Let's think about hypertoric variety, Gale duality.

Given a vector space in $\R^N$, we have $$ V \to \R^N \to (V^\perp)^* $$ $$ V^\perp \to (\R^N)^* \to V^* $$ Great. Given $\eta \in (V^\perp)^*$ and $\xi \in V^*$, we look at the fiber $V_\eta$ and $(V^\perp)_\xi$, they are partitioned by the restriction of the sign partitions in $\R^N$ and $(\R^N)^*$.