# Toric Varieties

- [Introduction and First Examples](introduction.md)

  - [The Projective Plane as a Toric Variety](introduction.md#the-projective-plane-as-a-toric-variety)

    - [`Example: Toric Example1`](introduction.md#example-ex-93014b)

  - [Resolution of a Nonprojective Toric Variety](introduction.md#resolution-of-a-nonprojective-toric-variety)

    - [`Example: Toric Example10`](introduction.md#example-ex-e700da)

    - [`Example: Toric Divisor Scroll Example`](introduction.md#example-ex-ee24e7)

    - [`Example: Toric Fano Index Example`](introduction.md#example-ex-9b8afb)

  - [The Cox Ring of a Toric Variety](introduction.md#the-cox-ring-of-a-toric-variety)

    - [`Example: Cox Ring Example`](introduction.md#example-ex-bb7120)

- [Fans in Toric Lattices](toric-fans.md)

  - [Construction of Fans](toric-fans.md#construction-of-fans)

    - [`Fan(Q): [TorCon] → TorFan`](toric-fans.md#function-fan-torcon)

    - [`Fan(R,S): [TorLatElt], [[RngIntElt]] → TorFan`](toric-fans.md#function-fan-torlatelt-rngintelt)

    - [`Fan(C): TorCon → TorFan`](toric-fans.md#function-fan-torcon-2)

    - [`FanOfAffineSpace(n): RngIntElt → TorFac`](toric-fans.md#function-fanofaffinespace-rngintelt)

    - [`FanOfWPS(W): SeqEnum → TorFan`](toric-fans.md#function-fanofwps-seqenum)

    - [`FanOfProjectiveSpace(n): RngIntElt → TorFac`](toric-fans.md#function-fanofprojectivespace-rngintelt)

    - [`FanOfFakeProjectiveSpace(W,Q): SeqEnum, SeqEnum → TorFan`](toric-fans.md#function-fanoffakeprojectivespace-seqenum-seqenum)

    - [`ZeroFan(L): TorLat → TorFan`](toric-fans.md#function-zerofan-torlat)

    - [`NormalFan(F,C): TorFan, TorCon → TorFan, Map`](toric-fans.md#function-normalfan-torfan-torcon)

    - [`NormalFan(P): TorPol → TorFan`](toric-fans.md#function-normalfan-torpol)

    - [`DualFan(P): TorPol → TorFan`](toric-fans.md#function-dualfan-torpol)

    - [`SpanningFan(P): TorPol → TorFan`](toric-fans.md#function-spanningfan-torpol)

    - [`Example: Toric Spanning Fan Example`](toric-fans.md#example-ex-f5bea2)

    - [`FanWithWeights(W): SeqEnum → TorFan`](toric-fans.md#function-fanwithweights-seqenum)

    - [`Blowup(F,v): TorFan, TorLatElt → TorFan`](toric-fans.md#function-blowup-torfan-torlatelt)

    - [`Example: Toric Fan With Weights Example`](toric-fans.md#example-ex-2a8220)

    - [`IsInSupport(v,F): TorLatElt, TorFan → BoolElt, RngIntElt`](toric-fans.md#function-isinsupport-torlatelt-torfan)

    - [`OneSkeleton(F): TorFan → TorFan`](toric-fans.md#function-oneskeleton-torfan)

    - [`Fan(F1,F2): TorFan, TorFan → TorFan`](toric-fans.md#function-fan-torfan-torfan)

    - [`F1 * F2: TorFan, TorFan → TorFan`](toric-fans.md#operation-op-times-torfan-torfan)

    - [`Fan(Q): [TorFan] → TorFan`](toric-fans.md#function-fan-torfan)

    - [`F ^ n: TorFan, RngIntElt → TorFan`](toric-fans.md#operation-op-pow-torfan-rngintelt)

    - [`F eq G: TorFan, TorFan → BoolElt`](toric-fans.md#operation-op-eq-torfan-torfan)

  - [Components of Fans](toric-fans.md#components-of-fans)

    - [`Skeleton(F,n): TorFan, RngIntElt → TorFan`](toric-fans.md#function-skeleton-torfan-rngintelt)

    - [`C in F: TorCon, TorFan → BoolElt`](toric-fans.md#operation-op-in-torcon-torfan)

    - [`Cones(F): TorFan → SeqEnum`](toric-fans.md#function-cones-torfan)

    - [`Cones(F,i): TorFan, RngIntElt → SeqEnum`](toric-fans.md#function-cones-torfan-rngintelt)

    - [`ConesOfCodimension(F,i): TorFan, RngIntElt → SeqEnum`](toric-fans.md#function-conesofcodimension-torfan-rngintelt)

    - [`MaxCones(F): TorFan → SeqEnum`](toric-fans.md#function-maxcones-torfan)

    - [`AllCones(F): TorFan → SeqEnum`](toric-fans.md#function-allcones-torfan)

    - [`Cone(F,i): TorFan, RngIntElt → TorCon`](toric-fans.md#function-cone-torfan-rngintelt)

