On semi-equivalence of generically-finite polynomial mappings

Let f,g:X→Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f,g:X \rightarrow Y$$\end{document} be continuous mappings. We say that f is topologically equivalent to g if there exist homeomorphisms Φ:X→X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi : X\rightarrow X$$\end{document} and Ψ:Y→Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi : Y\rightarrow Y$$\end{document} such that Ψ∘f∘Φ=g.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi \circ f\circ \Phi =g.$$\end{document} Moreover, we say that f is topologically semi-equivalent to g if there exist open, dense subsets U,V⊂X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U, V\subset X$$\end{document} and homeomorphisms Φ:U→V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi : U\rightarrow V$$\end{document} and Ψ:Y→Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi : Y\rightarrow Y$$\end{document} such that Ψ∘f∘Φ|U=g|U.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi \circ f\circ \Phi |_U=g|_{U}.$$\end{document} Let X, Y be smooth irreducible affine complex varieties. We show that every algebraic family F:M×X∋(m,x)↦F(m,x)=fm(x)∈Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F: M\times X\ni (m, x)\mapsto F(m, x)=f_m(x)\in Y$$\end{document} of polynomial mappings contains only a finite number of topologically non-equivalent proper mappings and only a finite number of topologically non-semi-equivalent generically-finite mappings. In particular there are only a finite number of classes of topologically non-equivalent proper polynomial mappings f:Cn→Cm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:{\mathbb {C}}^n\rightarrow {\mathbb {C}}^m$$\end{document} of a bounded (algebraic) degree. The same is true for a number of classes of topologically non-semi-equivalent generically-finite polynomial mappings f:Cn→Cm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:{\mathbb {C}}^n\rightarrow {\mathbb {C}}^m$$\end{document} of a bounded (algebraic) degree.


Introduction
Let f, g : X → Y be continuous mappings. We say that f is topologically equivalent to g if there exist homeomorphisms : X → X and : Y → Y such that • f • = g. Moreover, we say that f is topologically semi-equivalent to g if there exist open, dense subsets U, V ⊂ X and homeomorphisms : In the case X = C n and Y = C René Thom stated a Conjecture that there are only finitely many topological types of polynomials f : X → Y of bounded degree. This Conjecture was confirmed by Fukuda [2]. Also a more general problem was considered: how many The author is partially supported by the Narodowe Centrum Nauki Grant, number 2015/17/B/ST1/02637. B Zbigniew Jelonek najelone@cyf-kr.edu.pl 1 Instytut Matematyczny, Polska Akademia Nauk,Śniadeckich 8, 00-656 Warszawa, Poland topological types are there in the family P(n, m, k) of polynomial mapping f : C n → C m of degree bounded by k? Aoki and Noguchi [1] showed that there are only a finite number of topologically non-equivalent mappings in the family P (2, 2, k). Finally Nakai [8] showed that each familiy P(n, m, k), where n, m, k > 3, contains infinitely many different topological types even if we consider only generically-finite mappings. Hence the General Thom Conjecture is not true even for generically-finite mappings. However, we show in this paper that there are only a finite number of classes of topologically semi-equivalent generically-finite polynomial mappings f : C n → C m of a bounded (algebraic) degree. As a by product of our considerations we give a simple proof of the following interesting fact: for every n, m and k there are only a finite number of topological types of proper polynomial mappings f : C n → C m of (algebraic) degree bounded by k. Hence we can say that Thom Conjecture is true for proper polynomial mappings. We show also that if n ≤ m and n (d 1 , ..., d m ) denotes the family of all polynomial mappings F = ( f 1 , ..., f m ) : C n → C m of a multi-degree bounded by (d 1 , ..., d m ), then any two general member of this family are topologically equivalent.
In fact we prove more: if X, Y are smooth affine irreducible varieties, then every algebraic family F of polynomial mappings from X to Y contains only a finite number of topologically non-semi-equivalent (non-equivalent) generically-finite (proper) mappings. Moreover, if a family F is irreducible, then two generic members of F are in the same equivalence class.
Let us recall here, that a mapping f : X → Y is generically finite, if for general x ∈ X the set f −1 ( f (x)) is finite. Our proof goes as follows. Let M be a smooth affine irreducible variety and let F be a family of polynomial mappings induced be a regular mapping F : where we treat f m as a mapping f m : X → Z m := f m (X ), (2) for every m 1 , m 2 ∈ U the pairs ( f m 1 (X ), B( f m 1 )) and ( f m 2 (X ), B( f m 2 )) are equivalent via a homeomorphism, i.e., there is a homeomorphism In particular the group G = π 1 ( f m (X )\B( f m )) does not depend on m ∈ U. Using elementary facts from the theory of topological coverings, we show that the number of topological semi-types (types) of generically-finite (proper) mappings in the family F |U is bounded by the number of subgroups of G of index μ(F), hence it is finite. Then we conclude the proof by induction. Finally, the case of arbitrary M can be easily reduced to the smooth, irreducible, affine case. Remark 1.1 In this paper we use the term "polynomial mapping" for every regular mapping f : X → Y of affine varieties.

