The Experts below are selected from a list of 288 Experts worldwide ranked by ideXlab platform
Adrien Dubouloz - One of the best experts on this subject based on the ideXlab platform.
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rees algebras of Additive Group actions
arXiv: Algebraic Geometry, 2019Co-Authors: Adrien Dubouloz, Isac Heden, Takashi KishimotoAbstract: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.
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Rationally integrable vector fields and rational Additive Group actions
International Journal of Mathematics, 2016Co-Authors: Adrien Dubouloz, Alvaro LiendoAbstract: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. Our results lead 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 counterpart of the Makar–Limanov invariant for affine varieties and describe the structure of rational homogeneous Additive Group actions on toric varieties.
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On rational Additive Group actions
arXiv: Algebraic Geometry, 2014Co-Authors: Adrien Dubouloz, Alvaro LiendoAbstract: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.
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Additive Group actions on danielewski varieties and the cancellation problem
Mathematische Zeitschrift, 2006Co-Authors: Adrien DuboulozAbstract:Given complex algebraic varieties X and Y of the same dimension, the Cancellation Problem asks if an isomorphism between X × \(\mathbb{C}\) and Y × \(\mathbb{C}\) induces an isomorphism between X and Y. Iitaka and Fujita (J. Fac. Sci. Univ. 24:123–127, 1977) established that the answer is positive for a large class of varieties of any dimension. In 1989, Danielewski constructed a counterexample using smooth rational affine surfaces. His construction was further generalized by Fieseler (Comment. Math. Helvetici 69:5–27, 1994) and Wilkens (C.R. Acad. Sci. Paris Ser. I Math. 326(9):1111–1116, 1998) to describe a larger class of affine surfaces. Here we introduce higher-dimensional analogues of these surfaces. By studying algebraic actions of the Additive Group \(\mathbb{C}_{+}\) on certain of these varieties, we obtain new counterexamples to the Cancellation Problem in every dimension d ≥ 2.
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Additive Group actions on danielewski varieties and the cancellation problem
arXiv: Algebraic Geometry, 2005Co-Authors: Adrien DuboulozAbstract: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 .
Alvaro Liendo - One of the best experts on this subject based on the ideXlab platform.
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Rationally integrable vector fields and rational Additive Group actions
International Journal of Mathematics, 2016Co-Authors: Adrien Dubouloz, Alvaro LiendoAbstract: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. Our results lead 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 counterpart of the Makar–Limanov invariant for affine varieties and describe the structure of rational homogeneous Additive Group actions on toric varieties.
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Additive Group actions on affine t varieties of complexity one in arbitrary characteristic
Journal of Algebra, 2016Co-Authors: Kevin Langlois, Alvaro LiendoAbstract:Abstract Let X be a normal affine T -variety of complexity at most one over a perfect field k, where T = G m n stands for the split algebraic torus. Our main result is a classification of Additive Group actions on X that are normalized by the T -action. This generalizes the classification given by the second author in the particular case where k is algebraically closed and of characteristic zero. With the assumption that the characteristic of k is positive, we introduce the notion of rationally homogeneous locally finite iterative higher derivations which corresponds geometrically to Additive Group actions on affine T -varieties normalized up to a Frobenius map. As a preliminary result, we provide a complete description of these G a -actions in the toric situation.
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On rational Additive Group actions
arXiv: Algebraic Geometry, 2014Co-Authors: Adrien Dubouloz, Alvaro LiendoAbstract: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.
Shigeru Kuroda - One of the best experts on this subject based on the ideXlab platform.
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The automorphism theorem and Additive Group actions on the affine plane
arXiv: Commutative Algebra, 2016Co-Authors: Shigeru KurodaAbstract:Due to Rentschler, Miyanishi and Kojima, the invariant ring for a ${\bf G}_a$-action on the affine plane over an arbitrary field is generated by one coordinate. In this note, we give a new short proof for this result using the automorphism theorem of Jung and van der Kulk.
