Recall the adjunction formula $$ g(\alpha) = 1 + \frac{1}{2}\left( \alpha^2 -c_1(X)\cdot \alpha \right)$$ where $g(\alpha)$ is the genus of a pseudoholomorphic representative of the Poincaré dual of $\alpha$, $A=PD(\alpha)\in H^2(X;\Bbb Z)$, for a symplectic $4$-manifold $(X,\omega, J)$. The expected dimension of the space of such surfaces representing $\alpha$ is $$d(\alpha)=\frac{1}{2}\left(\alpha^2+c_1(X)\cdot \alpha\right) $$ In a survey of M. Usher on the Gromov-Taubes invariants, I read the following sentence: >from these formulas, one can verify that, for generic $J$, the only source of noncompactness of the moduli space arises from the fact that, for some $T=H_2(X;\Bbb Z)$ and $m>1$, a sequence of embedded square-zero tori representing a class $mT$ might converge to a double cover of a torus in class $T$. **so here's my question for you:** I don't quite see why this statement is a consequence of the adjunction formula: I can see that a square-zero ($\alpha^2=0$) torus must satisfy $c_1(X)\cdot \alpha=0$. Therefore $d(\alpha)=0$. Moreover I'm aware that a multiple cover of a torus must be a torus by looking at the Euler characteristics. Why is this the only possible source of non-compactness in our moduli space? why in other cases (other genus for example) the adjunction formula implies compactness? I originally posted this question on math.stackexchange but got no answer there so I moved it here.