Bifurcation set
Let X, Z be affine irreducible varieties of the same dimension and assume that X is smooth. Let f : X → Z be a dominant polynomial mapping. It is well known that there is a Zariski open non-empty subset U of Z such that for every have the same number μ( f ) of points. We say that μ( f ) is the topological degree of f. Recall the following (see [5,6]).

Definition 2.1
Let X, Z be as above and let f : X → Z be a dominant polynomial mapping.
We say that f is finite at a point z ∈ Z if there exists an open neighborhood U of z such that the mapping It is well-known that the set S f of points at which the mapping f is not finite is either empty or it is a hypersurface (see [5,6]). We say that S f is the set of non-properness of f. Definition 2.2 Let X be a smooth affine n-dimensional variety and let Z be an affine variety of the same dimension. Let f : X → Z be a generically finite dominant polynomial mapping of geometric degree μ( f ). The bifurcation set of f is

Remark 2.3
The same definition makes sense for those continuous mapping f : X → Z , for which we can define the topological degree μ( f ) and singularities of Z . In particular if Z 1 , Z 2 are affine algebraic varieties, f : X → Z 1 is a dominant polynomial mapping and : Z 1 → Z 2 is a homeomorphism which preserves singularities, then we can define B( • f ) as (B( f )). Moreover, the mapping • f behaves topologically as an analytic covering. We will use this facts in the proof of Theorem 3.5.
We have the following theorem (see also [7]). Proof Let us note that outside the set S f ∪ Sing(Z ) the mapping f is a (ramified) analytic covering of degree μ( f ). By Lemma 2.5 below, if z / ∈ Sing(Z ) we have # f −1 (z) ≤ μ( f ). Moreover, since f is an analytic covering outside S f ∪ Sing(Z ) we see that for y / ∈ S f ∪ Sing(Z ) the fiber f −1 (z) has exactly μ( f ) points counted with multiplicity. Take X 0 := Now let z ∈ S f \Sing(Z ). There are two possibilities: In case (b) we can assume that f −1 (z) = ∅. Let U be an affine neighborhood of z disjoint from Sing(z) over which the mapping f has finite fibers. Let V = f −1 (U ). By the Zariski Main Theorem in the version given by Grothendieck (see [3]), there exists a normal variety V and a finite mapping f : V → U such that We can choose a function h ∈ C[X ] which separates all x i (in particular we can take as h the equation of a general hyperplane section). Since f is finite, the minimal equation of h over the field C(Z ) is of the form: If we substitute f = z into this equation we get the desired result.

