Birational geometry of singular Fano double spaces of index two

TL;DR

This paper investigates the birational properties of high-dimensional Fano double spaces with quadratic singularities, establishing rigidity results and describing the structure of their birational maps.

Contribution

It provides new birational rigidity results for Fano double spaces of index two with specific singularities and general position conditions, expanding understanding of their birational geometry.

Findings

No rationally connected fiber space structures over bases of dimension ≥ 2.

Birational maps to Mori fiber spaces induce isomorphisms after blow-up.

Birational maps to Fano varieties with certain singularities are isomorphisms.

Abstract

In this paper we describe the birational geometry of Fano double spaces $V\stackrel{\sigma}{\to}{\mathbb P}^{M+1}$ of index 2 and dimension $\geqslant 8$ with at mostquadratic singularities of rank $\geqslant 8$, satisfying certain additional conditions of general position: we prove that these varieties have no structures of a rationally connected fibre space over a base of dimension $\geqslant 2$, that every birational map $\chi\colon V\dashrightarrow V'$ onto the total space of a Mori fibre space $V'/{\mathbb P}^1$ induces an isomorphism $V^+\cong V'$ of the blow up $V^+$ of the variety $V$ along $\sigma^{-1}(P)$, where $P\subset {\mathbb P}^{M+1}$ is a linear subspace of codimension 2, and that every birational map of the variety $V$ onto a Fano variety $V'$ with ${\mathbb Q}$-factorial terminal singularities and Picard number 1 is an isomorphism. We give an explicit lower estimate for the codimension of the set of varieties $V$ with worse singularities or not satisfying the conditions of general position, quadratic in $M$. The proof makes use of the method of maximal singularities and the improved $4n^2$-inequality for the self-intersection of a mobile linear system.