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 $P^1$ with $A$ itself a scroll and $B$ a fibre, 2)$(X,L)=(P^2\times P^2,{\mathcal{O}}_{P^2\times P^2}(1,1\left)\right)$ with $A\in \left|{\mathcal{O}}_{P^2\times P^2}\right(1,0\left)\right|,$$B\in \left|{\mathcal{O}}_{P^2\times P^2}\right(0,1\left)\right|,$$3)X=P_{P^2}(\mathcal{V})$ where $\mathcal{V}={\mathcal{O}}_{P^2}(1{)}^{\oplus 2}\oplus {\mathcal{O}}_{P^2}(2)$, $L$ is the tautological line bundle, $A=P_{P^2}$ $({\mathcal{O}}_{P^2}{(1)}^{\oplus 2}$, and $B\in {\pi }^{*}$$\left|{\mathcal{O}}_{P^2}\right(2\left)\right|$ , where $\pi$ : $X\to P^2$ is the scroll projection. This supplements a recent result of Chandler, Howard, and Sommese.