We give constructions of completions of the affine 3-space into total spaces of del Pezzo fibrations of every degree other than 7 over the projective line. We show in particular that every del Pezzo surface other than $\mathbb{P}^2$ blown-up in one or two points can appear as a closed fiber of a del Pezzo fibration $\pi: X\to \mathbb{P}^1$ whose total space $X$ is a $\mathbb{Q}$-factorial threefold with terminal singularities which contains $\mathbb{A}^3$ as the complement of the union of a closed fiber of $\pi$ and a prime divisor $B_h$ horizontal for $\pi$. For such completions, we also give a complete description of integral curves that can appear as general fibers of the induced morphism $\bar{\pi}:B_h \to \mathbb{P}^1$.

We study faithful actions with a dense orbit of abelian unipotent groups on quintic del Pezzo varieties over a field of characteristic zero. Such varieties are forms of linear sections of the Grassmannian of planes in a 5-dimensional vector space. We characterize which smooth forms admit these types of actions and show that in case of existence, the action is unique up to equivalence by automorphisms. We also give a similar classification for mildly singular quintic del Pezzo threefolds and surfaces.

We analyze the question of which motivic homotopy types admit smooth schemes as representatives. We show that given a pointed smooth affine scheme $X$ and an embedding into affine space, the affine deformation space of the embedding gives a model for the $\mathbb{P}^1$ suspension of $X$; we also analyze a host of variations on this observation. Our approach yields many examples of $\mathbb{A}^1-(n−1)$-connected smooth affine $2n$-folds and strictly quasi-affine $\mathbb{A}^1$-contractible smooth schemes.

We study toric $G$-solid Fano threefolds that have at most terminal singularities, where $G$ is an algebraic subgroup of the normalizer of a maximal torus in their automorphism groups.

We consider fibrations by affine lines on smooth affine surfaces obtained as complements of smooth rational curves $B$ in smooth projective surfaces $X$ defined over an algebraically closed field of characteristic zero. We observe that except for two exceptions, these surfaces $X\setminus B$ admit infinitely many families of $\mathbb{A}^1$-fibrations over the projective line with irreducible fibers and a unique singular fiber of arbitrarily large multiplicity. For $\mathbb{A}^1$-fibrations over the affine line, we give a new and essentially self-contained proof that the set of equivalence classes of such fibrations up to composition by automorphisms at the source and target is finite if and only if the self-intersection number of $B$ in $X$ is less than or equal to $6$.

We describe a method to construct completions of affine spaces into total spaces of $\mathbb{Q}$-factorial terminal Mori fiber spaces over the projective line. As an application we provide families of examples with non-rational, birationally rigid and non-stably rational general fibers.

The notion of commutator width of a group, defined as the smallest number of commutators needed to represent each element of the derived group as their product, has been extensively studied over the past decades. In particular, in 1992 Barge and Ghys discovered the first example of a simple group of commutator width greater than one among groups of diffeomorphisms of smooth manifolds. We consider a parallel notion of bracket width of a Lie algebra and present the first examples of simple Lie algebras of bracket width greater than one. They are found among the algebras of polynomial vector fields on smooth affine varieties.

We establish basic properties of a sheaf of graded algebras canonically associated to every relative affine scheme $~f:X\rightarrow S$ endowed with an action of the additive group scheme $\mathbb{G}_{a,S}$ over a base scheme or algebraic space $S$, which we call the (relative) Rees algebra of the $\mathbb{G}_{a,S}$-action. We illustrate these properties on several examples which played important roles in the development of the algebraic theory of locally nilpotent derivations and give some applications to the construction of families of affine threefolds with $\mathbb{G}_{a}$-actions.

We provide a complete description of normal affine algebraic varieties over the real numbers endowed with an effective action of the real circle, that is, the real form of the complex multiplicative group whose real locus consists of the unitary circle in the real plane. Our approach builds on the geometrico-combinatorial description of normal affine varieties with effective actions of split tori in terms of proper polyhedral divisors on semiprojective varieties due to Altmann and Hausen.

We give a general structure theorem for affine $\mathbb{A}^1$-fibrations on smooth quasi-projective surfaces. As an application, we show that every smooth $\mathbb{A}^1$-fibered affine surface non-isomorphic to the total space of a line bundle over a smooth affine curve fails the Zariski Cancellation Problem. The present note is an expanded version of a talk given at the Kinosaki Algebraic Geometry Symposium in October 2019.

