3 Toric varieties
  
  
  3.1 Toric variety: Category and Representations
  
  3.1-1 IsToricVariety
  
  IsToricVariety( M )  Category
  Returns:  true or false
  
  The GAP category of a toric variety.
  
  
  3.2 Toric varieties: Properties
  
  3.2-1 IsNormalVariety
  
  IsNormalVariety( vari )  property
  Returns:  true or false
  
  Checks if the toric variety vari is a normal variety.
  
  3.2-2 IsAffine
  
  IsAffine( vari )  property
  Returns:  true or false
  
  Checks if the toric variety vari is an affine variety.
  
  3.2-3 IsProjective
  
  IsProjective( vari )  property
  Returns:  true or false
  
  Checks if the toric variety vari is a projective variety.
  
  3.2-4 IsComplete
  
  IsComplete( vari )  property
  Returns:  true or false
  
  Checks if the toric variety vari is a complete variety.
  
  3.2-5 IsSmooth
  
  IsSmooth( vari )  property
  Returns:  true or false
  
  Checks if the toric variety vari is a smooth variety.
  
  3.2-6 HasTorusfactor
  
  HasTorusfactor( vari )  property
  Returns:  true or false
  
  Checks if the toric variety vari has a torus factor.
  
  3.2-7 HasNoTorusfactor
  
  HasNoTorusfactor( vari )  property
  Returns:  true or false
  
  Checks if the toric variety vari has no torus factor.
  
  3.2-8 IsOrbifold
  
  IsOrbifold( vari )  property
  Returns:  true or false
  
  Checks  if  the  toric  variety vari has an orbifold, which is, in the toric
  case, equivalent to the simpliciality of the fan.
  
  
  3.3 Toric varieties: Attributes
  
  3.3-1 AffineOpenCovering
  
  AffineOpenCovering( vari )  attribute
  Returns:  a list
  
  Returns  a  torus  invariant  affine  open covering of the variety vari. The
  affine open cover is computed out of the cones of the fan.
  
  3.3-2 CoxRing
  
  CoxRing( vari )  attribute
  Returns:  a ring
  
  Returns  the  Cox  ring  of  the  variety vari. The actual method requires a
  string  with  a  name for the variables. A method for computing the Cox ring
  without a variable given is not implemented. You will get an error.
  
  3.3-3 ListOfVariablesOfCoxRing
  
  ListOfVariablesOfCoxRing( vari )  attribute
  Returns:  a list
  
  Returns a list of the variables of the cox ring of the variety vari.
  
  3.3-4 ClassGroup
  
  ClassGroup( vari )  attribute
  Returns:  a module
  
  Returns the class group of the variety vari as factor of a free module.
  
  3.3-5 PicardGroup
  
  PicardGroup( vari )  attribute
  Returns:  a module
  
  Returns the Picard group of the variety vari as factor of a free module.
  
  3.3-6 TorusInvariantDivisorGroup
  
  TorusInvariantDivisorGroup( vari )  attribute
  Returns:  a module
  
  Returns the subgroup of the Weil divisor group of the variety vari generated
  by  the  torus invariant prime divisors. This is always a finitely generated
  free module over the integers.
  
  3.3-7 MapFromCharacterToPrincipalDivisor
  
  MapFromCharacterToPrincipalDivisor( vari )  attribute
  Returns:  a morphism
  
  Returns  a  map  which maps an element of the character group into the torus
  invariant  Weil  group  of  the  variety  vari. This has to viewn as an help
  method to compute divisor classes.
  
  3.3-8 Dimension
  
  Dimension( vari )  attribute
  Returns:  an integer
  
  Returns the dimension of the variety vari.
  
  3.3-9 DimensionOfTorusfactor
  
  DimensionOfTorusfactor( vari )  attribute
  Returns:  an integer
  
  Returns the dimension of the torus factor of the variety vari.
  
  3.3-10 CoordinateRingOfTorus
  
  CoordinateRingOfTorus( vari )  attribute
  Returns:  a ring
  
  Returns the coordinate ring of the torus of the variety vari. This method is
  not implemented, you need to call it with a second argument, which is a list
  of strings for the variables of the ring.
  
  3.3-11 IsProductOf
  
  IsProductOf( vari )  attribute
  Returns:  a list
  
  If  the  variety  vari is a product of 2 or more varieties, the list contain
  those  varieties.  If  it  is  not  a product or at least not generated as a
  product, the list only contains the variety itself.
  
  3.3-12 CharacterLattice
  
  CharacterLattice( vari )  attribute
  Returns:  a module
  
  The  method  returns  the character lattice of the variety vari, computed as
  the containing grid of the underlying convex object, if it exists.
  
  3.3-13 TorusInvariantPrimeDivisors
  
  TorusInvariantPrimeDivisors( vari )  attribute
  Returns:  a list
  
  The  method  returns  a  list  of  the torus invariant prime divisors of the
  variety vari.
  
  3.3-14 IrrelevantIdeal
  
  IrrelevantIdeal( vari )  attribute
  Returns:  an ideal
  
  Returns the irrelevant ideal of the cox ring of the variety vari.
  
