Lifting of the additive group scheme actions

Abstract

Let $B$ be a normal affine $\boldsymbol{C}$-domain and let $A$ be a $\boldsymbol{C}$-subalgebra of $B$ such that $B$ is a finite $A$-module. Let $\delta$ be a locally nilpotent derivation on $A$. Then $\delta$ lifts uniquely to the quotient field $L$ of $B$, which we denote by $\Delta$. We consider when $\Delta$ is a locally nilpotent derivation of $B$. This is a classical subject treated in [17, 19, 16]. We are interested in the case where $A$ is the $G$-invariant subring of $B$ when a finite group $G$ acts on $B$. As a related topic, we treat in the last section the finite coverings of log affine pseudo-planes in terms of the liftings of the $\boldsymbol{A}^1$-fibrations associated with locally nilpotent derivations.