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

Abstract

We prove the Kawamata-Viehweg vanishing theorem for a large class of divisors on surfaces in positive characteristic. By using this vanishing theorem, Reidertype 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. Contents 1. Introduction 1 2. Notations and terminology 4 3. Chain-connected divisors 4 4. Vanishing theorem on H 1 9 5. Adjoint linear systems for effective divisors 16 6. Extension theorems in positive characteristic 23 7. Applications to plane curves 24 Appendix A. Mumford's intersection form on a normal projective variety 25 References 27Key words and phrases. vanishing theorem, adjoint linear system, extension theorem, plane curve.[44] H. Terakawa, The k-very ampleness on a projective surface in positive characteristic,