The Picard group of a normal toric variety is a subgroup of the class group.
On a smooth normal toric variety, the Picard group is isomorphic to the class group, so the inclusion map is the identity.
PP3 = projectiveSpace 3; |
pic PP3 |
cl PP3 |
fromPicToCl PP3 |
FF7 = hirzebruchSurface 7; |
pic FF7 == cl FF7 |
fromPicToCl FF7 |
For weighted projective space, the inclusion corresponds to
l ℤ in
ℤ, where
l = lcm(q0,…, qd ).
X = weightedProjectiveSpace {1,2,3}; |
pic X |
cl X |
fromPicToCl X |
Y = weightedProjectiveSpace {1,2,2,3,4}; |
pic Y |
cl Y |
fromPicToCl Y |
The following examples illustrate some other possibilities.
C = normalToricVariety({{1,0,0},{0,1,0},{0,0,1},{1,1,-1}},{{0,1,2,3}}); |
pic C |
cl C |
fromPicToCl C |
X = normalToricVariety(id_(ZZ^3) | - id_(ZZ^3)); |
rays X |
max X |
pic X |
cl X |
fromPicToCl X |
prune cokernel fromPicToCl X |