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.
In the previous post we studied toric varieties, and in doing so we already saw that the combinatorial data of the fan defining a toric variety determines many of its properties. In this post we focus in particular on (torus-invariant) divisors defined on a toric variety.
Torus-Invariant Weil Divisors
We already know that a toric variety \(X_\Sigma\) contains the algebraic torus \(T_N \cong (\mathbb{C}^\ast)^n\) as an open dense subset. However, divisor theory on this torus is trivial: considering the coordinate ring of \(T_N\), it is the Laurent polynomial ring
\[\mathbb{C}[M] = \mathbb{C}[t_1^{\pm 1}, \ldots, t_n^{\pm 1}]\]and hence a UFD ([Ring Theory] §Polynomial Rings, ⁋Theorem 16). In a UFD every height-1 prime ideal is principal ([Commutative Algebra] §Primary Decomposition, ⁋Theorem 7), so every codimension-1 irreducible subvariety is the zero set of a single Laurent polynomial, and therefore every Weil divisor is principal. Consequently \(\Cl(T_N) = 0\), so there is no motivation to study divisors on the torus itself.
In other words, all nontrivial divisor information on \(X_\Sigma\) is concentrated in the boundary \(X_\Sigma \setminus T_N\), and this boundary consists exactly of the part fixed by the action of \(T_N\). Therefore, we develop the divisor theory on \(X_\Sigma\) by focusing on torus-invariant divisors.
For each 1-dimensional cone of the fan \(\Sigma\), i.e. each ray \(\rho \in \Sigma(1)\), consider its primitive generator \(v_\rho \in N\). The ray \(\rho\) corresponds to a codimension-1 torus orbit in \(X_\Sigma\), and its Zariski closure defines an irreducible divisor.
Definition 1 For each ray \(\rho \in \Sigma(1)\), we write \(D_\rho\) for the Zariski closure of the corresponding torus orbit. Then \(D_\rho\) is an irreducible Weil divisor on \(X_\Sigma\), and such a \(D_\rho\) is called a torus-invariant prime divisor.
This definition arises naturally from the orbit structure of a toric variety. As we saw in §Definition of Toric Varieties, ⁋Proposition 5, a \(d\)-dimensional cone corresponds to a torus orbit of dimension \(n-d\), so a 1-dimensional cone \(\rho\) corresponds to an orbit of dimension \(n-1\), i.e. a codimension-1 torus-invariant subvariety. The Zariski closure of this orbit naturally becomes a divisor.
Definition 2 We write \(\Div_T(X_\Sigma)\) for the free abelian group generated by the torus-invariant Weil divisors on \(X_\Sigma\). That is,
\[\Div_T(X_\Sigma) = \bigoplus_{\rho \in \Sigma(1)} \mathbb{Z} \cdot D_\rho.\]Any torus-invariant Weil divisor is uniquely expressed in the form \(D = \sum_{\rho \in \Sigma(1)} a_\rho D_\rho\).
We saw above that the divisor theory on \(T_N\) is trivial, and using this we can see that studying divisors on a toric variety reduces to studying torus-invariant divisors. Restricting an arbitrary divisor \(D \in \Div(X_\Sigma)\) to the open dense subset \(T_N\) gives a divisor \(D \cap T_N\) on \(T_N\), which is principal of the form \(\divisor(f)\); hence the support of \(D - \divisor(f)\) lies in the boundary \(X_\Sigma \setminus T_N = \bigcup_{\rho \in \Sigma(1)} D_\rho\) and is therefore torus-invariant. From this, \(D\) is linearly equivalent to some torus-invariant divisor, and more formally we can verify that
\[\Div_T(X_\Sigma) \hookrightarrow \Div(X_\Sigma) \twoheadrightarrow \Cl(X_\Sigma)\tag{1}\]is surjective. Thus, looking at \(\Div_T(X_\Sigma)\) instead of \(\Div(X_\Sigma)\) is not a trade-off that loses information.
