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Algebraic models of the line in the real affine plane

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Abstract

We study smooth rational closed embeddings of the real affine line into the real affine plane, that is algebraic rational maps from the real affine line to the real affine plane which induce smooth closed embeddings of the real euclidean line into the real euclidean plane. We consider these up to equivalence under the group of birational automorphisms of the real affine plane which are diffeomorphisms of its real locus. We show that in contrast with the situation in the categories of smooth manifolds with smooth maps and of real algebraic varieties with regular maps where there is only one equivalence class up to isomorphism, there are non-equivalent smooth rational closed embeddings up to such birational diffeomorphisms.

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Authors and Affiliations

  1. IMB UMR5584, CNRS, Univ. Bourgogne Franche-Comté, 21000, Dijon, France

    Adrien Dubouloz

  2. Laboratoire angevin de recherche en mathématiques (LAREMA), CNRS, Université d’Angers, 49045, Angers, France

    Frédéric Mangolte

Authors
  1. Adrien Dubouloz
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  2. Frédéric Mangolte
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Correspondence to Frédéric Mangolte.

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This research benefited at its early stage from the support of ANR Grant “BirPol” ANR-11-JS01-004-01. The second author is partially supported by the ANR Grant “ENUMGEOM” ANR-18-CE40-0009.

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Dubouloz, A., Mangolte, F. Algebraic models of the line in the real affine plane. Geom Dedicata 210, 179–204 (2021). https://doi.org/10.1007/s10711-020-00539-1

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