Mathematische Zeitschrift

, Volume 283, Issue 1–2, pp 133–142 | Cite as

On semi-equivalence of generically-finite polynomial mappings

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Article

Abstract

Let \(f,g:X \rightarrow Y\) be continuous mappings. We say that f is topologically equivalent to g if there exist homeomorphisms \(\Phi : X\rightarrow X\) and \(\Psi : Y\rightarrow Y\) such that \(\Psi \circ f\circ \Phi =g.\) Moreover, we say that f is topologically semi-equivalent to g if there exist open, dense subsets \(U, V\subset X\) and homeomorphisms \(\Phi : U\rightarrow V\) and \(\Psi : Y\rightarrow Y\) such that \(\Psi \circ f\circ \Phi |_U=g|_{U}.\) Let XY be smooth irreducible affine complex varieties. We show that every algebraic family \(F: M\times X\ni (m, x)\mapsto F(m, x)=f_m(x)\in Y\) 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 \(f:{\mathbb {C}}^n\rightarrow {\mathbb {C}}^m\) of a bounded (algebraic) degree. The same is true for a number of classes of topologically non-semi-equivalent generically-finite polynomial mappings \(f:{\mathbb {C}}^n\rightarrow {\mathbb {C}}^m\) of a bounded (algebraic) degree.

Mathematics Subject Classification

14 D 99 14 R 99 51 M 99 

1 Introduction

Let \(f,g:X \rightarrow Y\) be continuous mappings. We say that f is topologically equivalent to g if there exist homeomorphisms \(\Phi : X\rightarrow X\) and \(\Psi : Y\rightarrow Y\) such that \(\Psi \circ f\circ \Phi =g.\) Moreover, we say that f is topologically semi-equivalent to g if there exist open, dense subsets \(U, V\subset X\) and homeomorphisms \(\Phi : U\rightarrow V\) and \(\Psi : Y\rightarrow Y\) such that \(\Psi \circ f\circ \Phi |_U=g|_{U}.\)

In the case \(X={\mathbb {C}}^n\) and \(Y={\mathbb {C}}\) René Thom stated a Conjecture that there are only finitely many topological types of polynomials \(f: X\rightarrow Y\) of bounded degree. This Conjecture was confirmed by Fukuda [2]. Also a more general problem was considered: how many topological types are there in the family P(nmk) of polynomial mapping \(f:{\mathbb {C}}^n\rightarrow {\mathbb {C}}^m\) 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(nmk), 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 \(f: {\mathbb {C}}^n\rightarrow {\mathbb {C}}^m\) of a bounded (algebraic) degree. As a by product of our considerations we give a simple proof of the following interesting fact: for every nm and k there are only a finite number of topological types of proper polynomial mappings \(f: {\mathbb {C}}^n\rightarrow {\mathbb {C}}^m\) of (algebraic) degree bounded by k. Hence we can say that Thom Conjecture is true for proper polynomial mappings. We show also that if \(n\le m\) and \(\Omega _n(d_1,...,d_m)\) denotes the family of all polynomial mappings \(F=(f_1,...,f_m):{\mathbb {C}}^n\rightarrow {\mathbb {C}}^m\) 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 XY are smooth affine irreducible varieties, then every algebraic family \({\mathcal {F}}\) 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 \({\mathcal {F}}\) is irreducible, then two generic members of \({\mathcal {F}}\) are in the same equivalence class.

Let us recall here, that a mapping \(f:X\rightarrow Y\) is generically finite, if for general \(x\in X\) the set \(f^{-1}(f(x))\) is finite. Our proof goes as follows. Let M be a smooth affine irreducible variety and let \({\mathcal {F}}\) be a family of polynomial mappings induced be a regular mapping \(F: M\times X \rightarrow Y,\) i.e., \({\mathcal {F}}:=\{f_m: X\ni x\mapsto F(m,x)\in Y, \ m\in M\}.\) Let us recall that if \(f: X\rightarrow Z\) is a generically finite polynomial mapping of affine varieties, then the bifurcation set B(f) of f is the set \(\{ z\in Z: z\in Sing(Z) \ or\ \# f^{-1}(z)\not =\mu (f)\}\), where \(\mu (f)\) is the topological degree of f. The set B(f) is always closed in Z. We show that there exists a Zariski open, dense subset U of M such that
  1. (1)

    for every \(m\in U\) we have \(\mu (f_m)=\mu ({\mathcal {F}}),\) where we treat \(f_m\) as a mapping \(f_m: X\rightarrow Z_m:=\overline{f_{m}(X)},\)

     
  2. (2)

    for every \(m_1, m_2\in U\) the pairs \((\overline{f_{m_1}(X)}, B(f_{m_1}))\) and \((\overline{f_{m_2}(X)}, B(f_{m_2}))\) are equivalent via a homeomorphism, i.e., there is a homeomorphism \(\Psi : Y\rightarrow Y\) such that \(\Psi (\overline{f_{m_1}(X)})=\overline{f_{m_2}(X)}\) and \(\Psi (B(f_{m_1}))=B(f_{m_2}).\)

     
In particular the group \(G=\pi _1(\overline{f_{m}(X)}{\setminus } B(f_m))\) does not depend on \(m\in U.\) 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 \({\mathcal {F}}_{|U}\) is bounded by the number of subgroups of G of index \(\mu ({\mathcal {F}})\), 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 \(f: X\rightarrow Y\) of affine varieties.

