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Algebraic models of the line in the real affine plane

Adrien Dubouloz, Frédéric Mangolte

TL;DR

The paper investigates smooth embeddings of the real affine line into the real affine plane that arise from rational maps, classifying them up to birational diffeomorphisms of the plane. It introduces and distinguishes rectifiable and biddable rational smooth embeddings, linking their existence to the boundary geometry of $\mathbb{A}^{1}$-fibrations on algebraic models of $\mathbb{R}^{2}$ and to the real Kodaira dimension as a numerical obstruction. A key finding is that, unlike the complex/smooth setting, there are non-equivalent embeddings in the real rational setting; in particular, there exist non-rectifiable embeddings for degrees $d\ge 5$ constructed from nodal real rational plane curves. The paper provides a degree-by-degree classification for degrees up to four, showing that all embeddings with $\deg C\le 3$ are rectifiable and that quartics are typically rectifiable as well, with explicit exceptional configurations. Overall, the work reveals a richer birational-diffeomorphism landscape for real embeddings than in the purely algebraic or smooth categories, highlighting the roles of $\mathbb{A}^{1}$-fibrations, real Kodaira dimension, and real nodal plane curves in shaping equivalence classes.

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.

Algebraic models of the line in the real affine plane

TL;DR

The paper investigates smooth embeddings of the real affine line into the real affine plane that arise from rational maps, classifying them up to birational diffeomorphisms of the plane. It introduces and distinguishes rectifiable and biddable rational smooth embeddings, linking their existence to the boundary geometry of -fibrations on algebraic models of and to the real Kodaira dimension as a numerical obstruction. A key finding is that, unlike the complex/smooth setting, there are non-equivalent embeddings in the real rational setting; in particular, there exist non-rectifiable embeddings for degrees constructed from nodal real rational plane curves. The paper provides a degree-by-degree classification for degrees up to four, showing that all embeddings with are rectifiable and that quartics are typically rectifiable as well, with explicit exceptional configurations. Overall, the work reveals a richer birational-diffeomorphism landscape for real embeddings than in the purely algebraic or smooth categories, highlighting the roles of -fibrations, real Kodaira dimension, and real nodal plane curves in shaping equivalence classes.

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.

Paper Structure

This paper contains 17 sections, 19 theorems, 19 equations, 5 figures, 2 tables.

Table of Contents

  1. Preliminaries
  2. Euclidean topologies and birational diffeomorphisms
  3. Pairs, log-resolutions and SNC completions
  4. $\mathbb{A}^{1}$-fibered algebraic models of $\mathbb{R}^{2}$
  5. Rational closed embeddings of lines
  6. First examples and non-examples
  7. Rational embeddings of lines and projective rational plane curves
  8. Biddable and rectifiable lines
  9. Real Kodaira dimension: a numerical obstruction to biddability
  10. Rectifiability of biddable lines: a sufficient criterion
  11. Families of non-biddable affine lines
  12. Classification up to degree four
  13. Degree lower than or equal to $3$
  14. The quartic case
  15. Three singular points
  16. ...and 2 more sections

Key Result

Theorem 1

For every integer $d\geq5$ there exists a non-rectifiable rational smooth embedding of $\mathbb{A}_{\mathbb{R}}^{1}$ into $\mathbb{A}_{\mathbb{R}}^{2}$ whose associated projective curve $C\subset\mathbb{P}_{\mathbb{R}}^{2}$ is a rational nodal curve of degree $d$.

Figures (5)

  • Figure 4.1: Three singular points $A_{1}^*+2A_{1}^{i}$.
  • Figure 4.2: Three singular points, step 1: case $A_{2}+2A_{1}^{i}$ on the left and $A_{2}+2A_{2}^{i}$ on the right.
  • Figure 4.3: Three singular points, step 3: case $A_{2}+2A_{1}^{i}$ or $A_{2}+2A_{2}^{i}$.
  • Figure 4.4: Two singular points: case $A_{2}+A_{3}^{*}$ on the left and $A_{4}+A_{1}^{*}$ on the right.
  • Figure 4.5: One singular point: case $A_6$ on the left and $A_5^*$ on the right.

Theorems & Definitions (48)

  • Theorem
  • Definition 1
  • Theorem 2
  • Remark 3
  • Lemma 4
  • Theorem 5
  • Example 6
  • Definition 7
  • Definition 8
  • Example 9
  • ...and 38 more