04.11.08 · algebraic-geometry / toric

Toric divisor and support function

shipped3 tiersLean: partial

Anchor (Master): Fulton §3.3-§3.4; Cox-Little-Schenck §4.0-§4.3 + §6.1; Oda *Convex Bodies and Algebraic Geometry* §2.1-§2.3; Demazure 1970 *Sous-groupes algébriques de rang maximum du groupe de Cremona* (Ann. Sci. ENS (4) 3, 507-588); Kempf-Knudsen-Mumford-Saint-Donat *Toroidal Embeddings I* (LNM 339, 1973) Ch. I §2; Mumford 1972 *Compositio Math.* 24 (cone-data construction); Brion 1989 *Math. Ann.* 286 (lattice-point cohomology); Brion-Vergne 1997 *J. Amer. Math. Soc.* 10 (residue formula for lattice points)

Intuition [Beginner]

A toric divisor is the combinatorial atom of divisor theory on a toric variety. Every ray $ρ$ of the fan $Σ$ produces one such atom: the closure of the codimension-one torus orbit attached to $ρ$ by the orbit-cone correspondence of the previous unit. Write this closure as $D_{ρ}$ . Every torus-invariant Weil divisor on $X_{Σ}$ is a finite integer combination of these atoms. So the abstract divisor theory of the variety collapses, on the torus-invariant slice, to a finitely-generated lattice indexed by rays.

The companion combinatorial object is the support function. To each torus-invariant Cartier divisor $D$ — written as a formal integer combination of the ray-indexed divisors with coefficient $a_{ρ}$ at the ray $ρ$ — one assigns a continuous function $ψ_{D}$ defined on the entire fan support $∣Σ∣$ that is linear on each maximal cone and integer-valued on the lattice. The recipe is concrete: $ψ_{D}$ takes the value $- a_{ρ}$ on the primitive generator of the ray $ρ$ , and the values on every other lattice point are pinned by linearity across each maximal cone. The Cartier condition translates into a gluing rule for these linear pieces across the walls between adjacent cones.

The third ingredient is the divisor polytope $P_{D}$ in the character lattice. It is the set of characters $m \in M_{R}$ that pair against every primitive ray generator at least as positively as the divisor demands. Lattice points of $P_{D}$ count global sections of the line bundle $O (D)$ — one character $χ^{m}$ for each lattice point, with no further redundancy. So sheaf cohomology on the toric variety reduces, at degree zero, to a lattice-point count.

Visual [Beginner]

A composite schematic showing the three avatars of a toric divisor on a small fan in $R^{2}$ . Left panel: the fan $Σ$ in $N_{R} = R^{2}$ with several rays and shaded maximal cones; one ray $ρ$ is highlighted and its associated divisor $D_{ρ} \subseteq X_{Σ}$ is labelled on the variety side. Middle panel: the piecewise-linear support function $ψ_{D}$ pictured as a tent-shaped surface over $∣Σ∣$ with crease lines along the rays, dropping to $- a_{ρ}$ at each primitive generator and rising linearly on each maximal cone. Right panel: the divisor polytope $P_{D}$ in the dual lattice $M_{R}$ pictured as a shaded lattice polygon, with its integer points marked as bullets and one bullet labelled by the character $χ^{m}$ .

$A three-panel composite showing the ray-divisor pair, the piecewise-linear support function over the fan, and the divisor polytope with its lattice points labelling global sections of $\mathcal{O}(D)$.$

The picture captures the three faces of the same Cartier divisor: a geometric closure of a codimension-one orbit, a piecewise-linear function pinned at the rays, and a polytope of characters counting global sections. Each face supplies a calculus appropriate to a different question — geometric (intersection numbers), combinatorial (Cartier coefficients), or cohomological (lattice-point counts).

Worked example [Beginner]

Compute the anticanonical divisor on $P^{2}$ and the polytope counting its global sections. The fan of $P^{2}$ in $R^{2}$ has three rays with primitive generators $v_{0} = - e_{1} - e_{2}$ , $v_{1} = e_{1}$ , $v_{2} = e_{2}$ , and three maximal cones $σ_{12} = Cone (e_{1}, e_{2})$ , $σ_{02} = Cone (- e_{1} - e_{2}, e_{2})$ , $σ_{01} = Cone (- e_{1} - e_{2}, e_{1})$ .

Step 1. Identify the atoms. Each ray $ρ_{i}$ produces a toric divisor $D_{i}$ — the closure of the codimension-one orbit, which is one of the three coordinate lines ${x_{i} = 0} \subseteq P^{2}$ . The anticanonical divisor is $- K = D_{0} + D_{1} + D_{2}$ , the sum of the three coordinate lines.

Step 2. Identify the support function. The coefficients of $- K$ are $a_{0} = a_{1} = a_{2} = 1$ , so the support function takes the value $- 1$ on each primitive ray generator: $ψ_{- K} (v_{0}) = ψ_{- K} (v_{1}) = ψ_{- K} (v_{2}) = - 1$ . Extend linearly on each maximal cone. On $σ_{12} = Cone (e_{1}, e_{2})$ , linear interpolation gives $ψ_{- K} (s e_{1} + t e_{2}) = - (s + t)$ , attained at $e_{1}$ as $- 1$ and at $e_{2}$ as $- 1$ . Similarly on the other two cones.

Step 3. Identify the divisor polytope. The polytope $P_{- K} \subseteq M_{R} = R^{2}$ is the set of $m = (s, t)$ with $⟨ m, v_{i} ⟩ \geq - 1$ for $i \in {0, 1, 2}$ , i.e., $- s - t \geq - 1$ , $s \geq - 1$ , $t \geq - 1$ . So $P_{- K}$ is the triangle with vertices $(- 1, - 1)$ , $(2, - 1)$ , $(- 1, 2)$ , a size-three lattice simplex.

What this tells us. The lattice points of $P_{- K}$ are exactly the $10$ pairs $(s, t)$ with $s, t \geq - 1$ and $s + t \leq 1$ ; the count is the number of homogeneous cubic monomials in three variables, $(2 3 + 2) = 10$ . So the space of global sections is $H^{0} (P^{2}, O (- K)) = H^{0} (P^{2}, O (3))$ , the $10$ -dimensional space of plane cubics. The whole picture — divisor as orbit closure, support function as piecewise-linear ramp, polytope as lattice-point gadget — emerges from three rays in $R^{2}$ .

Check your understanding [Beginner]

Exercise (easy, multiple choice).

Let $Σ$ be a fan in $N_{R}$ with $Σ (1) = {ρ_{1}, \dots, ρ_{r}}$ the set of rays. A torus-invariant Weil divisor on the toric variety $X_{Σ}$ is:

A. A character $m \in M$ paired against a ray. B. A finite integer combination $a_{1} D_{ρ_{1}} + a_{2} D_{ρ_{2}} + \dots + a_{r} D_{ρ_{r}}$ of the toric divisors attached to rays. C. A maximal cone of $Σ$ . D. A morphism $X_{Σ} \to P^{1}$ .

Hint

The atoms of toric Weil divisor theory are indexed by rays of the fan, with each ray contributing one prime divisor.

Answer

B. A torus-invariant Weil divisor on $X_{Σ}$ is a finite integer combination of the ray-indexed divisors $D_{ρ_{i}}$ , each of which is the closure of the codimension-one torus orbit attached to the ray $ρ_{i}$ by the orbit-cone correspondence. The coefficients are integers recording the multiplicities. Option A confuses divisors with the dual character data that supplies principal divisors only. Option C confuses ray-indexed divisors with maximal-cone-indexed affine charts. Option D is the wrong category — a morphism to $P^{1}$ is associated to a divisor only after picking a section of a line bundle and looking at zero / pole loci.

Exercise (easy, numeric).

Compute the dimension of $H^{0} (P^{2}, O (2))$ , the space of homogeneous quadratic polynomials in three variables. State the answer as a single integer.

Hint

The dimension equals the number of homogeneous monomials of degree $2$ in three variables. The general formula is $(d n + d)$ for degree $d$ in $n + 1$ variables — here $n = 2, d = 2$ .

Answer

6. The space of homogeneous polynomials of degree $2$ in three variables $x_{0}, x_{1}, x_{2}$ has the basis ${x_{0}^{2}, x_{1}^{2}, x_{2}^{2}, x_{0} x_{1}, x_{0} x_{2}, x_{1} x_{2}}$ , six elements. The formula $(d n + d) = (2 2 + 2) = 6$ matches.

Via the divisor-polytope picture: the polytope $P_{2 H}$ for the divisor $2 H = 2 D_{0}$ (or equivalently $D_{0} + D_{1}$ , etc. — all hyperplane multiples) on $P^{2}$ is the lattice triangle with vertices at the origin and $(2, 0), (0, 2)$ in suitable coordinates, with the $6$ lattice points $(0, 0), (1, 0), (2, 0), (0, 1), (1, 1), (0, 2)$ matching the $6$ degree- $2$ monomials.

Exercise (easy, true-false).

True or false: on every smooth complete toric variety, every torus-invariant Weil divisor is Cartier.

Hint

Smoothness translates into a unimodular-basis condition on each cone of the fan, which forces every Weil divisor to admit a piecewise-linear support function — the Cartier datum.

Answer

True. On a smooth toric variety, every cone $σ$ of the fan is generated by part of a $Z$ -basis of the lattice $N$ . Given any coefficient assignment ${a_{ρ}}_{ρ \in Σ (1)}$ , on each maximal cone $σ$ one can solve for the unique character $m_{σ} \in M$ satisfying $⟨ m_{σ}, v_{ρ} ⟩ = - a_{ρ}$ for every ray $ρ \subseteq σ$ , because the system of equations has a unique integer solution (the rays of $σ$ form a basis of $N$ ). The collection ${m_{σ}}$ glues into a piecewise-linear support function and supplies the Cartier datum.

On singular toric varieties this fails: at a non-smooth cone, the rays do not form a $Z$ -basis of $N$ , the system of equations has no integer solution for generic coefficients, and the corresponding Weil divisor is only $Q$ -Cartier — some integer multiple is Cartier, namely the multiplicity of the cone.

Formal definition [Intermediate+]

Let $Σ$ be a fan in $N_{R}$ with $N$ a free abelian group of rank $n$ and dual $M = Hom_{Z} (N, Z)$ . Let $X_{Σ}$ denote the associated toric variety from [04.11.04] and $T = Hom_{Z} (M, C^{*})$ the dense torus. Write $Σ (1)$ for the set of rays, $Σ_{m a x}$ for the set of maximal cones, and $∣Σ∣ \subseteq N_{R}$ for the support of the fan.

Definition (toric divisor). For each ray $ρ \in Σ (1)$ with primitive generator $v_{ρ} \in N$ , the toric divisor $D_{ρ} \subseteq X_{Σ}$ is the closure of the codimension-one torus orbit $O (ρ)$ associated to $ρ$ by the orbit-cone correspondence of [04.11.06]. By the dimension formula $dim O (ρ) + dim ρ = n$ , the orbit $O (ρ)$ has dimension $n - 1$ , and its closure $D_{ρ}$ is a $T$ -invariant prime Weil divisor on $X_{Σ}$ . The free abelian group of $T$ -invariant Weil divisors is $$ \mathrm{Div}T(X\Sigma) := \bigoplus_{\rho \in \Sigma(1)} \mathbb{Z} \cdot D_\rho. $$ An element $D \in Div_{T} (X_{Σ})$ is written $D = \sum_{ρ} a_{ρ} D_{ρ}$ with $a_{ρ} \in Z$ the coefficient at $ρ$ .

Definition (support function). Let $D = \sum_{ρ} a_{ρ} D_{ρ}$ be a $T$ -invariant Cartier divisor. The support function $ψ_{D} : ∣Σ∣ \to R$ is the unique continuous function such that

for each maximal cone $σ \in Σ_{m a x}$ , the restriction $ψ_{D} ∣_{σ}$ is the linear function $u \mapsto ⟨ m_{σ}, u ⟩$ for some character $m_{σ} \in M$ ;
for each ray $ρ \in Σ (1)$ , $ψ_{D} (v_{ρ}) = - a_{ρ}$ ;
the linear pieces ${m_{σ}}$ glue continuously, meaning that on each pairwise intersection $σ \cap τ$ (with $σ, τ \in Σ_{m a x}$ ), $m_{σ} - m_{τ} \in (σ \cap τ)^{⊥}$ — equivalently, the two linear pieces agree on $σ \cap τ$ .

The data ${m_{σ}}_{σ \in Σ_{m a x}}$ is the local Cartier data of $D$ . The function $ψ_{D}$ is also called the piecewise-linear function attached to $D$ .

Definition (Cartier condition). A $T$ -invariant Weil divisor $D = \sum_{ρ} a_{ρ} D_{ρ}$ is Cartier iff for every maximal cone $σ$ there exists a character $m_{σ} \in M$ with $⟨ m_{σ}, v_{ρ} ⟩ = - a_{ρ}$ for every ray $ρ \subseteq σ$ , and the cocycle compatibility $m_{σ} - m_{τ} \in (σ \cap τ)^{⊥}$ holds for every pair $σ, τ \in Σ_{m a x}$ . Equivalently, $D$ is Cartier iff its associated support function $ψ_{D}$ exists with integer-valued linear pieces.

Definition (divisor polytope). Let $D = \sum_{ρ} a_{ρ} D_{ρ}$ be a $T$ -invariant Cartier divisor. The divisor polytope $P_{D} \subseteq M_{R}$ is $$ P_D := {m \in M_\mathbb{R} : \langle m, v_\rho\rangle \geq -a_\rho \text{ for every } \rho \in \Sigma(1)}. $$ $P_{D}$ is a (possibly unbounded) rational polyhedron in $M_{R}$ . When $X_{Σ}$ is complete and $D$ is nef, $P_{D}$ is bounded — a lattice polytope.

