Table Of ContentISTITUTO NAZIONALE DI ALTA MATEMATICA
INSTITUTTONES MATHEMATICAE
VOLUME IV
SILVIO GRECO
Normal varieties
Notes written with the collaboration
of A. DI SANTE
ACADEMIC PRESS LONDON AND NEW YORK 1978
ISTITUTO NAZIONALE DI ALTA MATEMATICA
INSTITUTIONS MATHEMATICAE
VOLUME IV
SILVIO GRECO
Normal varieties
Notes written with the collaboration of A. Di Sante
ACADEMIC PRESS LONDON AND NEW YORK 1978
Published by Istituto Nazionale di Alta Matematica Roma
© 1978 by Istituto Nazionale di Alta Matematica
Distributed throughout the world by
Academic Press inc. (London) Ltd.
24-28 Oval Road, London NW1 7DX
Printed in Italy
by “Monograf” - Bologna, via Collamarini 5 - 1978
Institutiones Mathematicae - Volume IV
Corso di lezioni delPIstituto Nazionale di Alta Matemática,
svolto dal Febbraio al Maggio 1976
INTRODUCTION
1) These are the notes of a course given at the University of
Napoli, on behalf of the Istitnto Nazionale di Alta Matemática.
The audience had a deep knowledge of classical Algebraic Geometry
and some of Commutative Algebra and the main purpose of the course
was to show how algebraic methods work in Algebraic Geometry.
I felt that the notion of normality is particularly suited for this
scope, since its origin is completely algebraic, but it has quite a geo
metrical taste.
Normal varieties were introduced by Zariski [43], who was the first
one to realize that the theory of integral dependence (started with Dede
kind in number theory and developed by various mathematicians up
to Krull in a purely algebraic contest) had a deep geometrical interest.
These notes are devoted to some of the most important results on
normal varieties, including two of the most striking results due to
Zariski, namely the analytic normality, and the arithmetic normality.
2) The first result we get here is the so called Krull-Serre cri
terion of normality, which shows that a variety is normal if and only
if it is non singular in codimension 1 and is « well behaved » with
respect to intersection with hypersurfaces. We give a complete (and
in part new) proof of this theorem, taking the chance, in doing it, to
discuss some algebraic facts (such as depth, extensions of rational
functions, the Hartshorne connectedness theorem, graded rings etc.)
which have a deep geometrical meaning.
This is covered in the first four sections. In section 5 we show how
an affine variety can be normalized and we discuss some property of
the conductor.
In section 6 we discuss the theorem of analytic normality and some
of its applications, such as the theory of branches and a theorem of
D. Ferrand [13] on «unibranch » varieties, and the « Zariski Main
Theorem ».
The last two sections follow, with a different language, the lecture
notes by Zariski [46], and deal with the theory of arithmetic normality.
6 Introduction
In section 7 the algebraic theory is given, while in section 8 we give
Zariski’s characterizations of arithmetically normal varieties in terms
of complete linear systems.
3) These notes are by no means self contained. Many proofs
are omitted or only sketched, in order to avoid technical complications
and/or repetitions of easily available material. Instead space is given
to the applications of elementary methods of Commutative Algebra,
including many examples and geometrical interpretations of alge
braic facts.
The only essentially complete proofs are the one of the Krull-Serre
criterion, and of almost all the theorem on arithmetic normality.
Occasionally we have added some proofs not easily available in the
literature. References are given for the omitted proofs.
The prerequisites are standard Commutative Algebra (e.g. Atiyah-
MacDonald [3]) and the basic facts in algebraic geometry listed in
section 1. We have avoided the language of schemes (although it
is there...) since their whole machinery was too heavy for our pur
poses. However we had to give some non standard definitions, for
which we refer to section 1. On the other hand some feeling for « clas
sical» algebraic geometry might help in understanding some facts
and examples.
4) To conclude I wish to thank the Istituto Nazionale di Alta
Matemática for giving me the opportunity (and the financial support)
to give this course and write these notes.
I wish to thank also the audience, first among all prof. A. Franchetta,
whose continuous encouragement and deep knowledge of Algebraic
Geometry were most valuable for me during the whole course. I am
also grateful to C. Ciliberto, G. Di Maio, N. Melone for attending
my lectures with particularly helpful criticism, and to A. Di Sante
who undertook the ungrateful task to take notes of the lectures,
filling up gaps, correcting mistakes, etc. His work was essential for
me to write the present notes.
