# NormalToricVariety ^** ZZ -- make the Cartesian power of a normal toric variety

## Synopsis

• Operator: ^**
• Usage:
X ^** i
• Inputs:
• Outputs:
• , the i-ary Cartesian product of X with itself

## Description

The $i$-ary Cartesian product of the variety $X$, defined over the ground field $k$, is the $i$-ary fiber product of $X$ with itself over $k$. For a normal toric variety, the fan of the $i$-ary Cartesian product is given by the $i$-ary Cartesian product of the cones.

 i1 : PP2 = toricProjectiveSpace 2; i2 : X = PP2 ^** 4; i3 : fromWDivToCl X o3 = | 1 1 1 0 0 0 0 0 0 0 0 0 | | 0 0 0 1 1 1 0 0 0 0 0 0 | | 0 0 0 0 0 0 1 1 1 0 0 0 | | 0 0 0 0 0 0 0 0 0 1 1 1 | 4 12 o3 : Matrix ZZ <--- ZZ

The factors are cached and can be accessed with components.

 i4 : factors = components X o4 = {PP2, PP2, PP2, PP2} o4 : List i5 : assert (# factors === 4) i6 : assert all (4, i -> factors#i === PP2)
 i7 : FF2 = hirzebruchSurface (2); i8 : Y = FF2 ^** 3; i9 : fromWDivToCl Y o9 = | 1 -2 1 0 0 0 0 0 0 0 0 0 | | 0 1 0 1 0 0 0 0 0 0 0 0 | | 0 0 0 0 1 -2 1 0 0 0 0 0 | | 0 0 0 0 0 1 0 1 0 0 0 0 | | 0 0 0 0 0 0 0 0 1 -2 1 0 | | 0 0 0 0 0 0 0 0 0 1 0 1 | 6 12 o9 : Matrix ZZ <--- ZZ
 i10 : X' = PP2 ** PP2; i11 : X'' = PP2 ^** 2; i12 : assert (rays X' == rays X'' and max X' == max X'')