On the existence of birational surjective parametrizations of affine surfaces

TL;DR

This paper investigates conditions under which affine rational complex surfaces can be parametrized birationally and surjectively, showing that many such surfaces do not admit these parametrizations due to geometric constraints.

Contribution

It establishes a necessary condition involving the infinity curve for the existence of birational surjective parametrizations and provides examples of surfaces that do not meet this condition.

Findings

Not all affine rational complex surfaces admit birational surjective parametrizations.

A necessary condition involves the presence of a rational component in the infinity curve.

Examples of surfaces without such parametrizations are provided.

Abstract

In this paper we show that not all affine rational complex surfaces can be parametrized birationally 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.