Approximating curves on real rational surfaces
János Kollár, Frédéric Mangolte
TL;DR
The paper investigates when a simple closed curve on the real locus of a smooth real rational surface can be approximated by the real points of a smooth rational curve. It develops a minimal model program for pairs $(S,(1-\\epsilon)C)$ to classify the possible topologies of the real locus $(S(\\mathbb{R}),C(\\mathbb{R}))$ arising from such approximations and delivers a precise equivalence: a curve is approximable by real-smooth rational curves exactly when the pair is real-diffeomorphic to a rational-model pair, or equivalently when it is not diffeomorphic to the torus with a null-homotopic loop. The work provides a complete topological taxonomy of feasible $(S(\\mathbb{R}),C(\\mathbb{R}))$ pairs via MMP steps, including RP$^2$-sums, and demonstrates how singularities can be managed while preserving the real part. It also discusses broader approximation problems, offering obstructions in higher dimensions and proposing conjectures for rationally connected varieties, thereby outlining a path to generalize these results beyond smooth rational surfaces.
Abstract
We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex self-intersection number.