2 Bifurcation set

Let XZ be affine irreducible varieties of the same dimension and assume that X is smooth. Let \(f: X\rightarrow Z\) 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 \(x_1,x_2\in U\) the fibers \(f^{-1}(x_1), f^{-1}(x_2)\) have the same number \(\mu (f)\) of points. We say that \(\mu (f)\) is the topological degree of f. Recall the following (see [5, 6]).

Definition 2.1

Let XZ be as above and let \(f : X \rightarrow Z\) be a dominant polynomial mapping. We say that f is finite at a point \(z \in Z\) if there exists an open neighborhood U of z such that the mapping \( f_{|f^{-1}(U)} :f^{-1} (U)\rightarrow U\) is proper.

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 \(f : X\rightarrow Z\) be a generically finite dominant polynomial mapping of geometric degree \(\mu (f).\) The bifurcation set of f is
$$\begin{aligned} B(f)=\{z\in Z : z\in Sing(Z) \ or \ \# f^{-1}(z)\not =\mu (f)\}. \end{aligned}$$

Remark 2.3

The same definition makes sense for those continuous mapping \(f: X\rightarrow Z\), for which we can define the topological degree \(\mu (f)\) and singularities of Z. In particular if \(Z_1, Z_2\) are affine algebraic varieties, \(f: X\rightarrow Z_1\) is a dominant polynomial mapping and \(\Phi : Z_1\rightarrow Z_2\) is a homeomorphism which preserves singularities, then we can define \(B(\Phi \circ f)\) as \(\Phi (B(f)).\) Moreover, the mapping \(\Phi \circ f\) 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]).

Theorem 2.4

Let XZ be affine irreducible complex varieties of the same dimension and suppose X is smooth. Let \(f: X \rightarrow Z\) be a polynomial dominant mapping. Then the set B(f) is closed and \(B(f)=K_0\cup S_f\cup Sing(Z).\)

Proof

Let us note that outside the set \(S_f\cup Sing(Z)\) the mapping f is a (ramified) analytic covering of degree \(\mu (f).\) By Lemma 2.5 below, if \(z\not \in Sing(Z)\) we have \(\# f^{-1}(z)\le \mu (f).\) Moreover, since f is an analytic covering outside \(S_f\cup Sing(Z)\) we see that for \(y \not \in S_f\cup Sing(Z)\) the fiber \(f^{-1}(z)\) has exactly \(\mu (f)\) points counted with multiplicity. Take \(X_0:=X {\setminus } f^{-1}(Sing(Z)\cup S_f).\) If \(z\in K_0(f_{|X_0})\), the set of critical values of \(f_{|X_0},\) then \(\# f^{-1}(z)<\mu (f).\)

Now let \(z\in S_f{\setminus } Sing(Z).\) There are two possibilities:
  1. (a)

    \(\# f^{-1}(z)=\infty .\)

     
  2. (b)

    \(\# f^{-1}(z)<\infty .\)

     
In case (b) we can assume that \(f^{-1}(z)\not =\emptyset .\) 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 \(\overline{V}\) and a finite mapping \(\overline{f}:\overline{ V}\rightarrow U\) such that
  1. (1)

    \(V\subset \overline{V}\),

     
  2. (2)

    \(\overline{f}_{|V}=f.\)

     
Since \(z\in \overline{f}(\overline{V}{\setminus } V)\), it follows from Lemma 2.5 below that \(\# f^{-1}(z)<\mu (f).\) Consequently, if \(z\in S_f,\) we have \(\# f^{-1}(z)<\mu (f).\) Finally, we have \(B(f)= K_0(f_{|X_0})\cup S_f\cup Sing(Z).\) However, the set \(K_0(f_{|X_0})\) is closed in \(Z{\setminus } (S_f\cup Sing(Z)).\) Hence B(f) is closed in Z. \(\square \)

Lemma 2.5

Let XZ be affine normal varieties of the same dimension. Let \(f:X\rightarrow Z\) be a finite mapping. Then for every \(z\in Z\) we have \(\# f^{-1}(z)\le \mu (f).\)