Principal Divisors and the Class Group
One of our ultimate goals is to compute \(\Cl(X_\Sigma)\) explicitly from the combinatorial data of the fan \(\Sigma\). Since (1) is a surjection, the first isomorphism theorem gives \(\Cl(X_\Sigma) \cong \Div_T(X_\Sigma) / \ker\), and \(\Div_T(X_\Sigma)\) is already completely described as the free abelian group \(\bigoplus_{\rho \in \Sigma(1)} \mathbb{Z} D_\rho\) determined by the rays of the fan. Therefore, to know \(\Cl(X_\Sigma)\) it remains only to describe the kernel of (1) in terms of the data of the fan.
By definition this kernel consists of those elements of \(\Div_T(X_\Sigma)\) that become \(0\) in \(\Cl(X_\Sigma)\), i.e. divisors that are simultaneously torus-invariant and principal. Thus the problem reduces to asking which rational functions on a toric variety define a torus-invariant principal divisor.
The most natural candidates are the characters \(\rchi^m\) corresponding to elements \(m \in M\) of the lattice. For each \(m \in M\), the character \(\rchi^m: T_N \to \mathbb{C}^\ast\) can be viewed as a rational function on \(X_\Sigma\) because \(T_N \subset X_\Sigma\) is open dense (§Affine Toric Varieties, ⁋Proposition 10). The character \(\rchi^m\) itself is not invariant under the action of \(T_N\), but for \(t \in T_N\) we have
\[(t \cdot \rchi^m)(x) = \rchi^m(t x) = \rchi^m(t) \cdot \rchi^m(x)\]by the convention of §Affine Toric Varieties, ⁋Proposition 10), so they differ only by a scalar multiple. Since scalar multiples do not change zeros and poles, even though the \(\rchi^m\) themselves are not torus-invariant, their divisors \(\divisor(\rchi^m)\) are torus-invariant and lie in \(\Div_T(X_\Sigma)\). Our first goal is to write this divisor explicitly in terms of the data of the fan.
Proposition 3 The principal divisor corresponding to the character \(\rchi^m\) is given by
\[\divisor(\rchi^m) = \sum_{\rho \in \Sigma(1)} \langle m, v_\rho \rangle D_\rho.\]Proof
Since \(X_\Sigma\) is normal (§Definition of Toric Varieties, ⁋Proposition 4), the local ring \(\mathcal{O}_{X_\Sigma, D_\rho}\) at the generic point of each prime divisor \(D_\rho\) is a DVR; write its valuation as \(v_{D_\rho}: \mathbb{C}(X_\Sigma)^\ast \to \mathbb{Z}\). By the definition of principal divisor,
\[\divisor(\rchi^m) = \sum_{\rho \in \Sigma(1)} v_{D_\rho}(\rchi^m) D_\rho\]so it suffices to show that \(v_{D_\rho}(\rchi^m) = \langle m, v_\rho \rangle\) for each \(\rho \in \Sigma(1)\). To verify this, consider the affine chart \(U_\rho = \Spec\mathbb{C}[\rho^\vee \cap M]\) corresponding to the ray \(\rho\). On \(U_\rho\), the prime ideal corresponding to the generic point of \(D_\rho\) is generated by \(\{u \in \rho^\vee \cap M : \langle u, v_\rho \rangle > 0\}\), and localizing at this prime yields a DVR where characters are evaluated by \(\rchi^u \mapsto \langle u, v_\rho \rangle\). Hence for any \(m \in M\) we have \(v_{D_\rho}(\rchi^m) = \langle m, v_\rho \rangle\).
From this we obtain a natural group homomorphism
\[M \to \Div_T(X_\Sigma);\qquad m \mapsto \divisor(\rchi^m).\]The image of this homomorphism is the set of principal divisors, and our claim is that this is exactly the kernel of (1).
