\(\Pi^ r\mathbf P^ 1\)-bundle from which a surjective morphism to \(\Pi^ m\mathbb{P}^ 1\) exists (Q1205450)

Some years ago \textit{E. Sato} [J. Math. Kyoto Univ. 25, 445-457 (1985; Zbl 0587.13004)] studied smooth projective varieties which admit two different projective space bundle structures. -- In the present paper the author deals with the similar problem to classify smooth projective varieties with two different \(\mathbb{P}^ 1 \times \cdots \times \mathbb{P}^ 1\)-bundle structures over some \(\mathbb{P}^ 1 \times \cdots \times \mathbb{P}^ 1\). More generally, he investigates varieties which admit a surjective morphism to some \(\mathbb{P}^ 1 \times \cdots \times \mathbb{P}^ 1\) and have the structure of \(\mathbb{P}^ 1 \times \cdots \times \mathbb{P}^ 1\)-bundle over a product of projective spaces and rational surfaces. The result is that the variety considered is isomorphic to the product of the targets of the two given morphisms.