Main result
We start with the following: Proof We can assume that Y is smooth. Since there exists a mapping π : Y l → C l which is generically etale, we can assume that Y = C l . Let us recall that if Z is an algebraic variety, then a point z ∈ Z is smooth if and only if the local Let y = (y 1 , ..., y l ) ∈ C l be a sufficiently generic point. Then by Sard's Theorem the fiber is the spectrum of a localization of C[X ] and so a domain. Since we are in characteristic zero, is necessarily geometrically reduced (i.e. stays reduced after extending to an algebraic closure of C(Y )). Since the property of fibres being geometrically reduced is open on the base, i.e. on Y , thus the fibres over an open subset of Y will be reduced. Consequently, there is a Zariski open, non-empty subset U ⊂ Y such that for y ∈ U the fiber f −1 (y) is reduced. Hence we can assume that Z is reduced. It is enough to show that every point z ∈ Z ∩ Sing(X ) is singular on Z .
Since the point z is smooth on Hence the point z is smooth on X , a contradiction.
We have: Proof Let X 1 be an algebraic completion of X and let Y be a smooth algebraic completion of Y. Take X 1 := graph( f ) ⊂ X 1 × Y and let X 2 be a desingularization of X 1 . We can assume that X ⊂ X 2 . We have an induced mapping f : There is a Whitney stratification S of X 2 which is compatible with R.
For any smooth strata S i ∈ S let B i be the set of critical values of the mapping f |S i and The restriction of the stratification S to X 3 gives a Whitney stratification which is compatible with the family R := R ∩ X 3 . We have a proper mapping f := f |X 3 : X 3 → Y \B which is a submersion on each stratum. By the Thom first isotopy theorem there is a trivialization of f which preserves the strata. It is an easy observation that this trivialization gives a trivialization of the mapping f : In particular the fibers f −1 (y 1 ) and f −1 (y 2 ) are homeomorphic via a stratum preserving homeomorphism. This means that the triples We also need the following: If G is generically finite, then by the topological degree μ(F) we mean the number μ(G). (2) Consider the projection π : Z (m, y) → m ∈ M. As we know from (1), the mapping π is dominant. By a well known result, after shrinking U 1 we can assume that every Now we are ready to prove our main result:   R, a). a f )) and G = π 1 (R, a). Hence [G : H f ] = k. It is well known that the fundamental group of a smooth algebraic variety is finitely generated. In particular the group G := π 1 (Q\B, a) is finitely generated. Let us recall the following result of M. Hall (see [4]): We show that f m := f is equivalent to f i . Let us consider two coverings f : (P f , a f ) → (R, a) and f i : (P f i , a f i ) → (R, a). Since f * (π 1 (P f , a f )) = f i * (π 1 (P f i , a f i )) we can lift the covering f to a homeomorphism φ : P f → P f i such that following diagram commutes: Hence for generically-finite mappings we have f m i )). Hence f m is semi-equivalent to f m i . In the case of proper mappings we show additionally that the mapping φ can be extended to a continuous mapping on the whole of X. Indeed, take a point x ∈ f −1 (B) and let Take small open disjoint neighborhoods W i (r ) of b i , such that W i (r ) shrinks to b i as r tends to 0. We can choose an open neighborhood V (r ) of y so small that f −1 i (V (r )) ⊂ s j=1 W i (r ). Now take a small connected neighborhood P x (r ) of x such that f (P x (r )) ⊂ V (r ). The set P x (r )\ f −1 (B) is still connected and it is transformed by φ into one particular set W i 0 (r ). We take (x) = b i 0 . It is easy to see that the mapping so defined is a continuous extension of φ. In fact φ(P x (r )\ f −1 (B)) shrinks to b i 0 if r goes to 0. Moreover, we still have f = f i • .
In a similar way the mapping determined by φ −1 is continuous. It is easy to see that • = • = identit y, hence is a homeomorphism. Consequently, the mapping This means that the family F |U contains only a finite number of topologically non-semiequivalent (non-equivalent) generically-finite (proper) mappings. In fact, the number of topological semi-types (types) of generically-finite (proper) mappings in F |U is bounded by the number of subgroups of G of index μ(F).
Let T = M\U. Hence dim T < dim M. By the induction the family F |T also contains only a finite number of topologically non-semi-equivalent (non-equivalent) generically-finite (proper) mappings. Consequently so does F .