Proposition 4 Assume that the rays \(\Sigma(1)\) of the fan \(\Sigma\) span \(N_\mathbb{R}\). Then the following sequence is exact:
\[0 \longrightarrow M \longrightarrow \Div_T(X_\Sigma) \longrightarrow \Cl(X_\Sigma) \longrightarrow 0.\]Here the first arrow is \(m \mapsto \divisor(\rchi^m)\), and the second arrow is \(D \mapsto [D]\).
Proof
First we show that \(M \to \Div_T(X_\Sigma)\) is injective. If \(\divisor(\rchi^m) = 0\), then \(\langle m, v_\rho \rangle = 0\) for all \(\rho \in \Sigma(1)\). Since the fan \(\Sigma\) spans \(N_\mathbb{R}\) by assumption, the primitive generators \(\{v_\rho\}\) span \(N_\mathbb{R}\), so \(m = 0\).
Next we show that the kernel of \(\Div_T(X_\Sigma) \to \Cl(X_\Sigma)\) coincides exactly with the divisors of characters. By definition the kernel is the set of divisors that are principal and simultaneously \(T\)-invariant. Suppose \(D = \divisor(f)\) is \(T\)-invariant. Since the support of a \(T\)-invariant divisor lies in the boundary \(X_\Sigma \setminus T_N = \bigcup_{\rho \in \Sigma(1)} D_\rho\), restricting \(f\) to the open dense subset \(T_N\) gives \(\divisor(f \rvert_{T_N}) = 0\). Hence \(f\rvert_{T_N}\) is a unit in the coordinate ring \(\mathbb{C}[M]\) of \(T_N\). The units of \(\mathbb{C}[M]\) are exactly of the form \(c \cdot \rchi^m\) (\(c \in \mathbb{C}^\ast\), \(m \in M\)), so \(f\rvert_{T_N} = c \cdot \rchi^m\), and by normality of \(X_\Sigma\) we have \(f = c \cdot \rchi^m\) on \(X_\Sigma\). Therefore \(D = \divisor(\rchi^m)\).
Finally we show surjectivity. For any divisor \(D \in \Div(X_\Sigma)\), its restriction \(D \cap T_N\) to \(T_N\) is a divisor on \(T_N \cong (\mathbb{C}^\ast)^n\). Since the coordinate ring \(\mathbb{C}[M]\) of \(T_N\) is a UFD, this is principal, and thus for some \(f \in \mathbb{C}(X_\Sigma)^\ast\) the support of \(D - \divisor(f)\) can be made to lie in \(X_\Sigma \setminus T_N = \bigcup_{\rho \in \Sigma(1)} D_\rho\). The right-hand side is expressed as a sum of torus-invariant divisors, so \([D] = [D - \divisor(f)]\) has a torus-invariant representative.
This exact sequence becomes a powerful tool for explicitly computing the class group of a toric variety. Under the assumption, \(\Div_T(X_\Sigma) \cong \mathbb{Z}^{\Sigma(1)}\) is a free abelian group of rank \(\lvert \Sigma(1) \rvert\) and \(M \cong \mathbb{Z}^n\) is a free abelian group of rank \(n\) embedded inside it, so \(\Cl(X_\Sigma) \cong \mathbb{Z}^{\Sigma(1)} / M\) is a finitely generated abelian group of rank \(\lvert \Sigma(1) \rvert - n\). However, note that it may have torsion in general—for instance, on \(\mathbb{P}(1,1,2)\) we have \(\Cl \cong \mathbb{Z} \oplus \mathbb{Z}/2\).
Cartier Divisors
On the other hand, to discuss line bundles via divisors we need to examine Cartier divisors. We studied Weil divisors above, and in the smooth case Weil and Cartier divisors coincide, so this part is already complete to some extent; but for toric varieties there exists a way to describe Cartier divisors even when they are not smooth.
