Reducible hyperplane sections I

In this article we begin the study of (X) over cap, an n-dimensional algebr aic submanifold of complex projective space P-N, in terms of a hyperplane s ection (A) over cap which is not irreducible. A number of general results a re given, including a Lefschetz theorem relating the cohomology of (X) over cap 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) over cap subset of P-N o f dimension at least five if the intersection of (X) over cap with some hyp erplane is a union of r greater than or equal to 2 smooth normal crossing d ivisors (A) over cap(1),...,(A) over cap(r), such that for each i, h(1) (O- (A) over cap i) equals the genus g((a) over cap(i)) of a curve section of ( A) over cap(i). Complete results are also given for the case of dimension f our when r = 2.