ag.algebraic geometry - Endomorphisms of bundles associated to codimension 2 subvarieties

The bundle constructed from the subvariety $Z subset X$ comes in exact triple
$$
0 to L to E to J_Z to 0,
$$
where $L$ is a line bundle on $X$ extending $det N_{Z/X}$. (In case $X = P^2$ and $Z$ is a set of points, $L$ can be chosen to be arbitrary (since each line bundle on $Z$ is trivial)). So, you can use this triple to compute any cohomological invariant of $E$. For example, if you are interested in $Gamma(P^2,End E)$ you can use the spectral sequence
with the first term having the following form
$$
begin{array}{ccccc}
Hom(L,J_Z) & to & Ext^1(J_Z,J_Z) oplus Ext^1(L,L) & to & Ext^2(J_Z,L) cr
& & Hom(J_Z,J_Z) oplus Hom(L,L) & to & Ext^1(J_Z,L) cr
& & & & Hom(J_Z,L)
end{array}
$$
and converging to $Ext^i(E,E) = H^i(P^2,End E)$. So, you see that the contributions to $Gamma(P^2,End E)$ come

1) from $Hom(J_Z,L)$;

2) from $Ker(Hom(J_Z,J_Z) oplus Hom(L,L) to Ext^1(J_Z,L))$; and

3) from $Ker(Hom(L,J_Z) to Ext^1(J_Z,J_Z) oplus Ext^1(L,L))$ (here one should also take into account the $d_2$ differential).

So, everything can be computed.