constraints are irreducible free curves, which in turn implies that it contributes positively to the Gromov-Witten invariant. To do this, we begin with curves meeting two very general points. If this curve is reducible and meet other constraints, then we show that it is smoothable. Also it will lie in a component whose general point parametrize a very free rational curve. In particular, this component has the expected dimension. Then we can choose the constraints such that only general curves (hence irreducible) in the components of expected dimension can meet all the constraints.

We first give $2$ sufficient conditions on when a reducible curve is smoothable.

\[lem:smoothing\] Let $X$ be a smooth projective variety. Let $C=C_1 \cup C_2 \cup C_3$ be a chain of ${{\mathbb}{P}}^1$s and $f: C \to X$ be a stable map. Assume that $C_1$ and $C_2$ are free curves and the Hom-scheme $Hom({{\mathbb}{P}}^1, X)$ has expected dimension $\dim X+(-K_X \cdot C_2)$ at the point $(f: C_2 \to X)$. Then the general point of any irreducible component of the Kontsevich moduli space containing $(C, f)$ parametrizes to an irreducible curve. If one of the free curves contains a very general point, then a very general deformation is free.

