components(NormalToricVariety) -- list the factors in a product

Synopsis

Function: components
Usage:

components X
Inputs:
- X, a normal toric variety,
Outputs:
- a list, of the factors from which the product was form or just \{X\} if X was not formed by a Cartesian product

Description

The Cartesian product of varieties $X_0, X_1, X_2, ...$, all defined over the same ground field $k$, is the fiber product $X_0 \times_k X_1 \times_k X_2 \times_k ...$. 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)

The factors are cached and can be accessed with components.

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)

Ways to use this method: