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In this text, we wish to provide the reader with a short guide to recent works on the theory of dilatations in Commutative Algebra and Algebraic Geometry. These works fall naturally into two categories: one emphasises foundational and theoretical aspects and the other applications to existing theories., arXiv admin note: text overlap with arXiv:2303.07712

We initiate a study of punctured tubular neighborhoods and homotopy theory at infinity in motivic settings. We use the six functors formalism to give an intrinsic definition of the stable motivic homotopy type at infinity of an algebraic variety. Our main computational tools include cdh-descent for normal crossing divisors, Euler classes, Gysin maps, and homotopy purity. Under-adic realization, the motive at infinity recovers a formula for vanishing cycles due to Rapoport-Zink; similar results hold for Steenbrink's limiting Hodge structures and Wildeshaus' boundary motives. Under the topological Betti realization, the stable motivic homotopy type at infinity of an algebraic variety recovers the singular complex at infinity of the corresponding topological space. We coin the notion of homotopically smooth morphisms with respect to a motivic ∞-category and use it to show a generalization to virtual vector bundles of Morel-Voevodsky's purity theorem, which yields an escalated form of Atiyah duality with compact support. Further, we study a quadratic refinement of intersection degrees, taking values in motivic cohomotopy groups. For relative surfaces, we show the stable motivic homotopy type at infinity witnesses a quadratic version of Mumford's plumbing construction for smooth complex algebraic surfaces. Our construction and computation of stable motivic links of Du Val singularities on normal surfaces is expressed entirely in terms of Dynkin diagrams. In characteristic p > 0, this improves Artin's analysis on Du Val singularities through étale local fundamental groups. The main results in the paper are also valid for-adic sheaves, mixed Hodge modules, and more generally motivic ∞-categories., Nous commençons une étude des voisinages tubulaires épointés et de la théorie de l'homotopie à l'infini dans un cadre motivique. Nous utilisons le formalisme des six foncteurs pour donner une définition intrinsèque du type d'homotopie motivique stable à l'infini d'une variété algébrique. Nos principaux outils de calcul incluent la descente cdh pour les diviseurs à croisement normaux, les classes d'Euler, les morphismes de Gysin et la pureté homotopique. Dans le cadre de la réalisation adique, le motif à l'infini redonne une formule pour les cycles évanescents due à Rapoport-Zink ; des résultats similaires s'appliquent aux structures de Hodge limites de Steenbrink et aux motifs bord de Wildeshaus. La réalisation topologique de Betti du type d'homotopie motivique stable à l'infini d'une variété algébrique redonne le complexe singulier à l'infini de l'espace topologique correspondant. Nous introduisons la notion de morphismes homotopiquement lisses par rapport à une ∞-catégorie motivique et nous l'utilisons pour montrer une généralisation du théorème de pureté de Morel-Voevodsky, qui donne une forme généralisée de la dualité d'Atiyah à support compact. De plus, nous étudions un raffinement quadratique des degrés d'intersection, prenant des valeurs dans les groupes de cohomotopies motiviques. Pour les surfaces relatives, nous montrons que le type d'homotopie motivique stable à l'infini témoigne d'une version quadratique de la construction de plomberie de Mumford pour les surfaces algébriques complexes lisses. Notre construction et le calcul des liens motiviques stables des singularités de Du Val sur les surfaces normales sont exprimés entièrement en termes de diagrammes de Dynkin. En caractéristique p > 0, cela améliore l'analyse d'Artin sur les singularités Du Val à travers les groupes fondamentaux locaux étales. Les principaux résultats de l'article sont également valables pour les faisceaux l-adiques, les modules de Hodge mixtes, et plus généralement les ∞-catégories motiviques.

We construct real rational quasi-projective surfaces with positive dimensional algebraic moduli of mutually non-isomorphic real forms.

