(Three free properties)
Let \(X_\Sigma\) be a toric variety of dimension \(n\).
\(X_\Sigma\) is rational: it contains the torus \(T \cong \GG_m^n\) as a dense open set, so \(k(X_\Sigma) = k(t_1, \ldots, t_n)\).
\(X_\Sigma\) is Cohen-Macaulay: every local ring \(R\) has \(\depth R = \dim R\).
Every vector bundle on an affine toric variety \(U_\sigma\) is trivial, equivalently every finitely generated projective \(k[S_\sigma]\)-module is free.
(Consequences)
Rationality implies that toric varieties cannot furnish unirational-but-nonrational examples: every toric variety is already rational.
Cohen-Macaulayness is the input that makes Serre duality and the vanishing theorems available without extra hypotheses, and it is why \(\omega_{X_\Sigma} = \OO(-\sum_\rho D_\rho)\) behaves as a dualising sheaf even at singular points. Note the contrast with Gorenstein, which is a genuine condition: \(X_\Sigma\) is Gorenstein exactly when \(K_X\) is Cartier, which for \(X_P\) means \(P\) reflexive.
The third statement is the toric case of a theorem on semigroup rings, and it says that \(\Pic(U_\sigma) = 0\) for every cone, which is what makes the class group of the cone over the rational normal curve a clean example of \(\Pic \subsetneq \Cl\).
Dependencies
PR-TORSMAFFSmooth affine toric varieties and why toric varieties are normaluses