Toric geometry

Toric geometry studies algebraic varieties containing an algebraic torus as a dense open subset, with the torus action extended to the whole variety. Rational polyhedral fans translate their geometry into lattice combinatorics.

A toric lattice is a finite-rank free abelian group $N$, embedded as a discrete full-rank subgroup of $N_{\mathbb{R}}=N{\otimes }_{\mathbb{Z}}\mathbb{R}$. Its dual is $M=\mathrm{Hom}(N,\mathbb{Z})$. This lattice of integral vectors is distinct from an order-theoretic lattice. It supplies the integral structure for toric cones and their dual monomials.

A lattice basis $v_1,\dots ,v_n$ expresses each vector of the toric lattice uniquely as ${\sum }_i{a}_i{v}_i$ with $a_i\in \mathbb{Z}$. Relative to any fixed integral basis, vectors form another lattice basis exactly when their determinant is $\pm 1$. A primitive vector is one whose integer span is saturated; it can be completed to such a basis by the Euclidean algorithm.

Given a dual pair of free lattices $N,M$ and a rational fan, glue $\mathrm{Spec}A[{\sigma }^{\vee }\cap M]$ over common-face charts. The monoid rings are semigroup algebras. A lattice map respecting cones gives a toric morphism. This construction works over a commutative ring $A$, including $\mathbb{Z}$, and also for an infinite fan in toric geometry.

// Target: toric-geometry.bigb

Take rays $(i,1)$ and cones generated by consecutive rays in ${\mathbb{Z}}^2$. Their dual generators $(1,-i)$ and $(-1,i+1)$ give $x_i=t{q}^{-i}$, $y_i=t^{-1}{q}^{i+1}$. Adjacent charts glue by $x_{i+1}=y_i^{-1}$, $y_{i+1}=x_i{y}_i^2$. The fibre $q=0$ is a chain of projective lines: each component has two affine-line charts joined by reciprocal coordinates, and consecutive components meet at one node. Inverting $q$ identifies all charts with the multiplicative group scheme over $\mathbb{Z}[q,q^{-1}]$. The local node algebra is free over $\mathbb{Z}[q]$, with basis $1,x_i^a,y_i^b$ for $a,b\ge 1$, so the morphism is flat.

// Target: toric-geometry.bigb

The lattice shear $(a,b)\mapsto (a+mb,b)$ shifts the chain charts by $m$ and preserves $q$. On the dense torus it acts as $(t,q)\mapsto (q^{m}t,q)$. For $m\ge 3$, translates of each chart of the formal completion of a scheme along $q=0$ are disjoint. Glue $m$ representative formal charts cyclically by the adjacent-chart formulas to obtain a formal scheme quotient. The two branch opens of each chart are disjoint because $D(x_i)\cap D(y_i)=D(q)$ is empty on the formal support. The quotient fibre is an $m$-gon of projective lines, and the quotient map is locally an isomorphism on each translated chart. Invariant morphisms descend uniquely through this gluing.

// Target: ringed-space.bigb

An algebraic torus of dimension $n$ is an algebraic group isomorphic over the base field to $({\mathbb{G}}_m{)}^n$. Its character and cocharacter lattices are dual free abelian groups $M$ and $N$.

A maximal algebraic torus is an algebraic torus maximal among the tori contained in an algebraic group. In a connected reductive algebraic group, all maximal algebraic tori are conjugate. A Borel subgroup contains maximal algebraic tori, and its unipotent radical gives a decomposition $B=T⋉U$.

A maximal algebraic torus $T$ is rational for $F$ when $F(T)=T$. Rationality does not require every point of $T$ to be fixed: the finite group $T^F$ is its rational-point group. Similarly a rational Borel subgroup or parabolic subgroup means an $F$-stable algebraic subgroup.

A rational maximal torus is elliptic when it is contained in no proper rational parabolic subgroup. This definition allows the split central torus of the ambient group; an absolute absence of every split subtorus is not the right condition when the centre is nontrivial.

The character lattice $M=\mathrm{Hom}(T,{\mathbb{G}}_m)$ of an algebraic torus is the free abelian group of algebraic group homomorphisms from $T$ to the multiplicative group.

