Covering rational surfaces with rational parametrization images

TL;DR

This paper introduces algorithms that generate a small number of rational maps whose images collectively cover rational surfaces and their affine parts, improving understanding of their parametrizations.

Contribution

It presents new algorithms for covering rational surfaces with a minimal number of rational parametrizations, under mild base locus assumptions.

Findings

Three rational maps cover the entire surface

Two rational maps cover the affine surface

Number of maps is generally optimal in the affine case

Abstract

Let $S$ be a rational projective surface given by means of a projective rational parametrization whose base locus satisfies a mild assumption. In this paper we present an algorithm that provides three rational maps $f,g,h:\mathbb{A}^2 --\to S\subset \mathbb{P}^n$ such that the union of the three images covers $S$. As a consequence, we present a second algorithm that generates two rational maps $f,\tilde{g}:\mathbb{A}^2 --\to S$, such that the union of their images covers the affine surface $S\cap \mathbb{A}^n$. In the affine case, the number of rational maps involved in the cover is in general optimal.