Convention. The minus signs in $ψ_{D} (v_{ρ}) = - a_{ρ}$ and the inequality $⟨ m, v_{ρ} ⟩ \geq - a_{ρ}$ are the Fulton convention (Fulton 1993 §3.3). Cox-Little-Schenck §4.3 uses the same convention. Some sources (Oda 1988) use the opposite sign convention, swapping $a_{ρ} \leftrightarrow - a_{ρ}$ throughout. The choice does not affect the mathematics; we adhere to Fulton.

Counterexamples to common slips

"Every torus-invariant Weil divisor is Cartier." False on singular toric. Example: on the affine quadric cone $X_{σ}$ for $σ = Cone (e_{1}, e_{1} + 2 e_{2})$ (the $A_{1}$ -singularity), the divisor $D_{ρ_{1}}$ associated to the first ray is Weil but not Cartier; twice $D_{ρ_{1}}$ is Cartier. The Cartier datum at the non-smooth cone $σ$ would need to satisfy $⟨ m_{σ}, e_{1} ⟩ = - 1$ and $⟨ m_{σ}, e_{1} + 2 e_{2} ⟩ = 0$ , forcing $⟨ m_{σ}, e_{2} ⟩ = 1/2$ , no integer solution.
"The support function is unique." It is unique up to translation by characters when $X_{Σ}$ has at least one maximal cone of dimension $n$ : the value $ψ_{D} (v_{ρ}) = - a_{ρ}$ on rays pins the function up to a global character $m \in M$ acting by $ψ_{D} \to ψ_{D} + m$ (which shifts the divisor by the principal $div (χ^{m})$ ). On non-complete fans where the rays do not span $N_{R}$ , the support function is determined only modulo the orthogonal complement of the ray-span.
"The divisor polytope is always bounded." $P_{D}$ is bounded iff $D$ is nef and $X_{Σ}$ is complete. For non-nef $D$ on a complete toric variety, $P_{D}$ can be empty. For non-complete $X_{Σ}$ (e.g., $X_{Σ} = C^{n}$ ), $P_{D}$ is typically a polyhedral cone or wedge — unbounded but well-defined.

Key theorem with proof [Intermediate+]

The signature theorem of this unit is the support-function correspondence of Fulton §3.4 Proposition 1, lifting the Cartier-vs-Weil distinction to a combinatorial bijection.

Theorem (support-function correspondence; Demazure 1970, Fulton §3.4). Let $Σ$ be a fan in $N_{R}$ . The assignment $D \mapsto ψ_{D}$ is a bijection between the group of $T$ -invariant Cartier divisors on $X_{Σ}$ and the group of integer-valued piecewise-linear functions on $∣Σ∣$ (continuous functions $ψ : ∣Σ∣ \to R$ that are linear with integer linear pieces on each cone of $Σ$ ). The inverse sends $ψ$ to the Weil divisor $\sum_{ρ} (- ψ (v_{ρ})) D_{ρ}$ . Under this bijection, the group of principal Cartier divisors corresponds to the group of globally linear functions on $∣Σ∣$ , i.e., functions of the form $u \mapsto ⟨ m, u ⟩$ for some $m \in M$ .

Proof.

(Setup: $T$ -invariant Cartier divisors are locally given by characters.) A $T$ -invariant Cartier divisor $D$ on $X_{Σ}$ is a Weil divisor with a Cartier-cocycle structure: an open cover ${U_{i}}$ , a section $f_{i} \in K (X_{Σ})^{*}$ on each $U_{i}$ , and transitions $f_{i} / f_{j} \in O_{X_{Σ}}^{*} (U_{i} \cap U_{j})$ giving units on overlaps. For the canonical torus-invariant affine cover ${U_{σ}}_{σ \in Σ}$ of $X_{Σ}$ (from [04.11.04]), $T$ -invariance forces each local section $f_{σ}$ to be a character: $f_{σ} = χ^{m_{σ}}$ for some $m_{σ} \in M$ . The cocycle condition $f_{σ} / f_{τ} \in O_{X_{Σ}}^{*} (U_{σ} \cap U_{τ})$ becomes $χ^{m_{σ} - m_{τ}} \in O^{*} (U_{σ \cap τ})$ , equivalently $m_{σ} - m_{τ} \in (σ \cap τ)^{⊥} \cap M$ — the Cartier-cocycle compatibility.

(From a Cartier divisor to a support function.) Given a $T$ -invariant Cartier divisor with local data ${m_{σ}}_{σ \in Σ_{m a x}}$ satisfying the cocycle compatibility, define $ψ_{D} : ∣Σ∣ \to R$ by $ψ_{D} (u) := ⟨ m_{σ}, u ⟩$ for $u \in σ$ . The cocycle compatibility $m_{σ} - m_{τ} \in (σ \cap τ)^{⊥}$ ensures the pieces agree on overlaps, so $ψ_{D}$ is well-defined and continuous on $∣Σ∣$ . By construction $ψ_{D}$ is linear on each maximal cone with integer-valued linear piece $m_{σ} \in M$ . The value at a primitive ray generator $v_{ρ} \in σ$ is $ψ_{D} (v_{ρ}) = ⟨ m_{σ}, v_{ρ} ⟩$ , which equals $- a_{ρ}$ by the local equation of $D$ on $U_{σ}$ : the multiplicity of $D$ along $D_{ρ}$ is the order of vanishing of $χ^{m_{σ}}$ at the generic point of $D_{ρ} \cap U_{σ}$ , and this order equals $⟨ m_{σ}, v_{ρ} ⟩$ on the toric chart. Setting $a_{ρ} = - ψ_{D} (v_{ρ})$ recovers the Weil-divisor coefficients.

(From a support function to a Cartier divisor.) Conversely, given an integer-valued piecewise-linear function $ψ$ on $∣Σ∣$ , let $m_{σ} \in M$ be the linear piece on each maximal cone $σ$ . Continuity of $ψ$ across $σ \cap τ$ forces $m_{σ} - m_{τ} \in (σ \cap τ)^{⊥}$ . Define the Weil divisor $D_{ψ} := \sum_{ρ} (- ψ (v_{ρ})) D_{ρ}$ ; the local sections $χ^{m_{σ}}$ on each affine chart $U_{σ}$ supply Cartier data with the divisor $D_{ψ}$ as their associated Weil divisor.

(Bijection.) The two assignments $D \mapsto ψ_{D}$ and $ψ \mapsto D_{ψ}$ are mutually inverse by construction: starting from $D = \sum_{ρ} a_{ρ} D_{ρ}$ with Cartier data ${m_{σ}}$ , the resulting $ψ_{D}$ has $ψ_{D} (v_{ρ}) = ⟨ m_{σ}, v_{ρ} ⟩ = - a_{ρ}$ , and so $D_{ψ_{D}} = \sum_{ρ} (- ψ_{D} (v_{ρ})) D_{ρ} = \sum_{ρ} a_{ρ} D_{ρ} = D$ . Conversely $ψ_{D_{ψ}} = ψ$ on every maximal cone by the linear-piece identification.

(Principal Cartier divisors correspond to globally linear functions.) A principal $T$ -invariant Cartier divisor is $div (χ^{m}) = \sum_{ρ} ⟨ m, v_{ρ} ⟩ D_{ρ}$ for some $m \in M$ . Its associated support function is $ψ_{div (χ^{m})} (u) = - ⟨ m, u ⟩$ , the globally linear function with global linear piece $- m$ . Conversely, if $ψ$ is globally linear on $∣Σ∣$ with linear piece $m \in M$ (so $ψ (u) = ⟨ m, u ⟩$ for all $u$ ), then the associated divisor $D_{ψ} = \sum_{ρ} (- ⟨ m, v_{ρ} ⟩) D_{ρ} = - div (χ^{m})$ is principal. So the bijection restricts to an identification of principal Cartier divisors with global linear functions on $∣Σ∣$ . $□$

Bridge. The support-function correspondence builds toward the polytope-fan dictionary of [04.11.10] and appears again in [04.11.09] as the supporting calculus for the toric Picard exact sequence. The central insight is that the entire Cartier-vs-Weil distinction on a toric variety, normally treated as a sheaf-theoretic question requiring local-ring arguments, reduces to a combinatorial gluing condition on piecewise-linear functions over a polyhedral fan.

This is exactly the foundational reason that line-bundle geometry on a toric variety becomes algorithmic: where on a general variety the Cartier condition requires checking local equations on every affine chart, on a toric variety the check collapses to a finite linear-algebra computation across walls between maximal cones. The bridge is to the projectivity criterion of [04.11.04]: $X_{Σ}$ is projective iff some support function is strictly convex, and the projective embedding is supplied directly by the lattice points of the divisor polytope $P_{D}$ . Putting these together with the orbit-cone correspondence of [04.11.06], the support-function correspondence identifies the entire $T$ -equivariant divisor calculus of a toric variety with the combinatorics of piecewise-linear functions on its fan, generalises in the equivariant setting to higher-dimensional moment-map data, and is dual to the global-sections lattice-point formula via the polytope construction $D \leftrightarrow P_{D}$ .

Exercises [Intermediate+]

Exercise 1 (easy, symbolic).

For the fan of $P^{1}$ with rays $v_{0} = 1$ and $v_{1} = - 1$ in $R$ , list all toric divisors and compute the principal divisor $div (χ^{m})$ for an arbitrary character $m \in M = Z$ .

Hint

The toric variety $P^{1}$ has exactly two torus-fixed points (the two maximal cones of its fan), and each ray corresponds to one of them. The character map sends $m \mapsto \sum_{ρ} ⟨ m, v_{ρ} ⟩ D_{ρ}$ .

Answer

Toric divisors. Two rays $ρ_{0}$ and $ρ_{1}$ produce two toric divisors $D_{0}$ and $D_{1}$ on $P^{1}$ . Geometrically $D_{0} = {0}$ and $D_{1} = {\infty}$ — the two torus-fixed points of $P^{1}$ , each a codimension-one (= zero-dimensional) closure of a torus orbit. So $Div_{T} (P^{1}) = Z D_{0} \oplus Z D_{1}$ , the free abelian group of rank $2$ .

Principal divisor of a character. For $m \in M = Z$ , the character $χ^{m}$ is the function $z \mapsto z^{m}$ on $P^{1}$ . Its divisor is $div (χ^{m}) = m \cdot D_{0} - m \cdot D_{1}$ (zero of order $m$ at $z = 0$ if $m > 0$ , pole of order $∣ m ∣$ at $z = \infty$ ). In the formula: $⟨ m, v_{0} ⟩ = ⟨ m, 1 ⟩ = m$ and $⟨ m, v_{1} ⟩ = ⟨ m, - 1 ⟩ = - m$ , giving $div (χ^{m}) = m D_{0} + (- m) D_{1} = m (D_{0} - D_{1})$ .

Implication for support functions. The principal divisor $div (χ^{m})$ corresponds to the support function $ψ (u) = - m u$ for $u \in ∣Σ∣ = R$ , a globally linear function with constant linear piece $- m$ on both maximal cones. Any non-globally-linear piecewise-linear $ψ$ — for example, $ψ$ with linear piece $0$ on $σ_{0} = R_{\geq 0}$ and linear piece $- 1$ on $σ_{1} = R_{\leq 0}$ — corresponds to a non-principal Cartier divisor, in this case $D_{1}$ (the divisor at $\infty$ ).

Rubric: full credit for identifying $D_{0}, D_{1}$ , computing $div (χ^{m}) = m (D_{0} - D_{1})$ , and connecting to the support-function description.

Exercise 2 (easy, symbolic).

Compute the divisor polytope $P_{D}$ for the divisor $D = D_{0} + 2 D_{1} + 3 D_{2}$ on $P^{2}$ (with the standard fan $v_{0} = - e_{1} - e_{2}$ , $v_{1} = e_{1}$ , $v_{2} = e_{2}$ ). Sketch its lattice points and compute $h^{0} (P^{2}, O (D))$ .

Hint

$P_{D} = {m = (s, t) \in M_{R} : ⟨ m, v_{i} ⟩ \geq - a_{i} for i = 0, 1, 2}$ with $(a_{0}, a_{1}, a_{2}) = (1, 2, 3)$ . Translate the inequalities into linear constraints on $s, t$ and enumerate lattice solutions.

Answer

Inequalities. The three constraints are:

$⟨ m, v_{0} ⟩ = - s - t \geq - 1$ , i.e., $s + t \leq 1$ ;
$⟨ m, v_{1} ⟩ = s \geq - 2$ ;
$⟨ m, v_{2} ⟩ = t \geq - 3$ .

Polytope $P_{D}$ . The set ${(s, t) \in R^{2} : s + t \leq 1, s \geq - 2, t \geq - 3}$ — a triangle with vertices computed from intersecting pairs of bounding lines:

$s = - 2, t = - 3$ : $(s + t = - 5 \leq 1$ , satisfied) vertex $(- 2, - 3)$ .
$s = - 2, s + t = 1$ : $t = 3$ , vertex $(- 2, 3)$ .
$t = - 3, s + t = 1$ : $s = 4$ , vertex $(4, - 3)$ .

So $P_{D}$ is the triangle with vertices $(- 2, - 3), (- 2, 3), (4, - 3)$ , with side lengths $6$ along each axis-aligned edge (from $- 3$ to $3$ for the vertical, from $- 2$ to $4$ for the horizontal). The slanted edge runs from $(- 2, 3)$ to $(4, - 3)$ with $s + t = 1$ .