An algebraic torus character is a morphism of algebraic groups $T\to {\mathbb{G}}_m$. For a split torus with character lattice $M$, these are precisely the Laurent monomials ${\chi }^m$, with ${\chi }^{m+m^{\prime }}={\chi }^m{\chi }^{m^{\prime }}$.

// Target: algebra.bigb

The cocharacter lattice $N=\mathrm{Hom}({\mathbb{G}}_m,T)$ is dual to the character lattice of an algebraic torus under composition:

A fan in toric geometry is a finite collection of rational strongly convex cones in toric geometry in a real vector space with a lattice, closed under taking faces of a polyhedral cone, such that the intersection of two cones is a face of each. A fan determines a toric variety.

A fan $\Sigma$ in $N_{\mathbb{R}}$ is a finite collection of strictly convex rational polyhedral cones closed under taking faces, such that the intersection of two cones is a face of each.

An infinite rational fan allows infinitely many cones in toric geometry while retaining the face and intersection axioms of a fan in toric geometry. Gluing its affine monoid charts constructs a possibly non-quasi-compact toric scheme over a base ring. Finiteness of the collection is unnecessary for the chart gluing, though it is needed for finite-type conclusions.

// Target: toric-geometry.bigb

The normal cone of a face $F$ of a polytope consists of linear functionals whose minimum is attained on all of $F$. These cones form the inward normal fan, complete in the dual of the polytope's affine span. For a full-dimensional lattice polytope, it is a complete rational fan in the full dual space.

// Target: geometry-and-topology.bigb

A cone in toric geometry is a convex cone generated by finitely many lattice vectors and containing no nonzero linear subspace. Such cones are the pieces of a fan in toric geometry.

A cone in toric geometry is the nonnegative real span of finitely many lattice vectors. It is rational when those generators lie in a lattice and strictly convex when it contains no nonzero linear subspace.

A face of a polyhedral cone $C$ is $C\cap \ker \ell$ for a linear functional nonnegative on $C$. Every convex face of a polyhedral cone is exposed in this way. Faces include the whole cone and its lineality space, and are closed under intersection.

// Target: geometry-and-topology.bigb

A ray of a fan is a one-dimensional cone in a fan in toric geometry. It has the form ${\mathbb{R}}_{\ge 0}v$, where $v$ is a primitive lattice vector, and occurs as a face of a polyhedral cone.

A ray of a fan is a one-dimensional cone. It has a unique primitive lattice generator $v_{\rho }$ pointing along it.

For a cone $\sigma \subseteq N_{\mathbb{R}}$, its dual cone is

For a closed polyhedral cone $C$, the maps $\tau \mapsto C^{\vee }\cap {\tau }^{\perp }$ and $\nu \mapsto C\cap {\nu }^{\perp }$ give inverse order-reversing bijections on faces. Choose a exposing functional in the relative interior of the dual face. The bipolar identity and an exposing functional for the original face show that applying the maps twice returns that face.

// Target: geometry-and-topology.bigb

For a rational cone $\sigma$, the lattice points ${\sigma }^{\vee }\cap M$ form a finitely generated additive monoid.

The lattice points of a rational polyhedral cone form a finitely generated additive monoid. Choose lattice generators for the cone. Subtracting integer parts of nonnegative coefficients leaves a lattice point in their bounded fundamental parallelepiped; its finitely many lattice points, together with the chosen generators, generate the monoid. This supplies finite generation of the coordinate ring of an affine toric variety.

// Target: geometry-and-topology.bigb

The Hilbert basis is the unique minimal generating set of the additive monoid of lattice points in a pointed rational cone.

A polyhedral cone is simplicial when its primitive ray generators are linearly independent. A smooth cone in a lattice must be simplicial, but simpliciality alone does not make its ray generators part of a lattice basis.

A fan is complete when the union of all its cones is the whole ambient real vector space. A toric variety is proper exactly when its fan is complete.

For a cone $\tau \in \Sigma$, its star is the fan in the quotient by $\mathrm{span}(\tau )$ formed from the images of all cones $\sigma$ containing $\tau$ as a face.

