Linear Determinants with Applications to the Picard Scheme of a Family of Algebraic Curves
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174 Birger Iversen MIT, Cambridge, MA/USA
Linear Determinants with Applications to the Picard Scheme of a Family of Algebraic Curves
Springer-Verlag Berlin· Heidelberg· NewYork 1970
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZOrich
174 Birger Iversen MIT, Cambridge, MA/USA
Linear Determinants with Applications to the Picard Scheme of a Family of Algebraic Curves
Springer-Verlag Berlin· Heidelberg· NewYork 1970
ISBN 3-540-05301-8 Springer-Verlag Berlin' Heidelberg' New York ISBN 0-387-05301-8 Springer-Verlag New York· Heidelberg' Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or si rnilar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1970. Library of Congress Catalog Card Number 70-143803 Printed in Germany. Offsetdruck:Julius Beltz, Weinheirn/Bergstr.
CONTENTS
Introduction.................. I-
The linear determinant
II.
Representation of n-fold sections by symmetric
IV .
1
prod ue ts ..................•.•..•..•...•..........
20
III. Invertibles sheaves and rational maps into C(g) •.
32
IV.
Construction of the Picard scheme of a family of
curves...........................................
52
Bi bliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
INTRODUCTION
This paper grew out of a study of A. Weil's construction of the Jacobian variety of an algebraic curve. The first delicate problem in Weil's construction is a rationality question connected with the symmetric product of a curve, which led
me to the theory of linear
determinants as exposed in Chapter I.
This theory studies
a l-dimensional integral representation, called M
n, Z (n)
l d , of
which denotes the n-fold symmetric product of
the n x n
matrices i.e. the algebra of tensors in (n factors)
Mn, z0 ••• 18Mn, Z
invariant under the standard
action of the symmetric group on
n
letters.
This repre-
sentation aside from its geometric aspects provides a linearization of Galois theory different from the one usually advocated.
A relation of
touched upon.
ld
to Azumaya algebras is just
I have kept this chapter entirely in the
notion of commutative algebra since I hope it has an independent interest. In chapter II is studied the geometric aspect of which is first of all the following: Let
f: X
Y
ld
be
a finite morphism of schemes whose fibers have constant rank
and let
n
of
X
of
X (n) Y
over
Y • over
X (n) y
By means of Y
to a geometric point
*
denote the n-fold symmetric product ld
is constructed a section
which underlies the geometric map which y
of
Y
associat
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