A virtually ample field that is not ample

TL;DR

The paper constructs an example of a field that becomes ample after a finite extension but is not ample itself, challenging the understanding of the properties of ample fields.

Contribution

It demonstrates the existence of a virtually ample field that is not ample, providing a counterexample to the assumption that virtual ampleness implies ampleness.

Findings

Existence of a virtually ample field that is not ample

Counterexample to the equivalence of ample and virtually ample fields

Clarifies the distinction between ample and virtually ample fields

Abstract

A field $K$ is called ample if for every geometrically integral $K$-variety $V$ with a smooth $K$-point, $V(K)$ is Zariski-dense in $V$. A field $K$ is virtually ample if some finite extension of $K$ is ample. We prove that there exists a virtually ample field that is not ample.