# Normal sheaf of non-reduced locally complete intersection space curves

Let $C$ be a non-reduced locally complete intersection curve on a smooth degree $d$ surface in $\mathbb{P}^3$ (for example a non-reduced Cartier divisor). For simplicity we can assume that $d>deg(C)$. Denote by $C_r$ the reduced scheme associated to $C$. It is clear the normal sheaves, $N_{C|\mathbb{P}^3}$ and $N_{C_r|\mathbb{P}^3}$ are locally free. Also, recall under the closed immersion, $i:C_r \hookrightarrow C$ we have the canonical maps, $i^*:H^0(N_{C|\mathbb{P}^3}) \to H^0(i^*N_{C|\mathbb{P}^3})$ and $H^0(N_{C_r|\mathbb{P}^3}) \to H^0(i^*N_{C|\mathbb{P}^3})$.

The question is whether it is true that the image of the latter map is contained in the image of $i^*$.

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What is the second map? –  Sasha Nov 27 '12 at 18:38
I don't understand why $N_{C/{\bf P}^3}$ is locally free in general, unless you always assume that $C$ is a Cartier divisor. –  Damian Rössler Nov 27 '12 at 23:31
@Rossler: Sorry, I meant to write a local complete intersection curve. Will update the question. –  Naga Venkata Nov 28 '12 at 14:20
@Sasha: The second map arises from the natural morphism: $N_{C_r|\mathbb{P}^3} \to i^*N_{C|\mathbb{P}^3}$ –  Naga Venkata Nov 28 '12 at 14:35
That is my question. How do you construct such a morphism? –  Sasha Nov 28 '12 at 18:26