- Introduction and First Examples
- Fans in Toric Lattices
- Construction of Fans
- Fan(Q) : [TorCon] -> TorFan
- Fan(R,S) : [TorLatElt],[[RngIntElt]] -> TorFan
- Fan(C) : TorCon -> TorFan
- FanOfAffineSpace(n) : RngIntElt -> TorFac
- FanOfWPS(W) : SeqEnum -> TorFan
- [Future release] FanOfProjectiveSpace(n) : RngIntElt -> TorFac
- FanOfFakeProjectiveSpace(W,Q) : SeqEnum, SeqEnum -> TorFan
- ZeroFan(L) : TorLat -> TorFan
- NormalFan(F,C) : TorFan,TorCon -> TorFan,Map
- NormalFan(P) : TorPol -> TorFan
- SpanningFan(P) : TorPol -> TorFan
- Example Toric_toric-spanning-fan-example (H126E6)
- FanWithWeights(W) : SeqEnum -> TorFan
- Blowup(F,v): TorFan,TorLatElt -> TorFan
- Example Toric_toric-fan-with-weights-example (H126E7)
- IsInSupport(v,F) : TorLatElt,TorFan -> BoolElt,RngIntElt
- OneSkeleton(F) : TorFan -> TorFan
- Fan(F1,F2) : TorFan,TorFan -> TorFan
- F eq G : TorFan,TorFan -> BoolElt
- Components of Fans
- Skeleton(F,n) : TorFan,RngIntElt -> TorFan
- C in F : TorCon,TorFan -> BoolElt
- Cones(F) : TorFan -> SeqEnum
- Cones(F,i) : TorFan,RngIntElt -> SeqEnum
- ConesOfCodimension(F,i) : TorFan,RngIntElt -> SeqEnum
- MaxCones(F) : TorFan -> SeqEnum
- AllCones(F) : TorFan -> SeqEnum
- Cone(F,i) : TorFan,RngIntElt -> TorCon
- Cone(F,S) : TorFan,[RngIntElt] -> TorCon
- NonSimplicialCones(F) : TorFan -> SeqEnum, SeqEnum
- SingularCones(F) : TorFan -> SeqEnum,SeqEnum
- Example Toric_toric-singular-cones-example (H126E8)
- ConeIndices(F) : TorFan -> SeqEnum
- ConeIndices(F,C) : TorFan, TorCon -> SeqEnum
- ConeIntersection(F,C1,C2) : TorFan,TorCon,TorCon -> TorCon
- Face(F,C) : TorFan,TorCon -> TorCon
- InnerNormals(F) : TorFan -> SeqEnum
- DualFaceInDualFan(P,Q) : TorPol,[RngIntElt] -> TorFan
- Rays(F) : TorFan -> SeqEnum
- Ray(F,i) : TorFan,RngIntElt -> TorLatElt
- AllRays(F) : TorFan -> SeqEnum
- PureRays(F) : TorFan -> SeqEnum
- PureRayIndices(F) : TorFan -> SeqEnum
- CreateVirtualRays(S) : [TorLatElt] -> SeqEnum
- VirtualRays(F) : TorFan -> SeqEnum
- VirtualRayIndices(F) : TorFan -> SeqEnum
- Properties of Fans
- Maps of Fans
- Geometrical Properties of Cones and Polyhedra
- Toric Varieties
- Constructors for Toric Varieties
- ToricVariety(k,n) : Fld,RngIntElt -> TorVar
- ToricVariety(k,Z) : Fld,[RngIntElt] -> TorVar
- ToricVariety(k,Z,Q) : Fld,[RngIntElt],[FldRatElt] -> TorVar
- ToricVariety(k,M,v) : Fld,[[RngIntElt]],[RngIntElt] -> TorVar
- Example Toric_toric-cox-example2 (H126E11)
- ToricVariety(k) : Fld -> TorVar
- ProjectiveSpace(k,n) : Fld,RngIntElt -> Prj
- ProjectiveSpace(k,W) : Fld,[RngIntElt] -> Prj
- AbsoluteRationalScroll(k,S) : Fld,[RngIntElt] -> TorVar
- RationalScroll(k,s,A) : Fld, RngIntElt, [RngIntElt] -> TorVar
- RuledSurface(k,n) : Fld, RngIntElt -> TorVar
- RuledSurface(k,a1,a2) : Fld, RngIntElt, RngIntElt -> TorVar
