4. Toric varieties X(Delta)
  
  This  chapter  concerns  toric  commands  which  deal  with  certain objects
  associated to the (non-affine) toric varieties X(Delta).
  
  
  4.1 Riemann-Roch spaces
  
  Let Delta denote a complete nonsingular fan.
  
  4.1-1 DivisorPolytope
  
  > DivisorPolytope( D, Rays ) _______________________________________function
  
  Input:  Rays is the list of smallest integer vectors in the rays for the fan
  Delta which determine the Weil divisors of X(Delta).
  D is the list of coefficients for the a Weil divisor.
  Output: the linear expressions in the affine coordinates of the space of the
  cone which must be positive for a point to be in the desired polytope.
  
  ---------------------------  Example  ----------------------------
    gap> DivisorPolytope([6,6,0],[[2,-1],[-1,2],[-1,-1]]);
    [ 2*x_1-x_2+6, -x_1+2*x_2+6, -x_1-x_2 ]
    
  ------------------------------------------------------------------
  
  See also Example 6.13 in [JV02].
  
  4.1-2 DivisorPolytopeLatticePoints
  
  > DivisorPolytopeLatticePoints( D, Delta, Rays ) ___________________function
  
  Input: Delta is the fan
  Rays is the ordered list of rays for Delta
  D is the list of coefficients for a Weil divisor.
  Output:  the list of points in P_D cap L_0^* which parameterize the elements
  in  the Riemann-Roch space L(D), where P_D is the polytope associated to the
  divisor D (see DivisorPolytope).
  
  ---------------------------  Example  ----------------------------
    gap> Div:=[6,6,0];; Rays:=[[2,-1],[-1,2],[-1,-1]];;
    gap> Delta0:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
    gap> P_Div:=DivisorPolytopeLatticePoints(Div,Delta0,Rays);
    [ [ -6, -6 ], [ -5, -5 ], [ -5, -4 ], [ -4, -5 ], [ -4, -4 ], [ -4, -3 ],
      [ -4, -2 ], [ -3, -4 ], [ -3, -3 ], [ -3, -2 ], [ -3, -1 ], [ -3, 0 ],
      [ -2, -4 ], [ -2, -3 ], [ -2, -2 ], [ -2, -1 ], [ -2, 0 ], [ -2, 1 ],
      [ -2, 2 ], [ -1, -3 ], [ -1, -2 ], [ -1, -1 ], [ -1, 0 ], [ -1, 1 ],
      [ 0, -3 ], [ 0, -2 ], [ 0, -1 ], [ 0, 0 ], [ 1, -2 ], [ 1, -1 ], [ 2, -2 ] ]
    gap>
    
  ------------------------------------------------------------------
  
  4.1-3 RiemannRochBasis
  
  > RiemannRochBasis( D, Delta, Rays ) _______________________________function
  
  Input: Delta is a complete and nonsingular fan
  D is the list of coefficients for the Weil divisor
  Rays is a list of rays for the fan used to describe the Weil divisors.
  Output:  A  basis  (a  list  of monomials) for the Riemann-Roch space of the
  divisor represented by D.
  
  For  details  on how the Weil divisors can be expressed in terms of the rays
  of the fan, please see section 3.3 in [Ful93]. This procedure does not check
  if the fan is complete and nonsingular.
  
  ---------------------------  Example  ----------------------------
    gap> Div:=[6,6,0];; Rays:=[[2,-1],[-1,2],[-1,-1]];;
    gap> Delta:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
    gap> RiemannRochBasis(Div,Delta,Rays);
    [ 1/(x_1^6*x_2^6), 1/(x_1^5*x_2^5), 1/(x_1^5*x_2^4), 1/(x_1^4*x_2^5),
      1/(x_1^4*x_2^4), 1/(x_1^4*x_2^3), 1/(x_1^4*x_2^2), 1/(x_1^3*x_2^4),
      1/(x_1^3*x_2^3), 1/(x_1^3*x_2^2), 1/(x_1^3*x_2), 1/x_1^3, 1/(x_1^2*x_2^4),
      1/(x_1^2*x_2^3), 1/(x_1^2*x_2^2), 1/(x_1^2*x_2), 1/x_1^2, x_2/x_1^2,
      x_2^2/x_1^2, 1/(x_1*x_2^3), 1/(x_1*x_2^2), 1/(x_1*x_2), 1/x_1, x_2/x_1,
      1/x_2^3, 1/x_2^2, 1/x_2, 1, x_1/x_2^2, x_1/x_2, x_1^2/x_2^2 ]
    
  ------------------------------------------------------------------
  
  
  4.2 Topological invariants
  
  Throughout this section, X(Delta) must be non-singular.
  
  4.2-1 EulerCharacteristic
  
  > EulerCharacteristic( Delta ) _____________________________________function
  
  Input:  Delta  is  a  nonsingular  fan  of cones, represented by its list of
  maximal cones.
  Output:  the Euler characteristic of the toric variety X(Delta), where Delta
  is a fan determined by Delta.
  
  ---------------------------  Example  ----------------------------
    gap> Cones:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
    gap> EulerCharacteristic(Cones);
    3
    
  ------------------------------------------------------------------
  
  Note: X(Delta) must be non-singular here.
  
  4.2-2 BettiNumberToric
  
  > BettiNumberToric( Delta, k ) _____________________________________function
  
  Input:  Delta  represents  a  nonsingular  fan Delta (represented by maximal
  cones),
  k is an integer.
  Output: the k-th Betti number of the toric variety X(Delta).
  
  The BettiNumberToric procedure does not check if Delta is nonsingular. It is
  possible  that this procedure outputs nonsense when Delta is not represented
  by maximal cones or is nonsingular.
  
  ---------------------------  Example  ----------------------------
    gap> Cones:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
    gap> BettiNumberToric(Cones,1);
    0
    gap> BettiNumberToric(Cones,2);
    1
    gap> Cones:=[[[2,-1],[-1,1]],[[-1,1],[-1,0]],[[-1,0],[2,-1]]];;
    gap> BettiNumberToric(Cones,1);
    0
    gap> BettiNumberToric(Cones,2);
    1
  ------------------------------------------------------------------
  
  Not  to  be  confused  with  the  Betti number of a polycyclically presented
  torsion free group, already available in GAP.
  
  
  4.3 Points over a finite field
  
  4.3-1 CardinalityOfToricVariety
  
  > CardinalityOfToricVariety( Cones, q ) ____________________________function
  
  Input:  Cones  is  the  list  of  maximal cones of a fan Delta, q is a prime
  power.
  Output:  The  size  of the set of GF(q)-rational points of the toric variety
  X(Delta).
  
  Note: X(Delta) must be non-singular here.
  
  ---------------------------  Example  ----------------------------
    gap> Cones:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
    gap> CardinalityOfToricVariety(Cones,3);
    13
    gap> CardinalityOfToricVariety(Cones,4);
    21
    gap> CardinalityOfToricVariety(Cones,5);
    31
    gap> CardinalityOfToricVariety(Cones,7);
    57
    
  ------------------------------------------------------------------