Rational polyhedral cone and dual cone

Intuition [Beginner]

Imagine you have a handful of arrows in the plane, all pointing outward from the origin. If you only let yourself add the arrows together — and scale them by nonnegative real numbers — the region you can reach is a wedge-shaped territory bounded by your outermost arrows. That wedge is a rational polyhedral cone: the set of all nonnegative real combinations of finitely many integer-vector arrows in the lattice $N={\mathbb{Z}}^n$. The word "rational" records that the generating arrows are integer vectors; "polyhedral" records that the cone has flat walls; and "cone" records that the region is closed under nonnegative scaling.

Every such cone has a partner living in the dual lattice $M={\mathbb{Z}}^n$. The dual cone ${\sigma }^{\vee }$ is the set of all integer-pairing functionals on $M$ that come out nonnegative on every vector in $\sigma$. In two dimensions, if $\sigma$ is the first quadrant, then ${\sigma }^{\vee }$ is also the first quadrant — same shape, sitting in the dual space. In general ${\sigma }^{\vee }$ records the linear inequalities that carve out $\sigma$: every wall of $\sigma$ corresponds to a generator of ${\sigma }^{\vee }$, and every wall of ${\sigma }^{\vee }$ corresponds to a generator of $\sigma$. Generators on one side; defining inequalities on the other. That is the duality.

Why bother? Because rational polyhedral cones are the combinatorial Lego pieces from which toric varieties are built. A cone in $N_{\mathbb{R}}$ produces an affine algebraic variety via the semigroup algebra of the lattice points in its dual cone. The whole dictionary that turns toric varieties into combinatorics flows through this single construction, and the duality $\sigma \leftrightarrow {\sigma }^{\vee }$ is the bookkeeping that makes the dictionary bidirectional.

Visual [Beginner]

A two-panel sketch in the plane. Left panel: the lattice $N_{\mathbb{R}}={\mathbb{R}}^2$ with the cone $\sigma$ drawn as the first quadrant — a shaded wedge bounded by the positive $x$-axis and the positive $y$-axis, with the two ray generators $e_1=(1,0)$ and $e_2=(0,1)$ marked. Right panel: the dual lattice $M_{\mathbb{R}}={\mathbb{R}}^2$ with the dual cone ${\sigma }^{\vee }$ drawn as the first quadrant of $M_{\mathbb{R}}$, bounded by the positive $x^{\ast }$-axis and the positive $y^{\ast }$-axis, with the two dual ray generators $e_1^{\ast }=(1,0)$ and $e_2^{\ast }=(0,1)$ marked. A double-arrow between the two panels labelled with the pairing inequality $⟨m,u⟩\ge 0$ — the defining condition of ${\sigma }^{\vee }$.

The picture captures the essential geometry: a cone sits on the $N$ side, its dual sits on the $M$ side, and the integer pairing $⟨-,-⟩:M\times N\to \mathbb{Z}$ is what makes the two pieces talk to each other.

Worked example [Beginner]

Take $N={\mathbb{Z}}^2$ with real form $N_{\mathbb{R}}={\mathbb{R}}^2$, and consider the cone generated by the two standard basis vectors $e_1=(1,0)$ and $e_2=(0,1)$. Compute the dual cone, verify the double-dual recovers the original, and identify the integer semigroup of lattice points in the dual cone.

Step 1. Form the cone $\sigma ={\mathbb{R}}_{\ge 0}{e}_1+{\mathbb{R}}_{\ge 0}{e}_2\subset {\mathbb{R}}^2$. A point $u=(u_1,u_2)$ lies in $\sigma$ if and only if $u_1,u_2\ge 0$. So $\sigma$ is the closed first quadrant.

Step 2. Compute the dual cone ${\sigma }^{\vee }$ inside $M_{\mathbb{R}}={\mathbb{R}}^2$. By definition, ${\sigma }^{\vee }=\{m=(m_1,m_2)\in {\mathbb{R}}^2:⟨m,u⟩\ge 0\text{ for all }u\in \sigma \}$, where the pairing is the integer dot product $⟨m,u⟩=m_1{u}_1+m_2{u}_2$.

It suffices to test the condition on the generators $e_1$ and $e_2$ (because the pairing is linear in $u$): $⟨m,e_1⟩=m_1\ge 0$ and $⟨m,e_2⟩=m_2\ge 0$. So ${\sigma }^{\vee }$ is the set of $(m_1,m_2)$ with $m_1\ge 0$ and $m_2\ge 0$, namely the closed first quadrant of $M_{\mathbb{R}}$. The dual cone is generated by the dual basis vectors $e_1^{\ast }=(1,0)$ and $e_2^{\ast }=(0,1)$.

Step 3. Verify the double dual. Apply the same recipe to ${\sigma }^{\vee }$: a point $u=(u_1,u_2)\in N_{\mathbb{R}}$ lies in $({\sigma }^{\vee }{)}^{\vee }$ iff $⟨m,u⟩\ge 0$ for all $m\in {\sigma }^{\vee }$. Testing on generators $e_1^{\ast }$ and $e_2^{\ast }$: $⟨e_1^{\ast },u⟩=u_1\ge 0$ and $⟨e_2^{\ast },u⟩=u_2\ge 0$. So $({\sigma }^{\vee }{)}^{\vee }$ is the closed first quadrant in $N_{\mathbb{R}}$, recovering $\sigma$.

Step 4. Compute the semigroup of lattice points in the dual cone. The set ${\sigma }^{\vee }\cap M=\{(m_1,m_2)\in {\mathbb{Z}}^2:m_1,m_2\ge 0\}={\mathbb{Z}}_{\ge 0}^2$. This is a commutative semigroup under addition, finitely generated by $e_1^{\ast }$ and $e_2^{\ast }$. The monoid algebra is $\mathbb{C}[{\sigma }^{\vee }\cap M]=\mathbb{C}[x,y]$, the polynomial ring in two variables, where the variable $x={\chi }^{e_1^{\ast }}$ corresponds to the generator $e_1^{\ast }$ and $y={\chi }^{e_2^{\ast }}$ to $e_2^{\ast }$.

What this tells us. The cone $\sigma$ produces, via the semigroup ${\sigma }^{\vee }\cap M$ and its monoid algebra, the polynomial ring $\mathbb{C}[x,y]$ — the coordinate ring of ${\mathbb{C}}^2$. This is the simplest example of the affine-toric-variety construction: the first-quadrant cone in $N_{\mathbb{R}}$ produces affine $2$-space ${\mathbb{C}}^2$ with its action of the torus $T=({\mathbb{C}}^{\ast }{)}^2$ by coordinate-wise multiplication. The duality computation is the bookkeeping that makes this construction work, and the semigroup step is exactly where Gordan's lemma will pay rent in the formal development below.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $N$ be a free abelian group of rank $n$ (a lattice) and let $M={\mathrm{Hom}}_{\mathbb{Z}}(N,\mathbb{Z})$ be its dual lattice, with integer pairing $⟨-,-⟩:M\times N\to \mathbb{Z}$. Write $N_{\mathbb{R}}=N{\otimes }_{\mathbb{Z}}\mathbb{R}$ and $M_{\mathbb{R}}=M{\otimes }_{\mathbb{Z}}\mathbb{R}$ for the real forms, dual real vector spaces under the $\mathbb{R}$-linear extension of the pairing. (For motivation, $N$ and $M$ are the cocharacter and character lattices of a split algebraic torus from 04.11.01.)

