# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1706.00492