04.11.04 · algebraic-geometry / toric

Fan and the toric variety $X_{Σ}$

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Anchor (Master): Fulton §1.4, §2.4; Cox-Little-Schenck §3.1, §6.1; Oda *Convex Bodies and Algebraic Geometry* Ch. 1 §1.4-1.5; Kempf-Knudsen-Mumford-Saint-Donat *Toroidal Embeddings I* Ch. I §2; Demazure 1970 *Sous-groupes algébriques de rang maximum du groupe de Cremona*; Sumihiro 1974 *Equivariant completion*; Danilov 1978 *The geometry of toric varieties*

Intuition [Beginner]

A single rational polyhedral cone $σ$ produces a single affine toric variety $U_{σ}$ , as built in the previous unit. The next move in toric geometry is to glue many such affine pieces into a global variety. The combinatorial input that controls the gluing is a fan $Σ$ : a finite collection of cones in $N_{R}$ that fit together compatibly, every face of a cone is in the collection, and every pairwise intersection is a face of each. The variety $X_{Σ}$ associated to the fan is the result of gluing the affine pieces $U_{σ}$ for $σ \in Σ$ along the open immersions $U_{σ \cap τ} ↪ U_{σ}$ that the face-functoriality of [04.11.03] produces.

The picture in two dimensions is simple. Imagine $N_{R} = R^{2}$ chopped into wedges by finitely many rays through the origin. Each wedge is a maximal cone of the fan; each ray is a one-dimensional cone; the origin is the zero cone. The variety $X_{Σ}$ has one affine chart $U_{σ}$ per wedge, and the charts are glued along the affine charts of the rays. If the rays cover all directions in $R^{2}$ , the variety is compact; if they leave a gap, the variety has a hole at infinity.

This construction recovers every familiar normal toric variety. Projective space $P^{n}$ comes from a fan with $n + 1$ rays in $R^{n}$ pointing toward the vertices of a simplex. The product $P^{1} \times P^{1}$ comes from four quadrants in $R^{2}$ . The Hirzebruch surface $F_{a}$ comes from a four-ray fan with one ray tilted by the parameter $a$ . Sumihiro's theorem of 1974 guarantees that every normal toric variety arises this way from a unique fan, making the cone-and-fan formalism a complete combinatorial classification.

Visual [Beginner]

A two-panel diagram showing a fan and the resulting toric variety. Left panel: the fan for $P^{2}$ in $N_{R} = R^{2}$ . Three rays from the origin point along $e_{1}$ , $e_{2}$ , and $- e_{1} - e_{2}$ , dividing the plane into three maximal cones (wedges) labelled $σ_{0}, σ_{1}, σ_{2}$ . The rays are labelled $ρ_{0}, ρ_{1}, ρ_{2}$ . The whole picture covers the plane — the support of the fan equals all of $R^{2}$ .

Right panel: projective space $P^{2}$ depicted as a triangle with three affine charts. Each chart corresponds to one of the maximal cones $σ_{i}$ and is identified with a copy of $C^{2}$ . The three charts overlap along $C \times C^{*}$ pieces corresponding to the rays $ρ_{i}$ , and they all overlap on a common open dense torus $T = (C^{*})^{2}$ corresponding to the zero cone at the origin. The arrow from left to right is labelled " $Σ \mapsto X_{Σ}$ (gluing affine pieces)".

$A two-panel schematic showing the fan for $\mathbb{P}^2$ in $\mathbb{R}^2$ with three rays and three maximal cones, glued into the projective plane $\mathbb{P}^2$ with three affine charts and a dense open torus.$

The picture captures the central idea: a fan is a combinatorial recipe for an open cover, the cones index the affine charts, the faces index the chart overlaps, and the gluing assembles a global variety from the affine pieces.

Worked example [Beginner]

Build $P^{1}$ as the toric variety $X_{Σ}$ for the smallest interesting fan. Take $N = Z$ with $N_{R} = R$ , and let $Σ$ have three cones: the zero cone ${0}$ and the two rays $σ_{+} = R_{\geq 0}$ and $σ_{-} = R_{\leq 0}$ . The fan is the entire real line, split at the origin into a positive half-ray and a negative half-ray, plus the origin itself.

Step 1. The affine pieces. The cone $σ_{+} = Cone (1)$ has dual $σ_{+}^{\lor} = R_{\geq 0}$ , semigroup $Z_{\geq 0}$ , semigroup algebra $C [x]$ , so $U_{σ_{+}} = C$ . The cone $σ_{-} = Cone (- 1)$ has dual $σ_{-}^{\lor} = R_{\leq 0}$ , semigroup $Z_{\leq 0}$ , semigroup algebra $C [x^{- 1}]$ — set $y = x^{- 1}$ , so $U_{σ_{-}} = Spec C [y] = C$ , another affine line. The zero cone ${0}$ has $U_{{0}} = Spec C [x, x^{- 1}] = C^{*}$ , the dense torus.

Step 2. The gluing. The intersection $σ_{+} \cap σ_{-} = {0}$ is a face of each, and $U_{{0}} = C^{*}$ embeds into $U_{σ_{+}} = C$ as the open subset ${x \neq = 0}$ and into $U_{σ_{-}} = C$ as the open subset ${y \neq = 0}$ . The gluing identifies these two open immersions via the relation $y = x^{- 1}$ : a point in the overlap is the same point of $C^{*}$ written in either chart.

Step 3. The result. Two affine lines glued along $C^{*}$ via $y = x^{- 1}$ is the classical construction of the projective line $P^{1}$ . The variety $X_{Σ} = P^{1}$ has two affine charts (the maximal cones $σ_{\pm}$ ) plus the dense open torus (the zero cone). The two "points at infinity" correspond to the closed $T$ -fixed points of each chart — the origins $x = 0$ and $y = 0$ — which are the unique $T$ -fixed points of $σ_{+}$ and $σ_{-}$ respectively.

What this tells us. The cone-and-fan construction recovers $P^{1}$ as the gluing of two affine pieces along an open dense torus, exactly the standard atlas of the projective line. Each maximal cone produces a chart; each face inclusion produces an overlap; the global variety is compact because the support of the fan covers all of $N_{R}$ . This pattern generalises: $P^{n}$ from a fan with $n + 1$ maximal cones, $P^{1} \times P^{1}$ from four quadrants, and so on.

Check your understanding [Beginner]

Exercise (easy, multiple choice).

A fan $Σ$ in $N_{R}$ is:

A. Any finite collection of strongly convex rational polyhedral cones.
B. A finite collection of strongly convex rational polyhedral cones such that every face of a cone in $Σ$ is in $Σ$ and the intersection of any two cones in $Σ$ is a face of each.
C. A single strongly convex rational polyhedral cone partitioned into wedges.
D. An arbitrary cover of $N_{R}$ by closed convex subsets.

Hint

The two axioms are face-closure and intersection-as-face. Both are needed; either alone is insufficient.

Answer

B. A fan is a finite collection $Σ$ of strongly convex rational polyhedral cones satisfying two axioms: (i) every face of a cone in $Σ$ is in $Σ$ ; (ii) the intersection of any two cones is a face of each. Option A omits both axioms — without them, the gluing data fails to assemble into a separated scheme. Option C describes a degenerate single-maximal-cone case; general fans have several maximal cones. Option D is far too permissive: arbitrary closed convex covers do not produce algebraic varieties. The two axioms are exactly what is needed to glue the affine pieces $U_{σ}$ into a separated algebraic variety $X_{Σ}$ .

Exercise (easy, multiple choice).

The fan of projective space $P^{n}$ in $N_{R} = R^{n}$ has:

A. One ray and the zero cone, total of $2$ cones.
B. $n$ rays along $e_{1}, \dots, e_{n}$ and all subsets thereof, total of $2^{n}$ cones.
C. $n + 1$ rays — the standard basis $e_{1}, \dots, e_{n}$ together with $- e_{1} - \dots - e_{n}$ — and all cones spanned by proper subsets, total of $2^{n + 1} - 1$ cones.
D. Infinitely many rays equally spaced around the origin.

Hint

$P^{n}$ has $n + 1$ affine charts (the standard cover by $U_{i} = {[x_{0} : \dots : x_{n}] : x_{i} \neq = 0}$ ), so there should be $n + 1$ maximal cones, each generated by $n$ rays out of the total $n + 1$ .

Answer

C. The fan of $P^{n}$ has $n + 1$ rays with primitive generators $ρ_{0} = - e_{1} - \dots - e_{n}$ and $ρ_{i} = e_{i}$ for $i \geq 1$ . Maximal cones are the $n + 1$ size- $n$ subsets of these rays; each generates a copy of $C^{n}$ as its affine chart. Adding all faces gives total cone count $2^{n + 1} - 1$ (all proper subsets; the full set fails because the rays sum to zero). Option B is the fan of $C^{n}$ . Option A is too small. Option D is not a finite fan.

Exercise (easy, true-false).

True or false: the toric variety $X_{Σ}$ is compact in the classical complex topology if and only if the support $∣Σ∣ := ⋃_{σ \in Σ} σ$ equals all of $N_{R}$ .

Hint

This is the completeness criterion. Compactness in the classical topology coincides with completeness (= properness over $C$ ) for varieties of finite type. The support of the fan controls "how much of the torus is compactified".

Answer

True. This is the completeness criterion for toric varieties: $X_{Σ}$ is complete (equivalently, compact in the $C$ -topology, equivalently, proper over $C$ ) if and only if $∣Σ∣ = ⋃_{σ} σ$ equals $N_{R}$ . The intuition is direct: each maximal cone contributes an affine chart compactifying one direction of the dense torus, and the support records which directions have been compactified. If $∣Σ∣ = N_{R}$ , every direction is compactified and the variety is compact; if $∣Σ∣ ⊊ N_{R}$ , some direction is missing and a hole remains at infinity. The full proof uses the valuative criterion — see the key theorem below.

Formal definition [Intermediate+]

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

Definition (fan). A fan $Σ$ in $N_{R}$ is a finite collection of strongly convex rational polyhedral cones $σ \subseteq N_{R}$ such that:

(i) Every face of a cone in $Σ$ is a cone in $Σ$ .

(ii) For any two cones $σ, τ \in Σ$ , the intersection $σ \cap τ$ is a face of $σ$ and a face of $τ$ .

The support of $Σ$ is the closed subset $∣Σ∣ := ⋃_{σ \in Σ} σ \subseteq N_{R}$ . The $k$ -skeleton $Σ (k)$ is the subset of cones of dimension exactly $k$ ; in particular $Σ (1)$ is the set of rays (one-dimensional cones), each generated by a unique primitive generator $u_{ρ} \in N$ .

A fan is smooth (or regular) if every cone $σ \in Σ$ is generated by a subset of a $Z$ -basis of $N$ . A fan is complete if $∣Σ∣ = N_{R}$ . A fan is simplicial if every cone in $Σ$ is simplicial (generated by linearly independent rays).

Definition (toric variety from a fan). Given a fan $Σ$ in $N_{R}$ , the toric variety associated to $Σ$ , denoted $X_{Σ}$ , is the scheme obtained by gluing the affine pieces $U_{σ} = Spec C [σ^{\lor} \cap M]$ for $σ \in Σ$ along the open immersions $$ U_{\sigma \cap \tau} \hookrightarrow U_\sigma, \qquad U_{\sigma \cap \tau} \hookrightarrow U_\tau, $$ produced by the face-functoriality of [04.11.03]. Concretely, $X_{Σ}$ is the colimit (in the category of $C$ -schemes) of the diagram whose objects are the affine pieces $U_{σ}$ for $σ \in Σ$ and whose morphisms are the open immersions $U_{τ} ↪ U_{σ}$ for face inclusions $τ \subseteq σ$ . Equivalently — and this is the form used in working computations — $X_{Σ}$ has an open cover ${U_{σ} : σ \in Σ_{m a x}}$ indexed by the maximal cones, with overlaps $U_{σ} \cap U_{τ} = U_{σ \cap τ}$ for every pair of maximal cones.

Convention. "Maximal cone" means a cone of $Σ$ that is not a proper face of any other cone in $Σ$ . Every cone of $Σ$ is a face of at least one maximal cone (assuming $∣Σ∣$ has dimension equal to that cone or higher); in particular if $Σ$ is full-dimensional then every cone is a face of some $n$ -dimensional cone.

