This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. The Korean original is the source of truth.
Toric geometry is, quite literally, the study of a special class of algebraic varieties called toric varieties. The advantage they hold over general algebraic varieties is that toric varieties are built essentially out of combinatorial data, making concrete computations far more tractable. In this post we begin by examining the simplest toric varieties, namely affine toric varieties.
Lattices and Cones
Intuitively, a toric variety can be thought of as a generalization of the process of gluing copies of \(\mathbb{A}^k_\mathbb{C}\) to obtain \(\mathbb{P}^k_\mathbb{C}\). The gluing instructions are now encoded inside lattices.
Definition 1 An abelian group \(N\) isomorphic to \(\mathbb{Z}^n\) is called a lattice. The dual lattice of a lattice \(N\) is given by the formula
\[M = \Hom(N, \mathbb{Z}).\]The natural evaluation pairing \(M\times N \to \mathbb{Z}\), \((m, v) \mapsto m(v)\) is then called the dual pairing.
For convenience we introduce the two notations
\[N_{\mathbb{R}} = N \otimes_{\mathbb{Z}} \mathbb{R},\qquad M_{\mathbb{R}} = M \otimes_{\mathbb{Z}} \mathbb{R}.\]We now define the following.
Definition 2 A subset \(\sigma\) of \(N_{\mathbb{R}}\) is called a strongly convex rational polyhedral cone if it satisfies the following conditions:
- \(\sigma\) is a cone with the origin as its vertex. That is, \(\lambda v \in \sigma\) for all \(v \in \sigma\) and \(\lambda \ge 0\).
- \(\sigma\) is generated by finitely many vectors \(v_1, \ldots, v_s \in N\) as \(\mathbb{R}_{\ge 0}\)-linear combinations. In other words, we may write \(\sigma = \mathbb{R}_{\ge 0} v_1 + \cdots + \mathbb{R}_{\ge 0} v_s\).
- \(\sigma\) is strongly convex. That is, \(\sigma \cap (-\sigma) = \{0\}\).
The first condition alone does not guarantee that \(\sigma\) is convex, but the \(\mathbb{R}_{\ge 0}\)-linear combination form in the second condition automatically implies that \(\sigma\) is a closed convex cone. The third condition means that \(\sigma\) contains no line passing through the origin. For each cone \(\sigma\) we can define the following.
Definition 3 A face \(\tau\) of a cone \(\sigma\) is obtained as
\[\tau = \sigma \cap u^{\perp} = \{ v \in \sigma \mid \langle u, v \rangle = 0 \}\]for some \(u \in M_{\mathbb{R}}\) such that \(\langle u, v \rangle \ge 0\) for all \(v \in \sigma\). When \(\tau\) is a face of \(\sigma\), we write \(\tau \prec \sigma\).
The toric varieties we shall define carry each cone as an affine chart, and the gluing data is determined by how these cones meet along faces. To make this precise, in this post we study affine toric varieties.
Affine Toric Varieties
Given a cone \(\sigma\), consider its dual cone
\[\sigma^\vee = \{ u \in M_{\mathbb{R}} \mid \langle u, v \rangle \ge 0 \text{ for all } v \in \sigma \}.\]Definition 4 For a cone \(\sigma\), define the semigroup
\[S_\sigma = \sigma^\vee \cap M = \{ u \in M \mid \langle u, v \rangle \ge 0 \text{ for all } v \in \sigma \},\]and through it define the semigroup algebra
\[\mathbb{C}[S_\sigma] = \mathbb{C}[\,{\rchi}^u \mid u \in S_\sigma].\]([Algebraic Structures] §Algebras, ⁋Definition 5) Here \(\rchi^u\) is the monomial corresponding to the element \(u \in M\).
