Birationally rigid Fano hypersurfaces

Abstract

We prove that a smooth Fano hypersurface $V=V_M\subset {\mathbb{P}}^M$, $M\ge 6$, is birationally superrigid. In particular, it cannot be fibered into uniruled varieties by a non-trivial rational map and each birational map onto a minimal Fano variety of the same dimension is a biregular isomorphism. The proof is based on the method of maximal singularities combined with the connectedness principle of Shokurov and Koll\' ar.

31 pages, LATeX

References in corpus (4)

• Essentials of the method of maximal singularities
• Fano's inequality is false for a simple Cremona transformation of five-dimensional projective space
• Fano's inequality is also false for three-dimensional quadric
• Fano's inequality is a mistake

Cited by in corpus (21)

• The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group)
• Birationally rigid iterated Fano double covers
• Birational superrigidity and K-stability of singular Fano complete intersections
• Birational superrigidity and K-stability of Fano complete intersections of index one (with an appendix written jointly with Charlie Stibitz)
• Parallel Kustin--Miller unprojection with an application to Calabi--Yau geometry
• The $4n^2$-inequality for complete intersection singularities
• Birational superrigidity is not a locally closed property
• On the sharpness of Tian's criterion for K-stability
• Birationally rigid hypersurfaces
• Log canonical thresholds and Monge-Ampere masses
• A sharp lower bound for the log canonical threshold
• Multiplicities and log canonical threshold
• On factoriality of nodal threefolds
• Birational rigidity of complete intersections
• Remarks on Type III Unprojection
• Double cubics and double quartics
• Higher index Fano varieties with finitely many birational automorphisms
• Birational superrigidity and K-stability of projectively normal Fano manifolds of index one
• On birational superrigidity and conditional birational superrigidity of certain Fano hypersurfaces
• Fano hypersurfaces and their birational geometry
• On nodal sextic fivefold