Preface The notion of singularity is basic to mathematics. In elementary algebra singularity appears as a multiple root of a polynomial. In geometry a point in a space is non-singular if it has a tangent space whose dimension is the same as that of the space. Both notions of singularity can be detected through the vanishing of derivitives. Over an algebraically closed field, a variety is non-singular at a point if there exists a tangent space at the point which has the same dimension as the variety. More generally, a variety is non-singular at a point if its local ring is a regular local ring. A fundamental problem is to remove a singularity by simple algebraic mappings. That is, can a given variety be desingularized by a proper, birational morphism from a non-singular variety? This is always possible in all dimensions, over fields of characteristic zero. We give a complete proof of this in Chapter 6. We also treat positive characteristic, developing the basic tools needed for this study, and giving a proof of resolution of surface singularities in positive characteristic in Chapter 7. In Section 2.5 we discuss important open problems, such as resolution of singularities in positive characteristic and local monomialization of mor- phisms. Chapter 8 gives a classification of valuations in algebraic function fields of surfaces, and a modernization of Zariski's original proof of local uni- formization for surfaces in characteristic zero. This book has evolved out of lectures given at the University of Mis- souri and at the Chennai Mathematics Institute, in Chennai, (also known as Madras), India. It can be used as part of a one year introductory sequence vn
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