F ) d 2 j=1 b 0j (F ). Calculating similar derivations with a 10 replaced by b 10 or a 11 replaced by b 11 we obtain that ∇w 1 , . . . , . Thus X 0 \ S has codimension 4 in Y and it is easy to see that X 0 ∩ S has also codimension at least 4.
To prove (2) consider the set Similarly as in (1) we compute that X has codimension at least 7. It follows that the projection of X on Ω * 2 (d 1 , d 2 ) has empty fibers on some open subset U ⊂ Ω * 2 (d 1 , d 2 ). Note that unlike in (1) there are two types of fibers of the projection onto C 6 : In both cases the computation is purely technical and similar to the computation in (1), so we leave the details to the reader.
First we compute the degree of the discriminant: We have the following method of computing the delta invariant (see [13], p. 92-93): Theorem 4.4. Let V 0 ⊂ C 2 be an irreducible germ of an analytic curve with the Puiseux parametrization of the form Let D j = gcd(a 0 , a 1 , . . . , a j−1 ). Then If V = r i=1 V i has r branches then where V · W denotes the intersection product.
The main result of this section will be based on the following: Proof. Letf (x, y, z) = z d 1 f x z , y z andg(x, y, z) = z d 2 g x z , y z be the homogenizations of f and g and let f (x, z) =f (x, 1, z) and g(x, z) =g(x, 1, z). For a generic mapping the curves C(F ) and {f = 0} have no common points at infinity (see Lemma 4.6). Moreover we may assume that (1 : 0 : 0) / ∈ C(F ). Thus F extends to a neighborhood of C(F ) ∩ L ∞ on which it is given by the formula .
Consider the set Note that if f (P ) = 0 or c = 0 then the fiber over P of the projection from X to Ω 2 (d 1 , d 2 ) is non-empty. Hence it suffices to prove that X has codimension at least 2.
Let p = (0 : 1 : 0), and q = (a : b : 0) ∈ L * ∞ . Let T (x, y, z) = (bx − ay, y, z) so that Let a i be the parameters in Ω 2 (d 1 , d 2 ) giving the coefficients off (and of f ) at x d 1 −i y i and let b i and c i describe respectively the coefficients ofg at x d 2 −i y i and x d 2 −i−1 y i z.
Finally note that if d 2 c = d 1 d then Hence we consider the set Similarly as above one can show that it has codimension 2, which concludes the proof.
Let C p be the branch of C(F ) at P . We find the Puiseux expansion of the branch F (C P ) of ∆(F ) at F (P ). We have If d 1 = d 2 then by Lemma 4.6 we have d − c = 0 and F (C P ) is smooth at F (P ).
So assume d 1 > d 2 . Since the function h(t) = f (P ) To proceed further we also need: Similarly as in Lemma 4.6 we will prove that X has codimension 3, so there is a dense open subset S ⊂ Ω(d 1 , d 2 ) such that the projection from X has empty fibers over F ∈ S.
Indeed, take p = (1 : 0 : 0), q = (0 : 1 : 0) and Y := {(p, q)} × Ω 2 (d 1 , d 2 ), it suffices to show that X 0 = X ∩ Y has codimension 3 in Y . Let a i and b i be the parameters in Ω 2 (d 1 , d 2 ) giving respectively the coefficients off at x d 1 −i y i and ofg at x d 2 −i y i .

The three equations describing
By the dynamical definition of intersection there exists a neighborhood U of 0, such that for small generic a, b we have This means that V i · V j is equal to the number of solutions of the following system: where a, b and S, T are sufficiently small. Take Thus we have V i · V j = mult 0 Q. Note that by Lemma 4.7 the minimal homogenous polynomials of the two components of Q have no nontrivial common zeroes, hence We can now prove the following: Proof Substituting z∈(∆\∆) Thus by Theorem 3.9 we get: . In particular branches V i are smooth if and only if d 1 = d 2 or d 1 = d 2 + 1.

The complex sphere
In the next two sections we show that our method can be easily generalized to the case when X is a complex sphere. Let φ = y 2 + 2xz and let S be a complex sphere: S = {(x, y, z) : φ = 1} (of course S is linearly equivalent with a standard sphere S ′ := {(x, y, z) : x 2 + y 2 + z 2 = 1}). Here we will study the set Ω S (d 1 , d 2 ). First we compute the critical set C(F ) of a generic mapping F = (f, g) ∈ Ω S (d 1 , d 2 ). Note that x ∈ C(F ) if rank (∇φ, ∇f, ∇g) < 3, hence C(F ) is the intersection of S and the surface given by In particular we have: Corollary 5.1. For a generic mapping F ∈ Ω S (d 1 , d 2 ) we have deg C(F ) = 2(d 1 +d 2 −1). Now we describe cusps of a generic mapping F : S → C 2 . Note that a tangent line to C(F ) is given by two equations: The mapping F has a cusp in a point (x, y, z) if (1) (x, y, z) ∈ C(F ) (2) the line given by the kernel of d (x,y,z) F is tangent to C(F ).
