(Classification)
If \(\sigma\) is a smooth cone of dimension \(k\) inside \(N \cong \ZZ^n\), then \[ U_\sigma \cong \AA^k \times \GG_m^{n-k} . \] Every smooth affine toric variety is of this form.
Extend the minimal generators of \(\sigma\) to a \(\ZZ\)-basis of \(N\), which is possible exactly because \(\sigma\) is smooth. In the dual basis \(\dualof{\sigma}\) is the cone spanned by \(e_1, \ldots, e_k\) and \(\pm e_{k+1}, \ldots, \pm e_n\), so \[ S_\sigma = \NN^k \oplus \ZZ^{n-k}, \qquad k[S_\sigma] = k[x_1, \ldots, x_k, x_{k+1}^{\pm 1}, \ldots, x_n^{\pm 1}] . \]
(Simplicial cones)
If \(\sigma\) is simplicial of dimension \(k\) inside \(N \cong \ZZ^n\), then \(U_\sigma\) is the quotient of \(\AA^k \times \GG_m^{n-k}\) by a finite abelian group, so a toric variety whose fan is simplicial has at worst finite quotient singularities. [@Ful93]
(Normality)
\(X_\Sigma\) is normal.
The reason is that \(S_\sigma\) is a saturated semigroup: if \(\ell m \in S_\sigma\) for some \(\ell > 0\) then \(m \in S_\sigma\), since the defining inequalities \(\inp{m}{u} \geq 0\) are homogeneous. Concretely, \(k[S_\sigma]\) is the intersection of the rings \(k[S_\tau]\) over the rays \(\tau \leq \sigma\), each of which is a Laurent polynomial ring \(k[x_1, x_2^{\pm 1}, \ldots, x_n^{\pm 1}]\) and so integrally closed, and an intersection of integrally closed domains with the same fraction field is integrally closed.
This is the standing caveat on the whole dictionary: fans produce normal toric varieties only. The nodal cubic has a torus acting with a dense orbit and is not on the list.
Dependencies
D-Q7Q2NFrom a fan to a toric varietyuses
Backlinks
D-FULSATSaturated semigroups and the normalisation of an affine toric varietyusesPR-FULCOTANThe cotangent space at a fixed point is the Hilbert basis of the semigroupusesPR-FULCMToric varieties are rational and Cohen-Macaulay, and affine ones have no nontrivial vector bundlesuses