Are there examples of Hilbert scheme $H$ of curves in $\mathbb{P}^3$ such that there exists an irreducible component $L$ of $H$ such that for any two points in $L$, there exist smooth projective curves $C_1,...,C_n$ such that $C_i \cap C_{i-1} \not= \emptyset$ and $\cup_{i=1}^n C_i$ passes through both of them?
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$\begingroup$ 1. I think you want to assume the curves are rational. : ) 2. Doesn't the Hilbert scheme of lines / degree 1 genus 0 curves, which is just the Grassmanian, which is rationally connected, provide an example? More generally, any degree and genus such that complete intersections form an irreducible component will work. $\endgroup$– Will SawinCommented Feb 13, 2014 at 23:14
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$\begingroup$ @Sawin: Thank you for your answer. In response to your comment 1, I know that my question clashes with the title, but I wanted to give slightly more freedom in the question. I wanted to include the bigger class of smooth curves rather than restricting to just rational curves although the title implies that. $\endgroup$– user46578Commented Feb 14, 2014 at 0:03
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$\begingroup$ All connected schemes are connected by smooth curves in this manner. At least all projective ones are, and Hilbert schemes are projective. $\endgroup$– Will SawinCommented Feb 14, 2014 at 2:12
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$\begingroup$ @user46578: As Will points out, your question as it stands doesn't make sense. Please edit. $\endgroup$– abxCommented Feb 14, 2014 at 6:25
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1$\begingroup$ Have you read Hartshorne's paper "Connectedness of the Hilbert scheme"? I seem to recall that he proves something along the lines of what you want. $\endgroup$– Daniel LoughranCommented Feb 14, 2014 at 11:09
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