The group of torus-invariant Weil divisors on a normal toric variety is the free abelian group generated by the torus-invariant irreducible divisors. The irreducible divisors correspond bijectively to rays in the associated fan. Since the rays are indexed in this package by 0, 1, …, n-1 the group of torus-invariant Weil divisors is canonically isomorphic to ℤn.
The examples illustrate various possible Weil groups.
i1 : PP2 = projectiveSpace 2; |
i2 : # rays PP2 o2 = 3 |
i3 : weilDivisorGroup PP2
3
o3 = ZZ
o3 : ZZ-module, free
|
i4 : FF7 = hirzebruchSurface 7; |
i5 : # rays FF7 o5 = 4 |
i6 : weilDivisorGroup FF7
4
o6 = ZZ
o6 : ZZ-module, free
|
i7 : U = normalToricVariety({{4,-1},{0,1}},{{0,1}});
|
i8 : # rays U o8 = 2 |
i9 : weilDivisorGroup U
2
o9 = ZZ
o9 : ZZ-module, free
|