Proof

Let \(\# f^{-1}(z)=\{ x_1, \ldots , x_r\}.\) We can choose a function \(h\in {\mathbb {C}}[X]\) 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 \({\mathbb {C}}(Z)\) is of the form:
$$\begin{aligned} T^s+a_1(f)T^{{s-1}}+\cdots +a_s(f)\in f^*{\mathbb {C}}[Z][T], \end{aligned}$$
where \( s\le \mu (f).\) If we substitute \(f=z\) into this equation we get the desired result. \(\square \)

3 Main result

We start with the following:

Lemma 3.1

Let \(f: X^k\rightarrow Y^l\) be a dominant polynomial mapping of affine irreducible varieties. There exists a Zariski open non-empty subset \(U\subset Y\) such that for any \(y\in U\) we have \(Sing(f^{-1}(y))=f^{-1}(y)\cap Sing(X).\)

Proof

We can assume that Y is smooth. Since there exists a mapping \(\pi : Y^l\rightarrow {\mathbb {C}}^l\) which is generically etale, we can assume that \(Y={\mathbb {C}}^l.\) Let us recall that if Z is an algebraic variety, then a point \(z\in Z\) is smooth if and only if the local ring \({\mathcal {O}}_z(Z)\) is regular, or equivalently \(\dim _{\mathbb {C}}{{\mathfrak {m}}}/{{\mathfrak {m}}}^2=\dim Z,\) where \({\mathfrak {m}}\) denotes the maximal ideal of \({\mathcal {O}}_z(Z)\).

Let \(y=(y_1,...,y_l)\in {\mathbb {C}}^l\) be a sufficiently generic point. Then by Sard’s Theorem the fiber \(Z=f^{-1}(y)\) is smooth outside Sing(X) and \(\dim Z=\dim X - l=k-l.\) Note that the generic (scheme-theoretic) fiber F of f is reduced. Indeed, this fiber \(F=Spec({\mathbb {C}}(Y)\otimes _{{\mathbb {C}}[Y]} {\mathbb {C}}[X])\) is the spectrum of a localization of \({\mathbb {C}}[X]\) and so a domain. Since we are in characteristic zero, the reduced \({\mathbb {C}}(Y)\)-algebra \({\mathbb {C}}(Y)\otimes _{{\mathbb {C}}[Y]} {\mathbb {C}}[X]\) is necessarily geometrically reduced (i.e. stays reduced after extending to an algebraic closure of \({\mathbb {C}}(Y)\)). 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 \(U\subset Y\) such that for \(y\in U\) the fiber \(f^{-1}(y)\) is reduced. Hence we can assume that Z is reduced. It is enough to show that every point \(z\in Z\cap Sing(X)\) is singular on Z.

Assume that \(z\in Z\cap Sing(X)\) is smooth on Z. Let \(f: X\rightarrow {\mathbb {C}}^l\) be given as \(f=(f_1,...,f_l)\), where \(f_i\in {\mathbb {C}}[X].\) Then \({\mathcal {O}}_z(Z)={\mathcal {O}}_z(X)/(f_1-y_1,...,f_l-y_l).\) In particular if \({\mathfrak {m}}'\) denotes the maximal ideal of \({\mathcal {O}}_z(Z)\) and \({\mathfrak {m}}\) denotes the maximal ideal of \({\mathcal {O}}_z(X)\) then \({\mathfrak {m}}'={\mathfrak {m}}/(f_1-y_1,...,f_l-y_l).\) Let \(\alpha _i\) denote the class of the polynomial \(f_i-y_i\) in \({{\mathfrak {m}}}/{{\mathfrak {m}}}^2.\) Let us note that
$$\begin{aligned} {{\mathfrak {m}}'}/{{\mathfrak {m}}'}^2 ={{\mathfrak {m}}}/({{\mathfrak {m}}}^2+(\alpha _1,...,\alpha _l)). \end{aligned}$$
(1)
Since the point z is smooth on Z we have \(\dim _{\mathbb {C}}{{\mathfrak {m}}'}/{{\mathfrak {m}}'}^2=\dim Z=\dim X-l.\) Take a basis \(\beta _1,...,\beta _{k-l}\) of the space \( {{\mathfrak {m}}'}/{{\mathfrak {m}}'}^2\) and let \(\overline{\beta _i}\in {{\mathfrak {m}}}/{{\mathfrak {m}}}^2\) correspond to \(\beta _i\) under the correspondence (1). Note that the vectors \(\overline{\beta _1},...,\overline{\beta _{k-l}}, \alpha _1,..., \alpha _l\) generate the space \({{\mathfrak {m}}}/{{\mathfrak {m}}}^2.\) This means that \(\dim _{\mathbb {C}}{{\mathfrak {m}}}/{{\mathfrak {m}}}^2\le k-l+l=k=\dim X.\) Hence the point z is smooth on X, a contradiction. \(\square \)

We have:

Lemma 3.2

Let XY be smooth complex irreducible algebraic varieties and \(f:X\rightarrow Y\) a regular dominant mapping. Let \(N\subset W\subset X\) be closed subvarieties of X. Then there exists a non-empty Zariski open subset \(U\subset Y\) such that for every \(y_1, y_2\in U\) the triples \((f^{-1}(y_1), W\cap f^{-1}(y_1), N\cap f^{-1}(y_1))\) and \((f^{-1}(y_2), W\cap f^{-1}(y_2), N\cap f^{-1}(y_2))\) are homeomorphic.

