‣ 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.
‣ IsMorphism( morph ) | ( property ) |
Returns: true or false
Checks if the grid morphism morph respects the fan structure.
‣ IsProper( morph ) | ( property ) |
Returns: true or false
Checks if the defined morphism morph is proper.
‣ SourceObject( morph ) | ( attribute ) |
Returns: a variety
Returns the source object of the morphism morph. This attribute is a must have.
‣ UnderlyingGridMorphism( morph ) | ( attribute ) |
Returns: a map
Returns the grid map which defines morph.
‣ 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.
‣ 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.
‣ 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.
‣ ClassGroup( morph ) | ( attribute ) |
Returns: a morphism
Returns the associated morphism between the class groups of source and range of the morphism morph
‣ MorphismOnCartierDivisorGroup( morph ) | ( attribute ) |
Returns: a morphism
Returns the associated morphism between the Cartier divisor groups of source and range of the morphism morph
‣ PicardGroup( morph ) | ( attribute ) |
Returns: a morphism
Returns the associated morphism between the class groups of source and range of the morphism morph
‣ UnderlyingListList( morph ) | ( attribute ) |
Returns: a list
Returns a list of list which represents the grid homomorphism.
‣ 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.
‣ 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.
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
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