Automorphisms of Danielewski varieties

Abstract

In 2007, Dubouloz introduced Danielewski varieties. Such varieties generalize Danielewski surfaces and provide counterexamples to generalized Zariski cancellation problem in arbitrary dimension. In the present work we describe the automorphism group of a Danielewski variety. This result is a generalization of a description of automorphisms of Danielewski surfaces due to Makar-Limanov.

Introduction

We assume that the base field $\mathbb{K}$ is an algebraically closed field of characteristic zero. Let V and W be affine algebraic varieties over $\mathbb{K}$. Generalized Zariski cancellation problem asks whether $V\times \mathbb{K}\cong W\times \mathbb{K}$ implies $V\cong W$. In 1989 Danielewski [6] introduced a class of surfaces given by $x{y}^n=P(z)$. Nowadays such varieties are called Danielewski surfaces. Danielewski surfaces corresponding to $n=1$ and $n=2$ provide a counterexample to the generalized Zariski cancellation problem. One of the possible ways to prove that varieties $\mathbb{V}(xy-P(z))$ and $\mathbb{V}(x{y}^2-P(z))$ are nonisomorphic is suggested by Makar-Limanov in [11], [12]. He computes the intersection of all kernels of locally nilpotent derivations in both cases and shows that, if $n=1$, the intersection is the field $\mathbb{K}$, and if $n=2$, the intersection is $\mathbb{K}[y]$. This intersection later was named Makar-Limanov invariant.

Moreover, Makar-Limanov in [11], [12] computes generators of automorphism groups of Danielewski surfaces. This inspired a lot of works on automorphisms of Danielewski surfaces and their generalizations. In [5] Crachiola investigated automorphism groups of surfaces of the form $\mathbb{V}(x{y}^n-z^2-h(y)z)$ over a field of arbitrary characteristic. In [8] Dubouloz and Poloni considered a more general class of surfaces $\mathbb{V}(x{y}^n-Q(y,z))$, where $n\ge 2$ and $Q(y,z)$ is a polynomial with coefficients in an arbitrary base field such that $Q(0,z)$ splits with $r\ge 2$ simple roots. Generators of their automorphism groups were computed. In [14] classifications of varieties of the form $\mathbb{V}(x{y}^n-Q(y,z))$ up to isomorphism and up to automorphism of ambient affine 3-space are obtained. In [4] Bianchi and Veloso considered one more generalization, surfaces $\mathbb{V}(xf(y)-Q(y,z))$, where $\mathrm{deg}f\ge 2$ and

$$Q(y,z)=z^d+s_{d-1}(y)z^{d-1}+\dots +s_0(y),d\ge 2.$$

For such surfaces it was proved that the Makar-Limanov invariant is $\mathbb{K}[y]$. This allowed to compute generators of the automorphism group of the varieties $\mathbb{V}(xf(y)-Q(z))$.

In 2007, Dubouloz [7] considered another generalization of Danielewski surfaces. He introduced varieties given by equations of the form

$$x{y}_1^{k_1}\dots y_m^{k_m}=z^d+s_{d-1}(y_1,\dots ,y_m)z^{d-1}+\dots +s_0(y_1,\dots y_m),$$

which was called Danielewski varieties. He proved that if $d\ge 2$ and all $k_i\ge 2$, then the Makar-Limanov invariant of such a variety is equal to $\mathbb{K}[y_1,\dots ,y_m]$. Using this Dubouloz found counterexamples to generalized Zariski cancellation problem in arbitrary dimension.

In this paper we describe automorphism groups of Danielewski varieties. To do this we need to investigate automorphism groups of other classes of varieties. So, we proceed in three steps.

Sections 2 Derivations, 3 , 4 contain preliminaries and lemmas, which we use later.

First of all, in Section 5, we consider varieties of the form $Z=\mathbb{V}(y_1^{k_1}\dots y_m^{k_m}-P(z))$. There exists an $(m-1)$-dimensional algebraic torus action on such a variety. We elaborate some technique, which shows that if all $k_i\ge 2$, then Z is rigid, that is it does not admit any nontrivial locally nilpotent derivations. Then we compute the generators of the automorphism group of Z, see Theorem 5.1. We prove that the automorphism group $\mathrm{Aut}(Z)$ is a semidirect product of two groups. The first group multiply variables by invertible functions and the second one is a finite group permuting variables. If Z does not admit any invertible regular functions, $\mathrm{Aut}(Z)$ is a linear algebraic group. Moreover, it is a finite extension of an algebraic torus. For generic variety Z, the automorphism group is a torus, but in special cases it has finite or discrete part, which can be rather large. We obtain criteria of commutativity, connectivity and solvability of $\mathrm{Aut}(Z)$.

Secondly, in Section 6, we investigate varieties of the form $Y=\mathbb{V}(x{y}_1^{k_1}\dots y_m^{k_m}-P(z))$ with all $k_i\ge 2$. Such varieties are particular cases of Danielewski varieties. In some sense these particular cases are the most interesting ones because they provide counterexamples to generalized Zariski cancellation problem. Using rigidity of Z we prove that the Makar-Limanov invariant of Y equals $\mathbb{K}[y_1,\dots ,y_m]$ and all locally nilpotent derivations are replicas of a canonical one, see Definition 6.1. We compute generators of $\mathrm{Aut}(Y)$, see Proposition 6.3. The automorphism group of Y is generated by exponents of replicas of the canonical locally nilpotent derivation, the quasitorus acting by multiplying variables by constants, and swappings of variables with coinciding $k_i$. Moreover, $\mathrm{Aut}(Y)$ is isomorphic to a semidirect products of these subgroups, see Theorem 6.4. We prove that $\mathrm{Aut}(Y)$ is never commutative and it is solvable if and only if there are no five variables $y_i$ with coinciding $k_i$.

In Section 7, we use technique inspired by [12] to compute the group of automorphisms of an arbitrary Danielewski variety X. The main idea is to consider a filtration on $\mathbb{K}[X]$ and to prove that the associated graduate algebra $\mathrm{Gr}(\mathbb{K}[X])$ is isomorphic to $\mathbb{K}[Y]$ for some Y considered in Section 6. Then we prove that every automorphism of X respects this filtration, see Proposition 2.10. Using this we introduce a homomorphism $\mathrm{\Phi }:\mathrm{Aut}(X)\to \mathrm{Aut}(Y)$. This allows to describe automorphisms of X using the description of automorphisms of Y, see Theorem 7.11. Again all locally nilpotent derivations on X are replicas of a canonical one, see Definition 7.1. The automorphism group of X is isomorphic to a semidirect product of commutative group consisting of exponents of all replicas of the canonical locally nilpotent derivation and a canonical group $\mathbb{G}$ of X. The group $\mathbb{G}$ is a finite extension of a torus. Every element of $\mathbb{G}$ permutes the variables and multiplies them by constants. For generic variety X the group $\mathbb{G}$ is trivial and $\mathrm{Aut}(X)$ is commutative. We prove that it is a criterium of commutativity of $\mathrm{Aut}(X)$. Also we give a sufficient condition of its solvability.

The author is grateful to Ivan Arzhantsev and Alexander Gaifullin for useful discussions. He is also grateful to the referee for useful comments and suggestions.