Birational geometry of Fano direct products

Abstract

We prove birational superrigidity of direct products $V=F_1\times ...\times F_K$ of primitive Fano varieties of the following two types: either $F_i\subset {\mathbb{P}}^M$ is a general hypersurface of degree $M$, $M\ge 6$, or $F_i\stackrel{\sigma }{\to }{\mathbb{P}}^M$ is a general double space of index 1, $M\ge 3$. In particular, each structure of a rationally connected fiber space on $V$ is given by a projection onto a direct factor. The proof is based on the connectedness principle of Shokurov and Koll\' ar and the technique of hypertangent divisors.

38 pages, LaTeX. This is the final enlarged version of the paper

Cited by in corpus (21)

• Log canonical thresholds of certain Fano hypersurfaces
• Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces
• Log canonical thresholds of smooth Fano threefolds. With an appendix by Jean-Pierre Demailly
• Birationally rigid varieties with a pencil of Fano double covers. III
• Birational geometry of del Pezzo fibrations with terminal quotient singularities
• Rationally connected non-Fano type varieties
• Super-rigid Affine Fano Varieties
• Birationally rigid Fano cyclic covers
• Birational geometry of Fano double covers
• The minimal log discrepancy
• On the sharpness of Tian's criterion for K-stability
• Log canonical thresholds on Gorenstein canonical del Pezzo surfaces
• Alpha invariants of birationally rigid Fano threefolds
• On the global log canonical threshold of Fano complete intersections
• Birational geometry of singular Fano varieties
• Two local inequalities
• Canonical and log canonical thresholds of Fano complete intersections
• Remarks on Galois rational covers
• Birational geometry of Fano double spaces of index two
• Exceptional del Pezzo hypersurfaces
• Worst singularities of plane curves of given degree