Algebraic models of the line in the real affine plane

TL;DR

This paper investigates algebraic rational embeddings of the real affine line into the real affine plane, revealing multiple non-equivalent classes under birational automorphisms that are diffeomorphisms.

Contribution

It demonstrates the existence of multiple non-equivalent smooth rational embeddings of the real line into the plane, contrasting with classical categories where only one class exists.

Findings

Multiple non-equivalent embeddings exist under birational automorphisms.

Contrast with smooth manifolds and algebraic varieties where only one class exists.

Provides algebraic models of the line in the real affine plane.

Abstract

We study smooth rational closed embeddings of the real affine line into the real affine plane, that is algebraic rational maps from the real affine line to the real affine plane which induce smooth closed embeddings of the real euclidean line into the real euclidean plane. We consider these up to equivalence under the group of birational automorphisms of the real affine plane which are diffeomorphisms of its real locus. We show that in contrast with the situation in the categories of smooth manifolds with smooth maps and of real algebraic varieties with regular maps where there is only one equivalence class up to isomorphism, there are non-equivalent smooth rational closed embeddings up to such birational diffeomorphisms.