On the birational geometry of conic bundles over the projective space

Let π:Z→Pn−1 be a general minimal n-fold conic bundle with a hypersurface BZ⊂Pn−1 of degree d as discriminant. We prove that if d≥4n+1 then −KZ is not pseudo-effective, and that if d=4n then none of the integral multiples of −KZ is effective. Finally, we provide examples of smooth unirational n-fold conic bundles π:Z→Pn−1 with discriminant of arbitrarily high degree.

Let (Formula presented.) be a general minimal n-fold conic bundle with a hypersurface (Formula presented.) of degree d as discriminant. We prove that if (Formula presented.), then (Formula presented.) is not pseudo-effective, and that if (Formula presented.), then none of the integral multiples of (Formula presented.) is effective. Finally, we provide examples of smooth unirational n-fold conic bundles (Formula presented.) with a discriminant of arbitrarily high degree.