Let $X$ be a smooth degree $d$ $(d \ge 5)$ surface in $\mathbb{P}^3$. Let $D$ be an effective Cartier divisor (hence locally of complete intersection) on $X$ and $D_{red}$ the associated reduced scheme which is also a Cartier divisor on $X$. Then, we have a natural inclusion of ideal sheaves, $$0 \to \mathcal{O}_X(-D) \to \mathcal{O}_X(-D_{red}).$$

Taking the dual we have

$$\mathcal{O}_X(D_{red}) \to \mathcal{O}_X(D) \to 0.$$

Since, $\mathcal{O}_X(D_{red})$ and $\mathcal{O}_X(D)$ are locally free sheaves of rank $1$, this would imply that the kernel of the latter map is zero. This would imply that $\mathcal{O}_X(D_{red}) \cong \mathcal{O}_X(D)$. This result is very surprising. Is there a mistake in the proof or is there an explaination for this behaviour?