Cubics

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Diagonal surface of Clebsch
A cubic with 27 straight lines
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Cayley's Cubic
A cubic with 4 A1-- singularities
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Cayley 1
A cubic with 3 A1+- singularities
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Cayley 2
A cubic with a D4-- singularity
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Cayley 3
A cubic with an A2 singularity
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Cayley 4
A cubic with a D6 singularity
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E6
A cubic with an E6 singularity. The most singular cubic surface.
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T[3,3,3]
A cubic with a T[3,3,3] singularity
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S20
A cubic with a line of A1 singularities
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S21
A cubic with two crosscaps
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S22
Yet another strange cubic
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S23
One way to unfold a cusp
 

See also unfolding of three planes for some more cubics.