Let $S$ be a smooth complex surface, If $c_1(S) \in N_1(S)$ is nef and non-torsion, then we know that this would imply some restrictions on the cone of effective curves (and surface itself)--see the discussion at the end of Borcea's paper on Horrocks-Mumford quintics--

In fact we will find that the cone of effective curves is generated by rational curves and possibly $c_1(S)$ itself, accumulating at most to $\mathbb{R}_{+} c_1(S)$.

My question is:

How much do we know about smooth surfaces with effective non-torion first Chen class?

Just remember that any effective curve has a unique Zariski decomposition

$$ H+N $$

where $H$ is nef, $N=\sum a_i \Gamma_i$, $a_i >0$, with neg-definite intersection matrix $(\Gamma_i\cdot \Gamma_j)$ and $H$ is orthogobal to all $\Gamma_i$.