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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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  • $\begingroup$ What is the second map? $\endgroup$
    – Sasha
    Commented Nov 27, 2012 at 18:38
  • $\begingroup$ 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. $\endgroup$
    – Damian Rössler
    Commented Nov 27, 2012 at 23:31
  • $\begingroup$ @Rossler: Sorry, I meant to write a local complete intersection curve. Will update the question. $\endgroup$
    – Naga Venkata
    Commented Nov 28, 2012 at 14:20
  • $\begingroup$ @Sasha: The second map arises from the natural morphism: $N_{C_r|\mathbb{P}^3} \to i^*N_{C|\mathbb{P}^3}$ $\endgroup$
    – Naga Venkata
    Commented Nov 28, 2012 at 14:35
  • $\begingroup$ That is my question. How do you construct such a morphism? $\endgroup$
    – Sasha
    Commented Nov 28, 2012 at 18:26

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