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A natural family of affine cubic surfaces arises from SL(2)-characters of the4-holed sphere and the 1-holed torus. The ideal locus is a tritangent planewhich is generic in the sense that the cubic curve at infinity consists ofthree lines pairwise intersecting in three double points. We show that everyaffine cubic surface which is smooth at infinity and whose ideal locus is ageneric tritangent plane arises as a relative SL(2)-character variety of the4-holed sphere. Every such affine cubic for which all the periodicautomorphisms of the tritangent plane extend to automorphisms of the cubicarises as a relative SL(2)-character variety of a 1-holed torus.

Affine cubic surfaces and relative SL(2)-character varieties of compact surfaces