Proof

Let \(X_1\) be an algebraic completion of X and let \(\overline{Y}\) be a smooth algebraic completion of Y. Take \(X_1':=\overline{graph(f)}\subset X_1\times \overline{Y}\) and let \(X_2\) be a desingularization of \(X_1'.\)

We can assume that \(X\subset X_2.\) We have an induced mapping \(\overline{f}: X_2\rightarrow \overline{Y}\) such that \(\overline{f}_{|X}=f.\) Let \(Z=X_2{\setminus } X.\) Denote by \(\overline{N}, \overline{W}\) the closures of N and W in \(X_2.\) Let \({\mathcal {R}}=\{\overline{N}\cap Z, \overline{W}\cap Z, \overline{N}, \overline{W}, Z\} ,\) a collection of algebraic subvarieties of \(X_2.\) There is a Whitney stratification \({\mathcal S}\) of \(X_2\) which is compatible with \({\mathcal {R}}.\)

For any smooth strata \(S_i\in \mathcal S\) let \(B_i\) be the set of critical values of the mapping \(\overline{f}_{|S_i}\) and denote \(B=\overline{ \bigcup B_i}.\) Take \(X_3=X_2{\setminus } \overline{f}^{-1}(B).\) The restriction of the stratification \(\mathcal S\) to \(X_3\) gives a Whitney stratification which is compatible with the family \({{\mathcal {R}}}':={{\mathcal {R}}}\cap X_3.\) We have a proper mapping \(f':=\overline{f}_{|X_3} : X_3\rightarrow \overline{Y}{\setminus } B\) 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 \(f: X{\setminus } f^{-1}(B)\rightarrow Y{\setminus } B:=U.\) 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 \((f^{-1}(y_1), W\cap f^{-1}(y_1), N\cap f^{-1}(y_1))\) and \((f^{-1}(y_2), W\cap f^{-1}(y_2), N\cap f^{-1}(y_2))\) are homeomorphic. \(\square \)

We also need the following:

Definition 3.3

Let XY be smooth affine varieties. By a family of regular mappings \({\mathcal {F}}_M(X,Y,F):={\mathcal {F}}\) we mean a regular mapping \(F: M\times X\rightarrow Y\), where M is an algebraic variety. The members of a family \({\mathcal {F}}\) are the mappings \(f_m: X\ni x\rightarrow F(m,x)\in Y.\) Let
$$\begin{aligned} G: M\times X\ni (m,x)\mapsto (m, F(m,x))\in Z=\overline{G(M\times X)}\subset M\times Y. \end{aligned}$$
If G is generically finite, then by the topological degree \(\mu ({\mathcal {F}})\) we mean the number \(\mu (G)\). Otherwise we put \(\mu ({\mathcal {F}})=0.\)

Later we will sometimes identify the mapping \(f_m\) with the mapping \(G(m,\cdot )=(m, f_m): X\rightarrow m\times Y.\) The following lemma is important:

Lemma 3.4

Let XY be smooth affine complex varieties. Let M be a smooth affine irreducible variety and let \({\mathcal {F}}\) be the family induced by a mapping \(F: M\times X \rightarrow Y,\) i.e., \({\mathcal {F}}=\{ f_m: X\ni x\mapsto F(m,x)\in Y, \ m\in M\}.\) Assume that \(\mu ({\mathcal {F}})>0.\) Take \(Z=\overline{G(M\times X)}\) and put \(Z_m= (m\times Y)\cap Z.\)

Then
  1. (1)

    There is an open non-empty subset \(U_1\subset M\) such that for every \(m\in U_1\) we have \(\mu (f_m)=\mu ({\mathcal {F}});\)

     
  2. (2)

    There is a non-empty open subset \(U_2\subset U_1\) such that for every \(m\in U_2\) we have \(\overline{f_m(X)}=Z_m:=(m\times Y)\cap Z\) and \(B(f_m)=B(G)_m:=(m\times Y)\cap B(G);\)

     
  3. (3)

    There is a non-empty open subset \(U_3\subset U_2\) such that for every \(m_1, m_2\in U_3\) the pairs \((\overline{f_{m_1}(X)}, B(f_{m_1}))\) and \((\overline{f_{m_2}(X)}, B(f_{m_2}))\) are equivalent by means of a homeomorphism, i.e., there is a homeomorphism \(\Psi : Y\rightarrow Y\) such that \(\Psi (\overline{f_{m_1}(X)})=\overline{f_{m_2}(X)}\) and \(\Psi (B(f_{m_1}))=B(f_{m_2}).\)

