low degree very free rational curve (i.e. a rational curve on which the pull back of the tangent bundle of $X$ is ample). Then an easy argument involving bend-and-break allows one to understand the boundary of the moduli space. When the variety has higher Picard number, we use Mori contractions to understand the geometry of the variety. Then we could either reduce this problem to lower dimensional cases or construct low degree very free curves.

Our technique also gives the following theorem.

Let $Y$ be a Fano $4$-fold and $f: C \to Y$ 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 $\langle [pt], [pt], [pt] \rangle^Y_{0, C}$ or $\langle [pt], [pt] \rangle^Y_{0, \tilde{C}}$.

\[rem:balance\] Let $X$ be a Fano $4$-fold of Picard number $1$ and $C$ be a free curve of minimal degree. Thus $-K_X \cdot C$ is $2, 3, 4$ or $5$. Then the quotient construction of Koll[á]{}r-Miyaoka-Mori gives an embedded very free curve of $-K_X$ degree $8$ except when $-K_X \cdot C=3, 5$. The question is whether or not the normal bundle of a general such 