Given a toric map $f : X \to Y$, this method returns the induced map of abelian groups from the Picard group of $Y$ to the Picard group of $X$. In other words, picardGroup is a contravariant functor on the category of 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' = picardGroup f o4 = | 1 | | 0 | 2 1 o4 : Matrix ZZ <--- ZZ |
i5 : assert (isWellDefined f and source f' == picardGroup Y and target f' == picardGroup X) |
The induced map between the Picard groups is compatible with the induced map between the groups of torus-invariant Cartier divisors.
i6 : f'' = cartierDivisorGroup f o6 = | 0 1 | | 0 0 | | 1 0 | | 0 0 | 4 2 o6 : Matrix ZZ <--- ZZ |
i7 : assert(f' * fromCDivToPic Y == fromCDivToPic X * f'') |
Neither the source nor the target of the toric map needs to be smooth.
i8 : W = weightedProjectiveSpace {1, 1, 2}; |
i9 : Z = toricBlowup({0, 1, 4}, (W ** toricProjectiveSpace 1), {0, -2, 1}); |
i10 : assert (not isSmooth W and not isSmooth Z) |
i11 : g = map(W, Z, matrix{{1,0,0},{0,1,0}}) o11 = | 1 0 0 | | 0 1 0 | o11 : ToricMap W <--- Z |
i12 : g' = picardGroup g o12 = | 0 | | -1 | | 0 | 3 1 o12 : Matrix ZZ <--- ZZ |
i13 : assert (isWellDefined g and source g' == picardGroup W and target g' == picardGroup Z) |
i14 : g'' = cartierDivisorGroup g o14 = | 0 0 0 | | 0 0 0 | | 1 0 0 | | 0 -1 0 | | 0 1 1 | | 0 0 0 | 6 3 o14 : Matrix ZZ <--- ZZ |
i15 : assert(g' * fromCDivToPic W == fromCDivToPic Z * g'') |