On reducible hyperplane sections of 4-folds

Abstract

We describe 4-dimensional complex projective manifolds X admitting a simple normal crossing divisor of the form A+B among their hyperplane sections, both components A and B having sectional genus zero. Let L be the hyperplane bundle. Up to exchanging the two components, (X,L,A,B) is one of the following: 1)(X,L) is a scroll over P1 with A itself a scroll and B a fibre, 2)(X,L)=(P2×P2,OP2×P2(1,1)) with A∈|OP2×P2(1,0)|,B∈|OP2×P2(0,1)|,3)X=PP2(V) where V=OP2(1)⊕2⊕OP2(2), L is the tautological line bundle, A=PP2 (OP2(1)⊕2, and B∈π*|OP2(2)| , where π : X→P2 is the scroll projection. This supplements a recent result of Chandler, Howard, and Sommese.