Conics on the complete intersection of two quadrics

Question

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 $\operatorname{Hom}(I_{c_1/S}, I_{c_2/S})\neq 0$?

Answer 1

If $f \colon I_{c_1} \to I_{c_2}$ is a non-trivial morphism then it is an isomorphism. Indeed, $f$ is injective because both $I_{c_1}$ and $I_{c_2}$ are torsion free, and since the Hilbert polynomials of $I_{c_1}$ and $I_{c_2}$ are equal, the Hilbert polynomial of the cokernel is zero, hence the cokernel vanishes.

Thus, $\mathrm{Hom}(I_{c_1},I_{c_2}) \ne 0$ if and only if $c_1$ and $c_2$ are linearly equivalent Weil divisors.