Toric Varieties
- Introduction and First Examples
- Fans in Toric Lattices
- Construction of Fans
Fan(Q): [TorCon] → TorFanFan(R,S): [TorLatElt], [[RngIntElt]] → TorFanFan(C): TorCon → TorFanFanOfAffineSpace(n): RngIntElt → TorFacFanOfWPS(W): SeqEnum → TorFanFanOfProjectiveSpace(n): RngIntElt → TorFacFanOfFakeProjectiveSpace(W,Q): SeqEnum, SeqEnum → TorFanZeroFan(L): TorLat → TorFanNormalFan(F,C): TorFan, TorCon → TorFan, MapNormalFan(P): TorPol → TorFanDualFan(P): TorPol → TorFanSpanningFan(P): TorPol → TorFanExample: Toric Spanning Fan ExampleFanWithWeights(W): SeqEnum → TorFanBlowup(F,v): TorFan, TorLatElt → TorFanExample: Toric Fan With Weights ExampleIsInSupport(v,F): TorLatElt, TorFan → BoolElt, RngIntEltOneSkeleton(F): TorFan → TorFanFan(F1,F2): TorFan, TorFan → TorFanF1 * F2: TorFan, TorFan → TorFanFan(Q): [TorFan] → TorFanF ^ n: TorFan, RngIntElt → TorFanF eq G: TorFan, TorFan → BoolElt
- Components of Fans
Skeleton(F,n): TorFan, RngIntElt → TorFanC in F: TorCon, TorFan → BoolEltCones(F): TorFan → SeqEnumCones(F,i): TorFan, RngIntElt → SeqEnumConesOfCodimension(F,i): TorFan, RngIntElt → SeqEnumMaxCones(F): TorFan → SeqEnumAllCones(F): TorFan → SeqEnumCone(F,i): TorFan, RngIntElt → TorConCone(F,S): TorFan, [RngIntElt] → TorConNonSimplicialCones(F): TorFan → SeqEnum, SeqEnumSingularCones(F): TorFan → SeqEnum, SeqEnumExample: Toric Singular Cones ExampleConeIndices(F): TorFan → SeqEnumConeIndices(F,C): TorFan, TorCon → SeqEnumConeIntersection(F,C1,C2): TorFan, TorCon, TorCon → TorConFace(F,C): TorFan, TorCon → TorConInnerNormals(F): TorFan → SeqEnumOuterNormals(F): TorFan → SeqEnumDualFaceInDualFan(P,Q): TorPol, [RngIntElt] → TorFanRays(F): TorFan → SeqEnumRay(F,i): TorFan, RngIntElt → TorLatEltAllRays(F): TorFan → SeqEnumPureRays(F): TorFan → SeqEnumPureRayIndices(F): TorFan → SeqEnumCreateVirtualRays(S): [TorLatElt] → SeqEnumVirtualRays(F): TorFan → SeqEnumVirtualRayIndices(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 → TorVarToricVariety(k,Z): Fld, [RngIntElt] → TorVarToricVariety(k,Z,Q): Fld, [RngIntElt], [FldRatElt] → TorVarToricVariety(k,M,v): Fld, [[RngIntElt]], [RngIntElt] → TorVarExample: Toric Cox Example2ToricVariety(k): Fld → TorVarProjectiveSpace(k,n): Fld, RngIntElt → PrjProjectiveSpaceAsToricVariety(k,n): Fld, RngIntElt → TorVarWeightedProjectiveSpace(k,n): Fld, RngIntElt → TorVarWPS(k,n): Fld, RngIntElt → TorVarProjectiveSpace(k,W): Fld, [RngIntElt] → PrjProjectiveSpaceAsToricVariety(k,W): Fld, [RngIntElt] → TorVarWeightedProjectiveSpace(k,W): Fld, [RngIntElt] → TorVarWPS(k,W): Fld, [RngIntElt] → TorVarFakeProjectiveSpace(k,W,Q): Fld, [RngIntElt], [FldRatElt] → TorVarAbsoluteRationalScroll(k,S): Fld, [RngIntElt] → TorVarRationalScroll(k,s,A): Fld, RngIntElt, [RngIntElt] → TorVarRuledSurface(k,n): Fld, RngIntElt → TorVarRuledSurface(k,a1,a2): Fld, RngIntElt, RngIntElt → TorVarHirzebruchSurface(k,n): Fld, RngIntElt → TorVarBigTorus(k,N): Rng, TorLat → TorVarBigTorus(X): TorVar → TorVar, TorMap, TorMapRestrictionToSubtorus(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 → RngCoxC1 eq C2: RngCox, RngCox → BoolEltBaseRing(C): RngCox → FldCoefficientRing(C): RngCox → FldUnderlyingRing(C): RngCox → RngMPolLength(C): RngCox → RngIntEltIrrelevantIdeal(C): RngCox → SeqEnumIrrelevantComponents(C): RngCox → SeqEnumIrrelevantGenerators(C): RngCox → SeqEnumGradings(C): RngCox → RngIntEltNumberOfGradings(C): RngCox → RngIntEltQuotientGradings(C): RngCox → RngIntEltNumberOfQuotientGradings(C): RngCox → RngIntEltC . i: RngCox, RngInt → RngMPolEltAssignNames(~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] → DivTorEltDivisor(X,i): TorVar, RngIntElt → DivTorEltDivisor(X,f): TorVar, RngMPolElt → DivTorEltDivisor(X,m): TorVar, TorLatElt → DivTorEltZeroDivisor(X): TorVar → DivTorEltRepresentative(X,m): TorVar, ModEDElt → DivTorEltRepresentative(X,m): TorVar, TorLatElt → DivTorEltCanonicalDivisor(X): TorVar → DivTorEltCanonicalClass(X): TorVar → DivTorEltD1 + D2: DivTorElt, DivTorElt → DivTorEltn * D: FldRatElt, DivTorElt → DivTorElt- D: DivTorElt → DivTorEltD1 - D2: DivTorElt, DivTorElt → DivTorEltD * v: DivTorElt, TorLatElt → RngIntEltExample: Toric Kawamata Blowup Example
- 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 → TorConNefCone(X): TorVar → TorConExtremalRays(X): TorVar → SeqEnumExtremalRayContractions(X): TorVar, RngIntElt → SeqEnumExtremalRayContraction(X,i): TorVar, RngIntElt → TorVar, TorMapExtremalRayContractionDivisor(X,i): TorVar, RngIntElt → DivTorEltTypeOfContraction(X,i): TorVar, RngIntElt → MonStgEltTypesOfContractions(X): TorVar → MonStgEltIsDivisorialContraction(X,i): TorVar, RngIntElt → BoolEltIsMoriFibreSpace(X,i): TorVar, RngIntElt → BoolEltIsFlipping(X,i): TorVar, RngIntElt → BoolEltFlip(X,i): TorVar, RngIntElt → TorVarFlip(D): DivTorElt → TorVarWeightsOfFlip(X,i): TorVar, RngIntElt → SeqEnumExample: Toric Flipwts ExampleExample: Toric Weights Of Flip ExampleMMP(X): TorVar → SeqEnum, SeqEnumExample: Toric Mmp Example1
- Decomposition of Toric Morphisms
- Schemes in Toric Varieties
