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?



linear system of non-reduced divisor and associated reduced divisors
The second short exact sequence is wrong. You should recognize this without knowing where the mistake is: $D\_{\mathrm{red}}\leq D$, so $\mathscr{O}\_X(D\_{\mathrm{red}}) \subseteq \mathscr{O}\_X(D)$ and so the map you have is surjective if and only if $D\_{\mathrm{red}}= D$. In fact that map is always injective as you discovered... The underlying point is that $\mathscr Hom$ is left exact, but not right exact.


The right computation would be that the dual of 
$$0 \to \mathscr{O}\_X(-D) \to \mathscr{O}\_X(-D\_{red})\to \mathscr F \to 0.$$
gives 
$$0 \to \mathscr Hom\_X(\mathscr F, \mathscr O\_X) \to \mathscr{O}\_X(D\_{\mathrm{red}}) \to \mathscr{O}\_X(D) \to \mathscr Ext^1\_X(\mathscr F, \mathscr O\_X)\to \mathscr Ext^1\_X(\mathscr O\_X(-D\_{\mathrm{red}}), \mathscr O\_X).$$


Now $\mathscr F$ is supported on $D$, so since $\mathscr O\_X$ is torsion free $\mathscr Hom\_X(\mathscr F, \mathscr O\_X)=0$ and $\mathscr O\_X(-D\_{\mathrm{red}})$ is locally free, so $\mathscr Ext^1\_X(\mathscr O\_X(-D\_{\mathrm{red}}), \mathscr O\_X)=0$ and hence you have a short exact sequence:
$$0 \to \mathscr{O}\_X(D\_{\mathrm{red}}) \to \mathscr{O}\_X(D) \to \mathscr Ext^1\_X(\mathscr F, \mathscr O\_X)\to 0.$$


