Toric variety

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.