Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-06-07T21:17:50.800Z Has data issue: false hasContentIssue false
  • English
  • Français

Möbius automorphisms of surfaces with many circles

Part of: Analytic and descriptive geometry Nonlinear incidence geometry Cycles and subschemes Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  02 September 2020

Niels Lubbes
Niels Lubbes*
Affiliation:
Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Vienna, Austria
*
e-mail: niels.lubbes@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We classify real two-dimensional orbits of conformal subgroups such that the orbits contain two circular arcs through a point. Such surfaces must be toric and admit a Möbius automorphism group of dimension at least two. Our theorem generalizes the classical classification of Dupin cyclides.

Keywords

Surface automorphismsweak del Pezzo surfacesMöbius geometrycirclesLie algebrastoric geometrylattice geometry

MSC classification

Primary: 14J50: Automorphisms of surfaces and higher-dimensional varieties
Secondary: 51B10: Möbius geometries 51N30: Geometry of classical groups 14C20: Divisors, linear systems, invertible sheaves
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 1 , February 2022 , pp. 42 - 71
Check for updates
Copyright
© Canadian Mathematical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Berndt, J., Console, S., and Olmos, C. E., Submanifolds and holonomy . 2nd ed., CRC Press, Boca Raton, FL, 2016.CrossRefGoogle Scholar
Cox, D. A., Little, J. B., and Schenck, H. K., Toric varieties. Vol. 124, American Mathematical Society, Providence, RI, 2011.Google Scholar
de Graaf, W. A., Pílniková, J., and Schicho, J., Parametrizing del Pezzo surfaces of degree 8 using Lie algebras. J. Symb. Comput. 44(2009), no. 1, 1–14.CrossRefGoogle Scholar
Douglas, A. and Repka, J.. The subalgebras of $so\left(4,\mathbb{C}\right)$ . Comm. Algebra 44(2016), no. 12, 5269–5286.CrossRefGoogle Scholar
Haase, C. and Schicho, J., An inequality for adjoint rational surfaces. Preprint, 2013. arXiv:1306.3389 Google Scholar
Hall, B., Lie groups, Lie algebras, and representations. Vol. 122, Springer, New York, NY, 2015.CrossRefGoogle Scholar
Kollár, J.. Quadratic solutions of quadratic forms . In: N. Budur, T. de Fernex, R. Docampo, and K. Tucker (eds.), Local and global methods in algebraic geometry, Contemp. Math., 12, American Mathematical Society, Providence, RI, 2018, pp. 211–249.CrossRefGoogle Scholar
Lubbes, N., Software library for computing Möbius automorphisms of surfaces. 2019. http://github.com/niels-lubbes/moebius_aut Google Scholar
Lubbes, N., Surfaces that are covered by two families of circles. Preprint, 2020. arXiv:1302.6710 Google Scholar
Lubbes, N. and Schicho, J., Lattice polygons and families of curves on rational surfaces. J. Algebraic Combin. 34(2011), no. 2, 213–236.CrossRefGoogle Scholar
Persistence of Vision Pty. Ltd. Persistence of vision raytracer version 3.6. 2004. www.povray.org Google Scholar
Pottmann, H., Shi, L., and Skopenkov, M., Darboux cyclides and webs from circles. Comput. Aided Geom. Design 29(2012), no. 1, 77–97.CrossRefGoogle Scholar
Serre, J. P., Topics in Galois theory . Jones and Bartlett Publishers, Boston, MA, 1992.Google Scholar
Silhol, R., Real algebraic surfaces . Springer, New York, NY, 1989.CrossRefGoogle Scholar
Skopenkov, M. and Krasauskas, R., Surfaces containing two circles through each point. Math. Ann. 373(2018), 1299–1327. http://org/10.1007/s00208-018-1739-z CrossRefGoogle Scholar
The Sage Developers, SageMath, the Sage mathematics software system. 2018. www.sagemath.org Google Scholar