Birational geometry of Fano hypersurfaces of index two

Abstract

We prove that every non-trivial structure of a rationally connected fibre space on a generic (in the sense of Zariski topology) hypersurface V of degree M in the (M+1)-dimensional projective space for M ≥ 16 is given by a pencil of hyperplane sections. In particular, the variety V is non-rational and its group of birational selfmaps coincides with the group of biregular automorphisms and for that reason is trivial. The proof is based on the techniques of the method of maximal singularities and inversion of adjunction.

Birational geometry of singular Fano hypersurfaces of index two

“…Hypersurfaces with at least one singular point form a divisor in the space F . Thus in the present paper we essentially improve the main result of [1]: we extend it to hypersurfaces with bounded singularities and give an effective estimate for the codimension of the complement to the set U of "correct" hypersurfaces (which grows as 1 2 M 2 when the dimension M grows). Theorem 1 immediately implies the standard set of facts about birational geometry of the variety V ∈ U. Corollary 1.…”

“…Proof. Let us assume the converse: B * ⊂ Sing V and show that the arguments of [1] exclude this option. Let us consider separately the three cases:…”

“…which is equivalent to the condition (R1.2) of the present paper, is sufficient. Thus all arguments of the paper [1], excluding the maximal singularity in Sections 4-6, work without modifications in our Case 3 and exclude the maximal singularity, the centre of which is not contained in Sing V . This completes the proof of Proposition 3.1.…”

“…For a Zariski general smooth hypersurface V ⊂ P the claim (iii) of Theorem 1 was shown for M 16 in [1]. Hypersurfaces with at least one singular point form a divisor in the space F .…”

“…Historical remarks and acknowledgements. The few attempts to study birational geometry of higher-dimensional Fano varieties of index higher than 1 are listed in the introduction to [1], see also the introduction to [4]. Here we note that, starting from the paper [3], the results about birational rigidity of particular classes of Fano varieties become effective in the sense that an explicit effective estimate for the codimension of the subset of non-rigid varieties in the natural parameter space of the given family is produced.…”