an:00147306
Zbl 0788.14038
Ohno, Masahiro
\(\Pi^ r\mathbf P^ 1\)-bundle from which a surjective morphism to \(\Pi^ m\mathbb{P}^ 1\) exists
EN
Geom. Dedicata 44, No. 3, 335-347 (1992).
00010595
1992
j
14J60 14M20 14F05
projective bundle; ruling; Hilbert scheme; Brauer group; different bundle structures over varieties
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.
B.Kreu??ler (Kaiserslautern)
Zbl 0587.13004