Consider a smooth projective variety $X$ and an exact sequence $$0\rightarrow\mathcal{H}\rightarrow\mathcal{O}_X\otimes V\rightarrow\mathcal{G}\rightarrow0$$ where $V:=H^0(X,\mathcal{G})$ and $\mathcal{H}$ is locally free. Pick a slop-semistable subsheaf $\mathcal{F}$ of $\mathcal{H}$ such that $\mu(\mathcal{F})>\mu(\mathcal{H})$ and $\mathcal{H}/\mathcal{F}$ is torsion free. Since $\mathcal{F}\subset\mathcal{O}_X\otimes V$, we know that $\mu(\mathcal{F})=0$.

Could the quotient $(\mathcal{O}_X\otimes V)/\mathcal{F}$ admit a non-trivial torsion subsheaf? If so, what would be the torsion sheaf?

Consider a smooth projective variety $X$ and an exact sequence $$0\rightarrow\mathcal{H}\rightarrow\mathcal{O}_X\otimes V\rightarrow\mathcal{G}\rightarrow0$$ where $V:=H^0(X,\mathcal{G})$ and $\mathcal{H}$ is locally free. Pick a slop-semistable subsheaf $\mathcal{F}$ of $\mathcal{H}$ such that $\mu(\mathcal{F})>\mu(\mathcal{H})$ and $\mathcal{H}/\mathcal{F}$ is torsion-free. Since $\mathcal{F}\subset\mathcal{O}_X\otimes V$, we know that $\mu(\mathcal{F})=0$.

Could the quotient $(\mathcal{O}_X\otimes V)/\mathcal{F}$ admit a non-trivial torsion subsheaf? If so, what would be the torsion sheaf?

Consider a smooth projective variety $X$ and an exact sequence $$0\rightarrow\mathcal{H}\rightarrow\mathcal{O}_X\otimes V\rightarrow\mathcal{G}\rightarrow0$$ where $V:=H^0(X,\mathcal{G})$ and $\mathcal{H}$ is locally free. Pick a slop-semistable subsheaf $\mathcal{F}$ of $\mathcal{H}$ such that $\mu(\mathcal{F})>\mu(\mathcal{H})$ and $\mathcal{H}/\mathcal{F}$ is torsion free. Since $\mathcal{F}\subset\mathcal{O}_X\otimes V$, we know that $\mu(\mathcal{F})=0$.

Could the quotient $(\mathcal{O}_X\otimes V)/\mathcal{F}$ admit a torsion subsheaf? If so, what would be the torsion sheaf?

Consider a smooth projective variety $X$ and an exact sequence $$0\rightarrow\mathcal{H}\rightarrow\mathcal{O}_X\otimes V\rightarrow\mathcal{G}\rightarrow0$$ where $V:=H^0(X,\mathcal{G})$ and $\mathcal{H}$ is locally free. Pick a slop-semistable subsheaf $\mathcal{F}$ of $\mathcal{H}$ such that $\mu(\mathcal{F})>\mu(\mathcal{H})$ and $\mathcal{H}/\mathcal{F}$ is torsion free. Since $\mathcal{F}\subset\mathcal{O}_X\otimes V$, we know that $\mu(\mathcal{F})=0$.

Could the quotient $(\mathcal{O}_X\otimes V)/\mathcal{F}$ admit a non-trivial torsion subsheaf? If so, what would be the torsion sheaf?

Consider a smooth projective variety $X$ and an exact sequence $$0\rightarrow\mathcal{H}\rightarrow\mathcal{O}_X\otimes V\rightarrow\mathcal{G}\rightarrow0$$ where $V:=H^0(X,\mathcal{G})$ and $\mathcal{H}$ is locally free. Pick a slop-semistable subsheaf $\mathcal{F}$ of $\mathcal{H}$ such that $\mu(\mathcal{F})>\mu(\mathcal{H})$ and $\mathcal{H}/\mathcal{F}$ is torsion free. Since $\mathcal{F}\subset\mathcal{O}_X\otimes V$, we know that $\mu(\mathcal{F})=0$.

Could the quotient $(\mathcal{O}_X\otimes V)/\mathcal{F}$ admit a torsion free? If so, what would be the torsion sheaf?

Consider a smooth projective variety $X$ and an exact sequence $$0\rightarrow\mathcal{H}\rightarrow\mathcal{O}_X\otimes V\rightarrow\mathcal{G}\rightarrow0$$ where $V:=H^0(X,\mathcal{G})$ and $\mathcal{H}$ is locally free. Pick a slop-semistable subsheaf $\mathcal{F}$ of $\mathcal{H}$ such that $\mu(\mathcal{F})>\mu(\mathcal{H})$ and $\mathcal{H}/\mathcal{F}$ is torsion free. Since $\mathcal{F}\subset\mathcal{O}_X\otimes V$, we know that $\mu(\mathcal{F})=0$.

Could the quotient $(\mathcal{O}_X\otimes V)/\mathcal{F}$ admit a torsion subsheaf? If so, what would be the torsion sheaf?