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ENRIQUES INVOLUTIONS AND BRAUER CLASSES

Part of: Surfaces and higher-dimensional varieties (Co)homology theory

Published online by Cambridge University Press:  13 December 2022

A. N. SKOROBOGATOV Open the ORCID record for A. N. SKOROBOGATOV [Opens in a new window]  and
D. VALLONI Open the ORCID record for D. VALLONI [Opens in a new window]
A. N. SKOROBOGATOV*
Affiliation:
Department of Mathematics South Kensington Campus Imperial College London SW7 2BZ London United Kingdom - and - Institute for the Information Transmission Problems Russian Academy of Sciences 19 Bolshoi Karetnyi Moscow 127994 Russia
D. VALLONI
Affiliation:
Leibniz Universität Hannover Riemann Center for Geometry and Physics Welfengarten 1 30167 Hannover Germany valloni@math.uni-hannover.de
*
e-mail: a.skorobogatov@imperial.ac.uk
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Abstract

We prove that every element of order 2 in the Brauer group of a complex Kummer surface X descends to an Enriques quotient of X. In generic cases, this gives a bijection between the set ${\mathcal Enr}(X)$ of Enriques quotients of X up to isomorphism and the set of Brauer classes of X of order 2. For some K3 surfaces of Picard rank $20,$ we prove that the fibers of ${\mathcal Enr}(X)\to \mathrm {{Br}}(X)[2]$ above the nonzero points have the same cardinality.

MSC classification

Primary: 14J28: $K3$ surfaces and Enriques surfaces 14F22: Brauer groups of schemes
Type
Article
Information
Nagoya Mathematical Journal , Volume 251 , September 2023 , pp. 606 - 621
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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Footnotes

D.V. was funded by the European Research Council under EU Horizon 2020 research and innovation program grant agreement no. 948066.

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