On surfaces with zero vanishing cycles
type: Item
instance of: scholarly article
title: On surfaces with zero vanishing cycles
zbMATH Open document ID: 1305.14003
author: Serge Lvovski
published in: Manuscripta Mathematica
publication date: 2014-11-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1305.2205
review text: 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.
Mathematics Subject Classification ID: 14D05
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14N99
zbMATH DE Number: 6369207
zbMATH Keywords: surface, smooth, projective, algebraic
zbMATH Keywords: Lefschetz pencil
zbMATH Keywords: triviality of vanishing cycles
reviewed by: Antonio Lanteri
MaRDI profile type: Publication
OpenAlex ID: W2113385578
arXiv ID: 1305.2205
cites work: Seminar of algebraic geometry du Bois-Marie 1963--1964. Topos theory and étale cohomology of schemes (SGA 4). Vol. 1: Topos theory. Exp. I--IV
cites work: La conjecture de Weil. II
cites work: La conjecture de Weil. I
cites work: Cohomologie des variétés algébriques
cites work: Q4143433
cites work: Q3701560
cites work: Q3735865
cites work: Q4740678
cites work: Algebraic surfaces containing a smooth curve of genus q(S) as an ample divisor
cites work: Q3698938
cites work: On Curves and Surfaces with Projectively Equivalent Hyperplane Sections
cites work: Q4145822
cites work: On the 2-connectedness of very ample divisors on a surface
cites work: Surfaces with zero Lefschetz cycles
DOI: 10.1007/S00229-014-0672-Z
description: scientific article; zbMATH DE number 6369207