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:A2-->S⊂Pn such that the union of the three images covers S. As a consequence, we present a second algorithm that generates two rational maps f,g˜:A2-->S, such that the union of its images covers the affine surface S∩An. In the affine case, the number of rational maps involved in the cover is in general optimal.

On the existence of finite surjective parametrizations of affine surfaces

“…Indeed, in [2] they establish necessary conditions for the existence of a birational surjective rational map from A 2 to an affine surface S, hence deducing plenty of examples of affine surfaces which do not admit a birational surjective parametrization from a Zariski open subset of A 2 . On the other hand, in [3] they construct an algorithm that, given any parametrization of a rational projective surface S, produces a cover of S by just three generically finite rational charts, provided that the input parametrization satisfy certain technical assumptions (namely, either its base locus is empty, or its Jacobian, specialized at each base point, has rank two). Here we generalize Theorem 3.1 and Corollary 3.5 of [2] to finite morphisms: Theorem 1.1.…”

A note about rational surfaces as unions of affine planes

“…In dimension two, as a consequence of the structure Theorem 1.2 below, all rational surfaces admit a covering of open subsets isomorphic to the affine plane. However, up to the authors' knowledge, no general results are known on the minimal number of open subsets of such a covering, while some advances are known by computer algebrists in terms of surjectivity of parametrizations [BR95,SSV17,CSSV18,CSSV21]. In this short note we prove that all projective smooth rational surfaces behave like the projective plane in this aspect.…”