Birational geometry of singular Fano hypersurfaces of index two

TL;DR

This paper classifies the birational geometry of general singular Fano hypersurfaces of index two in high dimensions, showing they are non-rational with a unique Fano fiber structure and identical automorphism groups.

Contribution

It provides a complete description of rationally connected fibrations on these hypersurfaces, establishing their non-rationality and automorphism group equality, extending known results to singular cases.

Findings

Hypersurfaces are non-rational.

Fano fiber structures are pencils of hyperplane sections.

Automorphism groups coincide with birational automorphism groups.

Abstract

For a Zariski general (regular) hypersurface $V$ of degree $M$ in the $(M+1)$-dimensional projective space, where $M$ is at least 16, with at most quadratic singularities of rank at least 13, we give a complete description of the structures of rationally connected (or Fano-Mori) fibre space: every such structure over a positive-dimensional base is a pencil of hyperplane sections. This implies, in particular, that $V$ is non-rational and its groups of birational and biregular automorphisms coincide. The set of non-regular hypersurfaces has codimension at least $\frac12(M-11)(M-10)-10$ in the natural parameter space.