Let $(M,\omega)$ be a symplectic manifold, and let $\mathscr{J}$ be the set of compatible almost complex structures on $M$.Finally let $A \in H^2(M,\mathbb{Z})$. Then we can consider the moduli space of parametrised J-holomorphic curves defined as follows.


$\mathscr{M}(A,J) = \{ (u,J)~|~ J\in \mathscr{J} ~ \text{ and u is J-holomorphic curve} ~\text{such that}~[u]= A\}$.We know that the automorphism group $G=\text{PSL}(2,\mathbb{C})$ of the sphere acts on $\mathscr{M}(A,J)$.


I read in a couple of papers the comments that $\mathscr{M}(A,J)/G$ can be compactified by adding cusp curves i.e connected unions of possible multi-covered J-holomorphic curves, but the "right" notion for the compactification of this space is by considering the finer notion of stable maps but this "simpler" notion suffices for all purposes in the paper. 


Could someone please explain why compactification by stable curves is "better" and under what conditions can we work with just cusp curves?



Moduli space of curves
Convergence to cusp curves is the original compactification by Gromov, whereas convergence to stable maps is the compactification by Kontsevich (a cusp curve corresponds to the image of a stable map). The latter is more accurate if you want to correctly model the topology of the moduli space (with a fixed number of marked points), so as to build ``virtual (pseudo)cycles'' associated with evaluation maps on the moduli spaces and hence use the full power of the Gromov-Witten invariants. I believe an explanation is in Chapters 5+7 of McDuff-Salamon's big book. (There they mention that if your manifold is semi-positive then the usual Gromov-Witten invariant can be built without the finer notion of stable maps.)  

Think of it this way: You could also take the one-point compactification, but that probably won't tell you a lot.


