On birational superrigidity and conditional birational superrigidity of certain Fano hypersurfaces

TL;DR

This paper establishes conditions under which certain Fano hypersurfaces and complete intersections are birationally superrigid, expanding the classification of these rigid varieties with specific singularities and base locus conditions.

Contribution

It proves birational superrigidity for a broad class of singular Fano hypersurfaces and introduces conditions for conditional superrigidity of smooth Fano hypersurfaces and complete intersections.

Findings

Birational superrigidity of hypersurfaces with quadratic singularities of rank ≥ N-s.

Completion of classification for hypersurfaces with only ordinary double points in most dimensions.

Conditional superrigidity results for certain smooth Fano hypersurfaces and complete intersections.

Abstract

We prove birational superrigidity of every hypersurface of degree N in P^N with singular locus of dimension s, under the assumption that N is at least 2s+8 and it has only quadratic singularities of rank at least N-s. Combined with the results of I. A. Chel'tsov and T. de Fernex, this completes the list of birationally superrigid singular hypersurfaces with only ordinary double points except in dimension 4 and 6. Further we impose an additional condition on the base locus of a birational map to a Mori fiber space. Then we prove conditional birational superrigidity of certain smooth Fano hypersurfaces of index larger or equal to 2, and birational superrigidity of smooth Fano complete intersections of index 1 in weak form.