Introduction and First Examples
The Projective Plane as a Toric Variety
Example Toric_toric-example1 (H118E1)
Resolution of a Nonprojective Toric Variety
Example Toric_toric-example10 (H118E2)
The Cox Ring of a Toric Variety
Example Toric_cox-ring-example (H118E3)
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 -> TorFan
ZeroFan(L) : TorLat -> TorFan
NormalFan(F,C) : TorFan,TorCon -> TorFan,Map
SpanningFan(P) : TorPol -> TorFan
DualFan(P) : TorPol -> TorFan
Example Toric_toric-spanning-fan-example (H118E4)
Blowup(F,v): TorFan,TorLatElt -> TorFan
IsInSupport(v,F) : TorLatElt,TorFan -> BoolElt,RngIntElt
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
AllCones(F) : TorFan -> SeqEnum
Cone(F,i) : TorFan,RngIntElt -> TorCon
Cone(F,S) : TorFan,[RngIntElt] -> TorCon
SingularCones(F) : TorFan -> SeqEnum,SeqEnum
Example Toric_toric-singular-cones-example (H118E5)
ConesOfMaximalDimension(F) : TorFan -> SeqEnum
ConeIndices(F) : TorFan -> SeqEnum
ConeIndices(F,C) : TorFan -> SeqEnum
ConeIntersection(F,C1,C2) : TorFan,TorCon,TorCon -> TorCon
Face(F,C) : TorFan,TorCon -> TorCon
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
VirtualRays(F) : TorFan -> SeqEnum
VirtualRayIndices(F) : TorFan -> SeqEnum
Properties of Fans
Ambient(F) : TorFan -> TorLat
IsComplete(F) : TorFan -> BoolElt
IsSingular(F) : TorFan -> BoolElt
IsNonsingular(F) : TorFan -> BoolElt
IsQFactorial(F) : TorFan -> BoolElt
IsTerminal(F) : TorFan -> BoolElt
IsCanonical(F) : TorFan -> BoolElt
IsGorenstein(F) : TorFan -> BoolElt
IsQGorenstein(F) : TorFan -> BoolElt
Maps of Fans
F @ f : TorFan,Map -> TorFan
SimplicialSubdivision(F) : TorFan -> TorFan
Example Toric_toric-simplicial-example (H118E6)
IsFanMap(F1,F2) : TorFan,TorFan -> BoolElt
IsFanMap(F1,F2,f) : TorFan,TorFan,Map -> BoolElt
ResolveFanMap(F1,F2) : TorFan,TorFan -> TorFan
Geometrical Properties of Cones and Polyhedra
IsSingular(C) : TorCon -> BoolElt
IsNonsingular(C) : TorCon -> BoolElt
IsSmooth(P) : TorPol -> BoolElt
IsGorenstein(C) : TorCon -> BoolElt
IsReflexive(P) : TorPol -> BoolElt
IsQGorenstein(C) : TorCon -> BoolElt
GorensteinIndex(C) : TorCon -> RngIntElt,TorLatElt
GorensteinIndex(P) : TorPol -> RngIntElt
IsQFactorial(C) : TorCon -> BoolElt
IsTerminal(C) : TorCon -> BoolElt
IsCanonical(C) : TorCon -> BoolElt
IsFano(P) : TorPol -> BoolElt
Example Toric_toric-terminal-polytope-example (H118E7)
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 (H118E8)
ToricVariety(k) : Fld -> TorVar
ProjectiveSpace(k,n) : Fld,RngIntElt -> Prj
ProjectiveSpace(k,W) : Fld,SeqEnum -> Prj
Toric Varieties and Their Fans
ToricVariety(k,F) : Fld,TorFan -> TorVar
Fan(X) : TorVar -> TorLat
Rays(X) : TorVar -> SeqEnum
OneParameterSubgroupsLattice(X) : TorVar -> TorLat
MonomialLattice(X) : TorVar -> TorLat
CoxMonomialLattice(X) : TorVar -> TorLat
DivisorClassLattice(X) : TorVar -> TorLat
IrrelevantIdeal(X) : TorVar -> SeqEnum
QuotientGradings(X) : TorVar -> SeqEnum
NumberOfQuotientGradings(X) : TorVar -> SeqEnum
Properties of Toric Varieties
IsSingular(X) : TorVar -> BoolElt
IsNonsingular(X) : TorVar -> BoolElt
IsGorenstein(X) : TorVar -> BoolElt
IsQGorenstein(X) : TorVar -> BoolElt
IsQFactorial(X) : TorVar -> BoolElt
IsTerminal(X) : TorVar -> BoolElt
IsCanonical(X) : TorVar -> BoolElt
IsComplete(X) : TorVar -> BoolElt
IsProjective(X) : TorVar -> BoolElt
IsFano(X) : TorVar -> BoolElt
IsFakeWeightedProjectiveSpace(X) : TorVar -> BoolElt
IsWeightedProjectiveSpace(X) : TorVar -> BoolElt
Affine Patches on Toric Varieties
ToricAffinePatch(X,i) : TorVar,RngIntElt -> TorVar,TorMap
ToricAffinePatch(X,S) : TorVar,[RngIntElt] -> TorVar,TorMap
The Cox Ring of a Toric Variety
