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NormalToricVarieties :: NormalToricVariety ^** ZZ

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

Synopsis

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'')

See also