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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?

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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.

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  • $\begingroup$ thanks so much, yes this answer my question! I also think the first term should be derived dual of I_L or I_C,(the character would be the same as I_L and I_C since there is no c_1 and ch_3 term!). Everything is nice. $\endgroup$
    – user41650
    Jul 16, 2021 at 6:56
  • $\begingroup$ I think one can easily show that the ideal sheaf I_C of a conic is not in the non-trivial component $\mathcal{A}_X$ of derived category if and only if I_C has such a resolution of vector bundles, which explain the P^1 family and P^2 family of conics on ordinary or special GM which is very clear. I found that directly project I_C into $\mathcal{A}_X$ using this short exact sequence would get rid of the ambiguity of the splitting of the vector bundle $\cE$ over conic, like (-1,-1) or (0,-2) etc, one can completely get rid of these because it is confused when C is not smooth. $\endgroup$
    – user41650
    Jul 18, 2021 at 6:52

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