Corollary 3.7
There is only a finite number of topologically non-semi-equivalent (nonequivalent) generically-finite (proper) polynomial mappings f : C n → C m of a bounded algebraic degree.

Families of proper mappings
In this section we slightly extend our previous result in the case of irreducible families of proper (or generically-finite) mappings. First we prove a following lemma: We construct a family of diffeomorphisms t , which are interpolation between translation x → x + v t and identity.
Let σ : Y → [0, 1] be a differentiable function such that σ = 1 on B(0, η) and σ = 0 outside B(0, ). Define a vector field V (x) = σ (x)v t . Integrating this vector field we get desired diffeomeorphisms t , for any t. Proof We follow the proof of Theorem 3.5 and we use here the same notation. By Lemma 3.4 there is a non-empty open subset U ⊂ M such that for every m 1 , m 2 ∈ U we have

Corollary 4.2 Let Y be a smooth manifold and Z be a smooth submanifold. For every point a ∈ Z and every open neighborhood V a of the point a, there is an open connected neighborhood U a of the point a, such that:
and (B( f t )) = B( f 0 ). It is enough to prove that mappings F t = t • f t are locally (in the sense of parameter t) equivalent.
(1) First step of the proof. Let C t ⊂ X denotes the preimage by F t of the set B (in fact C t = f −1 t (B( f t )) and put X t = X \C t . Put Q := Q\B.Assume that for all mappings F t there is a point a ∈ (X \ t∈I C t ) such that for all t ∈ I we have F t (a) = b.
Indeed let γ 1 , ..., γ s be generators of the group π 1 (X t 0 , a). Let U i be an open relatively compact neighborhoods of γ i such that U i ∩ C t 0 = ∅. For sufficiently small number > 0 and t ∈ (t 0 − , t 0 + ) we have U i ∩ C t = ∅. Let t ∈ (t 0 − , t 0 + ). Note that the loop F t (γ i ) is homotopic with the loop F t 0 (γ i ). In particular the group F t 0 * (π 1 (X t 0 , a)) is contained in the group F t * (π 1 (X t , a)). Since they have the same (finite!) index in π 1 (Y , b) they are equal. This means that the subgroup G t * (π 1 (X t , a)) ⊂ π 1 (Y , b) is locally constant, hence it is constant.
Let us consider two coverings F t : (X t , a) → (Q , b) and F 0 : (X 0 , a) → (Q , b). Since F t * π 1 (X t , a) = F 0 * π 1 (X 0 , a) we can lift the covering F t to a homeomorphism φ t : X t → X 0 . As before we can extend the homeomorphism φ t to the homeomorphism t : (2) The general case. Now we can prove Theorem 4.3. Since in general there is no a point a ∈ (X \ t∈I C t ) such that for all t ∈ I we have F t (a) = b, we have to modify our construction.
First we prove that for every t 0 ∈ I there exists > 0 and a family of homeomorphisms t : X → X , t ∈ (t 0 − , t 0 + ) such that F t = F t 0 • t for t ∈ (t 0 − , t 0 + ). Take a point a ∈ X t 0 and choose > 0 so small that a ∈ X t for t ∈ (t 0 − , t 0 + ). Put γ (t) t → F t (a) ∈ Y . We can take so small that the hypothesis of Corollary 4.2 is satisfied. Applying Corollary 4.2 with Y = Y \B and Z = Q\B we have a continuous family of diffeomeorphisms ψ t : Y → Y which preserves Q and B, t ∈ (t 0 − , t 0 + ) such that ψ t (F t (a)) = F 0 (a). Take G t = ψ t • F t . Arguing as in the first part of our proof all G t are topologically equivalent for t ∈ (t 0 − , t 0 + ). Hence also all F t are topologically equivalent for t ∈ (t 0 − , t 0 + ). Since F t are locally topologically equivalent, they are topologically equivalent for every t ∈ I . Proof Indeed, it is enough to note that a generic mapping f ∈ n (d 1 , ..., d m ) is proper.
Using the same method we can prove: Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.