On the unirationality of quadric bundles

Abstract

We prove that a general (n−1)-fold quadric bundle Qn−1→P1, over a number field, with (−KQn−1)n>0 and discriminant of odd degree δQn−1 is unirational, and that the same holds for quadric bundles over an arbitrary infinite field provided that Qn−1 has a point, is otherwise general and n≤5. As a consequence we get the unirationality of any smooth quadric surface bundle Q2→P2, over an algebraically closed field, with δQ2≤12.