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Sarti Dodecic


The dodecic surface defined by

 X_(12)=243S_(12)-22Q_(12)=0,
(1)

where

Q_(12)=(x^2+y^2+z^2+w^2)^6
(2)
S_(12)=33sqrt(5)(s_(2,3)^-+s_(3,4)^-+s_(4,2)^-)+19(s_(2,3)^++s_(3,4)^++s_(4,2)^+)+10s_(2,3,4)-14s_(1,0)+2s_(1,1)-6s_(1,2)-352s_(5,1)+336l_5^2l_1+48l_2l_3l_4
(3)
l_1=x^4+y^4+z^4+w^4
(4)
l_2=x^2y^2+z^2w^2
(5)
l_3=x^2z^2+y^2w^2
(6)
l_4=x^2w^2+y^2z^2
(7)
l_5=xyzw
(8)
s_(1,0)=l_1(l_2l_3+l_2l_4+l_3l_4)
(9)
s_(1,1)=l_1^2(l_2+l_3+l_4)
(10)
s_(1,2)=l_1(l_2^2+l_3^2+l_4^2)
(11)
s_(5,1)=l_5^2(l_2+l_3+l_4)
(12)
s_(2,3,4)=l_2^3+l_3^3+l_4^3
(13)
s_(2,3)^+/-=l_2^2l_3+/-l_2l_3^2
(14)
s_(3,4)^+/-=l_3^2l_4+/-l_3l_4^2
(15)
s_(4,2)^+/-=l_4^2l_2+/-l_4l_2^2.
(16)

Q_(12) and S_(12) are both invariants of order 12. It was discovered by A. Sarti in 1999.

SartiDodecic

The version with arbitrary w and has exactly 600 ordinary points (Endraß), and taking w=1 gives the surface with 560 real ordinary points illustrated above.

The Sarti surface is invariant under the bipolyhedral group.


See also

Algebraic Surface, Bipolyhedral Group, Dodecic Surface

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References

Endraß, S. "The Sarti Surface." http://enriques.mathematik.uni-mainz.de/docs/Esarti.shtml.

Cite this as:

Weisstein, Eric W. "Sarti Dodecic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SartiDodecic.html

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