For a normal toric variety, the class group has a presentation defined by the map from the group of torus-characters to group of torus-invariant Weil divisors induced by minimal nonzero lattice points on the rays of the associated fan. Hence, there is a surjective map from the group of torus-invariant Weil divisors to the class group. This method returns a matrix representing this map. Since the ordering on the rays of the toric variety determines a basis for the group of torus-invariant Weil divisors, this matrix is determined by a choice of basis for the class group.
The examples illustrate some of the possible maps from the group of torus-invariant Weil divisors to the class group.
i1 : PP2 = projectiveSpace 2;
|
i2 : A1 = fromWDivToCl PP2
o2 = | 1 1 1 |
1 3
o2 : Matrix ZZ <--- ZZ
|
i3 : assert( (target A1, source A1) === (classGroup PP2, weilDivisorGroup PP2) )
|
i4 : assert( A1 * matrix rays PP2 == 0)
|
i5 : X = weightedProjectiveSpace {1,2,2,3,4};
|
i6 : A2 = fromWDivToCl X
o6 = | 1 2 2 3 4 |
1 5
o6 : Matrix ZZ <--- ZZ
|
i7 : assert( (target A2, source A2) === (classGroup X, weilDivisorGroup X) )
|
i8 : assert( A2 * matrix rays X == 0)
|
i9 : Y = normalToricVariety( id_(ZZ^3) | - id_(ZZ^3));
|
i10 : A3 = fromWDivToCl Y
o10 = | 1 0 1 0 0 0 0 0 |
| 1 1 0 0 0 0 0 0 |
| 1 -1 -1 1 0 0 0 0 |
| -1 1 1 0 1 0 0 0 |
| 0 0 1 0 0 1 0 0 |
| 0 1 0 0 0 0 1 0 |
| 1 0 0 0 0 0 0 1 |
o10 : Matrix
|
i11 : classGroup Y
o11 = cokernel | 2 0 |
| 0 2 |
| 0 0 |
| 0 0 |
| 0 0 |
| 0 0 |
| 0 0 |
7
o11 : ZZ-module, quotient of ZZ
|
i12 : assert( (target A3, source A3) === (classGroup Y, weilDivisorGroup Y) )
|
i13 : assert( A3 * matrix rays Y == 0)
|
i14 : U = normalToricVariety({{4,-1},{0,1}},{{0,1}});
|
i15 : A4 = fromWDivToCl U
o15 = | 1 1 |
o15 : Matrix
|
i16 : classGroup U
o16 = cokernel | 4 |
1
o16 : ZZ-module, quotient of ZZ
|
i17 : assert( (target A4, source A4) === (classGroup U, weilDivisorGroup U) )
|
i18 : assert( A4 * matrix rays U == 0)
|
This matrix also induces the grading on the total coordinate ring of toric variety.
i19 : assert( transpose matrix degrees ring PP2 === fromWDivToCl PP2)
|
i20 : assert( transpose matrix degrees ring X === fromWDivToCl X)
|