# addBase -- changing the basis matrices of a toric vector bundle in Klyachko's description

## Synopsis

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

## Description

addBase replaces the basis matrices in E by the matrices in the List L. The matrices in L must be in GL($k,R$), where $k$ is the rank of the vector bundle E and $R$ is ZZ or QQ. 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 basis matrices in L will be assigned to the rays in that order. To see the order use rays(ToricVectorBundle).

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 = addBase(E,{matrix{{1,2},{3,1}},matrix{{-1,0},{3,1}},matrix{{1,2},{-3,-1}},matrix{{-1,0},{-3,-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 2 |, 0)} | 0 | | -3 -1 | | 0 | => (| 1 2 |, 0) | -1 | | 3 1 | | 0 | => (| -1 0 |, 0) | 1 | | 3 1 | | 1 | => (| -1 0 |, 0) | 0 | | -3 -1 | o4 : HashTable i5 : isVectorBundle F o5 = true