Suppose we are given a divisorial valuation $\mathcal{v}$ on a smooth toric variety $X_{\Sigma}$ (for a fan $\Sigma$), i.e. $\mathcal{v}$ is the valuation induced by a torus invariant divisor $D_{\tau}$ on some birational model $X_{\Sigma'}$ of $X_{\Sigma}$. Here, $\Sigma' \leq \Sigma$ is a fan subdivision and $\tau$ is a ray in $\Sigma'$.
Moreover, let $\sigma \in \Sigma(n)$ be an n-dimensional cone. We may suppose that $\sigma$ is spanned by the usual euclidean basis vectors $e_1, \cdots, e_n$. Let 0 denote the fixed point of the torus action corresponding to $\sigma$.
My question is:
(1) When is $\mathcal{v}$ centered at 0? (I'm looking for a condition in terms of $\tau$)
(2) When is $\pi \colon X_{\Sigma'} \to X_{\Sigma}$ an isomorphism over 0?