Definition 5 A function \(\psi: \lvert \Sigma \rvert \to \mathbb{R}\) defined on the support \(\lvert \Sigma \rvert = \bigcup_{\sigma \in \Sigma} \sigma\) of the fan \(\Sigma\) is called a piecewise linear function if for each cone \(\sigma \in \Sigma\), the restriction \(\psi\rvert_\sigma\) is of the form \(\psi(v) = \langle m_\sigma, v \rangle\) for some \(m_\sigma \in M_\mathbb{R}\). A piecewise linear function \(\psi\) is called integral if each \(m_\sigma\) lies in \(M\).
We write \(\PL(\Sigma, M_\mathbb{R})\) for the set of piecewise linear functions, and \(\PL(\Sigma, M)\) for the set of integral piecewise linear functions. The important point is that a torus-invariant Cartier divisor \(D\) determines a \(\psi_D \in \PL(\Sigma, M)\).
Proposition 6 For a torus-invariant Cartier divisor \(D = \sum_{\rho \in \Sigma(1)} a_\rho D_\rho\), using \(m_\sigma \in M\) determined by \(D\rvert_{U_\sigma} = \divisor(\rchi^{-m_\sigma})\) on each maximal cone \(\sigma \in \Sigma\), the function defined by
\[\psi_D: \lvert \Sigma \rvert \to \mathbb{R};\qquad \psi_D(v) = \langle m_\sigma, v \rangle \quad \text{for } v \in \sigma\]is an element of \(\PL(\Sigma, M)\), and in particular takes the value \(\psi_D(v_\rho) = -a_\rho\) at the primitive generator \(v_\rho\) of each ray \(\rho \in \Sigma(1)\).
Proof
Assume \(D\) is a Cartier divisor. The definition of a Cartier divisor is that it is locally principal, but on an affine toric variety \(U_\sigma\) a stronger fact holds: every \(T\)-invariant Cartier divisor on \(U_\sigma\) is globally principal. This is based on the fact that the coordinate ring \(\mathbb{C}[\sigma^\vee \cap M]\) carries an \(M\)-grading, so cocycle data for invertible \(\mathcal{O}_{U_\sigma}\)-modules trivializes. Accepting this, we have \(D\rvert_{U_\sigma} = \divisor(f_\sigma)\) for some rational function \(f_\sigma \in \mathbb{C}(X_\Sigma)^\ast\).
Since \(D\) is \(T\)-invariant, for any \(t \in T_N\) we have \(\divisor(t \cdot f_\sigma) = t \cdot \divisor(f_\sigma) = \divisor(f_\sigma)\), and two rational functions with the same divisor differ only by a scalar, so \(t \cdot f_\sigma = c(t) \cdot f_\sigma\) for some \(c(t) \in \mathbb{C}^\ast\). That is, \(f_\sigma\) is a weight vector for the \(T_N\)-action, and hence is of the form \(f_\sigma = c \cdot \rchi^{-m_\sigma}\) for some \(m_\sigma \in M\) and \(c \in \mathbb{C}^\ast\). (We write \(-m_\sigma\) in the exponent to make the sign calculation below neat.) Therefore on \(U_\sigma\),
\[D\rvert_{U_\sigma} = \divisor(\rchi^{-m_\sigma}) = \sum_{\rho \in \sigma(1)} \langle -m_\sigma, v_\rho \rangle D_\rho\]and \(a_\rho = -\langle m_\sigma, v_\rho \rangle\) for all \(\rho \in \sigma(1)\). Defining \(\psi_D(v) = \langle m_\sigma, v \rangle\), this is linear on \(\sigma\) and takes the desired value \(-a_\rho\) at \(v_\rho\). For \(\psi_D\) to be well-defined on the common face \(\tau = \sigma_1 \cap \sigma_2\) of two maximal cones \(\sigma_1, \sigma_2\), we need \(\langle m_{\sigma_1}, v\rangle = \langle m_{\sigma_2}, v\rangle\) (\(v \in \tau\)), which follows from the fact that on \(U_\tau\), \(\rchi^{m_{\sigma_2} - m_{\sigma_1}}\) must be a regular invertible function, i.e. a character \(\rchi^{m}\) (\(m \in \tau^\perp \cap M\)).
