Smoothness and completeness via fans

Intuition [Beginner]

A toric variety $X_{\Sigma }$ is built from a fan $\Sigma$ — a finite collection of cones in $N_{\mathbb{R}}$ that fit together compatibly. The variety inherits its geometry from the cones: every property of $X_{\Sigma }$ shows up as a combinatorial property of $\Sigma$. Two of the most important geometric properties are smoothness (the variety has no singular points, the local picture is a nice product of affine lines and units) and completeness (the variety is compact, the variety has no missing points at infinity). Both are governed by a single combinatorial input, and the dictionary is exact.

The smoothness rule is short. A cone is smooth when the primitive generators of its rays extend to a basis of the lattice $N$. The variety $X_{\Sigma }$ is then smooth when every cone in the fan is smooth. The intuition is that each cone produces one affine chart of the variety, the chart is a copy of affine space ${\mathbb{C}}^k$ times a torus piece exactly when the cone is smooth, and the whole variety is smooth when every chart is.

The completeness rule is even shorter. The support of the fan is the union of all its cones inside $N_{\mathbb{R}}$. The variety $X_{\Sigma }$ is complete (compact in the classical sense, proper over the base field) when this support is all of $N_{\mathbb{R}}$. The intuition is that each cone records one direction the variety has been compactified in; if the cones cover every direction, the variety is fully compact, and if a direction is missing, the variety has a hole at infinity in that direction.

Putting these together, the four classical examples assemble. Projective space ${\mathbb{P}}^n$ comes from a fan that is both smooth and complete — every chart is ${\mathbb{C}}^n$ and the cones cover ${\mathbb{R}}^n$. Weighted projective space $\mathbb{P}(1,2,3)$ comes from a fan that is complete but not smooth — the cones cover the plane but one chart has a singularity. The affine line $\mathbb{C}$ comes from a fan that is smooth but not complete — one chart is fine but a whole direction at infinity is missing. The dictionary between fans and varieties makes each of these read off from a finite combinatorial check.

Visual [Beginner]

A four-panel chart showing each combination of smooth/non-smooth and complete/non-complete in dimension two. Upper-left: the fan of ${\mathbb{P}}^2$ in ${\mathbb{R}}^2$, three rays toward the standard basis directions and one negative-sum direction, three maximal cones colouring all of ${\mathbb{R}}^2$, every cone unimodular — smooth and complete. Upper-right: the fan of weighted projective space $\mathbb{P}(1,2,3)$ in ${\mathbb{R}}^2$, three rays with primitive generators making non-unimodular cones, three maximal cones still covering ${\mathbb{R}}^2$ — complete but not smooth, with cyclic quotient singularities at the three torus-fixed points.

Lower-left: the fan of the affine line $\mathbb{C}$ inside $\mathbb{R}$, one ray pointing in the positive direction and the zero cone, smooth (the ray is a basis vector) but the support is only the half-line — smooth but not complete, missing the negative direction at infinity. Lower-right: a wedge of two non-unimodular cones in ${\mathbb{R}}^2$ not covering the plane — neither smooth nor complete, the worst case, used in the proof that some combination is needed.

The picture captures the two independent axes of the toric dictionary: smoothness is a per-cone unimodularity check, completeness is a global support-covers-everything check. Both can hold, neither can hold, or exactly one can hold; the four cases each have a textbook example in dimension two.

Worked example [Beginner]

Check that projective space ${\mathbb{P}}^2$ is both smooth and complete by reading off the smoothness and completeness criteria from its fan.

Step 1. Write down the fan of ${\mathbb{P}}^2$. Take $N={\mathbb{Z}}^2$ with $N_{\mathbb{R}}={\mathbb{R}}^2$. The fan $\Sigma$ has three rays with primitive generators $u_0=-e_1-e_2$, $u_1=e_1$, $u_2=e_2$. The maximal cones are the three two-dimensional wedges spanned by pairs of these rays.

Step 2. Check smoothness, cone by cone. For each maximal cone, compute the determinant of the matrix whose rows are the primitive ray generators. For the cone spanned by $u_1,u_2$, the matrix is the identity, determinant $1$. For the cone spanned by $u_1,u_0=e_1,-e_1-e_2$, the matrix is $\left[\begin{array}{cc}1& 0\\ -1& -1\end{array}\right]$, determinant $-1$. For the cone spanned by $u_2,u_0$, the matrix is $\left[\begin{array}{cc}0& 1\\ -1& -1\end{array}\right]$, determinant $1$. All three determinants are $\pm 1$, so every cone is unimodular — by definition smooth.

Step 3. Check completeness, by adding up the cones. The three maximal cones partition ${\mathbb{R}}^2$ into three wedges meeting only along rays and at the origin. The union of the three wedges is all of ${\mathbb{R}}^2$, so $|\Sigma |={\mathbb{R}}^2$ and the fan is complete.

Step 4. Conclude. By the smoothness criterion, ${\mathbb{P}}^2$ is a smooth variety. By the completeness criterion, ${\mathbb{P}}^2$ is a complete (compact) variety. Combining, ${\mathbb{P}}^2$ is a smooth complete toric surface — the simplest substantive example, and the prototype for the entire theory.

What this tells us. The two criteria are independent finite checks on a fan: count the determinants for smoothness, sum the cones to a region for completeness. A toric variety is smooth and complete when both checks pass; failing one or the other gives the other three boxes of the four-panel chart. The cone-by-cone determinant check is the universal model for testing smoothness throughout toric geometry.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $N$ be a lattice of rank $n$ with dual lattice $M={\mathrm{Hom}}_{\mathbb{Z}}(N,\mathbb{Z})$, as in [04.11.01]. Let $N_{\mathbb{R}}=N{\otimes }_{\mathbb{Z}}\mathbb{R}$.

Definition (smooth cone). A strongly convex rational polyhedral cone $\sigma \subseteq N_{\mathbb{R}}$ with primitive ray generators $u_1,\dots ,u_k\in N$ is smooth (also called regular or unimodular) when $\{u_1,\dots ,u_k\}$ extends to a $\mathbb{Z}$-basis of $N$. Equivalently, the $k\times n$ matrix with rows $u_1,\dots ,u_k$ extends to an $n\times n$ matrix of determinant $\pm 1$. Equivalently again, when $\sigma$ is simplicial of dimension $k$ (the $u_i$ are $\mathbb{Z}$-linearly independent and number exactly $k$), $\sigma$ is smooth iff the index $[N_{\sigma }:\mathbb{Z}⟨u_1,\dots ,u_k⟩]=1$, where $N_{\sigma }=N\cap {\mathrm{span}}_{\mathbb{R}}(\sigma )$ is the saturation of the $\mathbb{Z}$-span inside the linear hull.

Definition (smooth fan). A fan $\Sigma$ in $N_{\mathbb{R}}$ is smooth when every cone $\sigma \in \Sigma$ is smooth. By the face-closure axiom of [04.11.04], smoothness of every maximal cone suffices, because every face of a smooth cone is smooth (a subset of a $\mathbb{Z}$-basis is itself part of a $\mathbb{Z}$-basis).

Definition (complete fan). The support of a fan $\Sigma$ is $|\Sigma |:={\bigcup }_{\sigma \in \Sigma }\sigma \subseteq N_{\mathbb{R}}$. The fan is complete when $|\Sigma |=N_{\mathbb{R}}$.

Definition (simplicial fan). A fan $\Sigma$ is simplicial when every cone $\sigma \in \Sigma$ is simplicial (its primitive ray generators are $\mathbb{R}$-linearly independent and so number exactly the cone dimension). Smooth implies simplicial; the converse fails when ray generators are linearly independent but generate a sublattice of index larger than $1$ inside $N_{\sigma }$.

The four conditions — strongly convex, rational polyhedral, simplicial, smooth — form a chain of increasing strength: strongly convex rational polyhedral cones are the input data of the fan, simplicial cones rule out the "non-isolated" simplicial degeneracies, and smooth cones rule out the cyclic quotient singularities that simplicial-but-not-smooth cones produce.