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INITIAL FORMS OF STABLE INVARIANTS FOR Additive Group ACTIONS
Transformation Groups, 2014Co-Authors: Shigeru KurodaAbstract:The Derksen–Hadas–Makar-Limanov theorem (2001) says that the invariants for nontrivial actions of the Additive Group on a polynomial ring have no intruder. In this paper, we generalize this theorem to the case of stable invariants. We also prove a similar result for constants of locally finite higher derivations.
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Initial forms of stable invariants for Additive Group actions
arXiv: Commutative Algebra, 2013Co-Authors: Shigeru KurodaAbstract:The Derksen--Hadas--Makar-Limanov theorem (2001) says that the invariants for nontrivial actions of the Additive Group on a polynomial ring have no intruder. In this paper, we generalize this theorem to the case of stable invariants.
Emilie Dufresne - One of the best experts on this subject based on the ideXlab platform.
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Separating invariants for arbitrary linear actions of the Additive Group
Manuscripta Mathematica, 2013Co-Authors: Emilie Dufresne, Jonathan Elmer, Mufit SezerAbstract:Cataloged from PDF version of article.We consider an arbitrary representation of the Additive Group Ga over a field of characteristic zero and give an explicit description of a finite separating set in the corresponding ring of invariants. © 2013 Springer-Verlag Berlin Heidelberg
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the separating variety for the basic representations of the Additive Group
Journal of Algebra, 2013Co-Authors: Emilie Dufresne, Martin KohlsAbstract:Abstract For a Group G acting on an affine variety X, the separating variety is the closed subvariety of X × X encoding which points of X are separated by invariants. We concentrate on the indecomposable rational linear representations V n of dimension n + 1 of the Additive Group of a field of characteristic zero, and decompose the separating variety into the union of irreducible components. We show that if n is odd, divisible by four, or equal to two, the closure of the graph of the action, which has dimension n + 2 , is the only component of the separating variety. In the remaining cases, there is a second irreducible component of dimension n + 1 . We conclude that in these cases, there are no polynomial separating algebras.
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separating invariants for arbitrary linear actions of the Additive Group
arXiv: Commutative Algebra, 2013Co-Authors: Emilie Dufresne, Jonathan Elmer, Mufit SezerAbstract:We consider an arbitrary representation of the Additive Group over a field of characteristic zero and give an explicit description of a finite separating set in the corresponding ring of invariants.
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on the finite generation of Additive Group invariants in positive characteristic
Journal of Algebra, 2010Co-Authors: Emilie Dufresne, Andreas MaurischatAbstract:Abstract Roberts, Freudenburg, and Daigle and Freudenburg have given the smallest counterexamples to Hilbert's fourteenth problem as rings of invariants of algebraic Groups. Each is of an action of the Additive Group on a finite dimensional vector space over a field of characteristic zero, and thus, each is the kernel of a locally nilpotent derivation. In positive characteristic, Additive Group actions correspond to locally finite iterative higher derivations. We set up characteristic-free analogs of the three examples, and show that, contrary to characteristic zero, in every positive characteristic, the invariants are finitely generated.
Raphaël Achet - One of the best experts on this subject based on the ideXlab platform.
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The Picard Group of the forms of the affine line and of the Additive Group
Journal of Pure and Applied Algebra, 2017Co-Authors: Raphaël AchetAbstract:We obtain an explicit upper bound on the torsion of the Picard Group of the forms of the affine line and of the Additive Group, and a sufficient condition for this Picard Group to be non trivial. We also give examples of non trivial forms of the affine line with trivial Picard Groups.
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Picard Group of the forms of the affine line and of the Additive Group
arXiv: Algebraic Geometry, 2016Co-Authors: Raphaël AchetAbstract:We obtain an explicit upper bound on the torsion of the Picard Group of the forms of the affine line and their regular completions. We also obtain a sufficient condition for the Picard Group of the forms of the affine line to be non trivial and we give examples of non trivial forms of the affine line with trivial Picard Groups.