We construct smooth rational real algebraic varieties of every dimension ≥ 4 which admit infinitely many pairwise non-isomorphic real forms.

We study smooth rational closed embeddings of the real affine line into the real affine plane, that is algebraic rational maps from the real affine line to the real affine plane which induce smooth closed embeddings of the real euclidean line into the real euclidean plane. We consider these up to equivalence under the group of birational automorphisms of the real affine plane which are diffeomorphisms of its real locus. We show that in contrat with the situation in the categories of smooth manifolds with smooth maps and of real algebraic varieties with regular maps where there is only one equivalence class up to isomorphism, there are plenty of non-equivalent smooth rational closed embeddings up to such birational diffeomorphisms. Some of these are simply detected by the non-negativity of the real Kodaira dimension of the complement of their images. But we also introduce finer invariants derived from topological properties of suitable fake real planes associated to certain classes of such embeddings.

We give an algebro-geometric classification of smooth real affine algebraic surfaces endowed with an effective action of the real algebraic circle group $\mathbb{S}^1$ up to equivariant isomorphisms. As an application, we show that every compact differentiable surface endowed with an action of the circle $S^1$ admits a unique smooth rational real quasi-projective model up to $\mathbb{S}^1$-equivariant birational diffeomorphism.

We introduce Koras-Russell fiber bundles over algebraically closed fields of characteristic zero. After a single suspension, this exhibits an infinite family of smooth affine $\mathbb{A}^1$-contractible $3$-folds. Moreover, we give examples of stably $\mathbb{A}^1$-contractible smooth affine $4$-folds containing a Brieskorn-Pham surface, and a family of smooth affine $3$-folds with a higher dimensional $\mathbb{A}^1$-contractible total space.

Every deformed Koras-Russell threefold of the first kind $Y=\{x^nz=y^m−t^r+xh(x,y,t)\}$ in $\mathbb{A}^4$ is the algebraic quotient of proper Zariski locally trivial $\mathbb{G}_a$-action on $\mathrm{SL}_2×\mathbb{A}^1$.

Motivated by the study of the structure of algebraic actions the additive group on affine threefolds $X$, we consider a special class of such varieties whose algebraic quotient morphisms $X \rightarrow X/\!/\mathbb{G}_a$ restrict to principal homogeneous bundles over the complement of a smooth point of the quotient. We establish basic general properties of these varieties and construct families of examples illustrating their rich geometry. In particular, we give a complete classification of a natural subclass consisting of threefolds $X$ endowed with proper $\mathbb{G}_a$-actions, whose algebraic quotient morphisms $\pi : X \rightarrow X/\!/\mathbb{G}_a$ are surjective with only isolated degenerate fibers, all isomorphic to the affine plane $\mathbb{A}^2$ when equipped with their reduced structures.

Motivated by the general question of existence of open $\mathbb{A}^1$-cylinders in higher dimensional projective varieties, we consider the case of Mori Fiber Spaces of relative dimension three, whose general closed fibers are isomorphic to the quintic del Pezzo threefold $V_5$, the smooth Fano threefold of index two and degree five. We show that the total spaces of these Mori Fiber Spaces always contain relative $\mathbb{A}^2$-cylinders, and we characterize those admitting relative $\mathbb{A}^3$-cylinders in terms of the existence of certain special lines in their generic fibers.

We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real algebraic surfaces with trivial homology groups, whose real loci are diffeomorphic to $\mathbb{R}^2$, but which are pairwise not birationally diffeomorphic. There are thus infinitely many non-trivial models of the real affine plane, contrary to the compact case.

An algebraic variety is called $\mathbb{A}^{1}$-cylindrical if it contains an $\mathbb{A}^{1}$-cylinder, i.e. a Zariski open subset of the form $Z\times\mathbb{A}^{1}$ for some algebraic variety Z. We show that the generic fiber of a family $f:X\rightarrow S$ of normal $\mathbb{A}^{1}$-cylindrical varieties becomes $\mathbb{A}^{1}$-cylindrical after a finite extension of the base. Our second result is a criterion for existence of an $\mathbb{A}^{1}$-cylinder in X which we derive from a careful inspection of a relative Minimal Model Program ran from a suitable smooth relative projective model of X over S.

