A torus-invariant Weil divisor
D on a normal toric variety
X is Cartier if it is locally principal, meaning that
X has an open cover
{Ui} such that
D|Ui is principal in
Ui for every
i.
On a smooth variety, every Weil divisor is Cartier.
i1 : PP3 = projectiveSpace 3;
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i2 : all(3, i -> isCartier PP3_i)
o2 = true
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On a simplicial toric variety, every torus-invariant Weil divisor is
ℚ-Cartier --- every torus-invariant Weil divisor has a positive integer multiple that is Cartier.
i3 : W = weightedProjectiveSpace {2,5,7};
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i4 : isSimplicial W
o4 = false
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i5 : isCartier W_0
o5 = false
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i6 : isQQCartier W_0
o6 = false
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i7 : isCartier (35*W_0)
o7 = false
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In general, the Cartier divisors are only a subgroup of the Weil divisors.
i8 : X = normalToricVariety(id_(ZZ^3) | -id_(ZZ^3));
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i9 : isCartier X_0
o9 = false
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i10 : isQQCartier X_0
o10 = false
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i11 : K = toricDivisor X
o11 = - D - D - D - D - D - D - D - D
0 1 2 3 4 5 6 7
o11 : ToricDivisor on X
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i12 : isCartier K
o12 = true
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