On the birational geometry of conic bundles over the projective space

Let $\pi:Z\rightarrow\mathbb{P}^{n-1}$ be a general minimal $n$-fold conic bundle with a hypersurface $B_Z\subset\mathbb{P}^{n-1}$ of degree $d$ as discriminant. We prove that if $d\geq 4n+1$ then $-K_Z$ is not pseudo-effective, and that if $d = 4n$ then none of the integral multiples of $-K_{Z}$ is effective. Finally, we provide examples of smooth unirational $n$-fold conic bundles $\pi:Z\rightarrow\mathbb{P}^{n-1}$ with discriminant of arbitrarily high degree.Comment: 13 pages. We thank the referee for pointing out a mistake in a statement, about a birational version of Theorem 1.1 for 3-folds, that we removed. To appear in Mathematische Nachrichte