Finiteness problem for meromorphic mappings sharing \(n+3\) hyperplanes of \(\mathbb {P}^{n}(\mathbb {C})\) (Q2879371)

The paper under review deals with uniqueness theorems for meromorphic maps. The author improves the results of \textit{S. J. Drouilhet} [Trans. Am. Math. Soc. 265, 349--358 (1981; Zbl 0476.32034)] as follows. Let \(f:\mathbb{C}\to \mathbb{P}^n(\mathbb{C})\) be a differential non-degenerate meromorphic map and \(H_1,\ldots,H_q\) hyperplanes in general position in \(\mathbb{P}^n(\mathbb{C})\). Let \(d\) be a positive integer. We denote by \(\mathcal{G}(f, \{H_j\}_1^q, d)\) the family of differential non-degenerate meromorphic maps \(g:\mathbb{C}\to \mathbb{P}^n(\mathbb{C})\) such that NEWLINE\[NEWLINE\min\{\nu^0_f(H_j), d\}=\min\{\nu^0_g(H_j), d\}\quad {\text{ for all}}\,\, jNEWLINE\]NEWLINE and NEWLINE\[NEWLINEf=g \quad {\text{ on}} \quad \bigcup_j^q f^{-1}(H_j).NEWLINE\]NEWLINE Here we denote by \(\nu^0_f(H_j)\) the pull-back divisor of \(H_j\) by \(f\). The following theorem is the main results in the paper: If \(q=n+3\), then \(\mathcal{G}(f, \{H_j\}_1^q, 2)\) contains at most two elements.