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On rationality of nonsingular threefolds with a pencil of Del Pezzo surfaces of degree 4

Constantin Shramov
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Abstract

We prove a criterion of nonsingularity of a complete intersection of two fiberwise quadrics in a scroll over $P^1$. As a corollary we derive the following addition to the Alexeev theorem on rationality of standard Del Pezzo fibrations of degree 4 over $P^1$: we prove that any fibration of this kind with the topological Euler characteristic -4 is rational.

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Constantin Shramov. 2007-01-18. On rationality of nonsingular threefolds with a pencil of Del Pezzo surfaces of degree 4. https://arxiv.org/abs/math/0701521

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