Danielewski Surfaces Research Articles

Characterization of affine [formula omitted]-surfaces of hyperbolic type

In this note we extend the result from [14] and prove that if S is an affine non-toric Gm-surface of hyperbolic type that admits a Ga-action and X is an affine irreducible variety such that Aut(X) is isomorphic to Aut(S) as an abstract group, then X is a Gm-surface of hyperbolic type. Further, we show that a smooth Danielewski surface Dp={xy=p(z)}⊂A3, where p has no multiple roots, is determined by its automorphism group seen as an ind-group in the category of affine irreducible varieties.

Integrable Generators of Lie Algebras of Vector Fields on textrm{SL}_2(mathbb {C}) and on xy = z^2

For the special linear group textrm{SL}_2(mathbb {C}) and for the singular quadratic Danielewski surface x y = z^2 we give explicitly a finite number of complete polynomial vector fields that generate the Lie algebra of all polynomial vector fields on them. Moreover, we give three unipotent one-parameter subgroups that generate a subgroup of algebraic automorphisms acting infinitely transitively on x y = z^2.

Holomorphic Lie group actions on Danielewski surfaces

We prove that any Lie subgroup G (with finitely many connected components) of an infinite-dimensional topological group which is an amalgamated product of two closed subgroups can be conjugated to one factor. We apply this result to classify Lie group actions on Danielewski surfaces by elements of the overshear group (up to conjugation).

A general class of Danielewski algebras

Given a commutative field K of characteristic zero we study in this paper a class of K-algebras which contains the coordinate rings of the classical Danielewski surfaces. We describe their locally nilpotent K-derivations and compute their Makar-Limanov invariant. We also determine the K-algebras in our class that have a trivial Makar-Limanov invariant.

Automorphisms of Danielewski varieties

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.

Isomorphisms between cylinders over Danielewski surfaces

A special Danielewski surface is an affine surface which is the total space of a principal $$({{\mathbb {C}}},+)$$ -bundle over an affine line with a multiple origin. Using a fiber product trick introduced by Danielewski, it is known that cylinders over two such surfaces are always isomorphic provided that both bases have the same number of origins. The goal of this note is to give an explicit method to find isomorphisms between cylinders over special Danielewski surfaces. The method is based on the construction of appropriate locally nilpotent derivations.

On isotropy group of Danielewski surfaces

In the present work, we consider differential rings of the form where is a Danielewski surface and D is a locally nilpotent derivation on Influenced by several recent works, we describe the isotropy group of a locally nilpotent derivation, D, on Danielewski surfaces, in the cases and

Vector Fields and Automorphism Groups of Danielewski Surfaces

AbstractIn this paper, we study the Lie algebra of vector fields ${\operatorname{Vec}}(\textrm{D}_p)$ of a smooth Danielewski surface $\textrm{D}_p$. We prove that the Lie subalgebra $\langle{\operatorname{LNV}}(\textrm{D}_p) \rangle$ of ${\operatorname{Vec}}(\textrm{D}_p)$ generated by locally nilpotent vector fields is simple. Moreover, if the two Lie algebras $\langle{\operatorname{LNV}}(\textrm{D}_p) \rangle$ and $\langle{\operatorname{LNV}}(\textrm{D}_q) \rangle$ of two Danielewski surfaces $\textrm{D}_p$ and $\textrm{D}_q$ are isomorphic, then the surfaces $\textrm{D}_p$ and $\textrm{D}_q$ are isomorphic. As an application we prove that the ind-groups ${\operatorname{Aut}}(\textrm{D}_p)$ and ${\operatorname{Aut}}(\textrm{D}_q)$ are isomorphic if and only if $\textrm{D}_p \simeq \textrm{D}_q$ as a variety. We also show that any automorphism of the ind-group ${\operatorname{Aut}}^\circ (\textrm{D}_p)$ is inner.

ON THE CHARACTERIZATION OF DANIELEWSKI SURFACES BY THEIR AUTOMORPHISM GROUPS

In this note we show that if the automorphism group of a normal affine surface $S$ is isomorphic to the automorphism group of a Danielewski surface, then $S$ is isomorphic to a Danielewski surface.

Locally nilpotent derivations of double Danielewski surfaces

Extensive studies are being made on the family of Danielewski surfaces — they provide counter-examples to the Cancellation Problem. In [2], the authors investigated another family of non-cancellative surfaces which were named “double Danielewski surfaces”.In this note, we determine all the locally nilpotent derivations of a double Danielewski surface.

On double Danielewski surfaces and the Cancellation Problem

We study a two-dimensional family of affine surfaces which are counter-examples to the Cancellation Problem. We describe the Makar-Limanov invariant of these surfaces, determine their isomorphism classes and characterize the automorphisms of these surfaces.

Locally nilpotent derivations and automorphism groups of certain Danielewski surfaces

We describe the set of all locally nilpotent derivations of the quotient ring K[X,Y,Z]/(f(X)Y−φ(X,Z)) constructed from the defining equation f(X)Y=φ(X,Z) of a generalized Danielewski surface in K3 for a specific choice of polynomials f and φ, with 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 K-automorphisms of K[X,Y,Z]/(f(X)Y−φ(Z)) also for a specific choice of polynomials f and φ.

