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

Synopsis

Usage:

isLW = isLocallyWeil E
Inputs:
- E, a vector bundle on a toric variety (Klyachko's description)
Outputs:
- isLW, a Boolean value, 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.

Ways to use isLocallyWeil :

isLocallyWeil(ToricVectorBundleKlyachko)

For the programmer

The object isLocallyWeil is a method function with options.