On the existence of birational surjective parametrizations of affine surfaces

Abstract

In this paper we show that not all affine rational complex surfaces can be parametrized birational and surjectively. For this purpose, we prove that, if S is an affine complex surface whose projective closure is smooth, a necessary condition for S to admit a birational surjective parametrization from an open subset of the affine complex plane is that the infinity curve of S must contain at least one rational component. As a consequence of this result we provide examples of affine rational surfaces that do not admit birational surjective parametrizations.

Covering Rational Surfaces with Rational Parametrization Images

“…Taking into account the results in [24], the affine cover presented in this paper is, in general, optimal. Furthermore, it improves the results in [6] and extends the results in [7] to a much more general class of surfaces.…”

“…Corollary 2. (Corollary 2.5 [24]) Let X and f be as in Theorem 2. For any fundamental point P of f , h(g −1 (P)) is a connected finite union of rational curves.…”

“…. Such a subset consists of some rational curves and, if f contracts L ∞ , a closed point (see Corollary 2.5 [24]).…”

“…Also, in [23] the relation of the polynomiality of parametrizations to the surjectivity was analyzed. Nevertheless, in [24] it is shown that there exist rational surfaces that cannot be parametrized birationally and surjectively. As a consequence of this fact, the question of whether every rational surface can be covered by the union of the images of finitely many birational parametrizations is of interest.…”

“…In [24], it is proved that there exist affine surfaces that cannot be covered by means of a unique map from the affine plane. In fact, the surface in Example 5 is proved to be one of them.…”