in a surface otherwise we could choose the constraints so that $C$ and $C'$ avoid the surface). Again by bend-and-break, the curve $D$ cannot deform once $C$ and $C'$ are fixed. Thus the moduli space has expected dimension at this point, although possibly non-reduced. So the Gromov-Witten invariant is always positive.

The second case cannot happen if we choose the points to be general. In fact, the degree $1$ curve deforms in a divisor, and it has to connect all the three free curves. By the bend-and-break argument, it cannot deform once the two degree $2$ curves are fixed. But then the degree $3$ curve will miss this curve once the point is chosen to be general.

In the third case one of the degree $1$ curve will connect two degree $2$ curves. Then the union of them gives a non-zero Gromov-Witten invariant of the form $\langle [pt], [pt]\rangle^X_{0, \tilde{C}}$ by the same argument as the first case.

proof of Theorem \[thm:main\]
=============================

Throughout the section, let $X$ be a smooth projective Fano $4$-fold of pseudo-index at least $2$.

The following lemma is taken from [@AG_Fano], Lemma 7.2.16 (ii). The statement is due to Wi[ś]{}niewski [@Wis] but stated only for 