I was looking at a proof of the following theorem...
Let $S/\mathbb{C}$ be a smooth projective surface, let $C$ be a non-singular irreducible curve on $S$. Then for all $L \in \text{Pic}\,S$, the intersection number $\langle \mathcal{O}_S(C), L \rangle$ is equal to $\text{deg}(L|_C)$.
Proof: we have an exact sequence $0 \to \mathcal{O}_S(-C) \to \mathcal{O}_S \to \mathcal{O}_C \to 0$ which remains exact upon tensoring with $L^{-1}$, hence giving an exact sequence $0 \to L^{-1}(-C) \to L^{-1} \to L^{-1} \otimes \mathcal{O}_C \to 0$. By additivity of the Euler-Poincaré characteristic, we get $\chi(\mathcal{O}_S) - \chi(\mathcal{O}_S(-C)) = \chi(\mathcal{O}_C)$ and $\chi(L^{-1}) - \chi(L^{-1}(-C)) = \chi(L|_C^{-1})$. This allows us to write the intersection number as $\langle \mathcal{O}_S(C) , L \rangle = \chi(\mathcal{O}_C) - \chi(L|_C^{-1}) = -\text{deg}(L|_C^{-1}) = \text{deg}(L|_C)$, by Riemann-Roch.
Question
Where did we use the fact that $C$ is non-singular? It has to be essential, but I don't see why...
References: