scholarly journals STRATIFICATIONS ASSOCIATED TO REDUCTIVE GROUP ACTIONS ON AFFINE SPACES

2013 ◽  
Vol 65 (3) ◽  
pp. 1011-1047 ◽  
Author(s):  
V. Hoskins
Keyword(s):  
Group Actions ◽  
Reductive Group ◽  
Affine Spaces

scholarly journals The close relation between border and Pommaret marked bases

2021 ◽  
Author(s):  
Cristina Bertone ◽  
Francesca Cioffi
Keyword(s):  
Finite Order ◽  
Polynomial Ring ◽  
Group Actions ◽  
Close Relation ◽  
Open Covering ◽  
Order Ideal ◽  
Lower Dimension ◽  
Hilbert Schemes ◽  
Affine Spaces ◽  
The Ideal

AbstractGiven a finite order ideal $${\mathcal {O}}$$ O in the polynomial ring $$K[x_1,\ldots , x_n]$$ K [ x 1 , … , x n ] over a field K, let $$\partial {\mathcal {O}}$$ ∂ O be the border of $${\mathcal {O}}$$ O and $${\mathcal {P}}_{\mathcal {O}}$$ P O the Pommaret basis of the ideal generated by the terms outside $${\mathcal {O}}$$ O . In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among $$\partial {\mathcal {O}}$$ ∂ O -marked sets (resp. bases) and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked sets (resp. bases). We prove that a $$\partial {\mathcal {O}}$$ ∂ O -marked set B is a marked basis if and only if the $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked set P contained in B is a marked basis and generates the same ideal as B. Using a functorial description of these marked bases, as a byproduct we obtain that the affine schemes respectively parameterizing $$\partial {\mathcal {O}}$$ ∂ O -marked bases and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked bases are isomorphic. We are able to describe this isomorphism as a projection that can be explicitly constructed without the use of Gröbner elimination techniques. In particular, we obtain a straightforward embedding of border schemes in affine spaces of lower dimension. Furthermore, we observe that Pommaret marked schemes give an open covering of Hilbert schemes parameterizing 0-dimensional schemes without any group actions. Several examples are given throughout the paper.

scholarly journals Linearizing certain reductive group actions

1985 ◽  
Vol 292 (2) ◽  
pp. 463-463 ◽  
Author(s):  
H. Bass ◽  
W. Haboush
Keyword(s):  
Group Actions ◽  
Reductive Group

scholarly journals Reductive group actions on Kahler manifolds

1996 ◽  
pp. 85-92
Author(s):  
Eugene Lerman ◽  
Reyer Sjamaar
Keyword(s):  
Group Actions ◽  
Reductive Group ◽  
Kähler Manifolds ◽  
Kahler Manifolds

scholarly journals Reductive group actions on affine varieties and their doubling

1995 ◽  
Vol 45 (4) ◽  
pp. 929-950 ◽  
Author(s):  
Dmitri I. Panyushev
Keyword(s):  
Group Actions ◽  
Reductive Group

Deformations of reductive group actions

1989 ◽  
Vol 106 (1) ◽  
pp. 77-88 ◽  
Author(s):  
Gerd Müller
Keyword(s):  
Lie Group ◽  
Tangent Space ◽  
Group Actions ◽  
Reductive Group ◽  
Analytic Space ◽  
Precise Formulation ◽  
The Given

Consider actions of a reductive complex Lie group G on an analytic space germ (X, 0). In a previous paper [16] we proved that such an action is determined uniquely (up to conjugation with an automorphism of (X, 0)) by the induced action of G on the tangent space of (X, 0). Here it will be shown that every deformation of such an action, parametrized holomorphically by a reduced analytic space germ, is trivial, i.e. can be obtained from the given action by conjugation with a family of automorphisms of (X, 0) depending holomorphically on the parameter. (For a more precise formulation in terms of actions on analytic ℂ-algebras, see Theorem 2 below. An analogue for actions on formal ℂ-algebras is given there too.)

scholarly journals Reductive group actions with one-dimensional quotient

1992 ◽  
Vol 76 (1) ◽  
pp. 1-97 ◽  
Author(s):  
Hanspeter Kraft ◽  
Gerald W. Schwarz
Keyword(s):  
Group Actions ◽  
Reductive Group ◽  
One Dimensional
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