VI Contents §5.2. Embedded resolution of singularities 56 Chapter 6. Resolution of Singularities in Characteristic Zero 61 §6.1. The operator A and other preliminaries 62 §6.2. Hypersurfaces of maximal contact and induction in resolution 66 §6.3. Pairs and basic objects 70 §6.4. Basic objects and hypersurfaces of maximal contact 75 §6.5. General basic objects 81 §6.6. Functions on a general basic object 83 §6.7. Resolution theorems for a general basic object 89 §6.8. Resolution of singularities in characteristic zero 99 Chapter 7. Resolution of Surfaces in Positive Characteristic 105 §7.1. Resolution and some invariants 105 §7.2. r(q) = 2 109 §7.3. r(q) = 1 113 §7.4. Remarks and further discussion 130 Chapter 8. Local Uniformization and Resolution of Surfaces 133 §8.1. Classification of valuations in function fields of dimension 2 133 §8.2. Local uniformization of algebraic function fields of surfaces 137 §8.3. Resolving systems and the Zariski-Riemann manifold 148 Chapter 9. Ramification of Valuations and Simultaneous Resolution 155 Appendix. Smoothness and Non-singularity II 163 §A.l. Proofs of the basic theorems 163 §A.2. Non-singularity and uniformizing parameters 169 §A.3. Higher derivations 171 §A.4. Upper semi-continuity of vq(T) 174 Bibliography 179 Index 185
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