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. For more information, see Theorem 4.1.3 in Cox-Little-Schenck's Toric Varieties.
The following examples illustrate some possible class groups.
i1 : classGroup toricProjectiveSpace 1
1
o1 = ZZ
o1 : ZZ-module, free
|
i2 : classGroup hirzebruchSurface 7
2
o2 = ZZ
o2 : ZZ-module, free
|
i3 : classGroup affineSpace 3 o3 = 0 o3 : ZZ-module |
i4 : classGroup normalToricVariety ({{4,-1},{0,1}},{{0,1}})
o4 = cokernel | 4 |
1
o4 : ZZ-module, quotient of ZZ
|
i5 : classGroup normalToricVariety ( id_(ZZ^3) | - id_(ZZ^3))
o5 = cokernel | 2 0 |
| 0 2 |
| 0 0 |
| 0 0 |
| 0 0 |
| 0 0 |
| 0 0 |
7
o5 : ZZ-module, quotient of ZZ
|
The total coordinate ring of a toric variety is graded by its class group.
i6 : degrees ring toricProjectiveSpace 1
o6 = {{1}, {1}}
o6 : List
|
i7 : degrees ring hirzebruchSurface 7
o7 = {{1, 0}, {-7, 1}, {1, 0}, {0, 1}}
o7 : List
|
i8 : degrees ring affineSpace 3
o8 = {{}, {}, {}}
o8 : List
|
To avoid duplicate computations, the attribute is cached in the normal toric variety.