In dimension $d$, the fan whose rays are generated by $e_1,\dots ,e_d,-{\sum }_i{e}_i$ and whose cones are generated by proper subsets of those rays is complete. It is the fan of projective space ${\mathbb{P}}^d$.

A subdivision of a fan replaces its cones by smaller cones with the same support and compatible intersections. It induces a proper birational toric morphism.

This operation is a fan subdivision. A star subdivision inserts a ray through a lattice point in a cone and subdivides every cone containing that point. It induces a proper birational toric morphism.

On a smooth toric variety, the blowup along the orbit closure in a toric variety of a cone generated by part of a lattice basis $v_1,\dots ,v_k$ is the star subdivision at their sum. Locally the center has ideal $(x_1,\dots ,x_k)$; its blowup charts adjoin $x_j/x_i$, producing precisely these subdivided cones. This extends the toric blowup at a torus-fixed point to invariant centers of any codimension.

If a smooth maximal cone is generated by a lattice basis $e_1,\dots ,e_n$, its star subdivision through $e_1+\cdots +e_n$ is the fan of the blowup at the corresponding torus-fixed point.

For an ample invariant divisor $D$ and the blowup $\pi$ of a torus-fixed smooth point in dimension at least two, $c{\pi }^{\ast }D-E$ is ample for sufficiently large integers $c$. In the subdivided cone, its local characters are $c{m}_{\sigma }+u_i$, where $u_i$ is dual to the omitted ray generator. The new strict inequalities have gap one on the omitted ray and grow linearly with $c$ elsewhere, proving the claim by the toric ampleness criterion.

The blowup of ${\mathbb{A}}^n$ at the origin is the total space of the tautological line bundle ${\mathcal{O}}_{{\mathbb{P}}^{n-1}}(-1)$. Its exceptional divisor is the zero section ${\mathbb{P}}^{n-1}$, and projection to the direction of a line gives the bundle map to ${\mathbb{P}}^{n-1}$.

A toric resolution of singularities is obtained by subdividing a fan until every cone is generated by part of a lattice basis. The resulting smooth toric variety maps properly and birationally to the original one.

For this toric cone, one smooth fan subdivision has maximal cones $abc,acd$ and no exceptional divisors. Another has $abd,bce,cde,dbe$, with $e=a+c=(b+c+d)/2$. Its only exceptional divisor is a projective plane, since the quotient fan at $e$ has the three rays $\bar{b},\bar{c},\bar{d}$ with sum zero and unimodular neighboring cones. Both subdivisions admit no proper smooth coarsening. Their common smooth refinement is obtained from the first by adding $e$ in $ac$ and then $w=a+e$ in $ae$, and from the second by adding $w=b+d$ in $bd$. Each step is a toric blowup along an orbit closure.

First make a toric fan simplicial by compatible rational subdivisions. In a nonregular simplicial cone, choose a nonregular face minimal under inclusion. Its half-open fundamental parallelepiped contains a nonzero point of the toric lattice $w={\sum }_i{\alpha }_i{v}_i$, with every $0<{\alpha }_i<1$: a zero coefficient would put it in a regular proper face and force all coefficients to be integers. A star subdivision at the primitive point on this ray replaces a generator by $w$ and multiplies the affected lattice determinant by a number strictly between zero and one. The maximum multiplicity and the number of maximal cones realizing it decrease lexicographically, giving a finite toric resolution of singularities.

For the cyclic quotient surface singularity $\frac{1}{r}(1,q)$, expand $r/q=[b_1,\dots ,b_s{]}^{-}$ with $b_i\ge 2$. In the quotient lattice, start with $v_0=(0,1)$, $v_1=(1,q)/r$, and set $v_{i+1}=b_i{v}_i-v_{i-1}$. The last vector is $(1,0)$, each consecutive pair is a lattice basis, and subdivision by these rays is a resolution. Its exceptional chain has self-intersections $-b_i$. None is a minus-one curve, so the resolution is minimal. All coefficients are two exactly for $q=r-1$.

// Target: geometry-and-topology.bigb

In a smooth complete toric surface, if consecutive primitive rays are $u,w,v$ and $u+v=aw$, then the invariant divisor $D_w$ has $D_w^2=-a$. A ray introduced inside a cone by a smooth subdivision has $a>0$, so its divisor has negative self-intersection.

