Orbit closure in a toric variety
= Orbit closure in a toric variety
{title2=$V(\sigma)$}
For $\sigma\in\Sigma$, the closure of its torus orbit is
$$
V(\sigma)=\overline{O(\sigma)}
=\bigcup_{\tau\supseteq\sigma}O(\tau).
$$
Thus orbit-closure inclusion reverses face inclusion.