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