# Is this toric variety always smooth?

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 in $$\sigma$$. Let $$U_{\rho}$$ be defined as $$\operatorname{Spec}(k[\rho^{\vee}\cap\mathbb{Z}^3])$$. Is $$U_{\rho}$$ always a smooth toric variety? In the special case where $$\sigma$$ is the first octant and $$\rho$$ is the non-negative z-axis, we have $$U_{\rho}$$ isomorphic to $$\operatorname{Spec}(k[x,x^{-1},y,y^{-1},z])$$, so that $$U_{\rho}$$ is smooth in this case, since $$k[x,x^{-1},y,y^{-1},z])$$ is a regular ring and the field $$k$$ is algebraically closed.

• I don’t see the relevance of $\sigma$. Forgetting $\sigma$, the “special case” you discuss is not so special: there exists an automorphism of $\mathbb{Z}^3$ transforming $\rho$ into one of the axes. Commented Dec 21, 2023 at 7:30
• @PiotrAchinger Thank you very much for your kind help. Could you explain how to construct this automorphism of $\mathbb{Z}^3$? Thank you very much. Commented Dec 21, 2023 at 14:24

Anna is basically correct, except that $$U_{\sigma}$$ is smooth if and only if the set of its minimal ray generators form a subset of a $$\mathbb{Z}$$-basis.
Therefore, since $$\mathbb{Z}u_{\rho}$$ is linearly independent the variety $$U_{\rho}$$ is smooth.
$$U_\sigma$$ is smooth if and only if the generating set of $$\sigma$$, that is the minimal $$\{u_\rho| \rho \in \sigma(1)\}$$ is a $$\mathbb{Z}$$-basis for $$N$$, the lattice you are considering (which seems like here is $$\mathbb{Z}^3$$). You can find all the details you need on this in Cox, Schenck and Little's book "toric varieties".