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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

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.

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

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).

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 ] ]

4.1-3 RiemannRochBasis

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.

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> RiemannRochBasis(Div,Delta0,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

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.

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

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.

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

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.

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

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