First let us determine the kernel of d (x,y,z) F . If rank z y x f x f y f z = 2, then the kernel is given by a vector Otherwise it is a vector v(g) = ( y x g y g z , − z x g x g z , z y g x g y ). Let . Let C denote the set of cusps of F , for generic F we have from the construction: (and similarly open sets U x , U y ). In U z we have globally defined local coordinates x, y. Now the proof reduces to Lemma 3.7.
So to prove (1) it is sufficient to show that X has dimension strictly smaller than the dimension of H d 1 ,d 2 .
Let q = (1 : 0 : 0) ∈ P 2 , Y := {q} × H d 1 ,d 2 and X 0 = X ∩ Y . Note that all fibers of the projection X → Γ are isomorphic to X 0 , because the group GL(S) of linear transformations of S acts transitively on the conic at infinity of S. Thus dim(X) = dim(X 0 )+ dim(Γ) and to prove (1) it is sufficient to show that X 0 has codimension at least 2 in Y .
Let p = (q, F ) ∈ Y and let a i,j and b i,j be the parameters in H d 1 ,d 2 giving respectively the coefficients of f at x d 1 −i−j y i z j and of g at To conclude the proof of (1) we will show that the codimension of {a 1,0 b 1,0 = 0} ∩ X 0 in Y is at least 2 and ∇J and ∇J 1,1 are linearly independent outside {a 1,0 b 1,0 = 0} ∩ X 0 and thus the variety X 0 has codimension 2 in Y .

Lemma 5.3. There is a non-empty open subset
Proof. As in Lemma 5.2 (1) we consider the sets U x , U y , U z with globally defined local coordinates and reduce the proof to Lemmas 3.6 and Lemma 3.8.  Proof. We proceed similarly as in Lemma 5.2 (2).
Let Γ := {(x, y, z) ∈ P 2 : φ(x, y, z) = 0} ∼ = P 1 . Consider the set If {φ = 0} ∩ {v(f ) = 0} = ∅ then f belongs to the image of the projection of X on H d 1 . So to prove (1) it is sufficient to show that X has dimension strictly smaller than the dimension of H d 1 .
Let q = (1 : 0 : 0) ∈ P 2 , Y := {q} × H d 1 and X 0 = X ∩ Y . As before, all fibers of the projection X → Γ are isomorphic to X 0 , so dim(X) = dim(X 0 ) + dim(Γ) and it is sufficient to show that X 0 has codimension at least 2 in Y .
Lemma 5.5. There is a non-empty open subset V 2 ⊂ Ω S (d 1 , d 2 ) such that for all (f, g) ∈ V 2 the equations: (1) y 2 + 2xz = 1, Note that generically the curve and {x = f z = 0}. Thus by the Bezout Theorem deg{v(f ) = 0} = d 2 1 − d 1 + 1 and S ∩ {v(f ) = 0} has 2(d 2 1 − d 1 + 1) points. We leave checking that the intersections are transversal and there are no components at infinity to the reader. Now we are in a position to prove: Theorem 5.6. There is a Zariski open, dense subset U ⊂ Ω S (d 1 , d 2 ) such that for every mapping F ∈ U the mapping F has only folds and cusps as singularities and the number of cusps is equal to 2(d 2 1 + d 2 2 + 3d 1 d 2 − 3d 1 − 3d 2 + 1). Moreover the set C(F ) of critical points of F is a smooth connected curve, which is topologically equivalent to a sphere with g = (d 1 + d 2 − 2) 2 handles and 2(d 1 + d 2 − 1) points removed.
Proof. First assume that d 1 , d 2 ≥ 2. Note that every point a of the intersection of curves J(f, g) and J 1,1 (f, g) with v(f ) = 0 is a cusp. Moreover for a generic mapping F points with v(f ) = 0 are not cusps (Lemma 5.2). By Lemma 5.5 we have that in the set S ∩ {v(f ) = 0} there are exactly 2(d 2 1 − d 1 + 1) points and that the number of cusps of a generic mapping is equal to Moreover by Lemma 5.2 we have that C(F ) = S 1 (F ) is a smooth affine curve which is transversal to the plane at infinity. This means that J := C(F ) is also smooth at infinity, hence it is a smooth projective curve of degree 2(d 1 +d 2 −1). Note that Pic(S) = ZL 1 ⊕ZL 2 , where L 1 , L 2 are suitable lines in S. Moreover if H is a plane section then H ∼ L 1 + L 2 . Hence in Pic(S) we have C(F ) ∼ aL 1 + bL 2 where a + b = 2(d 1 + d 2 − 1). Take l i = L i ∩ S and note that Pic(S) is generated freely by l 1 or l 2 with the relation Consequently Γ 1 ∩ Γ 2 = ∅ and C(F ) is not smooth -a contradiction. This implies that C(F ) is connected.
