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

TL;DR

This paper characterizes affine cubic surfaces associated with SL(2)-character varieties of certain surfaces, showing how specific geometric conditions correspond to these algebraic structures and automorphisms.

Contribution

It establishes a correspondence between smooth affine cubic surfaces with a generic tritangent plane and relative SL(2)-character varieties of 4-holed spheres and 1-holed tori.

Findings

Every smooth affine cubic with a generic tritangent plane arises from a 4-holed sphere.

Automorphisms of the tritangent plane extend to the cubic for certain surfaces.

The geometric properties of the cubic surface determine its relation to surface character varieties.

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.