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Surfaces that are covered by two pencils of circles

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We list up to Möbius equivalence all possible degrees and embedding dimensions of real surfaces that are covered by at least two pencils of circles, together with the number of such pencils. In addition, we classify incidences between the contained circles, complex lines and isolated singularities. Such geometric characteristics are encoded in the Néron–Severi lattices of such surfaces and is of potential interest to geometric modelers and architects. As an application we confirm Blum’s conjecture in higher dimensional space and we address the Blaschke–Bol problem by classifying surfaces that are covered by hexagonal webs of circles. In particular, we find new examples of such webs that cannot be embedded in 3-dimensional space.

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Acknowledgements

I would like to thank R. Krasauskas, H. Pottmann, J. Schicho, M. Skopenkov and S. Zubė for the inspiring discussions, which have have been invaluable for this paper. In particular, the detailed comments of M. Skopenkov were extremely helpful. I thank J. Kollár for interesting historical remarks. The computations were done using [37, Sage] and [7, Magma]. The images were made using [29, Povray]. This research was supported by base funding of the King Abdullah University of Science and Technology (KAUST) and by the Austrian Science Fund (FWF) project P33003.

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  1. Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Vienna, Austria

    Niels Lubbes

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This research was supported by base funding of the King Abdullah University of Science and Technology (KAUST) and by the Austrian Science Fund (FWF) project P33003.

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Lubbes, N. Surfaces that are covered by two pencils of circles. Math. Z. 299, 1445–1472 (2021). https://doi.org/10.1007/s00209-021-02713-x

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