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We prove the Kawamata-Viehweg vanishing theorem for surfaces of del Pezzotype over perfect fields of positive characteristic $p>5$. As a consequence, weshow that klt threefold singularities over a perfect base field ofcharacteristic $p>5$ are rational. We show that these theorems are sharp byproviding counterexamples in characteristic $5$.

On the Kawamata-Viehweg vanishing theorem for log del Pezzo surfaces in positive characteristic