On rationality of nonsingular threefolds with a pencil of Del Pezzo surfaces of degree 4
Abstract
We prove a criterion of nonsingularity of a complete intersection of two fiberwise quadrics in a scroll over P1. As a corollary we derive the following addition to the Alexeev theorem on rationality of standard Del Pezzo fibrations of degree 4 over P1: we prove that any fibration of this kind with the topological Euler characteristic -4 is rational.
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