Projective manifolds whose topology is strongly reflected in their hyperplane sections
Let \(X\subset {\mathbb{P}}^ n\) be a complex connected closed smooth algebraic variety of dimension \(k\geq 2,\) let H be its general hyperplane section, \(i: H\to X\) being the inclusion. By Lefschetz' theorem the induced map on rational homology \(i_{*q}: H_ q(H)\to H_ q(X)\) is an isomorphism if \(q\leq k-2\) and is surjective if \(q=k-1\). The present paper is devoted to the study of the class \({\mathcal L}_ k\) of pairs (X,H) like above, such that \(i_{*k-1}\) is an isomorphism. This class is interesting, inasmuch as it is related to other geometrically relevant classes, like the class of varieties with degenerate dual variety and the class of scrolls. The known classification of \({\mathcal L}_ 2, {\mathcal L}_ 3, {\mathcal L}_ 5\) is briefly reviewed, and some remarks are added about \({\mathcal L}_ 4\). Then a few results about scrolls are proved: among other things it is shown that if X is a scroll in r-planes of dimension \(r+s\), with \(r\geq s\), then its defect (which is the measure of degeneracy of the dual variety) is exactly r-s, thus generalizing a theorem of A. Landmann and M. Reid to the effect that r-s is a lower bound for the defect.
scientific article; zbMATH DE number 3997970
scientific article