Birational rigidity of singular Fano hypersurfaces

TL;DR

This paper proves birational superrigidity for a broad class of singular Fano hypersurfaces, expanding understanding of their birational properties under various singularity conditions.

Contribution

It establishes birational superrigidity for singular Fano hypersurfaces with specific types of isolated singularities, a significant extension of prior results for smooth cases.

Findings

Birational superrigidity proven for hypersurfaces with semi-homogeneous singularities.

Superrigidity shown for hypersurfaces with singularities bounded by Tyurina numbers.

Results apply to hypersurfaces with dual varieties close to expected degree.

Abstract

We establish birational superrigidity for a large class of singular projective Fano hypersurfaces of index one. In the special case of isolated singularities, our result applies for instance to: (1) hypersurfaces with semi-homogeneous singularities of multiplicity asymptotically bounded by twice the square root of the dimension of the hypersurface, (2) hypersurfaces with isolated singularities whose Tyurina numbers satisfy a similar bound, and (3) hypersurfaces with isolated singularities whose dual variety is a hypersurface of degree sufficiently close to the expected degree.