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

Abstract

A natural family of affine cubic surfaces arises from SL(2)-characters of the 4-holed sphere and the 1-holed torus. The ideal locus is a tritangent plane which is generic in the sense that the cubic curve at infinity consists of three lines pairwise intersecting in three double points. We show that every affine cubic surface which is smooth at infinity and whose ideal locus is a generic tritangent plane arises as a relative SL(2)-character variety of the 4-holed sphere. Every such affine cubic for which all the periodic automorphisms of the tritangent plane extend to automorphisms of the cubic arises as a relative SL(2)-character variety of a 1-holed torus.