CoxRing(X) : TorVar -> RngCox
CoxRing(k,F) : Fld,TorFan -> RngCox
Example Toric_toric-cox-example1 (H118E9)
Example Toric_toric-cox-example2 (H118E10)
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
ToricVariety(C) : RngCox -> TorVar
Example Toric_toric-from-cox-example (H118E11)
Fan(C) : RngCox -> TorFan
CoxMonomialLattice(C) : RngCox -> TorLat
DivisorClassLattice(C) : RngCox -> TorLat
MonomialLattice(C) : RngCox -> TorLat
OneParameterSubgroupsLattice(C) : RngCox -> TorLat
RayLattice(C) : RngCox -> TorLat
DivisorClassGroup(C) : RngCox -> TorLat
RayLatticeMap(C) : RngCox -> Map
WeilToClassGroupsMap(C) : RngCox -> Map
Invariant Divisors and Riemann-Roch Spaces
Divisor Group
DivisorGroup(X) : TorVar -> DivTor
ToricVariety(G) : DivTor -> TorVar
G1 eq G2 : DivTor,DivTor -> BoolElt
Divisor(G,S) : DivTor,[RngIntElt] -> DivTorElt
Divisor(G,i) : DivTor,RngIntElt -> DivTorElt
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 (H118E12)
Properties of Divisors
Variety(D) : DivTorElt -> TorVar
Parent(D) : DivTorElt -> DivTor
Weil(D) : DivTorElt -> SeqEnum
Cartier(D) : DivTorElt -> SeqEnum[TorLatElt]
IsQCartier(D) : DivTorElt -> BoolElt
IsCartier(D) : DivTorElt -> BoolElt
IsWeil(D) : DivTorElt -> BoolElt
IsAmple(D) : DivTorElt -> BoolElt
IsNef(D) : DivTorElt -> BoolElt
IsBig(D) : DivTorElt -> BoolElt
PicardClass(D) : DivTorElt -> TorLatElt
MovablePart(D) : DivTorElt -> DivTorElt
Example Toric_toric-movable-example (H118E13)
ImageFan(D) : DivTorElt -> TorFan
Proj(D) : DivTorElt -> TorVar, PlcEnum
RelativeProj(D) : DivTorElt -> TorVar
IntersectionForm(X,C) : TorVar,TorCon -> TorLatElt
Linear Equivalence of Divisors
IsQPrincipal(D) : DivTorElt -> BoolElt
IsPrincipal(D) : DivTorElt -> BoolElt
IsLinearlyEquivalentToCartier(D) : DivTorElt -> BoolElt, DivTorElt
AreLinearlyEquivalent(D,E) : DivTorElt,DivTorElt -> BoolElt
DefiningMonomial(D) : DivTorElt -> RngMPolElt
LatticeElementToMonomial(D,v) : DivTorElt,TorLatElt -> RngMPolElt
Riemann--Roch Spaces of Invariant Divisors
RiemannRochPolytope(D) : DivTorElt -> TorPol
RiemannRochBasis(D) : DivTorElt -> [RngElt]
RiemannRochDimension(D) : DivTorElt -> RngIntElt
GradedCone(D) : DivTorElt -> TorCon
Polyhedron(D) : DivTorElt -> TorPol
Example Toric_toric-rr-example (H118E14)
HilbertSeries(D) : DivTor -> FldFunRatUElt
HilbertPolynomial(D) : DivTor -> [RngUPolElt]
HilbertCoefficients(D,l) : DivTor,RngIntElt -> [RngIntElt]
HilbertCoefficient(D,i) : DivTor,RngIntElt -> RngIntElt
Example Toric_toric-rr-by-hand (H118E15)
Maps from Lattice Maps
ToricVarietyMap(X,Y,f) : TorVar,TorVar,Map -> TorMap
Blowup(X,v) : TorVar,TorLatElt -> TorVar,TorMap
IdentityMap(X) : TorVar -> TorMap
Properties of Toric Maps
IsRegular(f) : TorMap -> BoolElt
IndeterminacyLocus(f) : TorMap -> [Sch]
Example Toric_toric-simplicial-example (H118E16)
The Geometry of Toric Varieties
Resolution of Singularities and Linear Systems
Resolution(X) : TorVar -> TorVar,TorMap
ResolveLinearSystem(D) : DivTorElt -> TorVar
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
IsMoriFibreSpace(X,i) : TorVar,RngIntElt -> BoolElt
IsDivisiorialContraction(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 (H118E17)
Example Toric_toric-weights-of-flip-example (H118E18)
MMP(X) : TorVar -> SeqEnum,SeqEnum
Example Toric_toric-mmp-example1 (H118E19)
Decomposition of Toric Morphisms
Example Toric_toric-decomposition-example (H118E20)
Construction of Subschemes
Scheme(X,f) : TorVar,RngMPolElt -> Sch
Scheme(X,Q) : TorVar,[RngMPolElt] -> Sch
Example Toric_toric-mmp-example1 (H118E21)
Bibliography
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Version: V2.19 of
Wed Apr 24 15:09:57 EST 2013