Unwinding this, we have the following. By definition \(S_\sigma\) is a semigroup under addition, and the semigroup algebra \(\mathbb{C}[S_\sigma]\) is most neatly described as the free \(\mathbb{C}\)-vector space on the set \(S_\sigma\) endowed with the multiplication
\[\rchi^u\cdot\rchi^{u'}=\rchi^{u+u'}.\]In other words, \(\mathbb{C}[S_\sigma]\) is the \(\mathbb{C}\)-algebra having \(\{\rchi^u \mid u\in S_\sigma\}\) as a basis and generated by the above multiplication. That \(S_\sigma\) is a finitely generated semigroup is known by Gordan’s lemma (Wikipedia); consequently \(\mathbb{C}[S_\sigma]\) is a finitely generated \(\mathbb{C}\)-algebra. This \(\mathbb{C}\)-algebra becomes the coordinate ring of the affine toric variety.
Definition 5 For a strongly convex rational polyhedral cone \(\sigma \subseteq N_{\mathbb{R}}\), we define the affine toric variety \(U_\sigma\) it determines by
\[U_\sigma = \Spec(\mathbb{C}[S_\sigma]).\]Example 6 When \(\sigma = \{0\}\), we have \(\sigma^\vee = M_{\mathbb{R}}\) and hence \(S_\sigma = M\). Therefore
\[\mathbb{C}[S_\sigma] = \mathbb{C}[M] = \mathbb{C}[\rchi^{\pm e_1^\ast}, \ldots, \rchi^{\pm e_n^\ast}],\]which is the coordinate ring of \((\mathbb{C}^\ast)^n\).
Example 7 Consider the case \(N = \mathbb{Z}^2\) where \(\sigma\) is generated by \(e_1\) and \(e_2\). Then
\[\sigma^\vee = \mathbb{R}_{\ge 0} e_1^\ast + \mathbb{R}_{\ge 0} e_2^\ast,\]so \(S_\sigma = \mathbb{Z}_{\ge 0}^2\) and
\[\mathbb{C}[S_\sigma] = \mathbb{C}[\rchi^{e_1^\ast}, \rchi^{e_2^\ast}] = \mathbb{C}[\z_1, \z_2].\]Hence \(U_\sigma = \mathbb{C}^2\).
Smoothness
In Example 6 with \(\sigma = \{0\}\) and in Example 7 with the standard quadrant \(\sigma = \mathrm{cone}(e_1, e_2)\), the varieties \(U_\sigma\) were respectively \(T_N\) and \(\mathbb{C}^2\), both smooth algebraic varieties. We shall now see that this is no accident: the smoothness of \(U_\sigma\) is completely determined by the combinatorial data of the cone \(\sigma\).
Definition 8 A strongly convex rational polyhedral cone \(\sigma \subset N_{\mathbb{R}}\) is called smooth (or regular, or nonsingular) if its primitive ray generators \(v_1, \ldots, v_k\) (with \(k = \dim \sigma\)) form part of a \(\mathbb{Z}\)-basis of the lattice \(N\).
Although the definition may look somewhat abstract, in practice it is equivalent to the conjunction of the following two conditions.
- \(\sigma\) is simplicial. That is, the number of rays equals the dimension.
- (When \(\sigma\) is full-dimensional) the determinant of the matrix \([v_1 \mid \cdots \mid v_n] \in \mathrm{Mat}_n(\mathbb{Z})\) whose columns are the primitive ray generators is \(\pm 1\).
In particular, for a two-dimensional cone in \(N = \mathbb{Z}^2\) it suffices to check that the \(2 \times 2\) matrix formed by the two ray generators has determinant \(\pm 1\), and indeed Example 6 and Example 7 satisfy this.
