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Preamble

I initially decided to post this question on math.stackexchange a few days ago, as I consider it to be much less of a research question and much more of "I'm learning" question. But there weren't any takers, and since then it's naturally slipped farther down the "Questions" list over there. So I'm trying my luck here instead.$^*$

Question

Does anyone know of a nice way to think about endomorphisms of vector bundles arising from the Serre construction/correspondence---that is, the vector bundles on projective varieties associated to codimension 2 subvarieties? I am interested in particular in the case of such bundles on $\mathbb{CP}^2$, where these bundles have sections vanishing on prescribed sets of points. Is there anything concrete we can say about elements of $\Gamma(\mathbb{CP}^2,\mbox{End} E)$ when E is one of these bundles?

General Motivation

I'm learning about vector bundle constructions, and I'm trying to compute the cohomologies of these constructions, namely $H^i(E)$ and $H^i(\mbox{End}(E))$.


$^*$ In order to maintain tidiness, I will remove the duplicate question from math.stackexchange if it seems to generate more activity here.

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2 Answers 2

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.

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Can you explain what is the general form of the spectral sequence you mentioned? Thanks! –  Fei YE Aug 9 '10 at 19:11
    
This is the hypercohomology spectral sequence. First, $E$ is quasiisomorphic to the complex $J_Z \to L[1]$. Second, $End E = E^*\otimes E$ is quasiisomorphic to $(L^*[-1] \to J_Z^*)\otimes(J_Z \to L[1]) \cong (L^*\otimes J_Z[-1] \to L^*\otimes L \oplus J_Z^*\otimes J_Z \to J_Z^*\otimes L[1])$. Now you apply the hypercohomology spectral sequence to this, and that's it. –  Sasha Aug 9 '10 at 19:47

This might be obvious, but one can sometimes argue via stability:

Notation as in Sasha's answer, with $L=\mathcal{O}(-n)$ and $n>0$. Then if you know that no curve of degree $\le n/2$ contains $Z$, then $E$ is stable (a destabilizing line bundle $\mathcal{O}(-m)\subset E$ has $-m \ge -n/2$).

And then of course $\Gamma(\mathbb{P}^2, \mathrm{End}(E))=k$.

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