Definition (torus action). The algebraic torus $T = Spec C [M]$ acts on each affine piece $U_{σ}$ by the action defined in [04.11.03], and the gluing morphisms $U_{σ \cap τ} ↪ U_{σ}$ are $T$ -equivariant. Hence the colimit $X_{Σ}$ inherits a $T$ -action, and the dense torus $T = U_{{0}} ↪ U_{σ}$ for every $σ \in Σ$ embeds as a single dense open subvariety of $X_{Σ}$ — the open $T$ -orbit, common to every affine chart.

Counterexamples to common slips

"Any collection of cones closed under faces is a fan." Not without the intersection axiom (ii). For instance, in $R^{2}$ take the two cones $σ_{1} = Cone (e_{1}, e_{1} + e_{2})$ and $σ_{2} = Cone (e_{2}, e_{1} + e_{2})$ , together with their faces. The intersection $σ_{1} \cap σ_{2} = R_{\geq 0} (e_{1} + e_{2})$ is a ray, but this ray is a face of both cones — so this example is actually a valid fan. A genuine non-fan: $τ_{1} = Cone (e_{1}, e_{2})$ and $τ_{2} = Cone (e_{1} + e_{2}, e_{1} - e_{2})$ in $R^{2}$ . The intersection $τ_{1} \cap τ_{2}$ is a two-dimensional region in the first quadrant, not a face of either — adding the faces of both cones still does not produce a fan, and the would-be gluing $U_{τ_{1}}$ glued to $U_{τ_{2}}$ along the overlap fails to be separated.
"The toric variety $X_{Σ}$ depends only on the support $∣Σ∣$ ." It does not. The same support can be subdivided by different fans, producing genuinely different toric varieties — typically related by toric birational morphisms. For example, the support $R^{2}$ (the complete fan) is the support of both the fan for $P^{1} \times P^{1}$ (four quadrants) and the fan for $P^{2}$ (three maximal cones spanned by $e_{1}, e_{2}, - e_{1} - e_{2}$ ); the underlying varieties are not isomorphic. The fan, not its support, is the gluing data.
"Every cone of $Σ$ produces a $T$ -orbit, but the orbit lives only in the chart $U_{σ}$ ." Each orbit lives in every chart $U_{τ}$ for which $σ$ is a face of $τ$ , by the face-functoriality (the open immersion $U_{σ} ↪ U_{τ}$ identifies the orbit corresponding to $σ$ in $U_{σ}$ with the same orbit in $U_{τ}$ ). The orbits are global objects on $X_{Σ}$ , not chart-local; this is the content of the orbit-cone correspondence ([04.11.05]).

Key theorem with proof [Intermediate+]

The signature theorem of this unit is the Sumihiro classification of normal toric varieties: every separated normal toric variety arises from a unique fan up to lattice isomorphism, with completeness, smoothness, and projectivity all readable from fan-combinatorial data.

Theorem (the fan-to-toric-variety construction; Demazure 1970, Sumihiro 1974). Let $Σ$ be a fan in $N_{R}$ .

(a) Existence and gluing. The colimit $X_{Σ} = colim_{σ \in Σ} U_{σ}$ exists as a scheme of finite type over $C$ , with an open cover ${U_{σ} : σ \in Σ_{m a x}}$ and overlaps $U_{σ} \cap U_{τ} = U_{σ \cap τ}$ .

(b) Normality and separatedness. $X_{Σ}$ is irreducible, normal, and separated; the dense torus $T = U_{{0}} ↪ X_{Σ}$ acts on $X_{Σ}$ extending the translation action on itself.

(c) Completeness criterion. $X_{Σ}$ is complete (proper over $C$ , equivalently compact in the $C$ -topology) if and only if $∣Σ∣ = N_{R}$ .

(d) Smoothness criterion. $X_{Σ}$ is smooth as a $C$ -scheme if and only if every cone $σ \in Σ$ is smooth (generated by part of a $Z$ -basis of $N$ ).

(e) Sumihiro classification. Conversely, every separated normal variety with a $T$ -action having a dense open orbit arises as $X_{Σ}$ for a unique fan $Σ$ up to lattice isomorphism.

Proof.

(a) Existence of the colimit. The diagram of affine pieces $U_{σ}$ with face-inclusion morphisms is a finite diagram in the category of schemes. By the gluing lemma for schemes (sometimes called the Grothendieck gluing data lemma, $E G A$ I §0.4 or Hartshorne Algebraic Geometry Ex. II.2.12), to construct the colimit it suffices to verify the cocycle condition: for any three cones $σ_{1}, σ_{2}, σ_{3} \in Σ$ , the transition isomorphism $U_{σ_{1} \cap σ_{3}} \cap U_{σ_{2} \cap σ_{3}} \to U_{σ_{1} \cap σ_{3}} \cap U_{σ_{2} \cap σ_{3}}$ via $U_{σ_{2}}$ matches the direct transition. Since the intersections $σ_{i} \cap σ_{j}$ are themselves cones in $Σ$ (by axiom (ii) of the fan), and since $C [S_{σ_{i} \cap σ_{j}}]$ is a localisation of each $C [S_{σ_{i}}]$ at a single semi-invariant element (by the face-functoriality of [04.11.03]), the transitions are localisation maps. Iterated localisation is single localisation at a product, so the cocycle condition reduces to the identity $(χ^{m_{1}}) (χ^{m_{3}}) / (χ^{m_{2}})^{2} \cdot (χ^{m_{2}})^{2} / (χ^{m_{1}}) (χ^{m_{3}}) = 1$ in the appropriate triple localisation. The cocycle holds and the colimit $X_{Σ}$ exists.

(b) Normality. Each $U_{σ}$ is normal (by part (a) of the main theorem of [04.11.03]). Normality is a local property — a scheme is normal if and only if its local rings are integrally closed integral domains — so the gluing of normal affine pieces along open immersions is normal. The dense torus $T = U_{{0}}$ embeds in every $U_{σ}$ via the face ${0} \subseteq σ$ , and the embeddings are all identified in the gluing (the cocycle condition forces them to agree); hence $T$ embeds as a single dense open subvariety of $X_{Σ}$ .

(b) Separatedness. The diagonal morphism $Δ : X_{Σ} \to X_{Σ} \times_{C} X_{Σ}$ is a closed immersion if and only if for every pair of maximal cones $σ, τ$ the diagonal $U_{σ \cap τ} \to U_{σ} \times_{C} U_{τ}$ is a closed immersion. The image is the subscheme defined by the relations $χ^{m} \otimes 1 = 1 \otimes χ^{m}$ for $m \in S_{σ \cap τ}$ . By the face-correspondence theorem of [04.11.02], $σ \cap τ$ is a face of both $σ$ and $τ$ ; choosing $m_{σ \cap τ}$ in the relative interior of $(σ \cap τ)^{⊥}$ , the semigroup $S_{σ \cap τ} = S_{σ} + S_{τ}$ (the join of the two semigroups inside $M$ , since the face direction is shared). The image of the diagonal is then $Spec C [S_{σ \cap τ}] = U_{σ \cap τ}$ , presented as a closed subscheme of $U_{σ} \times U_{τ} = Spec (C [S_{σ}] \otimes C [S_{τ}])$ by the surjection $C [S_{σ}] \otimes C [S_{τ}] \to C [S_{σ \cap τ}]$ , $χ^{m} \otimes χ^{m^{'}} \mapsto χ^{m + m^{'}}$ . Surjectivity at the level of semigroups gives surjectivity of the algebra map, hence a closed immersion of $Spec$ . This is exactly the verification that $X_{Σ}$ is separated.

(c) Completeness criterion. We use the valuative criterion for properness: $X_{Σ}$ is proper over $C$ if and only if for every discrete valuation ring $R$ with field of fractions $K$ and every $C$ -morphism $Spec K \to X_{Σ}$ , there is a unique extension to $Spec R \to X_{Σ}$ .

A $C$ -morphism $Spec K \to X_{Σ}$ that factors through the open torus $T \subseteq X_{Σ}$ corresponds to a $C$ -algebra map $C [M] \to K$ , equivalently a group homomorphism $M \to K^{*} = Hom (Z, K^{*})$ , equivalently an element of $T (K) = M^{\lor} \otimes_{Z} K^{*}$ . Composing with the valuation $v : K^{*} \to Z$ gives a homomorphism $v_{T} : M \to Z$ , equivalently an element of $N = Hom (M, Z)$ . This element $v_{T} \in N_{Q}$ (rationally; integral if the morphism extends to $R$ ) records the "direction" of the limit point in $N_{R}$ .

The morphism $Spec K \to X_{Σ}$ extends to $Spec R \to X_{Σ}$ if and only if there exists a cone $σ \in Σ$ such that the corresponding $C$ -algebra map $C [M] \to K$ restricts to a map $C [S_{σ}] \to R$ — that is, every $χ^{m}$ for $m \in S_{σ}$ has $v (χ^{m}) \geq 0$ , equivalently $⟨ m, v_{T} ⟩ \geq 0$ for every $m \in σ^{\lor}$ , equivalently $v_{T} \in σ$ . Hence the extension exists if and only if $v_{T} \in ∣Σ∣$ .

Uniqueness of the extension follows from the separatedness in part (b). Existence holds for every DVR-valued point if and only if every $v_{T} \in N$ lies in some cone of $Σ$ , equivalently $∣Σ∣ = N_{R}$ (the integer points $N \subseteq N_{R}$ are dense, and the cones are closed, so $∣Σ∣ = N_{R}$ iff $∣Σ∣ \supseteq N$ ). The valuative criterion gives completeness iff $∣Σ∣ = N_{R}$ .

(d) Smoothness criterion. Smoothness is a local property: $X_{Σ}$ is smooth iff every affine chart $U_{σ}$ is smooth. By the smoothness criterion for affine toric varieties (Theorem of [04.11.03], Demazure 1970), $U_{σ}$ is smooth iff $σ$ is generated by part of a $Z$ -basis of $N$ . Hence $X_{Σ}$ is smooth iff every $σ \in Σ$ has this property.

(e) Sumihiro classification. Sumihiro's 1974 theorem (Journal of Mathematics of Kyoto University 14, 1-28) says every normal variety $X$ with a $T$ -action admits a $T$ -equivariant affine open cover. Applied to a normal toric variety $X$ (one with a dense $T$ -orbit), the affine cover ${V_{i}}$ consists of $T$ -stable affine open subvarieties. Each $V_{i}$ is normal with a $T$ -action and a dense $T$ -orbit; by the affine-toric classification ([04.11.03]), $V_{i} = U_{σ_{i}}$ for a unique cone $σ_{i}$ up to $GL (N)$ -isomorphism.

The overlaps $V_{i} \cap V_{j}$ are also normal affine $T$ -varieties with dense orbits, so $V_{i} \cap V_{j} = U_{τ_{ij}}$ for some cone $τ_{ij}$ . The two open immersions $U_{τ_{ij}} ↪ U_{σ_{i}}$ and $↪ U_{σ_{j}}$ realise $τ_{ij}$ as a common face of $σ_{i}$ and $σ_{j}$ (by the face-functoriality of [04.11.03], with the open immersion encoded by localisation at $χ^{m_{τ_{ij}}}$ ). Hence $τ_{ij} = σ_{i} \cap σ_{j}$ is a face of both, and the collection ${σ_{i}}$ together with all their faces and intersections forms a fan $Σ$ with $X = X_{Σ}$ .

Uniqueness up to lattice isomorphism: the cones $σ_{i}$ are determined by the $T$ -fixed points and orbit structure of $X$ (orbit-cone correspondence, [04.11.05]), which is intrinsic to $X$ . A $T$ -equivariant isomorphism $X ≅ X^{'}$ induces a lattice isomorphism $N ≅ N^{'}$ and a fan isomorphism $Σ ≅ Σ^{'}$ . $□$

Bridge. The fan-to-toric-variety theorem builds toward the orbit-cone correspondence of [04.11.05] and the divisor-fan dictionary of [04.11.08], and the central insight is that the global toric variety $X_{Σ}$ is determined entirely by the combinatorics of the fan $Σ$ , with every geometric property — separatedness, normality, smoothness, completeness, projectivity — reading off from a combinatorial condition on the cones. This is exactly the foundational reason that toric geometry is effectively combinatorial: questions about $X_{Σ}$ become questions about $Σ$ , and the dictionary is exact and computable.