A compact invariant curve with consecutive rays $u,v,w$ on a smooth toric surface contracts torically to a smooth point exactly when $u+w=v$, equivalently when its self-intersection is $-1$. Deleting $v$ then reverses a toric blowup at a torus-fixed point.

// Target: geometry-and-topology.bigb

The product of fans ${\Sigma }_1\subseteq N_{1,\mathbb{R}}$ and ${\Sigma }_2\subseteq N_{2,\mathbb{R}}$ consists of the cones ${\sigma }_1\times {\sigma }_2$. Its toric variety is $X_{{\Sigma }_1}\times X_{{\Sigma }_2}$.

The fan $\{0,{\mathbb{R}}_{\ge 0}\}$ in $\mathbb{R}$ defines the affine line ${\mathbb{A}}^1$.

The fan $\{0,{\mathbb{R}}_{\ge 0},{\mathbb{R}}_{\le 0}\}$ defines the complex projective line ${\mathbb{P}}^1$.

A toric variety $X_{\Sigma }$ is obtained by gluing the affine toric varieties associated with the cones of a fan $\Sigma$. It contains an algebraic torus as a dense open orbit.

Two neighboring smooth cones can be normalized so that the three rays are $(1,0),(0,1),(-1,-r)$, with $r\ge 1$. Completeness forces the negative sign and their primitive relation has weights $(1,r,1)$. Thus these surfaces are precisely the weighted projective planes $\mathbb{P}(1,1,r)$. For $r=1$ the surface is the projective plane; for $r>1$ its unique singularity is the cyclic quotient surface singularity $\frac{1}{r}(1,1)$.

A toric surface is a two-dimensional normal toric variety, described by a toric fan in a rank-two toric lattice. Primitive consecutive rays bound its affine charts; those charts are smooth exactly when the two ray generators form a lattice basis.

The weighted projective space is $\mathrm{Proj}k[z_0,\dots ,z_n]$ with $\mathrm{deg}{z}_i=w_i>0$. Over an algebraically closed field of characteristic zero it is the geometric quotient of punctured affine space by $\lambda \cdot z_i={\lambda }^{w_i}{z}_i$. For well-formed weights, these weights also grade its Cox ring.

// Target: geometry-and-topology.bigb

A weighted projective space with positive integer weights $q_0,\dots ,q_n$ is well-formed when the greatest common divisor of the weights with any one weight omitted is one. The quotient toric lattice $N={\mathbb{Z}}^{n+1}/\mathbb{Z}(q_0,\dots ,q_n)$ is free and the images of the coordinate vectors are primitive toric rays. Its fan contains the cones on all proper subsets of these rays.

The toric divisor class sequence identifies the divisor class group with $\mathbb{Z}$ via $(a_i)\mapsto \sum q_i{a}_i$. Choose integers $a_i$ with $\sum q_i{a}_i=1$; then $D=\sum a_i{F}_i$ represents its positive generator $H$. This is an ample Q-Cartier divisor: for $\ell =\mathrm{lcm}(q_i)$ the local Cartier data of $\ell D$ on the chart omitting ray $j$ is $m_j=(\ell /q_j)e_j-\ell (a_i)$, which satisfies the strict toric ampleness criterion. The generator need not itself be a Cartier divisor; for example the positive generator on $\mathbb{P}(1,1,2)$ has Cartier index two.

With $D=\sum a_i{F}_i$ and $\sum q_i{a}_i=1$, the dual toric lattice is $M=\{m\in {\mathbb{Z}}^{n+1}:\sum q_i{m}_i=0\}$. A torus character ${\chi }^m$ is a section of $rD$ exactly when $b_i=m_i+r{a}_i\ge 0$. The vector $b$ has weighted degree $\sum q_i{b}_i=r$, and the correspondence ${\chi }^m\mapsto X^b$ is bijective on bases in every degree and respects multiplication. This gives the displayed graded-ring isomorphism, even when $D$ is a non-Cartier Weil divisor.

