On finite regular and holomorphic mappings

Abstract: Let $X, Y$ be smooth algebraic varieties of the same dimension. Let $f, g : X \to Y$ be finite polynomial mappings. We say that $f, g$ are equivalent if there exists a regular automorphism $\Phi \in Aut(X)$ such that $f = g\circ \Phi$. Of course if $f, g$ are equivalent, then they have the same discriminant and the same geometric degree. We show, that conversely there is only a finite number of non-equivalent proper polynomial mappings $f : X \to Y$, such that $D(f) = V$ and $\mu(f) = k.$ We prove the same statement in the local holomorphic situation. In particular we show that if $f : (\Bbb C^n, 0) \to (\Bbb C^n, 0)$ is a proper and holomorphic mapping of topological degree two, then there exist biholomorphisms $P,Q : (\Bbb C^n, 0) \to (\Bbb C^n, 0)$ such that $P\circ f\circ Q (x_1, x_2,..., x_n) = (x_1^2, x_2, ..., x_n)$. Moreover, for every proper holomorphic mapping $f : (\Bbb C^n, 0) \to (\Bbb C^n, 0)$ with smooth discriminant there exist biholomorphisms $P,Q : (\Bbb C^n, 0) \to (\Bbb C^n, 0)$ such that $P\circ f\circ Q (x_1, x_2,..., x_n) = (x_1^k, x_2, ..., x_n)$, where $k = \mu(f).$