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.