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

Abstract

In this paper we prove the birational superrigidity of Fano-Mori fibre spaces π: V → S, every fibre of which is a complete intersection of type d 1 • d 2 in the projective space P d 1 +d 2 , satisfying certain conditions of general position, under the assumption that the fibration V /S is sufficiently twisted over the base (in particular, under the assumption that the K-condition holds). The condition of general position for every fibre guarantees that the global log canonical threshold is equal to one. This condition bounds the dimension of the base S by a constant that depends on the dimension M of the fibre only (as the dimension M of the fibre grows, this constant grows as 1 2 M 2 ). The fibres and the variety V itself may have quadratic and bi-quadratic singularities, the rank of which is bounded from below.Bibliography: 34 items.

Remarks on Galois rational covers

“…In [9] the divisorial canonicity was established for Fano varieties of index 1 that are d-sheeted covers of P M , under the assumption that they have at worst quadratic singularities, the rank of which is bounded from below (the bound depends on the dimension M and the degree of the cover) and satisfy certain additional conditions of general position, and the varieties that are not divisorially canonical form a set, the codimension of which is bounded from below by an integer-valued function of the parameters d and M, which grows as 1 2 M 2 when M grows. Finally, for complete intersections of type d 1 • d 2 in P M +2 the divisorial canonicity was shown for the varieties with at worst quadratic and bi-quadratic singularities, the rank of which is bounded from below, in [10], under the assumption that certain additional conditions of general position are satisfied, and for the codimension of the set of complete intersections that do not satisfy those conditions, an estimate, similar to the estimates above, was obtained.…”