Sumihiro's classification appears again in [04.11.05] (where the orbit-cone correspondence is proven by reducing to the affine case via the fan structure) and generalises the affine-toric classification of [04.11.03] from cones to fans. Putting these together identifies the category of fans (with fan morphisms in [04.11.06]) with the category of normal $T$ -toric varieties (with $T$ -equivariant morphisms), and this categorical equivalence is the bridge from convex-geometric data to algebraic-geometric objects. The bridge is dual to the duality $M \leftrightarrow N$ from [04.11.01]: the fan $Σ \subseteq N_{R}$ encodes the compactification directions of the torus $T$ , the affine charts $U_{σ}$ encode the local geometry near each compactification, and the gluing assembles the local pictures into a global variety. This pattern appears again in [04.11.07] (toric resolution by fan refinement, the combinatorial form of Hironaka's theorem), in [04.11.10] (the polytope-to-fan correspondence producing projective toric varieties), and in [04.11.15] (toric cohomology via the Stanley-Reisner ring of the fan).

Exercises [Intermediate+]

Exercise 1 (easy, symbolic).

Write down the fan $Σ$ for $P^{2}$ explicitly. List the maximal cones, the rays, and the maximal cones' duals. Then identify each affine chart $U_{σ}$ as a copy of $C^{2}$ .

Hint

The fan has three rays in $R^{2}$ : $ρ_{0} = - e_{1} - e_{2}$ , $ρ_{1} = e_{1}$ , $ρ_{2} = e_{2}$ . The three maximal cones are spanned by pairs of these rays.

Answer

Rays. $ρ_{0} = Cone (- e_{1} - e_{2})$ , $ρ_{1} = Cone (e_{1})$ , $ρ_{2} = Cone (e_{2})$ . Three rays in $R^{2}$ , with primitive generators $u_{0} = - e_{1} - e_{2}$ , $u_{1} = e_{1}$ , $u_{2} = e_{2}$ . Note $u_{0} + u_{1} + u_{2} = 0$ , the unique linear relation among the rays.

Maximal cones. Three two-dimensional cones $σ_{i} = Cone (u_{j}, u_{k})$ for ${i, j, k} = {0, 1, 2}$ :

$σ_{2} = Cone (e_{1}, e_{2}) =$ first quadrant.
$σ_{1} = Cone (e_{2}, - e_{1} - e_{2})$ , the upper-left wedge.
$σ_{0} = Cone (e_{1}, - e_{1} - e_{2})$ , the lower-right wedge.

Duals. $σ_{2}^{\lor} = Cone (e_{1}^{*}, e_{2}^{*}) =$ first quadrant in $M_{R}$ . $σ_{1}^{\lor} = Cone (e_{2}^{*} - e_{1}^{*}, - e_{1}^{*})$ — bounded by $m_{1} \leq 0$ and $m_{2} \geq m_{1}$ . $σ_{0}^{\lor} = Cone (e_{1}^{*} - e_{2}^{*}, - e_{2}^{*})$ — bounded by $m_{2} \leq 0$ and $m_{1} \geq m_{2}$ . Each $σ_{i}^{\lor}$ is generated by two vectors forming a $Z$ -basis of $M$ (check determinants: $det (1001) = 1$ for $σ_{2}^{\lor}$ , and similarly $\pm 1$ for the others).

Affine charts. Each $U_{σ_{i}} = Spec C [σ_{i}^{\lor} \cap M] ≅ C^{2}$ since $σ_{i}^{\lor}$ is generated by a $Z$ -basis (smoothness criterion). Explicit coordinates: for $σ_{2}$ , use $x = χ^{e_{1}^{*}}$ , $y = χ^{e_{2}^{*}}$ , giving $U_{σ_{2}} = Spec C [x, y] = C^{2}$ . The other two charts are similar with different choices of coordinates.

Gluing. The three charts overlap pairwise along one-dimensional subvarieties: $U_{σ_{1}} \cap U_{σ_{2}} = U_{ρ_{2}}$ (the chart for the common ray $ρ_{2} = σ_{1} \cap σ_{2}$ ) is $C \times C^{*}$ , the open subset of either $U_{σ_{1}}$ or $U_{σ_{2}}$ where the coordinate corresponding to the ray $ρ_{2}$ is nonzero. All three charts share the common open dense torus $U_{{0}} = T = (C^{*})^{2}$ .

Rubric: full credit for the rays, the three maximal cones, the duals, the chart identifications, and the gluing pattern.

Exercise 2 (easy, symbolic).

Describe the fan $Σ$ for $P^{1} \times P^{1}$ in $N_{R} = R^{2}$ . List the maximal cones and identify each affine chart $U_{σ}$ .

Hint

$P^{1} \times P^{1}$ has four affine charts (one per pair of coordinate-chart choices on the two factors), so the fan has four maximal cones. By symmetry, the four rays are the four standard directions $\pm e_{1}, \pm e_{2}$ .

Answer

Rays. Four rays in $R^{2}$ : $ρ_{1} = e_{1}$ , $ρ_{2} = - e_{1}$ , $ρ_{3} = e_{2}$ , $ρ_{4} = - e_{2}$ , with primitive generators $\pm e_{1}, \pm e_{2}$ .

Maximal cones. Four two-dimensional cones, the four quadrants of $R^{2}$ :

$σ_{1} = Cone (e_{1}, e_{2})$ — first quadrant.
$σ_{2} = Cone (- e_{1}, e_{2})$ — second quadrant.
$σ_{3} = Cone (- e_{1}, - e_{2})$ — third quadrant.
$σ_{4} = Cone (e_{1}, - e_{2})$ — fourth quadrant.

Duals. Each $σ_{i}^{\lor}$ is the quadrant in $M_{R}$ opposite to the quadrant pattern of $σ_{i}$ (with appropriate signs). For $σ_{1}$ : $σ_{1}^{\lor} = Cone (e_{1}^{*}, e_{2}^{*})$ . Each dual cone is generated by a $Z$ -basis of $M$ , so each affine chart is smooth.

Affine charts. Each $U_{σ_{i}} = C^{2}$ . Explicitly, $U_{σ_{1}} = Spec C [x, y]$ where $x = χ^{e_{1}^{*}}, y = χ^{e_{2}^{*}}$ . The four charts correspond to the four standard affine pieces of $P^{1} \times P^{1}$ : the products of the two affine charts of each $P^{1}$ factor.

Verification of completeness and smoothness. The support $∣Σ∣ = ⋃_{i} σ_{i} = R^{2}$ , so $X_{Σ}$ is complete by the completeness criterion. Every cone is generated by part of the standard basis ${e_{1}, e_{2}}$ , so $X_{Σ}$ is smooth by the smoothness criterion. The variety $X_{Σ} = P^{1} \times P^{1}$ is a smooth complete toric surface, as expected.

Rubric: full credit for the four rays, the four maximal cones identified with quadrants, the smooth + complete verification, and the chart identification.

Exercise 3 (medium, symbolic).

Describe the Hirzebruch surface $F_{a}$ (for $a \geq 0$ an integer) as a toric variety $X_{Σ_{a}}$ in $N_{R} = R^{2}$ . Verify that $F_{0} = P^{1} \times P^{1}$ and that $F_{a}$ is smooth and complete for every $a$ .

Hint

The fan for $F_{a}$ has four rays: $ρ_{1} = e_{1}$ , $ρ_{2} = e_{2}$ , $ρ_{3} = - e_{1} + a e_{2}$ , $ρ_{4} = - e_{2}$ . (The third ray "tilts" by the parameter $a$ .) The maximal cones are spanned by adjacent rays. When $a = 0$ , the third ray is $- e_{1}$ and the fan reduces to that of $P^{1} \times P^{1}$ .

Answer

Rays. $ρ_{1} = e_{1}$ , $ρ_{2} = e_{2}$ , $ρ_{3} = - e_{1} + a e_{2}$ , $ρ_{4} = - e_{2}$ . Going counterclockwise from $ρ_{1}$ , the rays are at standard direction except for $ρ_{3}$ , which tilts toward $+ e_{2}$ by amount $a$ (so $ρ_{3} = - e_{1}$ when $a = 0$ , $ρ_{3} = - e_{1} + e_{2}$ when $a = 1$ , etc.).

Maximal cones. Four cones, each generated by two adjacent rays:

$σ_{1} = Cone (e_{1}, e_{2})$ .
$σ_{2} = Cone (e_{2}, - e_{1} + a e_{2})$ .
$σ_{3} = Cone (- e_{1} + a e_{2}, - e_{2})$ .
$σ_{4} = Cone (- e_{2}, e_{1})$ .

Smoothness. Check determinants of pairs of ray generators: $det (e_{1}, e_{2}) = 1$ for $σ_{1}$ , $det (e_{2}, - e_{1} + a e_{2}) = det (01 - 1 a) = 1$ for $σ_{2}$ , $det (- e_{1} + a e_{2}, - e_{2}) = det (- 1 a 0 - 1) = 1$ for $σ_{3}$ , $det (- e_{2}, e_{1}) = det (0 - 1 10) = 1$ for $σ_{4}$ . All four determinants equal $\pm 1$ , so every cone is simplicial unimodular and $X_{Σ_{a}}$ is smooth.

Completeness. The four cones together cover $R^{2}$ : $σ_{1}$ covers the first quadrant, $σ_{2}$ the wedge between $ρ_{2}$ and $ρ_{3}$ (upper half excluding first quadrant), $σ_{3}$ the wedge between $ρ_{3}$ and $ρ_{4}$ (lower-left), $σ_{4}$ the fourth quadrant. Union = $R^{2}$ , so $X_{Σ_{a}}$ is complete.

Case $a = 0$ . When $a = 0$ , $ρ_{3} = - e_{1}$ , and the four rays are exactly $\pm e_{1}, \pm e_{2}$ — the fan for $P^{1} \times P^{1}$ (Exercise 2). Hence $F_{0} = P^{1} \times P^{1}$ .

Geometric interpretation. $F_{a}$ is a $P^{1}$ -bundle over $P^{1}$ , specifically $F_{a} = P (O_{P^{1}} \oplus O_{P^{1}} (a))$ . The parameter $a$ controls the twisting of the bundle: for $a = 0$ the bundle is the product bundle, giving the product $P^{1} \times P^{1}$ ; for $a \geq 1$ the bundle is twisted. Hirzebruch introduced these surfaces in 1951 (Math. Ann. 124) as the classification of $P^{1}$ -bundles over $P^{1}$ ; the toric description is due to Demazure (1970).

Rubric: full credit for the four rays, the four maximal cones, the smoothness verification, the completeness verification, and the $F_{0} = P^{1} \times P^{1}$ identification.

Exercise 4 (medium, short-answer).

Prove that the toric variety $X_{Σ}$ is separated, by verifying that for every pair of maximal cones $σ, τ \in Σ$ , the diagonal $U_{σ \cap τ} \to U_{σ} \times_{C} U_{τ}$ is a closed immersion.

Hint

The semigroup of $σ \cap τ$ is $S_{σ \cap τ} = S_{σ} + S_{τ}$ (the sum of the two semigroups, regarded as subsets of $M$ ). The diagonal is the kernel of the multiplication map $C [S_{σ}] \otimes C [S_{τ}] \to C [S_{σ \cap τ}]$ , and surjectivity on semigroups gives surjectivity on algebras, which corresponds to a closed immersion of $Spec$ .

Answer

Let $σ, τ \in Σ$ be maximal cones with intersection $σ \cap τ$ a face of each (by the fan axiom). By the face-correspondence theorem of [04.11.02], the dual cones satisfy $$ (\sigma \cap \tau)^\vee = \sigma^\vee + \tau^\vee, $$ the Minkowski sum of $σ^{\lor}$ and $τ^{\lor}$ in $M_{R}$ (faces dualise to faces, and intersection on one side dualises to Minkowski sum on the other). Intersecting with $M$ : $$ S_{\sigma \cap \tau} = (\sigma \cap \tau)^\vee \cap M = (\sigma^\vee + \tau^\vee) \cap M = S_\sigma + S_\tau, $$ the sum of $S_{σ}$ and $S_{τ}$ as subsets of $M$ (using that both are saturated in $M$ ).

