Two meromorphic mappings having the same inverse images of moving hyperplanes

TL;DR

This paper proves that two meromorphic mappings sharing inverse images of moving hyperplanes with certain multiplicity conditions must be algebraically degenerated, generalizing and improving previous fixed hyperplane results.

Contribution

It extends Fujimoto's fixed hyperplane theorem to moving hyperplanes and provides explicit bounds for multiplicity levels ensuring algebraic degeneracy.

Findings

Maps are algebraically degenerated under specified conditions.

Explicit estimate for multiplicity level $l_0$ is provided.

Generalizes previous fixed hyperplane results.

Abstract

In this paper, we will show that if two meromorphic mappings $f$ and $g$ of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ have the same inverse images for $(2n+2)$ moving hyperplanes $\{a_i\}_{i=1}^{2n+2}$ with multiplicities counted to level $l_0$ then the map $f\times g$ must be algebraically degenerated over the field $\mathcal R\{a_i\}_{i=1}^{2n+2}$, where $l_0=3n^3(n+1)q(q-2)$ with $q=\binom{2n+2}{n+2}$. Our result generalizes the previous result for fixed hyperplanes case of Fujimoto and also improves his result by giving an explicit estimate for the number $l_0$.