On the unirationality of quadric bundles

TL;DR

This paper proves that general quadric bundles over certain fields are unirational under specific conditions related to their degree and discriminant, extending known results to higher dimensions and particular cases.

Contribution

It establishes new unirationality results for general quadric bundles over number fields and infinite fields, with explicit bounds on discriminant degrees and dimension.

Findings

Unirationality of general $n$-fold quadric bundles over number fields with positive canonical class.

Unirationality of quadric bundles over infinite fields with points, under certain conditions and dimension constraints.

Explicit bounds on discriminant degrees for unirationality of quadric bundles over algebraically closed fields.

Abstract

We prove that a general $n$-fold quadric bundle $\mathcal{Q}^{n-1}\rightarrow\mathbb{P}^{1}$, 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 $n\leq 5$. As a consequence we get the unirationality of a general $n$-fold quadric bundle $\mathcal{Q}^{h}\rightarrow\mathbb{P}^{n-h}$ with discriminant of odd degree $\delta_{\mathcal{Q}^{h}}\leq 3h+4$, and of any smooth $4$-fold quadric bundle $\mathcal{Q}^{2}\rightarrow\mathbb{P}^{2}$, over an algebraically closed field, with $\delta_{\mathcal{Q}^{2}}\leq 12$.