Table Of ContentSpringer Monographs in Mathematics
Springer-Verlag Berlin Heidelberg GmbH
S. S. Abhyankar
Resolution of Singularities
of
Embedded Aigebraic Surfaces
Second, Enlarged Edition
i
Springer
Shreeram S. Abhyankar
Purdue University
Department of Mathematies
West Lafayette, IN 47907
USA
The original edition was published in 1966 by Aeademie Press, Ine. in the
series Pure and Applied Mathematics, val. 24
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Abhyankar, Shrecram S.:
Resolution of singularitics of emhe!ldcd algebraic surfaces / S. S.
Abhyankar. -2., eu!. cd. -Berlin ; Heidelherg ; New York ; Barcelona
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Singapnrc ; Tokyo : Springer, I991!
(Springer moullgraphs in mathematics)
ISBN 978-3-642-08351-8 ISBN 978-3-662-03580-1 (eBook)
DOI 10.1007/978-3-662-03580-1
Mathematics 5ubject Classification (1991): 13F, 13H, I4E, 14J,325
ISBN 978-3-642-08351-8
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© Springer-Verlag Berlin Heidelberg 1998
Originally published by Spriuger-Verlag Berlin Heidelberg New York in 1998
Softcover repriot of the harocover 2nd edition 1998
The use of general descriptive names, registered names, trademarks etc. in this publication does not
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SPIN 10648305 44/3143-543210-Printedon acid-free paper
Dedicated to
Professor Oscar Zariski
Without his blessings who can resolve the singularities?
Preface to the Second Edition
The common solutions of a finite number of polynomial equations in
a finite number of variables constitute an algebraic variety. The degrees
of freedom of a moving point on the variety is the dimension of the
variety. A one-dimensional variety is a curve and a two-dimensional
variety is a surface. A three-dimensional variety may be called asolid.
Most points of a variety are simple points. Singularities are special
points, or points of multiplicity greater than one. Points of multiplicity
two are double points, points of multiplicity three are tripie points,
and so on. A nodal point of a curve is a double point where the curve
crosses itself, such as the alpha curve. A cusp is a double point where
the curve has a beak. The vertex of a cone provides an example of a
surface singularity. A reversible change of variables gives abirational
transformation of a variety. Singularities of a variety may be resolved
by birational transformations.
In the last century, resolution of singularities of curves was achieved
by Riemann, Noether and Dedekind by analytic, geometrie and alge
braic methods, respectively; for a historical overview of the resolution
problem see my expository article [AB]; here items [Al] to [A17] re
fer to the Additional Bibliography and items [1] to [26] refer to the
original Bibliography. Then, in case of characteristic zero, after sev
eral attempts by the Italian geometers such as Albanese [11] and Levi
[16] at the turn of the century, surface desingularization was achieved
by Zariski [A17] , who soon followed it up by solid desingularization
[25], which was brilliantly generalized by Hironaka [15] for higher
dimensional varieties. In my Ph.D. thesis [2], I gave a proof of surface
desingularization in case of characteristic p, which later on I extended
first to arithmetical surfaces [A2] and then to two-dimensional excellent
schemes [A3]. Briefly, when the coefficients of the defining equations
of an algebraic curve are integers, by reducing them modulo various
prime numbers we get a family of algebraic curves over fields of dif
ferent characteristics, and the resulting total object is called an arith-
vii
viü PREFACE TO THE SECOND EDITION
metieal surface. A two-dimensional scheme is a further generalization
of this concept.
The present book contains the geometrie part of the proof of solid
desingularization in characteristie p 1= 2,3,5 whieh I obtained in 1965;
the algorithmie part is contained in my four previous artieies [5] to [9];
the book does contain an alternative simple version of the algorithm for
characteristie zero; half of the book can also be used as an introduction
to birational algebraie geometry. I am thankful to Springer-Verlag for
reprinting this book which was first published in 1966 by Academie
Press. It may be hoped that this would stimulate other investigators
to settle the general desingularization problem for higher-dimensional
algebraic as weH as arithmetieal varieties. A discussion of this topie
addressed to scientists and engineers may be found in my 1990 book
[A7] whieh, according to the recent article [All] by Hauser, "provides
a description of the state of the art in resolution of singularities and
related problems."
There is also the quest ion of canonieal processes of desingulariza
tion. Such a process for algebraic as weH as arithmetieal curves is
described in my 1983 article [A4] and revisited in my 1997 article
[A9]. Moreover, a discussion of such processes for higher-dimensional
varieties in characteristic zero can be found in my monographs [A5]
and [A6] of 1983 and 1988, respectively. These discussions together
with various incarnations of the trick employed in item (10.24) of the
present book have recently led me to discover a short proof of analytie
desingularization in characteristie zero for any dimension on which I
gave a lecture in various places in 1996-97. The text ofthat lecture is
inserted as an Appendix to the present new edition of the book.
Shreeram S. Abhyankar
West Lafayette
16 July 1997
Preface to the First Edition
Some twenty years ago there appeared, in the Annals 0/ Mathe
matics, the marvelous memoir of Zariski entitled: Reduetion of
singularities of algebraie three-dimensional varieties. Not only was
a daring and ingenious solution of a diffieult problem given in it,
but so mueh of the teehnique invented for the solution has proved
to be of such signifieanee for algebraie geometry in general!
Hironaka's brilliantly energetie reeent solution of the general
resolution problem for zero eharaeteristie eonstitutes, indeed, a
high tribute to Zariski's memoir.
At present I am able to pay only a modest tribute to Zariski's
memoir by giving a self-eontained exposition of it. This then is
the primary aim of the monograph.
A seeondary aim is to partially extend some of the results to
nonzero eharaeteristie. The algorithm needed for such an extension
has already been published in four papers, and it will not be
repeated here. This monograph eontains the geometrie part of
the argument. However, we do include an alternative simple
version of the algorithm for zero eharaeteristie thereby making
the monograph self-eontained for that ease.
Finally, the matter is so arranged that ab out half of the monograph
ean be used as an introduction to certain foundational aspects of
algebraie geometry.
My thanks are due to Annette Wortman for an exeellent job of
typing the manuseript. The work on this monograph was partially
supported by the National Seienee Foundation under NSF-GP-
4248-50-395 at Purdue University; I am grateful for this support.
September, 1966 S.S.A.
Purdue University
ix
Contents
Preface to the Second Edition vii
Preface to the First Edition ix
o
Introduetion 1
CHAPTER 1. Local Theory
1 Terminology and preliminaries 7
2 Resolvers and principalizers 45
3 Dominant eharaeter of anormal sequenee 61
4 Unramified loeal extensions 108
5 Main results 148
CHAPTER 2. Global Theory
6 Terminology and preliminaries 155
7 Global resolvers 192
8 Global principalizers 223
9 Main results 233
CHAPTER 3. Some Cases of Three-Dimensional
Birational Resolution
10 Uniformization of points of small multiplicity 238
11 Three-dimensional birational resolution over a
ground field of eharaeteristic zero 261
12 Existenee of projective models having only points of
small multiplicity 262
13 Three-dimensional birational resolution over an
i=
algebraically closed ground field of eharateristic 2,3,5 283
xi
xii CONTENTS
Appendix on Analytic Desingularization in Chamcteristic Zero 285
Bibliogmphy 301
Additional Bibliogmphy 303
Index 0/ Notation 305
Index 0/ Definitions 307
List 0/ Corrections 311
