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# cartierInd -- computes the Cartier index

## Synopsis

• Usage:
i = cartierInd E
• Inputs:
• Outputs:
• i, an integer, Cartier index of E
• Consequences:
• The result of cartierInd will be stored as a cacheValue in E.cache#cartierInd.

## 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}}; i2 : D = {1,0,0}; i3 : L = toricVectorBundle(1, P112, toList(3: matrix{{1_QQ}}), apply( D, i-> matrix{{-i}})); i4 : details L o4 = HashTable{| -1 | => (| 1 |, | -1 |)} | -2 | | 0 | => (| 1 |, 0) | 1 | | 1 | => (| 1 |, 0) | 0 | o4 : HashTable i5 : cI = cartierInd L o5 = 2 i6 : isLocallyFree L o6 = false i7 : L2 = toricVectorBundle(1, P112, toList(3: matrix{{1_QQ}}), apply( cI*D, i-> matrix{{-i}})); i8 : isLocallyFree L2 o8 = true

## 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.