We initiate a study of punctured tubular neighborhoods and homotopy theory at infinity in motivic settings. We use the six functors formalism to give an intrinsic definition of the stable motivic homotopy type at infinity of an algebraic variety. Our main computational tools include cdh-descent for normal crossing divisors, Euler classes, Gysin maps, and homotopy purity. Under-adic realization, the motive at infinity recovers a formula for vanishing cycles due to Rapoport-Zink; similar results hold for Steenbrink's limiting Hodge structures and Wildeshaus' boundary motives. Under the topological Betti realization, the stable motivic homotopy type at infinity of an algebraic variety recovers the singular complex at infinity of the corresponding topological space. We coin the notion of homotopically smooth morphisms with respect to a motivic ∞-category and use it to show a generalization to virtual vector bundles of Morel-Voevodsky's purity theorem, which yields an escalated form of Atiyah duality with compact support. Further, we study a quadratic refinement of intersection degrees, taking values in motivic cohomotopy groups. For relative surfaces, we show the stable motivic homotopy type at infinity witnesses a quadratic version of Mumford's plumbing construction for smooth complex algebraic surfaces. Our construction and computation of stable motivic links of Du Val singularities on normal surfaces is expressed entirely in terms of Dynkin diagrams. In characteristic p > 0, this improves Artin's analysis on Du Val singularities through étale local fundamental groups. The main results in the paper are also valid for-adic sheaves, mixed Hodge modules, and more generally motivic ∞-categories.; Nous commençons une étude des voisinages tubulaires épointés et de la théorie de l'homotopie à l'infini dans un cadre motivique. Nous utilisons le formalisme des six foncteurs pour donner une définition intrinsèque du type d'homotopie motivique stable à l'infini d'une variété algébrique. Nos principaux outils de calcul incluent la descente cdh pour les diviseurs à croisement normaux, les classes d'Euler, les morphismes de Gysin et la pureté homotopique. Dans le cadre de la réalisation adique, le motif à l'infini redonne une formule pour les cycles évanescents due à Rapoport-Zink ; des résultats similaires s'appliquent aux structures de Hodge limites de Steenbrink et aux motifs bord de Wildeshaus. La réalisation topologique de Betti du type d'homotopie motivique stable à l'infini d'une variété algébrique redonne le complexe singulier à l'infini de l'espace topologique correspondant. Nous introduisons la notion de morphismes homotopiquement lisses par rapport à une ∞-catégorie motivique et nous l'utilisons pour montrer une généralisation du théorème de pureté de Morel-Voevodsky, qui donne une forme généralisée de la dualité d'Atiyah à support compact. De plus, nous étudions un raffinement quadratique des degrés d'intersection, prenant des valeurs dans les groupes de cohomotopies motiviques. Pour les surfaces relatives, nous montrons que le type d'homotopie motivique stable à l'infini témoigne d'une version quadratique de la construction de plomberie de Mumford pour les surfaces algébriques complexes lisses. Notre construction et le calcul des liens motiviques stables des singularités de Du Val sur les surfaces normales sont exprimés entièrement en termes de diagrammes de Dynkin. En caractéristique p > 0, cela améliore l'analyse d'Artin sur les singularités Du Val à travers les groupes fondamentaux locaux étales. Les principaux résultats de l'article sont également valables pour les faisceaux l-adiques, les modules de Hodge mixtes, et plus généralement les ∞-catégories motiviques.

We propose a framework to give a precise meaning to the intuitive notion of "family of real forms of a variety parametrised by a variety" and study some fundamental properties of this notion. As an illustration, for any $n \geq 1$, we construct the first example of a quasi-projective real surface whose mutually non-isomorphic real forms admit a moduli of dimension at least $n$, parametrised by the real points of an affine $n$-space. Expanding on these constructions, we can give quasi-projective real varieties of any dimension whose algebraic moduli of the non-isomorphic real forms has arbitrarily positive dimension., The article has a new title, since it now also discusses the concepts of real functors, relative real forms and the tautological complete relative form. This gives a clearer meaning to the notion of "family of real forms of a variety parametrised by a variety". The example of the quasi-projective surface with non-isomorphic real forms parametrised by affine n-space is recast in this new framework

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 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.

