= Smoothness criterion for a toric variety
A normal <affine toric variety> $U_\sigma$ is smooth exactly when the primitive <toric ray> generators of $\sigma$ extend to a <lattice basis> of the ambient <toric lattice> $N$. In that case a choice of basis gives
$$
U_\sigma\cong\mathbb A^{\dim\sigma}\times\mathbb G_m^{\operatorname{rank}N-\dim\sigma}.
$$
For a two-dimensional <toric cone> in a rank-two <toric lattice> with primitive generators $u,v$, the criterion becomes $|\det(u,v)|=1$. In higher rank, a rectangular coordinate <matrix> has no such single <determinant> test; the generators must span a saturated <subgroup> of the correct rank. For example $(1,0),(1,2)$ span a real basis but have <determinant> two, giving a singular chart. This distinguishes smoothness from <simplicial polyhedral cone>[simpliciality].<Toric Varieties by David A. Cox and others>[chapter1, smooth affine toric varieties]{ref}
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