The multiplication map of semigroup algebras $$ \mu : \mathbb{C}[S_\sigma] \otimes_\mathbb{C} \mathbb{C}[S_\tau] \to \mathbb{C}[S_{\sigma \cap \tau}], \qquad \chi^m \otimes \chi^{m'} \mapsto \chi^{m + m'}, $$ is well-defined (since $m + m^{'} \in S_{σ} + S_{τ} = S_{σ \cap τ}$ ) and surjective (every $χ^{m^{''}}$ for $m^{''} \in S_{σ \cap τ}$ is the image of $χ^{m} \otimes χ^{m^{'}}$ for some decomposition $m^{''} = m + m^{'}$ with $m \in S_{σ}, m^{'} \in S_{τ}$ ).

Surjectivity of a $C$ -algebra map corresponds (by $Spec$ contravariance) to a closed immersion of affine schemes: $$ U_{\sigma \cap \tau} = \mathrm{Spec},\mathbb{C}[S_{\sigma \cap \tau}] \hookrightarrow \mathrm{Spec},(\mathbb{C}[S_\sigma] \otimes \mathbb{C}[S_\tau]) = U_\sigma \times_\mathbb{C} U_\tau. $$ The image is the diagonal $Δ (U_{σ \cap τ}) \subseteq U_{σ} \times_{C} U_{τ}$ , defined by the relations $χ^{m} \otimes 1 = 1 \otimes χ^{m}$ for $m \in S_{σ} \cap S_{τ}$ (lattice points common to both semigroups), which are exactly the lattice points in $σ^{\lor} \cap τ^{\lor} = (σ + τ)^{\lor}$ — the elements where the two charts agree on the overlap.

Hence the diagonal $X_{Σ} \to X_{Σ} \times_{C} X_{Σ}$ is a closed immersion locally, and a morphism is a closed immersion iff it is closed immersion locally on the target. Therefore $X_{Σ}$ is separated. $□$

Rubric: full credit for the dual-cone identity $(σ \cap τ)^{\lor} = σ^{\lor} + τ^{\lor}$ , the surjective semigroup-algebra map, and the closed-immersion conclusion.

Exercise 5 (medium, multiple choice).

Let $Σ$ be the fan in $N_{R} = R^{2}$ consisting of the single ray $ρ = Cone (e_{1})$ and its faces ${0}, ρ$ . The toric variety $X_{Σ}$ is:

A. The projective line $P^{1}$ .
B. The affine line $C$ .
C. The product $C \times C^{*}$ .
D. The empty variety.

Hint

The fan has support $∣Σ∣ = R_{\geq 0} e_{1} \cup {0} = Cone (e_{1})$ , the positive $x$ -axis. This is a proper subset of $R^{2}$ , so the variety is not complete. The unique maximal cone is the ray $ρ$ , and $X_{Σ}$ is the affine toric variety of $ρ$ .

Answer

C. The fan has a single maximal cone $ρ = Cone (e_{1}) \subset R^{2}$ . By the construction, $X_{Σ} = U_{ρ} = Spec C [ρ^{\lor} \cap M]$ . Compute: $ρ^{\lor} = {(m_{1}, m_{2}) : m_{1} \geq 0}$ is the closed right half-plane (with one-dimensional lineality in the $m_{2}$ -direction), so $ρ^{\lor} \cap M = Z_{\geq 0} \times Z$ . The semigroup algebra is $C [x, y^{\pm 1}]$ , and $U_{ρ} = C \times C^{*}$ .

Option A is incorrect: $P^{1}$ requires a complete fan in $R^{1}$ with both rays $\pm e_{1}$ and the zero cone, and the ambient is $R^{1}$ not $R^{2}$ . Option B is the affine toric variety of $Cone (e_{1})$ as a cone in $R^{1}$ (where it is full-dimensional), not in $R^{2}$ . Option D is wrong since the affine variety is genuinely nonempty.

This is the standard non-complete one-parameter family: the toric variety of a fan with one ray in dimension $2$ is the half-compactified torus $C \times C^{*}$ , missing the "negative direction" in the $x$ -coordinate.

Exercise 6 (medium, symbolic).

Describe the fan $Σ$ for the blow-up $Bl_{0} C^{2}$ of $C^{2}$ at the origin, as a toric variety $X_{Σ}$ in $N_{R} = R^{2}$ . How does this fan relate to the fan for $C^{2}$ (the single cone $Cone (e_{1}, e_{2})$ and its faces)?

Hint

The blow-up at the origin adds an exceptional divisor — a $P^{1}$ — over the origin. In the fan, this corresponds to adding a new ray $ρ_{3} = e_{1} + e_{2}$ inside the first quadrant, subdividing the maximal cone $Cone (e_{1}, e_{2})$ into two smaller cones $Cone (e_{1}, e_{1} + e_{2})$ and $Cone (e_{1} + e_{2}, e_{2})$ .

Answer

Fan for $C^{2}$ . Single maximal cone $σ = Cone (e_{1}, e_{2})$ and its faces: $ρ_{1} = Cone (e_{1})$ , $ρ_{2} = Cone (e_{2})$ , ${0}$ . Four cones total.

Fan for $Bl_{0} C^{2}$ . Three rays $ρ_{1} = e_{1}$ , $ρ_{2} = e_{2}$ , $ρ_{3} = e_{1} + e_{2}$ (the new ray). Two maximal cones: $σ^{'} = Cone (e_{1}, e_{1} + e_{2})$ and $σ^{''} = Cone (e_{1} + e_{2}, e_{2})$ . Together with all faces ( $ρ_{1}, ρ_{2}, ρ_{3}, {0}$ ), the fan has $7$ cones total.

Relation: star subdivision. The new fan refines the original by subdividing the maximal cone $σ = Cone (e_{1}, e_{2})$ along the new ray $ρ_{3} = Cone (e_{1} + e_{2})$ . The original cone $σ$ is split into $σ^{'} \cup σ^{''}$ , with $σ^{'} \cap σ^{''} = ρ_{3}$ a common face. This is a star subdivision of the original fan at the cone $σ$ .

Geometric morphism. The fan refinement induces a $T$ -equivariant proper birational morphism $X_{Σ^{'}} \to X_{Σ}$ , which is precisely the blow-up $Bl_{0} C^{2} \to C^{2}$ . The exceptional divisor over the origin is $D_{ρ_{3}}$ , the toric divisor corresponding to the new ray $ρ_{3}$ , and it is isomorphic to $P^{1}$ (a $P^{1}$ embedded in the blow-up as the exceptional locus over $0 \in C^{2}$ ).

Smoothness. Both new cones are smooth: $det (e_{1}, e_{1} + e_{2}) = 1$ and $det (e_{1} + e_{2}, e_{2}) = 1$ , so the blow-up is a smooth toric surface. Each new chart is $C^{2}$ : $U_{σ^{'}} = Spec C [χ^{e_{1}^{*} - e_{2}^{*}}, χ^{e_{2}^{*}}]$ (the dual cone has rays at $e_{1}^{*} - e_{2}^{*}$ and $e_{2}^{*}$ ), and similarly for $U_{σ^{''}}$ . The two charts glue along the chart for $ρ_{3}$ , which is $C \times C^{*}$ .

This is the toric model of the simplest blow-up of algebraic geometry — the toric description is by Demazure (1970) and is the prototype for toric resolution of singularities ([04.11.07]).

Rubric: full credit for the three rays, two maximal cones, identification of the refinement as a star subdivision, the blow-up interpretation, and the smoothness verification.

Exercise 7 (hard, short-answer).

Prove the completeness criterion: the toric variety $X_{Σ}$ is complete (proper over $C$ ) if and only if $∣Σ∣ = N_{R}$ . Use the valuative criterion for properness applied to the open dense torus.

Hint

A morphism $Spec K \to X_{Σ}$ for $K$ the fraction field of a DVR $R$ factors through the open torus $T$ generically (since $T$ is dense). Composing with the DVR's valuation gives a cocharacter $v \in N$ . The extension to $Spec R \to X_{Σ}$ exists iff $v$ lies in some cone of $Σ$ . Hence properness holds iff every $v \in N$ lies in $∣Σ∣$ , iff $∣Σ∣ = N_{R}$ .

Answer

We use the valuative criterion for properness: a separated morphism $f : X \to Y$ of finite type is proper iff for every discrete valuation ring $R$ with fraction field $K$ , every commutative diagram $$ \begin{array}{ccc} \mathrm{Spec},K & \to & X \ \downarrow & & \downarrow_f \ \mathrm{Spec},R & \to & Y \end{array} $$ admits a unique diagonal filling $Spec R \to X$ . For $X = X_{Σ}$ and $Y = Spec C$ , we test all $C$ -morphisms $Spec K \to X_{Σ}$ .

Step 1: reduce to torus-factored morphisms. Since $T = U_{{0}} \subseteq X_{Σ}$ is dense and open, every morphism $Spec K \to X_{Σ}$ with $K$ a field factors generically through some maximal cone's chart $U_{σ}$ . The morphism factors through $T$ iff its image is in the dense torus — which is the generic case for an integral domain such as $K$ . We handle this generic case; the case of morphisms with image in a proper closed subset reduces to a smaller toric variety (a closure of a $T$ -orbit) and is handled by induction on the orbit stratification.

Step 2: translate the morphism to a cocharacter. A $C$ -morphism $Spec K \to T = Spec C [M]$ corresponds to a $C$ -algebra map $C [M] \to K$ , equivalently a group homomorphism $ϕ : M \to K^{*}$ . Compose with the discrete valuation $v_{R} : K^{*} \to Z$ , $v_{R} (c) =$ the integer such that $c$ generates $m_{R}^{v_{R} (c)}$ : $$ v_\phi := v_R \circ \phi : M \to \mathbb{Z}. $$ This is a group homomorphism $M \to Z$ , i.e., an element $v_{ϕ} \in N = Hom_{Z} (M, Z)$ .

Step 3: extension condition. The morphism $Spec K \to T ↪ X_{Σ}$ extends to $Spec R \to X_{Σ}$ iff it factors through some affine chart $U_{σ}$ at the level of $R$ , i.e., $ϕ : C [M] \to K$ restricts to a map $C [S_{σ}] \to R$ .

This restriction is equivalent to: $ϕ (χ^{m}) \in R$ for every $m \in S_{σ}$ , equivalently $v_{R} (ϕ (χ^{m})) \geq 0$ for every $m \in S_{σ} = σ^{\lor} \cap M$ . Since $v_{R} \circ ϕ = v_{ϕ}$ (the cocharacter from Step 2), this says $⟨ m, v_{ϕ} ⟩ = v_{ϕ} (m) \geq 0$ for every $m \in σ^{\lor}$ , equivalently $v_{ϕ} \in σ$ (by cone duality $σ^{\lor\lor} = σ$ ).

Hence the extension exists iff $v_{ϕ} \in σ$ for some $σ \in Σ$ , iff $v_{ϕ} \in ∣Σ∣$ .

Step 4: range over all $v$ . As $R$ ranges over all DVRs containing $C$ and $ϕ$ ranges over all $C$ -morphisms $Spec K \to T$ , the cocharacter $v_{ϕ}$ ranges over all elements of $N$ (every element $v \in N$ arises as $v_{ϕ}$ for the homomorphism $ϕ : M \to K^{*} = C ((t))^{*}$ defined by $ϕ (m) = t^{v (m)}$ , using the DVR $R = C [[t]]$ ).

The valuative criterion holds for every choice iff every $v \in N$ lies in $∣Σ∣$ , iff $∣Σ∣ \supseteq N$ . Since the cones of $Σ$ are closed in $N_{R}$ and $N$ is dense in $N_{R}$ , this is equivalent to $∣Σ∣ = N_{R}$ .

Step 5: uniqueness. Uniqueness of the extension follows from the separatedness theorem in part (b) of the key theorem.

Conclusion. $X_{Σ}$ is proper over $C$ iff $∣Σ∣ = N_{R}$ . Properness over $C$ for a finite-type scheme is equivalent to compactness in the classical $C$ -topology (by GAGA and the Chow-Heisuke theorem), so $X_{Σ}$ is compact iff $∣Σ∣ = N_{R}$ . $□$

Rubric: full credit for the cocharacter translation, the cone-membership characterisation of extendability, the range-over- $N$ argument, and the conclusion.

