We prove the Kawamata–Viehweg vanishing theorem for surfaces of del Pezzo type over perfect fields of positive characteristic $p>5$ . As a consequence, we show that klt threefold singularities over a perfect base field of characteristic $p>5$ are rational. We show that these theorems are sharp by providing counterexamples in characteristic $5$ .

We prove the Kawamata-Viehweg vanishing theorem on surfaces of del Pezzo type over perfect fields of positive characteristic p > 5. As a consequence, we show that klt threefold singularities over a perfect base field of characteristic larger than five are rational. We show that these theorems are sharp by providing counterexamples in characteristic five.

Let X be a smooth complex projective variety of dimension n. We prove bounds on Fujita's basepoint freeness conjecture that grow as n log log(n).

Let X be a smooth complex projective variety of dimension n and let A be an ample and basepoint free divisor. We prove K X + mA satisfies property N p for m n + 1 + p. We also show the graded ring of sections R(X, K X + mA) is Koszul for m n + 2.

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