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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 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

Take the weighted projective plane $X=\mathbb{P}(1,1,2)$. Its fan in $N={\mathbb{Z}}^2$ has rays generated byand all three two-dimensional cones spanned by adjacent rays. Their union is $N_{\mathbb{R}}$, so the fan is complete and the toric variety $X$ is proper.

The determinants of the cones $⟨v_1,v_2⟩$ and $⟨v_2,v_0⟩$ have absolute value one, whileThe smoothness criterion for a toric variety therefore shows that the affine chart for $⟨v_0,v_1⟩$ is singular; it is the cyclic quotient singularity of type $\frac{1}{2}(1,1)$. Thus $X$ is a singular proper toric surface.

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

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.