In this paper, we initiate a study of motivic homotopy theory at infinity. We use the six functor formalism to give an intrinsic definition of the stable motivic homotopy type at infinity of an algebraic variety. Our main computational tools include cdh-descent for normal crossing divisors, Euler classes, Gysin maps, and homotopy purity. Under $\ell$-adic realization, the motive at infinity recovers a formula for vanishing cycles due to Rapoport-Zink; similar results hold for Steenbrink's limiting Hodge structures and Wildeshaus' boundary motives. Under the topological Betti realization, the stable motivic homotopy type at infinity of an algebraic variety recovers the singular complex at infinity of the corresponding analytic space. We coin the notion of homotopically smooth morphisms with respect to a motivic $\infty$-category and use it to show a generalization to virtual vector bundles of Morel-Voevodsky's purity theorem, which yields an escalated form of Atiyah duality with compact support. Further we study a quadratic refinement of intersection degrees taking values in motivic cohomotopy groups. For relative surfaces, we show the stable motivic homotopy type at infinity witnesses a quadratic version of Mumford's plumbing construction for smooth complex algebraic surfaces. Our main results are also valid for $\ell$-adic sheaves, mixed Hodge modules, and more generally motivic $\infty$-categories., Comment: Draft, comments are welcome

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., 20 pages, new co-author and substantially revised version of the previous one

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., Comment: 30 pages

Affine ind-varieties are infinite dimensional generalizations of algebraic varieties which appear naturally in many different contexts, in particular in the study of automorphism groups of affine spaces. In this article we introduce and develop the basic algebraic theory of topologically integrable derivations of complete topological rings. We establish a bijective algebro-geometric correspondence between additive group actions on affine ind-varieties and topologically integrable derivations of their coordinate pro-rings which extends the classical fruitful correspondence between additive group actions on affine varieties and locally nilpotent derivations of their coordinate rings.

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., Comment: 24 pages

We give a general structure theorem for affine A1-fibrations on smooth quasi-projective surfaces. As an application, we show that every smooth A1-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., 於 城崎国際アートセンター(2019年10月21日-10月25日), 2019年度科学研究費補助金 基盤研究(S)(課題番号15H05738, 代表 金銅誠之), 2019年度科学研究費補助金 基盤研究(S)(課題番号16H06337, 代表 高橋篤史), 2019年度科学研究費補助金 基盤研究(S)(課題番号17H06127, 代表 齋藤政彦), Date : Oct. 21, 2019 (Mon) - Oct. 25, 2019 (Fri). Venue : Kinosaki International Arts Center., Kinosaki Algebraic Geometry Symposium 2019 is partially supported by Grantin-Aid for Scientific Research (S) 15H05738, (S) 16H06337, and (S) 17H06127. Organizers: Hiraku Kawanoue (Chubu University), Takashi Kishimoto (Saitama University), Shingo Taki (Tokai University)

We give a general structure theorem for affine A 1-fibrations on smooth quasi-projective surfaces. As an application, we show that every smooth 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.

International audience; Motivated by the general question of existence of open 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, the smooth Fano threefold of index two and degree five. We show that the total spaces of these Mori Fiber Spaces always contain a relative A(2)-cylinder, and we characterize those admitting relative A(3)-cylinders in terms of the existence of certain special lines in their generic fiber.; Dans le contexte général de l’étude de l’existence de A1-cylindresouverts dans les variétés projectives de dimension supérieure, nous considérons le cas des fibrations de Mori de dimension relative trois, dont les fibres générales sont isomorphes au volume quintique de del Pezzo, l’unique variété de Fano lisse de degré cinq et d’indice deux. Nous établissons que les espaces totaux de fibrations de Mori de ce type contiennent toujours unA2-cylindre relatif au-dessus de la base de la fibration. Nous donnons également une caractérisation reliant l’existence deA3-cylindres relatifs à l’existence de certaines droites spéciales dans la fibre générique de ces fibrations

International audience; 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 A1-contractible 3-folds. Moreover, we give examples of stably A1-contractible smooth affine 4-folds containing a Brieskorn–Pham surface, and a family of smooth affine 3-folds with a higher-dimensional A1-contractible total space.

