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Let $X$ be a smooth projective hypersurface in $\mathbb{P}^3$, $|\mathcal{O}_X(a)|$ be the complete linear system for some integer $a>0$. Ofcourse, a general element of the linear system is a smooth curve. Denote by $V$ the subvariety of $|\mathcal{O}_X(a)|$ parametrizing reducible curves (i.e., with at least 2 irreducible components).

Is any irreducible component of $V$ rationally connected? If nothing can be said in general, is there some standard method of approaching the problem. Any reference dealing with similar problems will be most welcome.

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    $\begingroup$ The connected components may be reducible. For instance, for $X$ a smooth quadric surface and $a=2$, your variety $V$ will have three irreducible components. The normalizations will be isomorphic to $\mathbb{P}^1\times \mathbb{P}^5$, $\text{Sym}^2(\mathbb{P}^3)$ and $\mathbb{P}^5\times \mathbb{P}^1$, respectively. The intersection of the components is the image of $\mathbb{P}^1\times \mathbb{P}^3\times \mathbb{P}^1$. Did you want to restrict to an irreducible component of $V$? $\endgroup$
    – Jason Starr
    Commented Sep 28, 2015 at 11:55
  • $\begingroup$ @JasonStarr I have edited the question to irreducible components. $\endgroup$
    – Ron
    Commented Sep 28, 2015 at 12:10
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    $\begingroup$ Since smooth surfaces in $\mathbb{P}^3$ have trivial $\text{Pic}^0$, every irreducible component $V_i$ of $V$ is the image of the product of the complete linear systems for the irreducible components of a curve in $X$ parameterized by a general point of $V_i$. Thus the component is unirational (possibly even rational). The example of curves in a smooth quadric illustrates this, for instance. $\endgroup$
    – Jason Starr
    Commented Sep 28, 2015 at 12:22
  • $\begingroup$ @JasonStarr Thank you very very much. This is very helpful. $\endgroup$
    – Ron
    Commented Sep 28, 2015 at 12:38

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