Let $X$ be a very general Gushel-Mukai fourfold, let $\mathcal{U}$ be the tautological sub-bundle and $\mathcal{Q}$ be the tautological quotient bundle. Let $C\subset X$ be a smooth conic such that $\mathcal{U}|_C\cong\mathcal{O}_C\oplus\mathcal{O}_C(-1)$ and $\mathcal{Q}|_C\cong\mathcal{O}_C(1)\oplus\mathcal{O}_C(1)\oplus\mathcal{O}_C$, such conic is called $\tau$-conic on $X$. Now, I am trying to compute left mutation $\mathrm{L}_{\mathcal{U}}I_C$, where $I_C$ is the ideal sheaf of $C$.

By definition, there is an exact triangle $\mathrm{RHom}(\mathcal{U},I_C)\otimes\mathcal{U}\rightarrow I_C\rightarrow\mathrm{L}_{\mathcal{U}}I_C$, it is easy to show that in this case, $\mathrm{RHom}(\mathcal{U},I_C)=k$, so that we have the triangle $\mathcal{U}\rightarrow I_C\rightarrow\mathrm{L}_{\mathcal{U}}I_C$. The image of $\pi:\mathcal{U}\rightarrow I_C$ should be ideal sheaf $I_{\Sigma}$ of a degree 4 del Pezzo surface $\Sigma$ and $\mathrm{Ker}\pi\cong\mathcal{O}_X(-H)$, so that there is a short exact sequence $$0\rightarrow I_{\Sigma}\rightarrow I_C\rightarrow F\rightarrow 0$$ My first question is how to describe the quotient sheaf $F$? What is this sheaf? The result of $\mathrm{L}_{\mathcal{U}}I_C$ is given by the triangle $$\mathcal{O}_X(-H)[1]\rightarrow\mathrm{L}_{\mathcal{U}}I_C\rightarrow F.$$ It is easy to see that for each conic $C$, there is a unique del Pezzo surface $\Sigma$ containing such $C$. But there could be different conics $C$ corresponds to the same $\Sigma$. In the current case, there is a family of conics parametrised by $\mathbb{P}^1$ corresponds to the same $\Sigma$. I expect that the object $E:=\mathrm{L}_{\mathcal{O}_X}(F\otimes\mathcal{O}_X(H))$ would be a fixed object for all those conics in the same $\Sigma$. But I don't know how to compute either $F$ or $E$. (I expect that this left mutation corresponds to a $\mathbb{P}^1$-bundle map from the locus of $\tau$-conics on $X$ to double dual EPW sextic $\widetilde{Y}_{A^{\perp}}$).

Let $C\subset X$ be a $\rho$-conic, then $\mathcal{U}|_C\cong\mathcal{O}_C(-1)\oplus\mathcal{O}_C(-1)$ and $\mathcal{Q}|_C\cong\mathcal{O}_C\oplus\mathcal{O}_C\oplus\mathcal{O}_C(2)$. Then when I compute $\mathrm{L}_{\mathcal{U}}I_C$, I get $\mathrm{RHom}(\mathcal{U},I_C)\cong k^2\oplus k[-1]$, then I would like to investigate whether the map $\pi':\mathcal{U}^2\rightarrow I_C$ is surjective. In the GM threefold case, the similar map as $\pi'$ is surjective since GM threefold does not contain any plane. I think for very general GM fourfold it is still surjective. Then we will have a short exact sequence $$0\rightarrow\mathrm{Ker}\pi'\rightarrow\mathcal{U}^2\rightarrow I_C\rightarrow 0.$$ Then $\mathrm{Ker}\pi'$ is a rank three torsion free sheaf, I guessed and tried many times, have no idea what this sheaf should be. (In dimension three case, the similar rank three sheaf is $\mathcal{Q}(-H)$, but in dimension 4, the character of $\mathrm{Ker}\pi'$ does not match that of $\mathcal{Q}(-H)$).

Let $X$ be a very general Gushel-Mukai fourfold, let $\mathcal{U}$ be the tautological sub-bundle and $\mathcal{Q}$ be the tautological quotient bundle. Let $C\subset X$ be a $\rho$-conic, then $\mathcal{U}|_C\cong\mathcal{O}_C(-1)\oplus\mathcal{O}_C(-1)$ and $\mathcal{Q}|_C\cong\mathcal{O}_C\oplus\mathcal{O}_C\oplus\mathcal{O}_C(2)$. 

