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Let $X$ be a projective connected scheme embedded into $\mathbb{P}^n$ for $n \ge 3$. Denote by $i$ the natural inclusion morphism from $X$ into $\mathbb{P}^n$. Denote by $X_H$ the intersection of $X$ with $H$, a general hyperplane in $\mathbb{P}^n$. Denote by $i_H:X_H \hookrightarrow \mathbb{P}^n$ the natural inclusion. Is it true that for a general hyperplane $H$ in $\mathbb{P}^n$, any $\mathcal{O}_{\mathbb{P}^n}$-morphism, $\mbox{Hom}_{\mathbb{P}^n}(\mathcal{I}_X,i_{H_*}\mathcal{O}_{X_H})$ be extended to an element in $\mbox{Hom}_{\mathbb{P}^n}(\mathcal{I}_X,i_*\mathcal{O}_{X})$? In other words, is the natural morphism from $\mbox{Hom}_{\mathbb{P}^n}(\mathcal{I}_X,i_*\mathcal{O}_{X})$ to $\mbox{Hom}_{\mathbb{P}^n}(\mathcal{I}_X,i_{H_*}\mathcal{O}_{X_H})$ surjective for a general choice of $H$?

EDIT Assume $X$ is local complete intersection in $\mathbb{P}^n$

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    $\begingroup$ Write down a long exact sequence and you will see where the obstruction lives. $\endgroup$
    – Sasha
    Feb 13, 2014 at 19:50
  • $\begingroup$ @Sasha: As far as I understand the obstruction lies in $H^1(\mathcal{H}om_{\mathbb{P}^n}(\mathcal{I}_X,i_*\mathcal{O}_X(-1))$. Is there any way of understanding when this vanishes? $\endgroup$
    – user46578
    Feb 21, 2014 at 18:44
  • $\begingroup$ Not quite correct. In fact it lies in $Ext^1(I_X,i_*O_X(-1))$ which is not quite the same as what you wrote. If $X$ is a l.c.i. you can write down Koszul resolutions for $I_X$ and for $i_*O_X(-1)$ and use them to compute the $Ext$-space. $\endgroup$
    – Sasha
    Feb 21, 2014 at 19:15

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