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NormalToricVarieties :: NormalToricVariety ** NormalToricVariety

NormalToricVariety ** NormalToricVariety -- make the Cartesian product of normal toric varieties

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 = toricProjectiveSpace 2;
i2 : FF2 = hirzebruchSurface 2;
i3 : X = FF2 ** PP2;
i4 : assert (# rays X == # rays FF2 + # rays PP2)
i5 : assert (matrix rays X == matrix rays FF2 ++ matrix rays PP2)
i6 : primaryDecomposition ideal X

o6 = {ideal (x , x ), ideal (x , x ), ideal (x , x , x )}
              0   2           1   3           4   5   6

o6 : List
i7 : flatten (primaryDecomposition \ {ideal FF2,ideal PP2})

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

o7 : List

The map from the torus-invariant Weil divisors to the class group is the direct sum of the maps for the factors.

i8 : assert (fromWDivToCl FF2 ++ fromWDivToCl PP2 == fromWDivToCl X)

See also