Proposition 9 The affine toric variety \(U_\sigma\) is a smooth algebraic variety if and only if \(\sigma\) is a smooth cone. More specifically, if \(\sigma\) is smooth and \(k = \dim \sigma\), then
\[U_\sigma \cong \mathbb{C}^k \times (\mathbb{C}^\ast)^{n-k}.\]Proof
(\(\Leftarrow\)) Suppose \(\sigma\) is smooth. By definition the ray generators \(v_1, \ldots, v_k\) form part of a basis \(\{v_1, \ldots, v_n\}\) of \(N\). Choose the dual basis \(\{v_1^\ast, \ldots, v_n^\ast\} \subset M\) of this basis; then
\[\sigma^\vee = \{u \in M_{\mathbb{R}} \mid \langle u, v_i\rangle \ge 0,\ i = 1, \ldots, k\} = \mathbb{R}_{\ge 0}\langle v_1^\ast, \ldots, v_k^\ast\rangle + \mathbb{R}\langle v_{k+1}^\ast, \ldots, v_n^\ast\rangle\]and therefore
\[S_\sigma = \sigma^\vee \cap M = \mathbb{Z}_{\ge 0}\langle v_1^\ast, \ldots, v_k^\ast\rangle \oplus \mathbb{Z}\langle v_{k+1}^\ast, \ldots, v_n^\ast\rangle.\]Hence
\[\mathbb{C}[S_\sigma] = \mathbb{C}[\rchi^{v_1^\ast}, \ldots, \rchi^{v_k^\ast}, \rchi^{\pm v_{k+1}^\ast}, \ldots, \rchi^{\pm v_n^\ast}] \cong \mathbb{C}[x_1, \ldots, x_k, x_{k+1}^{\pm 1}, \ldots, x_n^{\pm 1}],\]so \(U_\sigma \cong \mathbb{C}^k \times (\mathbb{C}^\ast)^{n-k}\) is smooth.
(\(\Rightarrow\)) For the converse, one starts from the fact that at the unique torus-fixed point of \(U_\sigma\) (corresponding to \(\sigma\) itself under the orbit-cone correspondence) the dimension of the cotangent space equals the number of rays \(\lvert \sigma(1) \rvert\). If \(U_\sigma\) is smooth, this dimension must be \(n = \dim U_\sigma\); hence the number of rays is \(n\), so \(\sigma\) is simplicial and moreover forms a \(\mathbb{Z}\)-basis of \(N\).
For example, take \(N = \mathbb{Z}^2\) and \(\sigma = \mathrm{cone}(e_2,\ 2e_1 - e_2)\). The matrix formed by the primitive generators of the two rays is
\[\begin{vmatrix} 0 & 2 \\ 1 & -1 \end{vmatrix} = -2,\]so \(\sigma\) is simplicial but not smooth. Indeed, computing the minimal generators of \(S_\sigma\) yields three elements \(u_1 = e_1^\ast\), \(u_2 = e_1^\ast + e_2^\ast\), \(u_3 = 2 e_1^\ast + e_2^\ast\), and the relation \(u_1 u_3 = u_2^2\) gives
\[\mathbb{C}[S_\sigma] \cong \mathbb{C}[x, y, z]/(xz - y^2).\]Thus \(U_\sigma\) is an affine toric variety having an \(A_1\) singularity at the origin. In general, if the determinant is \(\pm d\), then \(U_\sigma\) has a \(\mathbb{Z}/d\) quotient singularity at the origin.
Torus Action
To define the torus action on \(\Spec(\mathbb{C}[S_\sigma])\), we first review the familiar example of \(\mathbb{P}^N\). For \(\mathbb{P}^N = \Proj(\mathbb{C}[\x_0, \ldots, \x_N])\), the scaling action of \(\mathbb{C}^\ast\) decomposes the polynomial ring \(\mathbb{C}[\x_0, \ldots, \x_N]\) degree by degree. Each homogeneous component \(\mathbb{C}[\x_0, \ldots, \x_N]_d\) forms an eigenspace as a \(\mathbb{C}^\ast\)-module.