  3.3-15 MorphismFromCoxVariety
  
  MorphismFromCoxVariety( vari )  attribute
  Returns:  a morphism
  
  The method returns the quotient morphism from the variety of the Cox ring to
  the variety vari.
  
  3.3-16 CoxVariety
  
  CoxVariety( vari )  attribute
  Returns:  a variety
  
  The method returns the Cox variety of the variety vari.
  
  3.3-17 FanOfVariety
  
  FanOfVariety( vari )  attribute
  Returns:  a fan
  
  Returns the fan of the variety vari. This is set by default.
  
  3.3-18 CartierTorusInvariantDivisorGroup
  
  CartierTorusInvariantDivisorGroup( vari )  attribute
  Returns:  a module
  
  Returns  the the group of Cartier divisors of the variety vari as a subgroup
  of the divisor group.
  
  3.3-19 NameOfVariety
  
  NameOfVariety( vari )  attribute
  Returns:  a string
  
  Returns the name of the variety vari if it has one and it is known or can be
  computed.
  
  3.3-20 twitter
  
  twitter( vari )  attribute
  Returns:  a ring
  
  This  is  a dummy to get immediate methods triggered at some times. It never
  has a value.
  
  
  3.4 Toric varieties: Methods
  
  3.4-1 UnderlyingSheaf
  
  UnderlyingSheaf( vari )  operation
  Returns:  a sheaf
  
  The method returns the underlying sheaf of the variety vari.
  
  3.4-2 CoordinateRingOfTorus
  
  CoordinateRingOfTorus( vari, vars )  operation
  Returns:  a ring
  
  Computes  the  coordinate  ring  of  the  torus of the variety vari with the
  variables  vars.  The argument vars need to be a list of strings with length
  dimension or two times dimension.
  
  3.4-3 \*
  
  \*( vari1, vari2 )  operation
  Returns:  a variety
  
  Computes the categorial product of the varieties vari1 and vari2.
  
  3.4-4 CharacterToRationalFunction
  
  CharacterToRationalFunction( elem, vari )  operation
  Returns:  a homalg element
  
  Computes  the  rational function corresponding to the character grid element
  elem  or  to  the  list  of integers elem. To compute rational functions you
  first need to compute to coordinate ring of the torus of the variety vari.
  
  3.4-5 CoxRing
  
  CoxRing( vari, vars )  operation
  Returns:  a ring
  
  Computes  the  Cox  ring  of  the  variety  vari.  vars needs to be a string
  containing one variable, which will be numbered by the method.
  
  3.4-6 WeilDivisorsOfVariety
  
  WeilDivisorsOfVariety( vari )  operation
  Returns:  a list
  
  Returns a list of the currently defined Divisors of the toric variety.
  
  3.4-7 Fan
  
  Fan( vari )  operation
  Returns:  a fan
  
  Returns the fan of the variety vari. This is a rename for FanOfVariety.
  
  
  3.5 Toric varieties: Constructors
  
  3.5-1 ToricVariety
  
  ToricVariety( conv )  operation
  Returns:  a ring
  
  Creates a toric variety out of the convex object conv.
  
  
  3.6 Toric varieties: Examples
  
  
  3.6-1 The Hirzebruch surface of index 5
  
    Example  
    gap> H5 := Fan( [[-1,5],[0,1],[1,0],[0,-1]],[[1,2],[2,3],[3,4],[4,1]] );
    <A fan in |R^2>
    gap> H5 := ToricVariety( H5 );
    <A toric variety of dimension 2>
    gap> IsComplete( H5 );
    true
    gap> IsAffine( H5 );
    false
    gap> IsOrbifold( H5 );
    true
    gap> IsProjective( H5 );
    true
    gap> TorusInvariantPrimeDivisors(H5);
    [ <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> P := TorusInvariantPrimeDivisors(H5);
    [ <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> A := P[ 1 ] - P[ 2 ] + 4*P[ 3 ];
    <A divisor of a toric variety with coordinates [ 1, -1, 4, 0 ]>
    gap> A;
    <A divisor of a toric variety with coordinates [ 1, -1, 4, 0 ]>
    gap> IsAmple(A);
    false
    gap> CoordinateRingOfTorus(H5,"x");;
    Q[x1,x1_,x2,x2_]/( x2*x2_-1, x1*x1_-1 )
    gap> D:=CreateDivisor([0,0,0,0],H5);
    <A divisor of a toric variety with coordinates 0>
    gap> BasisOfGlobalSections(D);
    [ |[ 1 ]| ]
    gap> D:=Sum(P);
    <A divisor of a toric variety with coordinates [ 1, 1, 1, 1 ]>
    gap> BasisOfGlobalSections(D);
    [ |[ x1_ ]|, |[ x1_*x2 ]|, |[ 1 ]|, |[ x2 ]|,
      |[ x1 ]|, |[ x1*x2 ]|, |[ x1^2*x2 ]|, 
      |[ x1^3*x2 ]|, |[ x1^4*x2 ]|, |[ x1^5*x2 ]|, 
      |[ x1^6*x2 ]| ]
    gap> DivisorOfCharacter([1,2],H5);
    <A principal divisor of a toric variety with coordinates [ 9, 2, 1, -2 ]>
    gap> BasisOfGlobalSections(last);
    [ |[ x1_*x2_^2 ]| ]