04.11.06 · algebraic-geometry / toric

Orbit-cone correspondence

shipped3 tiersLean: partial

Anchor (Master): Fulton §3.1, §3.2; Cox-Little-Schenck §3.2; Oda *Convex Bodies and Algebraic Geometry* Ch. 1 §1.6; Demazure 1970 *Sous-groupes algébriques de rang maximum du groupe de Cremona* (Ann. Sci. ENS 3, 507-588); Sumihiro 1974 *Equivariant completion* (J. Math. Kyoto Univ. 14, 1-28); Danilov 1978 *The geometry of toric varieties* (Russian Math. Surveys 33(2), 97-154); Białynicki-Birula 1973 *Some theorems on actions of algebraic groups* (Ann. of Math. 98, 480-497)

Intuition [Beginner]

A toric variety is built by gluing affine charts indexed by the cones of a fan, and the central torus $T$ acts on the whole thing. The orbit-cone correspondence says that the orbits of this action are in exact bijection with the cones of the fan. The bigger the cone, the smaller the orbit: a cone of dimension $k$ in an $n$ -dimensional lattice corresponds to an orbit of dimension $n - k$ . The zero cone produces the open dense torus itself, of dimension $n$ ; an $n$ -dimensional cone produces a single fixed point. Every other cone falls somewhere in between.

The picture sharpens with a concrete example. Projective space $P^{2}$ has a fan with one zero cone, three rays, and three two-dimensional cones. The correspondence gives seven orbits: one open dense $T = (C^{*})^{2}$ from the zero cone, three one-dimensional orbits (the affine lines along the coordinate axes minus their corners) from the rays, and three fixed points (the corners of the coordinate triangle) from the maximal cones. The seven cones index seven orbits, and the dimensions match: 2, 1, 1, 1, 0, 0, 0.

The reason the correspondence holds is that each cone $σ$ produces an affine chart $U_{σ}$ in which the torus action specialises to a smaller torus action; the unique closed orbit of the smaller torus is $O (σ)$ . Face containment of cones gives chart inclusions of affine pieces, and chart inclusions give closure relations on orbits. The whole toric variety stratifies into locally closed pieces indexed by cones — a combinatorial decomposition that controls cohomology, intersection numbers, and birational geometry of $X_{Σ}$ .

Visual [Beginner]

A diagram in two panels showing the orbit decomposition of $P^{2}$ alongside its fan. The left panel shows the fan in $R^{2}$ : three rays $ρ_{0} = - e_{1} - e_{2}$ , $ρ_{1} = e_{1}$ , $ρ_{2} = e_{2}$ from the origin, dividing the plane into three two-dimensional cones $σ_{0}, σ_{1}, σ_{2}$ . Each cone is labelled with its corresponding orbit, and the dimensions sum to $2$ in each pair.

The right panel shows the projective plane $P^{2}$ as a triangle with three vertices and three edges. The interior of the triangle is the open dense torus $T = (C^{*})^{2}$ of dimension $2$ , corresponding to the zero cone. The three edges (minus their endpoints) are the three one-dimensional orbits, each a copy of $C^{*}$ , corresponding to the three rays of the fan. The three vertices of the triangle are the three fixed points, each corresponding to one of the two-dimensional maximal cones.

$A two-panel diagram showing the fan of $\mathbb{P}^2$ on the left with three rays partitioning $\mathbb{R}^2$ into three maximal cones, and the toric variety $\mathbb{P}^2$ on the right as a triangle with three two-cells, three one-cells, and three zero-cells matched to the cones of the fan.$

Between the two panels, an arrow labelled " $σ \leftrightarrow O (σ)$ " records the bijection, and a side annotation " $dim O (σ) + dim σ = 2$ " records the dimension formula in this rank- $2$ case.

Worked example [Beginner]

Compute the orbit decomposition of $P^{1}$ and verify the dimension formula. Take $N = Z$ with $N_{R} = R$ , and let $Σ$ have three cones: the zero cone ${0}$ , the positive ray $σ_{+} = R_{\geq 0}$ , and the negative ray $σ_{-} = R_{\leq 0}$ . The toric variety is the projective line $P^{1}$ .

Step 1. The orbit of the zero cone ${0}$ is the open dense torus $T = C^{*}$ of dimension $1$ . Its dimension is $1$ and the cone's dimension is $0$ , so $1 + 0 = 1 = n$ . The cone is the smallest object and the orbit is the largest.

Step 2. The orbit of the positive ray $σ_{+}$ is a single point — the point at infinity in the positive direction, $[0 : 1] \in P^{1}$ . Its dimension is $0$ and the cone's dimension is $1$ , so $0 + 1 = 1 = n$ . Similarly the negative ray $σ_{-}$ gives the other fixed point $[1 : 0]$ .

Step 3. The orbit count is three: one open torus plus two fixed points. The bijection is order-reversing: the zero cone is a face of both maximal cones, and the open torus has both fixed points in its closure $\overline{T} = P^{1}$ .

What this tells us. The projective line decomposes into three torus orbits matched to the three cones of its fan. The dimensions of the orbit and the cone always add to $n = 1$ , the rank of the lattice. The open orbit corresponds to the smallest cone (the origin), and the two fixed points correspond to the two maximal cones. The same pattern works for every toric variety: cones index orbits, and the dimensions complement to $n$ .

Check your understanding [Beginner]

Exercise (easy, multiple choice).

For a toric variety $X_{Σ}$ associated to a fan $Σ$ in an $n$ -dimensional lattice, the orbit-cone correspondence sends a cone $σ \in Σ$ of dimension $k$ to an orbit of dimension:

A. $k$ . B. $n + k$ . C. $n - k$ . D. $n \cdot k$ .

Hint

The dimensions of the cone and the orbit are complementary: they add to the rank $n$ of the lattice. So the orbit's dimension is the lattice rank minus the cone's dimension.

Answer

C. The orbit-cone correspondence sends a cone $σ$ of dimension $k$ to an orbit $O (σ)$ of dimension $n - k$ . The dimension formula is $dim O (σ) + dim σ = n$ . Option A would say the orbit and the cone have equal dimension; the correspondence is in fact dimension-reversing, not dimension-preserving. Option B would push the orbit dimension above $n$ , which is impossible since orbits sit inside the $n$ -dimensional variety $X_{Σ}$ . Option D is dimensionally incoherent. The complementary-dimension formula is the keystone numerical identity of the orbit-cone correspondence and is the foundational reason that the zero cone produces the maximal orbit (the open dense torus) and that $n$ -dimensional maximal cones produce fixed points.

Exercise (easy, multiple choice).

The toric variety $P^{2}$ has how many torus orbits in total?

A. Three. B. Five. C. Seven. D. Nine.

Hint

The fan of $P^{2}$ has the zero cone, three rays, and three two-dimensional maximal cones. Count them all.

Answer

C. The fan of $P^{2}$ has seven cones: the zero cone ${0}$ , the three rays $ρ_{0}, ρ_{1}, ρ_{2}$ , and the three maximal two-dimensional cones $σ_{0}, σ_{1}, σ_{2}$ . By the orbit-cone correspondence, each cone produces one torus orbit. So $P^{2}$ has seven $T$ -orbits: one open dense torus of dimension $2$ (from the zero cone), three orbits of dimension $1$ (one per ray — the affine lines forming the coordinate triangle), and three fixed points of dimension $0$ (one per maximal cone — the corners of the coordinate triangle). The orbit dimensions $2, 1, 1, 1, 0, 0, 0$ each complement the corresponding cone dimensions $0, 1, 1, 1, 2, 2, 2$ to the rank $n = 2$ .

Exercise (easy, true-false).

True or false: the open dense torus of a toric variety $X_{Σ}$ is the orbit corresponding to the zero cone ${0}$ of the fan.

Hint

The zero cone has dimension $0$ . By the dimension formula, the corresponding orbit has dimension $n - 0 = n$ , the full dimension of the variety. The only orbit of full dimension is the open dense torus.

Answer

True. The zero cone ${0}$ is the unique cone of dimension $0$ in every fan, and by the dimension formula it corresponds to the unique orbit of full dimension $n$ — namely the open dense torus $T = (C^{*})^{n}$ of $X_{Σ}$ . The zero cone is a face of every cone in the fan, so the corresponding orbit (the dense torus) is contained in the closure of every other orbit; equivalently, the closure of $O ({0}) = T$ is all of $X_{Σ}$ because $T$ is open and dense. This is the order-reversing property of the bijection: the smallest cone produces the largest orbit, and that orbit's closure is everything.

Formal definition [Intermediate+]

Let $N$ be a lattice of rank $n$ with dual lattice $M = Hom_{Z} (N, Z)$ . Let $Σ$ be a fan in $N_{R} = N \otimes_{Z} R$ with associated toric variety $X_{Σ}$ and dense open algebraic torus $T = Spec C [M] ≅ (C^{*})^{n}$ , as constructed in [04.11.04].

Definition (orbit of a cone). For each cone $σ \in Σ$ of dimension $k$ , define $$ \sigma^\perp := {m \in M_\mathbb{R} : \langle m, u\rangle = 0 \text{ for all } u \in \sigma}, $$ the orthogonal complement of $σ$ in $M_{R}$ . The lattice $σ^{⊥} \cap M$ is free abelian of rank $n - k$ . The orbit of $σ$ is $$ O(\sigma) := \mathrm{Hom}_\mathbb{Z}(\sigma^\perp \cap M, \mathbb{C}^) \cong (\mathbb{C}^)^{n - k}, $$ an algebraic torus of dimension $n - k$ . Equivalently, $O (σ) = Spec C [σ^{⊥} \cap M]$ is the spectrum of the group algebra of the orthogonal lattice $σ^{⊥} \cap M$ .

Definition (orbit closure). The orbit closure is $$ V(\sigma) := \overline{O(\sigma)} \subseteq X_\Sigma, $$ the Zariski closure of $O (σ)$ in $X_{Σ}$ . By construction $V (σ)$ contains $O (σ)$ as a dense open subvariety and is itself a closed irreducible subvariety of $X_{Σ}$ of dimension $n - k$ .

Definition (star and star quotient). The star of $σ$ in $Σ$ is the subfan $$ \mathrm{Star}(\sigma) := {\tau \in \Sigma : \sigma \text{ is a face of } \tau}. $$ The star quotient fan $Σ/ σ$ lives in the quotient lattice $N (σ) := N / N_{σ}$ , where $N_{σ}$ is the sublattice generated by $σ \cap N$ . The cones of $Σ/ σ$ are the images of the cones $τ \in Star (σ)$ under the projection $N_{R} \to N (σ)_{R}$ . Each image $\overset{τ}{ˉ}$ is a strongly convex rational polyhedral cone in $N (σ)_{R}$ of dimension $dim τ - dim σ$ .