Definition (rational polyhedral cone). A rational polyhedral cone in $N_{\mathbb{R}}$ is a subset of the form $$ \sigma = \mathrm{Cone}(u_1, \ldots, u_k) = {a_1 u_1 + \cdots + a_k u_k : a_i \in \mathbb{R}_{\geq 0}} $$ for some finite list of lattice vectors $u_1,\dots ,u_k\in N$. The vectors $u_1,\dots ,u_k$ are called generators of $\sigma$.

Definition (dual cone). Given any subset $S\subseteq N_{\mathbb{R}}$, the dual cone is $$ S^\vee = {m \in M_\mathbb{R} : \langle m, u\rangle \geq 0 \text{ for all } u \in S} \subseteq M_\mathbb{R}. $$ When $S=\sigma$ is a rational polyhedral cone generated by $u_1,\dots ,u_k$, the dual cone simplifies to ${\sigma }^{\vee }=\{m\in M_{\mathbb{R}}:⟨m,u_i⟩\ge 0\text{ for }i=1,\dots ,k\}$ — the intersection of finitely many closed half-spaces. The dual cone is thus a rational polyhedral cone in the half-space sense before we prove (via the Minkowski-Weyl identification) that it is also rational polyhedral in the generator sense.

Definition (strongly convex). A cone $\sigma \subseteq N_{\mathbb{R}}$ is strongly convex if $\sigma \cap (-\sigma )=\{0\}$, equivalently if $\sigma$ contains no positive-dimensional linear subspace. The intersection $\sigma \cap (-\sigma )$ is called the lineality space of $\sigma$ and is always a linear subspace (Exercise above).

Definition (face). A face of a rational polyhedral cone $\sigma$ is a subset $\tau =\sigma \cap H_m$ for some $m\in {\sigma }^{\vee }$, where $H_m=\{u\in N_{\mathbb{R}}:⟨m,u⟩=0\}$ is the supporting hyperplane to $\sigma$ at $m$. The cone $\sigma$ is itself a face (take $m=0$); the lineality space $\sigma \cap (-\sigma )$ is a face (it lies in $H_m$ for every $m\in {\sigma }^{\vee }\cap (-{\sigma }^{\vee })$); and every face of $\sigma$ is itself a rational polyhedral cone in $N_{\mathbb{R}}$.

Definition (dimension; ray; facet). The dimension of $\sigma$ is the real dimension of its linear span; $\sigma$ is full-dimensional if $\dim \sigma =n$. A ray of $\sigma$ is a one-dimensional face. A facet of $\sigma$ is a face of codimension one (i.e., of dimension $\dim \sigma -1$).

Counterexamples to common slips

• The set $\{(x,y)\in {\mathbb{R}}^2:x\ge 0,y\ge 0,y\le x^2\}$ is convex, closed under nonnegative scaling, but is not polyhedral — it has a curved boundary. "Polyhedral" rules out smooth boundary pieces and forces finitely many flat walls.

• The set generated by $(1,0)$ and $(\sqrt{2},1)$ in ${\mathbb{R}}^2$ is polyhedral (a wedge with two flat walls) but is not rational polyhedral in the standard lattice ${\mathbb{Z}}^2$ — the second generator is irrational. The "rational" hypothesis forces lattice generators, which is what lets us extract finiteness statements like Gordan's lemma below.

• The first quadrant $\sigma ={\mathbb{R}}_{\ge 0}^2\subset {\mathbb{R}}^2$ has dual ${\sigma }^{\vee }={\mathbb{R}}_{\ge 0}^2\subset {\mathbb{R}}^2$ — self-dual under the dot product. But take instead $\sigma =\mathrm{Cone}(e_1)=\{(x,0):x\ge 0\}\subset {\mathbb{R}}^2$. Its dual is ${\sigma }^{\vee }=\{(m_1,m_2):m_1\ge 0\}$, a closed half-plane — much larger than $\sigma$. The dual cone is generally not the same as the original; self-duality is a feature of full-dimensional simplicial cones with orthogonal generators.

• "Face" in convex-geometry usage is broader than "facet". The empty face (taking $m\in \mathrm{int}({\sigma }^{\vee })$ produces the face $\sigma \cap H_m=\sigma \cap (-\sigma )$, the lineality space, which may be just $\{0\}$) is a valid face, as is the whole cone $\sigma$ (take $m=0$). Facets are codimension-one faces; rays are one-dimensional faces; both are special cases.

Key theorem with proof [Intermediate+]

The signature theorem of this unit is the biduality of rational polyhedral cones — that the double dual recovers the original — together with Gordan's lemma on finite generation of the semigroup of lattice points in the dual cone. Both are proved here; the Master tier expands to full proofs of the equivalence chain for strong convexity and the face-correspondence theorem.

Theorem (cone duality / Farkas). Let $\sigma =\mathrm{Cone}(u_1,\dots ,u_k)\subseteq N_{\mathbb{R}}$ be a finitely generated rational polyhedral cone. Then ${\sigma }^{\vee \vee }=\sigma$, where on the left we apply duality once $N_{\mathbb{R}}\to M_{\mathbb{R}}$ via $\sigma \mapsto {\sigma }^{\vee }$ and again $M_{\mathbb{R}}\to N_{\mathbb{R}}$ via ${\sigma }^{\vee }\mapsto ({\sigma }^{\vee }{)}^{\vee }$, identifying $(N_\mathbb{R})^ = M_\mathbb{R}$and$(M_\mathbb{R})^* = N_\mathbb{R}$ via the pairing.*

Proof. The argument has two directions. First, $\sigma \subseteq {\sigma }^{\vee \vee }$ is straightforward. Second, ${\sigma }^{\vee \vee }\subseteq \sigma$ uses the finite-dimensional separating-hyperplane theorem.

Direction 1: $\sigma \subseteq {\sigma }^{\vee \vee }$. Take $u\in \sigma$. For any $m\in {\sigma }^{\vee }$, the defining condition of ${\sigma }^{\vee }$ gives $⟨m,u⟩\ge 0$. So $u$ satisfies the defining condition of ${\sigma }^{\vee \vee }=\{u^{\prime }\in N_{\mathbb{R}}:⟨m,u^{\prime }⟩\ge 0\text{ for all }m\in {\sigma }^{\vee }\}$. Hence $u\in {\sigma }^{\vee \vee }$.

Direction 2: ${\sigma }^{\vee \vee }\subseteq \sigma$. Take $u_0\in N_{\mathbb{R}}$ with $u_0\notin \sigma$. It suffices to find $m_0\in {\sigma }^{\vee }$ with $⟨m_0,u_0⟩<0$ — this exhibits $u_0\notin {\sigma }^{\vee \vee }$. The cone $\sigma$ is a closed convex subset of the finite-dimensional real vector space $N_{\mathbb{R}}$ (closed because it is the image of the closed orthant ${\mathbb{R}}_{\ge 0}^k$ under the continuous linear map $(a_1,\dots ,a_k)\mapsto \sum a_i{u}_i$; convex because nonnegative real combinations of generators stay in $\sigma$). By the finite-dimensional separating-hyperplane theorem (Hahn-Banach in finite dimensions, or directly: minimise the Euclidean distance from $u_0$ to the closed convex set $\sigma$ to find the foot $u^{\ast }\in \sigma$, and the vector $u_0-u^{\ast }$ together with its dual functional separates), there exists a linear functional $m_0\in M_{\mathbb{R}}$ such that $⟨m_0,u_0⟩<c\le ⟨m_0,u⟩$ for all $u\in \sigma$. Since $\sigma$ is a cone (closed under nonnegative scaling), the infimum $c={\inf }_{u\in \sigma }⟨m_0,u⟩$ is either $0$ (attained at $0\in \sigma$) or $-\infty$; the strict separation $⟨m_0,u_0⟩<c$ rules out $-\infty$, so $c=0$, giving $⟨m_0,u⟩\ge 0$ for all $u\in \sigma$ — that is, $m_0\in {\sigma }^{\vee }$ — and $⟨m_0,u_0⟩<0$. So $u_0\notin {\sigma }^{\vee \vee }$, proving ${\sigma }^{\vee \vee }\subseteq \sigma$. $\square$

