The hypersurface $V$ is disjoint from the points (i.e. the 0-dimensional toric varieties) corresponding to the vertices of $\Delta$, but intersects all the tori corresponding to any positive-dimensional face of $\Delta$.

Furthermore, this property characterizes $\mathbb{C}T_{\Delta}$ in the following sense. Let $\bar{\Delta}$ be a convex lattice polyhedron in $\mathbb{R}^{n+1}$ with a non-empty interior and $\bar{V}$ be the closure of ${V}^{\circ}$ in $\mathbb{C}T_{\bar{\Delta}}\supset(\mathbb{C}^{*})^{n+1}$. If a hypersurface $\bar{V}$ is disjoint from the points corresponding to the vertices of $\bar{\Delta}$ but intersects all the tori corresponding to positive-dimensional faces of $\bar{\Delta}$ then $\mathbb{C}T_{\bar{\Delta}}=\mathbb{C}T_{\Delta}$.