We study a wide class of affine varieties, which we call affine Fano varieties. By analogy with birationally super-rigid Fano varieties, we define super-rigidity for affine Fano varieties, and provide many examples and non-examples of super-rigid affine Fano varieties.

A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its etale endomorphisms are proper. We study the equivariant version of the conjecture for $\mathbb{Q}$-acyclic surfaces of negative Kodaira dimension and infinite algebraic groups. We show that it holds for groups other than $\mathbb{C}^*$, and for $\mathbb{C}^*$ we classify counterexamples relating them to Belyi-Shabat polynomials. Taking universal covers we get rational simply connected $\mathbb{C}^*$-surfaces of negative Kodaira dimension which admit non-proper $\mathbb{C}^*$-equivariant etale endomorphisms. We prove that for every integers $r\geq 1$, $k\geq2$, the $\mathbb{Q}$-acyclic rational hyperplane $u(1+u^rv)=w^k$, which has fundamental group $\mathbb{Z}/k\mathbb{Z}$ and negative Kodaira dimension, admits families of non-proper etale endomorphisms of arbitrarily high dimension and degree, whose members remain different after dividing by the action of the automorphism group by left and right composition.

We show that a del Pezzo fibration $\pi:V\rightarrow W$ of degre $d$ contains a vertical open cylinder, that is, an open subset whose intersection with the generic fiber of $\pi$ is isomorphic to $Z\times\mathbb{A}_{K}^{1}$ for some quasi-projective variety $Z$ defined over the function field $K$ of $W$, if and only if $d\geq5$ and $\pi:V\rightarrow W$ admits a rational section. We also construct twisted cylinders in total spaces of threefold del Pezzo fibrations $\pi:V\rightarrow\mathbb{P}^{1}$ of degree $d\leq4$.

We study real rational models of the euclidean affine plane $\mathbb{R}^{2}$ up to isomorphisms and up to birational diffeomorphisms. The analogous study in the compact case, that is the classification of real rational models of the real projective plane $\mathbb{R}\mathbb{P}^{2}$ is well known: up to birational diffeomorphisms, $\mathbb{P}^{2}(\mathbb{R})$ is the only model. A fake real plane is a smooth geometrically integral surface $S$ defined over $\mathbb{R}$ not isomorphic to $\mathbb{A}^2_\mathbb{R}$, whose real locus $S(\mathbb{R})$ is diffeomorphic to $\mathbb{R}^2$ and such that the complex surface $S_\mathbb{C}(\mathbb{C})$ has the rational homology type of $\mathbb{A}^2_\mathbb{C}$. We prove that fake planes exist by giving many examples and we tackle the question: does there exist fake planes $S$ such that $S(\mathbb{R})$ is not birationally diffeomorphic to $\mathbb{A}^2_\mathbb{R}(\mathbb{R})$ ?

We classify closed curves isomorphic to the affine line in the complement of a smooth rational projective plane conic $Q$. Over a field of characteristic zero, we show that up to the action of the subgroup of the Cremona group of the plane consisting of birational endomorphisms restricting to biregular auto-morphisms outside $Q$, there are exactly two such lines: the restriction of a smooth conic osculating $Q$ at a rational point and the restriction of the tangent line to $Q$ at a rational point. In contrast, we give examples illustrating the fact that over fields of positive characteristic, there exist exotic closed embeddings of the affine line in the complement of $Q$. We also determine an explicit set of birational endomorphisms of the plane whose restrictions generates the automorphism group of the complement of $Q$ over a field of arbitrary characteristic.

We construct algebraic families of exotic affine 3-spheres, that is, smooth affine threefolds diffeomorphic to a non-degenerate smooth complex affine quadric of dimension 3 but non algebraically isomorphic to it. We show in particular that for every smooth topologically contractible affine surface S with trivial automorphism group, there exists a canonical smooth family of pairwise non isomorphic exotic affine 3-spheres parametrized by the closed points of S.

We construct families of smooth affine surfaces with pairwise non isomorphic $\mathbb{A}^1$-cylinders but whose $\mathbb{A}^2$-cylinders are all isomorphic. These arise as complements of cuspidal hyperplane sections of smooth projective cubic surfaces.