Counterexamples to common slips

• "Simplicial implies smooth." No. The cone $\sigma =\mathrm{Cone}(e_1,2e_1+3e_2)$ in ${\mathbb{R}}^2$ is simplicial (its two generators are $\mathbb{R}$-linearly independent), but the determinant of the primitive matrix $\left[\begin{array}{cc}1& 0\\ 2& 3\end{array}\right]$ equals $3\ne \pm 1$, so the cone is not smooth. The corresponding affine toric variety $U_{\sigma }$ has a cyclic quotient singularity of type $(1/3)(1,2)$ at the origin.

• "Smooth implies complete." No, completeness and smoothness are independent. The affine line $\mathbb{C}$ is smooth but not complete (single-ray fan with support ${\mathbb{R}}_{\ge 0}$); the weighted projective space $\mathbb{P}(1,2,3)$ is complete but not smooth (three-ray fan covering ${\mathbb{R}}^2$ with non-unimodular cones).

• "Every cone of $\Sigma$ being smooth suffices for $X_{\Sigma }$ smooth even without considering the gluing." This holds because smoothness is local — the affine charts $U_{\sigma }$ are glued along open immersions, and a property local on a scheme is inherited under gluing. The smoothness criterion does not require any global compatibility beyond "every maximal cone is smooth."

• "The completeness of $\Sigma$ depends on the lattice $N$." It does not. The condition $|\Sigma |=N_{\mathbb{R}}$ depends only on the support of the cones in the real vector space, and is invariant under lattice change-of-basis. What does depend on $N$ is smoothness: a cone may be smooth with respect to one lattice and not another if the sublattice generated by ray generators has different index.

Key theorem with proof [Intermediate+]

The signature theorem of this unit is the joint statement of the two criteria, with a unified proof structure: smoothness is local (so a per-cone check), completeness is global (so a support-covers-all-of-$N_{\mathbb{R}}$ check via the valuative criterion).

Theorem (smoothness and completeness criteria; Demazure 1970, Fulton 1993). Let $\Sigma$ be a fan in $N_{\mathbb{R}}$, and let $X_{\Sigma }$ be the toric variety it produces (as in [04.11.04]).

(a) Affine smoothness. For a strongly convex rational polyhedral cone $\sigma$ with primitive ray generators $u_1,\dots ,u_k$, the affine toric variety $U_{\sigma }$ is smooth as a $\mathbb{C}$-scheme iff $\sigma$ is smooth (i.e. $\{u_1,\dots ,u_k\}$ is part of a $\mathbb{Z}$-basis of $N$).

(b) Global smoothness. $X_{\Sigma }$ is smooth iff every cone $\sigma \in \Sigma$ is smooth.

(c) Completeness criterion. $X_{\Sigma }$ is complete (proper over $\mathrm{Spec}\mathbb{C}$, equivalently compact in the classical complex topology) iff $|\Sigma |=N_{\mathbb{R}}$.

Proof.

(a) Affine smoothness, the $(\Leftarrow )$ direction. Suppose $\sigma$ is smooth, with primitive ray generators $u_1,\dots ,u_k$ extending to a $\mathbb{Z}$-basis $u_1,\dots ,u_n$ of $N$. Let $u_1^{\ast },\dots ,u_n^{\ast }$ be the dual basis of $M$. Then the dual cone is $$ \sigma^\vee = \mathrm{Cone}(u_1^, \ldots, u_k^, \pm u_{k+1}^, \ldots, \pm u_n^), $$ because $⟨u_i^{\ast },u_j⟩={\delta }_{ij}$ so $u_i^{\ast }\ge 0$ on $\sigma$ exactly for $i\le k$, and $u_j^{\ast }$ for $j>k$ pairs to zero with every $u_i$, $i\le k$, hence lies in ${\sigma }^{\perp }\subseteq {\sigma }^{\vee }$ together with its negative.

The semigroup is $$ S_\sigma = \sigma^\vee \cap M = \mathbb{Z}{\geq 0} u_1^* \oplus \cdots \oplus \mathbb{Z}{\geq 0} u_k^* \oplus \mathbb{Z} u_{k+1}^* \oplus \cdots \oplus \mathbb{Z} u_n^*, $$ the free commutative monoid on $k$ generators direct-summed with the free abelian group on $n-k$ generators. The semigroup algebra is therefore $$ \mathbb{C}[S_\sigma] = \mathbb{C}[x_1, \ldots, x_k, x_{k+1}^{\pm 1}, \ldots, x_n^{\pm 1}], $$ a Laurent polynomial ring in $n-k$ of the variables and an ordinary polynomial ring in the other $k$. This is a regular ring (polynomial rings are regular, localisations of regular rings are regular). Hence $U_{\sigma }=\mathrm{Spec}\mathbb{C}[S_{\sigma }]\cong {\mathbb{A}}^k\times {\mathbb{G}}_m^{n-k}$ is smooth as a $\mathbb{C}$-scheme.

(a) Affine smoothness, the $(\Rightarrow )$ direction. Suppose $U_{\sigma }$ is smooth. The unique torus-fixed point of $U_{\sigma }$ is the point corresponding to the maximal ideal ${\mathfrak{m}}_{\sigma }=({\chi }^m:m\in S_{\sigma }\setminus \{0\})\subseteq \mathbb{C}[S_{\sigma }]$. Smoothness at this point requires the local ring $\mathbb{C}[S_{\sigma }{]}_{{\mathfrak{m}}_{\sigma }}$ to be regular, which is equivalent to the cotangent space ${\mathfrak{m}}_{\sigma }/{\mathfrak{m}}_{\sigma }^2$ having dimension equal to the Krull dimension of $\mathbb{C}[S_{\sigma }]$, namely $n$.

The cotangent space has a basis indexed by the Hilbert basis of the semigroup $S_{\sigma }$ — the irreducible elements of $S_{\sigma }$ (those not expressible as a sum of two non-zero elements of $S_{\sigma }$). Hence ${\dim }_{\mathbb{C}}{\mathfrak{m}}_{\sigma }/{\mathfrak{m}}_{\sigma }^2=|\mathrm{Hilb}(S_{\sigma })|$. Smoothness forces this to equal $n$, the Krull dimension.

Now, the Hilbert basis of $S_{\sigma }$ for a smooth cone is the dual-basis vectors $u_1^{\ast },\dots ,u_n^{\ast }$ (with $u_j^{\ast }$ for $j>k$ counted twice, once with each sign, but only one of each contributes a substantive generator since the lineality factor contributes a free group). More carefully, when the cone $\sigma$ has dimension $k$, the Hilbert basis has exactly $n$ elements iff $\sigma$ is smooth — the dual semigroup is freely generated on the $k$ vectors $u_1^{\ast },\dots ,u_k^{\ast }$ together with the $n-k$ vectors $u_{k+1}^{\ast },\dots ,u_n^{\ast }$ that generate the lineality space. The combinatorial argument is: the Hilbert basis equals the number of facets of ${\sigma }^{\vee }$ plus its lineality rank, which equals $n$ iff the facets are dual to the rays of $\sigma$ in bijection — which happens exactly when $\sigma$ is simplicial and the ray generators form part of a $\mathbb{Z}$-basis. Hence smoothness of $U_{\sigma }$ implies smoothness of $\sigma$.

(b) Global smoothness. Smoothness of a scheme is a local property: $X_{\Sigma }$ is smooth iff every point has a smooth open neighbourhood. The affine open cover $\{U_{\sigma }:\sigma \in {\Sigma }_{\max }\}$ of $X_{\Sigma }$ provided by [04.11.04] is such a cover. Hence $X_{\Sigma }$ is smooth iff every $U_{\sigma }$ is smooth, iff every $\sigma \in {\Sigma }_{\max }$ is smooth (by part (a)), iff every $\sigma \in \Sigma$ is smooth (since faces of smooth cones are smooth, the maximal cones suffice).

(c) Completeness criterion. This is the valuative-criterion argument detailed in [04.11.04]. We rerun it cleanly.

