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