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Real rational surfaces

Frederic Mangolte

TL;DR

The paper surveys real rational surfaces with an emphasis on topology and birational geometry, linking classical results (Comessatti, Nash–Tognoli) to modern advances (Minimal Model Program, infinite transitivity of birational diffeomorphisms) and bridging to regulous geometry. It outlines how real rational models classify topological real loci, establishes infinite transitivity and generating sets for automorphism groups of real loci, and presents density results comparing algebraic symmetries with smooth diffeomorphisms. A new direction, regulous maps, is developed to unify algebraic and continuous perspectives, including a robust algebraic foundation and homotopy-approximation results. Collectively, the work advances understanding of Cremona-type groups over $\mathbb{R}$, the topology of real loci, and the interface between algebraic and topological mappings on real algebraic surfaces, with implications for both theory and applications in real algebraic geometry and topology.

Abstract

We survey some results on real rational surfaces focused on their topology and their birational geometry.

Real rational surfaces

TL;DR

The paper surveys real rational surfaces with an emphasis on topology and birational geometry, linking classical results (Comessatti, Nash–Tognoli) to modern advances (Minimal Model Program, infinite transitivity of birational diffeomorphisms) and bridging to regulous geometry. It outlines how real rational models classify topological real loci, establishes infinite transitivity and generating sets for automorphism groups of real loci, and presents density results comparing algebraic symmetries with smooth diffeomorphisms. A new direction, regulous maps, is developed to unify algebraic and continuous perspectives, including a robust algebraic foundation and homotopy-approximation results. Collectively, the work advances understanding of Cremona-type groups over , the topology of real loci, and the interface between algebraic and topological mappings on real algebraic surfaces, with implications for both theory and applications in real algebraic geometry and topology.

Abstract

We survey some results on real rational surfaces focused on their topology and their birational geometry.

Paper Structure

This paper contains 11 sections, 15 theorems, 27 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Real rational surfaces
  3. Examples of rational surfaces
  4. Rational models
  5. Automorphism groups of real loci
  6. Transitivity
  7. Generators
  8. Approximation of differentiable maps by algebraic maps
  9. Real algebraic models
  10. Automorphism groups
  11. Regulous maps

Key Result

Corollary 6

Let $S$ be a compact connected topological surface. If $S$ is nonorientable, or orientable of genus $0$ or $1$, then $S$ admits a real rational model.

Figures (6)

  • Figure 1: The real locus of the exceptional curve is depicted by the vertical line.
  • Figure 2: On the left: the real locus of the real quartic curve given by $8x^4+20x^2y^2-24x^2+8y^4-24y^2+16,25=0$; on the right: the double plane ramified over it.
  • Figure 3: The sphere $\sS^2$ with two sets of parallels.
  • Figure 4: The effect of the Dehn twist around $C$ on a curve.
  • Figure 5: The Cartan umbrella: $z(x^2+y^2)=x^3$.
  • ...and 1 more figures

Theorems & Definitions (38)

  • Example 1
  • proof
  • Example 2
  • Remark 3
  • Remark 4
  • Definition 5
  • Corollary 6
  • Theorem 7: Comessatti
  • Example 8
  • Theorem 9
  • ...and 28 more