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Izv. RAN. Ser. Mat., Forthcoming paper (Mi izv9146)  

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

A. V. Pukhlikov

Department of Mathematical Sciences, University of Liverpool

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 $V\slash S$ 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.

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


English version:
DOI: https://doi.org/10.1070/IM9146

UDC: 512.76
MSC: 14E05, 14E07
Received: 27.01.2021
Revised: 12.06.2021

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