Birational geometry of algebraic varieties, fibred into Fano double spaces

TL;DR

This paper introduces a quadratic technique to establish the birational rigidity of Fano-Mori fiber spaces, specifically for fibrations into Fano double spaces, with applications to higher-dimensional algebraic geometry.

Contribution

It develops a new quadratic method for proving birational rigidity and applies it to Fano double space fibrations over rationally connected bases.

Findings

Proves birational rigidity for generic fibrations into Fano double spaces of dimension ≥4.

Provides near-optimal estimates for the codimension of hypersurfaces with positive-dimensional singular sets.

Introduces a quadratic technique applicable to higher-dimensional algebraic varieties.

Abstract

We develop the quadratic technique of proving birational rigidity of Fano-Mori fibre spaces over a higher-dimensional base. As an application, we prove birational rigidity of generic fibrations into Fano double spaces of dimension $M\geqslant 4$ and index one over a rationally connected base of dimension at most $\frac12 (M-2)(M-1)$. An estimate for the codimension of the subset of hypersurfaces of a given degree in the projective space with a positive-dimensional singular set is obtained, which is close to the optimal one.