Moreover, using the same equations it is easy to check that S 1,1 is transversal to Y. Indeed we use the equations (1), (2) (3) for Y and standard equation for S 1,1 (see section 3). We differentiate equation for Y with respect to x, y and equation for S 1,1 with respect to f x , f y , f xx , f yy , g x , g y , g xx , g yy . I particular we get that S 1,1 ⋔ Ψ(Ω S (d 1 , 1) × U z ) for z = ±1. Similarly S 1,1 ⋔ Ψ(Ω S (d 1 , 1) × U x ) for x = ±1 and S 1,1 ⋔ Ψ(Ω S (d 1 , 1) × U y ) for y = ±1. Thus S 1,1 ⋔ Y. In the same way we can check that other conditions of our proof are satisfied.

The complex sphere: the discriminant
Here we analyze the discriminant of a generic mapping from Ω S (d 1 , d 2 ). Let us recall that the discriminant of the mapping F : S → C 2 is the curve ∆(F ) := F (C(F )), where C(F ) is the critical curve of F. We have: There is a non-empty open subset U ⊂ Ω S (d 1 , d 2 ) such that for every mapping F ∈ U : if |F −1 (p) ∩ C(F )| = 2 then the curve ∆(F ) has a normal crossing at p.
Proof. For this proof we will assume that S = {(x, y, z) : x 2 + y 2 + z 2 = 1}. We may assume that d 1 ≥ d 2 and since the assertion is trivial for d 1 = d 2 = 1 we may also assume that d 1 > 1.
To prove (1) consider the set X = {(p, q, F ) ∈ S × S × Ω S (d 1 , d 2 ) : p = q, F (p) = F (q), J(F )(p) = J(F )(q) = 0}. We will show that X has dimension not greater than dim Ω S (d 1 , d 2 ). So the projection of X on Ω S (d 1 , d 2 ) has finite fibers on some open subset U ⊂ Ω S (d 1 , d 2 ). Moreover if the fiber over F is finite then F |C(F ) is injective outside a finite set given by the fiber.
Let p = (0, 0, 1), q = (0, α, β) ∈ S \ p, Y q := {p} × {q} × Ω S (d 1 , d 2 ) and X q = X ∩ Y q . Note that for every pair (p ′ , q ′ ) ∈ S 2 there is a rotation T such that T (p ′ ) = p and T (q ′ ) = q. Moreover (p, q, F ) → (p ′ , q ′ , F • T ) is an isomorphism of the fiber over (p ′ , q ′ ) of the projection X → S × S with X q . Thus to prove (1) it is sufficient to show that every X q has codimension at least 4 in Y q .
To prove (2) consider the set X = {(p, q, r, F ) ∈ S × S × S × Ω S (d 1 , d 2 ) : p = q = r = p, F (p) = F (q) = F (r), J(F )(p) = J(F )(q) = J(F )(r) = 0}. Similarly as in (1) we compute that X has codimension at least 7. It follows that the projection of X on Ω 2 (d 1 , d 2 ) has empty fibers on some open subset U ⊂ Ω 2 (d 1 , d 2 ). The computation is purely technical and similar to the computation in (1), so we leave the details to the reader.
Consider the set Note that if f (P ) = 0 or c = 0 then the fiber over P of the projection from X to Ω S (d 1 , d 2 ) is non-empty. Hence it suffices to prove that X has codimension at least 2.
Let a i,j,k be the parameters in Ω S (d 1 , d 2 ) giving the coefficients off at x i y j z d 1 −i−j−k w k (i.e. of f at x i y j z d 1 −i−j−k ) and let b i,j,k describe the analogous coefficients ofg.
Let C p be the branch of C(F ) at P . Exactly as in the section 4 we have 2δ( To proceed further we also need: Similarly as in Lemma 6.4 we will prove that X has codimension 3, so there is a dense open subset U ⊂ Ω S (d 1 , d 2 ) such that the projection from X has empty fibers over F ∈ U .
