This book begins with an introduction to algebraic geometry in the language
of schemes. Then, the general theory is illustrated through the study of
arithmetic surfaces and the reduction of algebraic curves. The origin of
this work is notes distributed to the participants of a course on arithmetic
surfaces for graduate students. The aim of the course was to describe the
foundation of the geometry of arithmetic surfaces as presented in  and ,
and the theory of stable reduction
. In spite of the importance of recent developments in these subjects
and of their growing implications in number theory, unfortunately there does
not exist any book in the literature that treats these subjects in a systematic
manner, and at a level that is accessible to a student or to a mathematician
who is not a specialist in the field. The aim of this book is therefore to
gather together these results, now classical and indispensable in arithmetic
geometry, in order to make them more easily accessible to a larger audience.
The first part of the book presents general aspects of the theory of schemes.
It can be useful to a student of algebraic geometry, even if a thorough examination
of the subjects treated in the second part is not required. Let us briefly
present the contents of the first seven chapters that make up this first part.
I believe that we cannot separate
the learning of algebraic geometry from the study of commutative
algebra. That is the reason why the book starts with a chapter on the tensor
product, flatness, and formal completion. These notions will frequently recur
throughout the book. In the second chapter, we begin with Hilbert's
Nullstellensatz, in order to give an intuitive basis for the theory of schemes.
Next, schemes and morphisms of schemes, as well as other basic notions, are
defined. In Chapter 3, we study the fibered product of schemes and
the fundamental concept of base change. We examine the behavior of algebraic
varieties with respect to base change, before going on to proper morphisms
and to projective morphisms. Chapter 4 treats local properties of schemes
and of morphisms such as normality and smoothness. We conclude with an elementary
proof of Zariski's Main Theorem. The global aspect of schemes is approached
through the theory of coherent sheaves in Chapter 5. After studying coherent
sheaves on projective schemes, we define the Cech cohomology of sheaves,
and we look at some fundamental theorems such as Serre's finiteness
theorem, the theorem of formal functions, and as an application, Zariski's
connectedness principle. Chapter 6 studies particular coherent sheaves: the
sheaf of differentials, and, in certain favorable cases (local complete intersections),
the relative dualizing sheaf. At the end of that chapter, we present Grothendieck's
duality theory. Chapter 7 starts with a rather general study of divisors,
which is then restricted to the case of projective curves over a field. The
theorem of Riemann-Roch, as well as Hurwitz's theorem, are proven with the
help of duality theory. The chapter concludes with a detailed study of the
Picard group of a not necessarily reduced projective curve over an algebraically
closed field. The necessity of studying singular curves arises, among other
things, from the fact that an arithmetic (hence regular) surface in general
has fibers that are singular. These seven chapters can be used for
a basic course on algebraic geometry.
The second part of the book is made up of three chapters. Chapter 8 begins
with the study of blowing-ups. An intermediate section digresses towards
commutative algebra by giving, often without proof, some principal results
concerning Cohen-Macaulay, Nagata, and excellent rings. Next, we present
the general aspects of fibered surfaces over a Dedekind ring and the theory
of desingularization of surfaces. Chapter 9 studies intersection theory on
an arithmetic surface, and its applications. In particular, we show the adjunction
formula, the factorization theorem, Castelnuovo's criterion, and the existence
of the minimal regular model. The last chapter treats the reduction theory
of algebraic curves. After discussing general properties that essentially
follow from the study of arithmetic surfaces, we treat the different types
of reduction of elliptic curves in detail. The end of the chapter is devoted
to stable curves and stable reduction. We describe the proof of the stable
reduction theorem of Deligne-Mumford by Artin-Winters, and we give some
concrete examples of computations of the stable reduction.
From the outset, the book was written with arithmetic geometry in mind. In
particular, we almost never suppose that the base field is algebraically
closed, nor of characteristic zero, nor even perfect. Likewise, for the arithmetic
surfaces, in general we do not impose any hypothesis on the base (Dedekind)
rings. In fact, it does not demand much effort to work in general conditions,
and does not affect the presentation in an unreasonable way. The advantage
is that it lets us acquire good reflexes right from the beginning.
As far as possible, the treatment is self-contained. The prerequisites for
reading this book are therefore rather few. A good undergraduate student,
and in any case a graduate student, possesses, in principle, the background
necessary to begin reading the book. In addressing beginners, I have found
it necessary to render concepts explicit with examples, and above all exercises.
In this spirit, all sections end with a list of exercises. Some are simple
applications of already proven results, others are statements of results
which did not fit in the main text. All are sufficiently detailed to be solved
with a minimum of effort. This book should therefore allow the reader to
approach more specialized works such as  and  with more ease.
It is my great pleasure to thank Michel Matignon and Martin Taylor, who encouraged
me to write up my lecture notes. Reinie Erné
combined her linguistic and mathematical talents to translate this book from
French to English. I thank her for her patience and generous help. I thank
Philippe Cassou-Noguès, Reinie Erné, Arnaud Lacoume, Thierry
Sageaux, Alain Thiéry, and especially Dino Lorenzini, Sylvain Maugeais
for their careful reading of the manuscript. It is due to their vigilance
that many errors were found and corrected. My thanks also go to Jean Fresnel,
Dino Lorenzini, and Michel Matignon for mathematical discussions during the
preparation of the book. I thank the Laboratoire de Mathématiques
Pures de Bordeaux for providing me with such an agreeable environment for
the greatest part of the writing of this book.
I cannot thank my friends and family enough for their constant encouragement
and their understanding. I apologize for not being able to name them individually.
Finally, special thanks to Isabelle, who supported me and who put up with
me during the long period of writing. Without her sacrifices and the
encouragement that she gave me in moments of doubt, this book would probably
be far from being finished today.
The book is organized by chapter/section/subsection. Each section ends with
a series of exercises. The statements and exercises are numbered within each
section. References to results and definitions consist of the chapter number followed
by the section number and the reference number within the
section. The first one is omitted when the reference is to a result within
the same chapter. Thus a reference to Proposition 2.7; 3.2.7; means, respectively,
Section 2, Proposition 2.7 of the same chapter; and Chapter 3, Section 2,
Proposition 2.7. On the contrary, we always refer to sections and subsections
with the chapter number followed by the section number, and followed by the
subsection number for subsections: e.g., Section 3.2 and Subsection 3.2.4.
Future errata will be listed at