Abstract: Working over a ground field k of characteristic zero, this talk will discuss locally nilpotent derivations of rings of the form $B = R[z ]$, where $R$ is a commutative $k$-domain, and $z^n\in R$ for some positive integer $n$. Such a ring has a natural grading by $Z_n$ . We give basic properties of locally nilpotent derivations $D$ of $B$ which are homogeneous relative to this grading. In particular, $D$ is always a quasi-extension of a locally nilpotent derivation $\delta$ of $R$, and $D^2 z = 0$. This approach yields strong sufficient conditions for a ring of this type to be rigid, using in particular the absolute degree $|f |_R$ of elements of $R$. For example, we show that if $R$ is $Z$-graded, $f ∈ R$ is $Z$-homogeneous of degree coprime to $n$, and $|f |_R\ge 2$, then the ring $B = R[f^{1/n}]$ is rigid. The main idea is to study the locally nilpotent derivations of $B$ by looking at those of $R$.

Several applications of our results will be discussed. For example, we study $G_a$ -actions of Pham- Brieskorn surfaces and threefolds, with particular interest in questions of rigidity and stable rigidity. Our methods permit us to show rigidity for many cases which were previously open. This talk represents the speaker’s joint work with L. Moser-Jauslin.