A Noether-Lefschetz theorem for linked surfaces in \(\mathbb{P}{}^ 4\)
It is well-know from the Noether--Lefschetz theory that the Picard group of a ``general'' complete intersection of two hypersurfaces in \({\mathbf P}^ 4\) is generated by the hyperplane section (except for few known cases). The paper under review analyzes the same problem for the rest under a complete intersection of a fixed smooth surface \(S\). More precisely, the author proves that, if a smooth surface \(X\) is linked to \(S\) by a general complete intersection of two hypersurfaces of sufficiently high degree, then the Picard group of \(X\) is generated by the hyperplane section and \(X\cap S\). A precise estimate of how big the degrees of the hypersurfaces need to be is given in terms of the degree of \(S\). The method for the proof is to view \(X\) as the zero locus of a section of a vector bundle in the blow--up \({\mathbb{P}}^ 4\) along \(S\). The Noether-- Lefschetz type theorem for sections of regular vector bundles given by \textit{L. Ein} in Duke Math. J. 52, 689-706 (1985; Zbl 0589.14034) does not apply here. However, the same kind of techniques (theory of variations of Hodge structures) allows the author to conclude the proof.
scientific article; zbMATH DE number 57235
scientific article