Papers by Adrien Dubouloz
Rational quasi-projective surfaces with algebraic moduli of real forms
HAL (Le Centre pour la Communication Scientifique Directe), Jun 7, 2022
HAL (Le Centre pour la Communication Scientifique Directe), Sep 8, 2022
We study faithful actions with a dense orbit of abelian unipotent groups on quintic del Pezzo var... more 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 give families of examples of principal open subsets of the affine space \mathbb{A}^{3} which d... more 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
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... more 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 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
Le Centre pour la Communication Scientifique Directe - HAL - Université Paris Descartes, Feb 28, 2020
We give a general structure theorem for affine A 1-fibrations on smooth quasi-projective surfaces... more 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.
arXiv: Algebraic Geometry, Jun 15, 2011
Every A 1 −bundle over A 2 * , the complex affine plane punctured at the origin, is trivial in th... more Every A 1 −bundle over A 2 * , 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 S 3 C , given by z 2 1 + z 2 2 + z 2 3 + z 2 4 = 1, admits such a structure with an additional homogeneity property. Total spaces of nontrivial homogeneous A 1-bundles over A 2 * are classified up to Gm-equivariant algebraic isomorphism and a criterion for nonisomorphy is given. In fact S 3 C is not isomorphic as an abstract variety to the total space of any A 1-bundle over A 2 * 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.
American Mathematical Society, AMS, 2014
We analyze the question of which motivic homotopy types admit smooth schemes as representatives. ... more 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 P suspension of X ; we also analyze a host of variations on this observation. Our approach yields many examples of A-(n− 1)-connected smooth affine 2n-folds and strictly quasi-affine A-contractible smooth schemes.
We construct a smooth rational affine surface S with finite automorphism group but with the prope... more We construct a smooth rational affine surface S with finite automorphism group but with the property that the group of automorphisms of the cylinder SxA^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.
We describe a method to construct completions of affine spaces into total spaces of ℚ-factorial t... more We describe a method to construct completions of affine spaces into total spaces of 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.
We develop technics of birational geometry to study automorphisms of affine surfaces admitting ma... more 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.
We introduce Koras-Russell fiber bundles over algebraically closed fields of characteristic zero.... more 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.
We show that a del Pezzo fibration π : V → W of degre d contains a vertical open cylinder, that i... more We show that a del Pezzo fibration π : V → W of degre d contains a vertical open cylinder, that is, an open subset whose intersection with the generic fiber of π is isomorphic to Z×A_K^1 for some quasi-projective variety Z defined over the function field K of W , if and only if d > 5 and π : V → W admits a rational section. We also construct twisted cylinders in total spaces of threefold del Pezzo fibrations π : V → P 1 of degree d < 4.
Motivated by the study of the structure of algebraic actions the additive group on affine threefo... more 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 → 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 π : X → X//Ga are surjective with only isolated degenerate fibers, all isomorphic to the affine plane A 2 when equipped with their reduced structures.
Every deformed Koras-Russell threefold of the first kind Y = { x^nz=y^m-t^r + xh(x,y,t)} in A^4 i... more Every deformed Koras-Russell threefold of the first kind Y = { x^nz=y^m-t^r + xh(x,y,t)} in A^4 is the algebraic quotient of proper Zariski locally trivial G_a-action on SL_2 ×A^1.
We describe a family of rational affine surfaces S with huge groups of automorphisms in the follo... more 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.
Motivated by the general question of existence of open A1-cylinders in higher dimensional pro-jec... more Motivated by the general question of existence of open A1-cylinders in higher dimensional pro-jective 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 V5 , the smooth Fano threefold of index two and degree five. We show that the total spaces of these Mori Fiber Spaces always contain relative A2-cylinders, and we characterize those admitting relative A3-cylinders in terms of the existence of certain special lines in their generic fibers.
We construct two non isomorphic contractible affine threefolds X and Y with isomorphic cylinders,... more 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 establish basic properties of a sheaf of graded algebras canonically associated to every relat... more We establish basic properties of a sheaf of graded algebras canonically associated to every relative affine scheme f : X → S endowed with an action of the additive group scheme G_ a,S over a base scheme or algebraic space S, which we call the (relative) Rees algebra of the 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.
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Papers by Adrien Dubouloz