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# cartesianProduct(Sequence) -- make the Cartesian product of normal toric varieties

- Function: cartesianProduct
- Usage:
    `cartesianProduct Seq`
- Inputs:
  - Seq, a sequence, all of whose elements are normal toric varieties; a single normal toric variety is treated as a sequence with just one element
- Outputs:
  - a normal toric variety, the product of elements

## Description

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.

## See also

- making normal toric varieties -- information about the basic constructors
- NormalToricVariety ** NormalToricVariety -- make the Cartesian product of two normal toric varieties
- NormalToricVariety ^** ZZ -- make the Cartesian power of a normal toric variety
- normalToricVariety -- make a normal toric variety

## Ways to use this method:

- `cartesianProduct(NormalToricVariety)`
- cartesianProduct(Sequence) -- make the Cartesian product of normal toric varieties

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NormalToricVarieties/ToricVarietiesDocumentation.m2:1004:0.