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Chmutov Surface

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An algebraic surface with affine equation

P_d(x_1,x_2)+T_d(x_3)=0,
(1)

where T_d(x) is a Chebyshev polynomial of the first kind and P_d(x_1,x_2) is a polynomial defined by

P_d(x_1,x_2)=|x_1 1 0 ... 0 0 0; 2x_2 x_1 1 ... 0 0 0; 3 x_2 x_1 ... ... ... |; 0 1 x_2 ... 1 0 0; 0 0 1 ... x_1 1 0; | ... ... ... x_2 x_1 1; 0 0 0 ... 1 x_2 x_1|+|x_2 1 0 ... 0 0 0; 2x_1 x_2 1 ... 0 0 0; 3 x_1 x_2 ... ... ... |; 0 1 x_1 ... 1 0 0; 0 0 1 ... x_2 1 0; | ... ... ... x_1 x_2 1; 0 0 0 ... 1 x_1 x_2|,
(2)

where the matrices have dimensions d×d. These represent surfaces in CP^3 with only ordinary double points as singularities. The first few surfaces are given by

x+y+z=0
(3)
x^2+y^2+2z^2=1+2x+2y
(4)
6+x^3+y^3+4z^3=3(2xy+z).
(5)

The dth order such surface has

N(d)={1/(12)(5d^3-13d^2+12d) if d=0 (mod 6); 1/(12)(5d^3-13d^2+16d-8) if d=2,4 (mod 6); 1/(12)(5d^3-14d^2+13d-4) if d=1,5 (mod 6); 1/(12)(5d^3-14d^2+9d) if d=3 (mod 6)
(6)

singular points (Chmutov 1992), giving the sequence 0, 1, 3, 14, 28, 57, 93, 154, 216, 321, 425, 576, 732, 949, 1155, ... (OEIS A057870) for d=1, 2, .... For a number of orders d, Chmutov surfaces have more ordinary double points than any other known equations of the same degree.

Chmutov surfaces of order 4, 6, and 10
CmutovSurface

Based on Chmutov's equations, Banchoff (1991) defined the simpler set of surfaces

T_n(x)+T_n(y)+T_n(z)=0,
(7)

where n is even and T_n(x) is again a Chebyshev polynomial of the first kind. For example, the surfaces illustrated above have orders 2, 4, and 6 and are given by the equations

2(x^2+y^2+z^2)=3
(8)
3+8(x^4+y^4+z^4)=8(x^2+y^2+z^2)
(9)
2[x^2(3-4x^2)^2+y^2(3-4y^2)^2+z^2(3-4z^2)^2]=3.
(10)

See also

Goursat's Surface, Ordinary Double Point, Superellipse, Tanglecube

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References

Banchoff, T. F. "Computer Graphics Tools for Rendering Algebraic Surfaces and for Geometry of Order." In Geometric Analysis and Computer Graphics: Proceedings of a Workshop Held May 23-25, 1988 (Ed. P. Concus, R. Finn, and D. A. Hoffman). New York: Springer-Verlag, pp. 31-37, 1991.Chmutov, S. V. "Examples of Projective Surfaces with Many Singularities." J. Algebraic Geom. 1, 191-196, 1992.Hirzebruch, F. "Singularities of Algebraic Surfaces and Characteristic Numbers." In The Lefschetz Centennial Conference, Part I: Proceedings of the Conference on Algebraic Geometry, Algebraic Topology, and Differential Equations, Held in Mexico City, December 10-14, 1984 (Ed. S. Sundararaman). Providence, RI: Amer. Math. Soc., pp. 141-155, 1986.Sloane, N. J. A. Sequence A057870 in "The On-Line Encyclopedia of Integer Sequences."Trott, M. Graphica 1: The World of Mathematica Graphics. The Imaginary Made Real: The Images of Michael Trott. Champaign, IL: Wolfram Media, pp. 3 and 82, 1999.Trott, M. The Mathematica GuideBook for Graphics. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.

Cite this as:

Weisstein, Eric W. "Chmutov Surface." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ChmutovSurface.html

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