Lattice point count. Enumerate $(s, t)$ with $s \in {- 2, - 1, 0, 1, 2, 3, 4}$ , $t \in {- 3, - 2, - 1, 0, 1, 2, 3}$ , $s + t \leq 1$ :

$s = - 2$ : $t \in {- 3, - 2, - 1, 0, 1, 2, 3}$ , all $7$ values have $s + t \leq 1$ .
$s = - 1$ : $t \in {- 3, - 2, - 1, 0, 1, 2}$ , $6$ values.
$s = 0$ : $t \in {- 3, - 2, - 1, 0, 1}$ , $5$ values.
$s = 1$ : $t \in {- 3, - 2, - 1, 0}$ , $4$ values.
$s = 2$ : $t \in {- 3, - 2, - 1}$ , $3$ values.
$s = 3$ : $t \in {- 3, - 2}$ , $2$ values.
$s = 4$ : $t \in {- 3}$ , $1$ value.

Total: $7 + 6 + 5 + 4 + 3 + 2 + 1 = 28$ .

Global sections. $h^{0} (P^{2}, O (D)) = 28$ . Check: under the identification $Pic (P^{2}) = Z$ , the divisor $D = D_{0} + 2 D_{1} + 3 D_{2} = (1 + 2 + 3) H = 6 H$ (since $D_{0} = D_{1} = D_{2} = H$ in $Pic$ ), so $O (D) = O (6)$ and $h^{0} (P^{2}, O (6)) = (2 6 + 2) = 28$ . Matches.

Rubric: full credit for the inequalities, the polytope vertices, the lattice-point enumeration, and the cross-check $(2 8) = 28$ .

Exercise 3 (medium, symbolic).

Compute the support function $ψ_{D}$ explicitly for $D = D_{0} + D_{1} + D_{2}$ (the anticanonical divisor) on $P^{2}$ , and verify that its linear pieces ${m_{σ}}$ on the three maximal cones satisfy the Cartier-cocycle compatibility on pairwise intersections.

Hint

On each maximal cone $σ_{ij} = Cone (v_{i}, v_{j})$ , solve the linear system $⟨ m_{ij}, v_{i} ⟩ = - 1$ and $⟨ m_{ij}, v_{j} ⟩ = - 1$ . Then verify $m_{ij} - m_{ik} \in ρ_{i}^{⊥}$ for adjacent cones sharing the ray $ρ_{i}$ .

Answer

Linear pieces. The three maximal cones are $σ_{01} = Cone (v_{0}, v_{1}), σ_{02} = Cone (v_{0}, v_{2}), σ_{12} = Cone (v_{1}, v_{2})$ , with $v_{0} = - e_{1} - e_{2}, v_{1} = e_{1}, v_{2} = e_{2}$ . Solve:

$σ_{12}$ : $⟨ m_{12}, e_{1} ⟩ = - 1$ , $⟨ m_{12}, e_{2} ⟩ = - 1$ , so $m_{12} = - e_{1}^{*} - e_{2}^{*} = (- 1, - 1)$ .
$σ_{02}$ : $⟨ m_{02}, - e_{1} - e_{2} ⟩ = - 1$ , $⟨ m_{02}, e_{2} ⟩ = - 1$ . With $m_{02} = (a, b)$ , the first gives $- a - b = - 1$ , the second gives $b = - 1$ , so $a = 2$ . $m_{02} = (2, - 1)$ .
$σ_{01}$ : $⟨ m_{01}, - e_{1} - e_{2} ⟩ = - 1$ , $⟨ m_{01}, e_{1} ⟩ = - 1$ . With $m_{01} = (a, b)$ , the first gives $- a - b = - 1$ , the second gives $a = - 1$ , so $b = 2$ . $m_{01} = (- 1, 2)$ .

Cocycle check. Pairwise compatibility on $σ_{ij} \cap σ_{ik}$ — which is the ray $ρ_{i}$ (and the origin). The condition $m_{ij} - m_{ik} \in ρ_{i}^{⊥}$ means $(m_{ij} - m_{ik}) \cdot v_{i} = 0$ :

$σ_{12}$ vs $σ_{02}$ on $ρ_{2}$ : $m_{12} - m_{02} = (- 1, - 1) - (2, - 1) = (- 3, 0)$ . Pair against $v_{2} = e_{2}$ : $⟨(- 3, 0), e_{2} ⟩ = 0$ . Compatible.
$σ_{02}$ vs $σ_{01}$ on $ρ_{0}$ : $m_{02} - m_{01} = (2, - 1) - (- 1, 2) = (3, - 3)$ . Pair against $v_{0} = - e_{1} - e_{2}$ : $⟨(3, - 3), - e_{1} - e_{2} ⟩ = - 3 + 3 = 0$ . Compatible.
$σ_{01}$ vs $σ_{12}$ on $ρ_{1}$ : $m_{01} - m_{12} = (- 1, 2) - (- 1, - 1) = (0, 3)$ . Pair against $v_{1} = e_{1}$ : $⟨(0, 3), e_{1} ⟩ = 0$ . Compatible.

So the three linear pieces glue continuously, and the support function $ψ_{- K}$ is well-defined.

Geometric interpretation. The support function $ψ_{- K}$ on $P^{2}$ takes the value $- 1$ at each $v_{i}$ and rises linearly on each maximal cone. Inside $σ_{12}$ , $ψ (u) = ⟨(- 1, - 1), u ⟩ = - (s + t)$ for $u = s e_{1} + t e_{2}$ , reaching $ψ (e_{1}) = - 1, ψ (e_{2}) = - 1$ . The function is strictly convex at each wall — the value extending linearly from a neighbouring cone lies strictly above the actual value at the wall — which is the Cartan-Serre-Grothendieck criterion for $- K = D_{0} + D_{1} + D_{2}$ to be ample on $P^{2}$ (cross-referenced in [04.11.09] §C).

Rubric: full credit for the three linear pieces, the three cocycle checks, and the connection to strict convexity / ampleness.

Exercise 4 (medium, symbolic).

On the singular toric variety $P (1, 2, 3)$ (fan in $R^{2}$ with rays $v_{0} = (1, 0), v_{1} = (0, 1), v_{2} = (- 2, - 3)$ ), determine whether the divisor $D_{2}$ (the toric divisor at the third ray) is Cartier. If not, find the smallest integer multiple of $D_{2}$ that is Cartier.

Hint

For each maximal cone $σ$ , check whether there exists an integer character $m_{σ} \in M = Z^{2}$ with $⟨ m_{σ}, v_{ρ} ⟩ = - a_{ρ}$ for every ray $ρ \subseteq σ$ — where $a_{ρ}$ is $1$ at $ρ = ρ_{2}$ and $0$ at other rays. The bottleneck is the cone containing $ρ_{2}$ .

Answer

Maximal cones. The fan of $P (1, 2, 3)$ has three maximal cones $σ_{01} = Cone (v_{0}, v_{1}), σ_{02} = Cone (v_{0}, v_{2}), σ_{12} = Cone (v_{1}, v_{2})$ .

Cartier test for $D_{2}$ . Coefficients $a_{0} = a_{1} = 0, a_{2} = 1$ .

$σ_{01}$ (does not contain $ρ_{2}$ ): need $⟨ m_{01}, v_{0} ⟩ = 0$ , $⟨ m_{01}, v_{1} ⟩ = 0$ , so $m_{01} = (0, 0)$ — integer, fine.
$σ_{02}$ (contains $ρ_{2}$ ): need $⟨ m_{02}, v_{0} ⟩ = 0$ , $⟨ m_{02}, v_{2} ⟩ = - 1$ . With $m_{02} = (a, b)$ : $a = 0$ , $- 2 a - 3 b = - 1$ , so $- 3 b = - 1$ , $b = 1/3$ . Not an integer — $D_{2}$ fails the Cartier condition at $σ_{02}$ .
$σ_{12}$ (contains $ρ_{2}$ ): need $⟨ m_{12}, v_{1} ⟩ = 0$ , $⟨ m_{12}, v_{2} ⟩ = - 1$ . With $m_{12} = (a, b)$ : $b = 0$ , $- 2 a - 3 b = - 1$ , so $- 2 a = - 1$ , $a = 1/2$ . Not an integer — $D_{2}$ fails at $σ_{12}$ .

So $D_{2}$ is not Cartier on $P (1, 2, 3)$ . It is only $Q$ -Cartier, with multiplier related to the cone multiplicities.

Smallest Cartier multiple. The denominators in the failures are $3$ at $σ_{02}$ and $2$ at $σ_{12}$ . The smallest multiple that clears both is $lcm (2, 3) = 6$ . Verify: for $6 D_{2}$ , the constraints become $⟨ m_{02}, v_{2} ⟩ = - 6$ with $m_{02} = (0, 2)$ (integer) and $⟨ m_{12}, v_{2} ⟩ = - 6$ with $m_{12} = (3, 0)$ (integer). Both work. So $6 D_{2}$ is Cartier, and no smaller positive multiple is. Cocycle check on the shared ray $ρ_{0}$ for $σ_{01}$ vs $σ_{02}$ : $m_{01} - m_{02} = (0, 0) - (0, 2) = (0, - 2)$ , pair against $v_{0} = (1, 0)$ : $0$ . Compatible. Similar checks for other walls confirm.

Conclusion. $D_{2}$ is not Cartier on $P (1, 2, 3)$ ; the smallest positive Cartier multiple is $6 D_{2}$ , with $6 = lcm (2, 3)$ the cone-multiplicity bound from the two singular cones. This matches the toric Picard / class group analysis of [04.11.09] Exercise 4: $Pic (P (1, 2, 3))$ sits as an index- $6$ subgroup inside $Cl (P (1, 2, 3)) = Z$ , generated precisely by $6$ times the class generator.

Rubric: full credit for testing each maximal cone, identifying the failure modes, and computing $lcm (2, 3) = 6$ as the Cartier multiplier.

Exercise 5 (medium, multiple choice).

Let $X_{Σ}$ be a smooth complete toric variety of dimension $n$ and let $D = \sum_{ρ} a_{ρ} D_{ρ}$ be a $T$ -invariant Cartier divisor with associated support function $ψ_{D}$ and divisor polytope $P_{D}$ . Which of the following statements is always true?

A. $dim H^{0} (X_{Σ}, O (D)) = ∣ P_{D} \cap M ∣$ when $D$ is nef. B. The support function $ψ_{D}$ is constant on each maximal cone of $Σ$ . C. The divisor polytope $P_{D}$ is empty whenever $D$ has any negative coefficient $a_{ρ} < 0$ . D. Every torus-invariant Weil divisor on $X_{Σ}$ is principal.

Hint

One of these is the Demazure global-sections formula. The others have subtle counterexamples.

Answer

A. The Demazure global-sections formula (Fulton §3.4 Proposition 1, Cox-Little-Schenck Theorem 4.3.3) asserts that when $D$ is nef on a smooth complete toric $X_{Σ}$ , the lattice points of $P_{D}$ give a basis of $H^{0} (X_{Σ}, O (D))$ : one character $χ^{m}$ per lattice point $m \in P_{D} \cap M$ , with $dim H^{0} = ∣ P_{D} \cap M ∣$ exactly.

Option B is wrong: $ψ_{D}$ is linear (not constant) on each maximal cone, with linear piece $m_{σ} \in M$ . Constants come up only when $ψ_{D} \equiv 0$ , the divisor zero.

Option C is wrong: $P_{D}$ depends on all coefficients jointly, and a negative $a_{ρ} < 0$ tightens the inequality $⟨ m, v_{ρ} ⟩ \geq - a_{ρ}$ to $⟨ m, v_{ρ} ⟩ \geq ∣ a_{ρ} ∣ > 0$ , which can be satisfied. Example: on $P^{1}$ , the divisor $D = 2 D_{0} - D_{1}$ has $P_{D} = {m \in R : m \geq - 2, - m \geq 1} = [- 2, - 1]$ , with two lattice points ${- 2, - 1}$ , giving $h^{0} = 2$ (the divisor is $O (1)$ on $P^{1}$ , which has two sections).

Option D is wrong: principal divisors are exactly the image of $div : M \to Div_{T} (X_{Σ})$ , which has rank $n$ inside the rank- $∣Σ (1) ∣$ group $Div_{T}$ . For most $X_{Σ}$ , $∣Σ (1) ∣ > n$ , so there are non-principal Weil divisors. Example: on $P^{2}$ , the divisor $D_{0}$ is not principal; only $D_{0} - D_{1}$ (and integer multiples) are.

Rubric: full credit for identifying A and noting the counterexamples for B, C, D.

Exercise 6 (medium, symbolic).

On the Hirzebruch surface $F_{a}$ with fan $v_{1} = e_{1}, v_{2} = e_{2}, v_{3} = - e_{1} + a e_{2}, v_{4} = - e_{2}$ in $R^{2}$ , compute the divisor polytope $P_{D}$ for $D = D_{1} + D_{3} + b D_{2}$ and identify for which integer $b$ the divisor is nef.

Hint

Write out the four ray inequalities; the polytope is bounded iff the four constraints close up into a finite region. The nef condition translates into the polytope being non-empty and full-dimensional.

Answer

Inequalities. Coefficients $(a_{1}, a_{2}, a_{3}, a_{4}) = (1, b, 1, 0)$ . For $m = (s, t)$ :

$⟨ m, v_{1} ⟩ = s \geq - 1$ ;
$⟨ m, v_{2} ⟩ = t \geq - b$ ;
$⟨ m, v_{3} ⟩ = - s + a t \geq - 1$ , i.e., $s - a t \leq 1$ ;
$⟨ m, v_{4} ⟩ = - t \geq 0$ , i.e., $t \leq 0$ .

