Vanishing theorems and adjoint linear systems on normal surfaces in positive characteristic

We prove the Kawamata-Viehweg vanishing theorem for a large class of divisors on surfaces in positive characteristic. By using this vanishing theorem, Reider-type theorems and extension theorems of morphisms for normal surfaces are established. As an application of the extension theorems, we characterize non-singular rational points on any plane curve over an arbitrary base field in terms of rational functions on the curve.
Comment: 29 pages, v2: Example 3.8 added, some references added