On finite regular and holomorphic mappings

TL;DR

This paper classifies finite regular and holomorphic mappings between algebraic varieties, showing finiteness of equivalence classes under automorphisms for fixed discriminant and degree, and characterizing mappings of degree two as quadratic monomials.

Contribution

It establishes the finiteness of non-equivalent proper polynomial and holomorphic mappings with given discriminant and degree, and provides explicit normal forms for degree two holomorphic mappings.

Findings

Finite number of non-equivalent proper polynomial mappings with fixed discriminant and degree.

Proper holomorphic mappings of degree two are equivalent to quadratic monomials under biholomorphisms.

Mappings with smooth discriminant can be normalized to simple monomials.

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 $k = \mu(f).$