Algebraic models of the Euclidean plane

TL;DR

This paper introduces a new invariant for real algebraic surfaces, enabling classification up to birational diffeomorphism, and constructs infinite non-trivial models of the Euclidean plane with unique properties.

Contribution

It develops the real logarithmic-Kodaira dimension as a novel invariant and constructs infinite families of rational surfaces with real loci diffeomorphic to  but not birationally diffeomorphic.

Findings

Infinite families of rational surfaces with trivial homology.

Surfaces with real loci diffeomorphic to  but not birationally diffeomorphic.

Existence of infinitely many non-trivial models of the Euclidean plane.

Abstract

We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real algebraic surfaces with trivial homology groups, whose real loci are diffeomorphic to $\mathbb{R}^2$, but which are pairwise not birationally diffeomorphic. There are thus infinitely many non-trivial models of the euclidean plane, contrary to the compact case.