The isotropy group of a derivation on a Danielewski-type algebra

TL;DR

This work determines the automorphism group of Danielewski-type algebras $A_{c,q}=K[x,y,z]/(c(x)z-q(x,y))$ with $\deg c\ge2$ and $\deg_y q\ge2$, and proves that the isotropy group of any non-Locally Nilpotent Derivation is an algebraic group of dimension at most $3$. The automorphism group is shown to split as an inner semidirect product of the unipotent automorphism group by a torus-like subgroup ${\rm G}_{c,q}$, realized via a canonical homomorphism ${\rm Aut}_{K}(A_{c,q})\to {\rm G}_{c,q}$ and a corresponding embedding ${\rm G}_{c,q}\to {\rm Aut}_{K}(A_{c,q})$, yielding a complete abstract-group description. The isotropy result uses the ind-group structure on ${\rm Aut}_{K}(A_{c,q})$ and a case analysis of the induced action on the torus, showing either a torus-subgroup isotropy or a semidirect product with a ${\mathbb G}_a$-subgroup; a standard three-dimensional example demonstrates the bound is sharp. Collectively, the results advance understanding of automorphisms and isotropy in Danielewski-type algebras and related affine surfaces.

Abstract

Given an algebraically closed field $k$ of characteristic zero, we consider in this paper $k$-algebras of the form $$A_{c,q}=k[x,y,z]/\big(c(x)z-q(x,y)\big),$$ where $c(x)\in k[x]$ is a polynomial of degree at least two and $q(x,y)\in k[x,y]$ is a quasi-monic polynomial of degree at least two with respect to $y$. We give a complete description of the $k$-automorphism group of $A_{c,q}$ as an abstract group. Moreover, for every non-locally nilpotent $k$-derivation $δ$ of $A_{c,q}$ we prove that the isotropy group of $δ$ is a linear algebraic group of dimension at most three.

Theorems & Definitions (19)

• Proposition 1
• Theorem 2
• Proposition 3
• Proposition 4
• Lemma 5
• proof
• Lemma 6
• proof
• Theorem 7
• proof