Covering rational surfaces with rational parametrization images

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.

16 pages. Submitted