cartierDivisorGroup(NormalToricVariety) -- compute the group of torus-invariant Cartier divisors

Synopsis

Function: cartierDivisorGroup
Usage:

cartierDivisorGroup X
Inputs:
- X, a normal toric variety,
Outputs:
- a module, a finitely generated abelian group

Description

The group of torus-invariant Cartier divisors on $X$ is the subgroup of all locally principal torus-invariant 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 $sigma_0, sigma_1, \dots, sigma_{r-1}$, then we have $CDiv(X) = ker( \oplus_{i} M/M(sigma_i{}) \to{} \oplus_{i<j} M/M(sigma_i \cap sigma_j{})$. For more information, see Theorem 4.2.8 in Cox-Little-Schenck's Toric Varieties.

When $X$ is smooth, every torus-invariant Weil divisor is Cartier.

i1 : PP2 = toricProjectiveSpace 2;

i2 : cartierDivisorGroup PP2

       3
o2 = ZZ

o2 : ZZ-module, free

i3 : assert (isSmooth PP2 and weilDivisorGroup PP2 === cartierDivisorGroup PP2)

i4 : assert (id_(cartierDivisorGroup PP2) == fromCDivToWDiv PP2)

i5 : FF7 = hirzebruchSurface 7;

i6 : cartierDivisorGroup FF7

       4
o6 = ZZ

o6 : ZZ-module, free

i7 : assert (isSmooth FF7 and weilDivisorGroup FF7 === cartierDivisorGroup FF7)

i8 : assert (id_(cartierDivisorGroup FF7) == fromCDivToWDiv FF7)

On a simplicial toric variety, every torus-invariant Weil divisor is $\QQ$-Cartier; every torus-invariant Weil divisor has a positive integer multiple that is Cartier.

i9 : U = normalToricVariety ({{4,-1},{0,1}},{{0,1}});

i10 : assert (isSimplicial U and not isSmooth U and not isComplete U)

i11 : cartierDivisorGroup U

        2
o11 = ZZ

o11 : ZZ-module, free

i12 : weilDivisorGroup U

        2
o12 = ZZ

o12 : ZZ-module, free

i13 : prune coker fromCDivToWDiv U

o13 = cokernel | 4 |

                               1
o13 : ZZ-module, quotient of ZZ

i14 : assert ( (coker fromCDivToWDiv U) ** QQ == 0)

i15 : X = weightedProjectiveSpace {1,2,2,3,4};

i16 : assert (isSimplicial X and not isSmooth X and isComplete X)

i17 : cartierDivisorGroup X

        5
o17 = ZZ

o17 : ZZ-module, free

i18 : weilDivisorGroup X

        5
o18 = ZZ

o18 : ZZ-module, free

i19 : prune coker fromCDivToWDiv X

o19 = cokernel | 12 |

                               1
o19 : ZZ-module, quotient of ZZ

i20 : assert (rank coker fromCDivToWDiv X === 0)

In general, the Cartier divisors are only a subgroup of the Weil divisors.

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)

i23 : cartierDivisorGroup Q

        3
o23 = ZZ

o23 : ZZ-module, free

i24 : weilDivisorGroup Q

        4
o24 = ZZ

o24 : ZZ-module, free

i25 : prune coker fromCDivToWDiv Q

        1
o25 = ZZ

o25 : ZZ-module, free

i26 : assert (rank coker fromCDivToWDiv Q === 1)

i27 : Y = normalToricVariety (id_(ZZ^3) | -id_(ZZ^3));

i28 : assert (not isSimplicial Y and isComplete Y)

i29 : cartierDivisorGroup Y

        4
o29 = ZZ

o29 : ZZ-module, free

i30 : weilDivisorGroup Y

        8
o30 = ZZ

o30 : ZZ-module, free

i31 : prune cokernel fromCDivToWDiv Y

o31 = cokernel | 2 0 0 |
               | 0 2 0 |
               | 0 0 2 |
               | 0 0 0 |
               | 0 0 0 |
               | 0 0 0 |
               | 0 0 0 |

                               7
o31 : ZZ-module, quotient of ZZ

i32 : assert (rank coker fromCDivToWDiv Y === 4)

To avoid duplicate computations, the attribute is cached in the normal toric variety.

Ways to use this method: