On the unirationality of quadric bundles

We prove that a general $n$-fold quadric bundle ${\mathcal{Q}}^{n-1}\to {\mathbb{P}}^1$, over a number field, with $(-K_{{\mathcal{Q}}^{n-1}}{)}^n>0$ and discriminant of odd degree ${\delta }_{{\mathcal{Q}}^{n-1}}$ is unirational, and that the same holds for quadric bundles over an arbitrary infinite field provided that ${\mathcal{Q}}^{n-1}$ has a point, is otherwise general and $n\le 5$. As a consequence we get the unirationality of a general $n$-fold quadric bundle ${\mathcal{Q}}^h\to {\mathbb{P}}^{n-h}$ with discriminant of odd degree ${\delta }_{{\mathcal{Q}}^h}\le 3h+4$, and of any smooth $4$-fold quadric bundle ${\mathcal{Q}}^2\to {\mathbb{P}}^2$, over an algebraically closed field, with ${\delta }_{{\mathcal{Q}}^2}\le 12$.