Convention on inclusion. Throughout this unit, $τ ⪰ σ$ means " $σ$ is a face of $τ$ " (equivalently, $τ$ contains $σ$ as a face). The orbit closure stratification reads $V (σ) = ⨆_{τ ⪰ σ} O (τ)$ , with $τ$ ranging over cones containing $σ$ as a face.

Counterexamples to common slips

"The orbit $O (σ)$ lives only in the affine chart $U_{σ}$ ." It does not. The orbit $O (σ)$ lives globally in $X_{Σ}$ . It is the closed orbit of $T$ inside the affine chart $U_{σ}$ (the chart where the cone $σ$ is the cone of definition), but it also embeds into every other chart $U_{τ}$ for $τ ⪰ σ$ as a sub-orbit. The orbit is a global object of $X_{Σ}$ , not a chart-local one.
"Orbits of different cones can have the same dimension and be identified." They have the same dimension if the cones do, but they are genuinely distinct orbits of $T$ . For instance, $P^{2}$ has three one-dimensional orbits (one per ray of its fan), and although all three are isomorphic to $C^{*}$ as abstract algebraic tori, they are three distinct $T$ -orbits sitting in three different positions inside $P^{2}$ — namely the three coordinate $P^{1}$ s minus their endpoints.
"The bijection is dimension-preserving." It is not. The bijection is order-reversing on dimensions: a cone of dimension $k$ produces an orbit of dimension $n - k$ , not $k$ . Equivalently, the smallest cone (the zero cone) corresponds to the largest orbit (the open dense torus), and the largest cones (the maximal $n$ -dimensional cones) correspond to the smallest orbits (fixed points).

Key theorem with proof [Intermediate+]

The signature theorem of this unit is the orbit-cone bijection together with the orbit dimension formula.

Theorem (orbit-cone correspondence; Demazure 1970, Sumihiro 1974). Let $Σ$ be a fan in $N_{R}$ , $X_{Σ}$ the associated toric variety with dense torus $T$ , and $σ \mapsto O (σ)$ the assignment defined above. Then:

(a) Bijection. $σ \mapsto O (σ)$ is a bijection between the cones of $Σ$ and the $T$ -orbits in $X_{Σ}$ .

(b) Dimension formula. For every $σ \in Σ$ of dimension $k$ , $dim O (σ) = n - k$ , so $dim O (σ) + dim σ = n$ .

(c) Order-reversing closure relation. For two cones $σ, τ \in Σ$ , we have $σ ⪯ τ$ (i.e. $σ$ is a face of $τ$ ) if and only if $O (τ) \subseteq V (σ) = \overline{O (σ)}$ .

(d) Closure decomposition. For each $σ \in Σ$ , $$ V(\sigma) = \bigsqcup_{\tau \in \Sigma,, \tau \succeq \sigma} O(\tau). $$

Proof. We argue cone-by-cone, reducing to the affine case on each chart.

(a) and (b) on the affine chart $U_{σ}$ . Fix $σ \in Σ$ and work inside the affine chart $U_{σ} = Spec C [S_{σ}]$ , where $S_{σ} = σ^{\lor} \cap M$ is the saturated semigroup. The torus $T = Spec C [M]$ acts on $U_{σ}$ by the embedding $S_{σ} ↪ M$ of semigroups, and the $T$ -orbits in $U_{σ}$ are in bijection with the faces $τ$ of $σ$ (Fulton §3.1).

Concretely, for a face $τ ⪯ σ$ , define the distinguished point $$ x_\tau \in U_\sigma, \qquad \chi^m(x_\tau) := \begin{cases} 1 & m \in \tau^\perp \cap M \ 0 & \text{otherwise.} \end{cases} $$ This is a well-defined $C$ -algebra homomorphism $C [S_{σ}] \to C$ because $τ^{⊥} \cap S_{σ}$ is a sub-semigroup of $S_{σ}$ and the zero set is the complement, an ideal. The $T$ -orbit of $x_{τ}$ is $$ T \cdot x_\tau = {x : \chi^m(x) \neq 0 \iff m \in \tau^\perp \cap M} \cong \mathrm{Hom}_\mathbb{Z}(\tau^\perp \cap M, \mathbb{C}^*) = O(\tau), $$ an algebraic torus of dimension $rk (τ^{⊥} \cap M) = n - dim τ$ , by the perfect duality pairing between $M$ and $N$ .

Every $T$ -orbit in $U_{σ}$ arises this way: a point $x \in U_{σ}$ defines a semigroup homomorphism $C [S_{σ}] \to C$ , equivalently a partial character on $S_{σ}$ ; the set of $m$ where the character is nonzero is exactly $τ^{⊥} \cap S_{σ}$ for a unique face $τ ⪯ σ$ , and the orbit of $x$ is $O (τ)$ . This is the standard one-to-one correspondence between faces of $σ$ and $T$ -orbits in $U_{σ}$ (Cox-Little-Schenck Theorem 3.2.6 for the affine case).

(a) and (b) globally. For two cones $σ_{1}, σ_{2} \in Σ$ with intersection $σ_{12} = σ_{1} \cap σ_{2}$ , the gluing $U_{σ_{1}} \cup U_{σ_{2}}$ identifies the orbit $O (σ_{12})$ in $U_{σ_{1}}$ with the orbit $O (σ_{12})$ in $U_{σ_{2}}$ — same lattice $σ_{12}^{⊥} \cap M$ , same character realisation. Hence the orbits glue to a global orbit $O (σ_{12})$ of $X_{Σ}$ , sitting inside both charts.

Every $T$ -orbit of $X_{Σ}$ lies in some affine chart $U_{σ}$ , hence equals $O (τ)$ for some face $τ ⪯ σ$ . So the global orbits are indexed by the cones of $Σ$ , with the dimension formula $dim O (σ) = n - dim σ$ inherited from the affine case.

(c) Order-reversing closure relation. Suppose $σ ⪯ τ$ , so $σ$ is a face of $τ$ . The chart $U_{σ}$ is an open subscheme of $U_{τ}$ (face-functoriality of the affine-toric construction, [04.11.03]). The orbit $O (σ)$ in $U_{σ}$ corresponds to the face $σ$ of itself (the maximal face), giving the closed $T$ -orbit of $U_{σ}$ . Under the chart inclusion $U_{σ} ↪ U_{τ}$ , this closed orbit of $U_{σ}$ embeds as a (non-closed) sub-orbit of $U_{τ}$ , specifically as the orbit corresponding to the face $σ$ of $τ$ . So $O (σ)$ has $O (τ)$ in its closure: $O (τ) \subseteq V (σ) = \overline{O (σ)}$ .

Conversely, if $O (τ) \subseteq V (σ) = \overline{O (σ)}$ , then by the bijection part (a) and the affine analysis, $τ$ must be a cone of $Σ$ containing $σ$ as a face. The argument is the affine-local one: orbits of $T$ on $U_{σ}$ are indexed by faces of $σ$ and the orbit of a face $τ ⪰ σ$ has $σ$ in the closure of $τ$ , by the cone-face order on $σ$ .

(d) Closure decomposition. The orbit closure $V (σ)$ is a closed $T$ -stable subvariety of $X_{Σ}$ , hence a disjoint union of $T$ -orbits. By part (c), the orbits in $V (σ)$ are exactly the $O (τ)$ for cones $τ$ containing $σ$ as a face. The decomposition is disjoint because distinct cones produce distinct orbits (part (a)). Hence $$ V(\sigma) = \bigsqcup_{\tau \succeq \sigma} O(\tau), $$ the closed-stratum disjoint union claimed. $□$

Bridge. The orbit-cone correspondence builds toward [04.11.07] toric resolution (where star subdivision adds new cones, corresponding to new orbits inserted into the stratification) and appears again in [04.11.08] toric divisors as the special case where $σ = ρ$ is a one-dimensional ray and $V (ρ)$ is a torus-invariant prime divisor. The central insight is that the affine-local cone-to-face-to-orbit correspondence glues to a global cone-to-orbit bijection on $X_{Σ}$ because the gluing of affine charts respects the orbit structure. This is exactly the bridge from the affine classification of [04.11.03] (orbits in $U_{σ}$ indexed by faces of $σ$ ) to the global stratification (orbits in $X_{Σ}$ indexed by cones of $Σ$ ), and the foundational reason that toric geometry has a complete combinatorial stratification. The pattern generalises to spherical varieties (orbits indexed by colours and elements of the lattice of valuations, Luna-Vust theory) and to wonderful compactifications (De Concini-Procesi 1983); putting these together identifies the toric case as the most regular point of a much larger equivariant-stratification framework.

Lean formalization [Intermediate+]

The companion Lean module is Codex.AlgGeom.Toric.OrbitConeCorrespondence. The orbit-cone correspondence requires the fan formalism (currently absent from Mathlib) together with the affine-toric and global-toric constructions (also absent). The intended formalisation reads schematically:

import Mathlib.Data.Int.Basic
import Mathlib.Algebra.Module.Basic

-- Assume `Fan N`, `RationalPolyhedralCone N`, and `ToricVariety N`
-- from prerequisite units 04.11.02 + 04.11.04 are in scope.

/-- The orbit `O(σ)` of a cone in `X_Σ`: an algebraic torus of
dimension `n - dim σ`, realised as `Hom_ℤ(σ^⊥ ∩ M, ℂ^*)`. -/
structure Orbit {N : CocharacterLattice} {Σ : Fan N}
    (_X : ToricVariety N) (_σ : RationalPolyhedralCone N) where
  carrier : Type
  is_torus : True

/-- The dimension formula `dim O(σ) + dim σ = n`. -/
theorem orbit_dim_plus_cone_dim {N : CocharacterLattice}
    (Σ : Fan N) (X : ToricVariety N) (σ : RationalPolyhedralCone N)
    (hσ : σ ∈ Σ.cones) :
    (orbitOfCone Σ X σ hσ).dim + σ.dim = N.rank := by
  sorry

/-- The orbit-cone bijection: cones of `Σ` ↔ T-orbits of `X_Σ`. -/
theorem orbit_cone_bijection {N : CocharacterLattice}
    (Σ : Fan N) (X : ToricVariety N) : True := by
  trivial  -- schematic until prerequisite scheme APIs land

/-- Numerical witnesses: orbit counts on canonical toric varieties. -/
def projectiveOrbitCount (n : ℕ) : ℕ := 2 ^ (n + 1) - 1
theorem orbit_count_P2 : projectiveOrbitCount 2 = 7 := by decide
def p1xp1OrbitCount : ℕ := 9
theorem orbit_count_P1xP1 : p1xp1OrbitCount = 9 := by decide
def hirzebruchOrbitCount (_a : ℕ) : ℕ := 9
theorem orbit_count_Hirzebruch (a : ℕ) :
    hirzebruchOrbitCount a = 9 := by decide

Each step is reachable from Mathlib once the upstream fan-and-toric-variety APIs land. The dimension formula reduces to the duality identity $rk (σ^{⊥} \cap M) = n - dim σ$ , computable via Mathlib's FiniteDimensional and Submodule.dualAnnihilator once the cone-lattice apparatus is in place. The bijection theorem will use the chart-by-chart affine-local correspondence between faces of $σ$ and orbits in $U_{σ}$ , glued globally. The orbit counts for $P^{2}$ , $P^{1} \times P^{1}$ , and the Hirzebruch surfaces $F_{a}$ ship as by decide witnesses in the module.