The cone-duality theorem is the integer / rational shadow of the Minkowski-Weyl theorem: every finitely generated convex cone in a finite-dimensional real vector space is the intersection of finitely many closed half-spaces, and every such intersection is a finitely generated convex cone. The two descriptions — "$H$-representation" (intersection of half-spaces) and "$V$-representation" (nonnegative span of finitely many generators) — exchange under duality, so ${\sigma }^{\vee }$ has both descriptions, hence is itself a finitely generated rational polyhedral cone.

Theorem (Gordan's lemma). Let $\sigma \subseteq N_{\mathbb{R}}$ be a rational polyhedral cone and let $S={\sigma }^{\vee }\cap M$ be the additive semigroup of lattice points in the dual cone. Then $S$ is finitely generated as a commutative monoid: there exist $m_1,\dots ,m_r\in S$ such that every $m\in S$ is a nonnegative integer combination $m=b_1{m}_1+\cdots +b_r{m}_r$ with $b_i\in {\mathbb{Z}}_{\ge 0}$.

Proof. Since ${\sigma }^{\vee }$ is itself a finitely generated rational polyhedral cone (by the duality theorem above, applied to the cone $\sigma$ generated by lattice vectors: ${\sigma }^{\vee }$ is the intersection of finitely many lattice-rational half-spaces, hence the Minkowski-Weyl direction gives ${\sigma }^{\vee }=\mathrm{Cone}(m_1^{\prime },\dots ,m_s^{\prime })$ for lattice vectors $m_j^{\prime }\in M$ — one can in fact take the $m_j^{\prime }$ to be lattice vectors by choosing the half-space normals $u_i\in N$ and then constructing the extremal rays of ${\sigma }^{\vee }$, each of which is a rational ray and hence has a primitive lattice generator). Set $$ P = {a_1 m'1 + \cdots + a_s m's : a_j \in [0, 1] \cap \mathbb{R}} \subseteq M\mathbb{R}, $$ the parallelepiped spanned by the lattice generators of ${\sigma }^{\vee }$. $P$ is a bounded closed subset of $M\mathbb{R}$,sotheintersection$P \cap M$isafinitesetoflatticepoints.Take$m_1, \ldots, m_r$tobetheunionof(i)thegenerators$m'_1, \ldots, m'_s$themselves,and(ii)alllatticepointsin$P \cap M$.

Now let $m\in S={\sigma }^{\vee }\cap M$. Write $m={\sum }_j{a}_j{m}_j^{\prime }$ with $a_j\in {\mathbb{R}}_{\ge 0}$. Split each coefficient as $a_j=\lfloor a_j\rfloor +\{a_j\}$, with $\lfloor a_j\rfloor \in {\mathbb{Z}}_{\ge 0}$ and $\{a_j\}\in [0,1)$. Set $m^{\prime }={\sum }_j\lfloor a_j\rfloor m_j^{\prime }$ and $m^{\prime \prime }={\sum }_j\{a_j\}{m}_j^{\prime }$, so $m=m^{\prime }+m^{\prime \prime }$. The piece $m^{\prime }$ is a nonnegative integer combination of $m_1^{\prime },\dots ,m_s^{\prime }$. The piece $m^{\prime \prime }\in P$ (since each $\{a_j\}\in [0,1)$); moreover $m^{\prime \prime }=m-m^{\prime }\in M$ (as the difference of two lattice points), so $m^{\prime \prime }\in P\cap M$. Hence $m^{\prime \prime }\in \{m_1,\dots ,m_r\}$ by construction, and the decomposition $m=m^{\prime }+m^{\prime \prime }$ exhibits $m$ as a nonnegative integer combination of the generators in $\{m_1,\dots ,m_r\}$. The monoid $S$ is finitely generated. $\square$

The monoid algebra $\mathbb{C}[{\sigma }^{\vee }\cap M]$ — viewed as a $\mathbb{C}$-vector space with basis the formal symbols $\{{\chi }^m:m\in {\sigma }^{\vee }\cap M\}$ and multiplication ${\chi }^m{\chi }^{m^{\prime }}={\chi }^{m+m^{\prime }}$ — is therefore a finitely generated $\mathbb{C}$-algebra, hence a finitely generated commutative ring, hence the coordinate ring of an affine algebraic variety. This affine variety, denoted $U_{\sigma }=\mathrm{Spec}\mathbb{C}[{\sigma }^{\vee }\cap M]$, is the affine toric variety associated to $\sigma$, constructed in the next unit 04.11.03. Gordan's lemma is the foundational reason the construction produces a Noetherian object — without finite generation the $\mathrm{Spec}$ would not be of finite type and the dictionary between cones and varieties would not be algebraic in the usual sense.

Bridge. The duality ${\sigma }^{\vee \vee }=\sigma$ and Gordan's lemma build toward the entire affine-toric-variety construction, and the central insight is that a rational polyhedral cone in $N_{\mathbb{R}}$ has two interchangeable descriptions — by its generators on the $N$-side, and by its defining inequalities on the $M$-side — and the duality is exactly the dictionary between them. This is exactly the foundational reason that the toric construction $\sigma \mapsto U_{\sigma }=\mathrm{Spec}\mathbb{C}[{\sigma }^{\vee }\cap M]$ is well-defined: Gordan's lemma supplies the finite generation of the semigroup algebra, and cone duality identifies the cone $\sigma$ with the cone-of-cones ${\sigma }^{\vee \vee }$ so that the construction does not depend on the choice of generators. Putting these together, the rational polyhedral cone formalism generalises the everyday notion of "wedge in the plane" to a combinatorial gadget whose dual encodes the equational data of an affine variety. The bridge is dual to the duality of $M$ and $N$ from 04.11.01: every concept on the $N$-side has a corresponding concept on the $M$-side, and the cone duality identifies $\sigma$ with ${\sigma }^{\vee \vee }$ in the same way that the lattice duality identifies $M$ with ${\mathrm{Hom}}_{\mathbb{Z}}(N,\mathbb{Z})$. This pattern appears again in 04.11.03 (affine toric varieties from cones), in 04.11.04 (fans of cones and the global toric variety $X_{\Sigma }$), and in 04.11.05 (orbit-cone correspondence), each of which reads a piece of geometry out of the combinatorics developed here.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Mathlib has convex-cone infrastructure (Mathlib.Analysis.Convex.Cone.Basic, Mathlib.Analysis.Convex.Cone.Dual) covering convex cones and their dual cones in a real vector space under a continuous bilinear pairing. What is missing is the lattice-rational refinement that toric geometry requires. The intended formalisation reads schematically:

import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Analysis.Convex.Cone.Dual
import Mathlib.LinearAlgebra.FreeModule.Basic

variable {n : ℕ}

/-- A rational polyhedral cone in N_ℝ generated by finitely many
lattice vectors in N. -/
structure RationalPolyhedralCone (N : Type*) [AddCommGroup N] [Module ℤ N]
    [Module.Free ℤ N] where
  generators : Finset N
  -- the cone itself, as a subset of N ⊗ ℝ
  carrier : ConvexCone ℝ (N ⊗[ℤ] ℝ)
  spec : carrier = ConvexCone.span ℝ
    ((generators : Set N).image (· ⊗ₜ 1))

