The class group of a variety is the group of Weil divisors divided by the subgroup of principal divisors. 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.
The following examples illustrate various possible class groups.
i1 : cl projectiveSpace 2
1
o1 = ZZ
o1 : ZZ-module, free
|
i2 : cl hirzebruchSurface 7
2
o2 = ZZ
o2 : ZZ-module, free
|
i3 : AA3 = normalToricVariety({{1,0,0},{0,1,0},{0,0,1}},{{0,1,2}});
|
i4 : cl AA3
o4 = 0
o4 : ZZ-module
|
i5 : X = normalToricVariety({{4,-1},{0,1}},{{0,1}});
|
i6 : cl X
o6 = cokernel | 4 |
1
o6 : ZZ-module, quotient of ZZ
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i7 : C = normalToricVariety({{1,0,0},{0,1,0},{0,0,1},{1,1,-1}},{{0,1,2,3}});
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i8 : cl C
1
o8 = ZZ
o8 : ZZ-module, free
|
The total coordinate ring of a toric variety is graded by its class group.