On the family of holomorphic mappings into projective space with lacunary hypersurfaces (Q810211)

The authors study the behavior of sequences of holomorphic mappings \(D^ k\to {\mathbb{P}}^ n\setminus A\) \((D^ k=k\)-polydisk, \({\mathbb{P}}^ n=complex\) projective n-space, \(A=hypersurface\) with \(\ell\) \((\ell \geq n+2)\) distinct irreducible components]. The following results are presented: if the ``rank'' of \(({\mathbb{P}}^ n,A)\) is n then \(\nearrow^ n\setminus A\) is tautly imbedded modulo some algebraic set B in \({\mathbb{P}}^ n\); for \(n=2\), if the number of non-hyperbolic curves in \({\mathbb{P}}^ 2\) with respect to A is finite then there is a curve S in \({\mathbb{P}}^ 2\) such that \({\mathbb{P}}^ 2\setminus A\) is tautly imbedded modulo S in \({\mathbb{P}}^ 2\). Consequences are drawn on nonhyperbolic points and on the existence of certain regular rational functions on \({\mathbb{P}}^ 2\setminus A\).