/-- The dual cone of a rational polyhedral cone. -/
noncomputable def RationalPolyhedralCone.dual
    {N : Type*} [AddCommGroup N] [Module ℤ N] [Module.Free ℤ N]
    (σ : RationalPolyhedralCone N) :
    ConvexCone ℝ (Module.Dual ℤ N ⊗[ℤ] ℝ) := sorry

/-- Cone duality: the double dual recovers the original cone. -/
theorem RationalPolyhedralCone.dual_dual
    {N : Type*} [AddCommGroup N] [Module ℤ N] [Module.Free ℤ N]
    (σ : RationalPolyhedralCone N) :
    σ.dual.dual = σ.carrier := by
  -- via the finite-dimensional Hahn-Banach separating-hyperplane theorem
  sorry

/-- Gordan's lemma: the lattice points in the dual cone form a finitely
generated additive monoid. -/
theorem gordan_lemma
    {N : Type*} [AddCommGroup N] [Module ℤ N] [Module.Free ℤ N]
    (σ : RationalPolyhedralCone N) :
    AddMonoid.FG ((σ.dual : Set _) ∩ (Module.Dual ℤ N : Set _)) := by
  -- via the parallelepiped-of-fractional-parts argument
  sorry

/-- Strong convexity equivalence: σ is strongly convex iff σ^∨ spans M_ℝ. -/
theorem stronglyConvex_iff_dual_fullDimensional
    {N : Type*} [AddCommGroup N] [Module ℤ N] [Module.Free ℤ N]
    (σ : RationalPolyhedralCone N) :
    σ.carrier ∩ (- σ.carrier) = {0} ↔
      Submodule.span ℝ (σ.dual : Set _) = ⊤ := by
  sorry

Each step is reachable from current Mathlib but requires substantive new development. The separating-hyperplane theorem is in Mathlib (Mathlib.Analysis.Convex.SeparatingHyperplane), but the rational-cone specialisation, the Gordan-lemma parallelepiped argument, and the strong-convexity equivalence are all absent as named results. The expected route is a Mathlib.Combinatorics.Polyhedra.Rational namespace housing the rational-polyhedral-cone formalism as a contravariant functor to commutative monoids (via $\sigma \mapsto {\sigma }^{\vee }\cap M$), building toward the affine-toric-variety functor of the next unit.

Advanced results [Master]

Theorem (Carathéodory's theorem for cones). Let $\sigma =\mathrm{Cone}(u_1,\dots ,u_k)\subseteq N_{\mathbb{R}}$ be a rational polyhedral cone of dimension $d$. Every point $u\in \sigma$ can be written as a nonnegative combination of at most $d$ of the generators $u_i$. In particular, the cone $\sigma$ is the union of the cones spanned by all $d$-subsets of its generator set whose generators span a $d$-dimensional subspace.

Carathéodory for cones is the finite-dimensional version of the classical Carathéodory theorem for convex hulls; its proof goes by linear-dependence reduction. Given a representation $u={\sum }_i{a}_i{u}_i$ with $a_i\ge 0$ and more than $d$ nonzero coefficients, the corresponding generators $\{u_i:a_i>0\}$ are linearly dependent, so a relation ${\sum }_i{c}_i{u}_i=0$ with $c_i$ not all zero exists. Add a scalar multiple of this relation to the original representation to drive at least one coefficient to zero while keeping all others nonnegative; iterate until at most $d$ coefficients remain nonzero. The lemma is the foundational reason the rational-polyhedral-cone formalism is finite-dimensional in a strong sense: although the generator set can be arbitrarily large, the effective dimension at any point is bounded by the dimension of the cone.

Theorem (Minkowski-Weyl for cones; the equivalence of $V$- and $H$-representations). A subset $\sigma \subseteq N_{\mathbb{R}}$ is a rational polyhedral cone if and only if it is the intersection of finitely many closed rational half-spaces. The two representations exchange under duality $\sigma \mapsto {\sigma }^{\vee }$.

The Minkowski-Weyl theorem is the central duality result of finite-dimensional convex polyhedral geometry. The cone version proved in Exercise 8 is the lattice-rational specialisation; the polytope version (Minkowski's foundational 1896 result in Geometrie der Zahlen) is the cone-version applied to the cone over a polytope. Hermann Weyl established the second direction (half-space implies generator) in his 1935 paper Elementare Theorie der konvexen Polyeder (Commentarii Mathematici Helvetici 7), completing Minkowski's program.

Theorem (face lattice and orbit structure). Let $\sigma$ be a strongly convex rational polyhedral cone in $N_{\mathbb{R}}$. The face lattice $\mathrm{Face}(\sigma )$ — partially ordered by inclusion — is a finite graded lattice with rank function $\dim$. The map $\tau \mapsto \tau^ = \sigma^\vee \cap \tau^\perp$isanorder-reversingisomorphism$\mathrm{Face}(\sigma) \cong \mathrm{Face}(\sigma^\vee)^{\mathrm{op}}$,with$\dim \tau + \dim \tau^* = n$foreveryface$\tau$.*

The face-lattice correspondence is the combinatorial backbone of the toric dictionary. On the variety side it becomes the orbit-cone correspondence (developed in 04.11.05): the $T$-orbits in $U_{\sigma }$ are in inclusion-reversing bijection with the faces of $\sigma$, with the orbit corresponding to a face $\tau$ of dimension $d$ being a torus orbit of dimension $n-d$. The face $\sigma$ itself corresponds to the closed orbit, a fixed point of $T$; the zero face $\{0\}$ corresponds to the open dense $T$-orbit, isomorphic to $T$ itself. The combinatorial $\dim \tau +\dim {\tau }^{\ast }=n$ identity is exactly the orbit-dimension formula.

Theorem (Hilbert basis theorem for monoids; Hilbert 1890 / Gordan reformulation). The semigroup ${\sigma }^{\vee }\cap M$ has a unique minimal generating set, called the Hilbert basis of ${\sigma }^{\vee }$. It consists of the lattice points $m\in {\sigma }^{\vee }\cap M\setminus \{0\}$ that cannot be written as a sum of two other nonzero lattice points in ${\sigma }^{\vee }\cap M$.

The Hilbert basis of ${\sigma }^{\vee }$ is the canonical minimal generating set, and is what algorithmic toric-variety software (Macaulay2, Magma, Polymake, GAP) computes to present the affine toric variety $U_{\sigma }$. The minimal-generating-set computation reduces to enumerating the indecomposable lattice points in the parallelepiped of Gordan's lemma; the resulting set is finite (as Gordan's lemma proves) and uniquely determined, so the toric variety $U_{\sigma }$ comes with a canonical embedding in affine space of dimension equal to $|\mathrm{Hilb}({\sigma }^{\vee })|$.

Theorem (smoothness criterion; Demazure 1970). Let $\sigma$ be a strongly convex rational polyhedral cone of dimension $d$ in $N_{\mathbb{R}}$. The affine toric variety $U_{\sigma }=\mathrm{Spec}\mathbb{C}[{\sigma }^{\vee }\cap M]$ is smooth if and only if $\sigma$ is a simplicial unimodular cone — that is, $\sigma$ is generated by exactly $d$ lattice vectors that form part of a $\mathbb{Z}$-basis of $N$. In particular, $U_{\sigma }\cong {\mathbb{C}}^d\times T^{n-d}$ for some auxiliary torus $T^{n-d}$ when this condition holds.

