A general Hilbert-Mumford Criterion
نویسنده
چکیده
We provide a Hilbert-Mumford Criterion for the action of a reductive group G on a Q-factorial algebraic variety X, partially proving a conjecture stated by A. BiaÃlynicki-Birula. The result allows to construct G-invariant open subsets of X admitting a good quotient by G from certain open subsets admitting a good quotient by a maximal torus T of G. As an application, we indicate an explicit method to construct all G-invariant open subsets admitting a good quotient for a given G-action on a complete Q-factorial toric variety X. 1. Statement of the results Let the reductive group G act on a normal variety X over an algebraically closed field K. It is a central task of Geometric Invariant Theory to construct all open G-invariant subsets V ⊂ X admitting a good quotient, that means an affine Ginvariant morphism V → V/G onto an algebraic space such that locally, in the étale topology, V/G is the spectrum of the invariant functions, see [1, Chapter 7]. The problem splits into two parts: The first one is to find the open subsets U ⊂ X admitting a good quotient by a maximal torus T ⊂ G. For the combinatorial aspects of this step, see [1, Chapter 11]. The second part is to construct the open subsets admitting a good quotient by the action of G from the U ⊂ X found in part one. For this, one considers sets of the following form:
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