Similarly, there is a natural \(M\)-grading on \(\mathbb{C}[S_\sigma]\),
\[\mathbb{C}[S_\sigma] = \bigoplus_{u \in S_\sigma} \mathbb{C} \cdot \rchi^u,\]and when the \(n\)-dimensional algebraic torus \(T_N\) (to be defined shortly) acts on \(\mathbb{C}[S_\sigma]\), each one-dimensional subspace \(\mathbb{C}\cdot\rchi^u\) becomes the eigenspace of weight \(u \in M\). That is, for any \(t \in T_N\) the element \(\rchi^u\) is an eigenvector with eigenvalue \(\rchi^u(t) \in \mathbb{C}^\ast\). This \(M\)-grading is precisely the weight decomposition for the torus action.
We define \(T_N\) by
\[T_N = N \otimes_{\mathbb{Z}} \mathbb{C}^\ast.\]This is the tensor product of two abelian groups, but because the operation on \(\mathbb{C}^\ast\) is multiplication the notation can be a little confusing. As a set, \(T_N\) consists of finite sums of elements of the form \(v \otimes z\) with \(v \in N\) and \(z \in \mathbb{C}^\ast\), and it is an abelian group under addition. Bilinearity yields the following relations:
\[(v_1 + v_2) \otimes z = v_1 \otimes z + v_2 \otimes z,\qquad v \otimes (z_1 z_2) = v \otimes z_1 + v \otimes z_2,\qquad (k v) \otimes z = v \otimes z^k \quad (k \in \mathbb{Z})\]Note that in the second formula the left-hand side is multiplication in \(\mathbb{C}^\ast\) while the right-hand side is addition in \(T_N\). In other words, multiplication in \(\mathbb{C}^\ast\) becomes addition in \(T_N\) inside the tensor product.
Introducing a basis \(e_1, \ldots, e_n\) of \(N\), any element of \(T_N\) can be written as
\[t = e_1 \otimes z_1 + \cdots + e_n \otimes z_n = (z_1, \ldots, z_n), \qquad z_i \in \mathbb{C}^\ast.\]With this notation addition of two elements is given by
\[t + t' = e_1 \otimes (z_1 z_1') + \cdots + e_n \otimes (z_n z_n') = (z_1 z_1', \ldots, z_n z_n'),\]so \(T_N\) is naturally identified with the multiplicative group \((\mathbb{C}^\ast)^n\). Now for an element \(m \in M = \Hom(N, \mathbb{Z})\) we define a group homomorphism \(\rchi^m : T_N \to \mathbb{C}^\ast\) by the formula
\[\rchi^m(t) := z^{m(v)}, \qquad t = v \otimes z \in T_N, \; m(v) \in \mathbb{Z}.\]We claim that this is a well-defined group homomorphism. First, any element of \(T_N\) is a finite sum \(t = \sum_i v_i \otimes z_i\), and for such an element we define
\[\rchi^m(t) := \prod_i z_i^{m(v_i)};\]one checks that this gives a well-defined function extending the previous definition. Moreover, for any two elements
\[t = \sum_i v_i \otimes z_i,\qquad t' = \sum_j v_j' \otimes z_j'\]we have
\[\rchi^m(t + t') = \rchi^m\!\left(\sum_i v_i \otimes z_i + \sum_j v_j' \otimes z_j'\right) = \prod_i z_i^{m(v_i)} \prod_j (z_j')^{m(v_j')} = \rchi^m(t)\rchi^m(t'),\]so \(\rchi^m\) is indeed a group homomorphism. More concretely, introducing a basis \(e_1, \ldots, e_n\) of \(N\) and the dual basis \(e_1^\ast, \ldots, e_n^\ast \in M\), we can write \(m \in M\) as \(m = m_1 e_1^\ast + \cdots + m_n e_n^\ast\), and then
\[\rchi^m(t) = z_1^{m_1} \cdots z_n^{m_n}.\]Thus the character group \(\Hom(T_N, \mathbb{C}^\ast)\) is isomorphic to the dual lattice \(M\).