    - [`Cone(F,S): TorFan, [RngIntElt] → TorCon`](toric-fans.md#function-cone-torfan-rngintelt-2)

    - [`NonSimplicialCones(F): TorFan → SeqEnum, SeqEnum`](toric-fans.md#function-nonsimplicialcones-torfan)

    - [`SingularCones(F): TorFan → SeqEnum, SeqEnum`](toric-fans.md#function-singularcones-torfan)

    - [`Example: Toric Singular Cones Example`](toric-fans.md#example-ex-e75704)

    - [`ConeIndices(F): TorFan → SeqEnum`](toric-fans.md#function-coneindices-torfan)

    - [`ConeIndices(F,C): TorFan, TorCon → SeqEnum`](toric-fans.md#function-coneindices-torfan-torcon)

    - [`ConeIntersection(F,C1,C2): TorFan, TorCon, TorCon → TorCon`](toric-fans.md#function-coneintersection-torfan-torcon-torcon)

    - [`Face(F,C): TorFan, TorCon → TorCon`](toric-fans.md#function-face-torfan-torcon)

    - [`InnerNormals(F): TorFan → SeqEnum`](toric-fans.md#function-innernormals-torfan)

    - [`OuterNormals(F): TorFan → SeqEnum`](toric-fans.md#function-outernormals-torfan)

    - [`DualFaceInDualFan(P,Q): TorPol, [RngIntElt] → TorFan`](toric-fans.md#function-dualfaceindualfan-torpol-rngintelt)

    - [`Rays(F): TorFan → SeqEnum`](toric-fans.md#function-rays-torfan)

    - [`Ray(F,i): TorFan, RngIntElt → TorLatElt`](toric-fans.md#function-ray-torfan-rngintelt)

    - [`AllRays(F): TorFan → SeqEnum`](toric-fans.md#function-allrays-torfan)

    - [`PureRays(F): TorFan → SeqEnum`](toric-fans.md#function-purerays-torfan)

    - [`PureRayIndices(F): TorFan → SeqEnum`](toric-fans.md#function-purerayindices-torfan)

    - [`CreateVirtualRays(S): [TorLatElt] → SeqEnum`](toric-fans.md#function-createvirtualrays-torlatelt)

    - [`VirtualRays(F): TorFan → SeqEnum`](toric-fans.md#function-virtualrays-torfan)

    - [`VirtualRayIndices(F): TorFan → SeqEnum`](toric-fans.md#function-virtualrayindices-torfan)

  - [Properties of Fans](toric-fans.md#properties-of-fans)

    - [`Ambient(F): TorFan → TorLat`](toric-fans.md#function-ambient-torfan)

    - [`IsComplete(F): TorFan → BoolElt`](toric-fans.md#function-iscomplete-torfan)

    - [`IsSingular(F): TorFan → BoolElt`](toric-fans.md#function-issingular-torfan)

    - [`IsNonsingular(F): TorFan → BoolElt`](toric-fans.md#function-isnonsingular-torfan)

    - [`IsQFactorial(F): TorFan → BoolElt`](toric-fans.md#function-isqfactorial-torfan)

    - [`IsIsolated(F): TorFan → BoolElt`](toric-fans.md#function-isisolated-torfan)

    - [`IsTerminal(F): TorFan → BoolElt`](toric-fans.md#function-isterminal-torfan)

    - [`IsCanonical(F): TorFan → BoolElt`](toric-fans.md#function-iscanonical-torfan)

    - [`IsGorenstein(F): TorFan → BoolElt`](toric-fans.md#function-isgorenstein-torfan)

    - [`IsQGorenstein(F): TorFan → BoolElt`](toric-fans.md#function-isqgorenstein-torfan)

  - [Maps of Fans](toric-fans.md#maps-of-fans)

    - [`F @ f: TorFan, Map → TorFan`](toric-fans.md#operation-operation-torfan-map-torfan)

    - [`SimplicialSubdivision(F): TorFan → TorFan`](toric-fans.md#function-simplicialsubdivision-torfan)

    - [`SimplicialSubdivision(C): TorCon → TorFan`](toric-fans.md#function-simplicialsubdivision-torcon)

    - [`Example: Toric Simplicial Example`](toric-fans.md#example-ex-8ec30f)

    - [`IsFanMap(F1,F2): TorFan, TorFan → BoolElt`](toric-fans.md#function-isfanmap-torfan-torfan)

    - [`IsFanMap(F1,F2,f): TorFan, TorFan, Map → BoolElt`](toric-fans.md#function-isfanmap-torfan-torfan-map)

    - [`ResolveFanMap(F1,F2): TorFan, TorFan → TorFan`](toric-fans.md#function-resolvefanmap-torfan-torfan)

    - [`Resolution(F): TorFan → TorFan`](toric-fans.md#function-resolution-torfan)

    - [`Terminalisation(F): TorFan → TorFan`](toric-fans.md#function-terminalisation-torfan)

    - [`Canonicalisation(F): TorFan → TorFan`](toric-fans.md#function-canonicalisation-torfan)