     

Proof

  1. (1)

    Take \(G: M\times X\ni (m,x)\mapsto (m, F(m,x))\in Z.\) The mapping \(G: M\times X \ni (m,x)\mapsto (m, F(m,x))\in Z\) has a constant number of points in the fibers outside the bifurcation set \(B(G)\subset Z.\) Take \(U=Z{\setminus } B(G).\) By Theorem 2.4 the set U is open. Let \(\pi : Z \ni (m, y)\mapsto m\in M\) be the projection. We show that the constructible set \(\pi (U)\) is dense in M. Indeed, assume that \(\overline{\pi (U)}=N\) is a proper subset of M. Since U is dense in Z, we have \(\pi (Z)\subset N\), i.e., \(Z\subset N\times Y.\) This is a contradiction. In particular the set \(\pi (U)\) is dense in M and it contains a Zariski open, non-empty subset \(U_1\subset M.\) Of course \(\mu (f_m)=\mu ({\mathcal {F}})\) for \(m\in U_1.\)

     
  2. (2)

    Consider the projection \(\pi : Z\ni (m,y)\mapsto m\in M.\) As we know from (1), the mapping \(\pi \) is dominant. By a well known result, after shrinking \(U_1\) we can assume that every fiber \(Z_m\) of \(\pi \) (\(m\in U_2\subset U_1\)) is of pure dimension \(d=\mathrm{dim}\ Z- \mathrm{dim}\ M=\mathrm{dim}\ X.\) However, \(Z_m =\overline{f_m(X)}\cup B(G)_m.\) Generically the dimension of \(B(G)_m\) is less than d. Hence if we possibly shrink \(U_2\), we get \(Z_m=\overline{f_m(X)}\) for \(m\in U_2.\) Moreover, by Lemma 3.1 (after shrinking \(U_2\) if necessary), we can assume that \(Sing(Z_m)=Sing(Z)_m:=(m\times Y)\cap Sing(Z).\) Now it is easy to see that \(B(f_m)=B(G)_m.\)

     
  3. (3)

    We have \(\overline{f_m(X)}=Z_m\) and \(B(f_m)= B(G)_m\) for \(m\in U_2\). Now apply Lemma 3.2 with \(X=U_2\times Y,\) \(W=(U_2\times Y)\cap Z\), \(N=(U_2\times Y)\cap B(G)\) and \(f: U_1\times Y\ni (m,y)\mapsto m\in U_1.\)

     
\(\square \)

Now we are ready to prove our main result:

Theorem 3.5

Let XY be smooth affine irreducible varieties. Every algebraic family \({\mathcal {F}}\) of polynomial mappings from X to Y contains only a finite number of topologically non-semi-equivalent (non-equivalent) generically-finite (proper) mappings.

Proof

The proof is by induction on \(\dim M.\) We can assume that M is affine, irreducible and smooth. Indeed, M can be covered by a finite number of affine subsets \(M_i\), and we can consider the families \({\mathcal {F}}_{|M_i}\) separately. For the same reason we can assume that M is irreducible. Finally dim \(M{\setminus } Reg(M)<\) dim M and we can use induction to reduce the general case to the smooth one.

Assume that M is smooth and affine. If \(\mu ({\mathcal {F}})=0\), then \({\mathcal {F}}\) does not contain any generically-finite mapping. Hence we can assume that \(\mu ({\mathcal {F}})=k>0.\) By Lemma 3.4 there is a non-empty open subset \(U\subset M\) such that for every \(m_1, m_2\in U\) we have
  1. (1)

    \(\mu (f_{m_1})=\mu (f_{m_2})=k,\)

     
  2. (2)

    The pairs \((\overline{f_{m_1}(X)}, B(f_{m_1}))\) and \((\overline{f_{m_2}(X)}, B(f_{m_2}))\) are equivalent by means of a homeomorphism, i.e., there is a homeomorphism \(\Psi : Y\rightarrow Y\) such that \(\Psi (\overline{f_{m_1}(X)})=\overline{f_{m_2}(X)}\) and \(\Psi (B(f_{m_1}))=B(f_{m_2}).\)

     
Fix a pair \(Q=\overline{f_{m_0}(X)}, B=B(f_{m_0})\) for some \(m_0\in U_3.\) For \(m\in U_3\) the mapping \(f_m: X\rightarrow Y\) is topologically equivalent to the continuous mapping \(f'_m=\Psi _m\circ f_m\) with \(\overline{f'_{m}(X)}=Q\) and \(B(f'_m)=B\) (Lemma 3.4). Every mapping \(f'_m\) induces a topological covering \(f'_m :X{\setminus } {f'_m}^{-1}(B)=P_{f'_m}\rightarrow R=Q{\setminus } B\). Take a point \(a\in R\) and let \(a_{f'_m}\in {f'_m}^{-1}(a).\) We have an induced homomorphism
$$\begin{aligned} f_* :\pi _1(P_{f'_m}, a_{f'_m})\rightarrow \pi _1(R,a). \end{aligned}$$
Denote \(H_f=f_*(\pi _1(P_f, a_f))\) and \(G=\pi _1(R,a).\) 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 \(G:=\pi _1(Q{\setminus } B, a)\) is finitely generated. Let us recall the following result of M. Hall (see [4]):