Building on this understanding of the torus, we now define a \(T_N\)-action on \(U_\sigma = \Spec(\mathbb{C}[S_\sigma])\). Since \(\Spec\) is a contravariant functor, a geometric action \(T_N \times U_\sigma \to U_\sigma\) is encoded by a comodule structure on the coordinate ring \(\mathbb{C}[S_\sigma]\); how this contravariance manifests at the level of points will be checked directly in Example 14.
Specifically, define the \(\mathbb{C}\)-algebra homomorphism \(\rho\) on the coordinate ring \(\mathbb{C}[S_\sigma]\) by
\[\rho : \mathbb{C}[S_\sigma] \longrightarrow \mathbb{C}[S_\sigma] \otimes_{\mathbb{C}} \mathbb{C}[M], \qquad \rchi^u \longmapsto \rchi^u \otimes \rchi^u.\]Here \(\mathbb{C}[M] = \mathbb{C}[\rchi^m \mid m \in M]\) is the coordinate ring of \(T_N\), and \(\rchi^u \in \mathbb{C}[S_\sigma]\) is the eigenvector of weight \(u\). That \(\rho\) is a well-defined algebra homomorphism can be verified directly from bilinearity.
Strictly speaking, \(\mathbb{C}[S_\sigma] \otimes \mathbb{C}[M]\) corresponds geometrically, after taking \(\Spec\), to the map \(U_\sigma \times T_N \to U_\sigma\), so this is the comorphism of a right action of \(T_N\). However, because \(T_N\) is abelian we may treat right and left actions as the same, and we shall regard it as a left action \(T_N \times U_\sigma \to U_\sigma\). Then for each \(t \in T_N\), the action of \(t\) on \(\mathbb{C}[S_\sigma]\) is given by the composition
\[\mathbb{C}[S_\sigma] \xrightarrow{\rho} \mathbb{C}[S_\sigma] \otimes_{\mathbb{C}} \mathbb{C}[M] \xrightarrow{\id \otimes \ev_t} \mathbb{C}[S_\sigma] \otimes_{\mathbb{C}} \mathbb{C} \cong \mathbb{C}[S_\sigma].\]Applying this composition to \(\rchi^u\) yields
\[(\id \otimes \ev_t)(\rchi^u \otimes \rchi^u) = \rchi^u \otimes \rchi^u(t) = \rchi^u(t) \rchi^u,\]so the result of the action of \(t\) on \(\rchi^u \in \mathbb{C}[S_\sigma]\) is \(\rchi^u(t) \rchi^u\). Counitality \((\id \otimes \ev_e)\circ\rho = \id\) and coassociativity \((\rho \otimes \id)\circ\rho = (\id\otimes\Delta)\circ\rho\) are readily checked on the basis elements \(\rchi^u\), so \(\rho\) defines a \(\mathbb{C}[M]\)-comodule structure corresponding exactly to an algebraic \(T_N\)-action on \(U_\sigma\). Hence we have the following.
Proposition 10 The algebraic torus \(T_N\) acts on the affine toric variety \(U_\sigma = \Spec(\mathbb{C}[S_\sigma])\). First define the \(\mathbb{C}\)-algebra homomorphism \(\rho\) on the coordinate ring \(\mathbb{C}[S_\sigma]\) by
\[\rho : \mathbb{C}[S_\sigma] \longrightarrow \mathbb{C}[S_\sigma] \otimes_{\mathbb{C}} \mathbb{C}[M], \qquad \rchi^u \longmapsto \rchi^u \otimes \rchi^u.\]Then this \(\rho\) defines a \(T_N\)-action on \(U_\sigma\), and for each \(t \in T_N\) the result of the action of \(t\) on \(\rchi^u \in \mathbb{C}[S_\sigma]\) is
\[t \cdot \rchi^u = \rchi^u(t) \rchi^u.\]Here \(\rchi^u : T_N \to \mathbb{C}^\ast\) is the character corresponding to \(u \in M\), and \(\rchi^u(t) \in \mathbb{C}^\ast\) is its value.
