Affine toric variety ($U_{\sigma }$)

For a rational strongly convex cone $\sigma \subseteq N_{\mathbb{R}}$, the affine toric variety is

The coordinate ring of $U_{\sigma }$ is generated by the characters ${\chi }^m$ for lattice points $m\in {\sigma }^{\vee }\cap M$, with multiplication ${\chi }^m{\chi }^{m^{\prime }}={\chi }^{m+m^{\prime }}$.