On surfaces with zero vanishing cycles
Smooth non-degenerate projective surfaces \(X \subset \mathbb P^r\), over \(\mathbb C\), such that the vanishing cycles of any Lefschetz pencil of \(X\) are zero have been classified by \textit{F. L. Zak} [Math. Notes 13, 520--525 (1973); translation from Mat. Zametki 13, 869--880 (1973; Zbl 0275.14019)]. They are: \(\mathbb P^2\), any \(\mathbb P^1\)-bundle over a smooth curve embedded by the tautological line bundle, the Veronese surface in \(\mathbb P^5\), and any isomorphic projection of it into \(\mathbb P^4\). Zak's method of proof, however, does not seem to be applicable to the case of an algebraically closed field of finite characteristic. The paper under review contains an alternative proof, valid in any characteristic. The nice idea of the author is to use an argument of \textit{A. van de Ven} [Duke Math. J. 46, 403--407 (1979; Zbl 0458.14003)] to prove, in a purely geometric way, that if \(X\) admits a reducible, reduced, hyperplane section, smooth except at a single point where it has an ordinary quadratic singularity, and \(X\) is not \(\mathbb P^2\), then \(X\) is one of the last three surfaces in the above list.
scientific article; zbMATH DE number 6369207
scientific article