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.
PP2 = projectiveSpace 2; |
A = fromWDivToCl PP2 |
source A == wDiv PP2 |
target A == cl PP2 |
X = weightedProjectiveSpace {1,2,2,3,4}; |
fromWDivToCl X |
FF7 = hirzebruchSurface 7; |
A' = fromWDivToCl FF7 |
(source A', target A') == (wDiv FF7, cl FF7) |
U = normalToricVariety({{4,-1},{0,1}},{{0,1}}); |
fromWDivToCl U |
wDiv U |
cl U |
This matrix also induces the grading on the total coordinate ring of toric variety.
degrees ring PP2 |
degrees ring X |
degrees ring FF7 |
The optional argument
WeilToClass for the constructor
normalToricVariety allows one to specify a basis of the class group.