- HirzebruchSurface(k,n) : Fld, RngIntElt -> TorVar
- BigTorus(k,N) : Rng,TorLat -> TorVar
- BigTorus(X) : TorVar -> TorVar, TorMap, TorMap
- RestrictionToSubtorus(Z) : Sch -> Sch, TorMap
- Toric Varieties and their Fans
- Properties of Toric Varieties
- Affine Patches on Toric Varieties
- Cox Rings
- The Cox Ring of a Toric Variety
- Cox Rings in Their Own Right
- CoxRing(R,B,Z,Q) : RngMPol,SeqEnum,SeqEnum,SeqEnum -> RngCox
- C1 eq C2 : RngCox,RngCox -> BoolElt
- BaseRing(C) : RngCox -> Fld
- UnderlyingRing(C) : RngCox -> RngMPol
- Length(C) : RngCox -> RngIntElt
- IrrelevantIdeal(C) : RngCox -> SeqEnum
- IrrelevantComponents(C) : RngCox -> SeqEnum
- IrrelevantGenerators(C) : RngCox -> SeqEnum
- Gradings(C) : RngCox -> RngIntElt
- NumberOfGradings(C) : RngCox -> RngIntElt
- QuotientGradings(C) : RngCox -> RngIntElt
- NumberOfQuotientGradings(C) : RngCox -> RngIntElt
- C . i : RngCox, RngInt -> RngMPolElt
- AssignNames(~C, S) : RngCox, [MonStgElt] ->
- Name(C,i) : RngCox,RngIntElt -> RngMPolElt
- Recovering a Toric Variety From a Cox Ring
- Invariant Divisors and Riemann-Roch Spaces
- Divisor Group
- Constructing Invariant Divisors
- Divisor(X,S) : TorVar,[RngIntElt] -> DivTorElt
- Divisor(X,i) : TorVar,RngIntElt -> DivTorElt
- Divisor(X,f) : TorVar,RngMPolElt -> DivTorElt
- Divisor(X,m) : TorVar,TorLatElt -> DivTorElt
- ZeroDivisor(X) : TorVar -> DivTorElt
- Representative(X,m) : TorVar,ModEDElt -> DivTorElt
- Representative(X,m) : TorVar,TorLatElt -> DivTorElt
- CanonicalDivisor(X) : TorVar -> DivTorElt
- CanonicalClass(X) : TorVar -> DivTorElt
- Example Toric_toric-kawamata-blowup-example (H126E15)
- Properties of Divisors
- Linear Equivalence of Divisors
- Riemann--Roch Spaces of Invariant Divisors
- Maps of Toric Varieties
- The Geometry of Toric Varieties
- Resolution of Singularities and Linear Systems
- Mori Theory of Toric Varieties
- MoriCone(X) : TorVar -> TorCon
- NefCone(X) : TorVar -> TorCon
- ExtremalRays(X) : TorVar -> SeqEnum
- ExtremalRayContraction(X,i) : TorVar,RngIntElt -> TorVar,TorMap
- ExtremalRayContractionDivisor(X,i) : TorVar,RngIntElt -> DivTorElt
- TypeOfContraction(X,i) : TorVar,RngIntElt -> MonStgElt
- IsDivisorialContraction(X,i) : TorVar,RngIntElt -> BoolElt
- IsMoriFibreSpace(X,i) : TorVar,RngIntElt -> BoolElt
- IsFlipping(X,i) : TorVar,RngIntElt -> BoolElt
- Flip(X,i) : TorVar,RngIntElt -> TorVar
- Flip(D) : DivTorElt -> TorVar
- WeightsOfFlip(X,i) : TorVar,RngIntElt -> SeqEnum
- Example Toric_toric-flipwts-example (H126E20)
- Example Toric_toric-weights-of-flip-example (H126E21)
- MMP(X) : TorVar -> SeqEnum,SeqEnum
- Example Toric_toric-mmp-example1 (H126E22)
- Decomposition of Toric Morphisms
- Schemes in Toric Varieties
- Bibliography
V2.28, 13 July 2023