Let \(X, Y\) be smooth algebraic varieties of the same dimension.
Let \(f, g \colon X \longrightarrow Y\) be finite regular
mappings. We say that \(f, g\) are equivalent if there exists a
regular automorphism \(\Phi \in \operatorname{Aut}(X)\) such that
\(f = g \circ\Phi\). Of course if \(f, g\) are equivalent, then they
have the same discriminants (i.e., the same set of critical values)
and the same geometric degree. We show, that conversely there is
only a finite number of non-equivalent finite regular mappings \(f
\colon X \rightarrow Y,\) such that the discriminant \(D(f) = V\)
and \(\mu(f ) = k\). As one of applications we show that if
\(f\colon \Bbb C^n\to\Bbb C^n\) is a proper mapping with \(D(f)=\{
x\in \Bbb C^n : x_1=0\}\), then \(f\) is equivalent to the mapping
\(g\colon \Bbb C^n \ni (x_1,\dots, x_n)\mapsto (x_1^k, x_2,\dots,
x_n)\in\Bbb C^n,\) where \(k=\mu(f).\) Moreover, if \(f\colon X\to
Y\) is a finite mapping of topological degree two, then there exists
a regular automorphism \(\Phi\colon X\to X\) which acts transitively
on fibers of \(f.\) In particular for \(n>1\) there is no finite
mappings \(f\colon \Bbb P^n\to\Bbb P^n\) of topological degree
two.We prove the same statement in the local (and sometimes global)
holomorphic situation. In particular we show that if \(f \colon
(\mathbb C^{n} , 0) \rightarrow (\mathbb C^{n} , 0)\) is a proper
and holomorphic mapping of topological degree two, then there exist
biholomorphisms \(\Psi,\Phi \colon (\mathbb C^{n} , 0) \rightarrow
(\mathbb C^{n} , 0)\) such that \(\Psi\circ f\circ\Phi(x_{1} , x_{1}
, \dots, x_{n} ) = (x_{1}^{2} , x_{2} ,\dots, x_{n})\). Moreover,
for every proper holomorphic mapping \(f \colon (\mathbb C^{n} , 0)
\rightarrow (\mathbb C^{n} , 0)\) with smooth discriminant there
exist biholomorphisms \(\Psi,\Phi \colon (\mathbb C^{n} , 0)
\rightarrow (\mathbb C^{n}, 0)\) such that \(\Psi\circ
f\circ\Phi(x_{1} , x_{1} , \dots, x_{n} ) = (x_{1}^{k} , x_{2}
,\dots, x_{n})\), where $k = \mu(f )$.
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