I was having a look at Exercise V.21(1) of Beauville's "Complex Algebraic Surfaces", where it is asked to classify surfaces with ample anticanonical system. These are the surface $\mathbb{P}^1 \times \mathbb{P}^1$ and the blowups of $\mathbb{P}^2$ in $r$ points, with $r \leq 8$.
The hint is to use Castelnuovo's rationality criterion ($P_2 = q = 0$) in order to prove that such a surface is rational.
One can use Kodaira's vanishing theorem to prove that if the anticanonical system $|-K|$ is ample then $q = 0$. This theorem is beyond the scope of that book, though. Is there an easier method?
Moreover, this theorem requires the ground field to have characteristic zero. Is the classification the same also in positive characteristic?