The smoothness criterion identifies the smooth affine toric varieties combinatorially. The cone $\sigma =\mathrm{Cone}(e_1,e_2)$ in ${\mathbb{Z}}^2$ is simplicial unimodular (the generators $e_1,e_2$ form the standard $\mathbb{Z}$-basis), and $U_{\sigma }={\mathbb{C}}^2$ is smooth. The cone $\sigma =\mathrm{Cone}(e_1,e_1+2e_2)$ from Exercise 3 has generators not extending to a $\mathbb{Z}$-basis (the determinant of the matrix with columns $e_1$ and $e_1+2e_2$ is $2\ne \pm 1$), so $U_{\sigma }$ is singular — specifically, the $A_1$ Du Val singularity, as computed there. The smoothness criterion is the foundation of toric resolution of singularities: every singular toric variety can be resolved by subdividing its fan into simplicial unimodular subcones (a process called stellar subdivision or fan refinement, the toric implementation of Hironaka's theorem).

Theorem (compatibility with the algebraic torus). Let $T=({\mathbb{G}}_m{)}^n$ with cocharacter lattice $N$ and character lattice $M$. The zero cone $\sigma =\{0\}\subset N_{\mathbb{R}}$ has dual cone ${\sigma }^{\vee }=M_{\mathbb{R}}$, semigroup ${\sigma }^{\vee }\cap M=M={\mathbb{Z}}^n$, and monoid algebra $\mathbb{C}[M]=\mathbb{C}[x_1^{\pm 1},\dots ,x_n^{\pm 1}]$, recovering the coordinate ring of $T$ itself. The torus $T$ is the affine toric variety of the zero cone.

This is the simplest case of the toric dictionary, and the foundational reason every toric variety contains $T$ as an open dense subset: the zero cone is a face of every other cone, so its affine toric variety $U_{\{0\}}=T$ embeds as an open subset $T\hookrightarrow U_{\sigma }$ for every $\sigma$. The Hopf-algebra structure on $\mathbb{C}[M]$ from 04.11.01 extends to a $T$-action on $U_{\sigma }$ via the comorphism, and the orbit structure (via the face-lattice correspondence above) records exactly how the cone $\sigma$ decomposes the variety $U_{\sigma }$ into $T$-orbits.

Synthesis. The rational polyhedral cone formalism is the combinatorial bridge between the lattice $(M,N)$ of 04.11.01 and the algebraic-geometric construction of affine toric varieties in 04.11.03, and the central insight is that the two descriptions of a cone — by generators on the $N$-side and by defining inequalities on the $M$-side — are interchangeable via duality, ${\sigma }^{\vee \vee }=\sigma$. Three structural results — Minkowski-Weyl (the equivalence of $V$- and $H$-representations), Gordan's lemma (finite generation of ${\sigma }^{\vee }\cap M$), and the face lattice correspondence (inclusion-reversing bijection $\sigma \leftrightarrow {\sigma }^{\vee }$) — combine into one identification: a rational polyhedral cone is a single combinatorial object with two equivalent presentations, and the affine toric variety it produces inherits this duality at the level of $T$-equivariant geometry.

Putting these together, the rational-polyhedral-cone language organises the entire toric chapter, and the foundational reason is that every downstream construction (the affine toric variety $U_{\sigma }$ in 04.11.03, the fan $\Sigma$ and the global $X_{\Sigma }$ in 04.11.04, the orbit-cone correspondence in 04.11.05, $T$-invariant divisors and the Picard group via support functions in 04.11.08, the polytope-toric correspondence in 04.11.11) reads one piece of combinatorics from the lattice side of the dictionary developed in this unit. The bridge is dual to the duality of $M$ and $N$, and this pattern appears again every time a $T$-equivariant geometric notion on a toric variety is computed combinatorially: it has a description in terms of the cone $\sigma$, a description in terms of the dual cone ${\sigma }^{\vee }$, and a duality identifying the two. The smoothness criterion of Demazure 1970 is the foundational reason that resolution of singularities for toric varieties is purely combinatorial — the simplicial-unimodular condition is a $\mathbb{Z}$-arithmetic condition on the generators of $\sigma$, and refining $\sigma$ into simplicial unimodular subcones produces a toric resolution of $U_{\sigma }$. This identifies toric resolution with the combinatorial process of fan refinement, generalising classical Hilbert-basis-style computations to the full machinery of toric birational geometry.

The cone formalism also generalises in two directions. To non-rational polyhedral cones (where generators may be irrational), most of the geometry persists — Minkowski-Weyl, Carathéodory, and the face lattice are properties of finite-dimensional convex polyhedral geometry independent of any lattice — but Gordan's lemma and the affine-toric-variety construction require the rationality assumption: without lattice generators, ${\sigma }^{\vee }\cap M$ might collapse to the identity or fail to be finitely generated. To infinite-dimensional cones (in the analysis-flavoured generality of Mathlib.Analysis.Convex.Cone), the duality ${\sigma }^{\vee \vee }=\sigma$ requires either compactness or a weak-closed-cone hypothesis (Hahn-Banach in infinite dimensions involves topology); the finite-dimensional rational case used here is computationally tractable and is the version that drives toric geometry. The lattice formalism developed here is therefore the first nontrivial example of polyhedral geometry where the combinatorics, the analysis (separating-hyperplane theorem), and the algebraic geometry (affine toric varieties) all converge on a single object — the rational polyhedral cone $\sigma$ together with its dual ${\sigma }^{\vee }$.

Full proof set [Master]

Theorem (cone duality), proof. Given in the Intermediate-tier section: direction $\sigma \subseteq {\sigma }^{\vee \vee }$ is direct from the definition; direction ${\sigma }^{\vee \vee }\subseteq \sigma$ uses the finite-dimensional separating-hyperplane theorem to find, for each $u_0\notin \sigma$, a functional $m_0\in {\sigma }^{\vee }$ with $⟨m_0,u_0⟩<0$, exhibiting $u_0\notin {\sigma }^{\vee \vee }$. $\square$

Theorem (Gordan's lemma), proof. Given in the Intermediate-tier section: choose lattice ray generators $m_1^{\prime },\dots ,m_s^{\prime }$ of ${\sigma }^{\vee }$ (existence via Minkowski-Weyl); form the parallelepiped $P=\{\sum a_j{m}_j^{\prime }:a_j\in [0,1]\}$; observe $P\cap M$ is finite; show every $m\in {\sigma }^{\vee }\cap M$ decomposes as $m=m^{\prime }+m^{\prime \prime }$ with $m^{\prime }\in {\mathbb{Z}}_{\ge 0}⟨m_1^{\prime },\dots ,m_s^{\prime }⟩$ and $m^{\prime \prime }\in P\cap M$; combine the two finite sets to get a finite generating set for the semigroup. $\square$

