On the birational geometry of conic bundles over the projective space

Abstract

Let $\pi :Z\to {\mathbb{P}}^{n-1}$ 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 $d\ge 4n+1$ 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 $\pi :Z\to {\mathbb{P}}^{n-1}$ with discriminant of arbitrarily high degree.

arXiv comment13 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 Nachrichten