Birational geometry of Fano double spaces of index two

TL;DR

This paper investigates the birational geometry of Fano double spaces of index two, showing that their rationally connected fiber structures are limited to certain linear systems and that their birational and biregular automorphism groups are identical.

Contribution

It establishes the uniqueness of rationally connected fiber structures and the equality of birational and biregular automorphism groups for these Fano varieties.

Findings

Only pencils-subsystems of |−½K_V| serve as rationally connected fiber structures.

Birational and biregular automorphism groups of V are the same.

The results apply to double covers branched over generic hypersurfaces of degree 2(M−1).

Abstract

We study birational geometry of Fano varieties, realized as double covers $\sigma\colon V\to {\mathbb P}^M$, $M\geq 5$, branched over generic hypersurfaces $W=W_{2(M-1)}$ of degree $2(M-1)$. We prove that the only structures of a rationally connected fiber space on $V$ are the pencils-subsystems of the free linear system $|-\frac12 K_V|$. The groups of birational and biregular self-maps of the variety $V$ coincide.