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

## Synopsis

• Usage:
• Inputs:
• L, a list, with matrices over ZZ or QQ
• Outputs:

## 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