chap7.txt

  
  7 Toric morphisms
  
  
  7.1 Toric morphisms: Category and Representations
  
  7.1-1 IsToricMorphism
  
  IsToricMorphism( M )  Category
  Returns:  true or false
  
  The  GAP  category of toric morphisms. A toric morphism is defined by a grid
  homomorphism,  which  is  compatible  with  the  fan  structure  of  the two
  varieties.
  
  
  7.2 Toric morphisms: Properties
  
  7.2-1 IsMorphism
  
  IsMorphism( morph )  property
  Returns:  true or false
  
  Checks if the grid morphism morph respects the fan structure.
  
  7.2-2 IsProper
  
  IsProper( morph )  property
  Returns:  true or false
  
  Checks if the defined morphism morph is proper.
  
  
  7.3 Toric morphisms: Attributes
  
  7.3-1 SourceObject
  
  SourceObject( morph )  attribute
  Returns:  a variety
  
  Returns  the  source  object of the morphism morph. This attribute is a must
  have.
  
  7.3-2 UnderlyingGridMorphism
  
  UnderlyingGridMorphism( morph )  attribute
  Returns:  a map
  
  Returns the grid map which defines morph.
  
  7.3-3 ToricImageObject
  
  ToricImageObject( morph )  attribute
  Returns:  a variety
  
  Returns  the  variety  which is created by the fan which is the image of the
  fan  of  the source of morph. This is not an image in the usual sense, but a
  toric image.
  
  7.3-4 RangeObject
  
  RangeObject( morph )  attribute
  Returns:  a variety
  
  Returns  the range of the morphism morph. If no range is given (yes, this is
  possible), the method returns the image.
  
  7.3-5 MorphismOnWeilDivisorGroup
  
  MorphismOnWeilDivisorGroup( morph )  attribute
  Returns:  a morphism
  
  Returns  the  associated  morphism between the divisor group of the range of
  morph and the divisor group of the source.
  
  7.3-6 ClassGroup
  
  ClassGroup( morph )  attribute
  Returns:  a morphism
  
  Returns the associated morphism between the class groups of source and range
  of the morphism morph
  
  7.3-7 MorphismOnCartierDivisorGroup
  
  MorphismOnCartierDivisorGroup( morph )  attribute
  Returns:  a morphism
  
  Returns the associated morphism between the Cartier divisor groups of source
  and range of the morphism morph
  
  7.3-8 PicardGroup
  
  PicardGroup( morph )  attribute
  Returns:  a morphism
  
  Returns the associated morphism between the class groups of source and range
  of the morphism morph
  
  
  7.4 Toric morphisms: Methods
  
  7.4-1 UnderlyingListList
  
  UnderlyingListList( morph )  attribute
  Returns:  a list
  
  Returns a list of list which represents the grid homomorphism.
  
  
  7.5 Toric morphisms: Constructors
  
  7.5-1 ToricMorphism
  
  ToricMorphism( vari, lis )  operation
  Returns:  a morphism
  
  Returns  the  toric  morphism  with  source vari which is represented by the
  matrix lis. The range is set to the image.
  
  7.5-2 ToricMorphism
  
  ToricMorphism( vari, lis, vari2 )  operation
  Returns:  a morphism
  
  Returns  the  toric  morphism  with  source  vari  and  range vari2 which is
  represented by the matrix lis.
  
  
  7.6 Toric morphisms: Examples
  
  
  7.6-1 Morphism between toric varieties and their class groups
  
    Example  
    gap> P1 := Polytope([[0],[1]]);
    <A polytope in |R^1>
    gap> P2 := Polytope([[0,0],[0,1],[1,0]]);
    <A polytope in |R^2>
    gap> P1 := ToricVariety( P1 );
    <A projective toric variety of dimension 1>
    gap> P2 := ToricVariety( P2 );
    <A projective toric variety of dimension 2>
    gap> P1P2 := P1*P2;
    <A projective toric variety of dimension 3
     which is a product of 2 toric varieties>
    gap> ClassGroup( P1 );
    <A non-torsion left module presented by 1 relation for 2 generators>
    gap> Display(ByASmallerPresentation(last));
    Z^(1 x 1)
    gap> ClassGroup( P2 );
    <A non-torsion left module presented by 2 relations for 3 generators>
    gap> Display(ByASmallerPresentation(last));
    Z^(1 x 1)
    gap> ClassGroup( P1P2 );
    <A free left module of rank 2 on free generators>
    gap> Display( last );
    Z^(1 x 2)
    gap> PicardGroup( P1P2 );
    <A free left module of rank 2 on free generators>
    gap> P1P2;
    <A projective smooth toric variety of dimension 3 
     which is a product of 2 toric varieties>
    gap> P2P1:=P2*P1;
    <A projective toric variety of dimension 3 
     which is a product of 2 toric varieties>
    gap> M := [[0,0,1],[1,0,0],[0,1,0]];
    [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ]
    gap> M := ToricMorphism(P1P2,M,P2P1);
    <A "homomorphism" of right objects>
    gap> IsMorphism(M);
    true
    gap> ClassGroup(M);
    <A homomorphism of left modules>
    gap> Display(last);
    [ [  0,  1 ],
      [  1,  0 ] ]
    
    the map is currently represented by the above 2 x 2 matrix
    gap> ByASmallerPresentation(ClassGroup(M));
    <A non-zero homomorphism of left modules>
    gap> Display(last);
    [ [  0,  1 ],
      [  1,  0 ] ]
    
    the map is currently represented by the above 2 x 2 matrix