vm Preface in algebraic geometry, and would provide an exciting direction after the ba- sic notions of schemes and sheaves have been covered. A core course on resolution is covered in Chapters 2 through 6. The major ideas of resolution have been introduced by the end of Section 6.2, and after reading this far, a student will find the resolution theorems of Section 6.8 quite believable, and have a good feel for what goes into their proofs. Chapters 7 and 8 cover additional topics. These two chapters are inde- pendent, and can be chosen as possible followups to the basic material in the first 5 chapters. Chapter 7 gives a proof of resolution of singularities for surfaces in positive characteristic, and Chapter 8 gives a proof of local uniformization and resolution of singularities for algebraic surfaces. This chapter provides an introduction to valuation theory in algebraic geometry, and to the problem of local uniformization. The appendix proves foundational results on the singular locus that we need. On a first reading, I recommend that the reader simply look up the statements as needed in reading the main body of the book. Versions of almost all of these statements are much easier over algebraically closed fields of characteristic zero, and most of the results can be found in this case in standard textbooks in algebraic geometry. I assume that the reader has some familiarity with algebraic geometry and commutative algebra, such as can be obtained from an introductory course on these subjects. This material is covered in books such as Atiyah and MacDonald [13] or the basic sections of Eisenbud's book [37], and the first two chapters of Hartshorne's book on algebraic geometry [47], or Eisenbud and Harris's book on schemes [38]. I thank Professors Seshadri and Ed Dunne for their encouragement to write this book, and Laura Ghezzi, Tai Ha, Krishna Hanamanthu, Olga Kashcheyeva and Emanoil Theodorescu for their helpful comments on pre- liminary versions of the manuscript. For financial support during the preparation of this book I thank the National Science Foundation, the National Board of Higher Mathematics of India, the Mathematical Sciences Research Insititute and the University of Missouri. Steven Dale Cutkosky
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