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.

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 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.

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

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 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.

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.

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 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.

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.

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

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.

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.

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.

A divisor is basepoint-free when its global sections have no common zero. Its complete linear system therefore defines a Kodaira map everywhere.

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.

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.