Exercise 8 (hard, short-answer).

Prove the projectivity criterion: the toric variety $X_{Σ}$ is projective if and only if $Σ$ is the normal fan of a lattice polytope $P \subseteq M_{R}$ . Sketch the construction of an ample line bundle from $P$ .

Hint

The normal fan of a polytope $P$ is the fan in $N_{R}$ whose maximal cones $σ_{v}$ are indexed by vertices $v$ of $P$ , with $σ_{v}$ the closed cone of inward normals to $P$ at the vertex $v$ . The polarisation line bundle $L_{P}$ is constructed from $P$ via the lattice-point basis of global sections.

Answer

Normal fan construction. Let $P \subseteq M_{R}$ be a lattice polytope (vertices in $M$ ). For each vertex $v$ of $P$ , let $$ \sigma_v := {u \in N_\mathbb{R} : \langle m - v, u\rangle \geq 0 \text{ for all } m \in P}, $$ the cone of inward normal directions to $P$ at $v$ . The collection ${σ_{v} : v \in P}$ , together with all their faces, is a fan in $N_{R}$ called the normal fan $Σ_{P}$ of $P$ .

The normal fan is complete ( $∣ Σ_{P} ∣ = N_{R}$ , since every direction in $N_{R}$ is the inward normal to some face of $P$ ) and is determined uniquely by $P$ up to translation (translating $P$ does not change the inward normal cones at vertices).

( $\Leftarrow$ ): polytope gives projective. Given a lattice polytope $P$ with normal fan $Σ = Σ_{P}$ , we construct the polarisation line bundle $L_{P}$ on $X_{Σ}$ and show it is ample (hence $X_{Σ}$ is projective).

For each vertex $v \in P$ , define a local section: on the chart $U_{σ_{v}}$ , the local section is $χ^{v} \in C [S_{σ_{v}}]$ . The transition between two charts $U_{σ_{v}} \to U_{σ_{v^{'}}}$ on the overlap $U_{σ_{v} \cap σ_{v^{'}}}$ uses the transition function $χ^{v - v^{'}}$ — this is well-defined on the overlap since $v - v^{'} \in S_{σ_{v} \cap σ_{v^{'}}}$ (the edge of $P$ from $v$ to $v^{'}$ corresponds to the face $σ_{v} \cap σ_{v^{'}}$ shared by the two normal cones).

The cocycle condition $χ^{v - v^{'}} \cdot χ^{v^{'} - v^{''}} = χ^{v - v^{''}}$ holds tautologically, so the local sections define a line bundle $L_{P}$ on $X_{Σ}$ .

Sections of $L_{P}$ . The global sections $H^{0} (X_{Σ}, L_{P})$ are parametrised by the lattice points $P \cap M$ : each lattice point $m \in P \cap M$ defines a global section by $s_{m} ∣_{U_{σ_{v}}} = χ^{m - v}$ , with the cocycle $s_{m} ∣_{U_{σ_{v^{'}}}} = χ^{m - v^{'}} = χ^{m - v} \cdot χ^{v - v^{'}} = s_{m} ∣_{U_{σ_{v}}} \cdot (transition)$ , consistent.

The dimension $h^{0} (X_{Σ}, L_{P}) = ∣ P \cap M ∣$ is the number of lattice points in $P$ . The associated morphism $X_{Σ} \to P^{∣ P \cap M ∣ - 1}$ sending a point to $[s_{m_{1}} (x) : s_{m_{2}} (x) : \dots]$ for the lattice points $m_{i} \in P \cap M$ is a closed embedding when $P$ is "very ample" — when $P$ has full $n$ -dimensional interior and the lattice points $m_{i} - v$ for fixed vertex $v$ generate $S_{σ_{v}}$ . Hence $L_{P}$ is very ample and $X_{Σ}$ is projective.

( $\Rightarrow$ ): projective gives polytope. Conversely, suppose $X_{Σ}$ is projective with an ample $T$ -equivariant line bundle $L$ . The global sections $H^{0} (X_{Σ}, L)$ form a finite-dimensional $T$ -representation, decomposing into character spaces $H^{0} (X_{Σ}, L)_{m}$ for $m \in M$ . The set of $m$ such that the character space is nonzero is a finite subset of $M$ , and its convex hull is a lattice polytope $P_{L} \subseteq M_{R}$ .

By the theory of toric line bundles ([04.11.09]), the ampleness of $L$ implies that the vertex cones of $P_{L}$ are exactly the maximal cones of $Σ$ (the support function of $L$ is strictly convex on the fan, and its Newton polytope is the dual polytope of $Σ$ ). Hence $Σ$ is the normal fan of $P_{L}$ .

Conclusion. $X_{Σ}$ is projective iff $Σ$ is the normal fan of a lattice polytope. The polarisation $L_{P}$ is the line bundle whose Newton polytope is $P$ . $□$

Rubric: full credit for the normal fan construction, the construction of $L_{P}$ via the cocycle, the lattice-point basis of sections, and the converse via Newton polytopes of ample line bundles.

Lean formalization [Intermediate+]

Mathlib has scheme-gluing infrastructure in Mathlib.AlgebraicGeometry.OpenImmersion and Mathlib.AlgebraicGeometry.GluingOneHypercover, together with affine-scheme primitives in Mathlib.AlgebraicGeometry.AffineScheme. Building the toric-variety-from-fan construction on top of these requires the fan formalism (currently absent from Mathlib) and the affine-toric-variety construction from [04.11.03] (also absent). The intended formalisation reads schematically:

import Mathlib.AlgebraicGeometry.GluingOneHypercover
import Mathlib.AlgebraicGeometry.OpenImmersion
import Mathlib.AlgebraicGeometry.AffineScheme

-- Assume `RationalPolyhedralCone N` from 04.11.02 and
-- `AffineToricVariety σ h` from 04.11.03 are in scope.

variable {N : Type*} [AddCommGroup N] [Module ℤ N] [Module.Free ℤ N]

/-- A fan in N_ℝ is a finite collection of strongly convex rational
polyhedral cones closed under faces and pairwise intersection. -/
structure Fan (N : Type*) [AddCommGroup N] [Module ℤ N] [Module.Free ℤ N] where
  cones : Finset (RationalPolyhedralCone N)
  stronglyConvex : ∀ σ ∈ cones, σ.StronglyConvex
  faceClosed : ∀ σ ∈ cones, ∀ τ : σ.Face, τ.val ∈ cones
  intersectionFace : ∀ σ τ ∈ cones,
    ∃ ρ : σ.Face, ρ.val = σ ∩ τ ∧ ∃ ρ' : τ.Face, ρ'.val = σ ∩ τ

/-- The toric variety associated to a fan Σ, obtained by gluing the
affine pieces U_σ for σ ∈ Σ along the open immersions U_(σ ∩ τ) ↪ U_σ. -/
noncomputable def ToricVariety (Σ : Fan N) : AlgebraicGeometry.Scheme :=
  AlgebraicGeometry.Scheme.GlueData.glued
    { U := fun σ : Σ.cones => AffineToricVariety σ.val (Σ.stronglyConvex σ.val σ.property)
      V := fun (σ, τ) => AffineToricVariety (σ ∩ τ) sorry
      f := fun (σ, τ) => sorry  -- open immersion from face functoriality
      f_open := sorry
      t := sorry
      cocycle := sorry }

/-- Separatedness: X_Σ is separated as a ℂ-scheme. -/
theorem ToricVariety.separated (Σ : Fan N) :
    AlgebraicGeometry.IsSeparated (ToricVariety Σ) := by
  -- Diagonal U_(σ ∩ τ) ↪ U_σ ×_ℂ U_τ is a closed immersion via the
  -- surjective semigroup-algebra map ℂ[S_σ] ⊗ ℂ[S_τ] ↠ ℂ[S_(σ ∩ τ)].
  sorry

/-- Completeness criterion: X_Σ is proper over ℂ iff |Σ| = N_ℝ. -/
theorem ToricVariety.proper_iff_support_full (Σ : Fan N) :
    AlgebraicGeometry.IsProper (ToricVariety Σ) ↔
      Σ.support = (Set.univ : Set (N ⊗ ℝ)) := by
  -- Valuative criterion applied to torus-valued morphisms; the
  -- cocharacter of an R-valued point lies in some cone iff |Σ| = N_ℝ.
  sorry

/-- Smoothness criterion: X_Σ is smooth iff every σ ∈ Σ is smooth. -/
theorem ToricVariety.smooth_iff_all_cones_smooth (Σ : Fan N) :
    AlgebraicGeometry.IsSmooth (ToricVariety Σ) ↔
      ∀ σ ∈ Σ.cones, σ.SimplicialUnimodular := by
  -- Local smoothness: X_Σ smooth iff every chart U_σ smooth iff every
  -- σ simplicial unimodular by Demazure 1970.
  sorry

Each step is reachable from current Mathlib but requires substantive new development. The fan formalism is the primary missing prerequisite; once that is in place together with the affine-toric-variety construction from [04.11.03], the gluing follows from existing scheme-gluing infrastructure. The separatedness theorem reduces to surjectivity of semigroup-algebra maps, computable via AddMonoidAlgebra machinery. The completeness criterion via the valuative criterion uses Mathlib's IsProper together with DiscreteValuationRing. The smoothness criterion is the global glueing of the local Demazure criterion.

Advanced results [Master]

Theorem (functoriality of the fan-to-toric construction). The assignment $Σ \mapsto X_{Σ}$ extends to a functor from the category $Fan$ of fans (with fan morphisms $(N, Σ) \to (N^{'}, Σ^{'})$ : lattice maps $ϕ : N \to N^{'}$ such that for every $σ \in Σ$ there exists $σ^{'} \in Σ^{'}$ with $ϕ_{R} (σ) \subseteq σ^{'}$ ) to the category of normal $T$ -toric varieties (with $T$ -equivariant morphisms). The functor is fully faithful and essentially surjective (by Sumihiro 1974), making it an equivalence of categories.

The functoriality theorem packages the fan-to-toric correspondence as an equivalence of categories. The morphism side is the content of [04.11.06] (toric morphisms): a lattice map $ϕ : N \to N^{'}$ that sends each cone $σ$ into some cone $σ^{'}$ induces a $T$ -equivariant morphism $X_{Σ} \to X_{Σ^{'}}$ , with the morphism on the affine pieces $U_{σ} \to U_{σ^{'}}$ given by the comorphism on semigroup algebras $C [S_{σ^{'}}] \to C [S_{σ}]$ dual to $ϕ^{*} : (σ^{'})^{\lor} \cap M^{'} \to σ^{\lor} \cap M$ . The essential surjectivity is Sumihiro's classification: every normal $T$ -toric variety arises from a fan, with the fan recovered from the equivariant affine cover.

Theorem (Sumihiro's equivariant covering, refined). Let $X$ be a normal algebraic variety over $C$ with an action of an algebraic torus $T$ such that there is a dense $T$ -orbit. Then $X$ admits a unique-up-to-lattice-isomorphism fan $Σ$ such that $X ≅ X_{Σ}$ $T$ -equivariantly.

Sumihiro's 1974 theorem in its refined form for toric varieties (Journal of Mathematics of Kyoto University 14, 1-28, with refinement in Sumihiro 1975 Equivariant completion II) is the foundational classification theorem of toric geometry. It identifies the category of normal toric varieties with the category of fans on free $Z$ -modules. The proof has two parts: (i) every normal $T$ -variety admits a $T$ -equivariant affine open cover (general Sumihiro), and (ii) for normal toric varieties (with dense $T$ -orbit), each affine piece is an affine toric variety $U_{σ}$ for a unique cone $σ$ (classification of normal affine $T$ -varieties with dense orbit, due to Demazure 1970). Combining these gives the fan as the collection of cones from the affine cover, with faces and intersections coming from the gluing data.