We apply the valuative criterion for properness: $X_{\Sigma }$ is proper over $\mathrm{Spec}\mathbb{C}$ iff for every discrete valuation ring $R$ with fraction field $K$ and every $\mathbb{C}$-morphism $\mathrm{Spec}K\to X_{\Sigma }$, there is a unique extension to $\mathrm{Spec}R\to X_{\Sigma }$.

Step 1. A $\mathbb{C}$-morphism $\mathrm{Spec}K\to T\subseteq X_{\Sigma }$ factoring through the open dense torus $T=U_{\{0\}}$ corresponds to a $\mathbb{C}$-algebra map $\mathbb{C}[M]\to K$, equivalently a group homomorphism $\varphi :M\to K^{\ast }$. The discrete valuation $v_R:K^{\ast }\to \mathbb{Z}$ composed with $\varphi$ gives $v_{\varphi }:=v_R\circ \varphi :M\to \mathbb{Z}$, an element $v_{\varphi }\in N={\mathrm{Hom}}_{\mathbb{Z}}(M,\mathbb{Z})$.

Step 2. The morphism extends to $\mathrm{Spec}R\to X_{\Sigma }$ iff it factors through some affine chart $U_{\sigma }$ at the level of $R$, iff $\varphi ({\chi }^m)\in R$ for every $m\in S_{\sigma }={\sigma }^{\vee }\cap M$, iff $v_R(\varphi ({\chi }^m))=⟨m,v_{\varphi }⟩\ge 0$ for every $m\in {\sigma }^{\vee }$, iff $v_{\varphi }\in {\sigma }^{\vee \vee }=\sigma$ (cone duality). Hence the extension exists iff $v_{\varphi }\in \sigma$ for some $\sigma \in \Sigma$, iff $v_{\varphi }\in |\Sigma |$.

Step 3. As $R$ ranges over DVRs containing $\mathbb{C}$ and $\varphi$ ranges over $\mathbb{C}$-morphisms $\mathrm{Spec}K\to T$, the resulting cocharacter $v_{\varphi }$ ranges over all elements of $N$: every $v\in N$ arises from the homomorphism $\varphi (m)=t^{⟨m,v⟩}$ inside the DVR $R=\mathbb{C}[[t]]$ with fraction field $K=\mathbb{C}((t))$, v_R = \text{order-of-vanishing-at-t = 0}.

Step 4. The valuative criterion holds for every torus-factored point iff every $v\in N$ lies in $|\Sigma |$, iff $|\Sigma |\supseteq N$. Since the cones of $\Sigma$ are closed in $N_{\mathbb{R}}$ and the integer points $N\subseteq N_{\mathbb{R}}$ are dense, this is equivalent to $|\Sigma |=N_{\mathbb{R}}$.

Step 5. Uniqueness of the extension follows from the separatedness of $X_{\Sigma }$, established in [04.11.04]. For morphisms not factoring through the open torus, the morphism factors through the closure of some torus orbit (a smaller toric subvariety) and an induction on the orbit stratification reduces to the torus-factored case.

Step 6. Conversely, if $|\Sigma |\ne N_{\mathbb{R}}$, pick $v\in N\setminus |\Sigma |$. The morphism $\varphi (m)=t^{⟨m,v⟩}$ defines a $\mathbb{C}((t))$-point of $T$, but does not extend to a $\mathbb{C}[[t]]$-point of $X_{\Sigma }$ (since the extension would have to land in some chart $U_{\sigma }$, requiring $v\in \sigma$, contradicting $v\notin |\Sigma |$). Hence the valuative criterion fails and $X_{\Sigma }$ is not proper.

Combining steps 1 through 6, $X_{\Sigma }$ is proper iff $|\Sigma |=N_{\mathbb{R}}$, which (for a finite-type $\mathbb{C}$-scheme) is equivalent to compactness in the classical $\mathbb{C}$-topology. $\square$

Bridge. The smoothness and completeness criteria build toward the orbit-cone correspondence in [04.11.06] and the toric resolution of singularities in [04.11.07], and they appear again in [04.11.10] where the projectivity criterion strengthens completeness with the additional condition that $\Sigma$ is the normal fan of a lattice polytope. The foundational reason the criteria are this clean is that the fan-to-toric construction is functorial cone by cone: smoothness is a local question (per cone) and completeness is a global question (covering all of $N_{\mathbb{R}}$), and the fan formalism separates the two cleanly. This is exactly the bridge from convex geometry to algebraic geometry: a per-cone unimodularity check controls regular-local-ring conditions on the variety, and a global support-covers-everything check controls the valuative criterion of properness. Putting these together, every smooth complete toric variety is classified by a smooth complete fan, and identifies the category of smooth complete toric varieties with the category of smooth complete fans, generalising the classical correspondence between ${\mathbb{P}}^n$ and its standard fan.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Mathlib has the regular-ring infrastructure of Mathlib.RingTheory.RegularRing and the scheme-theoretic smoothness API of Mathlib.AlgebraicGeometry.Smooth. Mathlib also has the abstract valuative criterion via Mathlib.RingTheory.Valuation.ValuationRing and proper-morphism API in Mathlib.AlgebraicGeometry.Properness. Building the smoothness criterion and the completeness criterion on top of these requires (i) the rational-polyhedral-cone API from [04.11.02] (currently absent from Mathlib), (ii) the affine-toric-variety construction from [04.11.03], and (iii) the fan-and-toric-variety gluing from [04.11.04]. The intended formalisation reads schematically:

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

namespace Codex.AlgGeom.Toric

-- Lattice, cones, and fans as placeholder structures (full definitions
-- in `Codex.AlgGeom.Toric.AffineToricVarietyUSigma` and
-- `Codex.AlgGeom.Toric.FanAndToricVariety`).

structure Lattice where
  rank : ℕ
  is_free : True

structure RationalPolyhedralCone (N : Lattice) where
  rays : List (PrimitiveLatticeVector N)
  strongly_convex : True

def RationalPolyhedralCone.Smooth {N} (σ : RationalPolyhedralCone N) :
    Prop := σ.Simplicial ∧ True
  -- Primitive generators extend to a Z-basis of N.

structure Fan (N : Lattice) where
  cones : List (RationalPolyhedralCone N)
  faceClosed : True
  intersectionFace : True

def Fan.IsComplete {N} (Σ : Fan N) : Prop :=
  True  -- placeholder for |Σ| = N_R

def Fan.IsSmooth {N} (Σ : Fan N) : Prop :=
  ∀ σ ∈ Σ.cones, σ.Smooth

-- Affine and global smoothness criteria as named theorems.

theorem affineToricVariety_smooth_iff_cone_smooth {N}
    {σ : RationalPolyhedralCone N} (X : AffineToricVariety σ) :
    X.IsSmooth ↔ σ.Smooth := by
  -- (=>) Hilbert basis of S_σ has n elements iff primitive ray
  -- generators of σ form a Z-basis of N_σ.
  -- (<=) Choose dual basis u_1^*, ..., u_n^*; semigroup algebra is
  -- C[x_1, ..., x_k, x_{k+1}^±, ..., x_n^±], a regular ring.
  sorry

theorem toricVariety_smooth_iff_all_cones_smooth {N}
    {Σ : Fan N} (X : ToricVariety Σ) :
    X.IsSmooth ↔ ∀ σ ∈ Σ.cones, σ.Smooth := by
  -- Smoothness is local on a scheme; affine charts of X_Σ are the
  -- U_σ, smooth iff σ smooth by the affine theorem.
  sorry

theorem toricVariety_complete_iff_support_full {N}
    {Σ : Fan N} (X : ToricVariety Σ) :
    X.IsComplete ↔ Σ.IsComplete := by
  -- Valuative criterion applied to torus-valued points:
  -- (Spec K → T) ↝ (M → K^*) ↝ (M → ℤ) =: cocharacter v ∈ N;
  -- extension exists iff v ∈ |Σ|; ranges over all of N as data
  -- varies; closure of N is N_R, so the criterion is |Σ| = N_R.
  sorry

end Codex.AlgGeom.Toric

Each step is reachable from current Mathlib but requires substantive new development. The rational-polyhedral-cone formalism is the primary missing prerequisite; once that is in place together with the affine-toric-variety construction, the smoothness criterion reduces to the regular-ring decomposition of the semigroup algebra (computable via AddMonoidAlgebra), and the completeness criterion follows from the valuative criterion for properness applied to torus-valued points (via ValuationRing). The placeholder module Codex.AlgGeom.Toric.SmoothnessCompletenessFans declares the two theorems with sorry-stubbed proof bodies so downstream files (orbit-cone correspondence, toric resolution, divisor-fan correspondence) can reference the named statements.

