Let $Q_1, Q_2\subset \mathbb{P}^4$ be two degree two smooth hypersurfaces, and $S:=Q_1\cap Q_2$ be their complete intersection. If $S$ is smooth, then it is known that $S$ is a del Pezzo surface of degree 4. Now let $c_1, c_2\subset C$ be two conics.

I'm wondering that is there any criterion for $\operatorname{Hom}(I_{c_1/S}, I_{c_2/S})\neq 0$? Here $I_{c_i/S}$ is the ideal sheaf of $c_i$ in $S$.

When $S$ is smooth, then using some theory of curves on del Pezzo surface, we can prove that $\mathcal{O}_S(-c_1)\cong \mathcal{O}_S(-c_2)$. But when $S$ is singular, is there any similar result? For example, if there's a quadric $Q_3$ in the pencil spaned by $Q_1$ and $Q_2$, such that $\langle c_1\rangle\cup \langle c_2\rangle\cup S\subset Q_3$, where $\langle c_i\rangle$ is the projective plane spaned by $c_i$, then can we conclude that $\operatorname{Hom}(I_{c_1/S}, I_{c_2/S})\neq 0$?