- [Geometrical Properties of Cones and Polyhedra](toric-cones-and-polyhedra.md)

  - [`IsSingular(C): TorCon → BoolElt`](toric-cones-and-polyhedra.md#function-issingular-torcon)

  - [`IsNonsingular(C): TorCon → BoolElt`](toric-cones-and-polyhedra.md#function-isnonsingular-torcon)

  - [`IsSmooth(P): TorPol → BoolElt`](toric-cones-and-polyhedra.md#function-issmooth-torpol)

  - [`IsGorenstein(C): TorCon → BoolElt`](toric-cones-and-polyhedra.md#function-isgorenstein-torcon)

  - [`IsReflexive(P): TorPol → BoolElt`](toric-cones-and-polyhedra.md#function-isreflexive-torpol)

  - [`IsQGorenstein(C): TorCon → BoolElt`](toric-cones-and-polyhedra.md#function-isqgorenstein-torcon)

  - [`GorensteinIndex(C): TorCon → RngIntElt, TorLatElt`](toric-cones-and-polyhedra.md#function-gorensteinindex-torcon)

  - [`GorensteinIndex(P): TorPol → RngIntElt`](toric-cones-and-polyhedra.md#function-gorensteinindex-torpol)

  - [`IsIsolated(C): TorCon → BoolElt`](toric-cones-and-polyhedra.md#function-isisolated-torcon)

  - [`IsQFactorial(C): TorCon → BoolElt`](toric-cones-and-polyhedra.md#function-isqfactorial-torcon)

  - [`IsSimplicial(P): TorPol → BoolElt`](toric-cones-and-polyhedra.md#function-issimplicial-torpol)

  - [`IsTerminal(C): TorCon → BoolElt`](toric-cones-and-polyhedra.md#function-isterminal-torcon)

  - [`IsCanonical(C): TorCon → BoolElt`](toric-cones-and-polyhedra.md#function-iscanonical-torcon)

  - [`IsFano(P): TorPol → BoolElt`](toric-cones-and-polyhedra.md#function-isfano-torpol)

  - [`Example: Toric Terminal Polytope Example`](toric-cones-and-polyhedra.md#example-ex-ee8303)

- [Toric Varieties](toric-varieties.md)

  - [Constructors for Toric Varieties](toric-varieties.md#constructors-for-toric-varieties)

    - [`ToricVariety(k,n): Fld, RngIntElt → TorVar`](toric-varieties.md#function-toricvariety-fld-rngintelt)

    - [`ToricVariety(k,Z): Fld, [RngIntElt] → TorVar`](toric-varieties.md#function-toricvariety-fld-rngintelt-2)

    - [`ToricVariety(k,Z,Q): Fld, [RngIntElt], [FldRatElt] → TorVar`](toric-varieties.md#function-toricvariety-fld-rngintelt-fldratelt)

    - [`ToricVariety(k,M,v): Fld, [[RngIntElt]], [RngIntElt] → TorVar`](toric-varieties.md#function-toricvariety-fld-rngintelt-rngintelt)

    - [`Example: Toric Cox Example2`](toric-varieties.md#example-ex-70f3b4)

    - [`ToricVariety(k): Fld → TorVar`](toric-varieties.md#function-toricvariety-fld)

    - [`ProjectiveSpace(k,n): Fld, RngIntElt → Prj`](toric-varieties.md#function-projectivespace-fld-rngintelt)

    - [`ProjectiveSpaceAsToricVariety(k,n): Fld, RngIntElt → TorVar`](toric-varieties.md#function-projectivespaceastoricvariety-fld-rngintelt)

    - [`WeightedProjectiveSpace(k,n): Fld, RngIntElt → TorVar`](toric-varieties.md#function-weightedprojectivespace-fld-rngintelt)

    - [`WPS(k,n): Fld, RngIntElt → TorVar`](toric-varieties.md#function-wps-fld-rngintelt)

    - [`ProjectiveSpace(k,W): Fld, [RngIntElt] → Prj`](toric-varieties.md#function-projectivespace-fld-rngintelt-2)

    - [`ProjectiveSpaceAsToricVariety(k,W): Fld, [RngIntElt] → TorVar`](toric-varieties.md#function-projectivespaceastoricvariety-fld-rngintelt-2)

    - [`WeightedProjectiveSpace(k,W): Fld, [RngIntElt] → TorVar`](toric-varieties.md#function-weightedprojectivespace-fld-rngintelt-2)

    - [`WPS(k,W): Fld, [RngIntElt] → TorVar`](toric-varieties.md#function-wps-fld-rngintelt-2)

    - [`FakeProjectiveSpace(k,W,Q): Fld, [RngIntElt], [FldRatElt] → TorVar`](toric-varieties.md#function-fakeprojectivespace-fld-rngintelt-fldratelt)

    - [`AbsoluteRationalScroll(k,S): Fld, [RngIntElt] → TorVar`](toric-varieties.md#function-absoluterationalscroll-fld-rngintelt)