Advanced results [Master]

Smoothness criterion via the cone's generator data

Theorem (smoothness criterion; Demazure 1970, Fulton 1993 §2.1). Let $\sigma \subseteq N_{\mathbb{R}}$ be a strongly convex rational polyhedral cone with primitive ray generators $u_1,\dots ,u_k\in N$, where $k=\dim \sigma$. The affine toric variety $U_{\sigma }$ is smooth as a $\mathbb{C}$-scheme iff $\{u_1,\dots ,u_k\}$ extends to a $\mathbb{Z}$-basis of $N$. Equivalently, $\sigma$ is smooth iff (i) $\sigma$ is simplicial (the $u_i$ are $\mathbb{Z}$-linearly independent), and (ii) the index $[N_{\sigma }:\mathbb{Z}⟨u_1,\dots ,u_k⟩]=1$ inside the saturation $N_{\sigma }=N\cap {\mathrm{span}}_{\mathbb{R}}(\sigma )$.

The two equivalent formulations record different geometric perspectives. The first ("extends to a $\mathbb{Z}$-basis") emphasises the algebraic structure of the lattice — the dual basis $u_1^{\ast },\dots ,u_n^{\ast }$ exists, and the semigroup $S_{\sigma }$ decomposes as a free direct sum of monoid and group parts. The second ("simplicial + unit index") emphasises the lattice-index obstruction: even when the cone has the correct combinatorial shape (simplicial, dimension $k$), the primitive generators may span only a sub-lattice of $N_{\sigma }$, the index being the order of the cyclic quotient singularity at the torus-fixed point. The two perspectives meet at the Smith normal form of the matrix $[u_1,\dots ,u_k{]}^T$: the smooth case is when the Smith normal form has all elementary divisors equal to $1$, while a non-smooth simplicial cone has Smith normal form with at least one elementary divisor $>1$, that divisor recording the order of the quotient.

The history of the smoothness criterion is short and clean. Michel Demazure introduced it as Theorem 4 of his 1970 paper 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), where the criterion is stated as the condition for the affine toric variety to be "unimodular" — terminology that has shifted in modern usage to "smooth" or "regular." Fulton 1993 §2.1 codified the criterion as it appears in contemporary toric-geometry textbooks; Cox-Little-Schenck 2011 Proposition 1.3.18 gives the textbook-standard proof via the dual-basis decomposition of the semigroup algebra.

Completeness criterion via the fan support

Theorem (completeness criterion; Fulton 1993 §2.4, Cox-Little-Schenck 2011 Theorem 3.4.6). Let $\Sigma$ be a fan in $N_{\mathbb{R}}$. The toric variety $X_{\Sigma }$ is complete (proper over $\mathrm{Spec}\mathbb{C}$, equivalently compact in the classical complex topology) iff $|\Sigma |=N_{\mathbb{R}}$.

The completeness criterion identifies properness — a global cohomological condition (every coherent sheaf has finite-dimensional cohomology) — with a single combinatorial check on the fan's support. The proof is the valuative-criterion argument detailed above and in Exercise 8: a cocharacter $v\in N$ extracted from a discrete valuation must lie in some cone of $\Sigma$, and ranging over all DVRs sweeps out all of $N$, so properness is equivalent to $|\Sigma |\supseteq N$, equivalent to $|\Sigma |=N_{\mathbb{R}}$ by closure-of-$N$-equals-$N_{\mathbb{R}}$.

The criterion has three immediate consequences. First, the dimension hierarchy: the support $|\Sigma |\subseteq N_{\mathbb{R}}$ is a closed convex subset, and completeness is the maximal case where this subset fills the whole real vector space. Second, the maximal-cone count: a complete fan in $N_{\mathbb{R}}$ of rank $n$ has at least $n+1$ maximal cones of dimension $n$ (since fewer cones cannot fill all of $N_{\mathbb{R}}$ — the smallest complete fan in ${\mathbb{R}}^n$ is the fan of ${\mathbb{P}}^n$ with $n+1$ rays and $n+1$ maximal cones). Third, the GAGA correspondence: for $\mathbb{C}$-schemes of finite type, properness is equivalent to compactness in the classical topology, so the algebraic completeness criterion coincides with the topological compactness of $X_{\Sigma }^{an}$ as a complex analytic space.

Worked example: weighted projective space $\mathbb{P}(1,2,3)$

The weighted projective space $\mathbb{P}(a,b,c)=({\mathbb{C}}^3\setminus \{0\})/{\mathbb{C}}^{\ast }$ where ${\mathbb{C}}^{\ast }$ acts with weights $(a,b,c)$ is a complete toric variety when $\gcd (a,b,c)=1$, with simplicial fan whose maximal cones are determined by the weight data. For $(a,b,c)=(1,2,3)$, the fan in ${\mathbb{R}}^2$ has three primitive ray generators $v_0,v_1,v_2$ satisfying the weight relation $a{v}_0+b{v}_1+c{v}_2=0$.

The three torus-fixed points of $\mathbb{P}(1,2,3)$ are $[1:0:0]$, $[0:1:0]$, $[0:0:1]$, corresponding to the three maximal cones. The local geometry near each is determined by the isotropy subgroup of ${\mathbb{C}}^{\ast }$ at that point — the subgroup fixing the homogeneous coordinate that does not vanish there. At $[1:0:0]$ the isotropy is the identity group (weight $1$); at $[0:1:0]$ the isotropy is $\mathbb{Z}/2$ (weight $2$); at $[0:0:1]$ the isotropy is $\mathbb{Z}/3$ (weight $3$).

The local geometry of the variety near a torus-fixed point with isotropy $\mathbb{Z}/n$ is a cyclic quotient singularity of type $(1/n)(a_1,a_2)$ for some integers $a_1,a_2$ coprime to $n$ — the quotient of ${\mathbb{C}}^2$ by the action $(x,y)\mapsto (\zeta x,{\zeta }^{a_2/a_1}y)$ where $\zeta$ is a primitive $n$th root of unity. For $\mathbb{P}(1,2,3)$, the singularity types are:

• $[1:0:0]$: smooth (identity isotropy).
• $[0:1:0]$: $(1/2)(1,1)$ — the simplest substantive cyclic quotient, also known as the $A_1$ singularity or the ordinary double point, locally analytically the cone over a ${\mathbb{P}}^1$.
• $[0:0:1]$: $(1/3)(1,2)$ — a $\mathbb{Z}/3$ quotient, related to the $A_2$ singularity by a slightly different lattice realisation.

The variety $\mathbb{P}(1,2,3)$ is $\mathbb{Q}$-factorial in Mori's sense (Mori 1982 Annals of Mathematics 116): every Weil divisor becomes Cartier after multiplying by a positive integer (the integer being the LCM of the isotropy orders, here $\mathrm{lcm}(1,2,3)=6$). The fan is simplicial — each maximal cone has exactly two primitive ray generators in ${\mathbb{R}}^2$ — but not unimodular (the determinants of two of the three primitive matrices have absolute value $>1$).

The variety is the simplest substantive example of a toric $\mathbb{Q}$-factorial variety with cyclic quotient singularities, and it is a test case for the toric minimal model program (Reid 1980, Reid 1987, Cox-Little-Schenck Ch. 11). Its resolution of singularities is achieved by toric blow-up at the singular fixed points — adding an extra ray in each non-smooth cone subdivides the cone into smaller cones whose primitive matrices become unimodular. This is the toric realisation of Hirzebruch-Jung continued-fraction resolution (Hirzebruch 1953), and connects to the toric resolution theorem of [04.11.07].

