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Geometric classification of real ternary octahedral quartics. (English) Zbl 1452.14036

Summary: Ternary real-valued quartics in \(\mathbb{R}^{3}\) that are invariant under octahedral symmetry are considered. The geometric classification of these surfaces is given. A new type of surface emerges from this classification.

MSC:

14J25 Special surfaces
14P05 Real algebraic sets
14Q10 Computational aspects of algebraic surfaces
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[1] Barth, W, Two projective surfaces with many nodes, admitting the symmetries of the icosahedron, J. Algebraic Geom., 5, 173-186, (1996) · Zbl 0860.14032
[2] Bourbaki, N.: Lie Groups and Lie Algebras. Elements of Mathematics, Chapters 7-9. Springer, Berlin (2005) · Zbl 1139.17002
[3] Bromwich, TJ, The classification of quadrics, Trans. Am. Math. Soc., 6, 275-285, (1905) · JFM 36.0702.01 · doi:10.1090/S0002-9947-1905-1500711-6
[4] Buringthon, RS, A classification of quadrics in affine \(n\)-space by mean of arithmetic invariant, Am. Math. Monthly., 39, 527-532, (1932) · doi:10.1080/00029890.1932.11987364
[5] Clebsch, A, Ueber die anwendung der quadratischen substitution auf die gleichungen 5ten grades und die geometrische theorie des ebenen Fünfseits, Math. Ann., 4, 284-345, (1871) · JFM 03.0031.01 · doi:10.1007/BF01442599
[6] Endrass, S, Flaechen mit vielen doppelpunkten, DMV-Mitteilungen, 4, 17-20, (1995) · Zbl 0866.14032
[7] Goursat, É, Etude des surfaces qui admettent tous LES plans de symétrie d’un polyèdre régulier, Ann. Sci. Écol. Norm. Sup., 3, 159-200, (1887) · JFM 19.0781.03 · doi:10.24033/asens.295
[8] Klein, F.: Ueber Riemann’s Theorie der Algebraischen Functionen und ihrer Integrale: Einer Ergänzung der Gewöhnlichen Darstellungen. B.G. Teubner, Leipzig (1882) · JFM 14.0358.01
[9] Kraft, H., Procesi, C.: Classical invariant theory: a primer (1996). http://www.math.iitb.ac.in/ shripad/Wilberd/KP-Primer.pdf
[10] Kummer, E.: Über die Flächen vierten Grades mit sechzehn singulären Punkten, pp. 246-260. Monatsberichte der Königlichen Preußischen Akademie der Wissenschaften zu, Berlin (1864) · Zbl 06736753
[11] Mehlhorn, K., Yap, C.K.: Robust Geometric Computation. Courant Institute of Mathematical Sciences (under preparation) (2014)
[12] Sarti, A, Symmetrische flächen mit gewöhnlichen doppelpunkten, Math. Semesterber., 55, 1-5, (2008) · Zbl 1140.14311 · doi:10.1007/s00591-007-0030-2
[13] Schläfli, L, On the distribution of surfaces of the third order into species, in reference to the absence or presence of singular points, and the reality of their lines, Philos. Trans. R. Soc. London, 153, 193-241, (1863) · doi:10.1098/rstl.1863.0010
[14] Straten, D; Labs, O; Labs, O (ed.); etal., A visual introduction to cubic surfaces using the computer software SPICY, 225-238, (2003), Berlin · Zbl 1027.68129 · doi:10.1007/978-3-662-05148-1_12
[15] Wenninger, M.J.: Polyhedron Models. Cambridge University Press, New York (1971) · Zbl 0222.50010 · doi:10.1017/CBO9780511569746
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