greater than $3$, then we may deform the curve $C_2$ with $3$ points fixed. Then by bend-and-break, it breaks into a reducible or non-reduced curve, still passing through the fixed points. The $3$ points are intersection points of $C_1, C_3$ and $\Gamma$ with the divisor, and thus can be chosen to be general points in the divisor. Then we get a rational curve $D$, necessarily with $-K_X$ degree $2$, passing through $2$ general points in the divisor. The comb consisting of $C_1, C_3$ and $D$ can be smoothed to give a very free curve with smaller $-K_X$ degree by Lemma \[lem:smoothable\]. This is a contradiction since we assume $C$ is a very free curve with minimal $-K_X$ degree. Thus the Hom-scheme has dimension at most $9$. Then a similar argument using the estimate \[eq:1\] as in Lemma \[lem:smoothable\] shows that this reducible curve lies in a component whose general points parametrizes a free curve, in particular, has expected dimension. Thus the first case has no contributions to the Gromov-Witten invariant.

In the remaining cases, note the following.

The curve $C_2$ (resp. $C_3$) cannot deform in a surface.

If $C_2$ (resp. $C_3$) deforms in a surface, then it can meet neither 