affine variety/space vs. toric variety

Question

I think I'm not quite clear on the meaning of a toric variety...

Could someone explain the relation/difference between the affine variety/space and the toric variety?

I know that affine variety can be regarded as a subset of affine space, but not clear on their relation to the toric variety. Thanks!

Answer 1

The simplest answer is:

Not every affine variety is a toric variety. Also, not every toric variety is an affine variety.

The most fundamental examples are projective spaces $\Bbb P^n$. Projective spaces are toric varieties but not affine varieties.

Recall that affine varieties are of the form $X=\Bbb V(\mathfrak a)$ for an ideal $\mathfrak a\subseteq\Bbb C[x_1,\dotsc,x_n]$.

It would then make sense for affine toric varieties to be of the form $X=\Bbb V(\mathfrak a)$ for a certain kind of ideal $\mathfrak a\subseteq\Bbb C[x_1,\dotsc,x_n]$. This indeed turns out to be true.

An ideal $\mathfrak a\subseteq\Bbb C[x_1,\dotsc,x_n]$ is called toric if $\mathfrak a$ is prime and generated by binomials.

One can then prove that affine toric varieties are precisely the affine varieties of the form $X=\Bbb V(\mathfrak a)$ where $\mathfrak a\subseteq\Bbb C[x_1,\dotsc,x_n]$ is a toric ideal.

This gives a quick proof that the affine variety $\Bbb V(x^3-y^2-1)$ is not toric.

Obviously there's a lot more to say on the subject but this should get you started.