The existence of varieties whose hyperplane section is \({\mathbb{P}}^ r\)- bundle (Q809152)

There are two main results of the paper. In the first one the authors give sufficient conditions for a projective surface S and a vector bundle F on S, under which the projective bundle \({\mathbb{P}}(F)\) associated with F is an ample divisor in a smooth projective variety. - In the second one they also give sufficient conditions for a smooth projective variety S, under which any sequence \(\{A_ i\}\) of smooth projective varieties terminates, where \(A_ i\) is an ample divisor in \(A_{i+1}\), \(i=1,2,...\), and \(A_ 1\) is a \({\mathbb{P}}^ r\)-bundle over S. Although the authors work over an algebraically closed field of any characteristic, the above-mentioned results are in characteristic zero.