    - [`RationalScroll(k,s,A): Fld, RngIntElt, [RngIntElt] → TorVar`](toric-varieties.md#function-rationalscroll-fld-rngintelt-rngintelt)

    - [`RuledSurface(k,n): Fld, RngIntElt → TorVar`](toric-varieties.md#function-ruledsurface-fld-rngintelt)

    - [`RuledSurface(k,a1,a2): Fld, RngIntElt, RngIntElt → TorVar`](toric-varieties.md#function-ruledsurface-fld-rngintelt-rngintelt)

    - [`HirzebruchSurface(k,n): Fld, RngIntElt → TorVar`](toric-varieties.md#function-hirzebruchsurface-fld-rngintelt)

    - [`BigTorus(k,N): Rng, TorLat → TorVar`](toric-varieties.md#function-bigtorus-rng-torlat)

    - [`BigTorus(X): TorVar → TorVar, TorMap, TorMap`](toric-varieties.md#function-bigtorus-torvar)

    - [`RestrictionToSubtorus(Z): Sch → Sch, TorMap`](toric-varieties.md#function-restrictiontosubtorus-sch)

  - [Toric Varieties and their Fans](toric-varieties.md#toric-varieties-and-their-fans)

    - [`ToricVariety(k,F): Fld, TorFan → TorVar`](toric-varieties.md#function-toricvariety-fld-torfan)

    - [`Fan(X): TorVar → TorLat`](toric-varieties.md#function-fan-torvar)

    - [`Rays(X): TorVar → SeqEnum`](toric-varieties.md#function-rays-torvar)

    - [`OneParameterSubgroupsLattice(X): TorVar → TorLat`](toric-varieties.md#function-oneparametersubgroupslattice-torvar)

    - [`MonomialLattice(X): TorVar → TorLat`](toric-varieties.md#function-monomiallattice-torvar)

    - [`CoxMonomialLattice(X): TorVar → TorLat`](toric-varieties.md#function-coxmonomiallattice-torvar)

    - [`PicardLattice(X): TorVar → TorLat`](toric-varieties.md#function-picardlattice-torvar)

    - [`DivisorClassLattice(X): TorVar → TorLat`](toric-varieties.md#function-divisorclasslattice-torvar)

    - [`IrrelevantIdeal(X): TorVar → SeqEnum`](toric-varieties.md#function-irrelevantideal-torvar)

    - [`Gradings(X): Sch → SeqEnum`](toric-varieties.md#function-gradings-sch)

    - [`NumberOfGradings(X): Sch → RngIntElt`](toric-varieties.md#function-numberofgradings-sch)

    - [`QuotientGradings(X): TorVar → SeqEnum`](toric-varieties.md#function-quotientgradings-torvar)

    - [`NumberOfQuotientGradings(X): TorVar → RngIntElt`](toric-varieties.md#function-numberofquotientgradings-torvar)

  - [Properties of Toric Varieties](toric-varieties.md#properties-of-toric-varieties)

    - [`IsSingular(X): TorVar → BoolElt`](toric-varieties.md#function-issingular-torvar)

    - [`IsNonsingular(X): TorVar → BoolElt`](toric-varieties.md#function-isnonsingular-torvar)

    - [`IsGorenstein(X): TorVar → BoolElt`](toric-varieties.md#function-isgorenstein-torvar)

    - [`IsQGorenstein(X): TorVar → BoolElt`](toric-varieties.md#function-isqgorenstein-torvar)

    - [`IsQFactorial(X): TorVar → BoolElt`](toric-varieties.md#function-isqfactorial-torvar)

    - [`IsIsolated(X): TorVar → BoolElt`](toric-varieties.md#function-isisolated-torvar)

    - [`IsTerminal(X): TorVar → BoolElt`](toric-varieties.md#function-isterminal-torvar)

    - [`IsCanonical(X): TorVar → BoolElt`](toric-varieties.md#function-iscanonical-torvar)

    - [`IsComplete(X): TorVar → BoolElt`](toric-varieties.md#function-iscomplete-torvar)

    - [`IsProjective(X): TorVar → BoolElt`](toric-varieties.md#function-isprojective-torvar)

    - [`IsFano(X): TorVar → BoolElt`](toric-varieties.md#function-isfano-torvar)

    - [`IsWeakFano(X): TorVar → BoolElt`](toric-varieties.md#function-isweakfano-torvar)

    - [`IsFakeWeightedProjectiveSpace(X): TorVar → BoolElt`](toric-varieties.md#function-isfakeweightedprojectivespace-torvar)

    - [`IsWeightedProjectiveSpace(X): TorVar → BoolElt`](toric-varieties.md#function-isweightedprojectivespace-torvar)

  - [Affine Patches on Toric Varieties](toric-varieties.md#affine-patches-on-toric-varieties)

    - [`ToricAffinePatch(X,i): TorVar, RngIntElt → TorVar, TorMap`](toric-varieties.md#function-toricaffinepatch-torvar-rngintelt)