A word of caution is in order: the symbol \(\rchi^u\) has been used in two different senses in the discussion above. First, by definition it is a group homomorphism \(T_N \to \mathbb{C}^\ast\); but when regarded as an element of the vector space \(\mathbb{C}[S_\sigma]\), we think of it as the eigenvector of weight \(u\). The formula
\[t\cdot \rchi^u=\rchi^u(t)\rchi^u\]is exactly what relates these two meanings: when \(T_N\) acts on the vector space \(\mathbb{C}[S_\sigma]\), the element \(\rchi^u\) is an eigenvector for this action, and the corresponding eigenvalue is precisely \(\rchi^u(t)\).
Another subtlety is that the action of \(T_N\) on functions defined above, \(t\cdot f\), is more accurately a pullback,
\[(t\cdot f)(x) := f(t\cdot x).\]In general, for an arbitrary group algebra the standard left action on functions is defined by
\[(g\cdot f)(x) = f(g^{-1}\cdot x),\]but because \(T_N\) is abelian the definition without the inverse still yields a left action. The advantage of this convention is that \(\rchi^u\) becomes exactly the eigenvector of weight \(u\); with the standard convention the weight of \(\rchi^u\) would have been \(-u\). The same reason explains why the point-wise action to be examined shortly is \((t_1, t_2)\cdot(z_1, z_2) = (t_1 z_1, t_2 z_2)\) without an inverse.
In any case, the torus action defined above possesses an open dense torus orbit inside \(U_\sigma\), and this orbit is exactly the torus \(T_N\) itself. That is:
Proposition 11 Every affine toric variety \(U_\sigma\) contains the torus \(T_N\) as an open dense subset.
Proof
When \(\sigma = \{0\}\), we have \(\sigma^\vee = M_{\mathbb{R}}\) and hence \(S_\sigma = M\), so \(U_\sigma = \Spec(\mathbb{C}[M]) = T_N\).
In the general case \(S_\sigma \subseteq M\), so \(\mathbb{C}[S_\sigma] \subseteq \mathbb{C}[M]\), which induces the natural dominant morphism
\[T_N = \Spec(\mathbb{C}[M]) \longrightarrow \Spec(\mathbb{C}[S_\sigma]) = U_\sigma.\]Moreover, since \(\sigma\) is strongly convex, \(\sigma^\vee\) is a full-dimensional cone in \(M_{\mathbb{R}}\), and there exists a lattice point \(u_0 \in M\) in its interior. For any \(m \in M\), taking \(N\) sufficiently large makes both \(Nu_0\) and \(Nu_0 + m\) lie in \(S_\sigma\); hence \(m = (Nu_0 + m) - Nu_0\), showing that \(S_\sigma\) generates the entire group \(M\), and the fraction field of \(\mathbb{C}[S_\sigma]\) coincides with \(\mathbb{C}(M)\). Therefore \(T_N \hookrightarrow U_\sigma\) is an open dense subset.
We next examine how the face structure manifests on an affine toric variety. The following lemma is key.
Lemma 12 (Separation lemma) For a face \(\tau\) of a cone \(\sigma\) there exists an element \(u \in S_\sigma\) satisfying
\[\tau = \sigma \cap u^{\perp}.\]Moreover, for such \(u\) we have
\[\tau^\vee = \sigma^\vee + \mathbb{R}_{\ge 0}(-u).\]This yields the following proposition.
