If there are $2$ irreducible components, both of them are free. Thus the curve is smoothable to a very free curve. If there are $3$ irreducible components, by Lemma \[lem:smoothable\], it is also smoothable. This means that every irreducible component which parametrizes curves passing through two general points has the expected dimension and a general member is an irreducible free curve. Thus the Gromov-Witten invariant $\langle [pt], [pt], [A]^2, \ldots, [A]^2 \rangle^X_{0, \beta}$ is non-zero and enumerative, where $A$ is a very ample divisor.

Now we consider the case $-K_X \cdot C=8$. Since the normal bundle of a general such curve is not ${{\mathcal}{O}}(1)\oplus {{\mathcal}{O}}(1) \oplus {{\mathcal}{O}}(4)$, the deformation of such a curve with two points fixed sweeps out a divisor in $X$. Thus such curves make a positive contribution to the following Gromov-Witten invariant: $\langle [pt], [pt], [A]^3, [A]^2 \rangle^X_{0, C}$. Again $A$ is a very ample divisor as before. We will call the curve and the surface appearing in the constraints $\Gamma$ and $\Sigma$. Our next goal is to prove that such an invariant is non-zero by showing that no reducible curve can contribute to such invariant.

We first choose two very general points. Note that an irreducible 