Let $$𝕜$$ be a field of characteristic zero and G be a finite group of automorphisms of projective plane over $$𝕜$$ . Castelnuovo’s criterion implies that the quotient of projective plane by G is rational if the field $$𝕜$$ is algebraically closed. In this paper we prove that $${\mathbb{P}}_{𝕜}^2/\phantom{{\mathbb{P}}_{𝕜}^{2}G}G$$ is rational for an arbitrary field $$𝕜$$ of characteristic zero.

We give an overview of recent results concerning kernels of triangular derivations of polynomial rings. In particular, we examine the question of finite generation in dimensions 4, 5, 6, and 7.

We study $G$-actions of the form $(G:X\times X^{*})$, where $X^{*}$ is the dual (to $X$) $G$-variety. These actions are called the doubled ones. A geometric interpretation of the complexity of the action $(G:X)$ is given. It is shown that the doubled actions have a number of nice properties, if $X$ is spherical or of complexity one.