# Effective birational rigidity of Fano double hypersurfaces

## TL;DR

This paper establishes the birational superrigidity of certain Fano double hypersurfaces with singularities, providing explicit bounds on the parameter space and employing advanced geometric inequalities and hypertangent divisor techniques.

## Contribution

It introduces an effective criterion for birational rigidity of Fano double hypersurfaces with singularities, with explicit bounds and novel use of the $4n^2$-inequality.

## Key findings

- Proves birational superrigidity for a class of Fano double hypersurfaces.
- Provides a quadratic lower bound on the codimension of non-rigid varieties.
- Employs hypertangent divisors and the $4n^2$-inequality in the proof.

## Abstract

We prove birational superrigidity of Fano double hypersurfaces of index one with quadratic and multi-quadratic singularities, satisfying certain regularity conditions, and give an effective explicit lower bound for the codimension of the set of non-rigid varieties in the natural parameter space of the family. The lower bound is quadratic in the dimension of the variety. The proof is based on the techniques of hypertangent divisors combined with the recently discovered $4n^2$-inequality for complete intersection singularities.