Given a toric map $f : X \to Y$ where $Y$ a smooth toric variety, this method returns the induced map of abelian groups from the group of torus-invariant Weil divisors on $Y$ to the group of torus-invariant Weil divisors on $X$. For arbitrary normal toric varieties, the weilDivisorGroup is not a functor. However, weilDivisorGroup is a contravariant functor on the category of smooth normal toric varieties.
We illustrate this method on the projection from the first Hirzebruch surface to the projective line.
i1 : X = hirzebruchSurface 1; |
i2 : Y = toricProjectiveSpace 1; |
i3 : f = map(Y, X, matrix {{1, 0}})
o3 = | 1 0 |
o3 : ToricMap Y <--- X
|
i4 : f' = weilDivisorGroup f
o4 = | 0 1 |
| 0 0 |
| 1 0 |
| 0 0 |
4 2
o4 : Matrix ZZ <--- ZZ
|
i5 : assert (isWellDefined f and source f' == weilDivisorGroup Y and
target f' == weilDivisorGroup X)
|
The next example gives the induced map from the group of torus-invariant Weil divisors on the projective plane to the group of torus-invariant Weil divisors on the first Hirzebruch surface.
i6 : Z = toricProjectiveSpace 2; |
i7 : g = map(Z, X, matrix {{1, 0}, {0, -1}})
o7 = | 1 0 |
| 0 -1 |
o7 : ToricMap Z <--- X
|
i8 : g' = weilDivisorGroup g
o8 = | 0 1 0 |
| 1 1 0 |
| 1 0 0 |
| 0 0 1 |
4 3
o8 : Matrix ZZ <--- ZZ
|
i9 : assert (isWellDefined g and source g' == weilDivisorGroup Z and
target g' == weilDivisorGroup X)
|
The induced map between the groups of torus-invariant Weil divisors is compatible with the induced map between the class groups.
i10 : g'' = classGroup g
o10 = | 0 |
| 1 |
2 1
o10 : Matrix ZZ <--- ZZ
|
i11 : assert(g'' * fromWDivToCl Z == fromWDivToCl X * g') |