Description
addFiltration replaces the filtration matrices in
E by the matrices in the
List L. The matrices in
L must be $1$ by $k$ matrices over
ZZ, where $k$ is the rank of the vector bundle
E. The list has to contain one matrix for each ray of the underlying fan over which
E is defined. Note that in
E the rays are already sorted and that the filtration matrices in
L will be assigned to the rays in that order. To see the order, use
rays(ToricVectorBundle).
"The filtration on the vector bundle over a ray is given by the filtration matrix for this ray in the following way: The first index $j$, such that the $i$-th basis vector in the basis over this ray appears in the $j$-th step of the filtration, is the $i$-th entry of the filtration matrix. OR in other words, the $j$-th step step in the filtration is given by all columns of the basis matrix for which the corresponding entry in the filtration matrix is less or equal to $j$."
The matrices need not satisfy the compatibility condition. This can be checked with
isVectorBundle.
i1 : E = toricVectorBundle(2,pp1ProductFan 2)
o1 = {dimension of the variety => 2 }
number of affine charts => 4
number of rays => 4
rank of the vector bundle => 2
o1 : ToricVectorBundleKlyachko
|
i2 : details E
o2 = HashTable{| -1 | => (| 1 0 |, 0)}
| 0 | | 0 1 |
| 0 | => (| 1 0 |, 0)
| -1 | | 0 1 |
| 0 | => (| 1 0 |, 0)
| 1 | | 0 1 |
| 1 | => (| 1 0 |, 0)
| 0 | | 0 1 |
o2 : HashTable
|
i3 : F = addFiltration(E,{matrix{{1,3}},matrix{{-1,3}},matrix{{2,-3}},matrix{{0,-1}}})
o3 = {dimension of the variety => 2 }
number of affine charts => 4
number of rays => 4
rank of the vector bundle => 2
o3 : ToricVectorBundleKlyachko
|
i4 : details F
o4 = HashTable{| -1 | => (| 1 0 |, | 2 -3 |)}
| 0 | | 0 1 |
| 0 | => (| 1 0 |, | 1 3 |)
| -1 | | 0 1 |
| 0 | => (| 1 0 |, | -1 3 |)
| 1 | | 0 1 |
| 1 | => (| 1 0 |, | 0 -1 |)
| 0 | | 0 1 |
o4 : HashTable
|
i5 : isVectorBundle F
o5 = true
|
This means that for example over the first ray the first basis vector of the filtration of
F appears at the filtration step 1 and the second at 3.