    - [`ToricAffinePatch(X,S): TorVar, [RngIntElt] → TorVar, TorMap`](toric-varieties.md#function-toricaffinepatch-torvar-rngintelt-2)

    - [`ToricAffinePatch(X,S): TorVar, [RngMPolElt] → TorVar, TorMap`](toric-varieties.md#function-toricaffinepatch-torvar-rngmpolelt)

- [Cox Rings](cox-rings.md)

  - [The Cox Ring of a Toric Variety](cox-rings.md#the-cox-ring-of-a-toric-variety)

    - [`CoxRing(X): TorVar → RngCox`](cox-rings.md#function-coxring-torvar)

    - [`CoxRing(k,F): Fld, TorFan → RngCox`](cox-rings.md#function-coxring-fld-torfan)

    - [`Example: Toric Cox Example1`](cox-rings.md#example-ex-d5e182)

    - [`Example: Toric Cox Example2`](cox-rings.md#example-ex-7eeae1)

  - [Cox Rings in Their Own Right](cox-rings.md#cox-rings-in-their-own-right)

    - [`CoxRing(R,B,Z,Q): RngMPol, SeqEnum, SeqEnum, SeqEnum → RngCox`](cox-rings.md#function-coxring-rngmpol-seqenum-seqenum-seqenum)

    - [`C1 eq C2: RngCox, RngCox → BoolElt`](cox-rings.md#operation-op-eq-rngcox-rngcox)

    - [`BaseRing(C): RngCox → Fld`](cox-rings.md#function-basering-rngcox)

    - [`CoefficientRing(C): RngCox → Fld`](cox-rings.md#function-coefficientring-rngcox)

    - [`UnderlyingRing(C): RngCox → RngMPol`](cox-rings.md#function-underlyingring-rngcox)

    - [`Length(C): RngCox → RngIntElt`](cox-rings.md#function-length-rngcox)

    - [`IrrelevantIdeal(C): RngCox → SeqEnum`](cox-rings.md#function-irrelevantideal-rngcox)

    - [`IrrelevantComponents(C): RngCox → SeqEnum`](cox-rings.md#function-irrelevantcomponents-rngcox)

    - [`IrrelevantGenerators(C): RngCox → SeqEnum`](cox-rings.md#function-irrelevantgenerators-rngcox)

    - [`Gradings(C): RngCox → RngIntElt`](cox-rings.md#function-gradings-rngcox)

    - [`NumberOfGradings(C): RngCox → RngIntElt`](cox-rings.md#function-numberofgradings-rngcox)

    - [`QuotientGradings(C): RngCox → RngIntElt`](cox-rings.md#function-quotientgradings-rngcox)

    - [`NumberOfQuotientGradings(C): RngCox → RngIntElt`](cox-rings.md#function-numberofquotientgradings-rngcox)

    - [`C . i: RngCox, RngInt → RngMPolElt`](cox-rings.md#operation-operation-rngcox-rngint-rngmpolelt)

    - [`AssignNames(~C, S): RngCox, [MonStgElt]`](cox-rings.md#function-assignnames-rngcox-monstgelt-ref)

    - [`Name(C,i): RngCox, RngIntElt → RngMPolElt`](cox-rings.md#function-name-rngcox-rngintelt)

  - [Recovering a Toric Variety From a Cox Ring](cox-rings.md#recovering-a-toric-variety-from-a-cox-ring)

    - [`ToricVariety(C): RngCox → TorVar`](cox-rings.md#function-toricvariety-rngcox)

    - [`Example: Toric From Cox Example`](cox-rings.md#example-ex-e56de4)

    - [`Fan(C): RngCox → TorFan`](cox-rings.md#function-fan-rngcox)

    - [`CoxMonomialLattice(C): RngCox → TorLat`](cox-rings.md#function-coxmonomiallattice-rngcox)

    - [`BasisOfRationalFunctionField(X): TorVar → SeqEnum`](cox-rings.md#function-basisofrationalfunctionfield-torvar)

    - [`BasisOfDegree0CoxMonomials(X): TorVar → SeqEnum`](cox-rings.md#function-basisofdegree0coxmonomials-torvar)

    - [`DivisorClassLattice(C): RngCox → TorLat`](cox-rings.md#function-divisorclasslattice-rngcox)

    - [`MonomialLattice(C): RngCox → TorLat`](cox-rings.md#function-monomiallattice-rngcox)

    - [`OneParameterSubgroupsLattice(C): RngCox → TorLat`](cox-rings.md#function-oneparametersubgroupslattice-rngcox)

    - [`RayLattice(C): RngCox → TorLat`](cox-rings.md#function-raylattice-rngcox)

    - [`DivisorClassGroup(C): RngCox → TorLat`](cox-rings.md#function-divisorclassgroup-rngcox)

    - [`RayLatticeMap(C): RngCox → Map`](cox-rings.md#function-raylatticemap-rngcox)

    - [`WeilToClassGroupsMap(C): RngCox → Map`](cox-rings.md#function-weiltoclassgroupsmap-rngcox)