Lemma 3.6

Let G be a finitely generated group and let k be a natural number. Then there are only a finite number of subgroups \(H\subset G\) such that \([G:H]=k.\)

By Lemma 3.6 there are only a finite number of subgroups \(H_1,..., H_r\subset G\) with index k. Choose generically-finite (proper) mappings \(f_i=f'_{m_i}=\Psi _i\circ f_{m_i} : X\rightarrow Y\) such that \(H_{f_i}=H_i\) (of course only if such a mapping \(f_i\) does exist). We show that every generically-finite (proper) mapping \(f'_m\) (\(m\in U\)) is semi-equivalent (equivalent) to one of mappings \(f_i.\)

Indeed, let \(H_{f'_m}=H_{f_i}\) (here \(f'_m=\Psi _m\circ f_m\)). We show that \(f'_m:=f\) is equivalent to \(f_i.\) Let us consider two coverings \(f: (P_f, a_f)\rightarrow (R,a)\) and \(f_i: (P_{f_i}, a_{f_i})\rightarrow (R,a).\) Since \(f_*(\pi _1(P_f, a_f))={f_i}_*(\pi _1(P_{f_i}, a_{f_i}))\) we can lift the covering f to a homeomorphism \(\phi : P_f\rightarrow P_{f_i}\) such that following diagram commutes:
Hence for generically-finite mappings we have
$$\begin{aligned} (\Psi _i)^{-1}\circ \Psi _m\circ f_{m}\circ \phi ^{-1}|_U=f_{m_i}|_U, \end{aligned}$$
where \(V=X{\setminus } f_{m}^{-1}(B(f_{m}))\) and \(U=X{\setminus } f_{m_i}^{-1}(B(f_{m_i}))\). Hence \(f_{m}\) is semi-equivalent to \(f_{m_i}.\)

In the case of proper mappings we show additionally that the mapping \(\phi \) can be extended to a continuous mapping \(\Phi \) on the whole of X. Indeed, take a point \(x\in f^{-1}(B)\) and let \(y=f(x).\) The set \(f_i^{-1}(y)=\{ b_1,..., b_s\}\) is finite. 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 \(f_i^{-1}(V(r))\subset \bigcup ^s_{j=1} W_i(r).\) Now take a small connected neighborhood \(P_x(r)\) of x such that \(f(P_x(r))\subset V(r).\) The set \(P_x(r){\setminus } f^{-1}(B)\) is still connected and it is transformed by \(\phi \) into one particular set \(W_{i_0}(r).\) We take \(\Phi (x)=b_{i_0}.\) It is easy to see that the mapping \(\Phi \) so defined is a continuous extension of \(\phi .\) In fact \(\phi (P_x(r){\setminus } f^{-1}(B))\) shrinks to \(b_{i_0}\) if r goes to 0. Moreover, we still have \(f=f_i\circ \Phi .\)

In a similar way the mapping \(\Lambda \) determined by \(\phi ^{-1}\) is continuous. It is easy to see that \(\Lambda \circ \Phi =\Phi \circ \Lambda =identity\), hence \(\Phi \) is a homeomorphism. Consequently, the mapping \(f_i\circ \Phi =\Psi _i\circ f_{m_i}\circ \Phi \) is equal to \(f=\Psi _m\circ f_{m}.\) Finally, we get
$$\begin{aligned} (\Psi _i)^{-1}\circ \Psi _m\circ f_m\circ \Phi ^{-1}=f_{m_i}. \end{aligned}$$
This means that the family \({\mathcal {F}}_{|U}\) contains only a finite number of topologically non-semi-equivalent (non-equivalent) generically-finite (proper) mappings. In fact, the number of topological semi-types (types) of generically-finite (proper) mappings in \({\mathcal {F}}_{|U}\) is bounded by the number of subgroups of G of index \(\mu ({\mathcal {F}}).\)

Let \(T=M{\setminus } U.\) Hence dim \(T<\) dim M. By the induction the family \({\mathcal {F}}_{|T}\) also contains only a finite number of topologically non-semi-equivalent (non-equivalent) generically-finite (proper) mappings. Consequently so does \({\mathcal {F}}\). \(\square \)

Corollary 3.7

There is only a finite number of topologically non-semi-equivalent (non-equivalent) generically-finite (proper) polynomial mappings \(f: {\mathbb {C}}^n\rightarrow {\mathbb {C}}^m\) of a bounded algebraic degree. \(\square \)

