On the birational geometry of conic bundles over the projective space

TL;DR

This paper investigates the birational geometry of general minimal conic bundles over projective space, establishing conditions on the discriminant degree that influence the effectivity of the anti-canonical divisor and providing examples of unirational bundles with high-degree discriminants.

Contribution

It proves that for discriminant degree $d \,\geq\, 4n+1$, the anti-canonical divisor is not pseudo-effective, and for $d=4n$, no multiple of it is effective, plus constructs examples of high-degree discriminant unirational bundles.

Findings

If $d \geq 4n+1$, then $-K_Z$ is not pseudo-effective.

If $d=4n$, then no multiple of $-K_Z$ is effective.

Existence of unirational conic bundles with arbitrarily high discriminant degree.

Abstract

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.