Proposition (strong convexity equivalence), proof. Given in Exercise 4: identify the lineality space $W=\sigma \cap (-\sigma )$ of $\sigma$ as the annihilator of $\mathrm{span}{\sigma }^{\vee }$ in $N_{\mathbb{R}}$, equivalently $\dim W=n-\dim (\mathrm{span}{\sigma }^{\vee })$; identify the lineality space $W^{\prime }={\sigma }^{\vee }\cap (-{\sigma }^{\vee })$ of ${\sigma }^{\vee }$ as the annihilator of $\mathrm{span}\sigma$ in $M_{\mathbb{R}}$, giving $\dim W^{\prime }=n-\dim (\mathrm{span}\sigma )$. Then $\sigma$ strongly convex iff $W=0$ iff $\dim W=0$ iff $\dim (\mathrm{span}{\sigma }^{\vee })=n$ iff ${\sigma }^{\vee }$ is full-dimensional in $M_{\mathbb{R}}$ iff ${\sigma }^{\vee }$ spans $M_{\mathbb{R}}$. The chain of equivalences is complete. $\square$

Proposition (faces are nonnegative spans of generator subsets), proof. Given in Exercise 7: partition the generators of $\sigma$ by $⟨m,u_i⟩$ for the supporting hyperplane normal $m$; show that any $u\in \tau =\sigma \cap H_m$ written as $\sum a_i{u}_i$ has $a_i=0$ for $i$ with $⟨m,u_i⟩>0$ (by the nonnegativity of $⟨m,u⟩={\sum }_i{a}_i⟨m,u_i⟩$, forcing all positive summands to vanish since the sum is zero); conclude $\tau =\mathrm{Cone}(\{u_i:⟨m,u_i⟩=0\})$, a finitely generated rational polyhedral cone. $\square$

Proposition (Minkowski-Weyl for cones), proof. Given in Exercise 8: both directions follow from the duality theorem ${\sigma }^{\vee \vee }=\sigma$. The generator description of $\sigma$ implies ${\sigma }^{\vee }$ has a half-space description with normals the original generators, and $\sigma ={\sigma }^{\vee \vee }$ then has a generator description with the dual generators of ${\sigma }^{\vee }$. The half-space description direction is symmetric. The recursive structure makes the dimension count proceed by induction. $\square$

Proposition (face correspondence $\tau \leftrightarrow {\tau }^{\ast }$), proof. Let $\sigma$ be a strongly convex rational polyhedral cone in $N_{\mathbb{R}}$. The map $\Phi :\mathrm{Face}(\sigma )\to \mathrm{Face}({\sigma }^{\vee })$ defined by $\Phi(\tau) = \tau^ := \sigma^\vee \cap \tau^\perp$isaninclusion-reversingbijection,withinverse$\Phi^{-1}(\tau') = (\tau')^* := \sigma \cap (\tau')^\perp$,and$\dim \tau + \dim \tau^* = n$foreveryface$\tau$.*

Proof. Well-definedness: for $\tau \in \mathrm{Face}(\sigma )$, the set ${\tau }^{\perp }=\{m\in M_{\mathbb{R}}:⟨m,u⟩=0\text{ for all }u\in \tau \}$ is a linear subspace of $M_{\mathbb{R}}$; the intersection ${\sigma }^{\vee }\cap {\tau }^{\perp }$ is the face of ${\sigma }^{\vee }$ supported by the hyperplane ${\tau }^{\perp }$ (every $m\in {\sigma }^{\vee }\cap {\tau }^{\perp }$ has $⟨m,u⟩\ge 0$ for $u\in \sigma$ and $⟨m,u⟩=0$ for $u\in \tau$, identifying ${\tau }^{\ast }$ as the face of ${\sigma }^{\vee }$ at the supporting functional $u_{\tau }\in \mathrm{relint}(\tau )\subset \sigma$).

Inclusion-reversing: $\tau \subseteq {\tau }^{\prime }$ implies ${\tau }^{\prime \perp }\subseteq {\tau }^{\perp }$, hence ${\sigma }^{\vee }\cap {\tau }^{\prime \perp }\subseteq {\sigma }^{\vee }\cap {\tau }^{\perp }$, i.e., $({\tau }^{\prime }{)}^{\ast }\subseteq {\tau }^{\ast }$.

Bijection: the inverse map $\Psi :\mathrm{Face}({\sigma }^{\vee })\to \mathrm{Face}(\sigma )$ defined by $\Psi ({\tau }^{\prime })=({\tau }^{\prime }{)}^{\ast }:=\sigma \cap ({\tau }^{\prime }{)}^{\perp }$ is the same construction applied to ${\sigma }^{\vee }$, and the involutive identity $\Psi \Phi ={\mathrm{id}}_{\mathrm{Face}(\sigma )}$ follows from cone duality ${\sigma }^{\vee \vee }=\sigma$ and the double-annihilator identity $V^{\perp \perp }=V$ for any linear subspace $V$: $\Psi (\Phi (\tau ))=\sigma \cap ({\sigma }^{\vee }\cap {\tau }^{\perp }{)}^{\perp }$, and unpacking the annihilator gives $({\sigma }^{\vee }\cap {\tau }^{\perp }{)}^{\perp }={\sigma }^{\vee \perp }+{\tau }^{\perp \perp }=\mathrm{span}{\sigma }^{\vee \perp }+\mathrm{span}\tau =\tau$ (using that the linear span of $\sigma$ contains $\tau$ and intersecting with $\sigma$ recovers $\tau$ as the face of $\sigma$ along this subspace).

Dimension identity: $\dim {\tau }^{\ast }=\dim ({\sigma }^{\vee }\cap {\tau }^{\perp })$. The face ${\tau }^{\ast }$ of ${\sigma }^{\vee }$ lies in ${\tau }^{\perp }$, a linear subspace of $M_{\mathbb{R}}$ of dimension $n-\dim \tau$. Since ${\sigma }^{\vee }$ has full dimension $n$ in $M_{\mathbb{R}}$ (by strong convexity of $\sigma$, applied to the dual), the face ${\tau }^{\ast }$ has dimension equal to the codimension of $\mathrm{span}\tau$ in $\mathrm{span}{\sigma }^{\vee }=M_{\mathbb{R}}$, namely $\dim {\tau }^{\ast }=n-\dim \tau$. So $\dim \tau +\dim {\tau }^{\ast }=n$. $\square$

Proposition (Hilbert basis of ${\sigma }^{\vee }\cap M$), proof. Let $\sigma$ be a strongly convex rational polyhedral cone in $N_{\mathbb{R}}$ and let $S={\sigma }^{\vee }\cap M$. The set $\mathrm{Hilb}({\sigma }^{\vee }):=\{m\in S\setminus \{0\}:m\text{ is not a sum of two nonzero elements of }S\}$ is the unique minimal generating set of $S$, and is finite.

Proof. Finiteness: $\mathrm{Hilb}({\sigma }^{\vee })$ is contained in the bounded set $P\cap M$ of Gordan's lemma (every Hilbert-basis element fits inside the fundamental parallelepiped, since if not, write it as a sum of a lattice ray generator plus a residue and reduce). Hence $\mathrm{Hilb}({\sigma }^{\vee })$ is finite.

Generating: Take $m\in S$. Either $m=0$ (which is the empty combination of Hilbert-basis elements) or $m$ is a sum of two nonzero elements of $S$, say $m=m^{\prime }+m^{\prime \prime }$. Recursively decompose each $m^{\prime }$ and $m^{\prime \prime }$ that is not Hilbert-basis. The recursion terminates because each step strictly decreases a positive-integer measure (for example, the height $⟨m,u_{\sigma }⟩$ where $u_{\sigma }$ is a positive functional in $\mathrm{relint}(\sigma )$). The terminating elements are Hilbert-basis. So every $m\in S$ is a nonnegative integer combination of Hilbert-basis elements.

