Description
Contrary to what the name suggests,
ToricVectorBundle may well encode a toric reflexive sheaf that is not locally free.
cartierInd computes the Cartier index, that is, the smallest non-negative integer i such that the pullback of the bundle under the i-th toric Frobenius becomes locally free. In case of the reflexive sheaf of a Weil divisor D, this is the smallest i such that iD is Cartier. This method works well only on simplicial toric varieties.
cartierInd calls internally the method
toricChernCharacter.
i1 : P112 = ccRefinement transpose matrix {{1,0},{0,1},{-1,-2}};
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i2 : D = {1,0,0};
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i3 : L = toricVectorBundle(1, P112, toList(3: matrix{{1_QQ}}), apply( D, i-> matrix{{-i}}));
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i4 : details L
o4 = HashTable{| -1 | => (| 1 |, | -1 |)}
| -2 |
| 0 | => (| 1 |, 0)
| 1 |
| 1 | => (| 1 |, 0)
| 0 |
o4 : HashTable
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i5 : cI = cartierInd L
o5 = 2
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i6 : isLocallyFree L
o6 = false
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i7 : L2 = toricVectorBundle(1, P112, toList(3: matrix{{1_QQ}}), apply( cI*D, i-> matrix{{-i}}));
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i8 : isLocallyFree L2
o8 = true
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Caveat
This method works for a toric reflexive sheaf, which is locally Weil (see
isLocallyWeil), on a toric variety, whose fan is covered by cones of maximal dimension.