The Cartesian product of two varieties $X$ and $Y$, both defined over the same ground field $k$, is the fiber product $X \times_k Y$. For normal toric varieties, the fan of the product is given by the Cartesian product of each pair of cones in the fans of the factors.
i1 : PP2 = toricProjectiveSpace 2; |
i2 : FF2 = hirzebruchSurface 2; |
i3 : X = FF2 ** PP2; |
i4 : assert (# rays X == # rays FF2 + # rays PP2) |
i5 : assert (matrix rays X == matrix rays FF2 ++ matrix rays PP2) |
i6 : primaryDecomposition ideal X o6 = {ideal (x , x ), ideal (x , x ), ideal (x , x , x )} 2 0 1 3 6 5 4 o6 : List |
i7 : flatten (primaryDecomposition \ {ideal FF2,ideal PP2}) o7 = {ideal (x , x ), ideal (x , x ), ideal (x , x , x )} 2 0 1 3 2 1 0 o7 : List |
The map from the torus-invariant Weil divisors to the class group is the direct sum of the maps for the factors.
i8 : assert (fromWDivToCl FF2 ++ fromWDivToCl PP2 == fromWDivToCl X) |
The factors are cached and can be accessed with components.
i9 : factors = components X o9 = {FF2, PP2} o9 : List |
i10 : assert (# factors === 2) |
i11 : assert (factors#0 === FF2) |
i12 : assert (factors#1 === PP2) |