Hirzebruch surfaces $F_a$ and their fans

The Hirzebruch surface $F_a$ (sometimes also called ${\Sigma }_a$ in older literature, with no relation to fan-notation $\Sigma$) is the smooth complete toric surface defined as the ${\mathbb{P}}^1$-bundle $$ F_a = \mathbb{P}(\mathcal{O}{\mathbb{P}^1} \oplus \mathcal{O}{\mathbb{P}^1}(a)) $$ over ${\mathbb{P}}^1$, where the parameter $a\ge 0$ is an integer controlling the twist. Hirzebruch introduced these surfaces in 1951 (Mathematische Annalen 124, 77-86 ^{[Hirzebruch1951]}) as the canonical classification of ${\mathbb{P}}^1$-bundles over ${\mathbb{P}}^1$ up to isomorphism — every such bundle is $F_a$ for a unique $a\ge 0$.

The toric description of $F_a$ is from Demazure 1970 (the same paper that introduced the fan formalism). The fan ${\Sigma }_a$ in ${\mathbb{R}}^2$ has four primitive ray generators: $$ \rho_1 = e_1, \quad \rho_2 = e_2, \quad \rho_3 = -e_1 + a e_2, \quad \rho_4 = -e_2. $$ The four maximal cones are spanned by adjacent rays (going counterclockwise): $$ \sigma_1 = \mathrm{Cone}(\rho_1, \rho_2), \quad \sigma_2 = \mathrm{Cone}(\rho_2, \rho_3), \quad \sigma_3 = \mathrm{Cone}(\rho_3, \rho_4), \quad \sigma_4 = \mathrm{Cone}(\rho_4, \rho_1). $$ The parameter $a$ controls the "tilt" of the third ray ${\rho }_3$: when $a=0$, the ray is $-e_1$ and the fan reduces to that of ${\mathbb{P}}^1\times {\mathbb{P}}^1$; when $a\ge 1$, the ray tilts toward the upper half-plane, producing a substantively twisted bundle.

Smoothness of $F_a$. Each maximal cone has primitive matrix with determinant $\pm 1$. For ${\sigma }_2=\mathrm{Cone}(e_2,-e_1+a{e}_2)$, the matrix is $\left[\begin{array}{cc}0& 1\\ -1& a\end{array}\right]$ with determinant $1$. For ${\sigma }_3=\mathrm{Cone}(-e_1+a{e}_2,-e_2)$, the matrix is $\left[\begin{array}{cc}-1& a\\ 0& -1\end{array}\right]$ with determinant $1$. The other two cones are similarly unit-determinant by symmetric computation. Every ${\sigma }_i$ is smooth, so $F_a$ is smooth.

Completeness of $F_a$. The four maximal cones cover ${\mathbb{R}}^2$. To see this, follow the rays counterclockwise: ${\rho }_1$ to ${\rho }_2$ sweeps the first quadrant (${\sigma }_1$); ${\rho }_2$ to ${\rho }_3=-e_1+a{e}_2$ sweeps the wedge between $e_2$ and the upper-left tilted direction (${\sigma }_2$, the upper-left wedge); ${\rho }_3$ to ${\rho }_4=-e_2$ sweeps the lower-left (${\sigma }_3$); ${\rho }_4$ to ${\rho }_1$ sweeps the lower-right fourth quadrant (${\sigma }_4$). The four wedges together cover ${\mathbb{R}}^2$, so the support is full and $F_a$ is complete.

Picard number of $F_a$ is $2$. The Picard group $\mathrm{Pic}(F_a)={\mathbb{Z}}^2$ is generated by the fibre class $f$ and the negative section ${\sigma }_{-}$ of self-intersection $-a$. The relation is $f^2=0$, ${\sigma }_{-}^2=-a$, $f\cdot {\sigma }_{-}=1$. The intersection form is the matrix $\left[\begin{array}{cc}0& 1\\ 1& -a\end{array}\right]$ with determinant $-1$, so $\mathrm{Pic}(F_a)$ is a unimodular lattice of signature $(1,1)$ when $a>0$ and $(2,0)$ when $a=0$.

Topological invariants. $F_a$ is diffeomorphic to $S^2\times S^2$ when $a$ is even, and to ${\mathbb{CP}}^2\#\overline{{\mathbb{CP}}^2}$ (the connected sum of complex projective space with its conjugate orientation) when $a$ is odd. The Euler characteristic $\chi (F_a)=4$ for every $a\ge 0$, matching the four torus-fixed points of the toric structure.

Birational geometry. $F_a$ is birational to ${\mathbb{P}}^2$ via the elementary transformation that contracts the negative section ${\sigma }_{-}$ and blows up a point: $F_a\to {\mathbb{P}}^2$ contracts ${\sigma }_{-}$ to a singular point (which is then resolved by blowing up). This birational correspondence is the toric model for the classical surface birational geometry of Castelnuovo and Enriques.

The Hirzebruch surfaces are the prototype of the toric ${\mathbb{P}}^1$-bundle: a fibre bundle whose fibre and base are both toric varieties, with the bundle structure encoded by an integer parameter (the twist) and the total space recovered toric-geometrically as $X_{{\Sigma }_a}$. The pattern extends to ${\mathbb{P}}^k$-bundles over ${\mathbb{P}}^n$ (the generalised Hirzebruch varieties, sometimes called scroll-type varieties), and is the basic local model in the theory of toric vector bundles (Klyachko 1990, Payne 2008).

Synthesis. The smoothness and completeness criteria identify two independent combinatorial conditions on a fan $\Sigma$ — smoothness as a per-cone unimodularity check, completeness as a global support-covers-$N_{\mathbb{R}}$ check — that jointly classify the geometric type of $X_{\Sigma }$, and this is exactly the foundational reason the toric dictionary is effectively combinatorial: questions about regularity of the local rings, properness of the structure morphism, and compactness of the analytic variety all reduce to finite computations on the cones of the fan. The central insight is that the fan separates the local question (per-cone) from the global question (fan support), so the two criteria can be verified independently.

Putting these together with the affine-smoothness theorem and the global gluing of [04.11.04], the four cases of the smooth/complete chart all occur with explicit examples in dimension two: ${\mathbb{P}}^2$ smooth complete (fan with three unimodular cones covering ${\mathbb{R}}^2$); $\mathbb{P}(1,2,3)$ complete-but-not-smooth (three non-unimodular cones still covering ${\mathbb{R}}^2$); $\mathbb{C}$ smooth-but-not-complete (one unimodular ray in $\mathbb{R}$ not covering the negative direction); a generic non-fan-shaped cone arrangement neither smooth nor complete. This pattern recurs in higher dimensions: smoothness is always testable per cone, completeness always testable as support coverage, and the two checks are independent. The bridge is dual to the duality $M\leftrightarrow N$ of [04.11.01]: smoothness lives on the $M$-side (the semigroup algebra $\mathbb{C}[S_{\sigma }]$ being a regular ring is a condition on ${\sigma }^{\vee }\cap M$), completeness lives on the $N$-side (the cocharacters $v\in N$ lying in $|\Sigma |$ is a condition on cones in $N_{\mathbb{R}}$).

The criteria generalises in several directions worth recording. To non-rational fans (with irrational ray directions), the smoothness condition extends but the resulting variety is no longer algebraic — it lives in the category of analytic toric varieties (Oda 1988 Appendix). To non-strongly-convex fans (allowing cones with lineality space), the variety becomes a toric fibration over a torus quotient rather than a single toric variety. To infinite fans (countably many cones), the variety is an ind-toric variety used in affine Grassmannians (Faltings 2003) and Kac-Moody groups. To toric stacks (Borisov-Chen-Smith 2005, Geraschenko-Satriano 2011), the smoothness condition reads at the level of the stack structure rather than the underlying coarse space, allowing the simplicial-but-not-smooth simplicial fans to produce smooth Deligne-Mumford toric stacks.

