Birational geometry of algebraic varieties fibred into Fano double spaces

Abstract

We develop the quadratic technique of proving birational rigidity of Fano-Mori fibre spaces over a higher-dimensional base. As an application, we prove birational rigidity of generic fibrations into Fano double spaces of dimension M 4 and index one over a rationally connected base of dimension at most 1 2 (M − 2)(M − 1). An estimate for the codimension of the subset of hypersurfaces of a given degree in the projective space with a positive-dimensional singular set is obtained, which is close to the optimal one. Bibliography: 15 titles.14E05, 14E07

Birationally rigid singular double quadrics and double cubics

“…Proof. See Proposition 1 in [EP14] and Proposition 2.4 in [Puk15a]. Now as the linear system L defines precisely the double cover σ, for any point o ∈ B we get the inequality…”

The rationality problem for conic bundles

“…4, § 1], [Che05], [Sob02], [Gri00], [SC11], [Kry16]. A higher-dimensional generalization was discussed in a recent paper [Puk17]. It would be interesting to reformulate these results in terms of degeneracy loci and local invariants.…”

Factorial Hypersurfaces

“…§2 contains the proof of Theorem 1.1 by means of the technique developed in [3,4]. In §3 we obtain an estimate of the codimension of the set of non-factorial hypersurfaces, which implies Theorem 0.1.…”

“…The plan of the proof and the structure of the paper. Our proof is based on the famous Grothendieck's theorem [2] and the technique of estimating the codimension of the set of hypersurfaces in P N with a singular set of prescribed dimension, developed in [3,4]. Grothendieck's theorem claims that a variety with hypersurface singularities (in fact, with complete intersection singularities) is factorial, if the singular locus has codimension at least 4.…”