This is a two-dimensional weighted projective space. The weighted projective plane $\mathbb{P}(a,b,c)$ is the quotient of ${\mathbb{A}}^3\setminus \{0\}$ by $\lambda \cdot (x,y,z)=({\lambda }^{a}x,{\lambda }^{b}y,{\lambda }^{c}z)$. It is generally singular but is a proper toric surface.

For coprime positive integers $a,b$,and the natural map identifies the Picard group with $ab\mathbb{Z}$ inside the class group. The plane is smooth exactly when $a=b=1$.

For a lattice polytope $\Delta$, form the semigroup of lattice points $(m,l)$ with $l\ge 0$ and $m\in l\Delta$, using only $(0,0)$ in degree zero. Its semigroup algebra is graded by $l$, and its Proj construction gives ${\mathbb{P}}_{\Delta }$. The chart at a lattice vertex $v$ is $\mathrm{Spec}k[\mathrm{cone}(\Delta -v)\cap M]$; these charts give the normal fan of a polytope construction. For a polytope of smaller dimension, use its intrinsic affine lattice after translation.

// Target: geometry-and-topology.bigb

For a rational strongly convex cone $\sigma \subseteq N_{\mathbb{R}}$, the affine toric variety is

For $r>1$ and $\gcd (r,q)=1$, the quotient of ${\mathbb{A}}^2$ by $(x,y)\mapsto (\zeta x,{\zeta }^{q}y)$ is a cyclic quotient surface singularity. Its toric model uses the first-quadrant cone in ${\mathbb{Z}}^2+\mathbb{Z}(1,q)/r$. Conversely, every singular full-dimensional affine toric surface over an algebraically closed field of characteristic zero has this form. Smooth cones and cones with a torus factor require separate treatment.

// Target: geometry-and-topology.bigb

The cyclic quotient surface singularity $\frac{1}{r}(q,1)$ is the only singular point of a complete four-ray toric surface exactly under the displayed condition. Normalize the three smooth cones to the consecutive rays $(a,-1),(1,0),(0,1),(-1,b)$. The singular cone has determinant $1-ab=r$, so $ab=1-r$. In the basis $(-1,b),(0,-1)$, its other generator is $-a(-1,b)+r(0,-1)$, giving quotient parameter $a\mathrm{mod}r$. Thus $a=\pm s$ for a positive divisor $s$ of $r-1$. Conversely these rays construct the surface. Exchanging the two axes replaces $q$ by $q^{-1}$; replacing a factor by its complementary factor preserves the criterion under this equivalence.

For a strongly convex rational cone $\sigma$, the group generated by $S={\sigma }^{\vee }\cap M$ is $M$. Inverting its finitely many monomial generators gives $k[M]$, so $\mathrm{Spec}k[M]$ is a dense open algebraic torus in $\mathrm{Spec}k[S]$. The map ${\chi }^m\mapsto {\chi }^m\otimes {\chi }^m$ extends torus multiplication to the entire variety.

// Target: geometry-and-topology.bigb

The coordinate ring of $U_{\sigma }$ is generated by the characters ${\chi }^m$ for lattice points $m\in {\sigma }^{\vee }\cap M$, with multiplication ${\chi }^m{\chi }^{m^{\prime }}={\chi }^{m+m^{\prime }}$.

The cone generated by $(1,0)$ and $(-1,3)$ has semigroup ringwhere ${\mu }_3$ acts diagonally with weight one. Inserting the ray through $(0,1)$ resolves it, with one exceptional curve of self-intersection $-3$.

The orbit-cone correspondence assigns to every cone $\sigma \in \Sigma$ a torus orbit of codimension $\dim \sigma$. In particular, maximal cones correspond to torus-fixed points.

At a point $p$ of $\mathrm{Spec}k[C\cap M]$, the exponents $m$ with ${\chi }^m(p)\ne 0$ form a semigroup face. They are the lattice points of one face of $C$. Points with that same support face form a single algebraic torus orbit: their nonzero monomial evaluations extend to homomorphisms from the lattice generated by the face into $k^{\ast }$, and restriction from the full torus is surjective.