Now, I am trying to compute left mutation $\mathrm{L}_{\mathcal{U}}I_C$, where $I_C$ is the ideal sheaf of $C$.

get $\mathrm{RHom}(\mathcal{U},I_C)\cong k^2\oplus k[-1]$, then I would like to investigate whether the map $\pi':\mathcal{U}^2\rightarrow I_C$ is surjective. In the GM threefold case, the similar map as $\pi'$ is surjective since GM threefold does not contain any plane. I think for very general GM fourfold it is still surjective. Then we will have a short exact sequence $$0\rightarrow\mathrm{Ker}\pi'\rightarrow\mathcal{U}^2\rightarrow I_C\rightarrow 0.$$ Then $\mathrm{Ker}\pi'$ is a rank three torsion free sheaf, I guessed and tried many times, have no idea what this sheaf should be. (In dimension three case, the similar rank three sheaf is $\mathcal{Q}(-H)$, but in dimension 4, the character of $\mathrm{Ker}\pi'$ does not match that of $\mathcal{Q}(-H)$).

Let $X$ be a very general Gushel-Mukai fourfold, let $\mathcal{U}$ be the tautological sub-bundle and $\mathcal{Q}$ be the tautological quotient bundle. Let $C\subset X$ be a smooth conic such that $\mathcal{U}|_C\cong\mathcal{O}_C\oplus\mathcal{O}_C(-1)$ and $\mathcal{Q}|_C\cong\mathcal{O}_C(1)\oplus\mathcal{O}_C(1)\oplus\mathcal{O}_C$, such conic is called $\tau$-conic on $X$. Now, I am trying to compute left mutation $\mathrm{L}_{\mathcal{U}}I_C$, where $I_C$ is the ideal sheaf of $C$.

By definition, there is an exact triangle $\mathrm{RHom}(\mathcal{U},I_C)\otimes\mathcal{U}\rightarrow I_C\rightarrow\mathrm{L}_{\mathcal{U}}I_C$, it is easy to show that in this case, $\mathrm{RHom}(\mathcal{U},I_C)=k$, so that we have the triangle $\mathcal{U}\rightarrow I_C\rightarrow\mathrm{L}_{\mathcal{U}}I_C$. The image of $\pi:\mathcal{U}\rightarrow I_C$ should be ideal sheaf $I_{\Sigma}$ of a degree 4 del Pezzo surface $\Sigma$ and $\mathrm{Ker}\pi\cong\mathcal{O}_X(-H)$, so that there is a short exact sequence $$0\rightarrow I_{\Sigma}\rightarrow I_C\rightarrow F\rightarrow 0$$ My first question is how to describe the quotient sheaf $F$? What is this sheaf? The result of $\mathrm{L}_{\mathcal{U}}I_C$ is given by the triangle $$\mathcal{O}_X(-H)[1]\rightarrow\mathrm{L}_{\mathcal{U}}I_C\rightarrow F.$$ It is easy to see that for each conic $C$, there is a unique del Pezzo surface $\Sigma$ containing such $C$. But there could be different conics $C$ corresponds to the same $\Sigma$. In the current case, there is a family of conics parametrised by $\mathbb{P}^1$ corresponds to the same $\Sigma$. I expect that the object $E:=\mathrm{L}_{\mathcal{O}_X}(F\otimes\mathcal{O}_X(H))$ would be a fixed object for all those conics in the same $\Sigma$. But I don't know how to compute either $F$ or $E$. (I expect that this left mutation corresponds to a $\mathbb{P}^1$-bundle map from the locus of $\tau$-conics on $X$ to double dual EPW sextic $\widetilde{Y}_{A^{\perp}}$).

Let $C\subset X$ be a $\rho$-conic, then $\mathcal{U}|_C\cong\mathcal{O}_C(-1)\oplus\mathcal{O}_C(-1)$ and $\mathcal{Q}|_C\cong\mathcal{O}_C\oplus\mathcal{O}_C\oplus\mathcal{O}_C(2)$. Then when I compute $\mathrm{L}_{\mathcal{U}}I_C$, I get $\mathrm{RHom}(\mathcal{U},I_C)\cong k^2\oplus k[-1]$, then I would like to investigate whether the map $\pi':\mathcal{U}^2\rightarrow I_C$ is surjective. In the GM threefold case, the similar map as $\pi'$ is surjective since GM threefold does not contain any quadric surface. But for GM fourfold $X$, I am not sure whether it is true even if $X$ is very general. Suppose $\pi'$ is surjective. Then we will have a short exact sequence $$0\rightarrow\mathrm{Ker}\pi'\rightarrow\mathcal{U}^2\rightarrow I_C\rightarrow 0.$$ Then $\mathrm{Ker}\pi'$ is a rank three torsion free sheaf, I guessed and tried many times, have no idea what this sheaf should be. (In dimension three case, the similar rank three sheaf is $\mathcal{Q}(-H)$, but in dimension 4, the character of $\mathrm{Ker}\pi'$ does not match that of $\mathcal{Q}(-H)$).

