# zbMATH — the first resource for mathematics

$$\Pi^ r\mathbf P^ 1$$-bundle from which a surjective morphism to $$\Pi^ m\mathbb{P}^ 1$$ exists. (English) Zbl 0788.14038
Some years ago 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.
##### MSC:
 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14M20 Rational and unirational varieties 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
Full Text: