# Additive Group - Explore the Science & Experts | ideXlab

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### Adrien Dubouloz – 1st expert on this subject based on the ideXlab platform

• ##### rees algebras of AdditiveGroup actions
arXiv: Algebraic Geometry, 2019
Co-Authors: Adrien Dubouloz, Isac Heden, Takashi Kishimoto

Abstract:

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.

• ##### Rationally integrable vector fields and rational AdditiveGroup actions
International Journal of Mathematics, 2016

Abstract:

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.

• ##### On rational AdditiveGroup actions
arXiv: Algebraic Geometry, 2014

Abstract:

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.

### Alvaro Liendo – 2nd expert on this subject based on the ideXlab platform

• ##### Rationally integrable vector fields and rational AdditiveGroup actions
International Journal of Mathematics, 2016

Abstract:

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.

• ##### AdditiveGroup actions on affine t varieties of complexity one in arbitrary characteristic
Journal of Algebra, 2016
Co-Authors: Kevin Langlois, Alvaro Liendo

Abstract:

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.

• ##### On rational AdditiveGroup actions
arXiv: Algebraic Geometry, 2014

Abstract:

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 – 3rd expert on this subject based on the ideXlab platform

• ##### The automorphism theorem and AdditiveGroup actions on the affine plane
arXiv: Commutative Algebra, 2016
Co-Authors: Shigeru Kuroda

Abstract:

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.

• ##### INITIAL FORMS OF STABLE INVARIANTS FOR AdditiveGroup ACTIONS
Transformation Groups, 2014
Co-Authors: Shigeru Kuroda

Abstract:

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.

• ##### Initial forms of stable invariants for AdditiveGroup actions
arXiv: Commutative Algebra, 2013
Co-Authors: Shigeru Kuroda

Abstract:

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.