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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Conic bundle fourfolds with nontrivial unramified Brauer group

Authors: Asher Auel, Christian Böhning, Hans-Christian Graf von Bothmer and Alena Pirutka
Journal: J. Algebraic Geom. 29 (2020), 285-327
Published electronically: October 22, 2019
MathSciNet review: 4069651
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Abstract | References | Additional Information

Abstract: We derive a formula for the unramified Brauer group of a general class of rationally connected fourfolds birational to conic bundles over smooth threefolds. We produce new examples of conic bundles over $\mathbb {P}^3$ where this formula applies and which have nontrivial unramified Brauer group. The construction uses the theory of contact surfaces and, at least implicitly, matrix factorizations and symmetric arithmetic Cohen–Macaulay sheaves, as well as the geometry of special arrangements of rational curves in $\mathbb {P}^2$. We also prove the existence of universally $\operatorname {CH}_0$-trivial resolutions for the general class of conic bundle fourfolds we consider. Using the degeneration method, we thus produce new families of rationally connected fourfolds whose very general member is not stably rational.

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Additional Information

Asher Auel
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
MR Author ID: 932786

Christian Böhning
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Hans-Christian Graf von Bothmer
Affiliation: Fachbereich Mathematik der Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany
MR Author ID: 724323

Alena Pirutka
Affiliation: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012
MR Author ID: 934651

Received by editor(s): April 22, 2017
Received by editor(s) in revised form: April 4, 2019
Published electronically: October 22, 2019
Additional Notes: The first author was partially supported by the NSA grant H98230-16-1-032
Article copyright: © Copyright 2019 University Press, Inc.