A first look at toric varieties

This primer was written as an introduction to a talk given in the Junior Geometry graduate seminar at UT Austin. The talk covered most of the material found in Week 1: Cones, fans and toric varieties, but it felt pertinent to include this primer here too. Some participants had not seen schemes before; so I started with a lightning fast introduction taking people from vanishing sets of polynomials to $\operatorname{Spec}$. The main point of this writeup is to demonstrate that you can do algebraic geometry using arbitrary rings, though we fail to explain the Zariski topology.

An extraordinarily fast primer on algebraic geometry

Algebraic geometry studies, among other things, algebraic varieties. These are shapes cut out by polynomials. For instance, $X = V(x^2 + y^2 - 1)\subseteq K^2$ is an algebraic variety consisting of all points where the polynomial $x^2 + y^2 - 1$ vanishes. Here $K$ is the base field for my polynomial ring.

It also makes sense to consider the vanishing set of multiple polynomials at once, for instance $V(x^2 + y^2 - 1, x = 0)$ consists of the two points $(x,y) = (0, 1), (0,-1)$ since these are the only places where both polynomials simultaneously vanish.

Notice also that for two polynomials $f$ and $g$, $f + g$ vanishes whenever both $f$ and $g$ vanish and $fg$ vanishes wherever at least one of $f$ or $g$ vanishes. This means that if $I = (f_1,...,f_m)$ is the ideal generated by $f_1,...,f_m$ then $V(I) = V(f_1,...,f_m)$. We therefore can talk only about the vanishing of polynomial ideals without losing any generality.

Invertible polynomials, i.e. nonzero constants, don't vanish anywhere while the 0 polynomial vanishes everywhere:

• $V(c) = \emptyset$ if $c \in K$

• $V(c) = K^n$ if $c = 0$, where $n$ is the number of indeterminants in my polynomial ring.

Sometimes non-invertible polynomials don't vanish anywhere either. For instance, if $K = \mathbb{Q}$, then $x^2 - 2$ has no solutions. Similarly, if $K = \mathbb{R}$, $x^2 + y^2 + 1$ has no solutions. This does not happen when $K = \overline{K}$, that is, when $K$ is algebraically closed.

When $K = \overline{K}$ there is another notably nice property: every point $(a_1,...,a_n) \in K^n$ is the unique point where $x_1 - a_1,x_2 - a_2, ..., x_n - a_n$ all simultaneously vanish. This gives us a correspondence between points in $K^n$ and maximal ideals of the form $(x_1 - a_1,...,x_n - a_n)$ by taking vanishing sets. In fact, when $K = \overline{K}$, one can show that all maximal ideals are of this form, and so if we denote the set of all maximal ideals in $R = K[x_1,...,x_n]$ by $\operatorname{Maxspec}$, then

One can more generally consider the space of all prime ideals $\operatorname{Spec}(R)$ endowed with the Zariski toplogy. This contains $\operatorname{R}$ as a subset, but has additional "points" which prove useful whenever $K$ is not algebraically closed. In fact, $\operatorname{Spec}(R)$ makes sense for any ring $R$. By analogy with $K^n$ and $\operatorname{Maxspec}(K[x_1,...,x_n])$ for algebraically closed $K$, we denote by $\mathbb{A}^n_R$ the space $\operatorname{Spec}(R[x_1,...,x_n])$ and call it affine space over $R$.

Replacing $K^n$ by $\operatorname{Spec}(K[x_1,...,x_n])$ allows us to use ring operations to manipulate the underlying affine space. As alluded to earlier, invertible elements of our ring $R$ will not vanish anywhere. Thus, if we want to ignore portions of the space $\operatorname{Spec}(R)$ we can adjoin the multiplicative inverses of certain elements to our ring. This process of inverting a designated subset of elements in a ring is aptly called localization.

For example, consider one-dimensional affine space over $\mathbb{C}$; $\mathbb{A}^1_{\mathbb{C}}$ given by $\operatorname{Spec}$ of $R = \operatorname{Spec} \mathbb{C}[t]$. The origin of this space corresponds to the maximal ideal generated by $(t)$. If we adjoin $t^{-1}$ to our ring, then the ideal $(t)$ contains $1 = t\cdot t^{-1}$ and is hence no longer prime, but all other prime ideals remain. Thus $\mathbb{A}^1_{\mathbb{C}} \setminus \{0\} = \operatorname{Spec} \mathbb{C}[t,t^{-1}]$, and hence punctured affine space over $\mathbb{C}$ is in fact also an affine space.

Note: Despite the fact that $\mathbb{A}^1_{\mathbb{C}}$ is strictly larger than $\mathbb{C}$, it is common to interchange $\mathbb{A}^1_\mathbb{C}$ and $\mathbb{C}$ via the identification $\operatorname{Maxspec}(\mathbb{C}[t]) = \mathbb{C}$. This is not an egregious abuse of notation when working over an algebraically closed field; one loses nothing by working with $\operatorname{Maxspec}$ instead of $\operatorname{Spec}$ in this setting. We also denote by $\mathbb{C}^\times$ the space $\mathbb{A}^1_{\mathbb{C}} \setminus \{0\}$.

More generally, we see that $(\mathbb{C}^\times)^n = \operatorname{Spec} \mathbb{C}[x_1^{\pm},...,x_n^{\pm}]$. Because the punctured complex plane $\mathbb{C}^\times$ is homotopy equivalent to $S^1$, we call $(\mathbb{C}^\times)^n$the complex $n$-torus and denote it by $\mathbb{T}^n$. This is the torus in toric geometry.