chap6.txt

  
  6 Projective toric varieties
  
  
  6.1 Projective toric varieties: Category and Representations
  
  6.1-1 IsProjectiveToricVariety
  
  IsProjectiveToricVariety( M )  Category
  Returns:  true or false
  
  The GAP category of a projective toric variety.
  
  
  6.2 Projective toric varieties: Properties
  
  Projective  toric  varieties  have  no  additional properties. Remember that
  projective toric varieties are toric varieties, so every property of a toric
  variety is a property of an projective toric variety.
  
  
  6.3 Projective toric varieties: Attributes
  
  6.3-1 AffineCone
  
  AffineCone( vari )  attribute
  Returns:  a variety
  
  Returns the affine cone of the projective toric variety vari.
  
  6.3-2 PolytopeOfVariety
  
  PolytopeOfVariety( vari )  attribute
  Returns:  a polytope
  
  Returns  the polytope corresponding to the projective toric variety vari, if
  it exists.
  
  6.3-3 ProjectiveEmbedding
  
  ProjectiveEmbedding( vari )  attribute
  Returns:  a list
  
  Returns  characters  for  a  closed embedding in an projective space for the
  projective toric variety vari.
  
  
  6.4 Projective toric varieties: Methods
  
  6.4-1 Polytope
  
  Polytope( vari )  operation
  Returns:  a polytope
  
  Returns the polytope of the variety vari. Another name for PolytopeOfVariety
  for compatibility and shortness.
  
  
  6.5 Projective toric varieties: Constructors
  
  The  constructors  are  the same as for toric varieties. Calling them with a
  polytope will result in an projective variety.
  
  
  6.6 Projective toric varieties: Examples
  
  
  6.6-1 PxP1 created by a polytope
  
    Example  
    gap> P1P1 := Polytope( [[1,1],[1,-1],[-1,-1],[-1,1]] );
    <A polytope in |R^2>
    gap> P1P1 := ToricVariety( P1P1 );
    <A projective toric variety of dimension 2>
    gap> IsProjective( P1P1 );
    true
    gap> IsComplete( P1P1 );
    true 
    gap> CoordinateRingOfTorus( P1P1, "x" );
    Q[x1,x1_,x2,x2_]/( x2*x2_-1, x1*x1_-1 )
    gap> IsVeryAmple( Polytope( P1P1 ) );
    true
    gap> ProjectiveEmbedding( P1P1 );
    [ |[ x1_*x2_ ]|, |[ x1_ ]|, |[ x1_*x2 ]|, |[ x2_ ]|,
    |[ 1 ]|, |[ x2 ]|, |[ x1*x2_ ]|, |[ x1 ]|, |[ x1*x2 ]| ]
    gap> Length( last );
    9