6
$\begingroup$

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 $\mathbb{P}^n$.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.