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We present counterexamples to Fujita's conjecture in positivecharacteristics. Precisely, we show that over any algebraically closed field$k$ of characteristic $p>0$ and for any positive integer $m$, there exists asmooth projective surface $S$ with an ample Cartier divisor $A$ such that theadjoint linear system $|K_S+mA|$ is not free of base point. Our surface $S$ isa certain kind of generalization of Raynaud surfaces.

Counterexamples to Fujita's conjecture on surfaces in positive characteristic