both $C_1$ (or $C_4$) and $\Gamma$. Thus we may assume $C_2$ meet the surface $\Sigma$ and $C_1$. Then $C_2$ has to be a general point in the irreducible component of dimension $3$, thus a free curve. Note that $C_3$ cannot deform in a divisor otherwise $C_3$ cannot meet $\Gamma$. That is, $C_3$ deforms to a free curve and thus the dimension of the Hom-scheme has expected dimension at $C_3$. By Lemma \[lem:smoothing\], the union $C_2 \cup C_3 \cup C_4$ deforms to an irreducible free curve. But since $C_1$ is free, the union of $C_1, \ldots, C_4$ deforms to an irreducible free curve. So this reducible curve lies in an irreducible component of expected dimension. Then if we choose the constraints to general, such reducible curves will not contribute to the Gromov-Witten invariant.

We finish this section by another theorem about very free rational curves of low degree.

\[thm:BalancedCurve\] Let $X$ be a Fano $4$-fold and $f: C \to X$ be an embedded rational curve. Assume that the normal bundle of $C$ is isomorphic to ${{\mathcal}{O}}(2) \oplus {{\mathcal}{O}}(2) \oplus {{\mathcal}{O}}(2)$. Then there is a non-zero Gromov-Witten invariant of the form $\langle [pt], [pt], [pt] \rangle^X_{0, C}$ or $\langle [pt], [pt] \rangle^X_{0, 