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Two kinds of real lines on real del Pezzo surfaces of degree 1

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Abstract

We show how the real lines on a real del Pezzo surface of degree 1 can be split into two species, elliptic and hyperbolic, via a certain distinguished, intrinsically defined, \({\text {Pin}}^-\)-structure on the real locus of the surface. We prove that this splitting is invariant under real automorphisms and real deformations of the surface, and that the difference between the total numbers of hyperbolic and elliptic lines is always equal to 16.

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Acknowledgements

Our special thanks go to R. Rasdeaconu, discussions with whom were among the motivations for this study. For the artwork with the Hasse diagram of \(E_8\) on Fig. 2 we used Ringel’s sample in [19], and for the diagrams on Fig. 3, McKay’s Lie Hasse package [17]. The second author was partially funded by the grant ANR-18-CE40-0009 of Agence Nationale de Recherche.

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  1. Middle East Technical University, Ankara, Turkey

    S. Finashin

  2. Université de Strasbourg et IRMA (CNRS), Strasbourg, France

    V. Kharlamov

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  1. S. Finashin
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  2. V. Kharlamov
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Correspondence to V. Kharlamov.

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Finashin, S., Kharlamov, V. Two kinds of real lines on real del Pezzo surfaces of degree 1. Sel. Math. New Ser. 27, 83 (2021). https://doi.org/10.1007/s00029-021-00690-x

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