(The computation)
Let \(\sigma\) span \(N_\RR\), so \(\sigma^\perp = 0\) and the distinguished point \(x_\sigma \in U_\sigma\) is the torus-fixed point. Its maximal ideal and square are \[ \mfm = \gens{ \chi^u \st u \in S_\sigma \smz } , \qquad \mfm^2 = \gens{ \chi^u \st u \in (S_\sigma \smz) + (S_\sigma \smz) } , \] so \[ \dualof{T}_{x_\sigma} U_\sigma = \mfm/\mfm^2 = \bigoplus_{u \text{ irreducible}} k \cdot \chi^u , \] where \(u \in S_\sigma \smz\) is irreducible, or indecomposable, if it is not a sum of two nonzero elements of \(S_\sigma\). The irreducible elements are exactly the Hilbert basis of \(S_\sigma\), so \[ \dim T_{x_\sigma} U_\sigma = \size\ts{\text{Hilbert basis of } S_\sigma} . \]
(What this forces)
\(U_\sigma\) is smooth at \(x_\sigma\) exactly when the Hilbert basis has \(n = \dim N_\RR\) elements, and then \(\sigma\) is generated by a \(\ZZ\)-basis of \(N\).
Each extremal ray of \(\dualof{\sigma}\) contributes its primitive generator as an irreducible element, and distinct rays contribute distinct ones. So if \(U_\sigma\) is smooth at \(x_\sigma\) then \[ \size\ts{\text{rays of } \dualof{\sigma}} \leq \size\ts{\text{Hilbert basis}} = \dim T_{x_\sigma} U_\sigma = n . \] A pointed full-dimensional cone in \(\RR^n\) has at least \(n\) rays, so \(\dualof{\sigma}\) has exactly \(n\) of them, and the Hilbert basis is exactly those \(n\) primitive generators. They generate \(S_\sigma\), hence generate \(M\) as a group, and \(n\) generators of a rank-\(n\) lattice form a \(\ZZ\)-basis. Dualising, \(\sigma\) is generated by the dual basis, which is a \(\ZZ\)-basis of \(N\).
(Why the converse holds)
If the cone is generated by a lattice basis, the corresponding affine toric variety is visibly smooth. The converse is the substantive direction: smoothness forces the Hilbert basis to have exactly \(n\) elements. The mechanism is a count: the embedding dimension at the fixed point is the size of the Hilbert basis, smoothness caps that count at \(n\), and the cap squeezes the dual cone down to exactly \(n\) rays with no room left for a non-basis.
The count is also the practical statement. For \(\sigma = \Cone\big( (0,1), (2,-1) \big)\) the Hilbert basis of \(S_\sigma\) is \((1,0), (1,1), (1,2)\), three elements in dimension two, so \(U_\sigma\) embeds in \(\AA^3\) and not in \(\AA^2\): it is the quadric cone, singular at the fixed point and smooth everywhere else.
Dependencies
PR-TORSMAFFSmooth affine toric varieties and why toric varieties are normalusesPR-O8V3IThe orbit-cone correspondenceuses