Silvio Greco
Torino, December 1977
CONTENTS
Introduction.................................................................................................................pag. 5
0. Foreword................................................................................................................ * 9
1. Preliminaries......................................................................................................... » 9
A. Affine algebraic sets and varieties............................................................ » 9
B. Rational functions........................................................................................ » 12
C. Morphisms of affine varieties.................................................................... » 12
D. Projective varieties......................................................... *1 3
2. Normal varieties.................................................................................................... » 15
A. Normal rings and normal varieties........................................................ » 15
B. The structure theorem for normal rings andv arieties......................... » 16
C. Normality and finite morphisms............................................................... » 17
3. Depth and property S2 ........................................................................................ * 13
A. Regular sequences........................................................................................ * 18
B. Regular sequences and dimension. Completei ntersections . . . . » 19
C. The grade of an ideal................................................................................. * 21
D. Depth of a local rin g ................................................................................. » 24
E. Property Sn .................................................................................................... * 26
F. Normality and property S2 ........................................................................ » 27
G. Property S2 and extensions of rational functions................................ » 29
H. The connectedness theorem of Hartshome............................................ » 31
4. Simple subvarieties and property Rn ................................................................ » .33
A. Tangent cones and graded rings................................................................ » 33
B. Simple points and regular local rings.................................................... » 34
C. Normality and tangent cones.................................................................... » 36
D. Regularity in codimension n and property Rn .................................... » 37
E. The normality criterion of Krull-Serre.................................................... » 39
F. An application: double curves on a surface........................................... * 40
5. Normalization........................................................................................................ » 41
A. Normalization of an integral variety........................................................ » 41
B. The conductor................................................................................................ » 42
C. The normalization of an affine reduced variety................................... » 44
D. The prime ideals associated with the conductor................................ » 46
6. Analytic normality and applications................................................................ » 47
A. Analytic normality........................................................................................ » 47
B. Analytic branches......................................................................................... » 48
C. Unibranch points and a theorem of D. Ferrand................................ » 51
D. The Zariski Main Theorem........................................................................ » 52
8 Contents
7. Arithmetic normality.........................................................................................pag. 53
A. The method of the associated affine cone ............................................ » 54
B. Normal and arithmetically normal projective varieties....................... » 55
C. Normalization of a projective variety.................................................... » 58
D. The Veronese embeddings of a normalp rojective variety . . . . » 59
8. Arithmetic normality and linear series............................................................ » 62
A. Divisors............................................................................................................. » 63
B. The fundamental theorem ......................................................................... » 65
C. Proof of the fundamental theorem............................................................ » 67
D. Factorial projective varieties.................................................................... » 68
Bibliography................................................................................................................. » 71
0. Foreword.
All rings are assumed to be commutative and noetherian. Capital
X, Y, Z, T usually denote indeterminates.
1. Preliminaries.
We outline some fact on algebraic varieties, in order to fix some
nomenclature. Since we do not need the whole machinery of schemes,
we shall give some « ad hoc » definitions. We shall use these defini
tions without reference.
We shall assume that the reader is familiar with the basic facts
of Commutative Algebra. The book of Atiyah-MacDonald [3] provides
quite a good background.
As for algebraic varieties we refer to the standard literature, such
as Baldassarri [4], Dieudonné [12], Mumford [27], Samuel [38], Sha-
farevich [41], Zariski-Samuel [47].
We shall always work over an algebraically closed field fc, of arbi
trary characteristic.
A. Affine algebraic sets and varieties.
Let 1cn be the affine w-space, and let R = k[Xly ...,2TJ.
1.1. An (affine) algebraic set is a subset V of Tcn consisting of the
common zeros of a set ft = 0 of equations, where e R.
One can show that a finite number of equations is sufficient to
define V and that the algebraic subsets of Jcn correspond biunivocally
to the radical ideals of R. (Hilbert Nullstellensatz).
An algebraic set V ck n is irreducible if it is not the union of two
strictly smaller algebraic subsets of V. Equivalently, if the ideal cor
responding to V is prime.
In particular points of V are irreducible and correspond to maximal
ideals of R.
