Compactness as a consequence of the adjunction formula for genus second homology class

Question

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.

Answer 1

Assume your 4-manifold is minimal (otherwise there are multiply covered exceptional spheres which potentially give noncompactness). Then it's a computational check to see that: If $d(\alpha)=0$ with $\alpha=k[C]$ represented by a $J$-curve that is a $k$-fold cover of the underlying $J$-curve $C$ (assume connected for simplicity), then $k=1$ unless $C$ is a square-zero torus. Indeed, $d(k[C])=k\cdot d([C])+(k^2-k)[C]^2$ and note that $d([C])\ge0$ for generic $J$.

Now we can invoke some Gromov compactness. [And requiring that if $k>1$ and $J$ generic then the unbranched multiple covers of the square-zero tori are cut out transversally.]