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Let $X$ be a connected projective scheme in $\mathbb{P}^n$. Assume, $2 \le \dim X \le n-2$. Let $H$ be a general hyperplane in $\mathbb{P}^n$. Denote by $Z:=X.H$ and $Z'=X.H^m$ for $m \gg 0$. Then there exists an inclusion map, $i:Z \hookrightarrow Z'$. The question is: under what condition on $X$, given an infinitesimal deformation of $Z$ in $\mathbb{P}^n$, does there exist an infinitesimal deformation of $Z'$ in $\mathbb{P}^n$ which when restricted to $Z$ gives us the original deformation. I will elaborate on this.

Denote by $j:Z \hookrightarrow \mathbb{P}^n$ and $j_0:Z' \hookrightarrow \mathbb{P}^n$ the closed immersions. We know that the space of infinitesimal deformations of $Z, Z'$ in $\mathbb{P}^n$ are given by $H^0(\mathcal{N}_{Z|\mathbb{P}^n})$ and $H^0(\mathcal{N}_{Z'|\mathbb{P}^n})$, respectively. There are natural morphisms $\pi_1$ (resp. $\pi_2$) from $H^0(\mathcal{N}_{Z|\mathbb{P}^n})$ (resp. $H^0(\mathcal{N}_{Z'|\mathbb{P}^n})$) to $H^0(\mathcal{N}_{Z'|\mathbb{P}^n} \otimes i_*\mathcal{O}_Z)$ arising from applying $\mathcal{H}om_{\mathbb{P}^n}(-,j_*\mathcal{O}_{Z})$ (resp. $\mathcal{H}om_{\mathbb{P}^n}(\mathcal{I}_{Z'},-)$) to the morphisms $\mathcal{I}_{Z'} \hookrightarrow \mathcal{I}_{Z}$ (resp. $j_{0_*}\mathcal{O}_{Z'} \to j_*\mathcal{O}_Z$) and then applying the global section functor. The question above then translates into when can we say that the image of $\pi_1$ is contained in the image of $\pi_2$?

The motivation of the problem lies in the fact that not only are $Z$ and $Z'$ topologically the same, their structure sheaves differ in some sense by a multiple of a hyperplane section. I know that $Z$ and $Z'$ being topologically the same in not sufficient to expect a positive answer to the question.

Note We can assume that $X$ is a local complete intersection in $\mathbb{P}^n$.

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  • $\begingroup$ If your deformation of Z is a subscheme with ideal sheaf J, what happens when you consider the ideal J^m? $\endgroup$ – Vladimir Baranovsky Feb 20 '14 at 21:52
  • $\begingroup$ @Baranovsky: I do not understand the question. "What happens" is a very broad question. Is there any specific property you are inquisitive about? $\endgroup$ – user46578 Feb 21 '14 at 18:01
  • $\begingroup$ If your first order deformation of $Z$ is a subscheme of $X \otimes Spec (k[\varepsilon]/\varepsilon^2)$ with ideal sheaf $J$ then it seems that $J^m$ will define a subscheme deforming $Z'$. Unless I misunderstood the question. $\endgroup$ – Vladimir Baranovsky Feb 21 '14 at 18:10
  • $\begingroup$ @Baranovsky: May be I am getting this wrong. $X \otimes Spec(k[\epsilon]/(\epsilon^2)$ is not a first order deformation of $X$. May be you meant $X \times Spec(k[\epsilon]/(\epsilon^2)$ which is the trivial deformation. It is also not guarenteed that a first deformation of $Z$ comes from a first order deformation of $X$. Could you elaborate on your argument a bit more? $\endgroup$ – user46578 Feb 21 '14 at 18:21
  • $\begingroup$ Ok, if you say that the tangent spaces are $H^0(\mathcal{N}_{Z|\mathbb{P}^n})$ then you are deforming $Z$ and $Z'$ as subschemes of $\mathbb{P}^n$ not of $X$, as I misunderstood. Then taking the $m$-power of the deforming ideal does not quite work. $\endgroup$ – Vladimir Baranovsky Feb 22 '14 at 2:39

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