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NormalToricVarieties :: weilDivisorGroup(NormalToricVariety)

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

Synopsis

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