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Let $\mathbb{G}_m^r$ act on the affine space $\mathbb{A}^n$ through an embedding into the open dense torus. Is there a way to calculate the 1-parameter subgroups that determine the KN strata from the structure of the secondary fan?

Suppose our chosen linearization/character $\chi$ lies in a maximal cone $\sigma$ in the secondary fan. I would like to say these 1PS's are the primitive cocharacters that pair to zero with the walls and pair negatively with the characters in $\sigma$, however I am having trouble understanding it.

Hope the question makes sense to the MO community.

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I have an answer to this question now. The 1PS's for the KN strata have to depend on $\chi$ which determines the ordering, but one can also make noncanonical choices of 1PS's as in the question to simply describe the unstable loci as a union.

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