Extension of automorphisms of rational smooth affine curves

TL;DR

This paper constructs a linear action of the automorphism group of any complex rational smooth affine curve on three-dimensional affine space, embedding the curve equivariantly, and explains why lower dimensions are impossible.

Contribution

It demonstrates the existence of a specific equivariant embedding of rational smooth affine curves into three-dimensional space, establishing a dimension bound.

Findings

Existence of a linear automorphism action on a73 for all such curves

Construction of an equivariant embedding of the curve into a73

Identification of obstructions to lower-dimensional embeddings

Abstract

We provide the existence, for every complex rational smooth affine curve $\Gamma$, of a linear action of $\mathrm{Aut}(\Gamma)$ on the affine 3-dimensional space $\mathbb{A}^3$, together with a $\mathrm{Aut}(\Gamma)$-equivariant closed embedding of $\Gamma$ into $\mathbb{A}^3$. It is not possible to decrease the dimension of the target, the reason for this obstruction is also precisely described.