The pattern generalises further to the logarithmic toric world (Kato 1989, Olsson 2008): a logarithmic toric variety is one where each smooth chart ${\mathbb{A}}^k\times {\mathbb{G}}_m^{n-k}$ is equipped with the natural log structure from the smooth-divisor stratification, and the smoothness and completeness criteria translate to log-smoothness and log-properness criteria. This is the structural foundation of toric degenerations and Gross-Siebert mirror symmetry; the criteria of this unit are the building blocks of the log-toric framework that drives modern toric mirror symmetry research.

Full proof set [Master]

Proposition (smoothness criterion, affine direction), proof. Given in part (a) of the main theorem above. Briefly: for the $(\Leftarrow )$ direction, choose a dual basis $u_1^{\ast },\dots ,u_n^{\ast }$ of $M$ extending the dual to the smooth cone; the semigroup decomposes as $S_{\sigma }={\mathbb{Z}}_{\ge 0}^k\oplus {\mathbb{Z}}^{n-k}$ and the semigroup algebra factors as $\mathbb{C}[x_1,\dots ,x_k,x_{k+1}^{\pm },\dots ,x_n^{\pm }]$, a regular ring. For the $(\Rightarrow )$ direction, the cotangent space ${\mathfrak{m}}_{\sigma }/{\mathfrak{m}}_{\sigma }^2$ has dimension equal to the Hilbert basis size of $S_{\sigma }$; smoothness forces this to equal $n$, equivalent to the primitive ray generators forming part of a $\mathbb{Z}$-basis of $N_{\sigma }$, which extends to a $\mathbb{Z}$-basis of $N$. $\square$

Proposition (smoothness criterion, global), proof. Smoothness of a scheme is a local property — a scheme is smooth iff every point admits a smooth open neighbourhood. The fan-and-toric-variety construction [04.11.04] produces an open affine cover $\{U_{\sigma }:\sigma \in {\Sigma }_{\max }\}$ of $X_{\Sigma }$, so $X_{\Sigma }$ is smooth iff every $U_{\sigma }$ is smooth, iff every $\sigma$ is smooth by the affine theorem, iff every cone in $\Sigma$ is smooth (faces of smooth cones being smooth). $\square$

Proposition (completeness criterion), proof. Given in part (c) of the main theorem and in Exercise 8. Briefly: the valuative criterion for properness applied to torus-valued points translates a $\mathrm{Spec}K$-morphism into a cocharacter $v\in N$ via composition with the DVR's valuation; the extension to $\mathrm{Spec}R$ exists iff $v$ lies in some cone of $\Sigma$, iff $v\in |\Sigma |$; ranging over DVRs sweeps $v$ across all of $N$; closure of $N$ inside $N_{\mathbb{R}}$ is $N_{\mathbb{R}}$ itself, so properness iff $|\Sigma |=N_{\mathbb{R}}$. $\square$

Proposition (cyclic quotient singularities from simplicial-but-not-smooth cones), proof. Let $\sigma$ be a simplicial cone of dimension $k$ inside a lattice $N$ of rank $n$, with primitive ray generators $u_1,\dots ,u_k$. Assume $\sigma$ is not smooth: the index $d:=[N_{\sigma }:\mathbb{Z}⟨u_1,\dots ,u_k⟩]>1$ inside the saturation $N_{\sigma }$. Then the affine toric variety $U_{\sigma }$ has a cyclic quotient singularity of order $d$ at the unique torus-fixed point.

Proof. The sublattice $\mathbb{Z}⟨u_1,\dots ,u_k⟩\subseteq N_{\sigma }$ has index $d$, so the quotient $N_{\sigma }/\mathbb{Z}⟨u_1,\dots ,u_k⟩\cong \mathbb{Z}/d_1\oplus \cdots \oplus \mathbb{Z}/d_r$ for some integers $d_1|d_2|\cdots |d_r$ with $d_1\cdots d_r=d$ (the Smith normal form elementary divisors of the matrix $[u_1,\dots ,u_k{]}^T$ viewed as a map ${\mathbb{Z}}^k\to N_{\sigma }\cong {\mathbb{Z}}^k$).

By the toric covering construction (Cox-Little-Schenck §11.1; Reid 1980), there is a finite cover $U_{{\sigma }^{\prime }}\to U_{\sigma }$ where ${\sigma }^{\prime }$ is the cone with the same primitive generators viewed in the sublattice $\mathbb{Z}⟨u_1,\dots ,u_k⟩$ (so ${\sigma }^{\prime }$ is smooth in this finer lattice). The cover is a Galois cover with Galois group $N_{\sigma }/\mathbb{Z}⟨u_1,\dots ,u_k⟩$, the finite abelian group of order $d$.

Concretely, $U_{\sigma }$ is the GIT quotient ${\mathbb{C}}^k/(N_{\sigma }/\mathbb{Z}⟨u_1,\dots ,u_k⟩)$ where the finite abelian group acts on ${\mathbb{C}}^k$ via the character lattice quotient. The action has a unique fixed point at the origin $0\in {\mathbb{C}}^k$, and the quotient has a cyclic quotient singularity of type $(1/d)(a_1,\dots ,a_k)$ at this point, where $a_i$ are integers determined by the action's character data.

For $k=2$, the cyclic quotient singularity $(1/n)(1,q)$ where $q$ is coprime to $n$ is the model of the Hirzebruch-Jung singularity, resolved by a chain of ${\mathbb{P}}^1$s whose intersection numbers form the continued-fraction expansion of $n/q$. The resolution graph is a "string" of $-a_i$-curves where $n/q=[a_1,a_2,\dots ,a_r]$ is the continued fraction. $\square$

Proposition (Hirzebruch surface $F_a$ is smooth and complete), proof. Given in the worked example above. Each of the four maximal cones ${\sigma }_i$ has primitive matrix determinant $\pm 1$ (a direct calculation showing the tilt parameter $a$ does not affect the determinant due to the tilt being in one row only). The four cones together cover ${\mathbb{R}}^2$ (sweeping counterclockwise from ${\rho }_1$ through ${\rho }_4$ and back to ${\rho }_1$). Hence $F_a$ is smooth and complete. $\square$

Proposition (the smoothness criterion is sharp), proof. There exist simplicial cones $\sigma$ in ${\mathbb{R}}^2$ with $U_{\sigma }$ not smooth.

Proof. Take $\sigma =\mathrm{Cone}(e_1,e_1+2e_2)$ in ${\mathbb{Z}}^2$. The cone is simplicial (two linearly independent rays). The primitive matrix is $\left[\begin{array}{cc}1& 0\\ 1& 2\end{array}\right]$ with determinant $2\ne \pm 1$. The dual cone ${\sigma }^{\vee }$ has primitive generators determined by the dual basis condition; the semigroup $S_{\sigma }={\sigma }^{\vee }\cap {\mathbb{Z}}^2$ contains the additional lattice point $e_2^{\ast }$ beyond the two facet-normal generators, making the Hilbert basis size $3\ne 2$. Hence the cotangent space has dimension $3$ at the unique torus-fixed point of $U_{\sigma }$, exceeding the Krull dimension $2$. The variety $U_{\sigma }$ is singular at this point, with a $(1/2)(1,1)$ cyclic quotient singularity (the $A_1$ singularity).

Explicitly: $\mathbb{C}[S_{\sigma }]=\mathbb{C}[x,y,z]/(xz-y^2)$ where $x={\chi }^{e_1^{\ast }},z={\chi }^{2e_2^{\ast }-e_1^{\ast }},y={\chi }^{e_2^{\ast }}$, with the cone relation $xz=y^2$ visible as the toric ideal of the binomial generators. The corresponding variety is the quadric cone $\{xz=y^2\}\subset {\mathbb{C}}^3$, which has an isolated singularity at the origin — the ordinary double point. This is the canonical model of the simplest substantive toric singularity. $\square$

Proposition (the completeness criterion is sharp), proof. There exist fans $\Sigma$ with $|\Sigma |⊊N_{\mathbb{R}}$ and $X_{\Sigma }$ not complete.

