Birational geometry of Fano hypersurfaces of index two

TL;DR

This paper proves that general Fano hypersurfaces of degree M in (M+1)-space have a unique rationally connected fiber structure, are non-rational, and have trivial birational automorphism groups, using advanced birational geometry techniques.

Contribution

It establishes the uniqueness of fiber structures on general Fano hypersurfaces of index two for degrees M ≥ 14, revealing their non-rationality and automorphism group properties.

Findings

Any rationally connected fiber structure is given by a hyperplane pencil.

The hypersurface is non-rational.

The group of birational self-maps is trivial.

Abstract

We prove that every non-trivial structure of a rationally connected fibre space (and so every structure of a Mori-Fano fibre space) on a general (in the sense of Zariski topology) hypersurface of degree $M$ in the $(M+1)$-dimensional projective space for $M\geq 14$ is given by a pencil of hyperplane sections. In particular, the variety $V$ is non-rational and its group of birational self-maps coincide with the group of biregular automorphisms and is therefore trivial. The proof is based on the techniques of the method of maximal singularities and the inversion of adjunction.