# cokernel of the symmetric product of an injection.

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.

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.