Minimality: a Hilbert-basis element $h$ cannot be omitted, because by definition it is not a sum of other nonzero elements of $S$, hence cannot be written as a nonnegative integer combination of $S\setminus \{h\}$ without contradicting its indecomposability. The minimal generating set is therefore exactly the Hilbert basis. $\square$

Proposition (smoothness of $U_{\sigma }$ via simplicial unimodular cones), proof. Let $\sigma$ be a strongly convex rational polyhedral cone in $N_{\mathbb{R}}$ of dimension $d$. Then $U_{\sigma }=\mathrm{Spec}\mathbb{C}[{\sigma }^{\vee }\cap M]$ is a smooth complex algebraic variety if and only if $\sigma$ is simplicial unimodular: that is, $\sigma$ is generated by exactly $d$ lattice vectors $u_1,\dots ,u_d$ that form part of a $\mathbb{Z}$-basis of $N$.

Proof sketch. ($\Leftarrow$) If $\sigma =\mathrm{Cone}(u_1,\dots ,u_d)$ with $u_i$ extending to a $\mathbb{Z}$-basis $u_1,\dots ,u_n$ of $N$, change basis so that $u_i=e_i$ for $i=1,\dots ,d$ (the standard basis vectors). Then ${\sigma }^{\vee }=\{m\in M_{\mathbb{R}}:m_i\ge 0\text{ for }i\le d\}$, the closed half-space ${\mathbb{R}}_{\ge 0}^d\times {\mathbb{R}}^{n-d}$. The semigroup ${\sigma }^{\vee }\cap M={\mathbb{Z}}_{\ge 0}^d\times {\mathbb{Z}}^{n-d}$ is the free monoid generated by $e_1^{\ast },\dots ,e_d^{\ast }$ together with $\pm e_{d+1}^{\ast },\dots ,\pm e_n^{\ast }$. The monoid algebra is $\mathbb{C}[x_1,\dots ,x_d,x_{d+1}^{\pm 1},\dots ,x_n^{\pm 1}]$, the coordinate ring of ${\mathbb{C}}^d\times ({\mathbb{C}}^{\ast }{)}^{n-d}$, a smooth variety.

($\Rightarrow$) If $U_{\sigma }$ is smooth at the closed orbit (the $T$-fixed point given by the maximal ideal $⟨{\chi }^m:m\in {\sigma }^{\vee }\cap M\setminus \mathrm{lineality}⟩$), then the cotangent space at this fixed point has dimension equal to the dimension of $U_{\sigma }$, namely $n$. A direct computation identifies the cotangent space with $\mathrm{Hilb}({\sigma }^{\vee })\otimes \mathbb{C}$ modulo the relations from the semigroup; smoothness forces $|\mathrm{Hilb}({\sigma }^{\vee })|=n$ and the Hilbert basis to be a $\mathbb{Z}$-basis of $M$. Dualising, the generators of $\sigma$ form part of a $\mathbb{Z}$-basis of $N$, the simplicial-unimodular condition. $\square$

Connections [Master]

• Algebraic torus 04.11.01. The lattice $N$ and its dual $M$ in which rational polyhedral cones live are exactly the cocharacter and character lattices of the algebraic torus $T=({\mathbb{G}}_m{)}^n$. The zero cone $\sigma =\{0\}$ has dual cone ${\sigma }^{\vee }=M_{\mathbb{R}}$ and semigroup ${\sigma }^{\vee }\cap M=M$ itself, with monoid algebra $\mathbb{C}[M]=\mathbb{C}[x_1^{\pm 1},\dots ,x_n^{\pm 1}]$ — the coordinate ring of $T$. The torus is therefore the affine toric variety of the zero cone, and every other cone $\sigma$ produces an affine variety $U_{\sigma }$ containing $T$ as an open dense subset via the face inclusion $\{0\}\hookrightarrow \sigma$. The lattice formalism of the prerequisite unit supplies the dictionary $(M,N,⟨-,-⟩)$ in which the cones of this unit live, and the cone formalism here builds the first combinatorial datum from which toric varieties are constructed.

• Linear transformation / rank-nullity 01.01.05. The cone-duality theorem ${\sigma }^{\vee \vee }=\sigma$ is, at heart, a statement about the linear functional pairing $⟨-,-⟩:M_{\mathbb{R}}\times N_{\mathbb{R}}\to \mathbb{R}$ — every $m\in M_{\mathbb{R}}$ acts as a linear functional $N_{\mathbb{R}}\to \mathbb{R}$, and the cone ${\sigma }^{\vee }$ is the set of functionals nonnegative on $\sigma$. The dimension identities for lineality spaces and faces ($\dim W+\dim (\mathrm{span}{\sigma }^{\vee })=n$, $\dim \tau +\dim {\tau }^{\ast }=n$) are instances of the rank-nullity theorem applied to the pairing viewed as a linear map. The prerequisite unit on linear transformations supplies the kernel-image-rank machinery that underwrites the convex-geometric computations in this unit.

• Group 01.02.01. The lattice points ${\sigma }^{\vee }\cap M$ form a commutative monoid under addition (a semigroup containing $0$), and the cancellative-monoid structure on this semigroup is what makes the monoid algebra $\mathbb{C}[{\sigma }^{\vee }\cap M]$ a commutative ring. Gordan's lemma is the statement that this monoid is finitely generated, and the Hilbert-basis theorem identifies a unique minimal generating set. The prerequisite unit on groups supplies the abelian-group structure on $M$ that the additive semigroup ${\sigma }^{\vee }\cap M$ inherits.

• Affine toric variety 04.11.03. The next unit in the toric chapter constructs the affine algebraic variety $U_{\sigma }=\mathrm{Spec}\mathbb{C}[{\sigma }^{\vee }\cap M]$ from the cone $\sigma$. Gordan's lemma is the foundational reason this $\mathrm{Spec}$ is a Noetherian / finite-type object, and the cone-duality theorem makes the construction independent of the choice of generators of $\sigma$. The face-correspondence theorem developed here becomes the orbit-cone correspondence of 04.11.06: the $T$-orbits in $U_{\sigma }$ are in inclusion-reversing bijection with the faces of $\sigma$, with the dimension formula $\dim \tau +\dim \mathrm{orbit}(\tau )=n$ matching the face-lattice dimension identity proved here.

• Fan and toric variety $X_{\Sigma }$ 04.11.04. A fan in $N_{\mathbb{R}}$ is a finite collection of rational polyhedral cones closed under taking faces and intersections; the global toric variety $X_{\Sigma }$ is constructed by gluing the affine toric varieties $U_{\sigma }$ for $\sigma \in \Sigma$ along the affine pieces corresponding to face intersections. The cone formalism of this unit is the combinatorial atom from which fans are assembled, and the face-correspondence theorem ensures the gluing data (face inclusions) is functorial.

• Smoothness and completeness via fans 04.11.05. The downstream unit promotes the unimodular-basis condition on a single cone — $\sigma$ is generated by part of a $\mathbb{Z}$-basis of $N$ — into the global smoothness criterion for the toric variety $X_{\Sigma }$: smooth iff every cone of $\Sigma$ is simplicial unimodular. The lineality, face, and dimension calculus developed in this unit supplies the cone-level toolkit for the smoothness test; the completeness criterion $|\Sigma |=N_{\mathbb{R}}$ uses cone duality ${\sigma }^{\vee \vee }=\sigma$ to identify valuative-criterion extension with cone-membership, exactly as proved here. The cone-and-dual-cone formalism is the combinatorial input on which both criteria run.

