A generalization of Mumford's geometric invariant theory.

Hausen, Jürgen

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A general Hilbert-Mumford criterion

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Annales de l’institut Fourier

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Let a reductive group $G$ act on an algebraic variety $X$ . We give a Hilbert-Mumford type criterion for the construction of open $G$ -invariant subsets $V \subset X$ admitting a good quotient by $G$ .

Stable affine models for algebraic group actions.

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Geometric invariant theory for general algebraic groups

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Invariant theory for linear algebraic groups II (char k arbitrary)

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On orbits of the automorphism group on an affine toric variety

Ivan Arzhantsev, Ivan Bazhov (2013)

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Let X be an affine toric variety. The total coordinates on X provide a canonical presentation $\bar{X} \to X$ of X as a quotient of a vector space $\bar{X}$ by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut(X) on X coincide with the Luna strata defined by the canonical quotient presentation.

Quotients of an affine variety by an action of a torus

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Let X be an affine T-variety. We study two different quotients for the action of T on X: the toric Chow quotient X/C T and the toric Hilbert scheme H. We introduce a notion of the main component H 0 of H, which parameterizes general T-orbit closures in X and their flat limits. The main component U 0 of the universal family U over H is a preimage of H 0. We define an analogue of a universal family WX over the main component of X/C T. We show that the toric Chow morphism restricted on...

Reductive group actions with one-dimensional quotient

Hanspeter Kraft, Gerald W. Schwarz (1992)

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Equidimensional actions of algebraic tori

Haruhisa Nakajima (1995)

Annales de l'institut Fourier

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Let $X$ be an affine conical factorial variety over an algebraically closed field of characteristic zero. We consider equidimensional and stable algebraic actions of an algebraic torus on $X$ compatible with the conical structure. We show that such actions are cofree and the nullcones of $X$ associated with them are complete intersections.