 Cone in toric geometry

In the toric fan construction, a toric cone is a rational polyhedral convex conewith $v_i$ in a toric lattice $N$, satisfying $\sigma \cap (-\sigma )=\{0\}$. The last condition is strong convexity: the toric cone contains no nonzero linear subspace. The zero toric cone is allowed and corresponds to the dense algebraic torus chart.

A simplicial polyhedral cone has linearly independent primitive toric ray generators. For its affine toric variety to be smooth, those generators must extend to a lattice basis. The toric cone generated by $(1,0),(1,2)$ in ${\mathbb{Z}}^2$ is simplicial but not smooth, because the determinant is two rather than $\pm 1$. Thus real linear independence alone does not decide smoothness.^[ref]