# A short exact sequence on del Pezzo threefold and Gushel-Mukai

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?

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.
• 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. Jul 18 at 6:52