Proof. Take $\Sigma =\{\{0\},\mathrm{Cone}(e_1)\}$ inside $N=\mathbb{Z}$. The support is $|\Sigma |={\mathbb{R}}_{\ge 0}$, which is a proper subset of $\mathbb{R}$ (missing the negative ray). The variety $X_{\Sigma }=U_{\mathrm{Cone}(e_1)}=\mathrm{Spec}\mathbb{C}[x]$ is the affine line $\mathbb{C}$, which is not complete: the morphism $\mathrm{Spec}\mathbb{C}((t))\to \mathbb{C}$ given by $x=t^{-1}$ (so $v_{\varphi }(e_1^{\ast })=-1$, hence $v_{\varphi }=-1\notin |\Sigma |$) does not extend to $\mathrm{Spec}\mathbb{C}[[t]]\to \mathbb{C}$ because $t^{-1}\notin \mathbb{C}[[t]]$. The valuative criterion fails for this morphism, and $X_{\Sigma }$ is not proper. $\square$

Connections [Master]

• Affine toric variety 04.11.03. The local input to the smoothness criterion: the affine smoothness theorem ($U_{\sigma }$ smooth iff $\sigma$ smooth) is proved at the level of an individual cone, with the Hilbert-basis-equals-Krull-dimension count being the technical core. The global smoothness criterion for $X_{\Sigma }$ is the local criterion applied to every chart in the open cover supplied by the fan-to-toric construction. The prerequisite unit establishes the dual-basis decomposition of the semigroup algebra that powers the $(\Leftarrow )$ direction of the affine smoothness theorem.

• Fan and toric variety 04.11.04. The direct prerequisite. The completeness criterion uses the open affine cover of $X_{\Sigma }$ supplied by the fan to verify the valuative criterion chart by chart, and the global smoothness criterion combines the affine charts into a global verdict. Sumihiro's classification from [04.11.04] ensures the criteria apply to every normal toric variety, since every such variety arises from a fan.

• Rational polyhedral cone and dual cone 04.11.02. The combinatorial input. The duality ${\sigma }^{\vee \vee }=\sigma$ is the key step in the valuative-criterion argument for completeness: cocharacters $v\in N$ pair non-negatively with ${\sigma }^{\vee }$ iff they lie in $\sigma$, identifying support membership with extension existence. The face-correspondence theorem of the prerequisite controls how the Hilbert basis of $S_{\sigma }$ relates to the ray generators of $\sigma$, supplying the structural input to the smoothness argument.

• Orbit-cone correspondence 04.11.06. The next unit. The orbit-cone correspondence identifies $T$-orbits with cones of $\Sigma$ in inclusion-reversing bijection; the orbit closures form the toric stratification of $X_{\Sigma }$. Smoothness of $X_{\Sigma }$ is equivalent to smoothness of the open stratum (the dense torus, always smooth) and of every closed stratum (orbit closure), and the latter is governed by the orbit-cone-correspondence chain of fans on the quotient lattices.

• Toric resolution of singularities 04.11.07. The downstream unit. The smoothness criterion identifies which fans need resolving — those with at least one non-unimodular cone. The toric resolution theorem says every fan can be refined to a smooth fan by iterated star subdivision, producing a $T$-equivariant proper birational morphism $X_{{\Sigma }^{\prime }}\to X_{\Sigma }$. The completeness criterion is preserved under fan refinement, so resolution does not break completeness — every complete fan has a smooth complete refinement, giving a proper birational map from a smooth complete toric variety.

• Polytope-fan correspondence 04.11.10. The projective specialisation. A complete fan that is the normal fan of a lattice polytope $P$ produces a projective toric variety $X_{\Sigma }$, with $P$ supplying an explicit ample line bundle. The smoothness of $X_{\Sigma }$ corresponds (via the dual polytope) to a condition on the local geometry at each vertex of $P$ — the Delzant condition in symplectic geometry, requiring the edges at each vertex to form a $\mathbb{Z}$-basis of $N$.

• Toric Picard group 04.11.09. The unit that develops the Picard group of $X_{\Sigma }$ from divisor-class data. Smoothness affects the relationship between the Picard group (Cartier divisors modulo linear equivalence) and the class group (Weil divisors modulo linear equivalence): they coincide for smooth $X_{\Sigma }$ but differ for simplicial-but-not-smooth, by a quotient encoding the local cyclic singularities. For $\mathbb{P}(1,2,3)$, $\mathrm{Pic}(X)=\mathbb{Z}$ but $\mathrm{Cl}(X)=\mathbb{Z}$ also (since the variety is $\mathbb{Q}$-factorial), with the difference visible only in torsion data of higher cohomology.

• Toric divisor and support function 04.11.08. The sibling unit. The smoothness criterion of this unit translates at the divisor level into "every Weil divisor is Cartier" — equivalently, every piecewise-linear support function $\psi :|\Sigma |\to \mathbb{R}$ is integral on every ray generator. The completeness criterion controls when the divisor polytope $P_D$ of an effective $T$-invariant Cartier divisor is bounded — it is bounded iff $D$ is nef and $X_{\Sigma }$ is complete. The toric-divisor unit develops the support-function calculus in detail and depends on the smoothness / completeness verdict produced here.

• Resolution of singularities 04.06.02. The algorithmic precursor. Hironaka's 1964 general resolution theorem (proven for fields of characteristic zero) reduces in the toric setting to the combinatorial subdivision of a fan into a smooth refinement, an algorithmic special case. The smoothness criterion of this unit is the target condition for the refinement procedure: subdivide cones until every cone is unimodular. The toric algorithm was the first context where resolution was constructive, predating later constructive resolution algorithms (Bierstone-Milman 1997, Włodarczyk 2005).

• Symplectic toric manifolds via Delzant 05.09.01. The symplectic-side bridge. The Delzant theorem (Delzant 1988) classifies compact symplectic toric manifolds by Delzant polytopes — lattice polytopes whose edges at each vertex form a $\mathbb{Z}$-basis of $N$. This is the symplectic-side smoothness condition, and the corresponding algebraic toric variety is a smooth projective toric variety. The polytope's vertex condition (Delzant) matches the algebraic vertex condition (each vertex cone in the normal fan is smooth), giving a precise dictionary between symplectic smoothness and algebraic smoothness.

• Minimal model program (MMP) for toric varieties 04.11.13. The advanced application. Mori's minimal model program (Mori 1982 Annals of Mathematics 116) studies birational maps of varieties by following the extremal rays of the Mori cone. In the toric setting, the entire MMP is combinatorial: extremal rays correspond to rays of the secondary fan in the parameter space of fans, and the MMP steps (flips, flops, divisorial contractions) correspond to combinatorial moves on the fan. The smoothness criterion controls the terminal singularity condition central to the MMP, and the completeness criterion is preserved at each step. Toric MMP is the testing ground for the general MMP, with every step verifiable by direct fan computation.

• Tropicalisation 04.12.04. The downstream tropical-geometry connection. The fan $\Sigma$ of a toric variety $X_{\Sigma }$ is the tropicalisation of the open torus $T\subseteq X_{\Sigma }$ in the sense of Maclagan-Sturmfels: it records the directions in $N_{\mathbb{R}}$ along which a generic point of $T$ can degenerate. The smoothness and completeness criteria of this unit translate to tropical-geometric conditions on $\Sigma$ — smoothness as the unimodularity of the tropical cells, completeness as the tropical-fan support being all of $N_{\mathbb{R}}$. This bridge is the structural foundation of toric mirror symmetry (Batyrev-Borisov 1996) and of the Gross-Siebert program for mirror symmetry of Calabi-Yau toric hypersurfaces.

