CONTENTS Acknowledgements...................................................................................................5 Introduction..............................................................................................................6 Notations..................................................................................................................8 Chapter I. Zariski surfaces: definition and general properties................................10 Chapter II. The theory of adjoints..........................................................................14 Introduction..........................................................................................................14 §1. Behaviour of $ω_X$ under restriction to open subschemes............................15 §2. Behaviour of $ω_X$ under certain birational maps..........................................16 §3. Affine case, adjoints........................................................................................16 §4. Projective case................................................................................................20 §5. Theory of l-adjoints for l ≥ 1 (an outline).........................................................24 §6. Valuation theory for differentials.....................................................................25 Chapter III. An example relating to a question of O. Zariski...................................28 Introduction..........................................................................................................28 Part 1. Equation of the surface, singularities in characteristic 2..........................29 Part 2. .................................................................................................................33 Part 3. The singularity of F̅ at infinity....................................................................44 §1. Introductory remarks...................................................................................45 §2. Resolution of the singularity at infinity.........................................................47 §3. Behaviour of differentials at infinity under the resolution.............................52 Part 4. Conclusion................................................................................................57 Part 5. Appendix...................................................................................................58 Chapter IV. Generic Zariski surfaces..........................................................................64 Chapter V. Richness of the class of Zariski surfaces.................................................76 References................................................................................................................80
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