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Given a symplectic three-fold $(M,\omega)$ we
show that for a generic almost complex structure $J$
which is compatible with $\omega$,
there are finitely many $J$-holomorphic curves in $M$ of any genus $g\geq 0$
representing a homology class $\beta$ in $\Ht_2(M,\Z)$
with $c_1(M).\beta=0$, provided that the divisibility of $\beta$ is at most $4$ (i.e.
if $\beta=n\alpha$ with $\alpha\in\Ht_2(M,\Z)$ and $n\in \Z$ then $n\leq 4$).
Moreover, each such curve is embedded and $4$-rigid.
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