On reducible hyperplane sections of 4-folds

抄録

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 \bm{P}¹ with A itself a scroll and B a fibre, 2) (X, L)=(\bm{P}²× \bm{P}², \mathcal{O}_{\bm{P}²× \bm{P}²}(1, 1)) with A∈|\mathcal{O}_{\bm{P}²× \bm{P}²}(1, 0)|, B∈|\mathcal{O}_{\bm{P}²× \bm{P}²}(0, 1)|, 3) X=\bm{P}_{\bm{P}²}(\mathscr{V}) where \mathscr{V}=\mathcal{O}_{\bm{P}²}(1)^{oplus 2}oplus \mathcal{O}_{\bm{P}²}(2), L is the tautological line bundle, A=P_{\bm{P}²}(\mathcal{O}_{\bm{P}²}(1)^{oplus 2}), and B∈π^*|\mathcal{O}_{\bm{P}²}(2)|, where π: X→ \bm{P}² is the scroll projection. This supplements a recent result of Chandler, Howard, and Sommese.