Let $(X,x)$ be an isolated, Gorenstein singularity of dimension at least $2$ and $f: Y \to X$ be a resolution of singularities. Let $E_1, E_2$ be two distinct irreducible components of the exceptional divisor and $C, D \subset Y$ be two curves intersecting $E_1, E_2$, respectively at exactly one point and not intersecting any other component of the exceptional divisor. Is it possible that $f_*\mathcal{O}_C$ and $f_*\mathcal{O}_D$ define the same object in the derived category $D^b(X)$? In particular, I am looking for examples when $f_*\mathcal{O}_C$ and $f_*\mathcal{O}_D$ define the same object in the derived category.

Let $(X,x)$ be an isolated, Gorenstein singularity of dimension at least $2$ and $f: Y \to X$ be a resolution of singularities. Let $E_1, E_2$ be two distinct irreducible components of the exceptional divisor and $C, D \subset Y$ be two smooth curves intersecting $E_1, E_2$, respectively at exactly one point and not intersecting any other component of the exceptional divisor. Is it possible that $f_*\mathcal{O}_C$ and $f_*\mathcal{O}_D$ define the same object in the derived category $D^b(X)$? In particular, I am looking for examples when $f_*\mathcal{O}_C$ and $f_*\mathcal{O}_D$ define the same object in the derived category.