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nextstep

Joined: 06 Jan 2007
Posts: 539

Posted: Mon Sep 01, 2014 2:41 am    Post subject: Togliatti quintic (deg 5)

Togliatti (1940, 1949) showed that quintic surfaces having 31 ordinary double points exist, although he did not explicitly derive equations for such surfaces. Beauville (1978) subsequently proved that 31 double points are the maximum possible, and quintic surfaces having 31 ordinary double points are therefore sometimes called Togliatti surfaces. van Straten (1993) subsequently constructed a three-dimensional family of solutions and in 1994, Barth derived the example
 Quote: 64(x-w)[x^4-4x^3w-10x^2y^2-4x^2w^2+16xw^3-20xy^2w+5y^4+16w^4-20y^2w^2] -5sqrt(5-sqrt(5))(2z-sqrt(5-sqrt(5))w)[4(x^2+y^2+z^2)+(1+3sqrt(5))w^2]^2, where w is a parameter (Endraß 2003).

This surface is invariant under the group D_5 and contains exactly 15 lines. Five of these are the intersection of the surface with a D_5-invariant cone containing 16 nodes, five are the intersection of the surface with a D_5-invariant plane containing 10 nodes, and the last five are the intersection of the surface with a second D_5-invariant plane containing no nodes (Endraß 2003).
In MathMod:

TogliattiQuintics by taha_ab, on Flickr

 Quote: { "Iso3D": { "Cnd": [ "(x^2 + y^2 + z^2) > 100" ], "Component": [ "Togliatti" ], "Const": [ " w=1.3" ], "Fxyz": [ "64*(x -w)*(x^4 - 4*w*x^3 -1 0*x^2*y^2 - 4*x^2*w^2 + 16*w^3*x - 20*w*x*y^2 + 5*y^4 + 16*w^4 - 20*y^2*w^2) -5*sqrt(5 - sqrt(5))*(2*z - sqrt(5 - sqrt(5))*w)*(4*(x^2 + y^2 - z^2) + (1 + 3*sqrt(5))*w^2)^2" ], "Name": [ "Togliatti" ], "Xmax": [ "10" ], "Xmin": [ "-10" ], "Ymax": [ "10" ], "Ymin": [ "-10" ], "Zmax": [ "10" ], "Zmin": [ "-10" ] } }

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Abderrahman

Last edited by nextstep on Sat Sep 20, 2014 5:44 pm; edited 1 time in total
nextstep

Joined: 06 Jan 2007
Posts: 539

Posted: Sat Sep 13, 2014 6:11 pm    Post subject:

Hi,
Another Togliatti related surface, sometimes known as the "Dervish", can be defined by the MathMod's script :
 Quote: { "Iso3D": { "Name": [ "Dervish" ], "Component": [ " Dervish" ], "Cnd": [ "(x^2+y^2+z^2)>15" ], "Const": [ "r = (1/4)*(1+3*sqrt(5))", "a = -(8/5)*(1+1/(sqrt(5)))*sqrt(5-sqrt(5))", "c = (1/2)*sqrt(5-sqrt(5))" ], "Fxyz": [ "a*(x-z)*(cos((2*pi)/5)*x-sin((2*pi)/5)*y-z)*(cos((4*pi)/5)*x-sin((4*pi)/5)*y-z)*(cos((6*pi)/5)*x-sin((6*pi)/5)*y-z)*(cos((8*pi)/5)*x-sin((8*pi)/5)*y-z)+(1-c*z)*(x^2+y^2-1+r*z^2)^2" ], "Xmax": [ "4" ], "Xmin": [ "-4" ], "Ymax": [ "4" ], "Ymin": [ "-4" ], "Zmax": [ "4" ], "Zmin": [ "-4" ] } }

Dervish by taha_ab, on Flickr
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Abderrahman
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