Rationality problem of two-dimensional quasi-monomial group actions

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.

To appear in Transformation Groups, 29 pages. Theorem 1.6 is improved. Remark 1.7 is added. Theorem 2.7 is deleted. Section 3 (Proof of Theorem 1.6) is greatly improved by referee's valuable suggestions. Remarks 3.1, 3.2, 3.3, 3.4 are added. Some typos are corrected