On the birational geometry of conic bundles over the projective space

Abstract

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