### Toric Varieties: An Introduction

This is a blog post on toric varieties. We will be broadly following Christopher Eur’s Senior Thesis for the exposition.

A toric variety is an irreducible variety with a torus as an open dense subset. What does a dense subset of a variety look like? For instance, in $\Bbb{R}$ consider the set of integers. Or any infinite set of points for that matter. The closure of that set, under the Zariski Topology, is clearly the whole real line. Hence, a dense set under the Zariski topology looks nothing like a dense set under the standard topology.

An affine algebraic group $V$ is a variety with a group structure. The group operation is given by $\phi:V\times V\to V$, which is interpreted as a morphism of varieties (remember that the cartesian product of two varieties is a variety). The set of algebraic maps of two algebraic groups $V,W$, denoted as $\text{Hom}(V,W)$ is the set of group homomorphisms between $V$ and $W$ which are also morphisms between varieties. Are there variety morphisms which are not group homomorphisms? Yes. Consider the morphism $f:\Bbb{R}\to \Bbb{R}$ defined as $x\to x+1$.

The most important example for us is $(\Bbb{C}^*)^n\simeq \Bbb{C}^n-V(x_1x_2\dots x_n)$. This is the same as removing all the hyperplanes $x_i=0$ from $\Bbb{C}^n$. Again, this is the same as $V(1-x_1x_2\dots x_n y)\subset \Bbb{C}^{n+1}$, which is the same as embedding a variety in a higher dimensional space. The coordinate ring of $(\Bbb{C}^*)^n$ looks like $\Bbb{C}[x_1^{\pm},x_2^{\pm},\dots,x_n^{\pm}]\simeq \Bbb{C}[\Bbb{Z}^n]$. Why does the coordinate ring look like this? This is because in the ring $\Bbb{C}[x_1,x_2,\dots,x_n,y]/(1-x_1x_2\dots x_ny)$, all the $x_i's$ become invertible (in general, all of the $n+1$ variables become invertible. However, $y$ can be expressed in terms of the $x_i$‘s).

A torus is an affine variety isomorphic to $(\Bbb{C^*})^n$ for some $n$, whose group structure is inherited from that of $(\Bbb{C^*})^n$ through a group isomorphism.

Example: Let $V(x^2-y)\subset \Bbb{C}^2$, and consider $V_{xy}=V\cap (\Bbb{C}^*)^n$. We will now establish an isomorphism between $\Bbb{C}^*$ and $V_{xy}$. Consider the map $t\to (t,t^2)$ from $\Bbb{C}^*$ to $V_{xy}$. This map is bijective. How? If $t$ is non-zero, then so is each coordinate of $(t,t^2)$. Also, each point in $X_{xy}$ looks like $(t,t^2)$, where $t$ is a non-zero number, and each such point has been mapped to by $t\in\Bbb{C}^*$. Hence, we have a bijection. How does $V_{xy}$ inherit the group structure of $\Bbb{C}^*$? By the following relation: $(a,a^2).(b,b^2)=(ab,(ab)^2)$. Remember that $V_{xy}$ had no natural group structure before. Now it has one.

A map $\phi:(\Bbb{C}^*)^n\to (\Bbb{C}^*)^m$ is algebraic if and only if the map $\phi^*: \Bbb{C}[y_1^{\pm},y_2^{\pm},\dots,y_m^{\pm}]\to \Bbb{C}[x_1^{\pm},x_2^{\pm},\dots,x_n^{\pm}]$ is given by $y_i\to x^{\alpha_i}$ for $\alpha_i\in \Bbb{Z}^n$. In other words, the maps correspond bijectively to lattice maps $\Bbb{Z}^m\to \Bbb{Z}^n$. What does this mean? The condition that the variety morphism also be a group homomorphism was surely expected to place certain restrictions on the the nature of the nature of the morphism. The way that this condition places restrictions is that a unit can only map to a unit. And the only units in $\Bbb{C}[x_1^{\pm},x_2^{\pm},\dots,x_n^{\pm}]$ are monomials times a constant. Why’s that? Why isn’t an expression of the form $x_1+x_2$, for instance, a unit? Because $\Bbb{C}[x_1^{\pm},x_2^{\pm},\dots,x_n^{\pm}]$ is not a field! It is just a polynomial ring in which the variables happen to be invertible. Polynomials in those variables need not be! This is not the same as the rational field corresponding to the polynomial ring $\Bbb{C}[x_1,x_2,\dots,x_n]$. Returning to the proof, the constant is found to be $1$, and one side of the theorem is proved. The converse is trivial.

A character of a Torus $T$ is an element $\chi\in\text{Hom}(T,\Bbb{C}^*)$. An analogy that immediately comes to mind is that of a functional on an $n$-dimensional vector space. Characters are important in studying toric varieties.