Locally nilpotent derivations and automorphism groups of certain Danielewski surfaces

Abstract

We describe the set of all locally nilpotent derivations of the quotient ring $\mathbb{K}[X,Y,Z]/(f(X)Y-\phi (X,Z))$ constructed from the defining equation $f(X)Y=\phi (X,Z)$ of a generalized Danielewski surface in ${\mathbb{K}}^3$ for a specific choice of polynomials $f$ and $\phi$, with $\mathbb{K}$ an algebraically closed field of characteristic zero. As a consequence of this description we calculate the $ML$-invariant and the Derksen invariant of this ring. We also determine a set of generators for the group of $\mathbb{K}$-automorphisms of $\mathbb{K}[X,Y,Z]/(f(X)Y-\phi (Z))$ also for a specific choice of polynomials $f$ and $\phi$.

9 pages, comments are welcome