Reducible hyperplane sections I Dedicated to the memory of our friend and colleague, Michael Schneider
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Abstract
In this article we begin the study of \hat{X}, an ndimensional algebraic submanifold of complex projective space \bm{P}<SUP>N</SUP>, in terms of a hyperplane section 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}=\bm{P}<SUP>N</SUP> of dimension at least five if the intersection of \hat{X} with some hyperplane is a union of r≥q 2 smooth normal crossing divisors \hat{A<SUB>1</SUB>}, ..., \hat{A<SUB>r</SUB>}, such that for each i, h<SUP>1</SUP>(\mathcal{O}_{\hat{A<SUB>i</SUB>}}) equals the genus g(\hat{A<SUB>i</SUB>}) of a curve section of \hat{A<SUB>i</SUB>}. Complete results are also given for the case of dimension four when r=2.
Journal

 Tokyo Sugaku Kaisya Zasshi

Tokyo Sugaku Kaisya Zasshi 51(4), 887910, 19991001
The Mathematical Society of Japan