A normal toric variety corresponds to a strongly convex rational polyhedral fan. In this package, the fan associated to a normal $d$-dimensional toric variety lies in the rational vector space $\QQ^d$ with underlying lattice $N = \ZZ^d$. The fan is encoded by the minimal nonzero lattice points on its rays and the set of rays defining the maximal cones (a maximal cone is not properly contained in another cone in the fan). More information about the correspondence between normal toric varieties and strongly convex rational polyhedral fans appears in Subsection 3.1 of Cox-Little-Schenck.

To sidestep edge case issues, this package excludes some normal toric varieties. To avoid the $0$-dimensional vector space, all normal toric varieties in this package have positive dimension. Similarly, to circumvent the empty set of rays, all normal toric varieties in this package have an orbit other than the dense algebraic

- making normal toric varieties -- information about the basic constructors
- normalToricVariety -- make a normal toric variety
- rays(NormalToricVariety) -- get the rays of the associated fan
- max(NormalToricVariety) -- get the maximal cones in the associated fan
- expression(NormalToricVariety) -- get the expression used to format for printing

- affineSpace(ZZ) -- make an affine space as a normal toric variety
- hirzebruchSurface(ZZ) -- make any Hirzebruch surface
- kleinschmidt(ZZ,List) -- make any smooth normal toric variety having Picard rank two
- "makeSimplicial" -- see makeSimplicial(NormalToricVariety) -- make a birational simplicial toric variety
- "makeSmooth" -- see makeSmooth(NormalToricVariety) -- make a birational smooth toric variety
- NormalToricVariety ** NormalToricVariety -- make the Cartesian product of two normal toric varieties
- NormalToricVariety ^** ZZ -- make the Cartesian power of a normal toric variety
- "normalToricVariety" -- see normalToricVariety(List,List) -- make a normal toric variety
- normalToricVariety(Ring) -- get the associated normal toric variety
- smoothFanoToricVariety(ZZ,ZZ) -- get a smooth Fano toric variety from database
- source(ToricMap) -- get the source of the map
- target(ToricMap) -- get the target of the map
- "toricBlowup(List,NormalToricVariety)" -- see toricBlowup(List,NormalToricVariety,List) -- makes the toricBlowup of a normal toric variety along a torus orbit closure
- toricBlowup(List,NormalToricVariety,List) -- makes the toricBlowup of a normal toric variety along a torus orbit closure
- toricProjectiveSpace(ZZ) -- make a projective space as a normal toric variety
- "normalToricVariety(ToricDivisor)" -- see variety(ToricDivisor) -- get the underlying normal toric variety
- variety(ToricDivisor) -- get the underlying normal toric variety
- weightedProjectiveSpace(List) -- make a weighted projective space

