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

## Synopsis

• Function: weilDivisorGroup
• Usage:
weilDivisorGroup X
• Inputs:
• X, ,
• Outputs:
• , 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.