Let $S$ be a smooth projective surface and $i:C\subset S,j:D\subset S$ be two smooth curves on $S$ intersecting each other transversely. How can we say about the set $\text{Hom}_S(i_*\mathcal{O}_C,j_*\mathcal{O}_D)$?
I think $\text{Hom}_S(i_*\mathcal{O}_C,j_*\mathcal{O}_D)=0$ whenever $C\cap D=\varnothing$ by considering the stalks. But I do not know how to think about e.g. the case for $C\cap D=1$.
In fact $\mathrm{Hom}(i_*\mathcal{O}_C,j_*\mathcal{O}_D) = 0$ unless $C = D$. Indeed, if there is a non-zero morphism, its image is supported on $C \cap D$, hence it is a torsion subsheaf in $\mathcal{O}_D$, but $\mathcal{O}_D$ (considered as a sheaf on $D$) has no torsion.