Consider the surfaces $S$ for which the $p_g=0=q$ (of which there are many) and in addition that there is a surjection $\oplus_{D_i} H^0 (D_i;\mathbb{Q}) \to H^2(S; \mathbb{Q})$ by Gysin maps for $D_1, \dots, D_k$ a finite collection of rational curves on $S$, which will then generate the Picard group. I'm very tempted to say that such a surface must be rational, but I haven't been able to show this using combinations of the Noether/genus/RR formulas to modify the Castelnuovo rationality criterion.
Does anyone have an idea of a simple argument that might show that these surfaces are rational, or perhaps a counterexample?