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Bijective Cremona transformations of the plane

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Abstract

We study the birational self-maps of the projective plane over finite fields that induce permutations on the set of rational points. As a main result, we prove that no odd permutation arises over a non-prime finite field of characteristic two, which completes the investigation initiated by Cantat about which permutations can be realized this way. Main ingredients in our proof include the invariance of parity under groupoid conjugations by birational maps, and a list of generators for the group of such maps.

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Acknowledgements

We thank Brendan Hassett for suggesting us the problem in the present paper. We also thank Zinovy Reichstein for a quick proof that \({\mathrm {BCr}}_2(k)\) is not finitely generated when k is uncountable. We thank Julia Schneider for discussing with us on the key ideas that allowed us to attack the quintic transformations. Before we are able to prove our conjecture, Lian Duan assisted us designing a Magma code that can compute efficiently the parities of all possible quintic transformations over \(\mathbb {F}_q\) for \(q=4,8,16\). We are very grateful for his generous help. Finally, we thank the anonymous referee for their valuable suggestions. During this project, the first author was partially supported by funds from NSF Grant DMS-1701659. The second author is supported by the ERC Synergy Grant ERC-2020-SyG-854361-HyperK. The third author was supported by EPSRC grant EP/R021422/2. The last author was supported by FIBALGA ANR-18-CE40-0003-01, PEPS 2019 “JC/JC” and Étoiles Montantes de la Région Pays de la Loire.

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T1Z2, Canada

    Shamil Asgarli

  2. Mathematisches Institut der Universität Bonn, Endenicher Allee 60, 53121, Bonn, Germany

    Kuan-Wen Lai

  3. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Masahiro Nakahara

  4. Univ Angers, CNRS, LAREMA, SFR MATHSTIC, 49000, Angers, France

    Susanna Zimmermann

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  1. Shamil Asgarli
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  2. Kuan-Wen Lai
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  3. Masahiro Nakahara
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  4. Susanna Zimmermann
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Correspondence to Masahiro Nakahara.

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Asgarli, S., Lai, KW., Nakahara, M. et al. Bijective Cremona transformations of the plane. Sel. Math. New Ser. 28, 53 (2022). https://doi.org/10.1007/s00029-022-00768-0

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