Counterexamples to Fujita's conjecture on surfaces in positive characteristic

We present counterexamples to Fujita’s conjecture in positive characteristics. Precisely, we show that over any algebraically closed field $k$ of characteristic $p>0$ and for any positive integer $m$, there exists a smooth projective surface $S$ with an ample Cartier divisor $A$ such that the adjoint linear system $\mathrm{\mid }{K}_S+mA\mathrm{\mid }$ is not free of base point. Our surface $S$ is a certain kind of generalization of Raynaud surfaces.