Octics with 168 nodes

In 1995 S. Endraß gave the equations of two surfaces

X₈ and

X₈' of degree eight (octics) with 168 ordinary nodes. All nodes of

X₈ are real, whereas

X₈' has got 24 complex nodes.

$X8 = {64(x^2-w^2)(y^2-w^2)((x+y)^2-2w^2)((x-y)^2-2w^2) -[a(x^2+y^2)^2+(bz^2+cw^2)(x^2+y^2)-16z^4+dz^2w^2+fw^4]^2 = 0}, a = -4(1+sqrt(2)), b = 8(2+sqrt(2)), c = 2(2+7sqrt(2)), d = 8(1-2sqrt(2)), f = -(1+12sqrt(2))$

[120x120 pixel image of X8] 54K

$X8' = {64(x^2-w^2)(y^2-w^2)((x+y)^2-2w^2)((x-y)^2-2w^2) -[a(x^2+y^2)^2+(bz^2+cw^2)(x^2+y^2)-16z^4+dz^2w^2-fw^4]^2 = 0}, a = -4(1-sqrt(2)), b = 8(2-sqrt(2)), c = 2(2-7sqrt(2)), d = 8(1+2sqrt(2)), f = -(1-12sqrt(2))$

[120x120 pixel image of X8'] 62K

The surfaces X₈ and X₈' are not projectively isomorphic and share the following properties: they were found in a five-dimensional family of octics with 112 nodes and are invariant under the symmetry group D₈xZ₂. They are eightfold covers of projective quartics with 13 nodes under the map $(x:y:z:w)|-->(x$ ²:y²:z²:w²). The construction could not have been done without the help of a computer algebra system (Maple V): a system of five polynomial equations of degree up to 12 had to be solved to get the equations. The double solid branched over X₈ (resp. X₈') is a Calabi-Yau threefold with defect 19.

Stephan Endraß

Last update: Thursday, 06-Feb-2003 08:52:42 CET

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