    - [`WeilToClassLatticesMap(C): RngCox → Map`](cox-rings.md#function-weiltoclasslatticesmap-rngcox)

- [Invariant Divisors and Riemann-Roch Spaces](toric-divisors.md)

  - [Divisor Group](toric-divisors.md#divisor-group)

    - [`DivisorGroup(X): TorVar → DivTor`](toric-divisors.md#function-divisorgroup-torvar)

    - [`ToricVariety(G): DivTor → TorVar`](toric-divisors.md#function-toricvariety-divtor)

    - [`G1 eq G2: DivTor, DivTor → BoolElt`](toric-divisors.md#operation-op-eq-divtor-divtor)

    - [`Divisor(G,S): DivTor, [RngIntElt] → DivTorElt`](toric-divisors.md#function-divisor-divtor-rngintelt)

    - [`Divisor(G,S): DivTor, [FldRatElt] → DivTorElt`](toric-divisors.md#function-divisor-divtor-fldratelt)

    - [`Divisor(G,i): DivTor, RngIntElt → DivTorElt`](toric-divisors.md#function-divisor-divtor-rngintelt-2)

  - [Constructing Invariant Divisors](toric-divisors.md#constructing-invariant-divisors)

    - [`Divisor(X,S): TorVar, [RngIntElt] → DivTorElt`](toric-divisors.md#function-divisor-torvar-rngintelt)

    - [`Divisor(X,i): TorVar, RngIntElt → DivTorElt`](toric-divisors.md#function-divisor-torvar-rngintelt-2)

    - [`Divisor(X,f): TorVar, RngMPolElt → DivTorElt`](toric-divisors.md#function-divisor-torvar-rngmpolelt)

    - [`Divisor(X,m): TorVar, TorLatElt → DivTorElt`](toric-divisors.md#function-divisor-torvar-torlatelt)

    - [`ZeroDivisor(X): TorVar → DivTorElt`](toric-divisors.md#function-zerodivisor-torvar)

    - [`Representative(X,m): TorVar, ModEDElt → DivTorElt`](toric-divisors.md#function-representative-torvar-modedelt)

    - [`Representative(X,m): TorVar, TorLatElt → DivTorElt`](toric-divisors.md#function-representative-torvar-torlatelt)

    - [`CanonicalDivisor(X): TorVar → DivTorElt`](toric-divisors.md#function-canonicaldivisor-torvar)

    - [`CanonicalClass(X): TorVar → DivTorElt`](toric-divisors.md#function-canonicalclass-torvar)

    - [`D1 + D2: DivTorElt, DivTorElt → DivTorElt`](toric-divisors.md#operation-op-plus-divtorelt-divtorelt)

    - [`n * D: FldRatElt, DivTorElt → DivTorElt`](toric-divisors.md#operation-op-times-fldratelt-divtorelt)

    - [`- D: DivTorElt → DivTorElt`](toric-divisors.md#operation-operation-divtorelt-divtorelt)

    - [`D1 - D2: DivTorElt, DivTorElt → DivTorElt`](toric-divisors.md#operation-op-minus-divtorelt-divtorelt)

    - [`D * v: DivTorElt, TorLatElt → RngIntElt`](toric-divisors.md#operation-op-times-divtorelt-torlatelt)

    - [`Example: Toric Kawamata Blowup Example`](toric-divisors.md#example-ex-9d2674)

  - [Properties of Divisors](toric-divisors.md#properties-of-divisors)

    - [`Variety(D): DivTorElt → TorVar`](toric-divisors.md#function-variety-divtorelt)

    - [`Parent(D): DivTorElt → DivTor`](toric-divisors.md#function-parent-divtorelt)

    - [`Weil(D): DivTorElt → SeqEnum`](toric-divisors.md#function-weil-divtorelt)

    - [`Cartier(D): DivTorElt → SeqEnum[TorLatElt]`](toric-divisors.md#function-cartier-divtorelt)

    - [`IsQCartier(D): DivTorElt → BoolElt`](toric-divisors.md#function-isqcartier-divtorelt)

    - [`IsCartier(D): DivTorElt → BoolElt`](toric-divisors.md#function-iscartier-divtorelt)

    - [`IsWeil(D): DivTorElt → BoolElt`](toric-divisors.md#function-isweil-divtorelt)

    - [`IsAmple(D): DivTorElt → BoolElt`](toric-divisors.md#function-isample-divtorelt)

    - [`IsNef(D): DivTorElt → BoolElt`](toric-divisors.md#function-isnef-divtorelt)

    - [`IsBig(D): DivTorElt → BoolElt`](toric-divisors.md#function-isbig-divtorelt)

    - [`IsEffective(D): DivTorElt → BoolElt`](toric-divisors.md#function-iseffective-divtorelt)

    - [`PicardClass(D): DivTorElt → TorLatElt`](toric-divisors.md#function-picardclass-divtorelt)

    - [`MovablePart(D): DivTorElt → DivTorElt`](toric-divisors.md#function-movablepart-divtorelt)

    - [`Example: Toric Movable Example`](toric-divisors.md#example-ex-852f90)

