`‣ 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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