The Picard group of a normal toric variety is a subgroup of the class group. This function returns a matrix representing this map with respect to the chosen bases.
On a smooth normal toric variety, the Picard group is isomorphic to the class group, so the inclusion map is the identity.
i1 : PP3 = projectiveSpace 3; |
i2 : assert(isSmooth PP3 and isProjective PP3) |
i3 : fromPicToCl PP3 o3 = | 1 | 1 1 o3 : Matrix ZZ <--- ZZ |
i4 : assert(fromPicToCl PP3 === id_(classGroup PP3)) |
i5 : X = smoothFanoToricVariety(4,90); |
i6 : assert(isSmooth X and isProjective X and isFano X) |
i7 : fromPicToCl X o7 = | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 1 0 0 | | 0 0 0 1 0 | | 0 0 0 0 1 | 5 5 o7 : Matrix ZZ <--- ZZ |
i8 : assert(fromPicToCl X === id_(classGroup X)) |
i9 : U = normalToricVariety({{4,-1},{0,1}},{{0},{1}}); |
i10 : assert(isSmooth U and not isComplete U and # max U =!= 1) |
i11 : fromPicToCl U o11 = | 1 | o11 : Matrix |
i12 : assert(fromPicToCl U === id_(classGroup U)) |
For weighted projective space, the inclusion corresponds to l ℤ in ℤ where l = lcm(q0, q1, …, qd ).
i13 : P123 = weightedProjectiveSpace {1,2,3}; |
i14 : assert(isSimplicial P123 and isProjective P123) |
i15 : fromPicToCl P123 o15 = | 6 | 1 1 o15 : Matrix ZZ <--- ZZ |
i16 : assert(fromPicToCl P123 === lcm(1,2,3) * id_(classGroup P123)) |
i17 : P12234 = weightedProjectiveSpace {1,2,2,3,4}; |
i18 : assert(isSimplicial P12234 and isProjective P12234) |
i19 : fromPicToCl P12234 o19 = | 12 | 1 1 o19 : Matrix ZZ <--- ZZ |
i20 : assert(fromPicToCl P12234 === lcm(1,2,2,3,4) * id_(classGroup P12234)) |
The following examples illustrate some other possibilities.
i21 : Q = normalToricVariety({{1,0,0},{0,1,0},{0,0,1},{1,1,-1}},{{0,1,2,3}}); |
i22 : assert(not isSimplicial Q and not isComplete Q and # max Q === 1) |
i23 : fromPicToCl Q o23 = 0 1 o23 : Matrix ZZ <--- 0 |
i24 : assert(fromPicToCl Q == 0) |
i25 : Y = normalToricVariety(id_(ZZ^3) | -id_(ZZ^3)); |
i26 : assert(not isSimplicial Y and isProjective Y) |
i27 : fromPicToCl Y o27 = | 0 | | 0 | | 0 | | 2 | | 2 | | 2 | | 2 | o27 : Matrix |