The Cartesian product of varieties $X_0, X_1, X_2, \dots$, all defined over the same ground field $k$, is the fiber product $X_0 \times_k X_1 \times_k X_2 \times_k \dots$. For normal toric varieties, the fan of the product is given by the Cartesian product of the underlying fans of the factors.
i1 : X = toricProjectiveSpace 1; |
i2 : Y = toricProjectiveSpace 2; |
i3 : Z = toricProjectiveSpace 3; |
i4 : Seq = (X, Y, Z); |
i5 : P = cartesianProduct Seq; |
i6 : dim P o6 = 6 |
i7 : assert (dim P == 1+2+3) |
i8 : factors = components P
o8 = {X, Y, Z}
o8 : List
|
i9 : # factors o9 = 3 |
i10 : assert (factors#0 === X and factors#1 === Y and factors#2 === Z) |
This general method for constructing products is invoked by all of the other product constructors.