chap8.txt

  
  8 Toric divisors
  
  
  8.1 Toric divisors: Category and Representations
  
  8.1-1 IsToricDivisor
  
  IsToricDivisor( M )  Category
  Returns:  true or false
  
  The GAP category of torus invariant Weil divisors.
  
  
  8.2 Toric divisors: Properties
  
  8.2-1 IsCartier
  
  IsCartier( divi )  property
  Returns:  true or false
  
  Checks  if  the  torus  invariant Weil divisor divi is Cartier i.e. if it is
  locally principal.
  
  8.2-2 IsPrincipal
  
  IsPrincipal( divi )  property
  Returns:  true or false
  
  Checks  if  the  torus invariant Weil divisor divi is principal which in the
  toric invariant case means that it is the divisor of a character.
  
  8.2-3 IsPrimedivisor
  
  IsPrimedivisor( divi )  property
  Returns:  true or false
  
  Checks  if the Weil divisor divi represents a prime divisor, i.e. if it is a
  standard generator of the divisor group.
  
  8.2-4 IsBasepointFree
  
  IsBasepointFree( divi )  property
  Returns:  true or false
  
  Checks if the divisor divi is basepoint free. What else?
  
  8.2-5 IsAmple
  
  IsAmple( divi )  property
  Returns:  true or false
  
  Checks  if  the divisor divi is ample, i.e. if it is colored red, yellow and
  green.
  
  8.2-6 IsVeryAmple
  
  IsVeryAmple( divi )  property
  Returns:  true or false
  
  Checks if the divisor divi is very ample.
  
  
  8.3 Toric divisors: Attributes
  
  8.3-1 CartierData
  
  CartierData( divi )  attribute
  Returns:  a list
  
  Returns  the  Cartier  data of the divisor divi, if it is Cartier, and fails
  otherwise.
  
  8.3-2 CharacterOfPrincipalDivisor
  
  CharacterOfPrincipalDivisor( divi )  attribute
  Returns:  an element
  
  Returns the character corresponding to principal divisor divi.
  
  8.3-3 ToricVarietyOfDivisor
  
  ToricVarietyOfDivisor( divi )  attribute
  Returns:  a variety
  
  Returns  the  closure  of the torus orbit corresponding to the prime divisor
  divi.  Not  implemented  for other divisors. Maybe we should add the support
  here. Is this even a toric variety? Exercise left to the reader.
  
  8.3-4 ClassOfDivisor
  
  ClassOfDivisor( divi )  attribute
  Returns:  an element
  
  Returns the class group element corresponding to the divisor divi.
  
  8.3-5 PolytopeOfDivisor
  
  PolytopeOfDivisor( divi )  attribute
  Returns:  a polytope
  
  Returns the polytope corresponding to the divisor divi.
  
  8.3-6 BasisOfGlobalSections
  
  BasisOfGlobalSections( divi )  attribute
  Returns:  a list
  
  Returns  a basis of the global section module of the quasi-coherent sheaf of
  the divisor divi.
  
  8.3-7 IntegerForWhichIsSureVeryAmple
  
  IntegerForWhichIsSureVeryAmple( divi )  attribute
  Returns:  an integer
  
  Returns  an  integer  which,  to  be multiplied with the ample divisor divi,
  someone gets a very ample divisor.
  
  8.3-8 AmbientToricVariety
  
  AmbientToricVariety( divi )  attribute
  Returns:  a variety
  
  Returns the containing variety of the prime divisors of the divisor divi.
  
  8.3-9 UnderlyingGroupElement
  
  UnderlyingGroupElement( divi )  attribute
  Returns:  an element
  
  Returns an element which represents the divisor divi in the Weil group.
  
  8.3-10 UnderlyingToricVariety
  
  UnderlyingToricVariety( divi )  attribute
  Returns:  a variety
  
  Returns  the  closure  of the torus orbit corresponding to the prime divisor
  divi.  Not  implemented  for other divisors. Maybe we should add the support
  here. Is this even a toric variety? Exercise left to the reader.
  
