# Birational geometry of singular Fano hypersurfaces of index two

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## Abstract

For a Zariski general (regular) hypersurface *V* of degree *M* in the \((M+1)\)-dimensional projective space, where \(M\geqslant 16\), with at most quadratic singularities of rank \(\geqslant 13\), 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: \(\mathrm{Bir} V = \mathrm{Aut} V\). The set of non-regular hypersurfaces has codimension at least \(\frac{1}{2}(M-11)(M-10)-10\) in the natural parameter space.

## Mathematics Subject Classification

14E05 14E07## Notes

### Acknowledgements

The author thanks The Leverhulme Trust for the financial support of the present project (Research Project Grant RPG-2016-279). The author is also grateful to the members of the Divisions of Algebraic Geometry and Algebra of Steklov Mathematical Institute for the interest to this work and also to the colleagues—algebraic geometers at the University of Liverpool for the general support. Finally, the author thanks the referee for their work on the paper and the helpful remark about the statement of the main result.

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