 Solution

Let $v_1,\dots ,v_r$ be the primitive lattice generators of the rays of the strongly convex rational cone $\sigma \subseteq N_{\mathbb{R}}$. The smoothness criterion for a toric variety says that the affine toric variety $U_{\sigma }$ is smooth exactly when $v_1,\dots ,v_r$ form part of a $\mathbb{Z}$-basis of the cocharacter lattice of an algebraic torus $N$. Equivalently, there is a basis $e_1,\dots ,e_n$ of $N$ such thatIn particular, the cone must be a simplicial polyhedral cone and its ray generators must be primitive; for a full-dimensional cone the criterion says that those generators form a lattice basis.