Varieties which have two projective space bundle structures (Q1072603)

One of the most basic structures a variety may have is that of a \({\mathbb{P}}^ r\)-ruling. The interesting article under review deals with the general problem of classifying those smooth projective varieties M which admit two different such structures: more precisely which can be viewed as \({\mathbb{P}}^ r\)-bundles over some base variety in two different ways. Under a mild additional hypothesis this problem is solved whenever the base varieties are projective spaces, as well as for \(\dim (M)=3\) in general.