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;
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i2 : A = fromWDivToCl PP2
o2 = | 1 1 1 |
1 3
o2 : Matrix ZZ <--- ZZ
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i3 : source A == wDiv PP2
o3 = true
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i4 : target A == cl PP2
o4 = true
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i5 : X = weightedProjectiveSpace {1,2,2,3,4};
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i6 : fromWDivToCl X
o6 = | 1 2 2 3 4 |
1 5
o6 : Matrix ZZ <--- ZZ
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i7 : FF7 = hirzebruchSurface 7;
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i8 : A' = fromWDivToCl FF7
o8 = | 1 -7 1 0 |
| 0 1 0 1 |
2 4
o8 : Matrix ZZ <--- ZZ
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i9 : (source A', target A') == (wDiv FF7, cl FF7)
o9 = true
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i10 : U = normalToricVariety({{4,-1},{0,1}},{{0,1}});
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i11 : fromWDivToCl U
o11 = | 1 1 |
o11 : Matrix
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i12 : wDiv U
2
o12 = ZZ
o12 : ZZ-module, free
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i13 : cl U
o13 = cokernel | 4 |
1
o13 : ZZ-module, quotient of ZZ
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This matrix also induces the grading on the total coordinate ring of toric variety.
i14 : degrees ring PP2
o14 = {{1}, {1}, {1}}
o14 : List
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i15 : degrees ring X
o15 = {{1}, {2}, {2}, {3}, {4}}
o15 : List
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i16 : degrees ring FF7
o16 = {{1, 0}, {-7, 1}, {1, 0}, {0, 1}}
o16 : List
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The optional argument
WeilToClass for the constructor
normalToricVariety allows one to specify a basis of the class group.