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Does every irreducible component of a Hilbert scheme of curves in $\mathbb{P}^3$ contain a curve that is a union of lines (not necessarily reduced)? Furthermore, given a curve $C \subset \mathbb{P}^3$ does there exists an IRREDUCIBLE family of curves in $\mathbb{P}^3$ such that $C$ is a general fiber and there exists a fiber which is the union of lines (not necessarily reduced)?

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up vote 2 down vote accepted

I thought somebody had already answered this question for you in an earlier post. Assuming that by "union of lines (not necessarily reduced)" you mean "connected one-dimensional scheme whose irreducible components are lines", then then answer is yes. Take any codimension $2$ linear subspace $M$ that is disjoint from $C$, and let $L$ be a line that is disjoint from $M$. Consider a $\mathbb{G}_m$-action on $\mathbb{P}^n$ that has $L$ and $M$ as eigenspaces. For the induced action of $\mathbb{G}_m$ on the Hilbert scheme, one of the "limit points" of the orbit of $[C]$ will be a scheme supported on $L$.

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