Rationality problem of two-dimensional quasi-monomial group actions

TL;DR

This paper extends the solution to the rationality problem for two-dimensional quasi-monomial actions, focusing on actions defined within the base field, and employs Severi-Brauer varieties and conic bundle criteria.

Contribution

It generalizes previous results by solving the rationality problem for a broader class of two-dimensional quasi-monomial actions within the base field.

Findings

Complete solution for rationality of two-dimensional quasi-monomial actions within the base field

Use of Severi-Brauer varieties to analyze rationality

Application of conic bundle rationality criteria

Abstract

The rationality problem of two-dimensional purely quasi-monomial actions was solved completely by Hoshi, Kang and Kitayama [HKK]. As a generalization, we solve the rationality problem of two-dimensional quasi-monomial actions under the condition that the actions are defined within the base field. In order to prove the theorem, we give a brief review of the Severi-Brauer variety with some examples and rationality results. We also use a rationality criterion for conic bundles of $\mathbb{P}^1$ over non-closed fields.