Cones, fans and toric varieties
These notes correspond to the first meeting of the spring 2023 toric geometry learning seminar at UT Austin. For more information on the topics found here, please reference Fulton's An Introduction to Toric Varieties.
Definition of Toric Variety
Recall from the primer that \((C^\times)^n = \operatorname{Spec} \mathbb{C}[x_1,...,x_n]_{x_1x_2...x_n}\) is called the algebraic torus. Qualitatively, toric varieties are varieties over \(\mathbb{C}\) which 1. contain a copy of \((\mathbb{C}^\times)^n\) and 2. whose geometry is largely determined by the torus. Here is a definition:
Some authors, including Fulton, require that toric varieties be normal. This is because the category of normal toric varieties is equivalent to the category of rational polyhedral fans, and the study of toric varieties is really about moving between these two categories. By omitting the normal hypothesis we get a slightly weaker correspondence, discussed later.
Example 1: \((\mathbb{C}^\times)^n\) is of course itself an affine toric variety. The natural torus action is simply coordinate wise multiplication and the corresponding ring map is given by comultiplication:
\[T\times X\to X\hspace{1em};\hspace{1em}(t_1,...,t_n)\cdot (s_1,...,s_n)\mapsto (t_1s_1,...,t_ns_n)\] \[\mathbb{C}[t_1^{\pm},...,t_n^{\pm}] \to \mathbb{C}[y_1^{\pm},...,y_n^{\pm}]\otimes_{\mathbb{C}}\mathbb{C}[y_1^{\pm},...,y_n^{\pm}]\hspace{1em};\hspace{1em}t_i \mapsto x_i \otimes y_i\]Example 2: \(\mathbb{C}^n\) is an affine toric variety, where the torus action \(T\times X\to X\) is again given by pointwise multiplication on coordinates.
Example 3: \(\mathbb{P}^n_{\mathbb{C}}\) is a toric variety. The torus is \(T = \left\{[z_0:...:z_n] ~\mid~ z_i\neq 0\right\}\) and the torus action is again given by pointwise multiplication, this time on homogeneous coordinates.
It should be remarked that a toric variety is actually a package
\[(X, T \xrightarrow{\text{open}}X, T\times X \to X)\]consisting of the variety itself, a dominant open embedding of the torus and a specified torus action. There are typically many choices of embeddings and actions for any given toric variety, so it is standard practice to fix a choice of each before doing anything else.
Motivation for Fans
Fix \(T \cong (\mathbb{C}^\times)^n\) and consider the group of cocharacters or 1-parameter subgroups:
\[N = \operatorname{Hom}_{\text{Grp}}(\mathbb{C}^\times, T).\]It turns out that \(N \cong \mathbb{Z}^n\), as proved in Humphrey's Linear Algebraic Groups in section 16.2 for instance. Given \(u = (u_1,...,u_n) \in \mathbb{Z}^n = N\) the corresponding 1-parameter subgroup is
\[\lambda_u:\mathbb{C}^\times\to T, \hspace{1em} t \mapsto (t^{u_1},...,t^{u_{n}}).\]If we now take a toric variety \(X\), we can consider the extension of \(\lambda_u\) to \(X\) given by composing with the torus embedding:
\[\mathbb{C}^\times_t \xrightarrow{\lambda_u} T \hookrightarrow X\]where here the subscript on \(\mathbb{C}^\times_t\) is simply there to emphasize that we have chosen a coordinate \(t\) on \(\mathbb{C}^\times\) with respect to which \(\lambda_u\) is defined. If we now consider extending \(\lambda_u\) to \(\mathbb{C}\) we can ask: what is the limit as \(t\to 0\)?
Example: Suppose \(X = \mathbb{P}^2_{z_0,z_1,z_2}\) and choose \(u = (a,b) \in \mathbb{Z}^2\). We have several different cases to consider:
If \(u_0,u_1 > 0\) then \([t^{u_0}:t^{u_1}:1] \xrightarrow{t\to 0} [0:0:1]\).
If \(u_0 > u_1, u_1 < 0\) then \([t^{u_0}:t^{u_1}:1] = [t^{u_0 - u_1}:1:t^{-u_1}]\xrightarrow{t\to 0} [0:1:0]\)
If \(u_0 < u_1, u_0 < 0\) then \([t^{u_0}:t^{u_1}:1] = [1:t^{u_1 - u_0}:t^{-u_0}]\xrightarrow{t\to 0} [1:0:0]\)
If \(u_0 > u_1, u_1 = 0\) then \([t^{u_0}:t^{u_1}:1] = [t^{u_0}:1:1] \xrightarrow{t\to 0} [0:1:1]\)
If \(u_0 = 0, u_1 > u_0\) then \([t^{u_0}:t^{u_1}:1] = [1:t^{u_1}:1] \xrightarrow{t\to 0} [1:0:1]\)
If \(0 > u_0 = u_1\) then \([t^{u_0}:t^{u_1}:1] = [t^{u_0 - u_1}:t^{u_1 - u_1}:t^{-u_1}] \xrightarrow{t\to 0} [1:1:0]\)
If \(u_0 = u_1 = 0\) then \([t^{u_0}:t^{u_1}:1] = [1:1:1]\).