This correspondence is in fact an isomorphism. That is, the group \(\CaDiv_T(X_\Sigma)\) of torus-invariant Cartier divisors is isomorphic to \(\PL(\Sigma, M)\).
Line Bundles
Now that we have examined Cartier divisors to some extent, we will use them to study line bundles. When a Cartier divisor \(D\) is given, we defined the line bundle \(\mathcal{O}_{X_\Sigma}(D)\) in [Algebraic Varieties] §Line Bundles and Vector Bundles, ⁋Definition 17. Its sheaf of sections is given for each open set \(U \subseteq X_\Sigma\) by
\[\mathcal{O}_{X_\Sigma}(D)(U) = \{f \in \mathbb{C}(X_\Sigma)^\times \mid \divisor(f)\rvert_U + D\rvert_U \ge 0\} \cup \{0\},\]which intuitively means that \(\mathcal{O}_{X_\Sigma}(D)\) is the sheaf of rational functions on \(U\) that are allowed to have poles along \(D\) up to order \(D\). Even when \(D\) is a general Weil divisor that is not Cartier, the same formula on the right-hand side defines a well-defined \(\mathcal{O}_X\)-module sheaf (though not a line bundle, but a rank-\(1\) sheaf), and we shall adopt this formula as the definition of \(\mathcal{O}_{X_\Sigma}(D)\) below.
In the case of a toric variety, the global sections of this sheaf have an explicit basis given by characters \(\rchi^m\), and the condition is completely described by combinatorial data on the lattice \(M\).
Proposition 7 For a torus-invariant Weil divisor \(D = \sum_{\rho \in \Sigma(1)} a_\rho D_\rho\), the global section space of the sheaf \(\mathcal{O}_{X_\Sigma}(D)\) is given by:
\[H^0(X_\Sigma, \mathcal{O}_{X_\Sigma}(D)) = \bigoplus_{\substack{m \in M \\ \langle m, v_\rho \rangle \ge -a_\rho \text{ for all } \rho}} \mathbb{C} \cdot \rchi^m.\]Proof
Since \(D\) is \(T_N\)-invariant, \(T_N\) acts naturally on the sheaf \(\mathcal{O}_{X_\Sigma}(D)\), and this action defines a rational representation of the algebraic torus. Rational representations of an algebraic torus always decompose into weight spaces by characters, so the global section space also decomposes as
\[H^0(X_\Sigma, \mathcal{O}_{X_\Sigma}(D)) = \bigoplus_{m \in M} H^0(X_\Sigma, \mathcal{O}_{X_\Sigma}(D))_m.\]Here the weight-\(m\) component consists of sections that are multiplied by \(\rchi^m(t)\) under the action of \(t \in T_N\). Such sections are exactly of the form \(c \cdot \rchi^m\) (\(c \in \mathbb{C}\)). Indeed, if \(s \in H^0(\mathcal{O}_{X_\Sigma}(D))_m\), then \(s/\rchi^m\) is a \(T\)-invariant rational function on \(X_\Sigma\), and since \(T_N\) is open dense in \(X_\Sigma\) we have \(\mathbb{C}(X_\Sigma)^{T_N} = \mathbb{C}(T_N)^{T_N} = \Frac(\mathbb{C}[M])^{T_N} = \mathbb{C}\), so \(s = c \cdot \rchi^m\). That is, each weight space is 1-dimensional spanned by \(\rchi^m\) (for those \(m\) that give sections).
