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Mat. Sb., 2021, Volume 212, Number 4, Pages 113–130 (Mi msb9363)  

Birational geometry of singular Fano double spaces of index two

A. V. Pukhlikov

Department of Mathematical Sciences, University of Liverpool, Liverpool, UK

Abstract: We describe the birational geometry of Fano double spaces $V\stackrel{\sigma}{\to}{\mathbb P}^{M+1}$ of index 2 and dimension ${\geqslant 8}$ with at most quadratic singularities of rank ${\geqslant 8}$, satisfying certain additional conditions of general position: we prove that these varieties have no structures of a rationally connected fibre space over a base of dimension ${\geqslant2}$, that every birational map $\chi\colon V\dashrightarrow V'$ onto the total space of a Mori fibre space $V'/{\mathbb P}^1$ induces an isomorphism $V^+\cong V'$ of the blow-up $V^+$ of $V$ along $\sigma^{-1}(P)$, where $P\subset {\mathbb P}^{M+1}$ is a linear subspace of codimension 2, and that every birational map of $V$ onto a Fano variety $V'$ with ${\mathbb Q}$-factorial terminal singularities and Picard number 1 is an isomorphism. We give an explicit lower estimate, quadratic in $M$, for the codimension of the set of varieties $V$ that have worse singularities or do not satisfy the conditions of general position. The proof makes use of the method of maximal singularities and the improved $4n^2$-inequality for the self-intersection of a mobile linear system.
Bibliography: 20 titles.

Keywords: Fano variety, Mori fibre space, birational map, linear system, maximal singularity.

Funding Agency Grant Number
Leverhulme Trust RPG-2016-279
This research was carried out with the support of The Leverhulme Trust (grant RPG-2016-279).


DOI: https://doi.org/10.4213/sm9363

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English version:
Sbornik: Mathematics, 2021, 212:4, 551–566

Bibliographic databases:

UDC: 512.76
MSC: 14E05, 14E07
Received: 16.12.2019 and 10.08.2020

Citation: A. V. Pukhlikov, “Birational geometry of singular Fano double spaces of index two”, Mat. Sb., 212:4 (2021), 113–130; Sb. Math., 212:4 (2021), 551–566

Citation in format AMSBIB
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\by A.~V.~Pukhlikov
\paper Birational geometry of singular Fano double spaces of index two
\jour Mat. Sb.
\yr 2021
\vol 212
\issue 4
\pages 113--130
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\crossref{https://doi.org/10.4213/sm9363}
\transl
\jour Sb. Math.
\yr 2021
\vol 212
\issue 4
\pages 551--566
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