Birationally rigid hypersurfaces

TL;DR

This paper proves that all smooth hypersurfaces of degree N in projective N-space are birationally superrigid for N ≥ 4, confirming a long-standing conjecture and extending previous results.

Contribution

It establishes the birational superrigidity of smooth hypersurfaces in P^N for all N ≥ 4, using advanced techniques like maximal singularities and multiplier ideals.

Findings

All smooth hypersurfaces of degree N in P^N are birationally superrigid for N ≥ 4

Confirms Pukhlikov's conjecture for general N ≥ 4

Extends the known case N=4 to higher dimensions

Abstract

We prove that for N greater than or equal to 4, all smooth hypersurfaces of degree N in P^N are birationally superrigid. First discovered in the case N = 4 by Iskovskikh and Manin in a work that started this whole direction of research, this property was later conjectured to hold in general by Pukhlikov. The proof relies on the method of maximal singularities in combination with a delicate formula on restrictions of multiplier ideals.