Let $Y$ be degree 5 index two prime Fano threefold. Let $\mathcal{E}$ and $\mathcal{Q}$ be the tautological sub and quotient bundle on $Y$. It is not hard to show that there is a short exact sequence:
$$0\rightarrow\mathcal{E}\xrightarrow{p}\mathcal{Q}^{\vee}\rightarrow I\_L\rightarrow 0$$, where $L$ is a line on $Y$. This is because one can show the map $p$ is injective by slope stability of $\mathcal{E}$ and $\mathcal{Q}^{\vee}$ and slope of them are $-\frac{1}{2},-\frac{1}{3}.$


But I was wondering whether the following exact sequence also exist or not?
$$0\rightarrow I\_L\rightarrow\mathcal{Q}\xrightarrow{\pi}\mathcal{E}^{\vee}\rightarrow 0$$
The problem to show the map $\pi$ is surjective is that the cokernel of $\pi$ could be rank zero sheaf, for example $\mathcal{O}\_L$, then there is no contradiction by arguing with stability and slope.


The similar thing happened in Gushel-Mukai threefold, there is a short exact sequence $$0\rightarrow\mathcal{E}\xrightarrow{p}\mathcal{Q}^{\vee}\rightarrow I\_C\rightarrow 0$$ with $C$ being a conic by the same stability argument and note that the zero locus of section of $\mathcal{Q}$ is either two points or a conic(in both special and ordinary GM case, with the argument slightly different)


But I was wondering whether the following exact sequence exist?
$$0\rightarrow I\_C\rightarrow\mathcal{Q}\xrightarrow{\pi}\mathcal{E}^{\vee}\rightarrow 0.$$
I thought it exists, but then I located a mistake in the argument, say the cokernel of $\pi$ could be $\mathcal{O}\_L$ and the image of $pi$ can be a torsion free semistable sheaf $E\in M(2,1,5)$.


Maybe there is a very simple reason that such short exact sequence does not exist, which I miss it?



A short exact sequence on del Pezzo threefold and Gushel-Mukai
These sequences do not exist, because the kernel of an epimorphism of locally free sheaves is itself locally free, while the ideal sheaf of a curve on a threefold is not locally free.


Instead, there are distinguished triangles of the same form, but whose first terms are $I\_L^\vee$ and $I\_C^\vee$, the **derived** duals of the ideals. Note that these are complexes with two cohomology sheaves, which means that instead of a short exact sequence one has a four term exact sequence in each case.