4 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:

Lemma 4.1

Let \(Y={\mathbb {R}}^{n}\) and let \(Z\subset Y\) be a linear subspace of Y. Fix \(\epsilon >0\) and take \(\eta <\epsilon .\) Let \(B( 0, \eta )\) be a ball of radius \(\eta .\) Let \(\gamma : I\ni t\mapsto \gamma (t)\in B( 0, \eta )\cap Z\) be a smooth path. Then there exists a continuous family of diffeomorphisms \(\Phi _t : Y\rightarrow Y\), \(t\in [0,1]\) such that
  1. (1)

    \(\Phi _t(\gamma (t))=\gamma (0) \ \mathrm{{ and}}\ \ \Phi _t(z)=z \ \ \mathrm{{ for}}\ \ \Vert z\Vert \ge \epsilon .\)

     
  2. (2)

    \(\Phi _0=\mathrm{identity}.\)

     
  3. (3)

    \(\Phi _t(Z)=Z.\)

     

Proof

Let \(v_t=\gamma (0)-\gamma (t)\in T{\mathbb {R}}^n.\) We construct a family of diffeomorphisms \(\Phi _t\), which are interpolation between translation \(x\rightarrow x+v_t\) and identity.

Let \(\sigma : Y\rightarrow [0, 1]\) be a differentiable function such that \(\sigma =1\) on \(B(0, \eta )\) and \(\sigma =0\) outside \(B(0, \epsilon ).\) Define a vector field \(V(x)=\sigma (x)v_t.\) Integrating this vector field we get desired diffeomeorphisms \(\Phi _t,\) for any t. \(\square \)

Corollary 4.2

Let Y be a smooth manifold and Z be a smooth submanifold. For every point \(a\in Z\) and every open neighborhood \(V_a\) of the point a, there is an open connected neighborhood \(U_a\) of the point a, such that:
  1. (a)

    \(\overline{U_a}\subset V_a\),

     
  2. (b)

    if \(\gamma : I\ni t\mapsto \gamma (t)\in U_a\cap Z\) is a smooth path, then there is a continuous family of diffeomorphism \(\psi _t : Y\rightarrow Y\), \(t\in [0,1]\) such that

     
  1. (1)

    \(\psi _t(\gamma (t))= \gamma (0),\)

     
  2. (2)

    \(\psi _t(x)=x\) for \(x\not \in V_a\) and \(\psi _0=\mathrm{identity},\)

     
  3. (3)

    \(\psi _t(Z)=Z.\)

     

Now we are in a position to prove:

Theorem 4.3

Let XY be smooth affine irreducible varieties. Let \({\mathcal {F}}: M\times X\rightarrow Y\) be an algebraic family of proper polynomial mappings from X to Y. Assume that M is an irreducible variety. Then there exists a Zariski open dense subset \(U\subset M\) such that for every \(m,m'\in U\) mappings \(f_m\) and \(f_{m'}\) are topologically equivalent.

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 \(U\subset M\) such that for every \(m_1, m_2\in U\) we have
  1. (1)

    \(\mu (f_{m_1})=\mu (f_{m_2})=k,\)

     
  2. (2)

    The pairs \((\overline{f_{m_1}(X)}, B(f_{m_1}))\) and \((\overline{f_{m_2}(X)}, B(f_{m_2}))\) are equivalent by means of a homeomorphism, i.e., there is a homeomorphism \(\Psi : Y\rightarrow Y\) such that \(\Psi (\overline{f_{m_1}(X)})=\overline{f_{m_2}(X)}\) and \(\Psi (B(f_{m_1}))=B(f_{m_2}).\)

     
Fix a pair \((Q=\overline{f_{m_0}(X)}, B=B(f_{m_0}))\) for some \(m_0\in U.\) For \(m\in U\) the mappings \(f_m\) and \(f_{m_0}\) can be connected by a continuous path \(f_t, f_0=f_{m_0}, f_1=f_m.\) Moreover we have also a continuous family of homeomorphisms \(\Psi _t : Y\rightarrow Y\) such that \(\Psi _t(\overline{f_{t}(X)})=\overline{f_{0}(X)}\) and \(\Psi (B(f_{t}))=B(f_{0}).\) It is enough to prove that mappings \(F_t=\Psi _t\circ f_t\) are locally (in the sense of parameter t) equivalent.

(1) First step of the proof. Let \(C_t\subset X\) denotes the preimage by \(F_t\) of the set B (in fact \(C_t=f_t^{-1}(B(f_t)\)) and put \(X_t=X{\setminus } C_t.\) Put \(Q':=Q{\setminus } B.\)Assume that for all mappings \(F_t\) there is a point \(a\in (X{\setminus } \bigcup _{t\in I} C_t)\) such that for all \(t\in I\) we have \(F_t(a)=b.\)

We have an induced homomorphism \({G_t}_*: \pi _1(X_t,a)\rightarrow \pi _1(Q',b)\). We show that the subgroup \({F_t}_*( \pi _1(X_t,a))\subset \pi _1(Q',b)\) does not depend on t.

