Birational geometry of varieties fibred into complete intersections of codimension two

A. V. Pukhlikov

Izvestiya. Mathematics , Tome 86 (2022) no. 2, pp. 334-411

Résumé

In this paper we prove the birational superrigidity of Fano–Mori fibre spaces $\pi\colon V\to S$ all of whose fibres are complete intersections 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 also bounds the dimension of the base $S$ by a constant depending only on the dimension $M$ of the fibre (this constant grows like $M^2/2$ as $M\to\infty$). The fibres and the variety $V$ may have quadratic and bi-quadratic singularities whose rank is bounded below.

Détail

Keywords: Fano variety, birational rigidity, linear system, maximal singularity, quadratic singularity, bi-quadratic singularity.
Mots-clés : Mori fibre space, birational map