- abstractSheaf(NormalToricVariety,AbstractVariety,CoherentSheaf) -- make the corresponding abstract sheaf
- "abstractSheaf(NormalToricVariety,CoherentSheaf)" -- see abstractSheaf(NormalToricVariety,AbstractVariety,CoherentSheaf) -- make the corresponding abstract sheaf
- abstractSheaf(NormalToricVariety,AbstractVariety,ToricDivisor) -- make the corresponding abstract sheaf
- "abstractSheaf(NormalToricVariety,ToricDivisor)" -- see abstractSheaf(NormalToricVariety,AbstractVariety,ToricDivisor) -- make the corresponding abstract sheaf
- "abstractVariety(NormalToricVariety)" -- see abstractVariety(NormalToricVariety,AbstractVariety) -- make the corresponding abstract variety
- abstractVariety(NormalToricVariety,AbstractVariety) -- make the corresponding abstract variety
- "cartesianProduct(NormalToricVariety)" -- see cartesianProduct(Sequence) -- make the Cartesian product of normal toric varieties
- cartierDivisorGroup(NormalToricVariety) -- compute the group of torus-invariant Cartier divisors
- "intersectionRing(NormalToricVariety)" -- see Chow ring -- make the rational Chow ring
- "intersectionRing(NormalToricVariety,AbstractVariety)" -- see Chow ring -- make the rational Chow ring
- classGroup(NormalToricVariety) -- make the class group
- components(NormalToricVariety) -- list the factors in a product
- "cotangentSheaf(List,NormalToricVariety)" -- see cotangentSheaf(NormalToricVariety) -- make the sheaf of Zariski 1-forms
- cotangentSheaf(NormalToricVariety) -- make the sheaf of Zariski 1-forms
- "cotangentSheaf(ZZ,NormalToricVariety)" -- see cotangentSheaf(NormalToricVariety) -- make the sheaf of Zariski 1-forms
- "diagonalToricMap(NormalToricVariety)" -- see diagonalToricMap -- make a diagonal map into a Cartesian product
- "diagonalToricMap(NormalToricVariety,ZZ)" -- see diagonalToricMap -- make a diagonal map into a Cartesian product
- "diagonalToricMap(NormalToricVariety,ZZ,Array)" -- see diagonalToricMap -- make a diagonal map into a Cartesian product
- dim(NormalToricVariety) -- get the dimension of a normal toric variety
- expression(NormalToricVariety) -- get the expression used to format for printing
- fan(NormalToricVariety) -- make the 'Polyhedra' fan associated to the normal toric variety
- fromCDivToPic(NormalToricVariety) -- get the map from Cartier divisors to the Picard group
- fromCDivToWDiv(NormalToricVariety) -- get the map from Cartier divisors to Weil divisors
- fromPicToCl(NormalToricVariety) -- get the map from Picard group to class group
- fromWDivToCl(NormalToricVariety) -- get the map from the group of Weil divisors to the class group
- HH^ZZ(NormalToricVariety,CoherentSheaf) -- compute the cohomology of a coherent sheaf
- "HH^ZZ(NormalToricVariety,SheafOfRings)" -- see HH^ZZ(NormalToricVariety,CoherentSheaf) -- compute the cohomology of a coherent sheaf
- hilbertPolynomial(NormalToricVariety) -- compute the multivariate Hilbert polynomial
- "hilbertPolynomial(NormalToricVariety,Ring)" -- see hilbertPolynomial(NormalToricVariety) -- compute the multivariate Hilbert polynomial
- "hilbertPolynomial(NormalToricVariety,SheafOfRings)" -- see hilbertPolynomial(NormalToricVariety) -- compute the multivariate Hilbert polynomial
- hilbertPolynomial(NormalToricVariety,CoherentSheaf) -- compute the multivariate Hilbert polynomial
- "hilbertPolynomial(NormalToricVariety,Ideal)" -- see hilbertPolynomial(NormalToricVariety,CoherentSheaf) -- compute the multivariate Hilbert polynomial
- "hilbertPolynomial(NormalToricVariety,Module)" -- see hilbertPolynomial(NormalToricVariety,CoherentSheaf) -- compute the multivariate Hilbert polynomial
- ideal(NormalToricVariety) -- make the irrelevant ideal
- "monomialIdeal(NormalToricVariety)" -- see ideal(NormalToricVariety) -- make the irrelevant ideal
- isComplete(NormalToricVariety) -- whether a toric variety is complete
- isDegenerate(NormalToricVariety) -- whether a toric variety is degenerate
- isFano(NormalToricVariety) -- whether a normal toric variety is Fano
- isProjective(NormalToricVariety) -- whether a toric variety is projective
- isSimplicial(NormalToricVariety) -- whether a normal toric variety is simplicial
- isSmooth(NormalToricVariety) -- whether a normal toric variety is smooth
- isWellDefined(NormalToricVariety) -- whether a toric variety is well-defined
- makeSimplicial(NormalToricVariety) -- make a birational simplicial toric variety
- makeSmooth(NormalToricVariety) -- make a birational smooth toric variety
- map(NormalToricVariety,NormalToricVariety,Matrix) -- make a torus-equivariant map between normal toric varieties
- map(NormalToricVariety,NormalToricVariety,ZZ) -- make a torus-equivariant map between normal toric varieties
- max(NormalToricVariety) -- get the maximal cones in the associated fan
- nefGenerators(NormalToricVariety) -- compute generators of the nef cone
- NormalToricVariety ^ Array -- make a canonical projection map
- NormalToricVariety _ Array -- make a canonical inclusion into a product
- NormalToricVariety _ ZZ -- make an irreducible torus-invariant divisor
- orbits(NormalToricVariety) -- make a hashtable indexing the torus orbits (a.k.a. cones in the fan)
- orbits(NormalToricVariety,ZZ) -- get a list of the torus orbits (a.k.a. cones in the fan) of a given dimension
- picardGroup(NormalToricVariety) -- make the Picard group
- rays(NormalToricVariety) -- get the rays of the associated fan
- ring(NormalToricVariety) -- make the total coordinate ring (a.k.a. Cox ring)
- sheaf(NormalToricVariety,Module) -- make a coherent sheaf
- "OO _ NormalToricVariety" -- see sheaf(NormalToricVariety,Ring) -- make a coherent sheaf of rings
- "sheaf(NormalToricVariety)" -- see sheaf(NormalToricVariety,Ring) -- make a coherent sheaf of rings
- sheaf(NormalToricVariety,Ring) -- make a coherent sheaf of rings
- "todd(NormalToricVariety)" -- see todd(CoherentSheaf) -- compute the Todd class of a coherent sheaf
- toricDivisor(List,NormalToricVariety) -- make a torus-invariant Weil divisor
- toricDivisor(NormalToricVariety) -- make the canonical divisor
- weilDivisorGroup(NormalToricVariety) -- make the group of torus-invariant Weil divisors

The object NormalToricVariety is a type, with ancestor classes Variety < MutableHashTable < HashTable < Thing.