Locally nilpotent derivations and automorphism groups of certain Danielewski surfaces

TL;DR

This paper characterizes all locally nilpotent derivations and automorphism groups of certain generalized Danielewski surfaces, providing explicit invariants and generators for their automorphism groups.

Contribution

It offers a complete description of locally nilpotent derivations and automorphism groups for specific Danielewski surfaces, including calculations of invariants.

Findings

Computed the set of all locally nilpotent derivations.

Determined the ML and Derksen invariants of the ring.

Identified generators for the automorphism group.

Abstract

We describe the set of all locally nilpotent derivations of the quotient ring $\mathbb{K}[X,Y,Z]/(f(X)Y - \varphi(X,Z))$ constructed from the defining equation $f(X)Y = \varphi(X,Z)$ of a generalized Danielewski surface in $\mathbb K^3$ for a specific choice of polynomials $f$ and $\varphi$, 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 $\mathbb K[X,Y,Z]/(f(X)Y - \varphi(Z))$ also for a specific choice of polynomials $f$ and $\varphi$.