Birational geometry of singular Fano hypersurfaces of index two

Abstract

For a Zariski general (regular) hypersurface V of degree M in the (M + 1)-dimensional projective space, where M 16, with at most quadratic singularities of rank 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: Bir V = Aut V . The set of non-regular hypersurfaces has codimension at least 1 2 (M − 11)(M − 10) − 10 in the natural parameter space.Bibliography: 25 titles.

K-stability of birationally superrigid Fano varieties

“…By [Puk02, Proposition 5] again, we have mult x (M 2 ) ≤ 4 outside a subset Z ⊆ X of dimension at most 1, hence by [dFEM04], (X, M) is lc outside Z. On the other hand by the main result of [Puk17], the only possible center of maximal singularities of (X, M) is a linear section of X codimension 2. It follows that (X, M) is lc and every lc center of (X, M) is a linear section of codimension two.…”

“…Let be the set of regular hypersurfaces as defined in [Puk17, §0.2]. By [Puk17, Theorem 2], is non-empty and the complement of has codimension at least . Therefore, it suffices to show that every hypersurface in the set is K-stable.…”

“…On the other hand, let be a movable boundary; then, by the main result of [Puk17], the only possible center of maximal singularities of is a linear section of of codimension two. It follows that is lc and every lc center of is a linear section of codimension two.…”

Variation of stable birational types in positive characteristic

“…Birational rigidity has so far mostly been applied to Fano varieties of index one and two (see e.g. [Pu16,Pu20] for the index two case) and it is unknown whether the method applies to hypersurfaces X ⊂ P n+1 C of degree d n. Finally, the condition on the Picard rank seems to prevent applications to questions about stable birational equivalence.…”

Birational geometry of varieties, fibred into complete intersections of codimension two

“…Among the recent papers, where it is applied in the proof of birational rigidity, we mention [22,23,24]. In [25,5] both techniques were used. It seems that the quadratic technique can relax the upper bound for the dimension of the base S of a Fano-Mori fibre space.…”