  
  [1X7 [33X[0;0YToric morphisms[133X[101X
  
  
  [1X7.1 [33X[0;0YToric morphisms: Examples[133X[101X
  
  
  [1X7.1-1 [33X[0;0YMorphism between toric varieties and their class groups[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XP1 := Polytope([[0],[1]]);[127X[104X
    [4X[28X<A polytope in |R^1>[128X[104X
    [4X[25Xgap>[125X [27XP2 := Polytope([[0,0],[0,1],[1,0]]);[127X[104X
    [4X[28X<A polytope in |R^2>[128X[104X
    [4X[25Xgap>[125X [27XP1 := ToricVariety( P1 );[127X[104X
    [4X[28X<A projective toric variety of dimension 1>[128X[104X
    [4X[25Xgap>[125X [27XP2 := ToricVariety( P2 );[127X[104X
    [4X[28X<A projective toric variety of dimension 2>[128X[104X
    [4X[25Xgap>[125X [27XP1P2 := P1*P2;[127X[104X
    [4X[28X<A projective toric variety of dimension 3[128X[104X
    [4X[28X which is a product of 2 toric varieties>[128X[104X
    [4X[25Xgap>[125X [27XClassGroup( P1 );[127X[104X
    [4X[28X<A free left module of rank 1 on a free generator>[128X[104X
    [4X[25Xgap>[125X [27XDisplay(ByASmallerPresentation(ClassGroup( P1 )));[127X[104X
    [4X[28XZ^(1 x 1)[128X[104X
    [4X[25Xgap>[125X [27XClassGroup( P2 );[127X[104X
    [4X[28X<A free left module of rank 1 on a free generator>[128X[104X
    [4X[25Xgap>[125X [27XDisplay(ByASmallerPresentation(ClassGroup( P2 )));[127X[104X
    [4X[28XZ^(1 x 1)[128X[104X
    [4X[25Xgap>[125X [27XClassGroup( P1P2 );[127X[104X
    [4X[28X<A free left module of rank 2 on free generators>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( last );[127X[104X
    [4X[28XZ^(1 x 2)[128X[104X
    [4X[25Xgap>[125X [27XPicardGroup( P1P2 );[127X[104X
    [4X[28X<A free left module of rank 2 on free generators>[128X[104X
    [4X[25Xgap>[125X [27XP1P2;[127X[104X
    [4X[28X<A projective smooth toric variety of dimension 3 [128X[104X
    [4X[28X which is a product of 2 toric varieties>[128X[104X
    [4X[25Xgap>[125X [27XP2P1:=P2*P1;[127X[104X
    [4X[28X<A projective toric variety of dimension 3 [128X[104X
    [4X[28X which is a product of 2 toric varieties>[128X[104X
    [4X[25Xgap>[125X [27XM := [[0,0,1],[1,0,0],[0,1,0]];[127X[104X
    [4X[28X[ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XM := ToricMorphism(P1P2,M,P2P1);[127X[104X
    [4X[28X<A "homomorphism" of right objects>[128X[104X
    [4X[25Xgap>[125X [27XIsMorphism(M);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XClassGroup(M);[127X[104X
    [4X[28X<A homomorphism of left modules>[128X[104X
    [4X[25Xgap>[125X [27XDisplay(ClassGroup(M));[127X[104X
    [4X[28X[ [  0,  1 ],[128X[104X
    [4X[28X  [  1,  0 ] ][128X[104X
    [4X[28X[128X[104X
    [4X[28Xthe map is currently represented by the above 2 x 2 matrix[128X[104X
  [4X[32X[104X
  
  
  [1X7.2 [33X[0;0YThe GAP category[133X[101X
  
  [1X7.2-1 IsToricMorphism[101X
  
  [33X[1;0Y[29X[2XIsToricMorphism[102X( [3XM[103X ) [32X filter[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YThe  [3XGAP[103X  category of toric morphisms. A toric morphism is defined by a grid
  homomorphism,  which  is  compatible  with  the  fan  structure  of  the two
  varieties.[133X
  
  
  [1X7.3 [33X[0;0YProperties[133X[101X
  
  [1X7.3-1 IsMorphism[101X
  
  [33X[1;0Y[29X[2XIsMorphism[102X( [3Xmorph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YChecks if the grid morphism [3Xmorph[103X respects the fan structure.[133X
  
  [1X7.3-2 IsProper[101X
  
  [33X[1;0Y[29X[2XIsProper[102X( [3Xmorph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YChecks if the defined morphism [3Xmorph[103X is proper.[133X
  
