Let $X$ be a sympletic manifold and $A\in H\_2(X;\mathbb{Q})$. Let $g$ and $k$ be nonnegative integers.
Assume that $$\mathcal{M}\_{g,k}(X;A)$$


is dense in


$$\overline{\mathcal{M}}\_{g,k}(X;A).$$


Are the primary Gromov-Witten invariants corresponding to $X$ and $A$ enumerative?


If not, when does the condition $$\mathcal{M}\_{g,k}(X;A)$$


dense in


$$\overline{\mathcal{M}}\_{g,k}(X;A)$$ 


imply that the corresponding primary Gromov-Witten invariants are enumerative?



enumerative Gromov-Witten invariants
I am not sure what "primary" means. However, I believe the answer to your first question is "no". For a sufficiently general quintic hypersurface $X$ in $\mathbb{C}P^4$, for sufficiently small curve classes $A$, all genus $0$ curves in $X$ of class $A$ are pairwise disjoint and smooth with normal bundle $\mathcal{O}\_{\mathbb{P}^1}(-1)\oplus \mathcal{O}\_{\mathbb{P}^1}(-1)$. This implies that the moduli space $\mathcal{M}\_{0,k}(X;A)$ are dense in the compactified moduli spaces $\overline{\mathcal{M}}\_{0,k}(X;A)$. Nonetheless, the corresponding Gromov-Witten invariants are fractions, thus not enumerative. The issue is that some of the connected components of $\mathcal{M}\_{0,k}(X;A)$ have dimension that is larger than the "expected dimension", essentially because they parameterize multiple covers of embedded, smooth curves. Once one takes this into account via the multiple-covering formula, the corresponding instanton numbers are expected to be integers and enumerative (although, of course, the degree $10$ curves of Vainsencher - Pandharipande contradict enumerativity even of these instanton numbers). All of this is discussed in Cox-Katz, Example 7.4.4.1, pp. 205-206.