    - [`ImageFan(D): DivTorElt → TorFan`](toric-divisors.md#function-imagefan-divtorelt)

    - [`Proj(D): DivTorElt → TorVar, PlcEnum`](toric-divisors.md#function-proj-divtorelt)

    - [`RelativeProj(D): DivTorElt → TorVar`](toric-divisors.md#function-relativeproj-divtorelt)

    - [`IntersectionForm(X,C): TorVar, TorCon → TorLatElt`](toric-divisors.md#function-intersectionform-torvar-torcon)

    - [`IntersectionForms(X): TorVar → [TorLatElt]`](toric-divisors.md#function-intersectionforms-torvar)

    - [`CartierToWeilMap(X): TorVar → Map`](toric-divisors.md#function-cartiertoweilmap-torvar)

    - [`PicardToClassGroupsMap(X): TorVar → Map`](toric-divisors.md#function-picardtoclassgroupsmap-torvar)

    - [`PicardToClassLatticesMap(X): TorVar → Map`](toric-divisors.md#function-picardtoclasslatticesmap-torvar)

  - [Linear Equivalence of Divisors](toric-divisors.md#linear-equivalence-of-divisors)

    - [`IsQPrincipal(D): DivTorElt → BoolElt`](toric-divisors.md#function-isqprincipal-divtorelt)

    - [`IsPrincipal(D): DivTorElt → BoolElt`](toric-divisors.md#function-isprincipal-divtorelt)

    - [`IsLinearlyEquivalentToCartier(D): DivTorElt → BoolElt, DivTorElt`](toric-divisors.md#function-islinearlyequivalenttocartier-divtorelt)

    - [`AreLinearlyEquivalent(D,E): DivTorElt, DivTorElt → BoolElt`](toric-divisors.md#function-arelinearlyequivalent-divtorelt-divtorelt)

    - [`IsLinearlyEquivalent(D,E): DivTorElt, DivTorElt → BoolElt`](toric-divisors.md#function-islinearlyequivalent-divtorelt-divtorelt)

    - [`LinearlyEquivalentDivisorWithNoSupportOn(D,S): DivTorElt, [RngMPolElt] → DivTorElt`](toric-divisors.md#function-linearlyequivalentdivisorwithnosupporton-divtorelt-rngmpolelt)

    - [`DefiningMonomial(D): DivTorElt → RngMPolElt`](toric-divisors.md#function-definingmonomial-divtorelt)

    - [`LatticeElementToMonomial(D,v): DivTorElt, TorLatElt → RngMPolElt`](toric-divisors.md#function-latticeelementtomonomial-divtorelt-torlatelt)

  - [Riemann–Roch Spaces of Invariant Divisors](toric-divisors.md#riemann-roch-spaces-of-invariant-divisors)

    - [`RiemannRochPolytope(D): DivTorElt → TorPol`](toric-divisors.md#function-riemannrochpolytope-divtorelt)

    - [`RiemannRochBasis(D): DivTorElt → [RngElt]`](toric-divisors.md#function-riemannrochbasis-divtorelt)

    - [`RiemannRochDimension(D): DivTorElt → RngIntElt`](toric-divisors.md#function-riemannrochdimension-divtorelt)

    - [`GradedCone(D): DivTorElt → TorCon`](toric-divisors.md#function-gradedcone-divtorelt)

    - [`Polyhedron(D): DivTorElt → TorPol`](toric-divisors.md#function-polyhedron-divtorelt)

    - [`Example: Toric Rr Example`](toric-divisors.md#example-ex-35b77b)

    - [`HilbertSeries(D): DivTor → FldFunRatUElt`](toric-divisors.md#function-hilbertseries-divtor)

    - [`HilbertPolynomial(D): DivTor → [RngUPolElt]`](toric-divisors.md#function-hilbertpolynomial-divtor)

    - [`HilbertCoefficients(D,l): DivTor, RngIntElt → [RngIntElt]`](toric-divisors.md#function-hilbertcoefficients-divtor-rngintelt)

    - [`HilbertCoefficient(D,i): DivTor, RngIntElt → RngIntElt`](toric-divisors.md#function-hilbertcoefficient-divtor-rngintelt)

    - [`HilbertDeltaVector(D): DivTor → [RngIntElt]`](toric-divisors.md#function-hilbertdeltavector-divtor)

    - [`Example: Toric Rr By Hand`](toric-divisors.md#example-ex-ad5b0b)

- [Maps of Toric Varieties](toric-maps.md)

  - [Maps from Lattice Maps](toric-maps.md#maps-from-lattice-maps)

    - [`ToricVarietyMap(X,Y,f): TorVar, TorVar, Map → TorMap`](toric-maps.md#function-toricvarietymap-torvar-torvar-map)

    - [`ToricVarietyMap(X,Y): TorVar, TorVar → TorMap`](toric-maps.md#function-toricvarietymap-torvar-torvar)

    - [`Blowup(X,v): TorVar, TorLatElt → TorVar, TorMap`](toric-maps.md#function-blowup-torvar-torlatelt)