  8.3-11 DegreeOfDivisor
  
  DegreeOfDivisor( divi )  attribute
  Returns:  an integer
  
  Returns the degree of the divisor divi.
  
  8.3-12 MonomsOfCoxRingOfDegree
  
  MonomsOfCoxRingOfDegree( divi )  attribute
  Returns:  a list
  
  Returns the variety corresponding to the polytope of the divisor divi.
  
  8.3-13 CoxRingOfTargetOfDivisorMorphism
  
  CoxRingOfTargetOfDivisorMorphism( divi )  attribute
  Returns:  a ring
  
  A  basepoint  free  divisor divi defines a map from its ambient variety in a
  projective  space.  This  method  returns  the cox ring of such a projective
  space.
  
  8.3-14 RingMorphismOfDivisor
  
  RingMorphismOfDivisor( divi )  attribute
  Returns:  a ring
  
  A  basepoint  free  divisor divi defines a map from its ambient variety in a
  projective  space.  This method returns the morphism between the cox ring of
  this projective space to the cox ring of the ambient variety of divi.
  
  
  8.4 Toric divisors: Methods
  
  8.4-1 VeryAmpleMultiple
  
  VeryAmpleMultiple( divi )  operation
  Returns:  a divisor
  
  Returns  a  very  ample  multiple  of  the  ample divisor divi. Will fail if
  divisor is not ample.
  
  8.4-2 CharactersForClosedEmbedding
  
  CharactersForClosedEmbedding( divi )  operation
  Returns:  a list
  
  Returns  characters for closed embedding defined via the ample divisor divi.
  Fails if divisor is not ample.
  
  8.4-3 MonomsOfCoxRingOfDegree
  
  MonomsOfCoxRingOfDegree( vari, elem )  operation
  Returns:  a list
  
  Returns  the  monoms  of the Cox ring of the variety vari with degree to the
  class group element elem. The variable elem can also be a list.
  
  8.4-4 DivisorOfGivenClass
  
  DivisorOfGivenClass( vari, elem )  operation
  Returns:  a list
  
  Computes  a divisor of the variety divi which is member of the divisor class
  presented  by  elem.  The  variable  elem  can be a homalg element or a list
  presenting an element.
  
  8.4-5 AddDivisorToItsAmbientVariety
  
  AddDivisorToItsAmbientVariety( divi )  operation
  
  Adds the divisor divi to the Weil divisor list of its ambient variety.
  
  8.4-6 Polytope
  
  Polytope( divi )  operation
  Returns:  a polytope
  
  Returns the polytope of the divisor divi. Another name for PolytopeOfDivisor
  for compatibility and shortness.
  
  8.4-7 +
  
  +( divi1, divi2 )  operation
  Returns:  a divisor
  
  Returns the sum of the divisors divi1 and divi2.
  
  8.4-8 -
  
  -( divi1, divi2 )  operation
  Returns:  a divisor
  
  Returns the divisor divi1 minus divi2.
  
  8.4-9 *
  
  *( k, divi )  operation
  Returns:  a divisor
  
  Returns k times the divisor divi.
  
  
  8.5 Toric divisors: Constructors
  
  8.5-1 DivisorOfCharacter
  
  DivisorOfCharacter( elem, vari )  operation
  Returns:  a divisor
  
  Returns  the  divisor  of  the  toric  variety vari which corresponds to the
  character elem.
  
  8.5-2 DivisorOfCharacter
  
  DivisorOfCharacter( lis, vari )  operation
  Returns:  a divisor
  
  Returns  the  divisor  of  the  toric  variety vari which corresponds to the
  character which is created by the list lis.
  
  8.5-3 CreateDivisor
  
  CreateDivisor( elem, vari )  operation
  Returns:  a divisor
  
  Returns  the divisor of the toric variety vari which corresponds to the Weil
  group element elem.
  
  8.5-4 CreateDivisor
  
  CreateDivisor( lis, vari )  operation
  Returns:  a divisor
  
  Returns  the divisor of the toric variety vari which corresponds to the Weil
  group element which is created by the list lis.
  