We provide families of affine threefolds which are contractible in the unstable $\mathbb{A}^1$-homotopy category of Morel-Voevodsky and pairwise non-isomorphic, thus answering a question of A. Asok and B. Doran. As a particular case, we show that the Koras-Russell threefolds of the first kind are contractible, extending results of M. Hoyois, A. Krishna and P. A. Ostvaer.

We study certain pencils $\overline{f}:\mathbb{P}\dashrightarrow\mathbb{P}^{1}$ of del Pezzo surfaces generated by a smooth del Pezzo surface $S$ of degree less or equal to $3$ anti-canonically embedded into a weighted projective space $\mathbb{P}$ and an appropriate multiple of a hyperplane $H$. Our main observation is that every minimal model program relative to the morphism $\tilde{f}:\tilde{\mathbb{P}}\rightarrow\mathbb{P}^{1}$ lifting $\overline{f}$ on a suitable resolution $\sigma:\tilde{\mathbb{P}}\rightarrow\mathbb{P}$ of its indeterminacies preserves the open subset $\sigma^{-1}(\mathbb{P}\setminus H)\simeq\mathbb{A}^{3}$. As an application, we obtain projective completions of $\mathbb{A}^{3}$ into del Pezzo fibrations over $\mathbb{P}^{1}$ of every degree less or equal to $4$. We also obtain completions of $\mathbb{A}^{3}$ into Mori conic bundles, whose restrictions to $\mathbb{A}^{3}$ are twisted $\mathbb{A}_{*}^{1}$-fibrations over $\mathbb{A}^{2}$.

We describe a method to construct hypersurfaces of the complex affine $n$-space with isomorphic $\mathbb{ℂ}^∗$-cylinders. Among these hypersurfaces, we find new explicit counterexamples to the Laurent Cancellation Problem, i.e. hypersurfaces that are non isomorphic, although their $\mathbb{ℂ}^∗$-cylinders are isomorphic as abstract algebraic varieties. We also provide examples of non isomorphic varieties $X$ and $Y$ with isomorphic cartesian squares $X×X$ and $Y×Y$.

We characterize rational actions of the additive group on algebraic varieties defined over a field of characteristic zero in terms of a suitable integrability property of their associated velocity vector fields. This extends the classical correspondence between regular actions of the additive group on affine algebraic varieties and the so-called locally nilpotent derivations of their coordinate rings. This leads in particular to a complete characterization of regular additive group actions on semi-affine varieties in terms of their associated vector fields. Among other applications, we review properties of the rational counter-part of the Makar-Limanov invariant for affine varieties and describe the structure of rational homogeneous additive group actions on toric varieties.

We address a variant of Zariski Cancellation Problem, asking whether two varieties which become isomorphic after taking their product with an algebraic torus are isomorphic themselves. Such cancellation property is easily checked for curves, is known to hold for smooth varieties of log-general type by virtue of a result of Iitaka- Fujita and more generally for non $\mathbb{A}^1_*$-uniruled varieties. We show in contrast that for smooth affine factorial $\mathbb{A}^1_*$-ruled varieties, cancellation fails in any dimension bigger or equal to two.

In arXiv:1507.01574, we define and partially classify fake real planes, that is, minimal complex surfaces with conjugation whose real locus is diffeomorphic to the euclidean real plane $\mathbb{R}^2$. Classification results are given up to biregular isomorphisms and up to birational diffeomorphisms. In this note, we describe in an elementary way numerous examples of fake real planes and we exhibit examples of such planes of every Kodaira dimension $\kappa\in \{-\infty,0,1,2 \}$ which are birationally diffeomorphic to $\mathbb{R}^2$.

We show that the generic fiber of a family of smooth $\mathbb{A}^1$-ruled affine surfaces always carries an $\mathbb{A}^1$-fibration, possibly after a finite extension of the base. In the particular case where the general fibers of the family are irrational surfaces, we establish that up to shrinking the base, such a family actually factors through an $\mathbb{A}^1$-fibration over a certain scheme, induced by the MRC-fibration of a relative smooth projective model of the family. For affine threefolds fibered by irrational $\mathbb{A}^1$-ruled surfaces, this induced $\mathbb{A}^1$-fibration can also be obtained from a relative Minimal Model Program applied to a relative smooth projective model of the family.

We construct a smooth rational affine surface $S$ with finite automorphism group but with the property that the group of automorphisms of the cylinder $Sx\mathbb{A}^2$ acts infinitely transitively on the complement of a closed subset of codimension at least two. Such a surface $S$ is in particular rigid but not stably rigid with respect to the Makar-Limanov invariant.

