Ruled threefolds homeomorphic to \({\mathbb{P}}^ 2\times {\mathbb{P}}^ 1\) and their divisors (Q1262919)

Complex projective threefolds \(X=\mathbb{P}(E)\), where E is a topologically trivial rank-2 holomorphic vector bundle on \(\mathbb{P}^ 2\), are studied. Let \(p: X\to\mathbb{P}^ 2\) be the \(\mathbb{P}^ 1\)-bundle projection. The author shows that, except for \(\mathbb{P}^ 2\times \mathbb{P}^1\), p does not admit a holomorphic section, but there exists an irreducible surface \(S\subset X\) such that \(p|_ S:\quad S\to \mathbb{P}^ 2\) is a modification map, which is biholomorphic on \(S\cap p^{-1}(\mathbb{P}^ 2-Z)\), where \(Z\subset \mathbb{P}^2\) is a locally complete intersection of dimension 0 and degree \(d^ 2\), with d any positive integer. Then according to the configuration of Z and some related invariants, she finds conditions for a line bundle on X to have sections, to be ample, to be globally generated and to be very ample. In some instances, when Z consists of simple points, necessary and sufficient conditions for very ampleness are given.