chap5.txt

  
  5 Affine toric varieties
  
  
  5.1 Affine toric varieties: Category and Representations
  
  5.1-1 IsAffineToricVariety
  
  IsAffineToricVariety( M )  Category
  Returns:  true or false
  
  The  GAP category of an affine toric variety. All affine toric varieties are
  toric  varieties,  so everything applicable to toric varieties is applicable
  to affine toric varieties.
  
  
  5.2 Affine toric varieties: Properties
  
  Affine  toric  varieties have no additional properties. Remember that affine
  toric varieties are toric varieties, so every property of a toric variety is
  a property of an affine toric variety.
  
  
  5.3 Affine toric varieties: Attributes
  
  5.3-1 CoordinateRing
  
  CoordinateRing( vari )  attribute
  Returns:  a ring
  
  Returns   the  coordinate  ring  of  the  affine  toric  variety  vari.  The
  computation is mainly done in ToricIdeals package.
  
  5.3-2 ListOfVariablesOfCoordinateRing
  
  ListOfVariablesOfCoordinateRing( vari )  attribute
  Returns:  a list
  
  Returns a list containing the variables of the CoordinateRing of the variety
  vari.
  
  5.3-3 MorphismFromCoordinateRingToCoordinateRingOfTorus
  
  MorphismFromCoordinateRingToCoordinateRingOfTorus( vari )  attribute
  Returns:  a morphism
  
  Returns the morphism between the coordinate ring of the variety vari and the
  coordinate ring of its torus. This defines the embedding of the torus in the
  variety.
  
  5.3-4 ConeOfVariety
  
  ConeOfVariety( vari )  attribute
  Returns:  a cone
  
  Returns the cone ring of the affine toric variety vari.
  
  
  5.4 Affine toric varieties: Methods
  
  5.4-1 CoordinateRing
  
  CoordinateRing( vari, indet )  operation
  Returns:  a variety
  
  Computes  the  coordinate  ring  of  the  affine  toric  variety  vari  with
  indeterminates indet.
  
  5.4-2 Cone
  
  Cone( vari )  operation
  Returns:  a cone
  
  Returns  the  cone  of  the variety vari. Another name for ConeOfVariety for
  compatibility and shortness.
  
  
  5.5 Affine toric varieties: Constructors
  
  The  constructors  are  the same as for toric varieties. Calling them with a
  cone will result in an affine variety.
  
  
  5.6 Affine toric Varieties: Examples
  
  
  5.6-1 Affine space
  
    Example  
    gap> C:=Cone( [[1,0,0],[0,1,0],[0,0,1]] );
    <A cone in |R^3>
    gap> C3:=ToricVariety(C);
    <An affine normal toric variety of dimension 3>
    gap> Dimension(C3);
    3
    gap> IsOrbifold(C3);
    true
    gap> IsSmooth(C3);
    true
    gap> CoordinateRingOfTorus(C3,"x");
    Q[x1,x1_,x2,x2_,x3,x3_]/( x3*x3_-1, x2*x2_-1, x1*x1_-1 )
    gap> CoordinateRing(C3,"x");
    Q[x_1,x_2,x_3]
    gap> MorphismFromCoordinateRingToCoordinateRingOfTorus(C3);
    <A monomorphism of rings>
    gap> C3;
    <An affine normal smooth toric variety of dimension 3>
    gap> StructureDescription(C3);
    "|A^3"