Birationally trivial real smooth cubic surfaces

TL;DR

This paper characterizes when smooth real cubic surfaces are birationally trivial over the reals, linking this property to the connectedness of their real locus and the existence of skew lines, and provides a geometric parametrization method.

Contribution

It introduces a simple geometric method to parametrize birationally trivial real cubic surfaces using an existing algorithm.

Findings

Birational triviality linked to connected real locus

Existence of skew lines characterizes triviality

Provides a geometric parametrization method

Abstract

Smooth real cubic surfaces are birationally trivial (over $\R$) if and only if their real locus is connected or, equivalently, if and only if they have two skew real lines or two skew complex conjugate lines. In such a case a parametrization over the reals can be given by cubic polynomials. In this short note we provide a simple geometric method to obtain such parametrization based in an algorithm by I. Polo-Blanco and J. Top \cite{Po-Top}.