next | previous | forward | backward | up | top | index | toc | Macaulay2 website
OldToricVectorBundles :: addBaseChange

addBaseChange -- changing the transition matrices of a toric vector bundle

Synopsis

Description

addBaseChange replaces the transition matrices in E by the matrices in the List L. The matrices in L must be in GL($k$,ZZ) or GL($k$,QQ), where $k$ is the rank of the vector bundle T. The list has to contain one matrix for each maximal dimensional cone of the underlying fan over which E is defined. The fan can be recovered with fan(ToricVectorBundle). The vector bundle already has a list of pairs $(i,j)$ denoting the codim 1 intersections of two maximal cones with $i<j$ and they are ordered in lexicographic order. The matrices will be assigned to the pairs $(i,j)$ in that order. To see which codimension 1 cone corresponds to the pair $(i,j)$ use details(ToricVectorBundle). The matrix $A$ assigned to $(i,j)$ denotes the transition $(e_i^1,...,e_i^k) = (e_j^1,...,e_j^k)*A$. The matrices need not satisfy the regularity or the cocycle condition. These can be checked with regCheck and cocycleCheck.

i1 : E = toricVectorBundle(2,pp1ProductFan 2,"Type" => "Kaneyama")

o1 = {dimension of the variety => 2 }
      number of affine charts => 4
      rank of the vector bundle => 2

o1 : ToricVectorBundleKaneyama
i2 : details E

o2 = (HashTable{0 => (| 1 0 |, 0)  }, HashTable{(0, 1) => | 1 0 |})
                      | 0 1 |                             | 0 1 |
                1 => (| 1 0  |, 0)              (0, 2) => | 1 0 |
                      | 0 -1 |                            | 0 1 |
                2 => (| -1 0 |, 0)              (1, 3) => | 1 0 |
                      | 0  1 |                            | 0 1 |
                3 => (| -1 0  |, 0)             (2, 3) => | 1 0 |
                      | 0  -1 |                           | 0 1 |

o2 : Sequence
i3 : F = addBaseChange(E,{matrix{{1,2},{0,1}},matrix{{1,0},{3,1}},matrix{{1,-2},{0,1}},matrix{{1,0},{-3,1}}})

o3 = {dimension of the variety => 2 }
      number of affine charts => 4
      rank of the vector bundle => 2

o3 : ToricVectorBundleKaneyama
i4 : details F

o4 = (HashTable{0 => (| 1 0 |, 0)  }, HashTable{(0, 1) => | 1 2 | })
                      | 0 1 |                             | 0 1 |
                1 => (| 1 0  |, 0)              (0, 2) => | 1 0 |
                      | 0 -1 |                            | 3 1 |
                2 => (| -1 0 |, 0)              (1, 3) => | 1 -2 |
                      | 0  1 |                            | 0 1  |
                3 => (| -1 0  |, 0)             (2, 3) => | 1  0 |
                      | 0  -1 |                           | -3 1 |

o4 : Sequence
i5 : cocycleCheck F

o5 = true

See also

Ways to use addBaseChange :

For the programmer

The object addBaseChange is a method function.