On the unirationality of quadric bundles

We prove that a general (n−1)-fold quadric bundle Qn−1→P1, over a number field, with (−KQjavax.xml.bind.JAXBElement@34e98f3d)n>0 and discriminant of odd degree δQjavax.xml.bind.JAXBElement@6d762360 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 δQjavax.xml.bind.JAXBElement@7a8b1602≤12.