Theorem (projectivity ↔ normal fan of a polytope). Let $Σ$ be a complete fan in $N_{R}$ . Then $X_{Σ}$ is projective if and only if $Σ$ is the normal fan of a full-dimensional lattice polytope $P \subseteq M_{R}$ . In this case, $P$ is unique up to lattice translation, and the polarisation line bundle $L_{P}$ on $X_{Σ}$ has global sections $H^{0} (X_{Σ}, L_{P}) = ⨁_{m \in P \cap M} C χ^{m}$ with basis the lattice points of $P$ .

This is the polytope-fan correspondence on the projective side, proven in part above (Exercise 8) and developed in detail in [04.11.10] (Fulton §3.4; Cox-Little-Schenck §6.1). The bijection between polytopes and projective toric varieties is the algebraic side of the Delzant correspondence in symplectic geometry (Delzant 1988), which classifies symplectic toric manifolds by their moment polytopes. The two correspondences coincide on the projective-symplectic-toric overlap, and the moment map $μ : X_{Σ} \to P$ identifies the polytope with the image of the compact-torus action's moment map.

Theorem (toric resolution; Demazure 1970, KKMS 1973). Every toric variety $X_{Σ}$ admits a $T$ -equivariant resolution of singularities $X_{Σ^{'}} \to X_{Σ}$ , where $Σ^{'}$ is a smooth refinement of $Σ$ obtained by iterated star subdivision. The resolution is proper, birational, and constructive: a finite algorithm produces $Σ^{'}$ from $Σ$ by subdividing non-smooth cones along interior lattice points.

The toric resolution theorem is the combinatorial form of Hironaka's theorem on resolution of singularities (Hironaka 1964, Annals of Mathematics 79, 109-326), and is algorithmically explicit in the toric setting. For surfaces, the toric resolution is the Hirzebruch-Jung resolution via continued-fraction expansion of the singularity's index (Hirzebruch 1953, Mathematische Annalen 126, 1-22). For higher dimensions, the algorithm is more elaborate but still combinatorial (Knudsen 1973, Mumford 1976). The toric setting was the first context where resolution of singularities was understood algorithmically — a fact that motivated much of the development of toric geometry as a testbed for general resolution conjectures.

Theorem (intersection theory on toric varieties; Danilov 1978, Fulton-Sturmfels 1997). For a smooth complete toric variety $X_{Σ}$ , the Chow ring $A^(X_\Sigma) = \bigoplus_{k} A^k(X_\Sigma) $i s g e n er a t e d b y t h ec l a sseso f t or i c d i v i sor s$ D_\rho $f or$ \rho \in \Sigma(1) $, w i t h r e l a t i o n s t h e * * l in e a r e q u i v a l e n ce * * (f r o m t h ee x a c t se q u e n ce$ M \to \mathbb{Z}^{\Sigma(1)} \to \mathrm{Pic}(X_\Sigma) \to 0 $) an d t h e * * S t an l ey - R e i s n er r e l a t i o n s * *$ D_{\rho_1} \cdots D_{\rho_k} = 0 $f or$ {\rho_1, \ldots, \rho_k} $n o t g e n er a t in g a co n eo f$ \Sigma$.*

The Chow-ring formula for smooth complete toric varieties is the Stanley-Reisner ring of the fan $Σ$ , modulo linear relations from $M$ . Danilov's 1978 survey (Russian Math. Surveys 33, 97-154) established this for the rational-coefficient Chow ring; Fulton-Sturmfels 1997 (Topology 36, 335-353) extended it to integer coefficients and gave a combinatorial proof via the localisation theorem for $T$ -equivariant cohomology (Atiyah-Bott 1984, Topology 23, 1-28; Goresky-Kottwitz-MacPherson 1998, Inventiones Mathematicae 131, 25-83). The Stanley-Reisner description makes cohomology computations on toric varieties effective and combinatorial, including Hirzebruch-Riemann-Roch (Brion-Vergne 1997).

Theorem (Cox-quotient presentation; Cox 1995). Let $X_{Σ}$ be a toric variety with $Pic (X_{Σ})$ free abelian. Then $X_{Σ}$ is the GIT quotient $$ X_\Sigma = (\mathbb{C}^{\Sigma(1)} \setminus Z(\Sigma)) / G, $$ where $G = \mathrm{Hom}(\mathrm{Pic}(X_\Sigma), \mathbb{C}^) $a c t so n$ \mathbb{C}^{\Sigma(1)} $v ia t h ee x a c t se q u e n ce$ 0 \to M \to \mathbb{Z}^{\Sigma(1)} \to \mathrm{Pic}(X_\Sigma) \to 0 $, an d$ Z(\Sigma) $i s t h e * * i r r e l e v an tl oc u s * *$ \bigcup_{C \notin \Sigma} V(C) $f or$ C$ ranging over coordinate subspaces.*

Cox's 1995 paper (Journal of Algebraic Geometry 4, 17-50) reframed toric geometry through the homogeneous coordinate ring $C [x_{ρ} : ρ \in Σ (1)]$ — a polynomial ring with one variable per ray of the fan, graded by $Pic (X_{Σ})$ . The toric variety $X_{Σ}$ becomes a GIT quotient of an affine space by a torus, with the quotient stratification controlled by the fan combinatorics. This perspective unifies projective and toric geometry under a single functorial framework and has been the foundation for modern toric mirror symmetry (Batyrev-Borisov 1996), Cox-ring methods in algebraic geometry (Hu-Keel 2000, Michigan Math. Journal 48, 331-348), and computational toric algorithms (Macaulay2, Polymake, SageMath).

Synthesis. The toric variety from a fan $X_{Σ}$ is the combinatorial bridge from convex geometry to global algebraic geometry, and the central insight is that the fan $Σ$ — a finite collection of cones in $N_{R}$ subject to face-closure and intersection-as-face axioms — encodes a global normal $T$ -variety whose every geometric property reads off from the cones. Three structural results — gluing existence (the cocycle condition holds tautologically), separatedness (semigroup-algebra surjectivity at intersections), and Sumihiro's classification (every normal $T$ -toric variety arises this way) — combine into a single equivalence of categories: fans $\leftrightarrow$ normal $T$ -toric varieties. The bridge is exactly the duality of [04.11.01]: the fan in $N_{R}$ encodes which characters of $T$ extend to regular functions on which open piece of $X_{Σ}$ , the dual cones $σ^{\lor} \cap M$ supply the local coordinate rings, and the global variety is assembled from the local pictures along face inclusions.

Putting these together, the four criteria — separatedness, completeness, smoothness, projectivity — fully classify the geometric type of $X_{Σ}$ in terms of $Σ$ : separatedness holds automatically for fans (axiom (ii)), completeness iff $∣Σ∣ = N_{R}$ , smoothness iff every cone is simplicial unimodular, projectivity iff $Σ$ is the normal fan of a polytope. The foundational reason this combinatorial classification is exhaustive is Sumihiro's equivariant-covering theorem: every normal $T$ -variety with a dense orbit is captured by a fan, so no normal toric variety escapes the combinatorial framework. This is exactly what makes toric geometry an effectively combinatorial laboratory for global algebraic geometry, generalises the affine-local picture of [04.11.03] to a global picture, and is dual to the local-to-global passage from cones to fans.

The fan-to-toric construction generalises in three directions worth recording. To non-simplicial fans with non-rational cones, the construction $X_{Σ}$ may still exist as an algebraic space or stack rather than a scheme — the regime of toroidal embeddings (KKMS 1973) and logarithmic toric stacks (Olsson 2008). To infinite fans with countably many cones, the construction extends to ind-toric varieties used in affine Grassmannians and infinite-dimensional Lie theory (Kac-Moody groups, Mathieu 1988). To arbitrary base schemes $S$ instead of $C$ , the construction $X_{Σ} / S$ produces toric schemes over $S$ , specialising to toric varieties over each fibre and degenerating to combinatorial Stanley-Reisner schemes over $Z$ ; this is the foundation for integral models of toric varieties in arithmetic geometry, used by Demazure and Grothendieck in the development of reductive group schemes. The pattern recurs — each generalisation preserves the cone-to-variety dictionary developed here, with technical modifications recording the deviation from the rational case over $C$ .

Full proof set [Master]

Proposition (separatedness of $X_{Σ}$ ), proof. Given in Exercise 4: for any two maximal cones $σ, τ \in Σ$ , the face-correspondence theorem of [04.11.02] gives $(σ \cap τ)^{\lor} = σ^{\lor} + τ^{\lor}$ (Minkowski sum), hence $S_{σ \cap τ} = S_{σ} + S_{τ}$ ; the multiplication map $C [S_{σ}] \otimes C [S_{τ}] \to C [S_{σ \cap τ}]$ is surjective, dualising to a closed immersion of $Spec$ , which is the diagonal $U_{σ \cap τ} \to U_{σ} \times U_{τ}$ ; gluing these closed immersions gives the diagonal of $X_{Σ}$ , so $X_{Σ}$ is separated. $□$

Proposition (completeness criterion), proof. Given in Exercise 7: the valuative criterion applied to torus-valued points translates a $Spec K \to T$ to a cocharacter $v \in N$ ; the extension to $Spec R \to X_{Σ}$ exists iff $v$ lies in some cone of $Σ$ , iff $v \in ∣Σ∣$ ; ranging over all $v \in N$ (achievable by varying the DVR $R$ ), properness holds iff $∣Σ∣ \supseteq N$ , iff $∣Σ∣ = N_{R}$ . $□$

Proposition (smoothness criterion), proof. Given in part (d) of the main theorem: smoothness is local, $X_{Σ}$ is smooth iff every chart $U_{σ}$ is smooth, and $U_{σ}$ is smooth iff $σ$ is simplicial unimodular by the smoothness criterion of [04.11.03] (Demazure 1970). Hence $X_{Σ}$ is smooth iff every cone is simplicial unimodular. $□$

Proposition (projectivity criterion), proof. Given in Exercise 8: ( $\Leftarrow$ ) The normal fan $Σ_{P}$ of a lattice polytope $P$ gives a complete fan; the polarisation line bundle $L_{P}$ has lattice-point basis $H^{0} (X_{Σ}, L_{P}) = ⨁_{m \in P \cap M} C χ^{m}$ , and when $P$ has full $n$ -dimensional interior the associated morphism $X_{Σ} \to P^{∣ P \cap M ∣ - 1}$ is a closed embedding. ( $\Rightarrow$ ) Conversely, an ample $T$ -equivariant line bundle on $X_{Σ}$ has Newton polytope $P_{L}$ a lattice polytope, and the vertex cones of $P_{L}$ recover the maximal cones of $Σ$ , exhibiting $Σ$ as the normal fan of $P_{L}$ . $□$

Proposition (Sumihiro's classification), proof sketch. Let $X$ be a normal algebraic variety over $C$ with a $T$ -action having a dense open orbit. Then there is a fan $Σ$ in $N_{R}$ , unique up to lattice isomorphism, with $X ≅ X_{Σ}$ $T$ -equivariantly.

Proof. By Sumihiro's general equivariant covering theorem (1974, Journal of Mathematics of Kyoto University 14, 1-28), $X$ admits a $T$ -equivariant cover by $T$ -stable affine open subvarieties ${V_{i}}$ . Each $V_{i}$ is normal (subvariety of $X$ ), affine (by construction), and has a $T$ -action; the dense $T$ -orbit of $X$ restricts to a dense $T$ -orbit of $V_{i}$ (since $V_{i}$ is open and contains points of the dense orbit). Hence each $V_{i}$ is a normal affine $T$ -variety with dense $T$ -orbit.

By the classification of affine toric varieties from [04.11.03] (Demazure 1970), every normal affine $T$ -variety with dense orbit is of the form $U_{σ}$ for a unique strongly convex rational polyhedral cone $σ \subseteq N_{R}$ (where $N$ is the cocharacter lattice of $T$ ). Hence $V_{i} = U_{σ_{i}}$ for some cone $σ_{i}$ .

For the overlaps: $V_{i} \cap V_{j}$ is open in both $V_{i}$ and $V_{j}$ , $T$ -stable, and normal — hence also a normal affine $T$ -variety with dense orbit (the dense orbit of $X$ restricts to a dense orbit of the overlap). By the classification, $V_{i} \cap V_{j} = U_{τ_{ij}}$ for some cone $τ_{ij}$ . The two open immersions $U_{τ_{ij}} ↪ U_{σ_{i}}$ and $U_{τ_{ij}} ↪ U_{σ_{j}}$ correspond (by face-functoriality, [04.11.03]) to face inclusions $τ_{ij} ↪ σ_{i}$ and $τ_{ij} ↪ σ_{j}$ . The cone $τ_{ij}$ is therefore a common face of $σ_{i}$ and $σ_{j}$ , and by uniqueness of the affine cone for a normal affine $T$ -variety with dense orbit, $τ_{ij} = σ_{i} \cap σ_{j}$ (intersection as cones in $N_{R}$ ).

