On the rationality of non-singular threefolds with a pencil of Del Pezzo surfaces of degree 4

摘要

A criterion for the non-singularity of a complete intersection of two fibrewise quadrics in P-p1 (O(d(1)) circle plus...circle plus O (d(5))) is obtained. The following addition to Alexeev's theorem on the rationality of standard Del Pezzo fibrations of degree 4 over P-1 is deduced as a consequence: each fibration of this kind with topological Euler characteristic chi(X) = -4 is proved to be rational.