A classical result of Miyanishi-Sugie and Keel-McKernan asserts that for smooth affine surfaces, affine-uniruledness is equivalent to affine-ruledness, both properties being in fact equivalent to the negativity of the logarithmic Kodaira dimension. Here we show in contrast that starting from dimension three, there exists smooth affine varieties which are affine-uniruled but not affine-ruled.

We establish that the isomorphy type as an abstract algebraic variety of the complement of an ample hyperplane sub-bundle $H$ of a projective space bundle of rank $r-1$ over the projective line depends only on the the $r$-fold self-intersection of $H$. In particular it depends neither on the ambient bundle nor on a particular ample hyperplane sub-bundle with given $r$-fold self-intersection. Our proof exploits the unexpected property that every such complement comes equipped with the structure of a non trivial torsor under a vector bundle on the affine line with a double origin.

We describe a family of rational affine surfaces $S$ with huge groups of automorphisms in the following sense: the normal subgroup of $\mathrm{Aut}(S)$ generated by all its algebraic subgroups is not generated by any countable family of such subgroups, and the quotient of $\mathrm{Aut}(S)$ by this subgroup contains a free group over an uncountable set of generators.

Let $(S, B)$ be the log pair associated with a projective completion of a smooth quasi-projective surface $V$. Under the assumption that the boundary $B$ is irreducible, we obtain an algorithm to factorize any automorphism of $V$ into a sequence of simple birational links. This factorization lies in the framework of the log Mori theory, with the property that all the blow-ups and contractions involved in the process occur on the boundary. When the completion $S$ is smooth, we obtain a description of the automorphisms of $V$ which is reminiscent of a presentation by generators and relations except that the "generators" are no longer automorphisms. They are instead isomorphisms between different models of $V$ preserving certain rational fibrations. This description enables one to define normal forms of automorphisms and leads in particular to a natural generalization of the usual notions of affine and Jonquieres automorphisms of the affine plane. When $V$ is affine, we show however that except for a finite family of surfaces including the affine plane, the group generated by these affine and Jonquieres automorphisms, which we call the tame group of $V$, is a proper subgroup of $\mathrm{Aut}(V)$.

We describe the structure of geometric quotients for proper locally triangulable additve group actions on locally trivial $\mathbb{A}^3$-bundles over a noetherian normal base scheme $X$ defined over a field of characteristic $0$. In the case where $\mathrm{dim} X=1$, we show in particular that every such action is a translation with geometric quotient isomorphic to the total space of a vector bundle of rank $2$ over $X$. As a consequence, every proper triangulable $\mathbb{G}_a$-action on the affine four space $\mathbb{A}^4$ over a field of characteristic $0$ is a translation with geometric quotient isomorphic to $\mathbb{A}^3$.

Every $\mathbb{A}^1$−bundle over the complex affine plane punctured at the origin, is trivial in the differentiable category but there are infinitely many distinct isomorphy classes of algebraic bundles. Isomorphy types of total spaces of such algebraic bundles are considered; in particular, the complex affine $3$-sphere admits such a structure with an additional homogeneity property. Total spaces of nontrivial homogeneous $\mathbb{A}^1$-bundles over the punctured plane are classified up to $\mathbb{G}_m$-equivariant algebraic isomorphism and a criterion for nonisomorphy is given. In fact the affine $3$-sphere is not isomorphic as an abstract variety to the total space of any $\mathbb{A}^1$-bundle over the punctured plane of different homogeneous degree, which gives rise to the existence of exotic spheres, a phenomenon that first arises in dimension three. As a by product, an example is given of two biholomorphic but not algebraically isomorphic threefolds, both with a trivial Makar-Limanov invariant, and with isomorphic cylinders.

An additive group action on an affine $3$-space over a complex Dedekind domain $A$ is said to be twin-triangular if it is generated by a locally nilpotent derivation of $A[y,z,t]$ of the form $rd/dy+p(y)d/dz + q(y)d/dt$, where $r$ belongs to $A$ and $p,q$ belong to $A[y]$. We show that these actions are translations if and only if they are proper. Our approach avoids the computation of rings of invariants and focuses more on the nature of geometric quotients for such actions.