// Target: mathematical-optimization.bigb

If $\tau ⪯\sigma$, choose $m\in {\sigma }^{\vee }\cap M$ with $\tau =\sigma \cap m^{\perp }$. Then $U_{\tau }=D({\chi }^m)\subset U_{\sigma }$. Its complement is the union of orbit closures indexed by faces of $\sigma$ that are not faces of $\tau$.

// Target: geometry-and-topology.bigb

For $\sigma \in \Sigma$, the closure of its torus orbit isThus orbit-closure inclusion reverses face inclusion.

On the orbit of a face $\tau ⪯\sigma$, the monomial ${\chi }^{\mu }$, with $\mu \in {\sigma }^{\vee }\cap M$, vanishes exactly when $\mu \notin {\tau }^{\perp }$. Its reduced zero locus is consequently the union of those orbit closures. The principal subscheme need not be reduced: $V(x^2)$ and $V(x)$ have the same support but different ideals.

// Target: geometry-and-topology.bigb

The action of the algebraic torus on a toric variety is by automorphisms, and automorphisms preserve regular local rings. The smooth and singular loci are therefore unions of torus orbits.

A toric variety is proper exactly when its fan is complete.

An affine toric chart associated with a cone $\sigma$ is smooth exactly when the primitive ray generators of $\sigma$ extend to a lattice basis. For a two-dimensional cone generated by $u,v$, this is equivalent to $|\det (u,v)|=1$.

The Hirzebruch surface is the ruled surfaceIt is the smooth toric surface with rays $(1,0),(0,1),(-1,k),(0,-1)$.

The negative section $S\subset {\mathbb{F}}_k$ has self-intersection $S^2=-k$. Together with a fiber $F$ of the ruling, it generates the Picard group and satisfies $F^2=0$ and $S\cdot F=1$.

The fiber class $F$ is the divisor class of a fiber of ${\mathbb{F}}_k\to {\mathbb{P}}^1$. Its complete linear system gives the ruling.

For a toric fiber divisor $F$ on ${\mathbb{F}}_1$,The bundles $\mathcal{O}(kF)$ are basepoint-free for $k\ge 0$ but never ample, since $F^2=0$.

A lattice homomorphism that maps each cone of one fan into a cone of another induces an equivariant morphism of the corresponding toric varieties.

Every fan morphism from the fan of ${\mathbb{P}}^2$ to the fan of ${\mathbb{P}}^1$ is zero: the three source ray images sum to zero, while each adjacent pair must lie in one half-line. Composing a fan morphism to the product fan of ${\mathbb{P}}^1\times {\mathbb{P}}^1$ with both projections therefore makes it zero.

A lattice homomorphism always gives a homomorphism of dense algebraic tori and hence a rational map of their toric compactifications. It extends across the orbit of a source cone exactly when that cone maps into one cone of the target fan.

Each ray $\rho$ of a fan determines a torus-invariant prime divisor $D_{\rho }$. Integer combinations of these divisors encode line bundles and maps from a toric variety.

For a character ${\chi }^m$ of the dense torus,

For a toric variety whose rays span $N_{\mathbb{R}}$, the divisor class group is computed bywhere $m\mapsto (⟨m,v_{\rho }⟩{)}_{\rho }$.

For an invariant divisor $D={\sum }_{\rho }{a}_{\rho }{D}_{\rho }$, its lattice polytope isIts lattice points index torus-character sections of the associated line bundle.

The Cox construction presents a toric variety as a quotient of an open subset of affine space by a quasitorus determined by its divisor class group.

In the Cox ring $S=k[z_{\rho }]$, the irrelevant ideal is generated by ${\prod }_{\rho \notin \sigma (1)}{z}_{\rho }$ as $\sigma$ runs over maximal cones. The Cox construction takes the quotient of $\mathrm{Spec}S\setminus V(B_{\Sigma })$ by the diagonalizable group whose character group is the divisor class group.

// Target: geometry-and-topology.bigb

The Cox ring of a toric variety is the polynomial ring with one variable $x_{\rho }$ for each ray, graded by the divisor class group through $\mathrm{deg}{x}_{\rho }=[D_{\rho }]$.

A geometric quotient $X\to X/G$ has fibers equal to group orbits and carries precisely the invariant regular functions locally on the quotient.