next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
NormalToricVarieties :: cDiv

cDiv -- make the group of torus-invariant Cartier divisors

Synopsis

Description

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 -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

See also

Ways to use cDiv :