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isLocallyWeil -- checks whether a toric reflexive sheaf is locally Weil

Synopsis

• Usage:
isLW = isLocallyWeil E
• Inputs:
• Outputs:
• isLW, , true if the reflexive sheaf is locally Weil, false otherwise
• Consequences:
• The result of isLocallyWeil will be stored as a cacheValue in E.cache#isLW.

Description

Contrary to what the name suggests, ToricVectorBundle may well encode a toric reflexive sheaf that is not locally Weil (as soon as the toric variety has dimension at least three). isLocallyWeil permits to check whether a toric reflexive sheaf is locally Weil, that is, locally a direct sum of reflexive sheaves of rank one. If the toric variety is smooth, this is equivalent to being locally free, that is, a vector bundle.
isLocallyWeil calls internally the method compatibleBases, if all the bases have as many elements as the rank of the sheaf, then it is locally Weil.
 i1 : A3 = fan coneFromVData matrix {{1,0,0},{0,1,0},{0,0,1}}; i2 : filtMat = apply( { {{1,0,0},{0,1,0},{0,0,1}}, {{0,1,0},{1,0,0},{0,0,1}}, {{1,1,0},{1,0,0},{0,0,1}} }, matrix); i3 : filtStep = apply( { {{0,1,1}}, {{0,1,1}}, {{0,1,1}} }, matrix); i4 : E = toricVectorBundle (3,A3,filtMat,filtStep); i5 : details E o5 = HashTable{| 0 | => (| 1 0 0 |, | 0 1 1 |)} | 0 | | 0 1 0 | | 1 | | 0 0 1 | | 0 | => (| 0 1 0 |, | 0 1 1 |) | 1 | | 1 0 0 | | 0 | | 0 0 1 | | 1 | => (| 1 1 0 |, | 0 1 1 |) | 0 | | 1 0 0 | | 0 | | 0 0 1 | o5 : HashTable i6 : isLocallyWeil E o6 = false i7 : compatibleBases E o7 = HashTable{| 1 0 0 | => | 0 1 0 1 |} | 0 1 0 | | 0 0 1 1 | | 0 0 1 | | 1 0 0 0 | o7 : HashTable

Caveat

This method works for any toric reflexive sheaf on a toric variety, whose fan is covered by cones of maximal dimension.