Birationally rigid Fano fibre spaces. II

TL;DR

This paper establishes the birational rigidity of broad classes of Fano-Mori fibre spaces over arbitrary-dimensional bases, expanding understanding of their birational properties and fiber-wise behavior.

Contribution

It proves birational rigidity for large classes of Fano-Mori fibre spaces and shows that certain natural conditions ensure fiber-wise birational maps.

Findings

Birational rigidity holds for classes of Fano-Mori fibre spaces.

Fiber-wise birational maps occur under natural fiber conditions.

Constructs examples satisfying the conditions for rigidity.

Abstract

In this paper we prove birational rigidity of large classes of Fano-Mori fibre spaces over a base of arbitrary dimension, bounded from above by a constant that depends on the dimension of the fibre only. In order to do that, we first show that if every fibre of a Fano-Mori fibre space satisfies certain natural conditions, then every birational map onto another Fano-Mori fibre space is fibre-wise. After that we construct large classes of fibre spaces (into Fano double spaces of index one and into Fano hypersurfaces of index one) which satisfy those conditions. The paper is a follow up of my previous work: Pukhlikov A.V., Birationally rigid Fano fibrations, Izvestiya: Mathematics {\bf 64} (2000), No. 3, 563-581.