kernel(ToricVectorBundleKlyachko,Matrix) -- the kernel of a morphism to a vector bundle

Synopsis

Function: kernel
Usage:

E1 = ker(E,M)
Inputs:
- E, a vector bundle on a toric variety (Klyachko's description)
- M, a matrix, over ZZ or QQ
Optional inputs:
- SubringLimit => ..., default value infinity,
Outputs:
- E1, a vector bundle on a toric variety (Klyachko's description)

Description

M must be a matrix over ZZ or QQ where the source space is the space of the bundle, i.e., the matrix must have $k$ columns if the bundle has rank $k$. Then the new bundle is given on each ray $\rho$ by the following filtration of ker$(E,M)^\rho := $ ker$(M) \cap (E^\rho)$ :

ker$(E,M)^\rho(i) := $ ker$(M) \cap E^\rho(i)$.

i1 : E = tangentBundle hirzebruchFan 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 : E = E ** E

o2 = {dimension of the variety => 2 }
      number of affine charts => 4
      number of rays => 4
      rank of the vector bundle => 4

o2 : ToricVectorBundleKlyachko

i3 : M = matrix {{1,0,1,0},{0,1,0,1/1}}

o3 = | 1 0 1 0 |
     | 0 1 0 1 |

              2       4
o3 : Matrix QQ  <-- QQ

i4 : E1 = ker(E,M)

o4 = {dimension of the variety => 2 }
      number of affine charts => 4
      number of rays => 4
      rank of the vector bundle => 2

o4 : ToricVectorBundleKlyachko

i5 : details E1

o5 = HashTable{| -1 | => (| -1/2 1/2 |, | -1 0 |)}
               | 2  |     | 1    0   |
               | 0  | => (| 0 1 |, | -1 0 |)
               | -1 |     | 1 0 |
               | 0 | => (| 0 1 |, | -1 0 |)
               | 1 |     | 1 0 |
               | 1 | => (| 1 0 |, | -1 0 |)
               | 0 |     | 0 1 |

o5 : HashTable

Ways to use this method: