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Über dieses Buch

P. Dolbeault: Résidus et courants.- D. Mumford: Varieties defined by quadratic equations.- A. Néron: Hauteurs et théorie des intersections.- A. Seidenberg: Report on analytic product.- C.S. Seshadri: Moduli of p-vector bundles over an algebraic curve.- O. Zariski: Contributions to the problem of equi-singularity.

Inhaltsverzeichnis

Frontmatter

Residus et Courants

Soit X une surface de Riemann et soit g une fonction méromorphe sur un voisinage U d'un point P de X ayant pour seul pôle P; soit z une coordonnée locale sur U telle que z(P) = 0, alors g est holomorphe sur U\P et égale au voisinage de P à une série de Laurent dont le coefficient du terme en z

−1

est appelé le résidu ∝ de g en P; en fait ∝ est un invariant de la forme différentielle méromorphe fermée ω = g(z)dz.

Pierre Dolbeault

Varieties Defined by Quadratic Equations

First of all, let me fix my terminology and set-up. I will always be working over an algebraically closed ground field k. We will be concerned almost entirely with

projective varieties

over k (although many of our results generalize immediately to arbitrary projective schemes). By a projective variety, I will understand a topological space X all of whose points are closed, plus a sheaf β

X

of k-valued functions on X isomorphic to some subvariety of P

n

for some n. By a subvariety of ℙ

n

, I will mean the subset X ⊂ ℙ

n

(k) defined by some homogeneous prime ideal à ⊂ k[X

o

,…,X

n

], with its Zariski-topology and with the sheaf β

X

of functions from X to k induced locally by polynomials in the affine coordinates. Note that our varieties have only k-rational points — no generic points. In this, we depart slightly from the language of schemes. Note too that a projective variety can be isomorphic to many different subvarieties of ℙ

n

. An isomorphism of X with a subvariety of ℙ

n

will be called an

immersion

of X in ℙ

n

.

David Mumford

Hauteurs et Théorie des Intersections

Dans [6] sont étudiés divers problèmes se rattachant à la notion de hauteur d'un point rationnel d'une varieté algébrique définie sur un corps global.

Je me propose ici d'exposer, dans ses grandes lignes, les principaux résultats de ce travail, en commençant par traiter à part, et de façon plus détaillée, le cas particulier des corps de fonctions algébriques d'une variable, mettant en évidence le lien de cetťe théorie avec la notion de modèle minimal au sens de [5] Bien que les théorèmes fondamentaux du cas général puissent s'ob-tenir indépendamment de l'existence des modèles minimaux, l'intro-duction de ceux-ci permet de mieux préciser certains points du cas particulier envisagé, et de présenter la théorie sous une forme purement algébrique, sans recourir aux méthodes de majoration habituelles qui font intervenir la notion “grossière” de hauteur, et sans utiliser aucun passage à la limite.

A. Néron

Report on Analytic Products

1. For purposes of the present report, we shall feel free to consider situations as special as possible compatible with an exposition of the main issues.

We say that a complete local ring ϑ is an analytic product if it is of the form ϑ

1

[[u]] with u a non-unit analytically independent over the subring ϑ

1

.

A. Seidenberg

Moduli of π -Vector Bundles over an Algebraic Curve

Let X be a smooth algebraic curve, proper over ℓ the field complex numbers (or equivalently a compact Riemann surface) of genus g. Let J be the Jacobian of X; it is a group variety of dimension g and its underlying set of points is the set of divisor classes (or equivalently isomorphic classes of line bundles) of degree zero.

It is a classical result that the underlying topological space of J can be identified with the set of (

unitary

)

characters

of the fundamental group π

1

(X) into ℓ (i.e. homomorphisms of π

1

(X) into complex numbers of modulus one) and therefore J = S

1

× … × S

1

, g times, as a topological manifold S

1

being the unit circle in the complex plane.

The purpose of these lectures is to show how this result can be extended to the case of unitary representations of arbitrary rank of Fuchsian groups with compact quotient.

C. S. Seshadri

Contributions to the Problem of equisingularity

The general problem which we propose in these lectures is the following: given an irreducible subvariety W of the singular locus of an algebraic (or algebroid) variety V and given a simple point Q of W, give a precise meaning to the following intuitive statement: “

the singularity which

V

has at the point

Q is ‘not worse’ than (or is ‘of the same type’ as) the singularity which V has at the general point of W”. We briefly phrase this statement as follows: “V

is equisingular along

W,

at

Q.” It is understood that we require the solution to consist not merely of some plausible definition and some reasonable consequences, but above all of a body of

criteria

of various nature (algebro-geometric, topological and differentio-geometric) and of the proofs of

equivalence

of these various criteria.

In this lecture we give, in the first place, a complete solution of this problem in the special case of cod

V

W

= 1. We also treat a special type of equisingularity which we call

equisaturation

; we are led to this concept by our algebraic theory of

saturation and saturated local rings.

For both of these topics we need a thorough analysis of some old and new aspects of the concept of

equivalent

singularities of plane algebroid curves. This analysis is developed in Sections 1–6. In the last section we discuss connections with the differentiogeometric conditions A and B of Whitney-Thom.

Oscar Zariski
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