Historical & philosophical context [Master]

The smoothness criterion for affine toric varieties was first established 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) ^{[Demazure1970]}. Demazure's Theorem 4 stated the criterion in its modern form: the affine toric variety $U_{\sigma }$ is smooth (he wrote "régulière" in French) iff the cone $\sigma$ is generated by part of a basis of $N$. The proof in Demazure's paper proceeded via the dual-basis decomposition of the semigroup algebra, an argument that has been reproduced almost verbatim in every subsequent toric-geometry textbook. Demazure's motivation was the classification of algebraic subgroups of the Cremona group ${\mathrm{Cr}}_n=\mathrm{Bir}({\mathbb{P}}^n)$ of maximal rank — equivariant compactifications of the torus by combinatorial data — and the smoothness criterion arose as the necessary and sufficient condition for the compactification to be regular as a scheme.

The completeness criterion has a more distributed origin. Hideyasu Sumihiro's Equivariant completion (Journal of Mathematics of Kyoto University 14, 1974, pp. 1-28) ^{[Sumihiro1974]} established the foundational equivariant-covering theorem identifying every normal $T$-toric variety as $X_{\Sigma }$ for some fan, which made the completeness criterion well-posed: proper toric varieties are exactly the fans with full support. The valuative-criterion proof appeared in Tadao Oda's Convex Bodies and Algebraic Geometry (Ergebnisse 15, Springer 1988) ^[pending] as Theorem 1.11, and in William Fulton's Introduction to Toric Varieties (Princeton 1993) ^[Fulton1993] in §2.4 as the canonical textbook statement.

The pre-history of the smoothness criterion runs through several converging threads. Friedrich Hirzebruch's 1953 paper on continued-fraction resolutions (Mathematische Annalen 126, 1-22) ^[pending] gave the explicit Hirzebruch-Jung resolution algorithm for surface singularities of cyclic quotient type — the surface case of the toric resolution theorem, before the general toric formalism existed. David Mumford's An analytic construction of degenerating Abelian varieties (Compositio Mathematica 24, 1972, 239-272) ^[pending] introduced cone-and-lattice data as a tool for constructing degenerations, with the smoothness criterion appearing implicitly as the condition for the degenerating fibres to remain regular. Kempf, Knudsen, Mumford, and Saint-Donat's Toroidal Embeddings I (Lecture Notes in Mathematics 339, 1973) ^[pending] systematised the cone-and-fan formalism into a tool for resolution of singularities and compactification, with the smoothness criterion as the target of the resolution procedure.

The toric setting was the first algorithmic context for resolution of singularities — predating later constructive resolution algorithms by decades. Where Hironaka's 1964 theorem (Annals of Mathematics 79, 109-326) ^{[Hironaka1964]} established the general existence of resolution in characteristic zero non-constructively, the toric resolution theorem (Demazure 1970, KKMS 1973) gave a finite combinatorial algorithm producing the smooth refinement: subdivide cones until every cone is unimodular. The algorithm's simplicity made toric varieties the laboratory for understanding what constructive resolution should look like in general, a thread that motivated significant later research (Bierstone-Milman 1997 Inventiones Mathematicae 128, Włodarczyk 2005 Journal of the American Mathematical Society 18).

Shigefumi Mori's Threefolds whose canonical bundles are not numerically effective (Annals of Mathematics 116, 1982, 133-176) ^[Mori1982] introduced $\mathbb{Q}$-factorial varieties — the smoothness-relaxation matching simplicial-but-not-smooth fans — as central inputs to the minimal model program. Reid 1980 (Journées de géométrie algébrique d'Angers 1980, 273-310) ^[pending] classified the toric cyclic quotient singularities $(1/n)(a_1,\dots ,a_k)$ as the local model singularities of toric varieties, and the resulting canonical and terminal singularities in Reid's sense became the standard relaxation under which the MMP runs. The toric setting is the testing ground for the entire MMP, and the smoothness criterion of this unit is the strict-smooth endpoint of a chain of progressively weaker singularity conditions (smooth, $\mathbb{Q}$-factorial, terminal, canonical, log-terminal).

The Cox-quotient reframing (Cox 1995 Journal of Algebraic Geometry 4, 17-50) ^[Cox1995] gave the smoothness criterion a new functorial interpretation: a toric variety $X_{\Sigma }$ is smooth iff the homogeneous coordinate ring $\mathbb{C}[x_{\rho }:\rho \in \Sigma (1)]$ together with the irrelevant ideal $B(\Sigma )$ presents the variety as a quotient $({\mathbb{C}}^{\Sigma (1)}\setminus V(B(\Sigma )))/G$ where $G$ acts freely on the open locus — freeness is exactly the smooth condition translated to the Cox-quotient framework. This perspective unifies projective and toric smoothness conditions and has been foundational to toric mirror symmetry (Batyrev-Borisov 1996) and computational toric algorithms in Macaulay2, Polymake, and SageMath.

The pedagogical canon for the smoothness and completeness criteria is Fulton's 1993 Princeton book (§2.1 and §2.4) and Cox-Little-Schenck's 2011 AMS book (§3.1, §3.2, §3.4). Both volumes present the criteria with the canonical proofs (dual-basis decomposition for smoothness, valuative criterion for completeness) and the canonical examples (${\mathbb{P}}^n$ via the standard fan, $\mathbb{P}(a,b,c)$ via the weighted-quotient fan, $F_a$ via the four-ray tilted fan, $\mathbb{C}$ as the simplest non-complete example).

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}
}

@article{Sumihiro1974,
  author  = {Sumihiro, Hideyasu},
  title   = {Equivariant completion},
  journal = {Journal of Mathematics of Kyoto University},
  volume  = {14},
  year    = {1974},
  pages   = {1--28}
}

@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{Hirzebruch1953,
  author  = {Hirzebruch, Friedrich},
  title   = {{\"U}ber vierdimensionale {R}iemannsche {F}l{\"a}chen mehrdeutiger analytischer {F}unktionen von zwei komplexen {V}er{\"a}nderlichen},
  journal = {Mathematische Annalen},
  volume  = {126},
  year    = {1953},
  pages   = {1--22}
}

@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}
}

@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{Reid1980,
  author  = {Reid, Miles},
  title   = {Canonical $3$-folds},
  booktitle = {Journ{\'e}es de g{\'e}om{\'e}trie alg{\'e}brique d'Angers},
  publisher = {Sijthoff \& Noordhoff},
  year    = {1980},
  pages   = {273--310}
}

@article{Mori1982,
  author  = {Mori, Shigefumi},
  title   = {Threefolds whose canonical bundles are not numerically effective},
  journal = {Annals of Mathematics},
  volume  = {116},
  year    = {1982},
  pages   = {133--176}
}

@book{OdaConvexBodies,
  author    = {Oda, Tadao},
  title     = {Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties},
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  volume    = {15},
  year      = {1988}
}

@book{FultonToric,
  author    = {Fulton, William},
  title     = {Introduction to Toric Varieties},
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  year      = {1993}
}

@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{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{Hironaka1964,
  author  = {Hironaka, Heisuke},
  title   = {Resolution of singularities of an algebraic variety over a field of characteristic zero},
  journal = {Annals of Mathematics},
  volume  = {79},
  year    = {1964},
  pages   = {109--326}
}

@article{Delzant1988,
  author  = {Delzant, Thomas},
  title   = {Hamiltoniens p{\'e}riodiques et image convexe de l'application moment},
  journal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France},
  volume  = {116},
  year    = {1988},
  pages   = {315--339}
}

@article{BierstoneMilman1997,
  author  = {Bierstone, Edward and Milman, Pierre D.},
  title   = {Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant},
  journal = {Inventiones Mathematicae},
  volume  = {128},
  year    = {1997},
  pages   = {207--302}
}

@article{Wlodarczyk2005,
  author  = {W{\l}odarczyk, Jaros{\l}aw},
  title   = {Simple {H}ironaka resolution in characteristic zero},
  journal = {Journal of the American Mathematical Society},
  volume  = {18},
  year    = {2005},
  pages   = {779--822}
}

@article{BatyrevBorisov1996,
  author  = {Batyrev, Victor V. and Borisov, Lev A.},
  title   = {On {C}alabi-{Y}au complete intersections in toric varieties},
  booktitle = {Higher-dimensional complex varieties (Trento, 1994)},
  publisher = {de Gruyter},
  year    = {1996},
  pages   = {39--65}
}