    - [`ToricIdentityMap(X): TorVar → TorMap`](toric-maps.md#function-toricidentitymap-torvar)

  - [Properties of Toric Maps](toric-maps.md#properties-of-toric-maps)

    - [`IsRegular(f): TorMap → BoolElt`](toric-maps.md#function-isregular-tormap)

    - [`IndeterminacyLocus(f): TorMap → [Sch]`](toric-maps.md#function-indeterminacylocus-tormap)

    - [`Example: Toric Simplicial Example`](toric-maps.md#example-ex-cf6855)

- [The Geometry of Toric Varieties](toric-geometry.md)

  - [Resolution of Singularities and Linear Systems](toric-geometry.md#resolution-of-singularities-and-linear-systems)

    - [`Resolution(X): TorVar → TorVar, TorMap`](toric-geometry.md#function-resolution-torvar)

    - [`QFactorialisation(X): TorVar → TorVar, TorMap`](toric-geometry.md#function-qfactorialisation-torvar)

    - [`Terminalisation(X): TorVar → TorVar, TorMap`](toric-geometry.md#function-terminalisation-torvar)

    - [`Canonicalisation(X): TorVar → TorVar, TorMap`](toric-geometry.md#function-canonicalisation-torvar)

    - [`ResolveLinearSystem(D): DivTorElt → TorVar`](toric-geometry.md#function-resolvelinearsystem-divtorelt)

  - [Mori Theory of Toric Varieties](toric-geometry.md#mori-theory-of-toric-varieties)

    - [`MoriCone(X): TorVar → TorCon`](toric-geometry.md#function-moricone-torvar)

    - [`NefCone(X): TorVar → TorCon`](toric-geometry.md#function-nefcone-torvar)

    - [`ExtremalRays(X): TorVar → SeqEnum`](toric-geometry.md#function-extremalrays-torvar)

    - [`ExtremalRayContractions(X): TorVar, RngIntElt → SeqEnum`](toric-geometry.md#function-extremalraycontractions-torvar-rngintelt)

    - [`ExtremalRayContraction(X,i): TorVar, RngIntElt → TorVar, TorMap`](toric-geometry.md#function-extremalraycontraction-torvar-rngintelt)

    - [`ExtremalRayContractionDivisor(X,i): TorVar, RngIntElt → DivTorElt`](toric-geometry.md#function-extremalraycontractiondivisor-torvar-rngintelt)

    - [`TypeOfContraction(X,i): TorVar, RngIntElt → MonStgElt`](toric-geometry.md#function-typeofcontraction-torvar-rngintelt)

    - [`TypesOfContractions(X): TorVar → MonStgElt`](toric-geometry.md#function-typesofcontractions-torvar)

    - [`IsDivisorialContraction(X,i): TorVar, RngIntElt → BoolElt`](toric-geometry.md#function-isdivisorialcontraction-torvar-rngintelt)

    - [`IsMoriFibreSpace(X,i): TorVar, RngIntElt → BoolElt`](toric-geometry.md#function-ismorifibrespace-torvar-rngintelt)

    - [`IsFlipping(X,i): TorVar, RngIntElt → BoolElt`](toric-geometry.md#function-isflipping-torvar-rngintelt)

    - [`Flip(X,i): TorVar, RngIntElt → TorVar`](toric-geometry.md#function-flip-torvar-rngintelt)

    - [`Flip(D): DivTorElt → TorVar`](toric-geometry.md#function-flip-divtorelt)

    - [`WeightsOfFlip(X,i): TorVar, RngIntElt → SeqEnum`](toric-geometry.md#function-weightsofflip-torvar-rngintelt)

    - [`Example: Toric Flipwts Example`](toric-geometry.md#example-ex-33c790)

    - [`Example: Toric Weights Of Flip Example`](toric-geometry.md#example-ex-ddd3e8)

    - [`MMP(X): TorVar → SeqEnum, SeqEnum`](toric-geometry.md#function-mmp-torvar)

    - [`Example: Toric Mmp Example1`](toric-geometry.md#example-ex-cb6736)

  - [Decomposition of Toric Morphisms](toric-geometry.md#decomposition-of-toric-morphisms)

    - [`Example: Toric Decomposition Example`](toric-geometry.md#example-ex-444c76)

- [Schemes in Toric Varieties](toric-subschemes.md)

  - [Construction of Subschemes](toric-subschemes.md#construction-of-subschemes)

    - [`Scheme(X,f): TorVar, RngMPolElt → Sch`](toric-subschemes.md#function-scheme-torvar-rngmpolelt)

    - [`Scheme(X,Q): TorVar, [RngMPolElt] → Sch`](toric-subschemes.md#function-scheme-torvar-rngmpolelt-2)

    - [`BinomialToricEmbedding(Z): Sch → Sch, TorMap`](toric-subschemes.md#function-binomialtoricembedding-sch)

    - [`Example: Toric Mmp Example1`](toric-subschemes.md#example-ex-641627)