The collection $Σ := {all faces of σ_{i} for all i}$ is closed under faces (by definition) and intersection (since $σ_{i} \cap σ_{j}$ is a common face, hence belongs to $Σ$ ). Hence $Σ$ is a fan, and the gluing of the $V_{i}$ along the overlaps $V_{i} \cap V_{j} = U_{σ_{i} \cap σ_{j}}$ produces $X = X_{Σ}$ .

Uniqueness: the cones $σ_{i}$ are determined by the closed $T$ -fixed points of $X$ (each $σ_{i}$ has its unique $T$ -fixed point at the closed orbit) and the orbit structure (orbits in bijection with faces of $σ_{i}$ , orbit-cone correspondence [04.11.05]). A $T$ -equivariant isomorphism $X ≅ X^{'}$ induces an isomorphism of cocharacter lattices $N ≅ N^{'}$ and a bijection of cones $Σ ≅ Σ^{'}$ preserving the face structure — that is, a fan isomorphism. $□$

Proposition (functoriality $Σ \mapsto X_{Σ}$ ), proof. The assignment $Σ \mapsto X_{Σ}$ extends to a functor $Fan \to TorVar$ to the category of normal $T$ -toric varieties, with morphisms induced by lattice maps respecting cones.

Proof. Given a fan morphism $ϕ : (N, Σ) \to (N^{'}, Σ^{'})$ (a $Z$ -linear map $ϕ : N \to N^{'}$ such that for every $σ \in Σ$ there exists $σ^{'} \in Σ^{'}$ with $ϕ_{R} (σ) \subseteq σ^{'}$ ), construct the toric morphism $X_{ϕ} : X_{Σ} \to X_{Σ^{'}}$ on each affine piece by the comorphism on semigroup algebras $$ \phi^* : \mathbb{C}[S_{\sigma'}] \to \mathbb{C}[S_\sigma], \qquad \chi^{m'} \mapsto \chi^{\phi^(m')}, $$ where $\phi^ : M' \to M $i s t h e d u a l l a tt i ce ma p . T h e ma p$ \phi^ $se n d s$ S_{\sigma'} = (\sigma')^\vee \cap M' $in t o$ S_\sigma = \sigma^\vee \cap M $b y t h eco n d i t i o n$ \phi_\mathbb{R}(\sigma) \subseteq \sigma' $: f or$ m' \in (\sigma')^\vee $,$ \langle m', \phi_\mathbb{R}(u)\rangle \geq 0 $f or$ u \in \sigma $, e q u i v a l e n tl y$ \langle \phi^(m'), u\rangle \geq 0 $, e q u i v a l e n tl y$ \phi^*(m') \in \sigma^\vee$.

The local morphisms $U_{σ} \to U_{σ^{'}}$ are compatible with face inclusions (since $ϕ_{R}$ preserves face containment by linearity), so they glue to a global morphism $X_{ϕ} : X_{Σ} \to X_{Σ^{'}}$ . The functor laws (identity, composition) follow from the corresponding laws on lattice maps. $T$ -equivariance follows from the $M$ -grading being preserved by $ϕ^{*}$ .

The functor is faithful (lattice maps are determined by their action on $M$ , recoverable from the morphism of varieties via the $T$ -character group of the open orbit) and full (every $T$ -equivariant morphism $X_{Σ} \to X_{Σ^{'}}$ arises from a fan morphism, by the classification of $T$ -equivariant morphisms between normal toric varieties — Cox-Little-Schenck Theorem 3.3.4). Hence the functor is fully faithful.

Essential surjectivity is Sumihiro's classification: every normal $T$ -toric variety arises as $X_{Σ}$ for some fan. Combined with full faithfulness, the functor is an equivalence of categories. $□$

Connections [Master]

Affine toric variety 04.11.03. The local building blocks of $X_{Σ}$ . Each cone $σ \in Σ$ contributes an affine chart $U_{σ} = Spec C [σ^{\lor} \cap M]$ , and the face-functoriality $τ \subseteq σ \Rightarrow U_{τ} ↪ U_{σ}$ developed in the prerequisite unit supplies the open immersions along which the affine pieces glue. The toric variety $X_{Σ}$ is exactly the colimit of these affine pieces along their face-inclusion morphisms. Sumihiro's classification reduces the global toric problem to the affine case by the equivariant covering theorem, making [04.11.03] the technical core of this unit.
Rational polyhedral cone and dual cone 04.11.02. The combinatorial input of the fan-to-toric construction. The face-correspondence theorem of the prerequisite unit gives $(σ \cap τ)^{\lor} = σ^{\lor} + τ^{\lor}$ , which is the foundational reason that the diagonal $U_{σ \cap τ} \to U_{σ} \times U_{τ}$ is a closed immersion — the central step in the separatedness theorem for $X_{Σ}$ . Gordan's lemma supplies finite generation of the dual semigroups, hence finite-type-ness of $X_{Σ}$ .
Algebraic torus and character/cocharacter lattices 04.11.01. The torus $T = Spec C [M]$ acts on $X_{Σ}$ with a dense open orbit equal to $T$ itself (the chart $U_{{0}}$ corresponding to the zero cone, shared by every $U_{σ}$ ). The character lattice $M$ encodes the regular functions on $X_{Σ}$ that are $T$ -semi-invariant — namely those characters $m \in M$ such that $χ^{m}$ extends to a regular function on $U_{σ}$ for some $σ$ , equivalently $m \in σ^{\lor} \cap M$ for that cone. The cocharacter lattice $N$ encodes the one-parameter subgroups of $T$ , whose limit behaviour in $X_{Σ}$ records the orbit-cone correspondence of [04.11.06].
Projective space 04.07.01. The most fundamental toric variety beyond affine space: $P^{n} = X_{Σ_{P^{n}}}$ for the fan $Σ_{P^{n}}$ with $n + 1$ rays in $R^{n}$ (the standard basis together with $- e_{1} - \dots - e_{n}$ ) and all maximal cones spanned by proper subsets of these rays. The standard affine cover of $P^{n}$ by $n + 1$ copies of $C^{n}$ is recovered as the affine cover ${U_{σ}}$ for the maximal cones of $Σ_{P^{n}}$ . The Segre embedding of $P^{a} \times P^{b}$ corresponds to the product fan, illustrating the general pattern that products of toric varieties correspond to products of fans.
Smoothness and completeness via fans 04.11.05. The downstream sibling unit that turns the criteria stated in parts (c)–(d) of the key theorem into the two geometric tests of the fan-to-toric dictionary: $X_{Σ}$ is complete iff $∣Σ∣ = N_{R}$ , and $X_{Σ}$ is smooth iff every cone is simplicial unimodular. The valuative-criterion argument and the local Demazure criterion that appear here are developed at full length in [04.11.05], with the smoothness criterion extended to the global $X_{Σ}$ via the affine cover from this unit. The foundational reason these two tests are exhaustive is the fan-to-toric correspondence proved here.
Orbit-cone correspondence 04.11.06. The next sibling unit: $T$ -orbits in $X_{Σ}$ are in inclusion-reversing bijection with cones of $Σ$ , with the orbit corresponding to a cone $σ$ of dimension $k$ having dimension $n - k$ . The orbit-closure stratification of $X_{Σ}$ matches the face-inclusion stratification of $Σ$ . The proof uses the face-functoriality of [04.11.03] together with the gluing developed here to identify orbits as locally affine pieces of the global toric variety.
Toric resolution of singularities 04.11.07. Every toric variety $X_{Σ}$ admits a $T$ -equivariant resolution $X_{Σ^{'}} \to X_{Σ}$ by smooth refinement of the fan via iterated star subdivision (Demazure 1970, KKMS 1973). The combinatorial nature of toric resolution — algorithmic where Hironaka's general resolution is non-constructive — made toric geometry the first testbed for resolution-of-singularities conjectures, and remains a working laboratory for understanding the geometry of birational maps. For toric surfaces, the resolution is the Hirzebruch-Jung continued-fraction algorithm.
Toric divisor and support function 04.11.08. The $T$ -invariant prime divisors of $X_{Σ}$ are exactly the orbit closures $D_{ρ} = V (ρ)$ for rays $ρ \in Σ (1)$ , with the divisor lattice $⨁_{ρ} Z D_{ρ}$ generated by ray data of the fan. Cartier divisors on $X_{Σ}$ are encoded by support functions — piecewise-linear functions on $∣Σ∣$ that are linear on each cone — and the chart-cocycle compatibility uses precisely the overlap structure $U_{σ \cap τ} ↪ U_{σ}$ developed here. The toric-divisor unit builds the divisorial layer on top of the fan-to-toric construction.
Toric Picard group 04.11.09. The Picard exact sequence $0 \to M \to ⨁_{ρ} Z D_{ρ} \to Pic (X_{Σ}) \to 0$ presents the Picard group of $X_{Σ}$ as a finite quotient of free abelian groups, with the lattice data $(M, N, Σ)$ supplying every ingredient. The toric ampleness criterion (strict convexity of the support function on $Σ$ ) sharpens the projectivity criterion proved here, identifying the ample cone in $Pic (X_{Σ}) \otimes R$ as a rational polyhedral cone computable from fan data. The toric Picard unit is the line-bundle classification built directly on the fan-to-toric dictionary.
Polytope-fan correspondence 04.11.10. Projective toric varieties are classified by lattice polytopes via $P \mapsto Σ_{P}$ (normal fan) and $Σ \mapsto P$ (Newton polytope of an ample line bundle). This is the algebraic side of the Delzant correspondence on the symplectic side (Delzant 1988), and the polytope $P$ records both the toric variety and an ample polarisation simultaneously. The lattice points $P \cap M$ are an explicit basis of global sections of the polarisation $L_{P}$ , making polytopes a working tool for computing cohomology of projective toric varieties.
Scheme 04.02.01 and affine scheme 04.02.02. The toric variety $X_{Σ}$ is a separated normal scheme of finite type over $C$ , with the standard scheme-theoretic structure inherited from the gluing of affine schemes $U_{σ}$ . The gluing lemma for schemes (Grothendieck's EGA I §0.4 or Hartshorne Ex. II.2.12) is the foundational tool for the fan-to-toric construction, and the cocycle condition is verified automatically by the fan axioms together with the localisation structure of the affine pieces from [04.11.03].
Resolution of singularities 04.06.02. The toric resolution theorem (every $X_{Σ}$ admits a $T$ -equivariant resolution by fan refinement) is the toric implementation of Hironaka's 1964 theorem on resolution of singularities over fields of characteristic zero. The toric case is algorithmic: a constructive procedure produces the smooth refinement in finitely many steps, whereas Hironaka's general proof is a non-constructive Noetherian-induction argument. The toric setting was the first context where resolution was understood algorithmically, and the algorithmic perspective has since motivated significant research on constructive resolution (Bierstone-Milman 1997, Encinas-Villamayor 2003, Włodarczyk 2005).
Mikhalkin's correspondence theorem 04.12.05. The downstream tropical-enumerative-geometry unit takes the toric surface $X_{Σ}$ from the inner-normal fan of a Newton polygon $Δ \subseteq R^{2}$ as the algebraic-geometric home for complex algebraic curves of Newton polygon $Δ$ . The unbounded rays of a tropical curve in $R^{2}$ correspond bijectively to the rays of the fan $Σ$ , with the multiset of primitive unbounded directions matching the primitive normal vectors to the edges of $Δ$ . Mikhalkin's count of complex algebraic curves through generic points in $X_{Σ}$ equals the tropical count on the dual side, with each tropical curve weighted by a lattice-multiplicity product over its vertices. The fan-and-toric-variety dictionary of this unit is the structural backbone on which the surface-level correspondence sits.
Nishinou-Siebert correspondence 04.12.06. The higher-dimensional generalisation of Mikhalkin uses the toric variety $X_{Σ}$ from a complete simplicial fan $Σ$ in $N_{R}$ as the target of a toric degeneration $X \to Spec C {{t}}$ whose special fibre is the union of toric components indexed by maximal cones of $Σ$ . The dual intersection complex $B (Ξ)$ — built from the polyhedral subdivision $Ξ$ refining a polytope of an ample line bundle on $X_{Σ}$ — is the polyhedral base on which tropical curves are counted, and the Nishinou-Siebert lift assembles complex algebraic curves on $X_{Σ}$ from these tropical curves via log Gromov-Witten degeneration. The fan-to-toric-variety construction of this unit is the foundational input to the toric-degeneration mechanism that powers the higher-dimensional correspondence.
Toric degeneration of a Calabi-Yau variety 04.12.07. The downstream Calabi-Yau-degeneration unit builds its central fibre $X_{0} = ⋃_{i} X_{Δ_{i}}$ as a union of toric varieties indexed by cells of a polyhedral subdivision $P$ : each toric component $X_{Δ_{i}}$ is constructed from its cell $Δ_{i}$ via the polytope-fan dictionary specialised on the fan-to-toric construction of this unit. The polytope-fan construction is the foundational reason the toric-degeneration setup of [04.12.07] works: without the fan-to-toric dictionary developed here, there would be no way to assemble the central fibre from polyhedral subdivision data.
Dual intersection complex; tropical manifold $B$ 04.12.08. The downstream dual-intersection-complex unit builds $(B, P)$ from the dual cones $σ_{Δ_{i}}^{\lor}$ assembled into a (non-fan) cone complex; the building blocks are strongly convex rational polyhedral cones with face relations — exactly the fan-theoretic cones of this unit. The dual intersection complex is not itself a fan (it carries additional integral affine and singularity structure), but its combinatorial primitives are the fan-cones developed here, and the fan-to-toric construction is the foundational reason each maximal cell $σ \in P^{[n]}$ corresponds to an affine toric piece of the central fibre.
Gross-Siebert reconstruction theorem 04.12.09. The downstream reconstruction theorem reconstructs a toric degeneration of Calabi-Yau varieties whose central fibre $X_{0} = ⋃_{σ} X_{σ}$ is a union of toric varieties indexed by maximal cells of the tropical-manifold polyhedral decomposition $P$ . Each $X_{σ}$ is constructed from the cell $σ$ by the fan-to-toric construction of this unit, and the reconstruction theorem builds on this fan-and-toric primitive to produce the formal smoothing. The fan-and-toric dictionary is the structural backbone on which the Gross-Siebert reconstruction is built.
Slab function and structure of a tropical manifold 04.12.11. The downstream slab-function unit places, on every maximal cell $σ \in P^{[n]}$ of $(B, P)$ , the affine toric variety $X_{σ} = Spec C [Λ_{σ}]$ — the toric variety of the apex fan in $Λ_{σ}^{\lor}$ via the fan-to-toric construction of this unit. The slab functions paste these affine toric pieces into the central fibre $X_{0}$ exactly as the fan-gluing prescription developed here pastes affine pieces $U_{σ}$ into $X_{Σ}$ . The structure $S$ then deforms this fan-gluing pasting order-by-order in $t$ .

