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\(\Pi^ r\mathbf P^ 1\)-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.
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)
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