Lie Algebra generated by locally nilpotent derivations on Danielewski surfaces

We give a full description of the Lie algebra generated by locally nilpotent derivations (short LNDs) on smooth Danielewski surfaces Dp given by xy=p(z) . In case deg(p)≥3 it turns out to be not the whole Lie algebra VF ω alg (Dp) of volume preserving algebraic vector fields, thus answering a question posed by Lind and the first author. Also we show algebraic volume density property (short AVDP) for a certain homology plane, a homogeneous space of the form SL₂ (C)/N , where N is the normalizer of the maximal torus and another related example.

Complete algebraic vector fields on Danielewski surfaces

Nous donnons une classification de tous les champs de vecteurs algébriques complets sur les surfaces de Danielewski (surface lisse donnée par xy=p(z)). Nous utilisons le fait que pour chaque tel champ vectoriel, il existe une fibration préservée par son flot. Une classification des fonctions régulières avec fibre générique ℂ or ℂ * est requise pour obtenir la liste explicite des champs vectoriels. Nous présentons des résultats sur de telles fibrations définies sur des surfaces de Gizatullin et donnons une description précise de ces fibrés pour les surfaces de Danielewski.

Holomorphic Automorphisms of Danielewski Surfaces II: Structure of the Overshear Group

We apply Nevanlinna theory for algebraic varieties to Danielewski surfaces and investigate their group of holomorphic automorphisms. Our main result states that the overshear group which is known to be dense in the identity component of the holomorphic automorphism group, is a free amalgamated product.

Holomorphic automorphisms of Danielewski surfaces I — density of the group of overshears

We define the notion of shears and overshears on a Danielewski surface. We show that the group generated by shears and overshears is dense (in the compact open topology) in the path-connected component of the identity of the holomorphic automorphism group.

A Note on Noncancellable Varieties

Given two affine algebraic varieties X and Y such that X × ℂ ≅ Y × ℂ, it may happen that X and Y are not isomorphic. This is shown, for example, by so-called “Danielewski surfaces.” In this note, we give a simple construction of such varieties X, Y of an arbitrary dimension n > 1.

On a class of Danielewski surfaces in affine 3-space

In [L. Makar-Limanov, On groups of automorphisms of a class of surfaces, Israel J. Math. 69 (1990) 250–256] and [L. Makar-Limanov, On the group of automorphisms of a surface x n y = p ( z ) , Israel J. Math. 121 (2001) 113–123], L. Makar-Limanov computed the automorphism groups of surfaces in C 3 defined by the equations x n z − P ( y ) = 0 , where n ⩾ 1 and P ( y ) is a nonzero polynomial. Similar results have been obtained by A. Crachiola [A. Crachiola, On automorphisms of Danielewski surfaces, J. Algebraic Geom. 15 (2006) 111–132] for surfaces with equations x n z − y 2 − σ ( x ) y = 0 , where n ⩾ 2 and σ ( 0 ) ≠ 0 , defined over arbitrary base fields. Here we consider more general surfaces defined by equations x n z − Q ( x , y ) = 0 , where n ⩾ 2 and Q ( x , y ) is a polynomial with coefficients in an arbitrary base field k. We characterize among them the ones which are Danielewski surfaces in the sense of [A. Dubouloz, Danielewski–Fieseler surfaces, Transformation Groups 10 (2) (2005) 139–162], and we compute their automorphism groups. We study closed embeddings of these surfaces in affine 3-space. We show that in general their automorphisms do not extend to automorphisms of the ambient space. Finally, we give explicit examples of C ∗ -actions on a surface in A C 3 which can be extended holomorphically but not algebraically to C ∗ -actions on A C 3 .

Embeddings of Danielewski surfaces in affine space

We construct explicit embeddings of Danielewski surfaces [4] in affine spaces. The equations defining these embeddings are obtained from the 2\times2 minors of a matrix attached to a weighted rooted tree \gamma . We characterize those surfaces S_{\gamma} with a trivial Makar-Limanov invariant in terms of their associated trees. We prove that every Danielewski surface S with a nontrivial Makar-Limanov invariant admits a closed embedding in an affine space \mathbb{A}_{k}^{n} in such a way that every \mathbb{G}_{a,k} -action on S extends to an action on \mathbb{A}^{n} defined by a triangular derivation. We show that a Danielewski surface S with a trivial Makar-Limanov invariant and non-isomorphic to a hypersurface with equation xz-P(y)=0 in \mathbb{A}_{k}^{3} admits nonconjugated algebraically independent \mathbb{G}_{a,k} -actions.

Danielewski–Fieseler surfaces

We study a class of normal affine surfaces, with additive group actions, which contains in particular the Danielewski surfaces in A3 given by the equations xnz = P(y), where P is a nonconstant polynomial with simple roots. We call them Danielewski--Fieseler surfaces. We reinterpret a construction of Fieseler to show that these surfaces appear as the total spaces of certain torsors under a line bundle over a curve with an r fold point. We classify Danielewski-Fieseler surfaces through labelled rooted trees attached to such a surface in a canonical way. Finally, we characterize those surfaces which have a trivial Makar-Limanov invariant in terms of the associated trees.