Historical & philosophical context [Master]

The toric variety from a fan was first constructed in modern form 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 motivation was the classification of algebraic subgroups of the Cremona group $Cr_{n} = Bir (P^{n})$ of maximal rank, and the fan-to-toric construction arose as the combinatorial language for enumerating the $T$ -equivariant compactifications of the torus $T = (C^{*})^{n}$ . Demazure's paper established the gluing construction, the smoothness criterion (every cone simplicial unimodular), and the basic functoriality on fan morphisms. The term "espace torique" ("toric space" in English) is due to Demazure.

The scheme-theoretic foundations were systematised by George Kempf, Finn Faye Knudsen, David Mumford, and Bernard Saint-Donat in Toroidal Embeddings I (Lecture Notes in Mathematics 339, Springer 1973) ^[pending], where the fan formalism was recast as a tool for resolving singularities and compactifying equivariant varieties over arbitrary base schemes. KKMS's "toroidal embedding" is the local-étale model for varieties that look étale-locally like toric varieties; the global toric setting (where the variety is genuinely a fan-gluing) is the rigid combinatorial special case. The KKMS volume also established the toric resolution theorem — every fan admits a smooth refinement obtained by iterated star subdivision — making toric geometry the first testbed for resolution-of-singularities algorithms.

Hideyasu Sumihiro's Equivariant completion (Journal of Mathematics of Kyoto University 14, 1974, pp. 1-28) ^{[Sumihiro 1974]} proved the foundational equivariant-covering theorem: every normal $T$ -variety admits a $T$ -equivariant affine open cover. Applied to normal toric varieties, this classifies them as gluings $X_{Σ}$ from fans, completing the bijection between fans and normal toric varieties. Sumihiro's paper is the bridge from the affine-local picture of Demazure to the global classification — without Sumihiro's theorem, one would have only the affine-local picture; with it, every normal toric variety is captured by a fan. Sumihiro's 1975 follow-up Equivariant completion II (Journal of Mathematics of Kyoto University 15, 573-605) extended the theorem to non-toric reductive-group actions.

The Japanese-school perspective was consolidated by Tadao Oda in Convex Bodies and Algebraic Geometry (Ergebnisse 15, Springer 1988) ^[pending], an authoritative monograph emphasising the dual-cone formalism, the global toric construction, the orbit-cone correspondence, and the polytope-fan dictionary. Oda's book is the canonical mid-1980s reference and was the standard text until Fulton's 1993 volume superseded it for many uses. Vasily Danilov's 1978 survey The geometry of toric varieties (Russian Mathematical Surveys 33(2), 97-154) ^{[Danilov 1978]} is the canonical pre-Fulton English survey, explicitly cited by Fulton as the immediate predecessor; Danilov's article also established the Stanley-Reisner description of toric cohomology.

William Fulton's Introduction to Toric Varieties (Princeton 1993) ^{[Fulton 1993]} is the canonical short textbook of modern toric geometry, organising the cone-fan-variety dictionary into a coherent narrative covering smoothness, completeness, projectivity, divisors, line bundles, cohomology, and the polytope correspondence. David Cox, John Little, and Henry Schenck's Toric Varieties (American Mathematical Society 2011) ^[pending] is the modern thousand-page treatment, extending Fulton with additional material on Cox-quotients, toric mirror symmetry, GKZ-systems, and arithmetic-geometric applications.

The fan-to-toric construction took its modern functorial form in David Cox's 1995 paper The homogeneous coordinate ring of a toric variety (Journal of Algebraic Geometry 4, 17-50) ^{[Cox 1995]}, where the Cox quotient $X_{Σ} = (C^{Σ (1)} ∖ Z (Σ)) / G$ presents the toric variety as a GIT quotient of an affine space by a torus. This perspective unifies projective and toric geometry under a common functorial framework and has been the foundation for modern toric mirror symmetry (Batyrev-Borisov 1996), Cox-ring methods in algebraic geometry, and computational toric algorithms in Macaulay2, Polymake, and SageMath.

Friedrich Hirzebruch's introduction of the surfaces $F_{a}$ in Über eine Klasse von einfach-zusammenhängenden komplexen Mannigfaltigkeiten (Mathematische Annalen 124, 1951, pp. 77-86) ^{[Hirzebruch 1951]} predates the toric formalism by two decades but was identified as a toric example by Demazure (1970), illustrating the general pattern that classical examples from algebraic surface theory often have toric models with explicit fan presentations. The toric interpretation of the Hirzebruch surfaces $F_{a} = X_{Σ_{a}}$ (Exercise 3) is now a standard fixture of introductory toric-variety pedagogy.

Bibliography [Master]

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

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

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

@article{Sumihiro1975,
  author  = {Sumihiro, Hideyasu},
  title   = {Equivariant completion II},
  journal = {Journal of Mathematics of Kyoto University},
  volume  = {15},
  year    = {1975},
  pages   = {573--605}
}

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

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

@book{FultonToric,
  author    = {Fulton, William},
  title     = {Introduction to Toric Varieties},
  publisher = {Princeton University Press},
  series    = {Annals of Mathematics Studies},
  volume    = {131},
  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{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{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{FultonSturmfels1997,
  author  = {Fulton, William and Sturmfels, Bernd},
  title   = {Intersection theory on toric varieties},
  journal = {Topology},
  volume  = {36},
  year    = {1997},
  pages   = {335--353}
}

@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}
}

Prerequisites

04.02.01
04.02.02
04.07.01
04.11.01
04.11.02
04.11.03

Tier anchors

beginner: Fulton *Introduction to Toric Varieties* §1.4 informal — the fan as a finite collection of cones glued into a global toric variety
intermediate: Fulton *Introduction to Toric Varieties* §1.4 + §2.4; Cox-Little-Schenck *Toric Varieties* §3.1
master: Fulton §1.4, §2.4; Cox-Little-Schenck §3.1, §6.1; Oda *Convex Bodies and Algebraic Geometry* Ch. 1 §1.4-1.5; Kempf-Knudsen-Mumford-Saint-Donat *Toroidal Embeddings I* Ch. I §2; Demazure 1970 *Sous-groupes algébriques de rang maximum du groupe de Cremona*; Sumihiro 1974 *Equivariant completion*; Danilov 1978 *The geometry of toric varieties*

References

TODO_REF
Fulton — Introduction to Toric Varieties · §1.4 (fans and the variety $X_\Sigma$); §2.4 (completeness criterion); §3.4 (projectivity via polytopes); §2.1 (smoothness via regular cones)
TODO_REF
Cox-Little-Schenck — Toric Varieties · §3.1 (the toric variety from a fan); Theorem 3.1.5 (gluing affine pieces $U_\sigma$); §3.4 (completeness criterion); §6.1 (projectivity via normal fans of lattice polytopes)
TODO_REF
Oda — Convex Bodies and Algebraic Geometry · Ch. 1 §1.4 (fans); Theorem 1.13 ($X_\Sigma$ is separated); §1.5 (completeness, projectivity, smoothness criteria)
TODO_REF
Kempf-Knudsen-Mumford-Saint-Donat — Toroidal Embeddings I · Ch. I §2 (toroidal embeddings as gluings of affine $U_\sigma$ pieces); the scheme-theoretic gluing construction for arbitrary fans
TODO_REF
Demazure 1970 — Sous-groupes algébriques de rang maximum du groupe de Cremona · *Annales scientifiques de l'École normale supérieure* (4) 3, 507-588; the cone-and-fan formalism; the global toric variety $X_\Sigma$ as the gluing of $U_\sigma$ along face inclusions
TODO_REF
Sumihiro 1974 — Equivariant completion · *Journal of Mathematics of Kyoto University* 14, 1-28; theorem that every normal $T$-variety arises as $X_\Sigma$ for a unique fan up to lattice isomorphism
TODO_REF
Danilov 1978 — The geometry of toric varieties · *Russian Mathematical Surveys* 33(2), 97-154; the canonical pre-Fulton English survey of the cone-and-fan formalism; explicit cited by Fulton as the immediate predecessor
TODO_REF
Hirzebruch 1951 — Über eine Klasse von einfach-zusammenhängenden komplexen Mannigfaltigkeiten · *Mathematische Annalen* 124, 77-86; introduction of the Hirzebruch surfaces $F_a$ as $\mathbb{P}^1$-bundles over $\mathbb{P}^1$, recovered toric-geometrically as $X_{\Sigma_a}$ for the four-ray fan in $\mathbb{R}^2$
TODO_REF
Mumford 1972 — An analytic construction of degenerating Abelian varieties over complete rings · *Compositio Mathematica* 24, 239-272; precursor to KKMS using cone-and-lattice data to construct degenerations; the global gluing $X_\Sigma$ appears as the model fibre
TODO_REF
Cox 1995 — The homogeneous coordinate ring of a toric variety · *Journal of Algebraic Geometry* 4, 17-50; the global toric variety $X_\Sigma$ presented as a GIT quotient $(\mathbb{C}^{\Sigma(1)} \setminus Z(\Sigma)) / G$, the Cox-construction reframing of the gluing

Reviewer

TBD

Estimated time

beginner: 20m
intermediate: 50m
master: 85m