  
  8.6 Toric divisors: Examples
  
  
  8.6-1 Divisors on a toric variety
  
    Example  
    gap> H7 := Fan( [[0,1],[1,0],[0,-1],[-1,7]],[[1,2],[2,3],[3,4],[4,1]] );
    <A fan in |R^2>
    gap> H7 := ToricVariety( H7 );
    <A toric variety of dimension 2>
    gap> P := TorusInvariantPrimeDivisors( H7 );
    [ <A prime divisor of a toric variety with coordinates [ 1, 0, 0, 0 ]>, 
      <A prime divisor of a toric variety with coordinates [ 0, 1, 0, 0 ]>, 
      <A prime divisor of a toric variety with coordinates [ 0, 0, 1, 0 ]>, 
      <A prime divisor of a toric variety with coordinates [ 0, 0, 0, 1 ]> ]
    gap> D := P[3]+P[4];
    <A divisor of a toric variety with coordinates [ 0, 0, 1, 1 ]>
    gap> IsBasepointFree(D);
    true
    gap> IsAmple(D);
    true
    gap> CoordinateRingOfTorus(H7,"x");
    Q[x1,x1_,x2,x2_]/( x2*x2_-1, x1*x1_-1 )
    gap> Polytope(D);
    <A polytope in |R^2>
    gap> CharactersForClosedEmbedding(D);
    [ |[ 1 ]|, |[ x2 ]|, |[ x1 ]|, |[ x1*x2 ]|, |[ x1^2*x2 ]|, 
      |[ x1^3*x2 ]|, |[ x1^4*x2 ]|, |[ x1^5*x2 ]|, 
      |[ x1^6*x2 ]|, |[ x1^7*x2 ]|, |[ x1^8*x2 ]| ]
    gap> CoxRingOfTargetOfDivisorMorphism(D);
    Q[x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_10,x_11]
    (weights: [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ])
    gap> RingMorphismOfDivisor(D);
    <A "homomorphism" of rings>
    gap> Display(last);
    Q[x_1,x_2,x_3,x_4]
    (weights: [ [ 0, 0, 1, -7 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ])
      ^
      |
    [ x_3*x_4, x_1*x_4^8, x_2*x_3, x_1*x_2*x_4^7, x_1*x_2^2*x_4^6,
      x_1*x_2^3*x_4^5, x_1*x_2^4*x_4^4, x_1*x_2^5*x_4^3, 
      x_1*x_2^6*x_4^2, x_1*x_2^7*x_4, x_1*x_2^8 ]
      |
      |
    Q[x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_10,x_11]
    (weights: [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ])
    gap> ByASmallerPresentation(ClassGroup(H7));
    <A free left module of rank 2 on free generators>
    gap> Display(RingMorphismOfDivisor(D));
    Q[x_1,x_2,x_3,x_4]
    (weights: [ [ 1, -7 ], [ 0, 1 ], [ 1, 0 ], [ 0, 1 ] ])
      ^
      |
    [ x_3*x_4, x_1*x_4^8, x_2*x_3, x_1*x_2*x_4^7, x_1*x_2^2*x_4^6, 
      x_1*x_2^3*x_4^5, x_1*x_2^4*x_4^4, x_1*x_2^5*x_4^3, 
      x_1*x_2^6*x_4^2, x_1*x_2^7*x_4, x_1*x_2^8 ]
      |
      |
    Q[x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_10,x_11]
    (weights: [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ])
    gap> MonomsOfCoxRingOfDegree(D);
    [ x_3*x_4, x_1*x_4^8, x_2*x_3, x_1*x_2*x_4^7, x_1*x_2^2*x_4^6, 
      x_1*x_2^3*x_4^5, x_1*x_2^4*x_4^4, x_1*x_2^5*x_4^3, 
      x_1*x_2^6*x_4^2, x_1*x_2^7*x_4, x_1*x_2^8 ]
    gap> D2:=D-2*P[2];
    <A divisor of a toric variety with coordinates [ 0, -2, 1, 1 ]>
    gap> IsBasepointFree(D2);
    false
    gap> IsAmple(D2);
    false