International audience; The Zariski Cancellation problem for smooth affine surfaces asks whether two suchsurfaces whose products with the affine line are isomorphic are isomorphic themselves. Byresults of Iitaka-Fujita, the answer is positive for surfaces of non-negative Kodaira dimen-sion. By a characterization due to Miyanishi, surfaces of negative Kodaira dimension arefibered by the affine line, and by a celebrated result of Miyanishi-Sugie, the answer to theproblem is positive if one of the surfaces is the affine plane. On the other hand, exam-ples of non-isomorphicA1-fibered affine surfaces with isomorphicA1-cylinders were firstconstructed by Danielewski in 1989, and then by many other authors. All these counter-examples are essentially constructed by variants of the method employed by Danielewski,nowadays known as the ”Danielewski fiber product trick”. In this talk, I will explain thatthis method is actually more than a trick: re-interpreted in a suitable way, it providesa necessary and sufficient criterion for twoA1-fibered surfaces over a same affine basecurveCto have isomorphic relativeA1-cylinders overC. The characterization can beroughly stated as follows: two such surfaces have isomorphic relativeA1-cylinders overCif and only if their respectively relative (log)-canonical classes are equal when viewed ascertain naturally definedQ-divisors on a non-separated orbifold curve ̆CdominatingC,fully determined by the structure of the fibers of theA1-fibrations at hand. (Joint workin progress with S. Kaliman and M. Zaidenberg).

International audience; Every deformed Koras–Russell threefold of the first kind Y={xnz=ym−tr+xh(x,y,t)} in A4 is the algebraic quotient of proper Zariski locally trivial Ga-action on SL2×A1.

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 Ga-actions.

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 show that all complements of cuspidal hyperplane sections of smooth projective cubic surfaces have isomorphic $\mathbb{A}^{2}$ -cylinders. As a consequence, we derive that the $\mathbb{A}^{2}$ -cancellation problem fails in every dimension greater than or equal to $2$ .

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.

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

Every deformed Koras-Russell threefold of the first kind $Y=\left\{ x^{n}z=y^{m}-t^{r}+xh(x,y,t)\right\} $ in $\mathbb{A}^{4}$ is the algebraic quotient of proper Zariski locally trivial $\mathbb{G}_{a}$-action on $\mathrm{SL}_{2}\times\mathbb{A}^{1}$.

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, 16 pages

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

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 A1-contractible 3-folds. Moreover, we give examples of stably A1-contractible smooth affine 4-folds containing a Brieskorn-Pham surface, and a family of smooth affine 3-folds with a higher dimensional A1-contractible total space.

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//Ga 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 Ga-actions, whose algebraic quotient morphisms $\pi$ : X $\rightarrow$ X//Ga are surjective with only isolated degenerate fibers, all isomorphic to the affine plane A 2 when equipped with their reduced structures.

We give a criterion for a flat fibration with affine plane fibers over a smooth scheme defined over a field of characteristic zero to be a Zariski locally trivial $\mathbb{A}^2$-bundle. An application is a positive answer to a version of the Dolgachev-Weisfeiler Conjecture for such fibrations: a flat fibration $\mathbb{A}^m$ $\rightarrow$ $\mathbb{A}^n$ with all fibers isomorphic to $\mathbb{A}^2$ is the trivial $\mathbb{A}^2$-bundle.

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 ℚ-acyclic surfaces of negative Kodaira dimension and infinite algebraic groups. We show that it holds for groups other than ℂ∗, and for ℂ∗ we classify counterexamples relating them to Belyi-Shabat polynomials. Taking universal covers we get rational simply connected ℂ∗-surfaces of negative Kodaira dimension which admit non-proper ℂ∗-equivariant etale endomorphisms.We prove that for every integers r≥1, k≥2 the ℚ-acyclic rational hyperplane u(1+urv)=wk, which has fundamental group ℤ/kℤ 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.

