rational curve through a general points has $-K_X$ degree at least $2$. By Lemma \[lem:smoothable\], we only need to consider the following cases since in all the other cases, the reducible curve lies in an irreducible component whose general point parametrizes a very free curve.

1.  The curve has three irreducible component $C_1, C_2$ and $C_3$ with $-K_X$ degree $2, 4$, and $2$. The degree $2$ curves are free curves.

2.  The curve has four irreducible components $C_1$ through $C_4$ with $-K_X$ degree $2$ each. The curves $C_1$ and $C_4$ are free curves. And the deformation of $C_2$ or $C_3$ sweeps out a surface.

3.  The curve has four irreducible components $C_1$ through $C_4$ with $-K_X$ degree $2$ each. The curves $C_1$ and $C_4$ are free curves. And the curves $C_2$ and $C_3$ deform in divisors.

4.  The curve has four irreducible components $C_1$ through $C_4$ with $-K_X$ degree $2$ each. The curves $C_1$ and $C_4$ are free curves. And either $C_2$ or $C_3$ (or both) deforms to a free curve.

In all cases, the free curve $C_1$ (resp. $C_3$ in case one or $C_4$ in other cases) is fixed and cannot meet a general curve $\Gamma$ once the point 