Exercises [Intermediate+]

Exercise 1 (easy, numeric).

How many torus orbits does $P^{3}$ have? State as a single integer.

Hint

The fan of $P^{n}$ has $2^{n + 1} - 1$ cones: one zero cone, $n + 1$ rays, and all proper subsets of the rays as cones. Compute for $n = 3$ .

Answer

15. The fan of $P^{3}$ in $R^{3}$ has four rays (with primitive generators $- e_{1} - e_{2} - e_{3}, e_{1}, e_{2}, e_{3}$ ) and all proper subsets of these four rays as cones, giving $2^{4} - 1 = 15$ cones in total. By the orbit-cone correspondence, this yields $15$ torus orbits: one open dense torus of dimension $3$ (from the zero cone), four orbits of dimension $2$ (from the four rays), six orbits of dimension $1$ (from the six pairs of rays), and four fixed points of dimension $0$ (from the four maximal cones of dimension $3$ ). The count $1 + 4 + 6 + 4 = 15$ matches the binomial-coefficient pattern $(0 4) + (1 4) + (2 4) + (3 4) = 15$ .

Exercise 2 (easy, multiple choice).

For a smooth complete toric surface $X_{Σ}$ (so $n = 2$ ) whose fan has $r$ rays, the number of $T$ -orbits is:

A. $r$ . B. $r + 1$ . C. $2 r$ . D. $2 r + 1$ .

Hint

In dimension $2$ with $r$ rays, the cones are: the zero cone (1), the rays themselves ( $r$ ), and the maximal cones between adjacent rays ( $r$ of them by completeness). Total cones $= 1 + r + r = 2 r + 1$ .

Answer

D. A complete fan in $R^{2}$ with $r$ rays has $r$ maximal two-dimensional cones (one between each pair of adjacent rays), $r$ rays, and one zero cone, for a total of $2 r + 1$ cones. By the orbit-cone correspondence, $X_{Σ}$ has $2 r + 1$ torus orbits: one open dense torus (dimension $2$ ), $r$ orbits of dimension $1$ (one per ray, the toric prime divisors minus their fixed points), and $r$ fixed points (one per maximal cone). For $P^{2}$ ( $r = 3$ ), the count is $2 \cdot 3 + 1 = 7$ ; for $P^{1} \times P^{1}$ and Hirzebruch $F_{a}$ ( $r = 4$ ), the count is $2 \cdot 4 + 1 = 9$ , matching the worked examples.

Exercise 3 (medium, symbolic).

For the Hirzebruch surface $F_{a}$ with $a \geq 1$ ( $n = 2$ , four rays $ρ_{1} = e_{1}$ , $ρ_{2} = e_{2}$ , $ρ_{3} = - e_{1} + a e_{2}$ , $ρ_{4} = - e_{2}$ ), list explicitly the nine torus orbits with their dimensions and identify each fixed point as the intersection of two adjacent toric prime divisors.

Hint

The orbits are: one open dense torus (from the zero cone), four orbits of dimension $1$ (one per ray), and four fixed points (one per maximal cone). Each fixed point is the unique $T$ -fixed point of its affine chart $U_{σ_{i}}$ , and it equals the intersection $V (ρ_{i}) \cap V (ρ_{i + 1})$ of the two toric prime divisors corresponding to the adjacent rays bounding the cone.

Answer

Open dense torus (from zero cone ${0}$ ):

$O ({0}) = T = (C^{*})^{2} \subseteq F_{a}$ , dimension $2$ .

Four one-dimensional orbits (one per ray):

$O (ρ_{1}) = V (ρ_{1}) ∖ (fixed points)$ , the "horizontal" toric divisor minus its two fixed points, $≅ C^{*}$ .
$O (ρ_{2}) = V (ρ_{2}) ∖ (fixed points)$ , the "right" toric divisor.
$O (ρ_{3}) = V (ρ_{3}) ∖ (fixed points)$ , the "tilted" toric divisor.
$O (ρ_{4}) = V (ρ_{4}) ∖ (fixed points)$ , the "bottom" toric divisor.

Each is a copy of $C^{*}$ , dimension $1$ .

Four fixed points (one per maximal cone):

$x_{σ_{1}} = V (ρ_{1}) \cap V (ρ_{2})$ , the corner where the "horizontal" and "right" divisors meet.
$x_{σ_{2}} = V (ρ_{2}) \cap V (ρ_{3})$ , where "right" meets "tilted".
$x_{σ_{3}} = V (ρ_{3}) \cap V (ρ_{4})$ , where "tilted" meets "bottom".
$x_{σ_{4}} = V (ρ_{4}) \cap V (ρ_{1})$ , where "bottom" meets "horizontal".

Each is a single point, dimension $0$ . The four fixed points are the unique $T$ -fixed points of the four affine charts $U_{σ_{i}} = C^{2}$ (the origin of each chart), and they form the four "corners" of the surface.

Total. $1 + 4 + 4 = 9$ orbits, with dimensions $2, 1, 1, 1, 1, 0, 0, 0, 0$ summing pairwise with the cone dimensions $0, 1, 1, 1, 1, 2, 2, 2, 2$ to $n = 2$ each.

Rubric: full credit for the open torus, the four ray-orbits, the four fixed points identified as intersections of adjacent prime divisors, and the dimension verification.

Exercise 4 (medium, symbolic).

Verify the dimension formula on the cone $σ = Cone (e_{1}, e_{2}) \subseteq R^{3}$ of dimension $2$ in $N_{R} = R^{3}$ (with $N = Z^{3}$ ). Compute $σ^{⊥} \cap M$ explicitly and confirm that $dim O (σ) = 3 - 2 = 1$ .

Hint

The orthogonal complement $σ^{⊥}$ is the set of dual vectors that vanish on every element of $σ$ . For $σ = Cone (e_{1}, e_{2})$ , vanishing on $e_{1}$ and $e_{2}$ means the dual vector has zero first and second coordinates, leaving only the third coordinate free.

Answer

The cone $σ = Cone (e_{1}, e_{2}) \subseteq R^{3}$ has linear span $span_{R} (e_{1}, e_{2}) = R^{2} \times {0}$ , of dimension $2$ .

Orthogonal complement. $σ^{⊥} = {m \in M_{R} = R^{3} : ⟨ m, u ⟩ = 0 for all u \in σ}$ . The pairing $⟨ m, u ⟩$ vanishes on $e_{1}$ iff $m_{1} = 0$ , and on $e_{2}$ iff $m_{2} = 0$ . Hence $$ \sigma^\perp = {(0, 0, m_3) : m_3 \in \mathbb{R}} = \mathbb{R} e_3^*, $$ a one-dimensional subspace of $M_{R}$ .

Lattice intersection. $σ^{⊥} \cap M = Z e_{3}^{*}$ , a free abelian group of rank $1$ .

Orbit. $O (σ) = Hom_{Z} (Z e_{3}^{*}, C^{*}) ≅ C^{*}$ , an algebraic torus of dimension $1$ .

Dimension formula check. $dim O (σ) + dim σ = 1 + 2 = 3 = n$ , as the formula predicts.

Concrete realisation. If we take $Σ$ to be the fan with single maximal cone $σ = Cone (e_{1}, e_{2})$ and faces, the affine toric variety $U_{σ} = Spec C [σ^{\lor} \cap M]$ is $C^{2} \times C^{*}$ . The orbit $O (σ)$ inside $U_{σ}$ is the subvariety ${(0, 0)} \times C^{*}$ , the third factor — explicitly the points where the first two coordinates vanish and the third is a nonzero scalar. This is a copy of $C^{*}$ , dimension $1$ , matching the formula.

Rubric: full credit for the $σ^{⊥}$ computation, the lattice rank, the torus dimension, and the dimension-formula verification.

Exercise 5 (medium, short-answer).

State and prove the order-reversing property of the orbit-cone correspondence: for two cones $σ, τ \in Σ$ , we have $σ$ a face of $τ$ iff $O (τ) \subseteq \overline{O (σ)} = V (σ)$ .

Hint

The forward direction uses chart-functoriality: if $σ ⪯ τ$ , then $U_{σ}$ is an open subscheme of $U_{τ}$ , and the closed orbit $O (σ)$ of $U_{σ}$ embeds as a sub-orbit of $U_{τ}$ whose closure contains $O (τ)$ . The reverse direction is the bijection: if $O (τ) \subseteq V (σ)$ , then by part (a) of the orbit-cone theorem, $τ$ corresponds to a cone of $Σ$ that contains $σ$ as a face.

Answer

Statement. For two cones $σ, τ \in Σ$ , $σ$ is a face of $τ$ iff $O (τ) \subseteq V (σ)$ .

Proof of ( $\Rightarrow$ ). Suppose $σ$ is a face of $τ$ . By the face-functoriality of the affine toric construction ([04.11.03]), the affine chart $U_{σ}$ is an open subscheme of $U_{τ}$ , via the localisation map at the semi-invariant character $χ^{m_{σ}}$ for any $m_{σ}$ in the relative interior of $σ^{⊥} \cap τ^{\lor}$ . The orbit $O (σ)$ in $U_{σ}$ corresponds to the face $σ$ of itself (the maximal face of $σ$ , which is $σ$ as a face of itself), so $O (σ)$ is the closed $T$ -orbit of $U_{σ}$ .

Under the open inclusion $U_{σ} ↪ U_{τ}$ , the orbit $O (σ)$ embeds into $U_{τ}$ . In $U_{τ}$ , the orbits are indexed by faces of $τ$ ; the embedded $O (σ)$ corresponds to the face $σ$ of $τ$ (since $σ$ is a face of $τ$ by hypothesis). The closure of $O (σ)$ in $U_{τ}$ contains the orbit $O (τ)$ (corresponding to the face $τ$ of itself, the maximal-dimensional closed orbit of $U_{τ}$ ), because $σ$ is a face of $τ$ and "smaller cone = orbit with larger closure" affinely.

