In this article we begin the study of , an -dimensional algebraic submanifold of complex projective space , in terms of a hyperplane section which is not irreducible. A number of general results are given, including a Lefschetz theorem relating the cohomology of 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 of dimension at least five if the intersection of with some hyperplane is a union of smooth normal crossing divisors , . . . , , such that for each equals the genus of a curve section of . Complete results are also given for the case of dimension four when .
Karen A. CHANDLER. Alan HOWARD. Andrew J. SOMMESE. "Reducible hyperplane sections I." J. Math. Soc. Japan 51 (4) 887 - 910, October, 1999. https://doi.org/10.2969/jmsj/05140887