Polytope. The constraints $t \leq 0$ and $t \geq - b$ require $b \geq 0$ for the polytope to be non-empty (vertically). With $b \geq 0$ , $t$ ranges over $[- b, 0]$ . The horizontal extent: $s \geq - 1$ and $s \leq 1 + a t$ . The polytope is non-empty iff $1 + a t \geq - 1$ for some $t \in [- b, 0]$ , i.e., $a t \geq - 2$ . For $a \geq 0$ and $t \in [- b, 0]$ , $a t \in [- ab, 0]$ , so $min a t = - ab$ . The non-emptiness condition becomes $- ab \geq - 2$ , i.e., $ab \leq 2$ . But more importantly we want $P_{D}$ to be non-empty for every $t \in [- b, 0]$ , not just some — for the polytope to be full-dimensional, we need $- 1 \leq 1 + a t$ for all $t \in [- b, 0]$ , i.e., $a t \geq - 2$ for all $t \in [- b, 0]$ . With $a \geq 0$ this reduces to $- ab \geq - 2$ , i.e., $ab \leq 2$ .

Nef condition. The divisor $D = D_{1} + D_{3} + b D_{2}$ is nef iff its support function $ψ_{D}$ is convex on $∣Σ∣ = R^{2}$ . The function $ψ_{D}$ has linear pieces determined by the four maximal cones $σ_{12} = Cone (v_{1}, v_{2})$ and so on. Computing the linear pieces and checking convexity at the four walls gives the conditions: $b \geq 0$ (wall between $σ_{12}$ and $σ_{14}$ — actually $σ_{41}$ , depending on numbering — controlled by $a_{2} = b$ ) and $ab - 2 \leq 0$ (wall between $σ_{23}$ and $σ_{34}$ — controlled by the $a$ tilt). Combined: $0 \leq b$ and $ab \leq 2$ .

For $a = 0$ ( $F_{0} = P^{1} \times P^{1}$ ): nef iff $b \geq 0$ . For $a = 1$ ( $F_{1} = Bl_{p} P^{2}$ ): nef iff $0 \leq b \leq 2$ . For $a = 2$ ( $F_{2}$ ): nef iff $0 \leq b \leq 1$ . For $a \geq 3$ : nef iff $b = 0$ (only the boundary case).

Geometric interpretation. The constraint $ab \leq 2$ reflects the obstruction from the negative section $S$ of self-intersection $- a$ on $F_{a}$ : the divisor $D = D_{1} + D_{3} + b D_{2}$ has intersection $D \cdot S =$ certain combination involving $a$ and $b$ , and ampleness requires positivity. For large $a$ (large tilt), the nef cone shrinks.

Rubric: full credit for the four inequalities, the polytope-non-emptiness condition, and the nef-condition derivation.

Exercise 7 (hard, short-answer).

Prove the Demazure global-sections formula on a smooth complete toric variety $X_{Σ}$ : for a $T$ -invariant Cartier divisor $D = \sum_{ρ} a_{ρ} D_{ρ}$ , $$ H^0(X_\Sigma, \mathcal{O}(D)) = \bigoplus_{m \in P_D \cap M} \mathbb{C} \cdot \chi^m, $$ where $P_{D} = {m \in M_{R} : ⟨ m, v_{ρ} ⟩ \geq - a_{ρ} for all ρ \in Σ (1)}$ .

Hint

Decompose $H^{0} (X_{Σ}, O (D))$ into $T$ -weight spaces under the natural torus action. For each character $m \in M$ , identify when $χ^{m}$ extends to a section of $O (D)$ in terms of the divisor of $χ^{m}$ versus the divisor $D$ .

Answer

Step 1: weight decomposition. The torus $T$ acts on $X_{Σ}$ , hence on $O (D)$ ( $T$ -linearised), and on its space of global sections. By the standard theorem of finite-dimensional torus representations, $H^{0} (X_{Σ}, O (D))$ decomposes into a direct sum of weight spaces: $$ H^0(X_\Sigma, \mathcal{O}(D)) = \bigoplus_{m \in M} H^0(X_\Sigma, \mathcal{O}(D))m, $$ where $H^0(X\Sigma, \mathcal{O}(D))_m $i s t h es u b s p a ceo f$ \chi^m$-equivariant sections.

Step 2: each weight space is at most one-dimensional. A weight- $m$ global section of $O (D)$ corresponds to a $T$ -equivariant rational function $f$ on $X_{Σ}$ with $χ^{m}$ -equivariance and $div (f) + D \geq 0$ . Equivariance forces $f$ to be a scalar multiple of $χ^{m}$ as a rational function on the dense torus $T \subset X_{Σ}$ . So $H^{0} (X_{Σ}, O (D))_{m}$ is either $C \cdot χ^{m}$ or $0$ , depending on whether $χ^{m}$ extends to a section.

Step 3: extension condition. $χ^{m} \in H^{0} (X_{Σ}, O (D))$ iff $div (χ^{m}) + D \geq 0$ as a Weil divisor. Recall $div (χ^{m}) = \sum_{ρ} ⟨ m, v_{ρ} ⟩ D_{ρ}$ . Adding: $div (χ^{m}) + D = \sum_{ρ} (⟨ m, v_{ρ} ⟩ + a_{ρ}) D_{ρ}$ . The condition $div (χ^{m}) + D \geq 0$ becomes $⟨ m, v_{ρ} ⟩ + a_{ρ} \geq 0$ for every $ρ \in Σ (1)$ , i.e., $⟨ m, v_{ρ} ⟩ \geq - a_{ρ}$ . This is exactly the defining inequality of the divisor polytope $P_{D}$ .

Step 4: conclusion. $H^{0} (X_{Σ}, O (D))_{m} = C \cdot χ^{m}$ iff $m \in P_{D} \cap M$ , and zero otherwise. Summing over all weights $m \in M$ : $$ H^0(X_\Sigma, \mathcal{O}(D)) = \bigoplus_{m \in P_D \cap M} \mathbb{C} \cdot \chi^m. \qquad \square $$

Remark. The proof uses only the $T$ -action and the divisor calculus on the toric variety, not any specific smoothness or completeness. For singular toric $X_{Σ}$ the same formula applies with the same polytope $P_{D}$ ; for non-complete $X_{Σ}$ the polytope $P_{D}$ can be unbounded but the formula still holds (with the sum over lattice points of an unbounded polytope giving an infinite-dimensional cohomology when appropriate). The smoothness assumption is needed when one further wants to identify higher cohomology — the Demazure vanishing theorem $H^{i} (X_{Σ}, O (D)) = 0$ for $i \geq 1, D$ nef — but for $H^{0}$ alone the polytope formula is universal.

Rubric: full credit for the weight decomposition, the one-dimensionality of each weight space, the extension criterion, and the identification with $P_{D} \cap M$ .

Exercise 8 (hard, short-answer).

Prove that on a smooth toric variety $X_{Σ}$ , every $T$ -invariant Weil divisor is Cartier. (Equivalently: the inclusion $Pic (X_{Σ}) ↪ Cl (X_{Σ})$ is an equality.)

Hint

Smoothness means every cone $σ$ of $Σ$ is generated by part of a $Z$ -basis of $N$ . Use this to solve the Cartier-cocycle system on each maximal cone.

Answer

Step 1: reduce to maximal cones. A $T$ -invariant Weil divisor $D = \sum_{ρ} a_{ρ} D_{ρ}$ is Cartier iff it has a piecewise-linear support function $ψ_{D}$ with integer linear pieces. The linear pieces are pinned by the values at primitive ray generators: $ψ_{D} (v_{ρ}) = - a_{ρ}$ . The question is whether, on each maximal cone $σ$ , there exists a character $m_{σ} \in M$ realising these prescribed values on the rays of $σ$ .

Step 2: smoothness gives unique solution on each cone. Suppose $X_{Σ}$ is smooth, so every cone $σ$ is generated by part of a $Z$ -basis of $N$ . Let $σ$ be a maximal cone of dimension $k \leq n$ with primitive ray generators $v_{ρ_{1}}, \dots, v_{ρ_{k}}$ forming part of a $Z$ -basis ${v_{ρ_{1}}, \dots, v_{ρ_{k}}, w_{k + 1}, \dots, w_{n}}$ of $N$ . The dual basis ${v_{ρ_{1}}^{*}, \dots, v_{ρ_{k}}^{*}, w_{k + 1}^{*}, \dots, w_{n}^{*}}$ is a $Z$ -basis of $M$ . Define $$ m_\sigma := -\sum_{i=1}^{k} a_{\rho_i} \cdot v_{\rho_i}^*. $$ This is an element of $M$ (integer linear combination of integer-basis elements). It satisfies $⟨ m_{σ}, v_{ρ_{j}} ⟩ = - a_{ρ_{j}}$ for $j = 1, \dots, k$ by dual-basis duality, and $⟨ m_{σ}, w_{l} ⟩$ can be any integer (the rays of $σ$ do not constrain those coordinates). The choice above (setting the $w_{l}$ -components to zero) is one valid solution.

For maximal $σ$ (dimension $n$ ), the rays of $σ$ form a full $Z$ -basis of $N$ and the choice of $m_{σ}$ is unique.

Step 3: cocycle compatibility. On a pairwise intersection $σ \cap τ$ between two maximal cones, the common rays $ρ \subseteq σ \cap τ$ have $⟨ m_{σ}, v_{ρ} ⟩ = ⟨ m_{τ}, v_{ρ} ⟩ = - a_{ρ}$ by construction. So $⟨ m_{σ} - m_{τ}, v_{ρ} ⟩ = 0$ for every common ray, hence $m_{σ} - m_{τ} \in (σ \cap τ)^{⊥}$ . This is exactly the Cartier-cocycle compatibility.

Step 4: conclusion. Every $T$ -invariant Weil divisor $D = \sum_{ρ} a_{ρ} D_{ρ}$ on a smooth toric $X_{Σ}$ admits a piecewise-linear support function $ψ_{D}$ with integer linear pieces, hence is Cartier. Equivalently, $Pic (X_{Σ}) = Cl (X_{Σ})$ on smooth toric. $□$

Converse. If $X_{Σ}$ is not smooth, there exists a non-smooth cone $σ$ with primitive ray generators not forming a $Z$ -basis. Pick coefficients $a_{ρ}$ at the rays of $σ$ that have no integer solution for $m_{σ}$ (generic coefficients work). The corresponding Weil divisor is not Cartier. So the equality $Pic = Cl$ is equivalent to smoothness, a stronger statement going both directions.

Rubric: full credit for the cone-by-cone construction of $m_{σ}$ , the dual-basis decomposition, the cocycle compatibility, and the converse.

Lean formalization [Intermediate+]

The toric divisor and support function formalisation requires the Fan structure of [04.11.04] and the orbit-cone correspondence of [04.11.06], currently in their partial Lean status. Schematically:

import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Int.Defs
import Mathlib.LinearAlgebra.Basic

namespace Codex.AlgGeom.Toric.DivisorSupportFunction

universe u

/-- A lattice `N` (free abelian group of finite rank). -/
structure Lattice where
  carrier : Type u
  rank : ℕ

/-- A ray of a fan, given by its primitive generator. -/
structure Ray (N : Lattice) where
  primitive : True  -- schematic placeholder for the primitive generator

/-- A fan in `N_ℝ`. -/
structure Fan (N : Lattice) where
  rays : Finset (Ray N)
  /-- Maximal-cone data; placeholder. -/
  maximal_cones : True
  /-- Face-closure axiom from `04.11.04`. -/
  face_closed : True

/-- A T-invariant Weil divisor on `X_Σ`: a coefficient function `Σ(1) → ℤ`. -/
structure TInvariantWeilDivisor {N : Lattice} (Σ : Fan N) where
  coeff : Ray N → ℤ
  finite_support : True

/-- Addition of T-invariant Weil divisors is pointwise. -/
def TInvariantWeilDivisor.add {N : Lattice} {Σ : Fan N}
    (D₁ D₂ : TInvariantWeilDivisor Σ) : TInvariantWeilDivisor Σ where
  coeff := fun ρ => D₁.coeff ρ + D₂.coeff ρ
  finite_support := trivial

/-- The zero T-invariant Weil divisor. -/
def TInvariantWeilDivisor.zero {N : Lattice} (Σ : Fan N) :
    TInvariantWeilDivisor Σ where
  coeff := fun _ => 0
  finite_support := trivial

/-- A support function for a T-invariant Cartier divisor: a continuous
function `ψ : |Σ| → ℝ`, linear on each maximal cone with linear piece
`m_σ ∈ M`, satisfying the Cartier-cocycle compatibility
`m_σ - m_τ ∈ (σ ∩ τ)^⊥` on pairwise intersections. -/
structure SupportFunction {N : Lattice} (Σ : Fan N) where
  linear_pieces : True
  gluing : True
  integrality : True

/-- The coefficient at a ray: `a_ρ = -ψ(v_ρ)` (Fulton convention). -/
noncomputable def SupportFunction.coeffAtRay {N : Lattice} {Σ : Fan N}
    (_ψ : SupportFunction Σ) (_ρ : Ray N) : ℤ := 0  -- placeholder

/-- The divisor of a support function: ray coefficients `a_ρ = -ψ(v_ρ)`. -/
noncomputable def SupportFunction.toDivisor {N : Lattice} {Σ : Fan N}
    (ψ : SupportFunction Σ) : TInvariantWeilDivisor Σ where
  coeff := ψ.coeffAtRay
  finite_support := trivial

/-- The Cartier predicate on a T-invariant Weil divisor. -/
def TInvariantWeilDivisor.IsCartier {N : Lattice} {Σ : Fan N}
    (D : TInvariantWeilDivisor Σ) : Prop :=
  ∃ ψ : SupportFunction Σ, ∀ ρ ∈ Σ.rays, ψ.coeffAtRay ρ = D.coeff ρ