We can now prove the following: Theorem 6.6. There is a Zariski open, dense subset U ⊂ Ω S (d 1 , d 2 ) such that for every mapping F ∈ U the discriminant ∆(F ) = F (C(F )) has only cusps and nodes as singularities. Then the number of cusps is equal to c(F ) = 2(d 2 1 + d 2 2 + 3d 1 d 2 − 3d 1 − 3d 2 + 1) and the number of nodes is equal to Remark 6.7. If d 1 = d 2 = d then the discriminant has 4d − 2 smooth points at infinity and in each of these points it is tangent to the line L ∞ (at infinity) with multiplicity d. If d 1 > d 2 , then the discriminant has only one point at infinity with 2(d 1 + d 2 − 1) branches V 1 , ..., V 2(d 1 +d 2 −1) and each of these branches has delta invariant . In particular branches V i are smooth if and only if d 1 = d 2 or d 1 = d 2 + 1.

Generalized cusps
In this section our aim is to estimate the number of cusps of non-generic mappings. We start from: Definition 7.1. Let F : (C 2 , a) → (C 2 , F (a)) be a germ of a holomorphic mapping. We say that F has a generalized cusp at a if F a is proper, the curve J(F ) = 0 is reduced near a and the discriminant of F a is not smooth at F (a).
Remark 7.2. If F a is proper, J(F ) = 0 is reduced near a and J(F ) is singular at a then it follows from Corollary 1.11 from [9] that also the discriminant of F a is singular at F (a) and hence F has a generalized cusp at a. Now we introduce the index of generalized cusp: Definition 7.3. Let F = (f, g) : (C 2 , a) → (C 2 , F (a)) be a germ of a holomorphic mapping. Assume that F has a generalized cusp at a point a ∈ C 2 . Since the curve J(F ) = 0 is reduced near a, we have that the set {∇f = 0} ∩ {∇g = 0} has only isolated points near a. For a generic linear mapping T ∈ GL(2), if F ′ = (f ′ , g ′ ) = T • F then ∇f ′ does not vanish identically on any branch of {J(F ) = 0} near a. We say that the cusp of F at a has an index µ a := dim C O a /(J(F ′ ), Remark 7.4. We show below that the index µ a is well-defined and finite. Moreover, it is easy to see that a simple cusp has index one.
Hence our index coincides with the classical local number of cusps defined e.g. in [4].
We have (compare with [4], [5], [6]): Theorem 7.6. Let X ⊂ C m be a smooth surface. Let F = (f, g) ∈ Ω X (d 1 , d 2 ) and assume that F has a generalized cusp at a ∈ C 2 . If U a ⊂ X is a sufficiently small ball around a then µ a is equal to the number of simple cusps in U a of a generic mapping Proof. We can assume that X = C 2 and ∇f does not vanish identically on any branch of {J(F ) = 0} near a. In particular we have dim O a /(f x , f y ) = dim O a /(J(F ), f x , f y ) < ∞.
Since a is a cusp of F we have Φ(a) = 0. Moreover d a (Φ) < ∞, where d a (Φ) denotes the local topological degree of Φ at a. Indeed, if J 1,1 (F ) = 0 on some branch B of the curve J(F ) = 0 then the rank of F |B would be zero and by Sard theorem F has to contract B, which is a contradiction (F a is proper). By the Rouche Theorem (see [2], p. 86), we have that for large i the mapping Φ i has exactly d a (Φ) zeroes in U a and Ψ i has exactly d a (Ψ) zeroes in U a (counted with multiplicities, if Ψ(a) = 0 we put d a (Ψ) = 0). However, the mappings F i are generic, in particular all zeroes of Φ i and Ψ i are simple. Moreover the zeroes of Φ i which are not cusps of F i are zeroes of Ψ i . Hence µ a = d a (Φ) − d a (Ψ) is indeed the number of simple cusps of F i in U a . Corollary 7.7. Let F ∈ Ω X (d 1 , d 2 ) and d i > 1. Assume that F has generalized cusps at points a 1 , . . . , a r . Then r i=1 µ a i ≤ c X (d 1 , d 2 ). In particular the numbers of singular germs {F a , a ∈ X} which are finitely determined and are not folds, is bounded by the number c X (d 1 , d 2 ).
Proof. We prove only the last statement. Let F a be a singular germ which is finitely determined. Then the curve J(F a ) is reduced. There are two possibilities: In the case 1) we have by [9] that F a is equivalent to the germ (x, y) → (x k , y) and since J(F a ) is reduced we have k = 2, i.e., F a is a fold.
In the case 2) F a is a generalized cusp. Hence the number of such germs is bounded by the number of generalized folds.
Remark 7.8. Of course for X = C 2 or X = S the assumption d ′ i > 1 (or d i > 1) in Theorem 7.6 and Corollary 7.7 is not necessary.