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

TL;DR

This paper proves the birational superrigidity of certain Fano-Mori fibre spaces with fibers as complete intersections of codimension two, under general position and twisting conditions, extending understanding of their birational geometry.

Contribution

It establishes birational superrigidity for Fano-Mori fibre spaces with complete intersection fibers, under general position and twisting assumptions, including cases with singularities.

Findings

Global log canonical threshold equals one.

Bound on base dimension depending on fiber dimension.

Fibre and total space may have controlled quadratic singularities.

Abstract

In this paper we prove the birational superrigidity of Fano-Mori fibre spaces $\pi\colon V\to S$, every fibre of which is a complete intersection of type $d_1\cdot d_2$ in the projective space ${\mathbb P}^{d_1+d_2}$, 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 $\frac12 M^2$). The fibres and the variety $V$ itself may have quadratic and bi-quadratic singularities, the rank of which is bounded from below.