Let $X$ be a very general Gushel-Mukai fourfold, let $\mathcal{U}$ be the tautological sub-bundle and $\mathcal{Q}$ be the tautological quotient bundle. Let $C\subset X$ be a smooth conic such that $\mathcal{U}|_C\cong\mathcal{O}_C\oplus\mathcal{O}_C(-1)$ and $\mathcal{Q}|_C\cong\mathcal{O}_C(1)\oplus\mathcal{O}_C(1)\oplus\mathcal{O}_C$, such conic is called $\tau$-conic on $X$. Now, I am trying to compute left mutation $\mathrm{L}_{\mathcal{U}}I_C$, where $I_C$ is the ideal sheaf of $C$.

By definition, there is an exact triangle $\mathrm{RHom}(\mathcal{U},I_C)\otimes\mathcal{U}\rightarrow I_C\rightarrow\mathrm{L}_{\mathcal{U}}I_C$, it is easy to show that in this case, $\mathrm{RHom}(\mathcal{U},I_C)=k$, so that we have the triangle $\mathcal{U}\rightarrow I_C\rightarrow\mathrm{L}_{\mathcal{U}}I_C$. The image of $\pi:\mathcal{U}\rightarrow I_C$ should be ideal sheaf $I_{\Sigma}$ of a degree 4 del Pezzo surface $\Sigma$ and $\mathrm{Ker}\pi\cong\mathcal{O}_X(-H)$, so that there is a short exact sequence $$0\rightarrow I_{\Sigma}\rightarrow I_C\rightarrow F\rightarrow 0$$ My first question is how to describe the quotient sheaf $F$? What is this sheaf? The result of $\mathrm{L}_{\mathcal{U}}I_C$ is given by the triangle $$\mathcal{O}_X(-H)[1]\rightarrow\mathrm{L}_{\mathcal{U}}I_C\rightarrow F.$$ It is easy to see that for each conic $C$, there is a unique del Pezzo surface $\Sigma$ containing such $C$. But there could be different conics $C$ corresponds to the same $\Sigma$. In the current case, there is a family of conics parametrised by $\mathbb{P}^1$ corresponds to the same $\Sigma$. I expect that the object $E:=\mathrm{L}_{\mathcal{O}_X}(F\otimes\mathcal{O}_X(H))$ would be a fixed object for all those conics in the same $\Sigma$. But I don't know how to compute either $F$ or $E$. (I expect that this left mutation corresponds to a $\mathbb{P}^1$-bundle map from the locus of $\tau$-conics on $X$ to double dual EPW sextic $\widetilde{Y}_{A^{\perp}}$).

Let $C\subset X$ be a $\rho$-conic, then $\mathcal{U}|_C\cong\mathcal{O}_C(-1)\oplus\mathcal{O}_C(-1)$ and $\mathcal{Q}|_C\cong\mathcal{O}_C\oplus\mathcal{O}_C\oplus\mathcal{O}_C(2)$. Then when I compute $\mathrm{L}_{\mathcal{U}}I_C$, I get $\mathrm{RHom}(\mathcal{U},I_C)\cong k^2\oplus k[-1]$, then I would like to investigate whether the map $\pi':\mathcal{U}^2\rightarrow I_C$ is surjective. In the GM threefold case, the similar map as $\pi'$ is surjective since GM threefold does not contain any plane. I think for very general GM fourfold it is still surjective. Then we will have a short exact sequence $$0\rightarrow\mathrm{Ker}\pi'\rightarrow\mathcal{U}^2\rightarrow I_C\rightarrow 0.$$ Then $\mathrm{Ker}\pi'$ is a rank three torsion free sheaf, I guessed and tried many times, have no idea what this sheaf should be. (In dimension three case, the similar rank three sheaf is $\mathcal{Q}(-H)$, but in dimension 4, the character of $\mathrm{Ker}\pi'$ does not match that of $\mathcal{Q}(-H)$).