Reducible hyperplane sections. I. (English) Zbl 0973.14018

Chandler, Karen A.; Howard, Alan; Sommese, Andrew J.

J. Math. Soc. Japan 51, No. 4, 887-910 (1999).

Summary: In this article we begin the study of \(\widehat X\), an \(n\)-dimensional algebraic submanifold of complex projective space \(\mathbb{P}^N\), in terms of a hyperplane section \(\widehat A\) which is not irreducible. A number of general results are given, including a Lefschetz theorem relating the cohomology of \(\widehat X\) to the cohomology of the components of a normal crossing divisor which is ample, and a strong extension theorem for divisors which are high index Fano fibrations. As a consequence we describe \(\widehat X\subset\mathbb{P}^N\) of dimension at least five if the intersection of \(\widehat X\) with some hyperplane is a union of \(r\geq 2\) smooth normal crossing divisors \(\widehat A_1,\dots,\widehat A_r\), such that for each \(i\), \(h^1({\mathcal O}_{\widehat A_i})\) equals the genus \(g(\widehat A_i)\) of a curve section of \(\widehat A_i\). Complete results are also given for the case of dimension four when \(r=2\).

MSC:

14J40	\(n\)-folds (\(n>4\))
14C20	Divisors, linear systems, invertible sheaves
14J35	\(4\)-folds
14C17	Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry

Keywords:

hyperplane section; Lefschetz theorem; normal crossing divisor