Cubic surfaces with infinite, discrete automorphism group

TL;DR

This paper proves that certain affine cubic surfaces have automorphism groups containing an infinite cyclic subgroup of finite index, advancing understanding of their symmetry structures.

Contribution

It establishes that affine cubic surfaces defined by equations of the form xyz=g(x,y) have automorphism groups with a finite index subgroup isomorphic to the integers.

Findings

Automorphism group contains Z as a finite index subgroup

Cubic surfaces with specific equations have rich symmetry groups

Advances understanding of automorphism groups of affine cubic surfaces

Abstract

We prove that the automorphism group of an affine, cubic surface with equation $xyz=g(x,y)$ contains ${\mathbb Z}$ as a finite index subgroup. These equations were first studied by Mordell. v.2: small changes, references updated.