Reducible hyperplane sections I

In this article we begin the study of X, an n-dimensional algebraic submanifold of complex projective space P(powerN) in terms of a hyperplane section A which is not irreducible. A number of general results are given, including a Lefschetz theorem relating the cohornology of 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 X c P(powerN) of dimension at least five if the intersection of X with some hyperplane is a union of( r > 2 or r =2) smooth normal crossing divisors A1, . . ., Ar, such that for each i, h ʹ (0Ai ) equals the genus g(Ai) of a curve section of Ai. Complete results are also given for the case of dimension four when r = 2