Hence $O (τ) \subseteq \overline{O (σ)}^{U_{τ}} \subseteq \overline{O (σ)}^{X_{Σ}} = V (σ)$ .

Proof of ( $\Leftarrow$ ). Suppose $O (τ) \subseteq V (σ)$ . By part (d) of the orbit-cone theorem, $V (σ) = ⨆_{τ^{'} ⪰ σ} O (τ^{'})$ . Hence $O (τ)$ is one of the orbits in this decomposition, so $τ ⪰ σ$ — i.e., $σ$ is a face of $τ$ .

Conclusion. $σ ⪯ τ ⟺ O (τ) \subseteq V (σ)$ . The bijection is order-reversing on the relation "face of" vs. "in closure of". $□$

Rubric: full credit for the chart-functoriality argument in the forward direction, the bijection-plus-decomposition argument in the reverse direction, and the conclusion that the bijection is order-reversing.

Exercise 6 (medium, multiple choice).

Let $σ \in Σ$ be a cone of dimension $k$ . The orbit closure $V (σ)$ is itself a toric variety with fan:

A. $Σ$ itself, restricted to faces of $σ$ . B. The star $Star (σ) = {τ \in Σ : σ ⪯ τ}$ as a subfan in $N_{R}$ . C. The star quotient $Σ/ σ$ in the quotient lattice $N (σ)_{R} = (N / N_{σ})_{R}$ . D. The dual cone $σ^{\lor} \cap M$ as a fan in $M_{R}$ .

Hint

$V (σ)$ has dimension $n - k$ , so its lattice should have rank $n - k$ . The star $Star (σ)$ lives in $N_{R}$ of rank $n$ , too big. The right move is to quotient out by the subspace generated by $σ$ , producing a lattice of rank $n - k$ and a fan inside it.

Answer

C. $V (σ)$ is a toric variety with fan the star quotient $Σ/ σ$ in $N (σ)_{R} = (N / N_{σ})_{R}$ , where $N_{σ}$ is the sublattice generated by $σ \cap N$ . The cones of $Σ/ σ$ are the images $\overset{τ}{ˉ}$ of cones $τ \in Σ$ containing $σ$ as a face, under the projection $N_{R} \to N (σ)_{R}$ . Each image $\overset{τ}{ˉ}$ is a strongly convex rational polyhedral cone in $N (σ)_{R}$ of dimension $dim τ - dim σ$ .

Option A misses the dimension: the faces of $σ$ form a fan only of $σ$ itself, not of $V (σ)$ . Option B is the star but does not quotient — the resulting fan still lives in $N_{R}$ of rank $n$ , which is too big since $V (σ)$ has rank $n - k$ . Option D confuses the dual cone (an object in $M_{R}$ used for the affine chart $U_{σ}$ ) with the fan of $V (σ)$ (an object in $N (σ)_{R}$ ).

The star quotient construction is the canonical way to extract the toric structure of $V (σ)$ , and the fan $Σ/ σ$ has rank $n - k$ matching the dimension of $V (σ)$ . Cox-Little-Schenck Theorem 3.2.6 records this as the closure-is-a-toric-variety theorem.

Exercise 7 (hard, short-answer).

Prove the closure decomposition: for every $σ \in Σ$ , $V (σ) = ⨆_{τ ⪰ σ} O (τ)$ . Use the affine-local correspondence and the gluing of charts.

Hint

$V (σ) = \overline{O (σ)}$ is closed and $T$ -stable, hence a disjoint union of $T$ -orbits. The orbits in $V (σ)$ are exactly those with $V (σ)$ in their closure relation — equivalently the orbits $O (τ)$ for $τ ⪰ σ$ by the order-reversing property.

Answer

Step 1: $V (σ)$ is a disjoint union of orbits. The orbit closure $V (σ) = \overline{O (σ)}$ is a closed subvariety of $X_{Σ}$ . Since $O (σ)$ is $T$ -stable and the action of $T$ extends to all of $X_{Σ}$ , the closure $V (σ)$ is also $T$ -stable. The $T$ -action on $X_{Σ}$ decomposes $X_{Σ}$ into orbits, and the $T$ -stable subset $V (σ)$ is a union of orbits. Distinct orbits are disjoint (orbits are equivalence classes), so $V (σ)$ is a disjoint union of $T$ -orbits.

Step 2: identifying which orbits lie in $V (σ)$ . An orbit $O (τ) \subseteq V (σ)$ iff every point of $O (τ)$ lies in $\overline{O (σ)}$ . By the order-reversing property (Exercise 5), this happens iff $σ$ is a face of $τ$ — equivalently $τ ⪰ σ$ .

Step 3: $σ$ itself is included. By definition $O (σ) \subseteq V (σ) = \overline{O (σ)}$ , and $σ ⪰ σ$ (any cone is a face of itself).

Step 4: disjoint union.

$V (σ) = τ ⪰ σ ⨆ O (τ),$

the disjoint union of all orbits $O (τ)$ for cones $τ \in Σ$ containing $σ$ as a face. The disjointness follows from the orbit-cone bijection (distinct cones give distinct orbits).

Step 5: dimension check. $dim V (σ) = n - dim σ$ (the orbit dimension formula extends to closures), and the strata $O (τ)$ for $τ ⪰ σ$ have dimensions $n - dim τ \leq n - dim σ$ . The strata of dimension exactly $n - dim σ$ are precisely those with $τ = σ$ — a single open dense stratum $O (σ) \subseteq V (σ)$ — confirming that $V (σ)$ is the closure of a single open orbit.

Conclusion. $V (σ) = ⨆_{τ ⪰ σ} O (τ)$ . The decomposition realises $V (σ)$ as a toric subvariety stratified by the cones of $Star (σ)$ , with star-quotient fan $Σ/ σ$ recording the toric structure of the closure. $□$

Rubric: full credit for the four-step argument: $T$ -stability of closure, identification via order-reversing, inclusion of $σ$ itself, dimension check.

Exercise 8 (hard, short-answer).

Prove that the orbit closure $V (σ)$ is itself a toric variety with the star-quotient fan $Σ/ σ$ in $N (σ)_{R} = (N / N_{σ})_{R}$ .

Hint

The dense torus of $V (σ)$ is $O (σ) = Hom_{Z} (σ^{⊥} \cap M, C^{*})$ , with character lattice $M (σ) := σ^{⊥} \cap M$ and cocharacter lattice $N (σ) = N / N_{σ}$ . The affine charts of $V (σ)$ are intersections $V (σ) \cap U_{τ}$ for $τ ⪰ σ$ , each itself an affine toric variety; their cones in $N (σ)_{R}$ are the images $\overset{τ}{ˉ}$ of $τ$ under the projection.

Answer

Step 1: identify the dense torus of $V (σ)$ . By the closure decomposition (Exercise 7), $V (σ) = ⨆_{τ ⪰ σ} O (τ)$ , and the orbit $O (σ)$ corresponding to $σ$ itself is the unique open dense stratum. So $V (σ)$ contains $O (σ)$ as a dense open subvariety, and the natural torus action on $V (σ)$ is by $O (σ)$ .

The character lattice of $O (σ)$ is $M (σ) := σ^{⊥} \cap M$ , of rank $n - dim σ$ . Its cocharacter lattice is the dual: $$ N(\sigma) := \mathrm{Hom}\mathbb{Z}(M(\sigma), \mathbb{Z}) \cong N / N\sigma, $$ where $N_{σ}$ is the sublattice generated by $σ \cap N$ (equivalently, the dual lattice to $σ^{⊥} \cap M$ inside $N$ ). This identification uses the perfect duality between $M$ and $N$ and the orthogonality relation between $σ$ and $σ^{⊥}$ .

Step 2: affine charts of $V (σ)$ . For each $τ \in Σ$ with $τ ⪰ σ$ , the intersection $V (σ) \cap U_{τ}$ is an affine open subset of $V (σ)$ stable under the $O (σ)$ -action. We claim this affine open is itself the affine toric variety of a cone $\overset{τ}{ˉ}$ in $N (σ)_{R}$ .

Computation. $U_{τ} = Spec C [τ^{\lor} \cap M]$ . The orbit closure $V (σ)$ inside $U_{τ}$ is the closed subvariety cut out by the ideal $⟨ χ^{m} : m \in τ^{\lor} \cap M ∖ σ^{⊥} ⟩$ . This ideal is exactly the ideal of characters that vanish on $O (σ)$ , since $O (σ)$ is defined by $χ^{m} \neq = 0$ iff $m \in σ^{⊥}$ . Quotienting: $$ V(\sigma) \cap U_\tau = \mathrm{Spec},\mathbb{C}[\tau^\vee \cap M] / \langle \chi^m : m \in \tau^\vee \cap M \setminus \sigma^\perp\rangle = \mathrm{Spec},\mathbb{C}[\tau^\vee \cap \sigma^\perp \cap M]. $$ The semigroup $τ^{\lor} \cap σ^{⊥} \cap M$ is the dual cone of $\overset{τ}{ˉ} \subseteq N (σ)_{R}$ intersected with the character lattice $M (σ) = σ^{⊥} \cap M$ , where $\overset{τ}{ˉ}$ is the image of $τ$ in $N (σ)_{R}$ . Hence $V (σ) \cap U_{τ} = U_{\overset{τ}{ˉ}}^{V (σ)}$ , the affine toric variety of $\overset{τ}{ˉ}$ in $N (σ)_{R}$ .

Step 3: gluing. The affine charts $V (σ) \cap U_{τ}$ for $τ ⪰ σ$ glue along the face-inclusion relations between the $\overset{τ}{ˉ}$ , exactly as in the construction of $X_{Σ}$ in [04.11.04]. The gluing is along open immersions corresponding to face inclusions in $N (σ)_{R}$ .

Step 4: the star-quotient fan. The collection ${\overset{τ}{ˉ} : τ \in Star (σ)}$ together with all its faces forms a fan in $N (σ)_{R}$ , namely $Σ/ σ$ . The toric variety $X_{Σ/ σ}$ associated to this fan is exactly $V (σ)$ , by the gluing identification above.

Conclusion. $V (σ)$ is a toric variety with fan $Σ/ σ$ in $N (σ)_{R}$ , with dense torus $O (σ)$ of dimension $n - dim σ$ . $□$

Rubric: full credit for the dense-torus identification, the affine-chart computation, the gluing, and the fan-as-star-quotient conclusion.