Now the condition for \(c \cdot \rchi^m\) to be a section is \(\divisor(\rchi^m) + D \ge 0\). By Proposition 3,
\[\divisor(\rchi^m) + D = \sum_{\rho \in \Sigma(1)} (\langle m, v_\rho \rangle + a_\rho) D_\rho\]so the necessary and sufficient condition for this divisor to be effective is that \(\langle m, v_\rho \rangle + a_\rho \ge 0\) for all \(\rho\), i.e. \(\langle m, v_\rho \rangle \ge -a_\rho\). The direct sum of the 1-dimensional character spaces \(\mathbb{C} \cdot \rchi^m\) for such \(m \in M\) constitutes the entire global section space.
Thinking of the condition appearing in Proposition 7, \(\langle m, v_\rho\rangle \ge -a_\rho\) for all \(\rho \in \Sigma(1)\), we can collect all \(m\) satisfying this condition and define the polyhedron in \(M_\mathbb{R}\)
\[\Delta_D = \{m \in M_\mathbb{R} \mid \langle m, v_\rho \rangle \ge -a_\rho \text{ for all } \rho \in \Sigma(1)\}.\]Then the set of lattice points inside \(\Delta_D\) coincides exactly with the \(m\)’s appearing as summands.
On the other hand, this is not the first time we encounter a situation where lattice points inside a polyhedron carry geometric meaning: in §Definition of Toric Varieties, ⁋Proposition 9 we constructed a monomial map \(\phi_P: T_N \to \mathbb{P}^s\) from the lattice points \(P \cap M = \{m_0, \ldots, m_s\}\) of a lattice polytope \(P\), and showed that the Zariski closure of its image is isomorphic to \(X_P\). From the perspective of Proposition 7 above, these lattice points coincide exactly with the character basis of \(H^0(X_\Sigma, \mathcal{O}_{X_\Sigma}(D))\) corresponding to \(P = \Delta_D\). That is, the projective embedding determined by lattice points was in fact the standard linear system embedding determined by the global sections of the line bundle \(\mathcal{O}_{X_\Sigma}(D)\).
This perspective naturally leads to the question of how the geometric condition of ampleness translates into combinatorics of the fan for a toric variety. Stating the result first, this is completely described by the convexity of the piecewise linear function.
Definition 8 A piecewise linear function \(\psi: \lvert \Sigma \rvert \to \mathbb{R}\) is strictly convex if for any two distinct maximal cones \(\sigma_1, \sigma_2 \in \Sigma\) with corresponding \(m_{\sigma_1}, m_{\sigma_2} \in M_\mathbb{R}\),
\[\psi(v) = \langle m_{\sigma_1}, v \rangle \text{ for } v \in \sigma_1, \qquad \psi(v) = \langle m_{\sigma_2}, v \rangle \text{ for } v \in \sigma_2\]and for any point \(v\) in \(\sigma_1\) outside the relative interior of \(\sigma_1 \cap \sigma_2\),
\[\psi(v) < \langle m_{\sigma_2}, v \rangle\]always holds. Equivalently, the linear function \(\langle m_\sigma, \cdot\rangle\) defined by the data \(m_\sigma\) of each maximal cone \(\sigma\) is an upper bound for \(\psi\), with equality holding exactly when \(v \in \sigma\) (under the sign convention \(\psi_D(v_\rho) = -a_\rho\) of Proposition 6, for ample \(D\) the function \(\psi_D\) is convex in this sense).
A strictly convex piecewise linear function is a strong condition that completely determines the structure of the fan, and the following theorem shows that this condition corresponds exactly to the ampleness of a divisor.
Proposition 9 When the toric variety \(X_\Sigma\) is complete, a torus-invariant Cartier divisor \(D\) is ample if and only if the corresponding piecewise linear function \(\psi_D\) is strictly convex.
