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NormalToricVarieties :: NormalToricVariety ** NormalToricVariety

NormalToricVariety ** NormalToricVariety -- make the Cartesian product of two 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 $X \times_k Y$. 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 )}
              2   0           1   3           6   5   4

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

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

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)

The factors are cached and can be accessed with components.

i9 : factors = components X

o9 = {FF2, PP2}

o9 : List
i10 : assert (# factors === 2)
i11 : assert (factors#0 === FF2)
i12 : assert (factors#1 === PP2)

See also