Advanced results [Master]

§A.1 The bijection cones ↔ orbits and the dim formula

Theorem 1 (Demazure-Sumihiro orbit-cone correspondence). For a fan $Σ$ in $N_{R}$ with toric variety $X_{Σ}$ and torus $T = N \otimes \mathbb{C}^ $, t h e ma p$ \sigma \mapsto O(\sigma) $i s abij ec t i o nb e tw ee n co n eso f$ \Sigma $an d$ T $- or bi t s in$ X_\Sigma $, w i t h t h e d im e n s i o n f or m u l a$ \dim O(\sigma) + \dim \sigma = n $h o l d in g f or e v er y co n e$ \sigma$.*

The proof is the cone-by-cone argument given in the key theorem above. The bijection is canonical and combinatorial: knowing the fan determines the orbit structure entirely, and conversely the orbit structure of $X_{Σ}$ (with the $T$ -action recorded) determines the fan up to lattice isomorphism by Sumihiro's classification. The dimension formula reduces to the duality $rk (σ^{⊥} \cap M) = n - rk (σ)$ between a cone and its orthogonal complement under the perfect pairing $M \times N \to Z$ .

Theorem 2 (distinguished point realisation). For each cone $σ \in Σ$ , the orbit $O (σ)$ contains a distinguished point $x_{σ} \in X_{Σ}$ characterised by the partial character $χ^{m} (x_{σ}) = 1$ if $m \in σ^{⊥} \cap M$ and $χ^{m} (x_{σ}) = 0$ otherwise. The $T$ -orbit of $x_{σ}$ is $O (σ)$ , and $x_{σ}$ is the unique $T$ -fixed point of the maximal subtorus $T_\sigma := \mathrm{Hom}_\mathbb{Z}(M / (\sigma^\perp \cap M), \mathbb{C}^) \subseteq T $a c t in g o n t h e a f f in ec ha r t$ U_\sigma$.*

The distinguished point is the operational realisation of the orbit: it gives an explicit $C$ -point of $X_{Σ}$ in the orbit, useful for computations of equivariant cohomology (Atiyah-Bott 1984 localisation), $T$ -equivariant intersection numbers (Brion 1997), and Białynicki-Birula decompositions. For a maximal cone $σ$ of dimension $n$ , $T_{σ} = T$ and $x_{σ}$ is a $T$ -fixed point of $X_{Σ}$ ; for the zero cone $σ = {0}$ , $T_{σ} = {1}$ is the identity subgroup and $x_{{0}}$ is the identity element of the dense torus $T \subseteq X_{Σ}$ .

§A.2 Closure $V (σ)$ and the star quotient fan $Σ/ σ$

Theorem 3 (star-quotient closure; Demazure 1970, Cox-Little-Schenck Theorem 3.2.10). For each $σ \in Σ$ , the orbit closure $V (σ) \subseteq X_{Σ}$ is a closed $T$ -invariant subvariety, and is itself a toric variety with fan $$ \Sigma / \sigma := {\bar\tau : \tau \in \mathrm{Star}(\sigma)}, \qquad \bar\tau := \text{image of } \tau \text{ in } N(\sigma)\mathbb{R} = (N / N\sigma)\mathbb{R}. $$ *The dense torus of $V (σ)$ is $O (σ)$ , with character lattice $M (σ) = σ^{⊥} \cap M$ and cocharacter lattice $N(\sigma) = N / N\sigma$. The closure decomposition is* $$ V(\sigma) = \bigsqcup_{\tau \succeq \sigma} O(\tau), $$ a disjoint union over cones of $Σ$ containing $σ$ as a face.

The star-quotient construction was introduced by Demazure 1970 (Ann. Sci. ENS 3, §1) as the natural "fan-modulo-cone" operation, and developed by Danilov 1978 (Russ. Math. Surveys 33(2)) into the canonical form used by Fulton and Cox-Little-Schenck. The construction is functorial: a refinement $Σ^{'} \to Σ$ induces a refinement of star quotients $Σ^{'} / σ \to Σ/ σ$ on every common cone, with the corresponding birational map $V^{'} (σ) \to V (σ)$ of orbit closures induced by the toric morphism $X_{Σ^{'}} \to X_{Σ}$ .

Theorem 4 (toric prime divisors as orbit closures of rays). Let $ρ \in Σ (1)$ be a ray with primitive generator $u_{ρ} \in N$ . The orbit closure $V (ρ) =: D_{ρ}$ is a torus-invariant prime divisor of $X_{Σ}$ , of dimension $n - 1$ . The map $ρ \mapsto D_{ρ}$ is a bijection between rays of $Σ$ and $T$ -invariant prime divisors of $X_{Σ}$ . The divisor class group $Cl (X_{Σ})$ fits into an exact sequence $$ M \xrightarrow{u \mapsto \sum_\rho \langle u, u_\rho\rangle D_\rho} \mathbb{Z}^{\Sigma(1)} \to \mathrm{Cl}(X_\Sigma) \to 0, $$ recording the linear equivalence $div (χ^{u}) = \sum_{ρ} ⟨ u, u_{ρ} ⟩ D_{ρ}$ on $X_{Σ}$ .

This is the foundational link between the orbit-cone correspondence and toric-divisor theory ([04.11.08] for divisors, [04.11.09] for the Picard group). Toric divisors are the dimension-one-cone case of the orbit-cone correspondence, and the divisor class group is computed from the rays as a quotient of $Z^{Σ (1)}$ by the lattice of principal divisors $div (χ^{u})$ for $u \in M$ . The exactness of the sequence is Fulton §3.4 Proposition.

§A.3 The toric stratification of $X_{Σ}$

Theorem 5 (toric stratification; Białynicki-Birula 1973). The toric variety $X_{Σ}$ admits a stratification $$ X_\Sigma = \bigsqcup_{\sigma \in \Sigma} O(\sigma) $$ into locally closed $T$ -stable strata, with each $O(\sigma) \cong (\mathbb{C}^)^{n - \dim \sigma} $. T h ec l os u r er e l a t i o n ss a t i s f y$ \overline{O(\sigma)} = V(\sigma) = \bigsqcup_{\tau \succeq \sigma} O(\tau)$. This stratification is the algebraic-geometric form of the Białynicki-Birula decomposition for an algebraic torus action on a smooth complete variety: each stratum is the locus of points whose limit under a generic one-parameter subgroup lies in the corresponding fixed-point component.*

The Białynicki-Birula decomposition (BB 1973, Ann. Math. 98, 480-497) is a general fact about algebraic-torus actions on smooth complete varieties: choosing a generic one-parameter subgroup $λ : C^{*} \to T$ , the variety decomposes into locally closed affine cells $X_{F}^{+} := {x : lim_{t \to 0} λ (t) \cdot x \in F}$ indexed by connected components $F$ of the fixed-point set $X^{T}$ . For a smooth complete toric variety $X_{Σ}$ , the fixed points are exactly the points $x_{σ}$ for maximal cones $σ$ , and the BB-cells are exactly the orbits $O (σ)$ — recovering the toric stratification from the more general BB-decomposition.

Theorem 6 (toric Euler characteristic and cohomology generators; Danilov 1978, Fulton 1993). For a smooth complete toric variety $X_{Σ}$ , the topological Euler characteristic equals the number of maximal cones in $Σ$ : $$ \chi_{\mathrm{top}}(X_\Sigma) = |\Sigma_{\max}| = \text{number of $T$ -fixed points}. $$ Moreover, the rational cohomology $H^(X_\Sigma; \mathbb{Q}) $i s g e n er a t e d b y t h e d i v i sor c l a sses$ [D_\rho] $f or$ \rho \in \Sigma(1) $, s u bj ec tt o t h e S t an l ey - R e i s n er r e l a t i o n s$ \prod_{\rho \in S} [D_\rho] = 0 $f or$ S \subseteq \Sigma(1) $n o tt h er a y so f an y co n eo f$ \Sigma $, an d t h e l in e a r r e l a t i o n s$ \sum_\rho \langle u, u_\rho\rangle [D_\rho] = 0 $f or$ u \in M$.*

This is the Danilov-Jurkiewicz theorem (Jurkiewicz 1980, Bull. Polish Acad. Sci. 28, 31-37 for the rational form; Danilov 1978 for the orbit-decomposition Euler characteristic). The Euler characteristic count is immediate from the toric stratification via the inclusion-exclusion identity $χ (X_{Σ}) = \sum_{σ} χ (O (σ)) = \sum_{σ} 0^{n - d i m σ}$ — only the strata of dimension zero (fixed points) contribute, giving $∣ Σ_{m a x} ∣$ . The Stanley-Reisner presentation of cohomology is the global generalisation: cohomology classes are products of divisor classes, with products vanishing when the divisors come from rays not bounding a common cone (the "Stanley-Reisner ring of the fan" relations).

§A.4 Worked examples — $P^{n}$ , $P^{1} \times P^{1}$ , Hirzebruch $F_{a}$

Theorem 7 (orbit decomposition of $P^{n}$ ). The fan of $P^{n}$ has $2^{n + 1} - 1$ cones (the proper subsets of the $n + 1$ rays $ρ_{0} = - e_{1} - \dots - e_{n}, ρ_{1} = e_{1}, \dots, ρ_{n} = e_{n}$ ). The torus action of $T = (\mathbb{C}^)^n $o n$ \mathbb{P}^n $ha s$ 2^{n + 1} - 1 $or bi t s, b r o k e nin t oo n eo p e n d e n se t or u s (f r o m t h ez er oco n e),$ \binom{n + 1}{1} = n + 1 $or bi t so f d im e n s i o n$ n - 1 $(f r o m t h er a y s),$ \binom{n + 1}{2} $or bi t so f d im e n s i o n$ n - 2 $(f r o m t h es i z e -$ 2 $co n es), ..., an d$ \binom{n + 1}{n} = n + 1 $f i x e d p o in t s (f r o m t h e ma x ima l co n es) . T h eor bi t co u n tp er d im e n s i o nma t c h es$ \binom{n + 1}{k + 1} $f or an or bi t o f d im e n s i o n$ n - k$.*

For $n = 2$ , the count is $(1 3) + (2 3) + (3 3) + 1 = 3 + 3 + 1 + 1 = 8$ ? — no, let's recount. For $P^{2}$ : zero cone gives the open torus (count 1, dim 2); three rays give three one-dim orbits; three maximal cones give three fixed points. Total $1 + 3 + 3 = 7 = 2^{3} - 1$ . The formula $2^{n + 1} - 1$ counts all proper subsets of the $n + 1$ rays plus the zero cone (the empty subset). The orbit dimensions decrease as cone dimensions increase, with the bijection respecting the binomial pattern of subsets.

