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. For more information, see Theorem 4.1.3 in Cox-Little-Schenck's Toric Varieties.
The examples illustrate some of the possible maps from the group of torus-invariant Weil divisors to the class group.
i1 : PP2 = toricProjectiveSpace 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) |
The optional argument WeilToClass for the constructor normalToricVariety allows one to specify a basis of the class group.
This map is computed and cached when the class group is first constructed.