Proof
(\(\Rightarrow\)) Assume \(D\) is ample. By definition, for some \(k > 0\) the divisor \(kD\) is very ample, inducing a projective embedding \(\phi_{kD}: X_\Sigma \hookrightarrow \mathbb{P}^N\), and this embedding coincides with the monomial map defined by the lattice points of the polytope \(\Delta_{kD} \cap M\) (§Definition of Toric Varieties, ⁋Proposition 9). Since \(X_\Sigma\) is complete, the image \(\overline{\phi_{kD}(T_N)}\) is isomorphic to the projective toric variety obtained from the normal fan of \(\Delta_{kD}\) (see [CLS] Theorem 6.2.1), and the isomorphism with \(X_\Sigma\) yields that the normal fan of \(\Delta_{kD}\) coincides with \(\Sigma\). This means that \(\psi_{kD} = k\psi_D\) is strictly convex, and hence \(\psi_D\) is also strictly convex.
(\(\Leftarrow\)) Conversely, assume \(\psi_D\) is strictly convex. This means that \(\Sigma\) is the normal fan of some lattice polytope. Specifically, by strict convexity
\[\Delta_D = \{m \in M_\mathbb{R} \mid \langle m, v_\rho \rangle \ge -a_\rho \text{ for all } \rho \in \Sigma(1)\}\]is a bounded region and hence a lattice polytope. Since the normal fan of this polytope coincides with \(\Sigma\), by §Definition of Toric Varieties, ⁋Proposition 8 the variety \(X_\Sigma\) is projective. Moreover, the line bundle \(\mathcal{O}_{X_\Sigma}(D)\) corresponding to \(D\) defines a very ample line bundle, so \(D\) is ample.
The Picard Group of a Toric Variety
Finally, we describe the Picard group of a toric variety. The Picard group is the group of isomorphism classes of line bundles under tensor product, and it is isomorphic to the group of linear equivalence classes of Cartier divisors.
Proposition 10 For a toric variety \(X_\Sigma\), the following hold.
- The Picard group \(\Pic(X_\Sigma)\) is isomorphic to the group of linear equivalence classes of torus-invariant Cartier divisors, \(\CaDiv_T(X_\Sigma) / M\).
-
\(\Pic(X_\Sigma)\) is a subgroup of \(\Cl(X_\Sigma)\), and the following commutative diagram holds:
Here the vertical arrows represent inclusions.
Proof
(1) For a general algebraic variety, the group of Cartier divisors \(\CaDiv(X)\) is isomorphic to the group of line bundles, and quotienting by principal divisors gives the Picard group. For a toric variety, torus-invariant Cartier divisors alone suffice to represent every linear equivalence class, so \(\Pic(X_\Sigma) \cong \CaDiv_T(X_\Sigma) / M\) holds.
(2) The image of \(M \to \Div_T(X_\Sigma)\), namely \(\divisor(\rchi^m)\), consists of principal divisors and hence is automatically Cartier, so this arrow factors through \(M \to \CaDiv_T(X_\Sigma)\). Consequently the left two columns of the commutative diagram share the same \(M\), and the quotient \(\Pic(X_\Sigma) = \CaDiv_T(X_\Sigma)/M\) naturally sits inside \(\Cl(X_\Sigma) = \Div_T(X_\Sigma)/M\) as a subgroup. Injectivity follows from the fact that \(\CaDiv_T(X_\Sigma) \hookrightarrow \Div_T(X_\Sigma)\) is injective and both quotients are by the same image of \(M\).
The Picard group of a toric variety can be described explicitly in the language of piecewise linear functions. Quotienting \(\PL(\Sigma, M)\) by the globally linear functions, i.e. those given by \(\psi(v) = \langle m, v \rangle\) for a single \(m \in M\) on the whole space, yields the Picard group. This provides a powerful method for computing combinatorial invariants of a toric variety.
