Reducible hyperplane sections I

Abstract

In this article we begin the study of $\hat{X}$, an $n$-dimensional algebraic submanifold of complex projective space $P^N$, in terms of a hyperplane section $\hat{A}$ which is not irreducible. A number of general results are given, including a Lefschetz theorem relating the cohomology of $\hat{X}$ 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 $\hat{X}\subset P^N$ of dimension at least five if the intersection of $\hat{X}$ with some hyperplane is a union of $r\ge 2$ smooth normal crossing divisors $A_1$, . . . , $A_r$, such that for each $i,$$h^1\left({\mathcal{O}}_{{\hat{A}}_i}\right)$ equals the genus $g\left(A_i\right)$ of a curve section of $A_i$. Complete results are also given for the case of dimension four when $r=2$.