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;
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i2 : FF2 = hirzebruchSurface 2;
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i3 : X = FF2 ** PP2;
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i4 : #rays X == #rays FF2 + #rays PP2
o4 = true
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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
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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
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i7 : primaryDecomposition ideal X
o7 = {ideal (x , x ), ideal (x , x ), ideal (x , x , x )}
0 2 1 3 4 5 6
o7 : List
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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
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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
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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
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