# Why does every toric ideal correspond to an affine toric variety?

I'm reading the book Toric Varieties by Cox, Little and Schenk, and have a small question about the proof of their Proposition 1.1.11 (on page 16). The key part of the proposition is this:

Claim: Let $$I\subseteq\mathbb{C}[x_1,\ldots,x_s]$$ be a prime ideal generated by (pure) binomials. Then $$V(I)\subseteq\mathbb{C}^s$$ is an affine toric variety.

The idea of the proof (slightly reformulated to simplify the notation) is this:

• Use the fact that $$I$$ is both prime and binomial, to conclude that $$V(I)\cap(\mathbb{C}^*)^s$$ is a torus (since it's both an irreducible subvariety and a subgroup of $$(\mathbb{C}^*)^s$$).

• Suppose that the dimension of $$V(I)\cap(\mathbb{C}^*)^s$$ is $$n$$, and pick an isomorphism $$(\mathbb{C}^*)^n\to V(I)\cap(\mathbb{C}^*)^s$$.

• Form the composition $$\Phi\colon (\mathbb{C}^*)^n\to V(I)\cap(\mathbb{C}^*)^s\hookrightarrow (\mathbb{C}^*)^s$$, and note that it's an algebraic group homomorphism (concretely, this means that every component is given by a Laurent polynomial). Let $$Y$$ be the Zariski closure of $$\mathrm{Im}(\Phi)$$ in $$\mathbb{C}^s$$.

• Observe that $$V(I)=Y$$.

• Since we have already shown (see Proposition 1.1.8 on page 13) that every affine variety that arises as the closure of the image of an algebraic group homomorphism $$(\mathbb{C}^*)^n\to(\mathbb{C^*})^s$$ is an affine toric variety, the desired conclusion follows.

It's the second to last step that confuses me.

Since $$\mathrm{Im}(\Phi)=V(I)\cap(\mathbb{C}^*)^s$$, this essentially boils down to showing that $$\overline{V(I)\cap(\mathbb{C}^*)^s}=V(I)$$. The inclusion "$$\subseteq$$" is easy, because $$V(I)$$ certainly is a Zariski closed subset that contains $$V(I)\cap(\mathbb{C}^*)^s$$. But do we know that $$V(I)$$ is the smallest such subset?

Edit: I think I figured it out! See below.

If I'm not mistaken, this was easier than I thought. Simply note that $$V(I)\cap(\mathbb{C}^*)^s$$ is Zariski open in $$V(I)$$. Since $$V(I)$$ is irreducible, and every Zariski open subset of an irreducible affine variety is dense, it follows that the Zariski closure of $$V(I)\cap(\mathbb{C}^*)$$ in $$V(I)$$ is $$V(I)$$. And this, in turn, implies that the Zariski closure in $$\mathbb{C}^s$$ is $$V(I)$$, which is what we wanted to prorve.