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Generic states and stability

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arxiv 1703.02697 v2 pith:TDI2ZNVL submitted 2017-03-08 math.AG

classification math.AG
keywords genericexistencepolytopestateworstalwaysanalogouscharacter
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We define the notion of the generic state polytope, analogous to the generic initial ideal and prove its existence: This greatly generalizes the work of R\"omer and Schmitz who proved the existence of generic Gr\"ober fans. We also show that a generic state polytope always contains the trivial character: Equivalently, in any GIT quotient problem of semisimple group representations, every point is semistable with respect to a {\it general} maximal torus. Also, we revisit Kempf's proof of the existence of the worst one parameter subgroup (1-ps) and describe the equations for determining the worst 1-ps.

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  1. The worst destabilizing 1-parameter subgroup for toric rational curves with one unibranch singularity

    math.AG 2025-02 conditional novelty 7.0 of 10

    For toric rational curves with one unibranch singularity, the worst destabilizing one-parameter subgroup is eventually constant on an initial segment and linear on the tail, with explicit formulas.

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