/-- **Support-function correspondence** (Fulton §3.4 Prop 1; Demazure 1970).
The map `ψ ↦ D_ψ` is a bijection between integer-valued piecewise-linear
support functions on `Σ` and T-invariant Cartier divisors on `X_Σ`. -/
theorem support_function_correspondence {N : Lattice} (Σ : Fan N) :
    ∀ ψ : SupportFunction Σ, (ψ.toDivisor).IsCartier := by
  intro ψ
  exact ⟨ψ, fun _ _ => rfl⟩

/-- **Smoothness criterion at the divisor level** (Fulton §3.4 Prop 2).
Every T-invariant Weil divisor is Cartier iff `X_Σ` is smooth. -/
theorem smooth_iff_all_weil_cartier {N : Lattice} (_Σ : Fan N) :
    True := trivial  -- statement pending the Demazure smoothness API

/-- The divisor polytope `P_D ⊂ M_ℝ`. -/
structure DivisorPolytope {N : Lattice} {Σ : Fan N}
    (_D : TInvariantWeilDivisor Σ) where
  carrier : True

/-- **Demazure global-sections formula** (Fulton §3.4; CLS Theorem 4.3.3).
`dim H^0(X_Σ, O(D)) = |P_D ∩ M|`. -/
theorem global_sections_lattice_points {N : Lattice} {Σ : Fan N}
    (_D : TInvariantWeilDivisor Σ) :
    True := trivial  -- statement pending the O(D)-sheaf API

/-- The anticanonical divisor `-K = Σ_ρ D_ρ`. -/
noncomputable def anticanonicalDivisor {N : Lattice} (Σ : Fan N) :
    TInvariantWeilDivisor Σ where
  coeff := fun ρ => if ρ ∈ Σ.rays then 1 else 0
  finite_support := trivial

/-- **Anticanonical lattice-point count on ℙ²** (Fulton §3.3 worked
example). The polytope `P_{-K}` on `ℙ²` is the size-three simplex
with exactly `10` lattice points. -/
theorem anticanonical_p2_h0_eq_ten :
    (3 + 1) * (3 + 2) / 2 = 10 := by decide

/-- **Anticanonical lattice-point count on ℙ^n**: `|P_{-K} ∩ M| =
binomial(2n+1, n)`. Numerical witness for small `n`. -/
theorem anticanonical_pn_count :
    (Nat.choose 5 2 = 10) ∧ (Nat.choose 7 3 = 35) ∧ (Nat.choose 9 4 = 126) := by
  decide

/-- **O(d) on ℙ^n lattice-point count**: `h^0(ℙ^n, O(d)) = binomial(n+d, n)`.
Numerical witness for `ℙ^2` and `d ∈ {1, 2, 3, 4}`. -/
theorem o_d_on_p2_dimensions :
    (Nat.choose 3 2 = 3) ∧
    (Nat.choose 4 2 = 6) ∧
    (Nat.choose 5 2 = 10) ∧
    (Nat.choose 6 2 = 15) := by
  decide

/-- **Cartier multiplier on `ℙ(1, 2, 3)`**: the smallest positive
Cartier multiple of `D_2` is `6 · D_2`, with `6 = lcm(1, 2, 3)`. -/
theorem weighted_projective_123_cartier_multiplier :
    Nat.lcm 2 3 = 6 := by decide

end Codex.AlgGeom.Toric.DivisorSupportFunction

The skeleton above declares the toric-divisor-and-support-function correspondence (statement only, proof reducing to Fulton §3.4 Proposition 1 once the Cartier-divisor sheaf and chart-by-chart description of toric divisors are in Mathlib), the smooth-iff-Weil-is-Cartier theorem, the Demazure global-sections formula, the anticanonical divisor on $P^{n}$ , and decidable numerical witnesses for the lattice-point counts on $P^{2}$ and $P^{n}$ , as well as the Cartier-multiplier computation on $P (1, 2, 3)$ . The placeholder-stubbed statements are reachable from current Mathlib once the Fan formalism and toric-divisor infrastructure of [04.11.04] and [04.11.06] ship.

Advanced results [Master]

§A. T-invariant prime divisors and the support-function correspondence

Theorem (support-function correspondence — full form; Demazure 1970, Fulton §3.4 Proposition 1). Let $Σ$ be a fan in $N_{R}$ and let $X_{Σ}$ be the associated toric variety. There is a canonical isomorphism of abelian groups $$ \mathrm{CDiv}T(X\Sigma) \cong \mathrm{PL}\mathbb{Z}(\Sigma), $$ *where $CDiv_{T} (X_{Σ})$ denotes the group of $T$ -invariant Cartier divisors on $X\Sigma $an d$ \mathrm{PL}\mathbb{Z}(\Sigma) $d e n o t es t h e g r o u p o f in t e g er - v a l u e d p i ece w i se - l in e a r f u n c t i o n so n$ |\Sigma| $t ha t a r e l in e a r o n e a c h co n eo f$ \Sigma $w i t h l in e a r p i ece in$ M $. T h e i so m or p hi s m se n d s$ D = \sum\rho a_\rho D_\rho \mapsto \psi_D $w i t h$ \psi_D(v_\rho) = -a_\rho $an d l in e a r p i eces$ {m_\sigma}{\sigma \in \Sigma{\max}}$ satisfying the Cartier-cocycle compatibility.*

The full Demazure correspondence reorganises Cartier-divisor theory on $X_{Σ}$ around the combinatorial object $PL_{Z} (Σ)$ . The key conceptual move is that local Cartier data — a chart-by-chart choice of $χ^{m_{σ}}$ giving the local equation of the divisor — is encoded globally by a single continuous piecewise-linear function. Continuity is the cocycle compatibility; integrality is the Cartier integrality.

Theorem (principal Cartier divisors $\leftrightarrow$ linear functions; Fulton §3.4). Under the isomorphism above, the subgroup of principal $T$ -invariant Cartier divisors ${div (χ^{m}) : m \in M}$ corresponds to the subgroup $M \subseteq PL_{Z} (Σ)$ of globally linear functions $u \mapsto ⟨ m, u ⟩$ .

The principal-vs-Cartier distinction translates, via the correspondence, into the elementary observation that globally linear functions form a sub-lattice of piecewise-linear functions. The quotient $PL_{Z} (Σ) / M$ is the Picard group $Pic (X_{Σ})$ for smooth $X_{Σ}$ , and a finer Cartier-cocycle-graded quotient for singular $X_{Σ}$ (cross-reference [04.11.09]).

Theorem (orbit-cone refinement; Cox-Little-Schenck §4.1). Let $D_{ρ} \subseteq X_{Σ}$ be the closure of the codimension-one orbit attached to the ray $ρ$ under the orbit-cone correspondence of [04.11.06]. Then $D_{ρ}$ is a $T$ -invariant prime Weil divisor, and the assignment $ρ \mapsto D_{ρ}$ is a bijection between rays of $Σ$ and $T$ -invariant prime Weil divisors on $X_{Σ}$ . The orbit-closure decomposition $$ D_\rho = \bigsqcup_{\tau \succeq \rho} O(\tau) $$ expresses $D_{ρ}$ as the disjoint union of all torus orbits indexed by cones $τ \in Σ$ containing $ρ$ .

The orbit-closure decomposition supplies the geometric meaning of the divisor $D_{ρ}$ : it is the closure of a single orbit and stratifies into smaller orbits indexed by cones in the star of $ρ$ . This is the divisor-level statement of the orbit-cone correspondence and supplies the bridge from the cone combinatorics of $Σ$ to the divisor geometry of $X_{Σ}$ .

§B. Cartier vs Weil — when they agree

Theorem (smoothness criterion at the divisor level; Fulton §3.4 Proposition 2). Let $X_{Σ}$ be a toric variety. Then $X_{Σ}$ is smooth iff every $T$ -invariant Weil divisor on $X_{Σ}$ is Cartier. Equivalently, the inclusion $Pic (X_{Σ}) ↪ Cl (X_{Σ})$ is an equality iff $X_{Σ}$ is smooth iff every cone $σ \in Σ$ is unimodular (generated by part of a $Z$ -basis of $N$ ).

The proof in the forward direction follows from the smoothness criterion of [04.11.05] (the cones are unimodular) together with the dual-basis Cartier construction sketched in Exercise 8 above: every Weil divisor admits a piecewise-linear support function by solving the Cartier-cocycle system on each cone using the dual basis. The reverse direction follows by exhibiting, at any non-smooth cone, a Weil divisor whose Cartier system has no integer solution.

Theorem ( $Q$ -Cartier refinement; Cox-Little-Schenck §4.2). Let $X_{Σ}$ be a simplicial toric variety (every cone has $dim$ -many rays — possibly not forming a $Z$ -basis). Then every $T$ -invariant Weil divisor on $X_{Σ}$ is $Q$ -Cartier: some positive integer multiple is Cartier. The smallest such multiplier is the multiplicity of the worst non-smooth cone, defined as $mult (σ) = ∣ det (v_{ρ_{1}}, \dots, v_{ρ_{n}}) ∣$ for the primitive ray generators of $σ$ .

The simplicial case interpolates between smooth and general singular. For $Q$ -factorial toric (= simplicial), the Picard group is a finite-index sublattice of the divisor class group, with index controlled by cone multiplicities. For non-simplicial toric, even rational Cartier divisors are obstructed, and one requires the full divisor class group machinery of [04.11.09].

Theorem (Cartier-cocycle exact sequence; Fulton §3.4, KKMS Ch. I §2). There is a short exact sequence of abelian groups $$ 0 \to M \to \mathrm{PL}\mathbb{Z}(\Sigma) \to \mathrm{Pic}(X\Sigma) \to 0, $$ with the first map sending $m \mapsto ψ_{m}$ (the globally linear function $u \mapsto ⟨ m, u ⟩$ , corresponding to the principal divisor $- div (χ^{m})$ ) and the second the natural quotient (a piecewise-linear function to its associated Cartier-divisor class modulo principal).

This is the support-function-version of the toric Picard exact sequence in [04.11.09]. The middle term $PL_{Z} (Σ)$ replaces the abstract divisor group, supplying a combinatorial parameterisation of Cartier divisors that is more directly amenable to ampleness analysis (where strict convexity of the piecewise-linear function characterises ample divisors, per [04.11.09] §C).

§C. Sections via lattice-point inequalities — the polytope of sections

Theorem (Demazure global-sections formula; Fulton §3.4 Proposition 1, Cox-Little-Schenck Theorem 4.3.3). Let $X_{Σ}$ be a toric variety and $D = \sum_{ρ} a_{ρ} D_{ρ}$ a $T$ -invariant Cartier divisor. Then $$ H^0(X_\Sigma, \mathcal{O}(D)) = \bigoplus_{m \in P_D \cap M} \mathbb{C} \cdot \chi^m, $$ where $P_{D} = {m \in M_{R} : ⟨ m, v_{ρ} ⟩ \geq - a_{ρ} for all ρ \in Σ (1)}$ is the divisor polytope. In particular, $$ \dim H^0(X_\Sigma, \mathcal{O}(D)) = |P_D \cap M| $$ — the number of lattice points of $P_{D}$ .

The proof is the $T$ -equivariant decomposition argument from Exercise 7: $H^{0}$ decomposes into weight spaces by the torus action, each weight space is at most one-dimensional (spanned by a $χ^{m}$ ), and the extension criterion $div (χ^{m}) + D \geq 0$ reduces to the polytope-defining inequalities. The smoothness or completeness of $X_{Σ}$ is not needed for $H^{0}$ ; the same formula holds for arbitrary toric varieties.

Theorem (Demazure vanishing; Fulton §3.4, Cox-Little-Schenck Theorem 9.2.3). Let $X_{Σ}$ be a smooth complete toric variety and $D$ a $T$ -invariant Cartier divisor that is nef (i.e., $ψ_{D}$ is convex on $∣Σ∣$ ). Then $H^{i} (X_{Σ}, O (D)) = 0$ for every $i \geq 1$ .

Demazure vanishing supplies the higher-cohomology counterpart to the global-sections formula: when $D$ is nef, the Euler characteristic $χ (O (D)) = \sum_{i} (- 1)^{i} dim H^{i} (X_{Σ}, O (D))$ reduces to $dim H^{0} (X_{Σ}, O (D)) = ∣ P_{D} \cap M ∣$ , a lattice-point count. This realises Hirzebruch-Riemann-Roch on a smooth complete toric variety as a purely combinatorial Ehrhart-polynomial computation (Brion-Vergne 1997 J. Amer. Math. Soc. 10).

Theorem (Ehrhart polynomial; Ehrhart 1962, Brion-Vergne 1997). Let $X_{Σ}$ be a projective smooth toric variety and $L = O (D)$ a nef line bundle with divisor polytope $P = P_{D}$ . For each integer $k \geq 1$ , the lattice-point count $$ L(P, k) := |kP \cap M| $$ is a polynomial in $k$ of degree $n = dim X_{Σ}$ , called the Ehrhart polynomial of $P$ . Its leading coefficient is the lattice volume of $P$ , $vol (P) / n!$ , and the polynomial satisfies the Hilbert-function identity $$ L(P, k) = \dim H^0(X_\Sigma, L^{\otimes k}) = \chi(X_\Sigma, L^{\otimes k}). $$

The Ehrhart polynomial is the toric realisation of the Hilbert polynomial in classical projective geometry. Its leading coefficient (the volume) recovers the degree of $X_{Σ}$ under the polarisation $L$ ; its lower-order coefficients encode the Todd class times the Chern character, exhibiting Hirzebruch-Riemann-Roch as a polynomial identity in $k$ .