Proposition 13 For a face \(\tau\) of a cone \(\sigma\), the variety \(U_\tau\) is a principal open subset of \(U_\sigma\). ([Algebraic Varieties] §Affine Varieties, ⁋Definition 5) Specifically, choosing \(u \in S_\sigma\) with \(\tau = \sigma \cap u^{\perp}\), we have
\[U_\tau = \{ x \in U_\sigma \mid \rchi^u(x) \neq 0 \}.\]Proof
Let \(u \in S_\sigma\) with \(\tau = \sigma \cap u^\perp\). By Lemma 12 we have \(\tau^\vee = \sigma^\vee + \mathbb{R}_{\ge 0}(-u)\), so
\[S_\tau = \tau^\vee \cap M = (\sigma^\vee + \mathbb{R}_{\ge 0}(-u)) \cap M = S_\sigma + \mathbb{Z}_{\ge 0}(-u).\]The last equality follows from the facts that \(u \in S_\sigma\) and \(-u \in M\): one checks that any lattice point of \(\sigma^\vee + \mathbb{R}_{\ge 0}(-u)\) can be written as \(w - ku\) for some \(w \in S_\sigma\) and \(k \in \mathbb{Z}_{\ge 0}\). From this we obtain the localization of the semigroup algebra
\[\mathbb{C}[S_\tau] = \mathbb{C}[S_\sigma + \mathbb{Z}_{\ge 0}(-u)] = \mathbb{C}[S_\sigma][\rchi^{-u}] = \mathbb{C}[S_\sigma]_{\rchi^u}.\]Here \(\rchi^u \in \mathbb{C}[S_\sigma]\) and \(\rchi^{-u}\) is the inverse of \(\rchi^u\). Therefore
\[U_\tau = \Spec(\mathbb{C}[S_\tau]) = \Spec(\mathbb{C}[S_\sigma]_{\rchi^u}) = (U_\sigma)_{\rchi^u} = \{ x \in U_\sigma \mid \rchi^u(x) \neq 0 \}.\]For instance, in the two-dimensional cone \(\sigma\) of Example 7, the faces \(\tau_1=\mathbb{R}_{\geq 0}e_1\) and \(\tau_2=\mathbb{R}_{\geq 0}e_2\) are themselves one-dimensional cones, and their faces are the origin. In this structure, each of \(U_{\tau_1}\) and \(U_{\tau_2}\) is a principal open subset inside the whole \(U_\sigma\), and according to this face structure \(U_\sigma\) acquires a stratification reminiscent of a CW complex.
We claim that the open embedding \(U_\tau \hookrightarrow U_\sigma\) is \(T_N\)-equivariant. That is, the actions of \(T_N\) on \(U_\tau\) and on \(U_\sigma\) are compatible with the inclusion. This follows from a straightforward computation, so the above inclusion is also an inclusion of toric varieties.
Example 14 In Example 7 we saw that when \(N = \mathbb{Z}^2\) and \(\sigma = \mathbb{R}_{\geq 0}e_1+ \mathbb{R}_{\geq 0}e_2\), we have \(U_\sigma = \mathbb{C}^2\). Let us now examine concretely how the torus \(T_N = (\mathbb{C}^\ast)^2\) acts on \(U_\sigma = \mathbb{C}^2\).
First, as we saw in Example 7, \(\mathbb{C}[S_\sigma] = \mathbb{C}[\z_1, \z_2]\) (with \(\z_i = \rchi^{e_i^\ast}\)), and \(U_\sigma = \Spec(\mathbb{C}[\z_1, \z_2]) = \mathbb{C}^2\). By Proposition 10, an element \(t = (t_1, t_2) \in T_N = (\mathbb{C}^\ast)^2\) acts on the coordinate ring by
\[t \cdot \z_i = \rchi^{e_i^\ast}(t) \z_i = t_i \z_i \qquad i = 1, 2.\]To read off the action on points, recall that a point \((z_1, z_2) \in \mathbb{C}^2\) corresponds to the maximal ideal \(\mathfrak{m} = (\z_1 - z_1, \z_2 - z_2)\) of \(\mathbb{C}[\z_1, \z_2]\). Applying the ring map \(\sigma_t^\ast : \z_i \mapsto t_i \z_i\) to the generators of \(\mathfrak{m}\) yields the ideal
\[(t_1\z_1 - z_1, t_2\z_2 - z_2) = (\z_1 - z_1/t_1, \z_2 - z_2/t_2),\]which corresponds to the point \((z_1/t_1, z_2/t_2)\). Because \(\Spec\) is contravariant, this is interpreted as \((z_1/t_1, z_2/t_2)\) being carried to \((z_1,z_2)\); therefore computing at the level of points gives the torus action
\[(t_1, t_2) \cdot (z_1, z_2) = (t_1 z_1, t_2 z_2).\]This is the most natural action, in which each component of \((\mathbb{C}^\ast)^2\) scales the corresponding coordinate of \(\mathbb{C}^2\).