Indeed let \(\gamma _1,..., \gamma _s\) be generators of the group \(\pi _1(X_{t_0}, a).\) Let \(U_i\) be an open relatively compact neighborhoods of \(\gamma _i\) such that \(\overline{U_i}\cap C_{t_0}=\emptyset .\) For sufficiently small number \(\epsilon >0\) and \(t\in (t_0-\epsilon ,t_0+\epsilon )\) we have \(\overline{U_i}\cap C_{t}=\emptyset .\) Let \(t\in (t_0-\epsilon ,t_0+\epsilon )\). Note that the loop \(F_t(\gamma _i)\) is homotopic with the loop \(F_{t_0}(\gamma _i).\) In particular the group \({F_{t_0}}_*( \pi _1(X_{t_0},a))\) is contained in the group \({F_t}_*( \pi _1(X_t,a)).\) Since they have the same (finite!) index in \(\pi _1(Y',b)\) they are equal. This means that the subgroup \({G_t}_*( \pi _1(X_t,a))\subset \pi _1(Y',b)\) is locally constant, hence it is constant.

Let us consider two coverings \(F_t: (X_t, a)\rightarrow (Q',b)\) and \(F_0: (X_0, a)\rightarrow (Q',b)\). Since \({F_t}_* \pi _1(X_t, a)={F_0}_*\pi _1 (X_0, a)\) we can lift the covering \(F_t\) to a homeomorphism \(\phi _t: X_t\rightarrow X_0.\) As before we can extend the homeomorphism \(\phi _t\) to the homeomorphism \(\Phi _t: X\rightarrow X\), such that \(F_0\circ \Phi _t =F_t.\)

(2) The general case. Now we can prove Theorem 4.3. Since in general there is no a point \(a\in (X{\setminus } \bigcup _{t\in I} C_t)\) such that for all \(t\in I\) we have \(F_t(a)=b,\) we have to modify our construction.

First we prove that for every \(t_0\in I\) there exists \(\epsilon >0\) and a family of homeomorphisms \(\Phi _t: X\rightarrow X\), \(t\in (t_0-\epsilon , t_0+\epsilon )\) such that \(F_t=F_{t_0}\circ \Phi _t\) for \(t\in (t_0-\epsilon , t_0+\epsilon ).\) Take a point \(a\in X_{t_0}\) and choose \(\epsilon >0\) so small that \(a\in X_t\) for \(t\in (t_0-\epsilon , t_0+\epsilon )\). Put \(\gamma (t)\ni t\mapsto F_t(a)\in Y'.\) We can take \(\epsilon \) so small that the hypothesis of Corollary 4.2 is satisfied. Applying Corollary 4.2 with \(Y'=Y{\setminus } B\) and \(Z=Q{\setminus } B\) we have a continuous family of diffeomeorphisms \(\psi _t: Y\rightarrow Y\) which preserves Q and B, \(t\in (t_0-\epsilon , t_0+\epsilon )\) such that \(\psi _t(F_t(a))=F_{0}(a).\) Take \(G_t=\psi _t\circ F_t.\) Arguing as in the first part of our proof all \(G_t\) are topologically equivalent for \(t\in (t_0-\epsilon , t_0+\epsilon )\). Hence also all \(F_t\) are topologically equivalent for \(t\in (t_0-\epsilon , t_0+\epsilon )\). Since \(F_t\) are locally topologically equivalent, they are topologically equivalent for every \(t\in I\). \(\square \)

Corollary 4.4

Let \(n\le m\) and let \(\Omega _n(d_1,...,d_m)\) denotes the family of all polynomial mappings \(F=(f_1,...,f_m):{\mathbb {C}}^n\rightarrow {\mathbb {C}}^m\) of a multi-degree bounded by \((d_1,...,d_m).\) Then any two general members of this family are topologically equivalent.

Proof

Indeed, it is enough to note that a generic mapping \(f\in \Omega _n(d_1,...,d_m)\) is proper. \(\square \)

Using the same method we can prove:

Theorem 4.5

Let XY be smooth affine irreducible varieties. Let \({\mathcal {F}}: M\times X\rightarrow Y\) be an algebraic family of generically-finite polynomial mappings from X to Y. Assume that M is an irreducible variety. Then there exists a Zariski open dense subset \(U\subset M\) such that for every \(m,m'\in U\) the mappings \(f_m\) and \(f_{m'}\) are topologically semi-equivalent.

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Authors and Affiliations

  1. 1.Instytut MatematycznyPolska Akademia NaukWarszawaPoland

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