Simple derivations and isotropy on Danielewski-type algebras

Abstract

Let $\K$ be an algebraically closed field of characteristic zero. We study isotropy groups of simple derivations on Danielewski-type algebras \[ A_{c,q}=\K[x,y,z]/(c(x)z-q(x,y)). \] More precisely, motivated by the recent result of Mendes and Pan for simple derivations of $\K[x,y]$ (\cite[Theorem 1]{MP}), we ask whether every simple derivation $D$ of $A_{c,q}$ has trivial isotropy group. We prove that this property holds generically: for each fixed pair of degrees $°(c)\ge 2$ and ${°}_y(q)\ge 2$, there exists a nonempty Zariski open subset of the parameter space of reduced pairs $(c,q)$ such that every simple derivation of $A_{c,q}$ has trivial isotropy group. On the other hand, we construct a special Danielewski-type algebra admitting a simple derivation with nontrivial isotropy. Thus, the Mendes-Pan phenomenon holds generically for Danielewski-type algebras, but fails in full generality. Over $\K=\mathbb C$, we also discuss the associated foliations and formulate questions about their polynomial and birational symmetries.