The Picard group of a variety is the group of Cartier divisors divided by the subgroup of principal divisors. For a normal toric variety, the Picard group has a presentation defined by the map from the group of torus-characters to the group of torus-invariant Cartier divisors.
When the normal toric variety is smooth, the Picard group is isomorphic to the class group.
i1 : PP3 = projectiveSpace 3; |
i2 : assert(isSmooth PP3 and isProjective PP3) |
i3 : picardGroup PP3 1 o3 = ZZ o3 : ZZ-module, free |
i4 : assert(picardGroup PP3 === classGroup PP3 and isFreeModule picardGroup PP3) |
i5 : X = smoothFanoToricVariety(4,90); |
i6 : assert(isSmooth X and isProjective X and isFano X) |
i7 : picardGroup X 5 o7 = ZZ o7 : ZZ-module, free |
i8 : assert(fromCDivToPic X === fromWDivToCl X and isFreeModule picardGroup X) |
i9 : U = normalToricVariety({{4,-1},{0,1}},{{0},{1}}); |
i10 : assert(isSmooth U and not isComplete U and # max U =!= 1) |
i11 : picardGroup U o11 = cokernel | 4 | 1 o11 : ZZ-module, quotient of ZZ |
i12 : assert(classGroup U === picardGroup U and not isFreeModule picardGroup U) |
For an affine toric variety, the Picard group is trivial.
i13 : AA3 = affineSpace 3 o13 = AA3 o13 : NormalToricVariety |
i14 : assert(isSimplicial AA3 and isSmooth AA3 and # max AA3 === 1) |
i15 : picardGroup AA3 o15 = 0 o15 : ZZ-module |
i16 : assert(picardGroup AA3 == 0 and isFreeModule picardGroup AA3) |
i17 : Q = normalToricVariety({{1,0,0},{0,1,0},{0,0,1},{1,1,-1}},{{0,1,2,3}}); |
i18 : assert(not isSimplicial Q and not isComplete Q and # max Q === 1) |
i19 : picardGroup Q o19 = 0 o19 : ZZ-module |
i20 : assert(picardGroup Q == 0 and isFreeModule picardGroup Q) |
If the fan associated to X contains a cone of dimension dim(X), then the Picard group is free.
i21 : Y = normalToricVariety(id_(ZZ^3) | -id_(ZZ^3)); |
i22 : assert(not isSimplicial Y and isProjective Y) |
i23 : picardGroup Y 1 o23 = ZZ o23 : ZZ-module, free |
i24 : assert(rank picardGroup Y === 1 and isFreeModule picardGroup Y) |