On the birational geometry of conic bundles over the projective space

Alex Massarenti and Massimiliano Mella

Mathematische Nachrichten, 2024, vol. 297, issue 4, 1208-1220

Abstract: Let π:Z→Pn−1$\pi :Z\rightarrow \mathbb {P}^{n-1}$ be a general minimal n$n$‐fold conic bundle with a hypersurface BZ⊂Pn−1$B_Z\subset \mathbb {P}^{n-1}$ of degree d$d$ as discriminant. We prove that if d≥4n+1$d\ge 4n+1$, then −KZ$-K_Z$ is not pseudo‐effective, and that if d=4n$d = 4n$, then none of the integral multiples of −KZ$-K_{Z}$ is effective. Finally, we provide examples of smooth unirational n$n$‐fold conic bundles π:Z→Pn−1$\pi :Z\rightarrow \mathbb {P}^{n-1}$ with a discriminant of arbitrarily high degree.

Date: 2024
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