If we partition \(N = \mathbb{Z}^2\) into regions where choices of \(u\) give the same limit point, then we get a fan:
We can dually write this object as a convex polytope: 
Now consider the torus orbit of each of these points. We write down three of them:
\(T\cdot [0:0:1] = \{[0:0:z_0] \mid z_0 \neq 0\} = [0:0:1]\),
\(T \cdot [1:0:1] = \{[z_0:0:z_2] \mid z_0, z_2 \neq 0\}\),
\(T \cdot [1:1:1] = \{[z_0:z_1:z_2] \mid z_0, z_1, z_2 \neq 0\}\) etc.
In particular, the torus orbits of these seven points do not intersect and their union covers \(\mathbb{P}^2_{\mathbb{C}}\). The fan tells us how the torus action partitions the underlying toric variety!
There is one other thing I'd like to point out here. Notice how there are three "2-dimensional" components of the fan, three "1-dimensional" components and one "0-dimensional" component namely the origin. These components are called cones, and their dimension relates to the dimension of their torus orbits in the following way. If \(\sigma\) is a \(d\)-dimensional cone with a corresponding point \(P_{\sigma}\), then \(T\cdot P_{\sigma}\) is a \(\operatorname{dim} X - \operatorname{dim} \sigma\) subspace of \(X\). In our case, the orbits of the points corresponding to the 2-dimensional cones are just points, the orbits of the 1-dimensional cones become copies of \(\mathbb{P}^1_{\mathbb{C}}\) and the action of \(T\) on the origin recovers the entire torus. This phenomenon will be explored further when we discuss the "orbit cone correspondence".
Rational Polyhedral Cones
I hope that you are now adequately convinced that fans connect in some interesting way to toric varieties. In this section, we'll cover the basics of the polyhedral geometry needed to understand this connection.
Fix a lattice \(N \cong \mathbb{Z}^n\) and let \(N_{\mathbb{R}} = N \otimes_{\mathbb{Z}} \mathbb{R}\cong \mathbb{R}^n\). We first define polyhedral cones:
Definition: A convex polyhedral cone in \(N_{\mathbb{R}}\) is a set
\[\sigma = \{a_1v_1+...+a_sv_s \in N_{\mathbb{R}} ~\mid~ a_i \geq 0\}\]generated by any finite set of vectors \(v_1,...,v_s \in N_{\mathbb{R}}\). Such vectors are called the generators of \(\sigma\), and we sometimes write \(\text{span}_{\mathbb{R}_{\geq 0}}(v_1,...,v_s)\) to denote \(\sigma\). In addition, \(\sigma\) is said to be
rational if each generator has integer coordinates, i.e. if \(v_i \in N\) for all \(1\leq i\leq s\)
strictly convex if \(\sigma\) does not contain a line The dimension of \(\sigma\) is denoted \(\operatorname{dim} \sigma\) and is defined to be the dimension of \(\mathbb{R} \cdot \sigma = \sigma + (-\sigma)\) as a linear space over \(\mathbb{R}\).
Note that the word "polyhedra" refers to the finite generating set of \(\sigma\): this assumption disallows cones which look like ice cream cones.
In the above image, only the left cone is polyhedral.
We also make frequent use of the dual lattice \(M = \operatorname{Hom}_{\mathbb{Z}}(N,\mathbb{Z})\cong \mathbb{Z}^n\) and the dual vector space \(M_\mathbb{R} = M\otimes_\mathbb{Z} \mathbb{R}\cong \mathbb{R}^n\). This is where the dual cone lives:
The following fact from the theory of convex sets is crucial:
From here we can immediately get the duality theorem.
We now run through a few more basic definitions and facts regarding polyhedral cones.
Definition: Let \(\sigma\) be a polyhedral cone. A face of \(\sigma\) is the intersection of \(\sigma\) with any supporting hyperplane:
\[\tau = \sigma \cap u^{\perp} = \{v \in \sigma ~\mid~ \langle u,v \rangle = 0\}\]for some \(u\in \sigma^\vee\).
Any face \(\tau\) of \(\sigma\) is again a polyhedral cone. A cone \(\sigma\) is a face of itself, as seen by taking \(u = 0\) in the above definition, while every other face is referred to as a proper face.
Proof:
\[\bigcap (\sigma \cap u_i^{\perp}) = \sigma \cap \left(\sum u_i\right)^\perp.\]The final fact we will need is the following:
This is not difficult to prove; one can easily find a procedure by which to find generators for \(\sigma^\vee\) given the generators of \(\sigma\) and use this to prove both rationality and convexity. We refer the reader to page 12 of Fulton's book.
This gives us Gordan's lemma, which will be critical to our construction of toric varieties from cones.