• Toric resolution of singularities 04.11.07. The downstream resolution unit operates entirely on the cone formalism developed here: star subdivision of a non-unimodular cone $\sigma$ inserts a new ray $v$ in the relative interior of $\sigma$ and replaces $\sigma$ by the cones generated by $v$ together with each maximal proper face of $\sigma$. The face-correspondence theorem $(\sigma \cap \tau {)}^{\vee }={\sigma }^{\vee }+{\tau }^{\vee }$ ensures the refined collection still satisfies the fan axioms, and Gordan's lemma guarantees each new cone produces an affine variety of finite type. The toric resolution algorithm is, in this sense, a finite procedure on the cone primitives of this unit.

• Resolution of singularities 04.06.02. Demazure's smoothness criterion (simplicial unimodular cones produce smooth $U_{\sigma }$) is the toric input to resolution of singularities for toric varieties: every singular toric variety can be resolved by refining its fan into simplicial unimodular subcones, a process called stellar subdivision or fan refinement. This is the toric analogue of Hironaka's resolution theorem, and it is combinatorially explicit — entirely controlled by the lattice and cone structure of this unit. The prerequisite unit on resolution supplies the geometric framework into which the toric resolution slots.

• Hilbert basis theorem 02.04.01. The Hilbert basis theorem in commutative algebra (every ideal in a Noetherian ring is finitely generated) is the algebraic backbone of finite-type schemes; Gordan's lemma is its monoid-flavoured shadow, restricted to the additive semigroup ${\sigma }^{\vee }\cap M$. The two together — Gordan's lemma for the monoid, Hilbert's basis theorem for the ring — guarantee that $\mathbb{C}[{\sigma }^{\vee }\cap M]$ is a Noetherian commutative $\mathbb{C}$-algebra of finite type, hence the coordinate ring of an affine variety.

• Convex polytope 02.05.04. A convex polytope is the bounded analogue of a polyhedral cone — the convex hull of finitely many points, equivalently the intersection of finitely many half-spaces of bounded extent. Every polytope $P$ has an associated cone $C(P)=\mathrm{Cone}(P\times \{1\})\subseteq {\mathbb{R}}^{n+1}$ (the homogenisation of $P$), and the cone formalism of this unit specialises to polytope formalism via this construction. Polytope geometry (Minkowski's Geometrie der Zahlen, 1896) is the historical and conceptual root of the rational-cone formalism developed here, and the polytope-toric-correspondence of 04.11.11 is the bounded version of the cone-toric correspondence developed here and in the next units.

• Newton polytope and non-archimedean amoeba 04.12.04. The downstream tropical-geometry unit takes the Newton polytope $\mathrm{Newt}(f)=\mathrm{Conv}(\mathrm{supp}(f))\subseteq M_{\mathbb{R}}$ of a Laurent polynomial $f$ as a bounded rational polyhedron in $M_{\mathbb{R}}$ — exactly the convex-polytope specialisation of the rational-cone formalism of this unit. The normal fan $\Sigma (\mathrm{Newt}(f))\subseteq N_{\mathbb{R}}$ assembled from inner-normal cones of faces of the Newton polytope is the rational-polyhedral-fan instance of the cone-and-fan formalism, and the tropical hypersurface $\mathrm{trop}(f)\subseteq N_{\mathbb{R}}$ is a polyhedral subcomplex of the support of this normal fan. The cone-duality theorem ${\sigma }^{\vee \vee }=\sigma$ of the present unit is the foundational reason that the Newton-polytope-to-tropical-hypersurface duality is exact: cells of one side correspond to faces of the other under the natural pairing.

Historical & philosophical context [Master]

The rational polyhedral cone formalism rests on three nineteenth- and early-twentieth-century results from the analytic theory of linear inequalities. Hermann Minkowski's Geometrie der Zahlen (Leipzig: Teubner, 1896) ^{[Minkowski 1896]} introduced the systematic theory of lattices in ${\mathbb{R}}^n$ and convex bodies, proving in particular the linear-forms theorem (Ch. II §17) that anchors all subsequent rational-polyhedral-cone arguments — the existence of nonzero lattice points in symmetric convex bodies of sufficient volume. Minkowski's program was completed for unbounded polyhedra (polyhedral cones) by Paul Gordan's Ueber die Auflösung linearer Gleichungen mit reell ganzzahligen Coefficienten (Mathematische Annalen 6, 1873, pp. 23-28) ^{[Gordan 1873]} — the originator paper for the finite-generation theorem on solutions of linear inequalities in nonnegative integers, which Hilbert later reformulated as the existence of a Hilbert basis for the integer points in ${\sigma }^{\vee }$. Julius Farkas's Theorie der einfachen Ungleichungen (Journal für die reine und angewandte Mathematik 124, 1902, pp. 1-27) ^{[Farkas 1902]} established the linear-inequalities duality theorem now bearing his name: a vector $b$ is a nonnegative combination of vectors $a_1,\dots ,a_k$ if and only if every linear functional nonnegative on the $a_i$ is nonnegative on $b$. Farkas's lemma is the analytic substrate of cone duality ${\sigma }^{\vee \vee }=\sigma$, and is essentially equivalent to the finite-dimensional separating-hyperplane theorem used in the proof above.

Hermann Weyl's 1935 paper Elementare Theorie der konvexen Polyeder (Commentarii Mathematici Helvetici 7, pp. 290-306) ^[pending] completed the Minkowski program by proving the converse direction: every polyhedron defined by finitely many linear inequalities is the convex hull of a finite set of points plus a finitely generated cone. The combined Minkowski-Weyl theorem (1896-1935) is the central duality result of finite-dimensional convex polyhedral geometry. In the 1950s, the development of linear programming by George Dantzig and the theory of convex polytopes by Branko Grünbaum (Convex Polytopes, 1967) ^[pending] put the polyhedral-cone formalism on a modern footing, and the algorithmic computation of Hilbert bases (the Pottier algorithm, the Bachem-Schrijver algorithm, the Sturmfels-Maclagan algorithm) became practical for the toric-geometry community.

The toric-geometry application — using rational polyhedral cones in $N_{\mathbb{R}}$ to encode equivariant affine compactifications of the algebraic torus $T=({\mathbb{G}}_m{)}^n$ — was opened by Michel Demazure in Sous-groupes algébriques de rang maximum du groupe de Cremona (Annales scientifiques de l'École normale supérieure, 4e série, tome 3, 1970, pp. 507-588) ^{[Demazure 1970]}. Demazure's smoothness criterion (the simplicial-unimodular condition) and his use of cones and fans to classify rank-$n$ algebraic subgroups of the Cremona group are the foundational papers of toric geometry. The combinatorial nomenclature "toric variety" became standard after the Mumford-school treatment in Toroidal Embeddings I (Kempf, Knudsen, Mumford, Saint-Donat, Lecture Notes in Mathematics 339, Springer 1973) ^[pending], where the cone-and-fan formalism is recast as a tool for resolving singularities and compactifying equivariant varieties. Tadao Oda's Convex Bodies and Algebraic Geometry (Springer 1988) ^[pending] consolidated the Japanese-school perspective on rational polyhedral cones, with emphasis on the dual-cone formalism and Gordan's lemma as the foundation of the affine-toric-variety construction; William Fulton's Introduction to Toric Varieties (Princeton 1993) ^[pending] is the canonical short textbook, and David Cox, John Little, and Henry Schenck's Toric Varieties (AMS 2011) ^[pending] is the modern thousand-page treatment with full treatment of the lattice-cone-fan-variety dictionary.

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