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NormalToricVarieties :: NormalToricVariety ^** ZZ

NormalToricVariety ^** ZZ -- Cartesian power

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 = projectiveSpace 2;
i2 : X = PP2 ^** 4;
i3 : degrees ring X

o3 = {{1, 0, 0, 0}, {1, 0, 0, 0}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 1, 0, 0},
     ------------------------------------------------------------------------
     {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 1, 0}, {0, 0, 1, 0}, {0, 0, 0, 1},
     ------------------------------------------------------------------------
     {0, 0, 0, 1}, {0, 0, 0, 1}}

o3 : List
i4 : fromWDivToCl X

o4 = | 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
o4 : Matrix ZZ  <--- ZZ
i5 : FF2 = hirzebruchSurface(2) ;
i6 : Y = FF2 ^** 3;
i7 : degrees ring Y

o7 = {{1, 0, 0, 0, 0, 0}, {-2, 1, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0}, {0, 1, 0,
     ------------------------------------------------------------------------
     0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, -2, 1, 0, 0}, {0, 0, 1, 0, 0, 0},
     ------------------------------------------------------------------------
     {0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, -2, 1}, {0, 0, 0,
     ------------------------------------------------------------------------
     0, 1, 0}, {0, 0, 0, 0, 0, 1}}

o7 : List
i8 : fromWDivToCl Y

o8 = | 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
o8 : Matrix ZZ  <--- ZZ
i9 : X' = PP2 ** PP2;
i10 : X'' = PP2 ^** 2;
i11 : rays X' == rays X''

o11 = true
i12 : max X' == max X''  

o12 = true

See also