§D. Worked example — anticanonical on $P^{2}$

Theorem (anticanonical divisor on $P^{n}$ ; Fulton §3.5). Let $P^{n}$ have its standard toric fan with rays $v_{0} = - e_{1} - \dots - e_{n}, v_{1} = e_{1}, \dots, v_{n} = e_{n}$ in $R^{n}$ . Then:

The canonical class is $K_{P^{n}} = - (D_{0} + D_{1} + \dots + D_{n}) \in Pic (P^{n})$ .
The anticanonical divisor $- K = D_{0} + D_{1} + \dots + D_{n}$ has divisor polytope $P_{- K}$ the size- $(n + 1)$ simplex in $M_{R} = R^{n}$ with vertices $(n, - 1, - 1, \dots, - 1), (- 1, n, - 1, \dots, - 1), \dots, (- 1, \dots, - 1, n)$ and $(- 1, - 1, \dots, - 1)$ .
The lattice-point count is $∣ P_{- K} \cap M ∣ = (n 2 n + 1)$ .
Equivalently, $O (- K_{P^{n}}) = O_{P^{n}} (n + 1)$ , with global sections of dimension $(n 2 n + 1)$ — the space of homogeneous degree- $(n + 1)$ polynomials in $n + 1$ variables.

For $n = 2$ : $- K_{P^{2}} = D_{0} + D_{1} + D_{2} = 3 H$ , polytope is the size-three simplex with $(2 5) = 10$ lattice points, dimension of plane cubics. For $n = 3$ : $- K_{P^{3}} = 4 H$ , polytope is the size-four simplex with $(3 7) = 35$ lattice points, dimension of homogeneous quartics in four variables.

The anticanonical example is the cleanest illustration of the entire divisor-and-support-function framework. The divisor $- K_{P^{2}}$ is an honest geometric object (the sum of the three coordinate lines); the support function $ψ_{- K}$ is a piecewise-linear ramp pinned at $- 1$ on each ray; the divisor polytope $P_{- K}$ is a lattice triangle with exactly ten integer points; and the resulting global sections are the ten monomials forming a basis of the cubic plane curves linear system.

Theorem (anticanonical on Hirzebruch surfaces; Fulton §3.4). On the Hirzebruch surface $F_{a} = X_{Σ_{a}}$ with the four-ray fan $v_{1} = e_{1}, v_{2} = e_{2}, v_{3} = - e_{1} + a e_{2}, v_{4} = - e_{2}$ , the canonical class is $K_{F_{a}} = - (D_{1} + D_{2} + D_{3} + D_{4}) + (a - 2) D_{2}$ , the anticanonical divisor is $- K = D_{1} + (2 - a) D_{2} + D_{3} + D_{4}$ , and $- K$ is ample iff $a \leq 1$ (i.e., $F_{0} = P^{1} \times P^{1}$ and $F_{1} = Bl_{p} P^{2}$ are del Pezzo surfaces; $F_{a}$ for $a \geq 2$ is not Fano).

The Hirzebruch-surface anticanonical example illustrates that the support-function framework distinguishes ample from non-ample with a single combinatorial check (strict convexity vs convexity of the piecewise-linear function on the fan). The transition $a = 1 \to a = 2$ — where $- K$ drops from ample to merely nef — corresponds to a wall-crossing in the support-function space, with the Hirzebruch surface losing its Fano status as the tilt $a$ exceeds the del Pezzo threshold.

Theorem (anticanonical on toric Calabi-Yau hypersurfaces; Batyrev 1994). Let $X_{Σ}$ be a projective smooth toric variety such that the anticanonical class $- K_{X_{Σ}}$ is associated to a reflexive polytope $P^{\circ}$ (the polar dual of the polytope $P = P_{- K}$ defining $X_{Σ}$ ). Then generic anticanonical hypersurfaces $Y \subset X_{Σ}$ are smooth Calabi-Yau varieties, and the polar duality $P \leftrightarrow P^{\circ}$ supplies the Batyrev mirror Calabi-Yau $Y^{\circ} \subset X_{Σ^{\circ}}$ . The 473,800,776 reflexive 4-polytopes classified by Kreuzer-Skarke 2000-2002 account for the bulk of explicitly known Calabi-Yau 3-folds and their mirrors.

The Batyrev construction lifts the anticanonical divisor from a worked example to a mirror-symmetry generator. The reflexive-polytope condition on $P_{- K}$ — that the polar dual is also a lattice polytope — is the combinatorial signature of Gorenstein toric Calabi-Yau geometry, and the polar-duality involution becomes the mirror-symmetry exchange of A-side and B-side Calabi-Yau geometries (cross-reference [04.11.10] for the polytope-fan dictionary and [04.12.09] for the Gross-Siebert reconstruction generalisation).

Synthesis. The toric divisor and support function form the foundational reason that line-bundle and section geometry on a toric variety reduces to combinatorial calculations on a polyhedral fan: prime Weil divisors are atoms indexed by rays, Cartier divisors are encoded by piecewise-linear functions, and global sections are enumerated by lattice points of a polytope. The central insight is that the three different presentations — divisor, support function, polytope — are three avatars of the same Cartier-divisor object, with the bijection furnished by Fulton §3.4 Proposition 1 making the whole calculus algorithmic.

This is exactly the foundational fact that identifies toric divisor theory with algorithmic algebraic geometry: where a general variety's divisor theory requires sheaf-cohomological computation via spectral sequences or derived categories, on a toric variety the entire calculus reduces to lattice arithmetic. The bridge is the global-sections formula $H^{0} (X_{Σ}, O (D)) = ⨁_{m \in P_{D} \cap M} C \cdot χ^{m}$ — sheaf cohomology becomes lattice-point counting, and Hirzebruch-Riemann-Roch becomes a polynomial identity in Ehrhart polynomials.

Putting these together with the toric Picard exact sequence of [04.11.09], the support-function correspondence is the bridge between fan combinatorics and divisor / line-bundle geometry. The pattern recurs in three generalisations. To higher cohomology, Demazure vanishing supplies $H^{i} (X_{Σ}, O (D)) = 0$ for nef $D$ and $i \geq 1$ , reducing Euler characteristics to Ehrhart polynomials and Hirzebruch-Riemann-Roch to lattice-point identities (Brion-Vergne 1997). To singular toric, the Cartier-Weil distinction widens; one still has support functions, but the Cartier subgroup is a finite-index sublattice of the piecewise-linear function lattice, with index controlled by cone multiplicities (Cox-Little-Schenck §4.2). To equivariant cohomology, the support-function description lifts to a description of equivariant Cartier divisors in terms of $M$ -graded sheaves on the fan, and the Demazure formula becomes an equivariant Riemann-Roch theorem in $T$ -equivariant K-theory (Vergne 1999, Brion-Vergne 1997). Each generalisation preserves the divisor-support-function-polytope triad and identifies the line-bundle theory of the toric variety with a piece of combinatorial polytope theory.

Full proof set [Master]

Proposition (support-function correspondence), proof. Given in the key theorem. The forward direction sends a $T$ -invariant Cartier divisor to its piecewise-linear support function by reading off local Cartier data ${m_{σ}}$ from each affine chart $U_{σ}$ and gluing into a continuous $ψ_{D}$ . The reverse direction reads off Weil-divisor coefficients from $- ψ (v_{ρ})$ at each primitive ray generator and reconstructs the Cartier data ${m_{σ}}$ from the integer linear pieces. Mutual inverses are confirmed by the chart-by-chart identification $a_{ρ} = - ψ_{D} (v_{ρ}) = - ⟨ m_{σ}, v_{ρ} ⟩$ for $ρ \subseteq σ$ . $□$

Proposition (smoothness criterion at divisor level; Fulton §3.4 Prop 2), proof. On a smooth toric $X_{Σ}$ , every $T$ -invariant Weil divisor is Cartier; conversely, if every $T$ -invariant Weil divisor on $X_{Σ}$ is Cartier, then $X_{Σ}$ is smooth.

Proof. Given in Exercise 8. Forward direction: on a smooth cone $σ$ , the primitive ray generators $v_{ρ_{1}}, \dots, v_{ρ_{k}}$ extend to a $Z$ -basis of $N$ . The dual basis ${v_{ρ_{i}}^{*}}$ is a $Z$ -basis of $M$ , and one constructs $m_{σ} = - \sum_{i} a_{ρ_{i}} v_{ρ_{i}}^{*} \in M$ as an explicit integer character satisfying $⟨ m_{σ}, v_{ρ_{j}} ⟩ = - a_{ρ_{j}}$ for every $j$ . Cocycle compatibility follows on shared rays.

Reverse direction: if $X_{Σ}$ has a non-smooth cone $σ$ , the primitive ray generators of $σ$ do not form a $Z$ -basis of $N \cap R ⟨ σ ⟩$ . Pick coefficients $a_{ρ}$ at the rays of $σ$ such that the linear system $⟨ m_{σ}, v_{ρ} ⟩ = - a_{ρ}$ has no integer solution (generic coefficients work — the system has a unique $Q$ -solution, and irrationality of denominators in the basis matrix of the rays forces some $a_{ρ}$ to fail integer-divisibility). The resulting Weil divisor $D = \sum_{ρ} a_{ρ} D_{ρ}$ is not Cartier. So smoothness at the cone level is equivalent to all-Weil-Cartier. $□$

Proposition (Demazure global-sections formula; Fulton §3.4 Prop 1), proof. Given in Exercise 7. The weight-decomposition argument shows $H^{0} (X_{Σ}, O (D)) = ⨁_{m} H_{m}^{0}$ with each $H_{m}^{0}$ at most one-dimensional, spanned by $χ^{m}$ when $m \in P_{D} \cap M$ (the extension criterion $div (χ^{m}) + D \geq 0$ rewrites as the polytope inequalities $⟨ m, v_{ρ} ⟩ \geq - a_{ρ}$ ) and zero otherwise. $□$

Proposition (Demazure vanishing; Cox-Little-Schenck Theorem 9.2.3), proof sketch. On a smooth complete toric $X_{Σ}$ , $H^{i} (X_{Σ}, O (D)) = 0$ for $i \geq 1$ when $D$ is nef.

Proof sketch. The cohomology $H^{i} (X_{Σ}, O (D))$ decomposes into $T$ -weight pieces $H_{m}^{i}$ . For each $m \in M$ , the weight piece is computed by the cellular cohomology of a complex of sub-fans of $Σ$ depending on $m$ — concretely, the simplicial complex $Σ_{m} := {σ \in Σ : m \in / σ^{\lor} + \sum_{ρ} a_{ρ} m_{ρ}^{σ}}$ where $m_{ρ}^{σ}$ is the local Cartier datum. When $D$ is nef (equivalently $ψ_{D}$ is convex), $Σ_{m}$ is either empty (giving zero contribution to all $H^{i}$ ) or contractible (giving zero contribution to $H^{i}$ for $i \geq 1$ by the cellular cohomology of a contractible complex). Either way, $H_{m}^{i} = 0$ for $i \geq 1$ , hence $H^{i} (X_{Σ}, O (D)) = 0$ for $i \geq 1$ . The full argument uses the toric Čech-cohomology computation of Demazure 1970 refined by Mavlyutov 2000 Compositio Math. 124. $□$

Proposition (Ehrhart polynomiality; Ehrhart 1962, Brion-Vergne 1997), proof sketch. For a lattice polytope $P \subseteq M_{R}$ of dimension $n$ , the function $L (P, k) = ∣ k P \cap M ∣$ is a polynomial in $k$ of degree $n$ with rational coefficients.

Proof sketch. Two complementary arguments. The classical Ehrhart 1962 C. R. Acad. Sci. 254 proof uses the generating function $\sum_{k \geq 0} L (P, k) t^{k} = h^{*} (P, t) / (1 - t)^{n + 1}$ with $h^{*} (P, t)$ a polynomial of degree $\leq n$ , giving polynomiality of $L (P, k)$ by direct power-series manipulation. The Brion-Vergne 1997 proof uses the Atiyah-Bott-Berline-Vergne equivariant localisation on the toric variety $X_{P}$ : the Euler characteristic $χ (X_{P}, L_{P}^{\otimes k})$ localises to a sum over toric-fixed points, each contribution being a rational function of $k$ , with the cancellations giving polynomiality. Equivalently, by the Demazure global-sections formula and Demazure vanishing, $L (P, k) = χ (X_{P}, L_{P}^{\otimes k})$ is the Hilbert polynomial of the polarised toric variety $(X_{P}, L_{P})$ in degree $k$ — polynomial in $k$ by Hilbert's theorem on Hilbert polynomials. $□$

Proposition (anticanonical on $P^{n}$ ), proof. On $P^{n}$ with the standard toric fan, $- K_{P^{n}} = D_{0} + D_{1} + \dots + D_{n}$ has $∣ P_{- K} \cap M ∣ = (n 2 n + 1)$ .

Proof. The fan has $n + 1$ rays $v_{0} = - e_{1} - \dots - e_{n}, v_{1} = e_{1}, \dots, v_{n} = e_{n}$ , with the sum identity $v_{0} + v_{1} + \dots + v_{n} = 0$ . The anticanonical polytope $P_{- K}$ is defined by $⟨ m, v_{i} ⟩ \geq - 1$ for $i = 0, 1, \dots, n$ . The constraints $⟨ m, v_{i} ⟩ = m_{i} \geq - 1$ for $i = 1, \dots, n$ give $m_{i} \geq - 1$ . The constraint $⟨ m, v_{0} ⟩ = - m_{1} - \dots - m_{n} \geq - 1$ gives $m_{1} + \dots + m_{n} \leq 1$ . So $P_{- K} = {m \in R^{n} : m_{i} \geq - 1 for i = 1, \dots, n, and \sum_{i} m_{i} \leq 1}$ , the size- $(n + 1)$ lattice simplex translated.

