= Solution
Let $v_1,\ldots,v_r$ 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 $v_1,\ldots,v_r$ form part of a $\mathbb Z$-basis of the <cocharacter lattice of an algebraic torus> $N$. Equivalently, there is a basis $e_1,\ldots,e_n$ of $N$ such that
$$
\sigma=\operatorname{Cone}(e_1,\ldots,e_r).
$$
In 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.
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