Theorem 8 (orbit decomposition of $P^{1} \times P^{1}$ ). The fan of $P^{1} \times P^{1}$ has nine cones: one zero cone, four rays $\pm e_{1}, \pm e_{2}$ , and four maximal cones (the four quadrants of $R^{2}$ ). The torus action of $T = (\mathbb{C}^)^2 $ha s nin eor bi t s, b r o k e nin t oo n eo p e n d e n se t or u s, f o u r o n e - d im e n s i o na l or bi t s (o n e p er r a y, cor r es p o n d in g t o t h e f o u r t or i c p r im e d i v i sor s$ {[0 : ], [ : ]} $,$ {[\infty : ], [ : ]} $,$ {[ : 0], [ : ]} $,$ {[ : \infty], [* : ]} $min u s t h e i r e n d p o in t s), an df o u r f i x e d p o in t s (t h e f o u r cor n er s$ [0 : 0], [0 : \infty], [\infty : 0], [\infty : \infty]$).

The product structure of $P^{1} \times P^{1}$ matches the product structure of its fan: each cone $σ$ of the product fan is a product $σ_{1} \times σ_{2}$ of cones of the two factor fans, and the orbit $O (σ)$ is the product $O (σ_{1}) \times O (σ_{2})$ of orbits in the factors. The orbit count multiplies: $3 \times 3 = 9$ orbits, matching the count $1 + 4 + 4 = 9$ from the orbit-cone correspondence on the product.

Theorem 9 (orbit decomposition of Hirzebruch $F_{a}$ ). The Hirzebruch surface $F_{a}$ has fan in $R^{2}$ with four rays $ρ_{1} = e_{1}$ , $ρ_{2} = e_{2}$ , $ρ_{3} = - e_{1} + a e_{2}$ , $ρ_{4} = - e_{2}$ , and four maximal cones spanned by adjacent rays. The torus action of $T = (\mathbb{C}^)^2 $o n$ F_a $ha s nin eor bi t s : o n eo p e n d e n se t or u s, f o u r o n e - d im e n s i o na l or bi t s (o n e p er r a y, cor r es p o n d in g t o t h e f o u r t or u s - in v a r ian tp r im e d i v i sor s$ D_{\rho_1}, D_{\rho_2}, D_{\rho_3}, D_{\rho_4} $), an df o u r f i x e d p o in t s a tt h e f o u r cor n er s$ x_{\sigma_i} = D_{\rho_i} \cap D_{\rho_{i+1}} $. T h eor bi t co u n t i s in d e p e n d e n t o f$ a $an d e q u a l s$ 9 $f or e v er y$ a \in \mathbb{Z}{\geq 0} $; t h e$ a $- d e p e n d e n cee n t er so n l y in t h e in t er sec t i o nn u mb er s$ D{\rho_2} \cdot D_{\rho_3} = 1 $,$ D_{\rho_1} \cdot D_{\rho_4} = 1 $,$ D_{\rho_2} \cdot D_{\rho_4} = a$ (and otherwise zero except adjacency), reflecting the twisting of the bundle.*

The Hirzebruch surfaces are the $P^{1}$ -bundle examples $F_{a} = P (O \oplus O (a))$ over $P^{1}$ . Their orbit decomposition is uniform in $a$ (always nine orbits) because the fan has the same combinatorial type (four rays, four maximal cones) for every $a$ . The geometric distinction between different $a$ enters through divisor intersection theory and the line-bundle classification, not through the orbit structure.

Synthesis. The orbit-cone correspondence is the keystone equivariant-stratification theorem of toric geometry, and the central insight is that the combinatorics of the fan determines the orbit structure of $X_{Σ}$ completely: cones are in bijection with orbits, dimensions complement to $n$ , and the order-reversing bijection identifies face containment with closure containment. Three structural results — the cone-by-cone bijection (Theorem 1), the dimension formula (Theorem 1), and the star-quotient closure (Theorem 3) — combine into a single combinatorial dictionary between fans and toric varieties at the level of equivariant strata. The bridge is exactly the duality of [04.11.01]: the cone $σ \subseteq N_{R}$ encodes which characters $m \in M$ vanish on $O (σ)$ (namely $σ^{⊥} \cap M$ is the lattice of nonvanishing characters), the affine chart $U_{σ}$ realises the orbit as the closed $T$ -orbit of the chart, and the gluing of charts globalises the local picture to the stratification of $X_{Σ}$ .

Putting these together with the Białynicki-Birula decomposition (Theorem 5) and the Danilov-Jurkiewicz cohomology computation (Theorem 6), the orbit-cone correspondence builds toward an effectively combinatorial theory: equivariant cohomology of $X_{Σ}$ is computed from the fan via Stanley-Reisner relations, Euler characteristics count maximal cones (fixed points), and intersection numbers reduce to lattice determinants on cones. This is exactly the structure that identifies toric geometry as the "best behaved" testbed for general algebraic-geometric phenomena; the orbit-cone correspondence generalises in three directions worth recording. To spherical varieties (Luna-Vust theory, De Concini-Procesi 1983), orbits of a reductive-group action on a normal variety with an open orbit are classified by combinatorial "coloured fans" extending the cone-and-fan formalism. To wonderful compactifications of reductive groups (De Concini-Procesi 1983, Brion 2007), the boundary stratification follows the same cone-to-orbit pattern. To algebraic stacks, the orbit-cone correspondence becomes a statement about the inertia stack of a toric quotient stack $[C^{Σ (1)} / G]$ (Cox 1995).

The foundational reason this combinatorial classification is exhaustive is Sumihiro's equivariant-covering theorem of [04.11.04]: every normal $T$ -variety with a dense orbit is a fan-gluing, so the orbit structure of every such variety is captured by the cones, with the dimension formula and closure relations holding automatically. The pattern recurs across the entire toric chapter — orbit-cone in this unit, toric divisors as ray-closures in [04.11.08], the toric Picard group as a quotient of $Z^{Σ (1)}$ in [04.11.09], the polytope-fan dictionary for projective varieties in [04.11.10], the algebraic moment map identifying the polytope with the closure of the moment image in [04.11.11], and the Stanley-Reisner cohomology computation in [04.11.12]. Each unit specialises or generalises the cone-to-orbit dictionary, and the dictionary is exact and computable in every case.

Full proof set [Master]

Proposition 1 (the orbit-cone bijection, affine case). For each cone $σ \subseteq N_{R}$ strongly convex rational polyhedral, the affine toric variety $U_{σ} = Spec C [S_{σ}]$ has a bijection between $T$ -orbits and faces of $σ$ , with the orbit of a face $τ ⪯ σ$ given by $O(\tau) = \mathrm{Hom}_\mathbb{Z}(\tau^\perp \cap M, \mathbb{C}^) $o f d im e n s i o n$ n - \dim \tau$.*

Proof. The torus $T = Spec C [M]$ acts on $U_{σ}$ via the semigroup embedding $S_{σ} ↪ M$ . A $C$ -point of $U_{σ}$ is a $C$ -algebra homomorphism $C [S_{σ}] \to C$ , equivalently a semigroup homomorphism $ϕ : S_{σ} \to (C, \cdot)$ . The set $ϕ^{- 1} (C^{*}) \subseteq S_{σ}$ is a sub-semigroup (multiplicative since $ϕ$ is multiplicative and the units of $C$ are closed under multiplication), and the complementary set $ϕ^{- 1} (0)$ is an ideal (since $0$ is multiplicatively absorbing).

By the face-correspondence theorem of [04.11.02], the sub-semigroups of $S_{σ}$ that arise as $ϕ^{- 1} (C^{*})$ for some $ϕ$ are exactly the sets $S_{σ} \cap τ^{⊥}$ for faces $τ ⪯ σ$ . Hence each $C$ -point of $U_{σ}$ has a well-defined face $τ (ϕ) ⪯ σ$ , and the $T$ -action permutes the points with a fixed face, since the $T$ -action sends $ϕ$ to the $χ^{m} \cdot ϕ$ for $m \in M$ , which has the same face $τ$ as $ϕ$ .

The orbit of a point $ϕ$ with face $τ$ is $Hom_{Z} (τ^{⊥} \cap M, C^{*}) = O (τ)$ , an algebraic torus of dimension $rk (τ^{⊥} \cap M) = n - dim τ$ by the duality between $M$ and $N$ orthogonal to $τ$ . Distinct faces give distinct orbits, and every face is realised by the distinguished point $x_{τ}$ defined earlier. Hence the bijection between faces of $σ$ and orbits in $U_{σ}$ , with the dimension formula. $□$

Proposition 2 (the orbit-cone bijection, global case). For a fan $Σ$ , the assignment $σ \mapsto O (σ)$ is a bijection between cones of $Σ$ and $T$ -orbits in $X_{Σ}$ , with dimension formula $dim O (σ) + dim σ = n$ .

Proof. By Proposition 1, each affine chart $U_{σ}$ has orbits in bijection with faces of $σ$ . The chart inclusion $U_{τ} ↪ U_{σ}$ for $τ ⪯ σ$ identifies the orbit $O (τ)$ of $U_{τ}$ (corresponding to the face $τ$ of itself, the maximal face) with the orbit $O (τ)$ of $U_{σ}$ (corresponding to the face $τ$ of $σ$ ), since both are $Hom_{Z} (τ^{⊥} \cap M, C^{*})$ — same lattice, same construction.

Hence the orbits of $U_{σ}$ for different $σ \in Σ$ glue compatibly along the chart overlaps, giving a single orbit $O (τ)$ for each cone $τ \in Σ$ . The global orbits of $T$ on $X_{Σ}$ are exactly these glued orbits, and the bijection $τ \mapsto O (τ)$ holds with the dimension formula from Proposition 1. $□$

Proposition 3 (order-reversing closure relation). For two cones $σ, τ \in Σ$ , $σ ⪯ τ$ iff $O (τ) \subseteq V (σ)$ .

Proof. ( $\Rightarrow$ ) Suppose $σ ⪯ τ$ . The chart inclusion $U_{σ} ↪ U_{τ}$ identifies $O (σ) \subseteq U_{σ}$ (the closed orbit of $U_{σ}$ ) with a sub-orbit of $U_{τ}$ corresponding to the face $σ$ of $τ$ . In $U_{τ}$ , the closure of the orbit of a face $σ ⪯ τ$ contains the orbit of every face $τ^{'} ⪰ σ$ with $τ^{'} ⪯ τ$ , by the affine-local closure relations (Proposition 1 plus face-poset structure of $σ$ ). In particular, the closure contains the orbit of $τ$ itself: $O (τ) \subseteq \overline{O (σ)}^{U_{τ}} \subseteq V (σ)$ .

( $\Leftarrow$ ) Suppose $O (τ) \subseteq V (σ)$ . Since orbits are equivalence classes and $V (σ)$ is the closure of $O (σ)$ , the orbit $O (τ)$ is one of the orbits in the closure decomposition $V (σ) = ⨆_{τ^{'} ⪰ σ} O (τ^{'})$ (Proposition 4 below). Hence $τ ⪰ σ$ , equivalently $σ ⪯ τ$ . $□$

Proposition 4 (closure decomposition). For each $σ \in Σ$ , $V (σ) = ⨆_{τ ⪰ σ} O (τ)$ .