Translate by $m \mapsto m + (1, 1, \dots, 1)$ : in new coordinates $m_{i}^{'} = m_{i} + 1 \geq 0$ , $\sum_{i} m_{i}^{'} \leq n + 1$ . So $P_{- K}$ becomes the standard simplex ${m^{'} \in R^{n} : m_{i}^{'} \geq 0, \sum_{i} m_{i}^{'} \leq n + 1}$ . The lattice points in this simplex are tuples $(m_{1}^{'}, \dots, m_{n}^{'}) \in Z_{\geq 0}^{n}$ with $m_{1}^{'} + \dots + m_{n}^{'} \leq n + 1$ , equivalent to $(m_{0}^{'}, m_{1}^{'}, \dots, m_{n}^{'}) \in Z_{\geq 0}^{n + 1}$ with $m_{0}^{'} + m_{1}^{'} + \dots + m_{n}^{'} = n + 1$ (introducing $m_{0}^{'} := n + 1 - \sum_{i} m_{i}^{'}$ ).

The count of non-negative integer tuples summing to $n + 1$ is the multinomial coefficient $(n ( n + 1 ) + n) = (n 2 n + 1)$ . So $∣ P_{- K} \cap M ∣ = (n 2 n + 1)$ . For $n = 2$ : $(2 5) = 10$ . For $n = 3$ : $(3 7) = 35$ . For $n = 4$ : $(4 9) = 126$ . $□$

Proposition ( $Q$ -Cartier on $P (1, 2, 3)$ ), proof. On the singular weighted projective space $P (1, 2, 3)$ with fan rays $v_{0} = (1, 0), v_{1} = (0, 1), v_{2} = (- 2, - 3)$ , the toric divisor $D_{2}$ is not Cartier, but $6 D_{2}$ is.

Proof. Given in Exercise 4. The Cartier-cocycle system at $σ_{02} = Cone (v_{0}, v_{2})$ requires $⟨ m_{02}, v_{0} ⟩ = 0$ and $⟨ m_{02}, v_{2} ⟩ = - 1$ . With $m_{02} = (a, b)$ : $a = 0, - 2 a - 3 b = - 1$ , giving $b = 1/3$ , not integer. Similarly $σ_{12} = Cone (v_{1}, v_{2})$ requires $b = 0, - 2 a - 3 b = - 1$ , giving $a = 1/2$ , not integer. So $D_{2}$ fails Cartier at both singular cones. For $6 D_{2}$ , the constraints become $⟨ m_{02}, v_{2} ⟩ = - 6$ with $m_{02} = (0, 2)$ (integer) and $⟨ m_{12}, v_{2} ⟩ = - 6$ with $m_{12} = (3, 0)$ (integer). Cocycle compatibility on shared rays is direct. So $6 D_{2}$ is Cartier, and $6 = lcm (2, 3)$ is the minimal positive multiplier matching the toric Picard / class group index calculation from [04.11.09]. $□$

Connections [Master]

Fan and toric variety $X_{Σ}$ 04.11.04. The structural prerequisite. The toric divisor $D_{ρ}$ for $ρ \in Σ (1)$ is defined relative to the gluing $X_{Σ} = ⋃_{σ} U_{σ}$ from [04.11.04], and the Cartier-cocycle compatibility of a support function uses the chart-overlap structure $U_{σ \cap τ} ↪ U_{σ}$ from the fan construction. The smoothness criterion that lifts smoothness from $X_{Σ}$ to "every Weil divisor is Cartier" passes through the unimodular-cone condition of [04.11.04].
Orbit-cone correspondence 04.11.06. The geometric realisation of $D_{ρ}$ as the closure of the codimension-one torus orbit $O (ρ)$ uses the orbit-cone correspondence directly. The dimension formula $dim O (ρ) + dim ρ = n$ identifies $D_{ρ}$ as a divisor (dimension $n - 1$ ) since rays are one-dimensional cones. The orbit-closure decomposition $D_{ρ} = ⨆_{τ ⪰ ρ} O (τ)$ exhibits $D_{ρ}$ itself as a toric variety with its own fan, generalising the divisor-level statement to a divisor-stratified picture.
Toric Picard group 04.11.09. The companion sibling unit in this batch. The toric Picard exact sequence $0 \to M \to ⨁_{ρ} Z D_{ρ} \to Pic (X_{Σ}) \to 0$ is the abelian-group formulation of the Cartier-divisor calculus developed here. The support-function description in §A is the piecewise-linear-function avatar of the middle term, and the strict-convexity ampleness criterion in [04.11.09] §C uses the support function defined in §B above. The Cartier-vs-Weil refinement of §B is the input to the Picard-vs-class-group distinction in [04.11.09].
Polytope-fan dictionary 04.11.10. The downstream unit. The divisor polytope $P_{D}$ defined in §C is the polytope-side of the polytope-fan dictionary: a projective toric variety $X_{Σ}$ with an ample divisor $D$ gives a lattice polytope $P_{D}$ , and the polytope-fan dictionary recovers $(X_{Σ}, O (D))$ from $P_{D}$ as the projective toric variety of the polytope. The Demazure global-sections formula $H^{0} (X_{P}, L_{P}) = ⨁_{m \in P \cap M} C χ^{m}$ is the polytope-side reading of §C's global-sections formula.
Algebraic moment map and the polytope 04.11.11. The downstream unit. The divisor polytope $P_{D}$ for an ample divisor $D$ on a Kähler toric manifold coincides with the image of the algebraic moment map $μ : X_{Σ} \to M_{R}$ , providing the symplectic-side description of the support-function calculus. Delzant 1988 Bull. SMF 116 supplies the symplectic classification of compact toric manifolds by their Delzant polytopes, with the polytope-side data exactly the divisor polytope.
Algebraic torus and character/cocharacter lattices 04.11.01. The lattice duality $M \leftrightarrow N$ underpinning the support function. The pairing $⟨ m, v ⟩$ that defines $ψ_{D} (v_{ρ}) = - a_{ρ}$ on rays and the inequality $⟨ m, v_{ρ} ⟩ \geq - a_{ρ}$ defining $P_{D}$ uses the perfect pairing $M \times N \to Z$ from [04.11.01]. The character map $m \mapsto χ^{m}$ that identifies global sections with lattice points is the algebraic-torus character classification.
Smoothness and completeness via fans 04.11.05. The smoothness criterion at the divisor level (§B) is the divisor-side counterpart of the smoothness criterion of [04.11.05]. Smoothness translates into unimodular cones, which translates into "every Weil is Cartier", which translates into $Pic = Cl$ . The completeness criterion controls when the divisor polytope $P_{D}$ is bounded — bounded iff $D$ is nef and $X_{Σ}$ is complete.
Toric resolution of singularities 04.11.07. The sibling unit. A toric resolution $f : X_{Σ^{'}} \to X_{Σ}$ via star subdivision adds one exceptional divisor $D_{ρ}$ per new ray $ρ$ , with the corresponding support-function values $ψ (u_{ρ})$ encoding the discrepancy data along the resolution. The pullback $f^{*} D$ of a Cartier divisor $D$ on $X_{Σ}$ has support function $ψ_{f^{*} D} = ψ_{D} \circ ϕ_{R}$ for the lattice map $ϕ$ of the refinement, an explicitly computable transform on piecewise-linear functions. The divisor language developed here is the bookkeeping framework for the resolution algorithm of [04.11.07].
Ample and very ample line bundle 04.05.05. The general Cartan-Serre-Grothendieck criterion for ampleness specialises on a smooth complete toric variety to the strict-convexity criterion on the support function $ψ_{D}$ (cross-reference [04.11.09] §C). The ample cone in $Pic (X_{Σ})_{R}$ is the open polyhedral cone of strictly convex piecewise-linear functions modulo characters.
Toric cohomology and the Chow ring 04.11.12. The downstream unit. The toric divisors $D_{ρ}$ form the degree-one generators of the Chow ring $A^{*} (X_{Σ})$ of a smooth complete toric variety, with the Stanley-Reisner relations supplying the higher-degree structure and the toric Picard exact sequence supplying the linear relations. The global-sections formula via $P_{D}$ is the degree-zero piece of the toric Hirzebruch-Riemann-Roch theorem of [04.11.13].
Cox-ring and GIT quotient 04.11.15 pending. The Cox ring $C [x_{ρ} : ρ \in Σ (1)]$ is graded by the Picard group via the toric Picard exact sequence, and a monomial $\prod_{ρ} x_{ρ}^{a_{ρ}}$ corresponds to the toric divisor $\sum_{ρ} a_{ρ} D_{ρ}$ . The polytope $P_{D}$ has the interpretation as the polytope of degree- $D$ monomials in the Cox ring, recovering the global-sections decomposition.
Picard group of a general variety 04.05.02. Toric divisor theory is the sharp combinatorial case of general divisor theory, where the Cartier vs Weil distinction reduces to a unimodular-basis check at each cone, the support function classifies Cartier divisors, and the divisor polytope enumerates global sections. The general theory requires cohomological / Hodge-theoretic machinery; the toric case dispatches it with lattice arithmetic.

Historical & philosophical context [Master]

The toric divisor and support function formalism in its modern form is due to Michel Demazure in Sous-groupes algébriques de rang maximum du groupe de Cremona (Annales scientifiques de l'École normale supérieure (4) 3, 1970, pp. 507-588) ^{[Demazure 1970]}. Demazure's paper established the bijection between $T$ -invariant Cartier divisors on $X_{Σ}$ and integer-valued piecewise-linear functions on $∣Σ∣$ , the smoothness criterion at the divisor level (every Weil is Cartier iff every cone is unimodular), the strict-convexity ampleness criterion, and the global-sections lattice-point formula. The term "fonction d'appui" ("support function") and the divisor-polytope construction $P_{D}$ are Demazure's coinages.

The Cartier-Weil refined formulation, distinguishing $Pic$ from $Cl$ on singular toric, was systematised in Toroidal Embeddings I by Kempf, Knudsen, Mumford, and Saint-Donat (Lecture Notes in Mathematics 339, Springer-Verlag 1973) ^[pending], where the polytope-and-fan formalism was developed as an exposition of Demazure's results within the broader scheme-theoretic toroidal-embeddings framework. KKMS Chapter I §2 contains the first systematic treatment of $T$ -invariant Cartier divisors and their support functions on singular toric. The work was motivated by Mumford's earlier An analytic construction of degenerating Abelian varieties over complete rings (Compositio Mathematica 24, 1972, pp. 239-272) ^{[Mumford 1972]}, where cone-and-lattice data was used to construct degenerations of abelian varieties, with the support function emerging as the natural object recording the Cartier-divisor structure of the degenerate fibre.

Tadao Oda's Convex Bodies and Algebraic Geometry (Ergebnisse der Mathematik 15, Springer-Verlag 1988) ^[pending] is the canonical mid-1980s monograph, with §2.1-§2.3 the standard reference for the toric divisor and support function correspondence, the Cartier-Weil distinction, and the ample-line-bundle / strict-convexity dictionary. Oda's exposition emphasises the convex-body language and was the main pre-Fulton reference. Vasily Danilov's earlier 1978 survey The geometry of toric varieties (Russian Mathematical Surveys 33(2), 97-154) ^{[Danilov 1978]} gave the first English-language treatment of the toric divisor theory and developed the support-function description in connection with the Stanley-Reisner perspective on the Chow ring.

William Fulton's Introduction to Toric Varieties (Princeton University Press 1993) ^{[Fulton 1993]} §3.3-§3.4 is the canonical short textbook reference for the support-function correspondence, the smoothness criterion at the divisor level, and the global-sections lattice-point formula. Fulton's exposition is the standard pedagogical anchor for the material of this unit, with §3.3 introducing $T$ -invariant Weil divisors and §3.4 developing the Cartier formulation via support functions. David Cox, John Little, and Henry Schenck's Toric Varieties (American Mathematical Society 2011) ^[pending] is the modern thousand-page treatment, extending Fulton's exposition with extensive material on the divisor polytope, the global-sections formula (Theorem 4.3.3), and the strict-convexity ampleness criterion (Theorem 6.1.14).

Michel Brion's Groupe de Picard et nombres caractéristiques des variétés sphériques (Duke Mathematical Journal 58(2), 1989, pp. 397-424) ^{[Brion 1989]} extended the toric divisor-and-support-function calculus to the broader class of spherical varieties, with the toric case appearing as the rank-zero special case. The Brion paper supplies the lattice-point cohomology formula in the spherical setting and unifies the toric divisor theory with the moment-map perspective of Atiyah 1982 and Guillemin-Sternberg 1982.

Michel Brion and Michèle Vergne's Residue formulae, vector partition functions and lattice points in rational polytopes (Journal of the American Mathematical Society 10(4), 1997, pp. 797-833) ^{[Brion-Vergne 1997]} supplied the equivariant residue formula extending the Demazure global-sections formula to higher cohomology and arbitrary toric divisors. The Brion-Vergne theorem realises Hirzebruch-Riemann-Roch on a smooth complete toric variety as an explicit polynomial identity in Ehrhart polynomials of polytopes — an algorithmic-cohomology formula that has been the foundation for modern toric computational algorithms in Macaulay2, Polymake, and SageMath.

Miles Reid's Decomposition of toric morphisms (in Arithmetic and Geometry II, Progress in Mathematics 36, Birkhäuser 1983, pp. 395-418) ^[pending] established piecewise-linear functions on a fan as the universal Cartier-divisor object in the toric Minimal Model Program. Reid's analysis used the support-function description to control extremal contractions and small modifications, anticipating the general MMP under Kawamata, Kollár, Mori, Reid, and Shokurov by several years. The toric MMP is the proof-of-concept algorithmic version, and the support-function calculus of this unit is its combinatorial backbone.

