next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
NormalToricVarieties :: NormalToricVariety ** NormalToricVariety

NormalToricVariety ** NormalToricVariety -- Cartesian product

Synopsis

Description

The Cartesian product of two varieties X and Y, both defined over the same ground field k, is the fiber product X ×k Y. For normal toric varieties, the fan of the product is given by the Cartesian product of each pair of cones in the fans of the factors.
i1 : PP2 = projectiveSpace 2;
i2 : FF2 = hirzebruchSurface 2;
i3 : X = FF2 ** PP2;
i4 : #rays X == #rays FF2 + #rays PP2

o4 = true
i5 : transpose matrix rays X

o5 = | 1 0 -1 0  0  0 0 |
     | 0 1 2  -1 0  0 0 |
     | 0 0 0  0  -1 1 0 |
     | 0 0 0  0  -1 0 1 |

              4        7
o5 : Matrix ZZ  <--- ZZ
i6 : transpose matrix rays FF2 ++ transpose matrix rays PP2

o6 = | 1 0 -1 0  0  0 0 |
     | 0 1 2  -1 0  0 0 |
     | 0 0 0  0  -1 1 0 |
     | 0 0 0  0  -1 0 1 |

              4        7
o6 : Matrix ZZ  <--- ZZ
i7 : primaryDecomposition ideal X

o7 = {ideal (x , x ), ideal (x , x ), ideal (x , x , x )}
              0   2           1   3           4   5   6

o7 : List
i8 : flatten (primaryDecomposition \ {ideal FF2,ideal PP2})

o8 = {ideal (x , x ), ideal (x , x ), ideal (x , x , x )}
              0   2           1   3           0   1   2

o8 : List
The map from the torus-invariant Weil divisors to the class group is the direct sum of the maps for the factors
i9 : fromWDivToCl FF2 ++ fromWDivToCl PP2

o9 = | 1 -2 1 0 0 0 0 |
     | 0 1  0 1 0 0 0 |
     | 0 0  0 0 1 1 1 |

              3        7
o9 : Matrix ZZ  <--- ZZ
i10 : fromWDivToCl X

o10 = | 1 -2 1 0 0 0 0 |
      | 0 1  0 1 0 0 0 |
      | 0 0  0 0 1 1 1 |

               3        7
o10 : Matrix ZZ  <--- ZZ

See also