Finally, let us apply the tools we have developed so far to the most familiar toric variety, projective space \(\mathbb{P}^n\). (§Definition of Toric Varieties, ⁋Example 10)
Example 11 The rays of the fan defining \(\mathbb{P}^n\) are \(\rho_0, \rho_1, \ldots, \rho_n\) generated by \(v_0 = -e_1 - \cdots - e_n\) and \(v_i = e_i\) (\(1 \le i \le n\)), and each maximal cone \(\sigma_i\) is the \(n\)-dimensional cone generated by all but \(\rho_i\), corresponding to the standard affine chart \(U_{\sigma_i} = \{\x_i \neq 0\}\) of \(\mathbb{P}^n\) (§Definition of Toric Varieties, ⁋Example 10). Each torus-invariant prime divisor \(D_i = D_{\rho_i}\) corresponds in homogeneous coordinate representation to the coordinate hyperplane
\[D_0 = \{\x_0 = 0\}, \quad D_1 = \{\x_1 = 0\}, \quad \ldots, \quad D_n = \{\x_n = 0\}.\]These are each hyperplane divisors on \(\mathbb{P}^n\), and from basic algebraic geometry we know that they determine a single hyperplane class. For convenience let us choose the representative \(H = D_0\).
Let us compute the piecewise linear function \(\psi_H\) corresponding to \(H\). Under the monomial map convention \((t_1, \ldots, t_n) \mapsto [1 : t_1 : \cdots : t_n]\) of §Definition of Toric Varieties, ⁋Example 10, we have \(t_i = \x_i/\x_0\), so the local equation of \(H = \{\x_0 = 0\}\) on each affine chart is:
- When \(i = 0\), on \(U_{\sigma_0} = \{\x_0 \neq 0\}\) the divisor \(H\) is empty, so \(H\rvert_{U_{\sigma_0}} = \divisor(1) = \divisor(\rchi^0)\), i.e. \(m_{\sigma_0} = 0\);
- When \(i \ge 1\), the local equation of \(H \cap U_{\sigma_i}\) is \(\x_0/\x_i = t_i^{-1} = \rchi^{-e_i^\ast}\), so from the convention \(H\rvert_{U_{\sigma_i}} = \divisor(\rchi^{-m_{\sigma_i}})\) of Proposition 6 we have \(m_{\sigma_i} = e_i^\ast\).
From this, \(\psi_H\) is given by \(0\) on \(\sigma_0\) and by \(\langle e_i^\ast, -\rangle\) on \(\sigma_i\) (\(i \ge 1\)), and its values on the rays are \(\psi_H(v_0) = -1\) and \(\psi_H(v_i) = 0\) (\(i \ge 1\)), consistent with \(\psi_H(v_\rho) = -a_\rho\) of Proposition 6.
That this \(\psi_H\) is strictly convex can be verified by direct application of Definition 8: taking a point \(v\) in the interior of each maximal cone \(\sigma_i\) (i.e. writing it as a positive combination of generators of \(\sigma_i = \mathrm{cone}(v_l : l \neq i)\), so \(v = \sum_{l \neq i} a_l v_l\) with \(a_l > 0\) for the generators \(\{v_l\}_{l \neq i}\) of \(\sigma_i\)) and evaluating with \(m_{\sigma_k}\) of another maximal cone \(\sigma_k\), one can directly compute that \(\psi_H(v) = \langle m_{\sigma_i}, v\rangle\) is strictly smaller than \(\langle m_{\sigma_k}, v\rangle\) in every case. For instance, when \(i \ge 1\) and \(k = 0\) we have \(\langle e_i^\ast, v\rangle = -a_0 < 0 = \langle 0, v\rangle\), and when \(i = 0\) and \(k \ge 1\) we have \(0 = \langle 0, v\rangle < a_k = \langle e_k^\ast, v\rangle\). Therefore by Proposition 9, \(H\) is ample, which is a toric reconfirmation of the familiar fact that the hyperplane divisor on \(\mathbb{P}^n\) is ample.
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.
[Oda] Tadao Oda, Convex Bodies and Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, 1988.
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