International audience; We study certain pencils 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 P and an appropriate multiple of a hyperplane $H$. Our main observation is that every minimal model program relative to the morphism lifting such pencil on a suitable resolution of its indeterminacies preserves the open subset $P \ H ≃ A^3$. As an application, we obtain projective completions of $A^3$ into del Pezzo fibrations over $P^1$ of every degree less or equal to 4. We also obtain completions of $A^3$ into Mori conic bundles, whose restrictions to $A^3$ are twisted $C*$-fibrations over $A^2$.

International audience; We study real rational models of the euclidean 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, there is only one model. A fake real plane is a nonsingular affine surface defined over the reals with homologically trivial complex locus and real locus diffeomorphic to $\mathbb{R}^2$ but which is not isomorphic to the real affine plane. We prove that fake planes exist by giving many examples and we tackle the question: does there exist fake planes whose real locus is not birationally diffeomorphic to the real affine plane?

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., 10 pages, comments are welcome

We study certain pencils 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 P and an appropriate multiple of a hyperplane H. Our main observation is that every minimal model program relative to the morphism lifting such pencil on a suitable resolution of its indeterminacies preserves the open subset P H \^a A^3. As an application, we obtain projective completions of A^3 into del Pezzo fibrations over P^1 of every degree less or equal to 4. We also obtain completions of A^3 into Mori conic bundles, whose restrictions to A^3 are twisted C*-fibrations over A^2 .

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

Let $(S, B_{S})$ be the log pair associated with a projective completion of a smooth quasi-projective surface $V$. Under the assumption that the boundary $B_{S}$ is irreducible, we obtain an algorithm to factorize any automorphism of $V$ into a sequence of simple 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 Jonquières 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 Jonquières automorphisms, which we call the tame group of $V$, is a proper subgroup of $\mathrm{Aut}(V)$.

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.

Affine surfaces in $\mathbb{C}^{3}$ defined by an equation of the form $x^{n}z-Q(x,y)=0$ have been increasingly studied during the past 15 years. Of particular interest is the fact that they come equipped with an action of the additive group $\mathbb{C}_{+}$ induced by such an action on the ambient space. The litterature of the last decade may lead one to believe that there are essentially no other of rational surfaces in $\mathbb{C}^{3}$ with this property. In this note, we construct an explicit example of a surface nonisomorphic to a one of the above type but equipped with a free $\mathbb{C}_{+}$-action induced by an action on $\mathbb{C}^{3}$. We give an elementary and self-contained proof of this fact. As an application, we construct a wild but stably-tame automorphisme of $\mathbb{C}^{3}$ which seems to be new., Comment: This article is written in an expositional style and focuses on examples. It could also serve as an introduction to recent results concerning the classification of affine surfaces with additive group actions

We describe the structure of geometric quotients for proper locally triangulable [math] -actions on locally trivial [math] -bundles over a nœtherian normal base scheme [math] defined over a field of characteristic [math] . In the case where [math] , 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 [math] over [math] . As a consequence, every proper triangulable [math] -action on the affine four space [math] over a field of characteristic [math] is a translation with geometric quotient isomorphic to [math] .

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., To appear in Proceedings of the GABAG2012 conference

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

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.

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 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.

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 admitts 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.

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.

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.

The affine cancellation problem, which asks whether complex affine varieties with isomorphic cylinders are themselves isomorphic, has a positive solution for two dimensional varieties whose coordinate rings are unique factorization domains, in particular for the affine plane, but counterexamples are found within normal surfaces Danielewski surfaces and factorial threefolds of logarithmic Kodaira dimension equal to 1. The latter are therefore remote from the affine three-space, the first unknown case where the base of one cylinder is an affine space. Locally trivial Ga-actions play a significant role in these examples. Threefolds admitting free Ga-actions are discussed, especially a class of varieties with negative logarithmic Kodaira dimension which are total spaces of nonisomorphic Ga-bundles. Some members of the class are shown to be isomorphic as abstract varieties, but it is unknown whether any members of the class constitute counterexamples to cancellation.