I am primarily interested in the hypertoric variety $\mathfrak M(\mathcal B_d)$ associated to the braid arrangement.
Any hypertoric variety $X$, say of complex dimension $2n$, comes equipped with an action of a complex torus $T = (\mathbb C^\times)^n$. We have a moment map $p\colon X\to \mathbb C^n$, and the action of $T$ on $X$ preserves the fibers of $p$.
What can one say about the structure of $T$-orbits in each fiber? For example, is each fiber a toric variety? Is the number of $T$-orbits finite? Can one say interesting things about this structure using the combinatorial data from the hyperplane arrangement? Thanks!
Edit: The fibers are not necessarily toric varieties, as the fiber $p^{-1}(0)$ contains the core of the hypertoric (which is a union of toric varieties, in general). For smooth hypertoric varieties associated to hyperplane arrangements, I've managed to show that the number of $T$-orbits is finite, but I would still like to know more about the structure of these orbits.
