Let $k$ be an algebraically closed field. Let $\sigma$ be a 3-dimensional simplicial cone in $\mathbb{R}^3$ and let $\rho$ be a ray of $\sigma$. Say $\rho=\sigma\cap H_m$, where $H_m$ is the plane in $\mathbb{R}^3$ passing through the origin, having $m\in\sigma^{\vee}\cap\mathbb{Z}^3$ as its normal vector. Then $U_{\rho}=\operatorname{Spec}(k[\rho^{\vee}\cap\mathbb{Z}^3])$ is a principal open subset of the affine toric variety $U_{\sigma}$ associated to the cone $\sigma$. Let $Z$ be the complement of $U_{\rho}$ in $U_{\sigma}$. Is $Z$ an affine toric variety? Why? I am thinking now that $Z$ might be the Zariski closure $V(\rho)$ of the torus orbit $O(\rho)$ corresponding to the ray $\rho$. Is this correct?
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