F=2$ and $n=2$.

5.  $a=4, c=2$. Then $-K_X \cdot F=2$, $3$ or $4$ and $n=0$. But in this case the section $D$ is a moving curve. So $-K_X \cdot D \geq 2$. Thus $-K_X \cdot F$ cannot be $4$.

In the cases that has not been ruled out, $c\geq n+1$. Thus there is a very free curve (in the surface $\tilde{\Sigma}$) of the class $D+(c-1)F$ since the pair $(\tilde{\Sigma}, C)$ is equivalent to $({{\mathbb}{F}}_n, \sigma)$ . It deforms to a very free curve in $X$ with smaller $-K_X$ degree.

So we have proved the following.

\[lem:balance\] Let $X$ be a smooth projective Fano $4$-fold with pseudo-index at least $2$. Let $C$ be a very free curve with minimal $-K_X$ degree and general in an irreducible component of the moduli space. Assume that $-K_X \cdot C$ is $7$ or $8$. Then the normal bundle is not ${{\mathcal}{O}}(1)\oplus {{\mathcal}{O}}(1)\oplus{{\mathcal}{O}}(a), a=3$ or $4$.

Now we are ready to finish the proof of Theorem \[thm:lowdegree\].

Let $C$ be a very free curve with minimal $-K_X$ degree. If $-K_X \cdot C \leq 7$, then any reducible curve in the curve class $[C]$ and passing through $2$ very general points has at most $3$ irreducible components. 