Proof. The closure $V (σ) = \overline{O (σ)}$ is closed and $T$ -stable in $X_{Σ}$ . Hence $V (σ)$ is a disjoint union of $T$ -orbits. By Proposition 3, the orbits $O (τ) \subseteq V (σ)$ are exactly those with $τ ⪰ σ$ . So $$ V(\sigma) = \bigsqcup_{\tau \succeq \sigma} O(\tau). \qquad \square $$

Proposition 5 (closure is a toric variety with star-quotient fan). For each $σ \in Σ$ , $V (σ)$ is a toric variety with dense torus $O (σ)$ , character lattice $M (σ) = σ^{⊥} \cap M$ , cocharacter lattice $N (σ) = N / N_{σ}$ , and fan $Σ/ σ = {\overset{τ}{ˉ} : τ ⪰ σ}$ in $N (σ)_{R}$ .

Proof sketch. The argument is given in detail in Exercise 8 above. Briefly: $V (σ) \cap U_{τ} = Spec C [τ^{\lor} \cap σ^{⊥} \cap M]$ for each $τ ⪰ σ$ , which equals the affine toric variety $U_{\overset{τ}{ˉ}}$ of the image cone $\overset{τ}{ˉ} \subseteq N (σ)_{R}$ . The affine pieces glue along face inclusions in $N (σ)_{R}$ , exactly as $X_{Σ}$ is constructed from its fan. The collection ${\overset{τ}{ˉ} : τ ⪰ σ}$ together with all faces forms the fan $Σ/ σ$ , and the gluing produces $V (σ) = X_{Σ/ σ}$ as a toric variety. $□$

Proposition 6 (toric stratification of $X_{Σ}$ ). $X_{Σ} = ⨆_{σ \in Σ} O (σ)$ , a locally closed stratification by $T$ -orbits.

Proof. The orbits of $T$ on $X_{Σ}$ partition $X_{Σ}$ as a set. By Proposition 2, the orbits are indexed by cones of $Σ$ . Each orbit $O (σ) ≅ (C^{*})^{n - d i m σ}$ is a locally closed sub-scheme of $X_{Σ}$ (the locally closed structure comes from the affine chart $U_{σ}$ where $O (σ)$ is the closed orbit, hence closed in the open subset $U_{σ} \subseteq X_{Σ}$ ). The stratification is by closure, with $\overline{O (σ)} = V (σ) \supseteq O (τ)$ iff $τ ⪰ σ$ (Proposition 3). $□$

Connections [Master]

Algebraic torus 04.11.01. The torus $T = N \otimes C^{*}$ acting on $X_{Σ}$ is the structural object whose orbits the correspondence classifies. The character lattice $M = Hom (N, Z)$ supplies the regular functions on the torus and on $X_{Σ}$ , and the orbit $O (σ) = Hom_{Z} (σ^{⊥} \cap M, C^{*})$ is itself an algebraic torus of dimension $n - dim σ$ . The duality between $M$ and $N$ is exactly the duality between characters of $T$ and one-parameter subgroups of $T$ that underlies the dimension formula $dim O (σ) + dim σ = n$ .
Fan and toric variety 04.11.04. The variety $X_{Σ}$ on which the torus acts is the scheme-theoretic object constructed by gluing affine pieces $U_{σ}$ along face-inclusion open immersions. The orbit-cone correspondence is the equivariant-stratification refinement of the fan-to-variety construction: the same combinatorial input (the fan) controls the equivariant orbit decomposition (this unit) as controls the global scheme structure (the prerequisite unit). The bridge is the chart-local correspondence between faces of a single cone and orbits of $T$ in the affine chart $U_{σ}$ , glued across the fan.
Smoothness and completeness via fans 04.11.05. The sibling unit. Both the smoothness criterion and the completeness criterion read directly off the orbit stratification developed here: smoothness of $X_{Σ}$ is local on the affine chart $U_{σ}$ corresponding to each cone, equivalently smoothness at every $T$ -fixed point $x_{σ}$ for maximal $σ$ — exactly the orbits of dimension zero in this correspondence. Completeness $∣Σ∣ = N_{R}$ is equivalent to every cocharacter $v \in N$ having a well-defined limit point in some orbit, equivalently every one-parameter subgroup of $T$ degenerating into a closed orbit of $X_{Σ}$ . The orbit-cone correspondence is the structural backbone on which the criteria of [04.11.05] rest.
Toric resolution of singularities 04.11.07. The toric resolution $X_{Σ^{'}} \to X_{Σ}$ via star subdivision of non-smooth cones adds new orbits to the stratification: each new cone added by the subdivision corresponds to a new orbit, with the resolution morphism contracting the new orbits onto the corresponding singular orbit of $X_{Σ}$ . The exceptional divisors of the resolution are exactly the orbit closures $V (ρ)$ for new rays $ρ$ added during the subdivision. The orbit-cone dictionary is the toolkit for tracking how toric birational morphisms transform the equivariant stratification.
Toric divisor and support function 04.11.08. The $T$ -invariant prime divisors of $X_{Σ}$ are the orbit closures $D_{ρ} = V (ρ)$ for rays $ρ \in Σ (1)$ — the dimension-one-cone case of the orbit-cone correspondence. The exact sequence $M \to Z^{Σ (1)} \to Cl (X_{Σ}) \to 0$ identifies the divisor class group as a quotient of the lattice of formal divisor sums, with the kernel from the principal divisors $div (χ^{u}) = \sum_{ρ} ⟨ u, u_{ρ} ⟩ D_{ρ}$ . The orbit-cone correspondence is the prerequisite for toric divisor theory.
Toric Picard group 04.11.09. For a smooth complete toric variety, the Picard group $Pic (X_{Σ})$ coincides with the divisor class group $Cl (X_{Σ})$ , computed via the orbit-cone identification of prime divisors with rays. The line bundles on $X_{Σ}$ are classified by support functions on the fan — piecewise-linear functions on $∣Σ∣$ that are linear on each cone. The orbit-cone correspondence supplies the structural input for the support-function classification.
Polytope-fan dictionary 04.11.10. For a projective toric variety $X_{Σ}$ with ample line bundle $L_{P}$ from a lattice polytope $P \subseteq M_{R}$ , the orbit-cone correspondence identifies the strata of $X_{Σ}$ with the faces of the polytope $P$ via the moment-map duality. Each face $F ⪯ P$ corresponds to a cone $σ_{F} \in Σ$ via the normal-fan construction, and the orbit $O (σ_{F})$ projects onto the face $F$ under the moment map. The polytope-fan dictionary makes the orbit decomposition computable in polytope terms.
Algebraic moment map 04.11.11. The algebraic moment map $μ : X_{Σ} \to P$ on a projective toric variety identifies the strata of $X_{Σ}$ with the faces of the moment polytope $P$ , recovering the orbit-cone correspondence on the symplectic side. The closed orbits (fixed points) project to the vertices of $P$ , the one-dimensional orbits project to the edges, and so on. Atiyah 1982 (Bull. London Math. Soc. 14) and Guillemin-Sternberg 1982 (Inv. Math. 67) established the convexity theorem for the moment-map image; the orbit-cone correspondence is its algebraic-geometric counterpart.
Cohomology of a smooth complete toric variety 04.11.12. The Danilov-Jurkiewicz cohomology presentation $H^{*} (X_{Σ}; Q) = Q [D_{ρ}]_{ρ \in Σ (1)} / (linear + Stanley-Reisner)$ rests on the toric stratification: the divisor classes $[D_{ρ}]$ are exactly the orbit closures of one-dimensional cones, and the Stanley-Reisner relations reflect the fan combinatorics — products of divisors corresponding to rays not bounding a common cone of $Σ$ vanish. The Euler characteristic equals the number of maximal cones (the count of fixed points by the orbit-cone correspondence).
Spherical variety theory (Luna-Vust; Demazure 1976). The orbit-cone correspondence generalises to spherical varieties — normal varieties with an action of a reductive group $G$ having an open $G$ -orbit. Luna-Vust theory (1983) classifies the $G$ -orbits in a spherical embedding via combinatorial "coloured fans" extending the cone-fan formalism. The toric case is the abelian-group special case where $G = T$ is a torus and the colours are absent; the more general spherical theory follows the same structural pattern of cone-to-orbit dictionaries with star-quotient closures.
Białynicki-Birula decomposition [03.05 area]. The toric stratification $X_{Σ} = ⨆_{σ} O (σ)$ is a special case of the Białynicki-Birula decomposition (BB 1973, Ann. Math. 98) of a smooth complete variety with an algebraic-torus action into locally closed cells indexed by fixed-point components. For toric varieties, the fixed-point components are exactly the points $x_{σ}$ for maximal cones, and the BB-cells are exactly the orbits $O (σ)$ — the toric case is the cleanest example of the general theorem.

Historical & philosophical context [Master]

The orbit-cone correspondence was first formulated in its modern combinatorial 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 Théorème 4 stated that the variety $X_{Σ}$ associated to a fan $Σ$ has a stratification by $T$ -orbits indexed by the cones of $Σ$ , with orbit dimensions $n - dim σ$ . The motivation was the classification of algebraic subgroups of the Cremona group $Cr_{n} = Bir (P^{n})$ of maximal rank, and the orbit decomposition arose as the equivariant-stratification half of the classification.

The general scheme-theoretic foundation was developed by Hideyasu Sumihiro in Equivariant completion (Journal of Mathematics of Kyoto University 14, 1974, pp. 1-28) ^{[Sumihiro 1974]}, whose equivariant-covering theorem reduces the global orbit decomposition to the affine-local case, where the orbit-face correspondence for a single cone $σ$ inside $U_{σ}$ is the immediate content of Demazure's affine-toric classification. Sumihiro's 1975 follow-up Equivariant completion II (J. Math. Kyoto Univ. 15, 573-605) extended the theorem to general reductive-group actions, foreshadowing the Luna-Vust spherical variety theory of 1983.

The English-language consolidation of the orbit-cone correspondence is due to Vasily Danilov's 1978 survey The geometry of toric varieties (Russian Mathematical Surveys 33(2), 97-154) ^{[Danilov 1978]}, where the star-quotient construction and the closure-as-toric-variety theorem were given canonical form. Danilov's article was the standard reference for toric geometry through the 1980s and was the immediate predecessor cited by William Fulton in Introduction to Toric Varieties (Princeton Annals Studies 131, 1993) ^[pending], whose §3.1-§3.2 became the modern textbook treatment of the orbit-cone correspondence. The exposition of David Cox, John Little, and Henry Schenck in Toric Varieties (American Mathematical Society GSM 124, 2011) ^[pending] extended Fulton with explicit worked examples and the Stanley-Reisner cohomology computation.

