The group of torus-invariant Cartier divisors on
X is the subgroup of all locally principal torus-invarient Weil divisors. On a normal toric variety, the group of torus-invariant Cartier divisors can be computed as an inverse limit. More precisely, if
M denotes the lattice of characters on
X and the maximal cones in the fan of
X are
s0,…,sr-1 then we have
CDiv(X) = ker( ⊕i M/M(si) → ⊕i<j M/M(si ∩sj).
When
X is smooth, every torus-invariant Weil divisor is Cartier.
i1 : PP2 = projectiveSpace 2;
|
i2 : Div = wDiv PP2
3
o2 = ZZ
o2 : ZZ-module, free
|
i3 : Div == cDiv PP2
o3 = true
|
i4 : id_Div == fromCDivToWDiv PP2
o4 = true
|
i5 : isSmooth PP2
o5 = true
|
i6 : FF1 = hirzebruchSurface 1;
|
i7 : cDiv FF1
4
o7 = ZZ
o7 : ZZ-module, free
|
i8 : isIsomorphism fromCDivToWDiv FF1
o8 = true
|
i9 : isSmooth FF1
o9 = true
|
On a simplicial toric variety, every torus-invariant Weil divisor is
ℚ-Cartier --- every torus-invariant Weil divisor has a positive integer multiple that is Cartier.
i10 : U = normalToricVariety({{4,-1},{0,1}},{{0,1}});
|
i11 : cDiv U
2
o11 = ZZ
o11 : ZZ-module, free
|
i12 : wDiv U
2
o12 = ZZ
o12 : ZZ-module, free
|
i13 : fromCDivToWDiv U
o13 = | 4 -1 |
| 0 1 |
2 2
o13 : Matrix ZZ <--- ZZ
|
i14 : prune cokernel fromCDivToWDiv U
o14 = cokernel | 4 |
1
o14 : ZZ-module, quotient of ZZ
|
i15 : isSimplicial U
o15 = true
|
i16 : U' = normalToricVariety({{4,-1},{0,1}},{{0},{1}});
|
i17 : cDiv U'
2
o17 = ZZ
o17 : ZZ-module, free
|
i18 : wDiv U'
2
o18 = ZZ
o18 : ZZ-module, free
|
i19 : fromCDivToWDiv U'
o19 = | 1 0 |
| 0 1 |
2 2
o19 : Matrix ZZ <--- ZZ
|
i20 : isSmooth U'
o20 = true
|
In general, the Cartier divisors are only a subgroup of the Weil divisors.
i21 : C = normalToricVariety({{1,0,0},{0,1,0},{0,0,1},{1,1,-1}},{{0,1,2,3}});
|
i22 : cDiv C
3
o22 = ZZ
o22 : ZZ-module, free
|
i23 : wDiv C
4
o23 = ZZ
o23 : ZZ-module, free
|
i24 : prune coker fromCDivToWDiv C
1
o24 = ZZ
o24 : ZZ-module, free
|
i25 : isSimplicial C
o25 = false
|
i26 : X = normalToricVariety(id_(ZZ^3) | -id_(ZZ^3));
|
i27 : wDiv X
8
o27 = ZZ
o27 : ZZ-module, free
|
i28 : cDiv X
4
o28 = ZZ
o28 : ZZ-module, free
|
i29 : prune cokernel fromCDivToWDiv X
o29 = cokernel | 2 0 0 |
| 0 2 0 |
| 0 0 2 |
| 0 0 0 |
| 0 0 0 |
| 0 0 0 |
| 0 0 0 |
7
o29 : ZZ-module, quotient of ZZ
|
i30 : isSimplicial X
o30 = false
|