On the existence of finite surjective parametrizations of affine surfaces

TL;DR

This paper explores the limitations of finite surjective parametrizations of affine surfaces, demonstrating that many such surfaces cannot be covered by a finite surjective morphism from the affine plane.

Contribution

It provides explicit constructions of affine surfaces that do not admit finite surjective parametrizations, advancing understanding of rational parametrization constraints.

Findings

Many affine surfaces lack finite surjective parametrizations

Explicit examples of non-parametrizable surfaces are constructed

The work extends previous studies on rational algebraic variety parametrizations

Abstract

We investigate surjective parametrizations of rational algebraic varieties, in the vein of recent work by Jorge Caravantes, J. Rafael Sendra, David Sevilla, and Carlos Villarino. In particular, we show how to construct plenty of examples of affine surfaces $S$ not admitting a finite surjective morphism $f: \mathbb{A}^2 \to S$.