David Cox's 1995 paper The homogeneous coordinate ring of a toric variety (Journal of Algebraic Geometry 4, 17-50) ^{[Cox 1995]} reframed toric divisors as monomial classes in the Cox ring $C [x_{ρ} : ρ \in Σ (1)]$ graded by $Pic (X_{Σ})$ , with the divisor polytope $P_{D}$ becoming the polytope of degree- $D$ monomials. The Cox-ring perspective unifies divisor theory, line-bundle theory, and the GIT-quotient construction $X_{Σ} = (C^{Σ (1)} ∖ Z (Σ)) / G$ under a single functorial framework and has been the foundation for modern computational toric algorithms.

Anvar Mavlyutov's Semiample hypersurfaces in toric varieties (Compositio Mathematica 124(1), 2000, pp. 7-24) ^[pending] refined the Demazure vanishing theorem and the support-function description for singular toric varieties, extending the smooth-case results to the $Q$ -factorial and simplicial settings. Mavlyutov's work is the canonical reference for the Cartier-multiplier analysis on singular toric and supplies the higher-cohomology refinements of the Demazure global-sections formula.

Friedrich Hirzebruch's introduction of the surfaces $F_{a}$ in Über eine Klasse von einfach-zusammenhängenden komplexen Mannigfaltigkeiten (Mathematische Annalen 124, 1951, pp. 77-86) ^{[Hirzebruch 1951]} predates the toric formalism by two decades but supplied the prototype example of a smooth complete toric variety with a non-toric Cartier divisor (the negative section $S$ of self-intersection $- a$ ). The Hirzebruch surfaces remain the standard pedagogical examples of Cartier divisors beyond $P^{n}$ , with the support function $ψ_{S}$ on the four-ray fan recovering the negative section's geometry combinatorially.

Bibliography [Master]

@article{Demazure1970,
  author  = {Demazure, Michel},
  title   = {Sous-groupes alg{\'e}briques de rang maximum du groupe de Cremona},
  journal = {Annales scientifiques de l'{\'E}cole normale sup{\'e}rieure (4)},
  volume  = {3},
  year    = {1970},
  pages   = {507--588}
}

@book{KempfKnudsenMumfordSaintDonat1973,
  author    = {Kempf, George and Knudsen, Finn Faye and Mumford, David and Saint-Donat, Bernard},
  title     = {Toroidal Embeddings I},
  publisher = {Springer-Verlag},
  series    = {Lecture Notes in Mathematics},
  volume    = {339},
  year      = {1973}
}

@article{Mumford1972,
  author  = {Mumford, David},
  title   = {An analytic construction of degenerating {A}belian varieties over complete rings},
  journal = {Compositio Mathematica},
  volume  = {24},
  year    = {1972},
  pages   = {239--272}
}

@article{Danilov1978,
  author  = {Danilov, Vasily I.},
  title   = {The geometry of toric varieties},
  journal = {Russian Mathematical Surveys},
  volume  = {33},
  number  = {2},
  year    = {1978},
  pages   = {97--154}
}

@book{OdaConvexBodies,
  author    = {Oda, Tadao},
  title     = {Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties},
  publisher = {Springer-Verlag},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
  volume    = {15},
  year      = {1988}
}

@incollection{Reid1983,
  author    = {Reid, Miles},
  title     = {Decomposition of toric morphisms},
  booktitle = {Arithmetic and Geometry, Volume II},
  publisher = {Birkh{\"a}user},
  series    = {Progress in Mathematics},
  volume    = {36},
  year      = {1983},
  pages     = {395--418}
}

@article{Brion1989,
  author  = {Brion, Michel},
  title   = {Groupe de {P}icard et nombres caract{\'e}ristiques des vari{\'e}t{\'e}s sph{\'e}riques},
  journal = {Duke Mathematical Journal},
  volume  = {58},
  number  = {2},
  year    = {1989},
  pages   = {397--424}
}

@book{FultonToric,
  author    = {Fulton, William},
  title     = {Introduction to Toric Varieties},
  publisher = {Princeton University Press},
  series    = {Annals of Mathematics Studies},
  volume    = {131},
  year      = {1993}
}

@article{Cox1995,
  author  = {Cox, David A.},
  title   = {The homogeneous coordinate ring of a toric variety},
  journal = {Journal of Algebraic Geometry},
  volume  = {4},
  year    = {1995},
  pages   = {17--50}
}

@article{BrionVergne1997,
  author  = {Brion, Michel and Vergne, Mich{\`e}le},
  title   = {Residue formulae, vector partition functions and lattice points in rational polytopes},
  journal = {Journal of the American Mathematical Society},
  volume  = {10},
  number  = {4},
  year    = {1997},
  pages   = {797--833}
}

@article{Mavlyutov2000,
  author  = {Mavlyutov, Anvar R.},
  title   = {Semiample hypersurfaces in toric varieties},
  journal = {Compositio Mathematica},
  volume  = {124},
  number  = {1},
  year    = {2000},
  pages   = {7--24}
}

@book{CoxLittleSchenck,
  author    = {Cox, David A. and Little, John B. and Schenck, Henry K.},
  title     = {Toric Varieties},
  publisher = {American Mathematical Society},
  series    = {Graduate Studies in Mathematics},
  volume    = {124},
  year      = {2011}
}

@article{Hirzebruch1951,
  author  = {Hirzebruch, Friedrich},
  title   = {{\"U}ber eine {K}lasse von einfach-zusammenh{\"a}ngenden komplexen {M}annigfaltigkeiten},
  journal = {Mathematische Annalen},
  volume  = {124},
  year    = {1951},
  pages   = {77--86}
}

@article{Ehrhart1962,
  author  = {Ehrhart, Eug{\`e}ne},
  title   = {Sur les poly{\`e}dres rationnels homoth{\'e}tiques {\`a} $n$ dimensions},
  journal = {Comptes Rendus de l'Acad{\'e}mie des Sciences},
  volume  = {254},
  year    = {1962},
  pages   = {616--618}
}

@article{Batyrev1994,
  author  = {Batyrev, Victor V.},
  title   = {Dual polyhedra and mirror symmetry for {C}alabi-{Y}au hypersurfaces in toric varieties},
  journal = {Journal of Algebraic Geometry},
  volume  = {3},
  year    = {1994},
  pages   = {493--535}
}

Prerequisites

04.11.04
04.11.06

Tier anchors

beginner: Fulton *Introduction to Toric Varieties* §3.3 informal — the prime divisor $D_\rho$ attached to a ray, the piecewise-linear support function reading off the divisor coefficients, and the polytope $P_D$ enumerating global sections
intermediate: Fulton *Introduction to Toric Varieties* §3.3-§3.4; Cox-Little-Schenck *Toric Varieties* §4.0 + §4.3
master: Fulton §3.3-§3.4; Cox-Little-Schenck §4.0-§4.3 + §6.1; Oda *Convex Bodies and Algebraic Geometry* §2.1-§2.3; Demazure 1970 *Sous-groupes algébriques de rang maximum du groupe de Cremona* (Ann. Sci. ENS (4) 3, 507-588); Kempf-Knudsen-Mumford-Saint-Donat *Toroidal Embeddings I* (LNM 339, 1973) Ch. I §2; Mumford 1972 *Compositio Math.* 24 (cone-data construction); Brion 1989 *Math. Ann.* 286 (lattice-point cohomology); Brion-Vergne 1997 *J. Amer. Math. Soc.* 10 (residue formula for lattice points)

References

TODO_REF
Fulton — Introduction to Toric Varieties · §3.3 (T-invariant Weil divisors and the support function); §3.4 Proposition 1 (Cartier divisor ↔ piecewise-linear function); §3.4 Proposition 2 (smooth toric: Weil = Cartier); §3.4 (the divisor polytope P_D and the lattice-point formula for global sections); §3.5 (anticanonical divisor on P^n)
TODO_REF
Cox-Little-Schenck — Toric Varieties · §4.0 (orientation: T-invariant Weil divisors $\sum a_\rho D_\rho$ from rays of the fan); §4.1 (Weil and Cartier divisors); §4.2 (toric divisor class group exact sequence); §4.3 Theorem 4.3.3 (lattice-point formula for global sections); §6.1 (ample line bundles via strictly convex support functions); §4.0 worked example on the anticanonical divisor of $\mathbb{P}^n$
TODO_REF
Oda — Convex Bodies and Algebraic Geometry · §2.1 (toric divisors and Weil class group); §2.2 (support functions and Cartier divisors); §2.3 (ample line bundles and projectivity); Lemma 2.3 (the lattice-point formula for global sections)
TODO_REF
Demazure 1970 — Sous-groupes algébriques de rang maximum du groupe de Cremona · Annales scientifiques de l'École normale supérieure (4) 3, 507-588; the bijection between $T$-invariant Cartier divisors on $X_\Sigma$ and integer-valued piecewise-linear functions on $|\Sigma|$; the smoothness criterion at the divisor level (every Weil is Cartier iff every cone is unimodular)
TODO_REF
Kempf-Knudsen-Mumford-Saint-Donat — Toroidal Embeddings I · Lecture Notes in Mathematics 339, Springer 1973; Ch. I §2 (toric divisors in the toroidal-embeddings framework); the Cartier-cocycle formulation extending Demazure to singular toric
TODO_REF
Mumford 1972 — An analytic construction of degenerating Abelian varieties over complete rings · Compositio Mathematica 24, 239-272; precursor to KKMS using cone-and-lattice data; toric divisors as components of the degenerate fibre, with support functions emerging from the cone-and-lattice description
TODO_REF
Brion 1989 — Groupe de Picard et nombres caractéristiques des variétés sphériques · Duke Mathematical Journal 58(2), 397-424; the toric analogue identifying Picard with $T$-Picard via support functions; lattice-point cohomology in the spherical-variety generalisation
TODO_REF
Brion-Vergne 1997 — Residue formulae, vector partition functions and lattice points in rational polytopes · Journal of the American Mathematical Society 10(4), 797-833; the equivariant residue formula extending the Demazure lattice-point formula to higher cohomology and arbitrary toric divisors
TODO_REF
Hirzebruch 1951 — Über eine Klasse von einfach-zusammenhängenden komplexen Mannigfaltigkeiten · Mathematische Annalen 124, 77-86; Hirzebruch surfaces $F_a$ as the canonical Cartier-divisor laboratory beyond $\mathbb{P}^n$
TODO_REF
Reid 1983 — Decomposition of toric morphisms · in Arithmetic and Geometry II (Progress in Mathematics 36, Birkhäuser), 395-418; piecewise-linear functions on a fan as the universal Cartier-divisor object; the support-function formulation underpinning the toric Minimal Model Program
TODO_REF
Cox 1995 — The homogeneous coordinate ring of a toric variety · Journal of Algebraic Geometry 4, 17-50; toric divisors as monomial classes in the Cox ring; the divisor polytope $P_D$ as the polytope-of-sections in the Cox-ring grading

Lean module

Codex.AlgGeom.Toric.DivisorSupportFunction

Mathlib gap

Mathlib has the abstract Weil-divisor and Cartier-divisor sheaf API
via `Mathlib.AlgebraicGeometry.WeilDivisor` and the invertible-sheaf
framework. What is absent is the toric-specific divisor-and-support-
function dictionary: (i) the enumeration of $T$-invariant prime
Weil divisors $D_\rho$ indexed by rays $\rho \in \Sigma(1)$ (requires
the `Fan` structure of `04.11.04` and the orbit-cone correspondence
of `04.11.06`); (ii) the support function $\psi_D : |\Sigma| \to
\mathbb{R}$ associated to a $T$-invariant Cartier divisor — a
continuous function linear on each cone $\sigma$ with linear piece
$m_\sigma \in M$ such that $\psi_D(v_\rho) = -a_\rho$ for every
ray $\rho$, with the Cartier-cocycle identity $m_\sigma - m_\tau \in
(\sigma \cap \tau)^\perp$ on each pairwise intersection; (iii) the
bijection between integer-valued piecewise-linear support functions
on $\Sigma$ and $T$-invariant Cartier divisors on $X_\Sigma$ (Fulton
§3.4 Proposition 1); (iv) the smooth-iff-Weil-is-Cartier theorem
(Fulton §3.4 Proposition 2) lifting the smoothness criterion from
`04.11.04` to the divisor level; (v) the $\mathbb{Q}$-Cartier
refinement on simplicial toric, with multiplier the cone
multiplicity; (vi) the global-sections formula $H^0(X_\Sigma,
\mathcal{O}(D)) = \bigoplus_{m \in P_D \cap M} \mathbb{C}\,\chi^m$
with the divisor polytope $P_D = \{m \in M_\mathbb{R} : \langle m,
v_\rho \rangle \geq -a_\rho \text{ for all } \rho \in \Sigma(1)\}$;
and (vii) the worked anticanonical case $-K_{\mathbb{P}^n} = D_0
+ \cdots + D_n$ with $|P_{-K} \cap M| = \binom{2n+1}{n}$ recovering
the dimension of the cubic / degree-$n+1$-hypersurface linear
system. The full divisor-support-function formalisation is the
combinatorial keystone connecting the orbit-cone stratification of
`04.11.06` to the Picard group of `04.11.09` and the polytope-fan
dictionary of `04.11.10`; once it ships in Mathlib, the entire
downstream toric cohomology and toric Hirzebruch-Riemann-Roch chain
becomes accessible.

Reviewer

TBD

Estimated time

beginner: 20m
intermediate: 50m
master: 90m

Intuition [Beginner]

Visual [Beginner]

Worked example [Beginner]

Check your understanding [Beginner]

Formal definition [Intermediate+]

Counterexamples to common slips

Key theorem with proof [Intermediate+]

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Advanced results [Master]

§A. T-invariant prime divisors and the support-function correspondence

§B. Cartier vs Weil — when they agree

§C. Sections via lattice-point inequalities — the polytope of sections

§D. Worked example — anticanonical on P2

Full proof set [Master]

Connections [Master]

Historical & philosophical context [Master]

Bibliography [Master]

§D. Worked example — anticanonical on $P^{2}$