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# weilDivisorGroup(NormalToricVariety) -- make the group of torus-invariant Weil divisors

- Function: weilDivisorGroup
- Usage:
    `weilDivisorGroup X`
- Inputs:
  - X, a normal toric variety,
- Outputs:
  - a module, a finitely generated free abelian group

## Description

The group of torus-invariant Weil divisors on a normal toric variety is the free abelian group generated by the torus-invariant irreducible divisors. The irreducible divisors correspond bijectively to rays in the associated fan. Since the rays are indexed in this package by $0, 1, \dots, n-1$ the group of torus-invariant Weil divisors is canonically isomorphic to $\ZZ^n$. For more information, see Theorem 4.1.3 in Cox-Little-Schenck's Toric Varieties.

The examples illustrate various possible Weil groups.

    i1 : PP2 = toricProjectiveSpace 2;

    i2 : # rays PP2

    o2 = 3

    i3 : weilDivisorGroup PP2

           3
    o3 = ZZ

    o3 : ZZ-module, free

    i4 : FF7 = hirzebruchSurface 7;

    i5 : # rays FF7

    o5 = 4

    i6 : weilDivisorGroup FF7

           4
    o6 = ZZ

    o6 : ZZ-module, free

    i7 : U = normalToricVariety ({{4,-1},{0,1}},{{0,1}});

    i8 : # rays U

    o8 = 2

    i9 : weilDivisorGroup U

           2
    o9 = ZZ

    o9 : ZZ-module, free

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

## See also

- working with divisors -- information about toric divisors and their related groups
- NormalToricVariety _ ZZ -- make an irreducible torus-invariant divisor
- fromCDivToWDiv(NormalToricVariety) -- get the map from Cartier divisors to Weil divisors
- fromWDivToCl(NormalToricVariety) -- get the map from the group of Weil divisors to the class group
- ToricDivisor -- the class of all torus-invariant Weil divisors
- weilDivisorGroup(ToricMap) -- make the induced map between groups of Weil divisors

## Ways to use this method:

- weilDivisorGroup(NormalToricVariety) -- make the group of torus-invariant Weil divisors

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NormalToricVarieties/DivisorsDocumentation.m2:240:0.