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 $f:({\mathbb{C}}^n,0)\to ({\mathbb{C}}^n,0)$ is a proper and holomorphic mapping of topological degree two, then there exist biholomorphisms $P,Q:({\mathbb{C}}^n,0)\to ({\mathbb{C}}^n,0)$ 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 $f:({\mathbb{C}}^n,0)\to ({\mathbb{C}}^n,0)$ with smooth discriminant there exist biholomorphisms $P,Q:({\mathbb{C}}^n,0)\to ({\mathbb{C}}^n,0)$ such that $P\circ f\circ Q(x_1,x_2,...,x_n)=(x_1^k,x_2,...,x_n)$, where $k=\mu (f).$