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Fix an integer $N$, $X$ a (smooth) complete intersection subvariety in $\mathbb{P}^N$. Denote by $P$ the Hilbert polynomial of $X$ (as a subvariety in $\mathbb{P}^N$). Consider the Hilbert scheme $\mbox{Hilb}_P$, parametrizing all subscheme in $\mathbb{P}^N$ with Hilbert polynomial $P$. Let $H$ be an irreducible component of $\mbox{Hilb}_P$ containing the point corresponding to $X$. Then,

1) Do all closed points in $H$ correspond to complete intersection subschemes in $\mathbb{P}^N$ with Hilbert polynomial $P$?

2) If a general element of $H$ correspond to a complete intersection subscheme in $\mathbb{P}^N$ with Hilbert polynomial $P$, then is there a positive answer to question $1$?

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    $\begingroup$ No to both questions. Take for $X$ a complete intersection of 3 quadrics in $\mathbb{P}^4$. A general element of $H$ is of the same type (it is a smooth curve of genus 5 canonically embedded). But $H$ contains closed points corresponding to trigonal curves of genus 5, and these are not complete intersections. $\endgroup$
    – abx
    Commented Jun 21, 2017 at 12:59
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    $\begingroup$ Easier, I think. Consider the Hilbert scheme of $4$ points in $\mathbb{P}^2$. Four general points are the complete intersection of two conics, but if three of the points become colinear, they are not a complete intersection. If all four become colinear, they are a complete intersection but of a different sort: line intersect degree four. $\endgroup$
    – David E Speyer
    Commented Jun 21, 2017 at 18:50

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