Let us now examine the orbit structure of this action. The open dense orbit provided by Proposition 11 is
\[(\mathbb{C}^\ast)^2 = \{(z_1, z_2) \mid z_1 \neq 0, z_2 \neq 0\},\]which is the two-dimensional torus \(T_N\) itself. More interesting are the lower-dimensional orbits: the coordinate axes \((\mathbb{C}^\ast) \times \{0\}\) and \(\{0\} \times (\mathbb{C}^\ast)\). These are orbits in which one coordinate is zero, so the freedom of the torus action drops by one. Likewise, the origin \((0, 0)\) is a fixed point of the torus action, which can be thought of formally as a zero-dimensional orbit.
We close this post by proving some basic properties of affine toric varieties.
Proposition 15 For an affine toric variety \(U_\sigma\) the following hold:
- \(U_\sigma\) is a normal variety.
- \(U_\sigma\) is irreducible.
- \(U_\sigma\) is an \(n\)-dimensional variety.
Proof
- For normality, Gordan’s lemma implies that \(S_\sigma = \sigma^\vee \cap M\) is a finitely generated semigroup. Moreover \(S_\sigma\) is saturated: if \(k \cdot u \in S_\sigma\) for some \(k \in \mathbb{Z}_{>0}\) and \(u \in M\), then \(u \in S_\sigma\). This means that \(\langle u, v \rangle \ge 0\) holds on \(\sigma\). An affine semigroup algebra is a normal domain if and only if its semigroup is saturated; therefore \(\mathbb{C}[S_\sigma]\) is a normal domain, and its spectrum \(U_\sigma\) is normal.
- For irreducibility, since \(S_\sigma\) is a sub-semigroup of the torsion-free abelian group \(M\), the algebra \(\mathbb{C}[S_\sigma]\) has no zero divisors. Hence \(\mathbb{C}[S_\sigma]\) is an integral domain, and \(U_\sigma = \Spec(\mathbb{C}[S_\sigma])\) is irreducible.
- Since \(\mathbb{C}[S_\sigma] \subseteq \mathbb{C}[M]\), its fraction field is contained in \(\mathbb{C}(M)\). On the other hand, because \(\sigma\) is strongly convex, \(\sigma^\vee\) is a full-dimensional cone in \(M_{\mathbb{R}}\); hence there exists a lattice point \(u_0 \in M\) in its interior. For any \(m \in M\), taking \(N\) sufficiently large makes both \(Nu_0\) and \(Nu_0 + m\) lie in \(S_\sigma\). Thus \(m = (Nu_0 + m) - Nu_0\), showing that \(S_\sigma\) generates the whole group \(M\), and the fraction field of \(\mathbb{C}[S_\sigma]\) coincides exactly with \(\mathbb{C}(M)\). Since \(M \cong \mathbb{Z}^n\), the transcendence degree of \(\mathbb{C}(M)\) is \(n\), and therefore \(\dim U_\sigma = n\).
References
[Ful] William Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, Princeton University Press, 1993.
[CLS] David Cox, John Little, Hal Schenck, Toric Varieties, Graduate Studies in Mathematics, American Mathematical Society, 2011.
댓글남기기