classGroup(NormalToricVariety) -- make the class group

Synopsis

Function: classGroup
Usage:

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

Description

The class group of a variety is the group of Weil divisors divided by the subgroup of principal divisors. For a normal toric variety, the class group has a presentation defined by the map from the group of torus-characters to group of torus-invariant Weil divisors induced by minimal nonzero lattice points on the rays of the associated fan. For more information, see Theorem 4.1.3 in Cox-Little-Schenck's Toric Varieties.

The following examples illustrate some possible class groups.

i1 : classGroup toricProjectiveSpace 1

       1
o1 = ZZ

o1 : ZZ-module, free

i2 : classGroup hirzebruchSurface 7

       2
o2 = ZZ

o2 : ZZ-module, free

i3 : classGroup affineSpace 3

o3 = 0

o3 : ZZ-module

i4 : classGroup normalToricVariety ({{4,-1},{0,1}},{{0,1}})

o4 = cokernel | 4 |

                              1
o4 : ZZ-module, quotient of ZZ

i5 : classGroup normalToricVariety ( id_(ZZ^3) | - id_(ZZ^3))

o5 = cokernel | 2 0 |
              | 0 2 |
              | 0 0 |
              | 0 0 |
              | 0 0 |
              | 0 0 |
              | 0 0 |

                              7
o5 : ZZ-module, quotient of ZZ

The total coordinate ring of a toric variety is graded by its class group.

i6 : degrees ring toricProjectiveSpace 1

o6 = {{1}, {1}}

o6 : List

i7 : degrees ring hirzebruchSurface 7

o7 = {{1, 0}, {-7, 1}, {1, 0}, {0, 1}}

o7 : List

i8 : degrees ring affineSpace 3

o8 = {{}, {}, {}}

o8 : List

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

Ways to use this method: