Fan in toric geometry ($\Sigma$)

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 ${\mathbb{P}}^1$.