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

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

2010-06-19

arxiv.org/abs/1006.3838v2

Mathematics

Geometric Topology

10 pages, no figures, various corrections and a shorter proof; some literature references added; manuscript now submitted for

Scientific paper

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.

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