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The lattice points of a rational polyhedral cone form a finitely generated additive monoid. Choose lattice generators for the cone. Subtracting integer parts of nonnegative coefficients leaves a lattice point in their bounded fundamental parallelepiped; its finitely many lattice points, together with the chosen generators, generate the monoid. This supplies finite generation of the coordinate ring of an affine toric variety.

// Target: geometry-and-topology.bigb

For $m=(u,v)\in M\cong {\mathbb{Z}}^2$, membership in the dual cone is equivalent toThe Hilbert basis of a rational cone is $(0,1),(1,1),(2,1),(3,1)$. Therefore the coordinate ring of an affine toric variety is

Write $m=a{e}_1^{\ast }+b{e}_2^{\ast }\in M$. The inequalities defining the dual cone areThusIts Hilbert basis of a rational cone isIndeed, after subtracting copies of $(2,3)$ one reduces to $b=0,1,$ or $2$, and the remaining point is generated by $(1,0)$ and $(1,1)$. The coordinate ring of an affine toric variety is thereforePutting $A=x$, $B=xy$, and $C=x^2{y}^3$ gives the alternative presentation