Clearly, my question can be asked more generally, but suppose for simplicity that $X$ is a smooth surface and that $0 \to E \to F \to K \to 0 $ is exact with $E ,F$ rank two bundles on $X$ and $K$ a line bundle on a (smooth?) divisor $D \subset X$. Can one give information on the cokernel of the injection $Sym^n(E) \to Sym^n(F) $ ? For example is this cokernel some obvious sum of linear algebra constructions (various symmetric powers) of $E$ , $F$, and $K$ ?
I can only say that I have thought about this question for a while and no answer has occurred to me. 


Adam



cokernel of the symmetric product of an injection. 
Hi Adam,


Say $n=2$. Let $EF\subset Sym^2(F)$ be the subsheaf generated locally by products of sections of $E$ and $F$. Then a diagram chase should give an extension
$$0\to (EF)/Sym^2(E)\to Sym^2(F)/Sym^2(E)\to Sym^2(F)/EF\to 0$$
of sheaves supported on $D$. 
To get a better sense of this, say $E=O\_X\oplus O\_X$, 
$F=O\_X\oplus O\_X(D)$ and $K=O\_D(D)$ (which is locally what's going on). Then 
$$Sym^2(E) = O\_X^3,\ EF=O\_X\oplus O\_X(D)^2,\ Sym^2(F) = O\_X\oplus O\_X(D)\oplus O\_X(2D)$$
so that the above sequence is an extension of $O\_D(D)$ by $O\_D(D)$. So perhaps in general, it is an extension of $K$ by $K$, but I admit I haven't checked this. In which case, the pattern
for $n>2$ shouldn't be hard.


