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RATIONALITY OVER NONCLOSED FIELDS OF FANO THREEFOLDS WITH HIGHER GEOMETRIC PICARD RANK

Part of: Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  05 August 2022

Alexander Kuznetsov  and
Yuri Prokhorov Open the ORCID record for Yuri Prokhorov [Opens in a new window]
Alexander Kuznetsov
Affiliation:
Steklov Mathematical Institute of the Russian Academy of Sciences, 8 Gubkin Street, Moscow 119991, Russia and Laboratory of Algebraic Geometry, National Research University Higher School of Economics, 6 Usachev Street, Moscow, 119048, Russia ([email protected])
Yuri Prokhorov*
Affiliation:
Steklov Mathematical Institute of the Russian Academy of Sciences, 8 Gubkin Street, Moscow 119991, Russia; Laboratory of Algebraic Geometry, National Research University Higher School of Economics, 6 Usachev Street, Moscow, 119048, Russia and Department of Algebra, Moscow State University, Moscow 119992, Russia
*
[email protected]
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Abstract

We prove rationality criteria over nonclosed fields of characteristic $0$ for five out of six types of geometrically rational Fano threefolds of Picard number $1$ and geometric Picard number bigger than $1$. For the last type of such threefolds, we provide a unirationality criterion and construct examples of unirational but not stably rational varieties of this type.

Keywords

Fano varietiesrationalityMori contractionsHilbert schemes

MSC classification

Primary: 14E08: Rationality questions
Secondary: 14J45: Fano varieties 14E30: Minimal model program (Mori theory, extremal rays)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 23 , Issue 1 , January 2024 , pp. 207 - 247
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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