Minimal degree rational curves on real surfaces

TL;DR

This paper classifies real families of minimal degree rational curves covering embedded rational surfaces, providing a new algorithm to find such curves from surface parametrizations and revealing geometric constraints related to the surface's projective closure.

Contribution

It offers a classification of minimal degree rational curves on real surfaces and introduces an algorithm for identifying these curves from surface parametrizations.

Findings

If the surface's projective closure isn't isomorphic to the sphere, the covering rational curves are limited in dimension.

Almost all minimal degree rational curves over the reals are smooth when not minimal over the complex numbers.

The developed algorithm efficiently finds all minimal degree rational curves covering a given real surface.

Abstract

We classify real families of minimal degree rational curves that cover an embedded rational surface. A corollary is that if the projective closure of a smooth surface is not biregular isomorphic to the projective closure of the unit-sphere, then the set of minimal degree rational curves that cover the surface is either empty or of dimension at most two. Moreover, if these curves are of minimal degree over the real numbers, but not over the complex numbers, then almost all the curves are smooth. Our methods lead to an algorithm that takes as input a real surface parametrization and outputs all real families of rational curves of lowest possible degree that cover the image surface.