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
Steven Dale Cutkosky
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