Gordan's lemma

In convex geometry, Gordan's lemma states that the semigroup of integral points in the dual cone of a rational convex polyhedral cone is finitely generated. In algebraic geometry, the prime spectrum of the semigroup algebra of such a semigroup is, by definition, an affine toric variety; thus, the lemma says an affine toric variety is indeed an algebraic variety. The lemma is named after the German mathematician Paul Gordan (1837–1912).

There are topological and algebraic proofs.

Let ${\displaystyle \sigma }$ be the cone as given in the lemma. Let ${\displaystyle u_{1},\dots ,u_{r}}$ be the integral vectors so that ${\displaystyle \sigma =\{x\mid \langle u_{i},x\rangle \geq 0,1\leq i\leq r\}.}$ Then the ${\displaystyle u_{i}}$'s generate the dual cone ${\displaystyle \sigma ^{\vee }}$; indeed, writing C for the cone generated by ${\displaystyle u_{i}}$'s, we have: ${\displaystyle \sigma \subset C^{\vee }}$, which must be the equality. Now, if x is in the semigroup

...
Wikipedia