Birational geometry of varieties fibred into complete intersections of codimension two

Pukhlikov, AV (2022) Birational geometry of varieties fibred into complete intersections of codimension two. IZVESTIYA MATHEMATICS, 86 (2). pp. 334-411.

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Official URL: http://dx.doi.org/10.1070/im9146

Abstract

<jats:title>Abstract</jats:title><jats:p>In this paper we prove the birational superrigidity of Fano–Mori fibre spaces<jats:inline-formula><jats:tex-math><?CDATA $\pi\colon V\to S$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="IZV_86_2_334ieqn1.gif" xlink:type="simple" /></jats:inline-formula>all of whose fibres are complete intersections of type<jats:inline-formula><jats:tex-math><?CDATA $d_1\cdot d_2$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="IZV_86_2_334ieqn2.gif" xlink:type="simple" /></jats:inline-formula>in the projective space<jats:inline-formula><jats:tex-math><?CDATA ${\mathbb P}^{d_1+d_2}$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="IZV_86_2_334ieqn3.gif" xlink:type="simple" /></jats:inline-formula>satisfying certain conditions of general position, under the assumption that the fibration<jats:inline-formula><jats:tex-math><?CDATA $V/S$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="IZV_86_2_334ieqn4.gif" xlink:type="simple" /></jats:inline-formula>is sufficiently twisted over the base (in particular, under the assumption that the<jats:inline-formula><jats:tex-math><?CDATA $K$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="IZV_86_2_334ieqn5.gif" xlink:type="simple" /></jats:inline-formula>-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<jats:inline-formula><jats:tex-math><?CDATA $S$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="IZV_86_2_334ieqn6.gif" xlink:type="simple" /></jats:inline-formula>by a constant depending only on the dimension<jats:inline-formula><jats:tex-math><?CDATA $M$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="IZV_86_2_334ieqn7.gif" xlink:type="simple" /></jats:inline-formula>of the fibre (this constant grows like<jats:inline-formula><jats:tex-math><?CDATA $M^2/2$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="IZV_86_2_334ieqn8.gif" xlink:type="simple" /></jats:inline-formula>as<jats:inline-formula><jats:tex-math><?CDATA $M\to\infty$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="IZV_86_2_334ieqn9.gif" xlink:type="simple" /></jats:inline-formula>). The fibres and the variety<jats:inline-formula><jats:tex-math><?CDATA $V$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="IZV_86_2_334ieqn10.gif" xlink:type="simple" /></jats:inline-formula>may have quadratic and bi-quadratic singularities whose rank is bounded below.</jats:p>

Item Type:	Article
Uncontrolled Keywords:	Fano variety, Mori fibre space, birational map, birational rigidity, linear system, maximal singularity, quadratic singularity, bi-quadratic singularity
Divisions:	Faculty of Science and Engineering > School of Physical Sciences
Depositing User:	Symplectic Admin
Date Deposited:	07 Jul 2021 09:12
Last Modified:	28 Jun 2023 12:07
DOI:	10.1070/IM9146
Open Access URL:	https://arxiv.org/abs/2101.10830
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3129035