## Resolution of SingularitiesThe notion of singularity is basic to mathematics. In algebraic geometry, the resolution of singularities by simple algebraic mappings is truly a fundamental problem. It has a complete solution in characteristic zero and partial solutions in arbitrary characteristic. The resolution of singularities in characteristic zero is a key result used in many subjects besides algebraic geometry, such as differential equations, dynamical systems, number theory, the theory of $\mathcal{D}$-modules, topology, and mathematical physics. This book is a rigorous, but instructional, look at resolutions. A simplified proof, based on canonical resolutions, is given for characteristic zero. There are several proofs given for resolution of curves and surfaces in characteristic zero and arbitrary characteristic. Besides explaining the tools needed for understanding resolutions, Cutkosky explains the history and ideas, providing valuable insight and intuition for the novice (or expert). There are many examples and exercises throughout the text. The book is suitable for a second course on an exciting topic in algebraic geometry. A core course on resolutions is contained in Chapters 2 through 6. Additional topics are covered in the final chapters. The prerequisite is a course covering the basic notions of schemes and sheaves. |

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### Contents

1 | |

3 | |

Chapter 3 Curve Singularities | 17 |

Chapter 4 Resolution Type Theorems | 37 |

Chapter 5 Surface Singularities | 45 |

Chapter 6 Resolution of Singularities in Characteristic Zero | 61 |

Chapter 7 Resolution of Surfaces in Positive Characteristic | 105 |

Chapter 8 Local Uniformization and Resolution of Surfaces | 133 |

Chapter 9 Ramification of Valuations and Simultaneous Resolution | 155 |

Appendix Smoothness and Nonsingularity II | 163 |

179 | |

185 | |

Back Cover | 189 |

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### Common terms and phrases

affine algebraically closed field assume assumption basic object birational blow-up blowing called Chapter characteristic zero closed point completion conclusions condition consider construct contained Corollary curve Dedekind domain defined Definition dimension dominates embedded equation example exceptional divisor exists extension field of characteristic finite follows further give given hold hypersurface ideal sheaf implies induction infinite integer irreducible isomorphism Lemma locus maximal ideal Maxw-ord monoidal transform monomial morphism multiplicity natural neighborhood non-singular non-singular curve normal notation pairs permissible polynomial positive possible prepared prime principal projective proof proper prove rank reduced regular local ring regular parameters Remark residue resolution of singularities satisfy separable sequence simple Sing SNCs spec(R strict transform subset subvariety Suppose surface Theorem uniformization unique valuation ring variety