The Russell cubic is a smooth contractible affine complex threefold which is not isomorphic to affine three-space. In previous articles, we discussed the structure of the automorphism group of this variety. Here we review some consequences of this structure and generalize some results to other hypersurfaces which arise as deformations of Koras-Russell threefolds.

We give families of examples of principal open subsets of the affine space $\mathbb{A}^{3}$ which do not have the cancellation property. We show as a by-product that the cylinders over Koras-Russell threefolds of the first kind have a trivial Makar-Limanov invariant.

For automorphisms of a polynomial ring in two variables over a domain $R$, we show that local tameness implies global tameness provided that every $2$-generated invertible $R$-module is free. We give many examples illustrating this property.

We construct two non isomorphic contractible affine threefolds $X$ and $Y$ with isomorphic cylinders, showing that the generalized Cancellation Problem has a negative answer in general for contractible affine threefolds. We also establish that $X$ and $Y$ are actually biholomorphic as complex analytic varieties, providing the first example of a pair of biholomorphic but not isomorphic exotic affine $3$-spaces.

We develop technics of birational geometry to study automorphisms of affine surfaces admitting many distinct rational fibrations, with a particular focus on the interactions between automorphisms and these fibrations. In particular, we associate to each surface $S$ of this type a graph encoding equivalence classes of rational fibrations from which it is possible to decide for instance if the automorphism group of $S$ is generated by automorphisms preserving these fibrations.

The Koras-Russell threefold is the hypersurface $X$ of the complex affine four-space defined by the equation $x^2y+z^2+t^3+x=0$. It is well-known that $X$ is smooth contractible and rational but that it is not algebraically isomorphic to affine three-space. The main result of this article is to show that there exists another hypersurface $Y$ of the affine four-space, which is isomorphic to $X$ as an abstract variety, but such that there exists no algebraic automorphism of the ambient space which restricts to an isomorphism between $X$ and $Y$. In other words, the two hypersurfaces are inequivalent. The proof of this result is based on the description of the automorphism group of $X$. We show in particular that all algebraic automorphisms of $X$ extend to automorphisms of the ambient space.

We show that the Makar-Limanov invariant of the cylinder over the Koras-Russell cubic affine threefold is trivial. This means that regular functions which are invariant under all algebraic actions of the additive group on this variety are constants.

L. Makar-Limanov computed the automorphisms groups of surfaces in $\mathbb{C}^3$ defined by the equations $x^nz−P(y)=0$, where $n\geq 1$ and $P(y)$ is a nonzero polynomial. Similar results have been obtained by A. Crachiola for surfaces defined by the equations $x^nz−y^2−h(x)y=0$, where $n\geq 2$ and $h(0)=\neq 0$, defined over an arbitrary base field. Here we consider the more general surfaces defined by the equations $x^nz−Q(x,y)=0$, where $n\geq 2$ and $Q(x,y)$ is a polynomial with coefficients in an arbitrary base field $k$. Among these surfaces, we characterize the ones which are Danielewski surfaces 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 the ambient space. Finally, we give explicit examples of $\mathbb{C}^*$-actions on a surface in $\mathbb{C}^3$ which can be extended holomorphically but not algebraically to a $\mathbb{C}^*$-action on $\mathbb{C}^3$.

The cancellation problem asks if two complex algebraic varieties $X$ and $Y$ of the same dimension such that $X\times\mathbb{C}$ and $Y\times\mathbb{C}$ are isomorphic are isomorphic. Iitaka and Fujita established that the answer is positive for a large class of varieties of any dimension. In 1989, Danielewski constructed a famous counter-example using smooth affine surfaces with additive group actions. His construction was further generalized by Fieseler and Wilkens to describe a larger class of affine surfaces. Here we construct higher dimensional analogues of these surfaces. We study algebraic actions of the additive group $\mathbb{C}_{+}$ on certain of these varieties, and we obtain counter-examples to the cancellation problem in any dimension $n\geq2$.

We construct explicit embeddings of generalized Danielewski surfaves in affine spaces. The equations defining these embeddings are obtained from the $2\times 2$ minors of a matrix attached to a labelled rooted tree. Then we describe more precisely those surfaces with a trivial Makar-Limanov invariant.

We study a class of normal affine surfaces with additive group actions which contains in particular the Danielewski surfaces in $\mathbb{A}^{3}$ given by the equations $x^nz=P(y)$, 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.