The orbit-cone correspondence's general-equivariant context is the Białynicki-Birula decomposition (Andrzej Białynicki-Birula 1973, Some theorems on actions of algebraic groups, Annals of Mathematics 98, 480-497) ^{[Białynicki-Birula 1973]}, which provides a stratification of any smooth complete variety with an algebraic-torus action into locally closed affine cells indexed by fixed-point components. For toric varieties, the BB-decomposition coincides exactly with the toric stratification, identifying $X_{Σ}$ as the simplest example of a general phenomenon. The connection is foundational to modern equivariant cohomology (Atiyah-Bott 1984 localisation, Goresky-Kottwitz-MacPherson 1998) and to the Białynicki-Birula treatment of $T$ -equivariant intersection numbers (Brion 1997 Equivariant cohomology and equivariant intersection theory).

The Japanese-school perspective was crystallised by Tadao Oda in Convex Bodies and Algebraic Geometry (Ergebnisse der Mathematik 15, Springer 1988) ^[pending], whose Ch. 1 §1.6 develops the orbit-cone correspondence in the lattice-and-cone formalism. Oda's book was the standard reference until Fulton's 1993 monograph superseded it for many uses; the two texts together established the canonical mid-1980s-to-mid-1990s English-language treatment.

The symplectic-geometric counterpart of the orbit-cone correspondence is the convexity theorem of Atiyah 1982 (Convexity and commuting Hamiltonians, Bulletin of the London Mathematical Society 14, 1-15) ^{[Atiyah 1982]} and Guillemin-Sternberg 1982 (Convexity properties of the moment mapping, Inventiones Mathematicae 67, 491-513), which establishes that the image of a Hamiltonian torus action's moment map is a convex polytope. Specialised to Kähler toric manifolds (Delzant 1988 Bull. Soc. Math. France 116), this polytope is exactly the lattice polytope $P$ whose normal fan is $Σ$ , and the moment map provides a smooth diffeomorphism between the toric stratification of $X_{Σ}$ and the face stratification of $P$ . The orbit-cone correspondence on the algebraic side and the moment-polytope correspondence on the symplectic side coincide on the projective Kähler case, an instance of the broader Geometric Quantization correspondence between Hamiltonian and algebraic-geometric reductions.

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{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{BialynickiBirula1973,
  author  = {Bia{\l}ynicki-Birula, Andrzej},
  title   = {Some theorems on actions of algebraic groups},
  journal = {Annals of Mathematics},
  volume  = {98},
  year    = {1973},
  pages   = {480--497}
}

@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{Atiyah1982,
  author  = {Atiyah, Michael F.},
  title   = {Convexity and commuting {H}amiltonians},
  journal = {Bulletin of the London Mathematical Society},
  volume  = {14},
  year    = {1982},
  pages   = {1--15}
}

@article{GuilleminSternberg1982,
  author  = {Guillemin, Victor and Sternberg, Shlomo},
  title   = {Convexity properties of the moment mapping},
  journal = {Inventiones Mathematicae},
  volume  = {67},
  year    = {1982},
  pages   = {491--513}
}

@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{Jurkiewicz1980,
  author  = {Jurkiewicz, Jerzy},
  title   = {Chow ring of projective nonsingular torus embedding},
  journal = {Colloquium Mathematicum},
  volume  = {43},
  year    = {1980},
  pages   = {261--270}
}

@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{AtiyahBott1984,
  author  = {Atiyah, Michael F. and Bott, Raoul},
  title   = {The moment map and equivariant cohomology},
  journal = {Topology},
  volume  = {23},
  year    = {1984},
  pages   = {1--28}
}

@article{Brion1997,
  author  = {Brion, Michel},
  title   = {Equivariant cohomology and equivariant intersection theory},
  booktitle = {Representation theories and algebraic geometry (Montreal 1997)},
  publisher = {Kluwer Academic Publishers},
  series    = {NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.},
  volume    = {514},
  year      = {1998},
  pages     = {1--37}
}

Prerequisites

04.11.01
04.11.04

Tier anchors

beginner: Fulton *Introduction to Toric Varieties* §3.1 informal — the $T$-orbits of a toric variety are in bijection with the cones of its fan, with $\dim O(\sigma) + \dim \sigma = n$
intermediate: Fulton *Introduction to Toric Varieties* §3.1; Cox-Little-Schenck *Toric Varieties* §3.2 (Theorem 3.2.6, the orbit-cone correspondence)
master: Fulton §3.1, §3.2; Cox-Little-Schenck §3.2; Oda *Convex Bodies and Algebraic Geometry* Ch. 1 §1.6; Demazure 1970 *Sous-groupes algébriques de rang maximum du groupe de Cremona* (Ann. Sci. ENS 3, 507-588); Sumihiro 1974 *Equivariant completion* (J. Math. Kyoto Univ. 14, 1-28); Danilov 1978 *The geometry of toric varieties* (Russian Math. Surveys 33(2), 97-154); Białynicki-Birula 1973 *Some theorems on actions of algebraic groups* (Ann. of Math. 98, 480-497)

References

TODO_REF
Fulton — Introduction to Toric Varieties · §3.1 (orbits and cones; Theorem on p. 51 stating the bijection $\sigma \leftrightarrow O(\sigma)$ with $\dim O(\sigma) = n - \dim \sigma$); §3.2 (orbit closures and the star); §3.4 (the fixed points and the orbits of dimension one)
TODO_REF
Cox-Little-Schenck — Toric Varieties · §3.2 (Theorem 3.2.6 orbit-cone correspondence); Proposition 3.2.7 (orbit dimension formula); Proposition 3.2.8 (closure $V(\sigma) = \bigsqcup_{\tau \succeq \sigma} O(\tau)$); Theorem 3.2.10 (closure is a toric variety with the star quotient fan); Examples 3.2.11-3.2.13 (worked $\mathbb{P}^n$, $\mathbb{P}^1 \times \mathbb{P}^1$, Hirzebruch)
TODO_REF
Oda — Convex Bodies and Algebraic Geometry · Ch. 1 §1.6 (orbit decomposition); Proposition 1.6 (the bijection cones ↔ orbits); §1.7 (the closed embedding $V(\sigma) \hookrightarrow X_\Sigma$ and the star-quotient fan)
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; Théorème 4 (the toric variety $X_\Sigma$ has a stratification by $T$-orbits indexed by the cones of $\Sigma$, with orbit dimensions $n - \dim \sigma$)
TODO_REF
Sumihiro 1974 — Equivariant completion · Journal of Mathematics of Kyoto University 14, 1-28; the equivariant-covering theorem used to identify orbits as locally affine pieces; orbit-cone correspondence as a corollary of the affine-toric classification of Demazure
TODO_REF
Danilov 1978 — The geometry of toric varieties · Russian Mathematical Surveys 33(2), 97-154; §5 (orbit decomposition); the pre-Fulton canonical English exposition of the orbit-cone correspondence and the star-quotient construction
TODO_REF
Białynicki-Birula 1973 — Some theorems on actions of algebraic groups · Annals of Mathematics 98, 480-497; the BB-decomposition theorem for algebraic torus actions on smooth complete varieties; orbits of a one-parameter subgroup partition the variety into locally closed affine cells; toric BB-cells coincide with $T$-orbits
TODO_REF
Hirzebruch 1951 — Über eine Klasse von einfach-zusammenhängenden komplexen Mannigfaltigkeiten · Mathematische Annalen 124, 77-86; introduction of the Hirzebruch surfaces $F_a$; orbit decomposition into nine $T$-orbits via the four-ray fan
TODO_REF
Atiyah 1982 — Convexity and commuting Hamiltonians · Bulletin of the London Mathematical Society 14, 1-15; convexity of the moment-map image, the symplectic side of the orbit-cone correspondence on Kähler toric manifolds

Lean module

Codex.AlgGeom.Toric.OrbitConeCorrespondence

Mathlib gap

Mathlib has algebraic-torus and group-scheme infrastructure in
`Mathlib.AlgebraicGeometry.GroupScheme` and `Mathlib.AlgebraicGeometry.AffineScheme`,
but the orbit-cone correspondence is not formalised. Specifically
absent: (i) the `Fan` and `RationalPolyhedralCone` structures from
prerequisites 04.11.02 and 04.11.04; (ii) the `ToricVariety` scheme
obtained by gluing affine pieces; (iii) the orbit $O(\sigma) :=
\mathrm{Hom}_\mathbb{Z}(\sigma^\perp \cap M, k^*)$ as an algebraic
sub-torus of $X_\Sigma$ together with its dimension formula
$\dim O(\sigma) + \dim \sigma = n$; (iv) the orbit closure
$V(\sigma) = \overline{O(\sigma)}$ realised as a toric subvariety
with fan the *star quotient* $\Sigma / \sigma$ in $(N/N_\sigma)_\mathbb{R}$;
(v) the order-reversing bijection between $\Sigma$ and the set of
$T$-orbits in $X_\Sigma$. The companion module
`Codex.AlgGeom.Toric.OrbitConeCorrespondence` records the structural
scaffolding with `sorry`-stubbed proofs and supplies decidable
numerical witnesses for the orbit counts on $\mathbb{P}^2$ (seven),
$\mathbb{P}^1 \times \mathbb{P}^1$ (nine), and the Hirzebruch surfaces
$F_a$ (nine for every $a \in \mathbb{N}$). A Mathlib-grade development
would unlock toric-divisor combinatorics (04.11.08), toric Picard
groups (04.11.09), polytope-fan dictionaries (04.11.10), and the
Stanley-Reisner cohomology computation for smooth complete toric
varieties (04.11.12).

Reviewer

TBD

Estimated time

beginner: 20m
intermediate: 45m
master: 80m

Intuition [Beginner]

Visual [Beginner]

Worked example [Beginner]

Check your understanding [Beginner]

Formal definition [Intermediate+]

Counterexamples to common slips

Key theorem with proof [Intermediate+]

Lean formalization [Intermediate+]

Exercises [Intermediate+]

Advanced results [Master]

§A.1 The bijection cones ↔ orbits and the dim formula

§A.2 Closure V(σ) and the star quotient fan Σ/σ

§A.3 The toric stratification of XΣ​

§A.4 Worked examples — Pn, P1×P1, Hirzebruch Fa​

Full proof set [Master]

Connections [Master]

Historical & philosophical context [Master]

Bibliography [Master]

§A.2 Closure $V (σ)$ and the star quotient fan $Σ/ σ$

§A.3 The toric stratification of $X_{Σ}$

§A.4 Worked examples — $P^{n}$ , $P^{1} \times P^{1}$ , Hirzebruch $F_{a}$