  
  [1X7.4 [33X[0;0YAttributes[133X[101X
  
  [1X7.4-1 SourceObject[101X
  
  [33X[1;0Y[29X[2XSourceObject[102X( [3Xmorph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya variety[133X
  
  [33X[0;0YReturns  the  source  object of the morphism [3Xmorph[103X. This attribute is a must
  have.[133X
  
  [1X7.4-2 UnderlyingGridMorphism[101X
  
  [33X[1;0Y[29X[2XUnderlyingGridMorphism[102X( [3Xmorph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya map[133X
  
  [33X[0;0YReturns the grid map which defines [3Xmorph[103X.[133X
  
  [1X7.4-3 ToricImageObject[101X
  
  [33X[1;0Y[29X[2XToricImageObject[102X( [3Xmorph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya variety[133X
  
  [33X[0;0YReturns  the  variety  which is created by the fan which is the image of the
  fan  of  the source of [3Xmorph[103X. This is not an image in the usual sense, but a
  toric image.[133X
  
  [1X7.4-4 RangeObject[101X
  
  [33X[1;0Y[29X[2XRangeObject[102X( [3Xmorph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya variety[133X
  
  [33X[0;0YReturns  the range of the morphism [3Xmorph[103X. If no range is given (yes, this is
  possible), the method returns the image.[133X
  
  [1X7.4-5 MorphismOnWeilDivisorGroup[101X
  
  [33X[1;0Y[29X[2XMorphismOnWeilDivisorGroup[102X( [3Xmorph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism[133X
  
  [33X[0;0YReturns  the  associated  morphism between the divisor group of the range of
  [3Xmorph[103X and the divisor group of the source.[133X
  
  [1X7.4-6 ClassGroup[101X
  
  [33X[1;0Y[29X[2XClassGroup[102X( [3Xmorph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism[133X
  
  [33X[0;0YReturns the associated morphism between the class groups of source and range
  of the morphism [3Xmorph[103X[133X
  
  [1X7.4-7 MorphismOnCartierDivisorGroup[101X
  
  [33X[1;0Y[29X[2XMorphismOnCartierDivisorGroup[102X( [3Xmorph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism[133X
  
  [33X[0;0YReturns the associated morphism between the Cartier divisor groups of source
  and range of the morphism [3Xmorph[103X[133X
  
  [1X7.4-8 PicardGroup[101X
  
  [33X[1;0Y[29X[2XPicardGroup[102X( [3Xmorph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism[133X
  
  [33X[0;0YReturns  the  associated  morphism  between  the Picard groups of source and
  range of the morphism [3Xmorph[103X[133X
  
  [1X7.4-9 Source[101X
  
  [33X[1;0Y[29X[2XSource[102X( [3Xmorph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya variety[133X
  
  [33X[0;0YReturn the source of the toric morphism [3Xmorph[103X.[133X
  
  [1X7.4-10 Range[101X
  
  [33X[1;0Y[29X[2XRange[102X( [3Xmorph[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya variety[133X
  
  [33X[0;0YReturns the range of the toric morphism [3Xmorph[103X if specified.[133X
  
  [1X7.4-11 MorphismOnIthFactor[101X
  
  [33X[1;0Y[29X[2XMorphismOnIthFactor[102X( [3Xmorph[103X ) [32X attribute[133X
  
  
  [1X7.5 [33X[0;0YMethods[133X[101X
  
  [1X7.5-1 UnderlyingListList[101X
  
  [33X[1;0Y[29X[2XUnderlyingListList[102X( [3Xmorph[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns a list of list which represents the grid homomorphism.[133X
  
  
  [1X7.6 [33X[0;0YConstructors[133X[101X
  
  [1X7.6-1 ToricMorphism[101X
  
  [33X[1;0Y[29X[2XToricMorphism[102X( [3Xvari[103X, [3Xlis[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism[133X
  
  [33X[0;0YReturns  the  toric  morphism  with  source [3Xvari[103X which is represented by the
  matrix [3Xlis[103X. The range is set to the image.[133X
  
  [1X7.6-2 ToricMorphism[101X
  
  [33X[1;0Y[29X[2XToricMorphism[102X( [3Xvari[103X, [3Xlis[103X, [3Xvari2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism[133X
  
  [33X[0;0YReturns  the  toric  morphism  with  source  [3Xvari[103X  and  range [3Xvari2[103X which is
  represented by the matrix [3Xlis[103X.[133X
  
