Geometric Invariant Theory

248 Pages·1982·12.414 MB·English

by  David Mumford, John Fogarty

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Ergebnisse der Mathematik
und ihrer Grenzgebiete 34
A Series of Modern Surveys in Mathematics
Editorial Board: P. R. Halmos_ P. J. Hilton (Chairman)
R. Remmert B. Szdkefalvi-Nagy
Advisors: L. V. Ahlfors F. L. Bauer A. Dold
J.L.Doob_ S. Eilenberg K. W. Gruenberg M. Kneser
G. H. Miiller M. M. Postnikov E. Sperner

D. Mumford — J. Fogarty
Geometric
Invariant Theory
Second Enlarged Edition
Berlin Heidelberg New York 1982
Professor David Mumford
Department of Mathematics
Harvard University
1 Oxford Street
Cambridge, MA 02138
USA
Professor John Fogarty
Department of Mathematics and Statistics
University of Massachusetts
Amherst, MA 01003
USA
AMS Subject Classification (1980): 14D xx
ISBN 3-540-11290-1 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-11290-1 Springer-Verlag New York Heidelberg Berlin
ISBN 3-540-03284-3 1. Aufl. Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-03284-3 Ist ed. Springer-Verlag New York Heidelberg Berlin
Library of Congress Cataloging in Publication Data. Mumford, David. Geometric
invariant theory. (Ergebnisse der Mathematik und ihrer Grenzgebiete; 34).
Bibliography: p. Includes index. 1. Geometry, Algebraic. 2. Invariants. 3. Moduli
theory. I. Fogarty, John, 1934-. II. Title. III. Series. QA564.M85. 1982. 516.35. 81-21370.
ISBN 0-387-11290-1 (U.S.) AACR2
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 similar 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 “Verwertungsgesellschaft Wort”,
Munich.
© Springer-Verlag Berlin Heidelberg 1965, 1982
Printed in Germany
Offset printing: Beltz Offsetdruck, Hemsbach/Bergstr. Bookbinding: J. Schaffer
oHG, Griinstadt.
2141/3140-543210
Preface to second edition
In the 16 years since this book was published, there has been an ex-
plosion of activity in algebraic geometry. In particular, there has been
great progress in invariant theory and the theory of moduli. Reprinting
this monograph gave us the opportunity of making various revisions.
The first edition was written primilarly as a research monograph on a
geometric way to formulate invariant theory and its applications to the
theory of moduli. In this edition we have left the original text essentially
intact, but have added appendices, which sketch the progress on the
topics treated in the text, mostly without proofs, but with discussions
and references to all the original papers.
In the preface to the 1** edition, it was explained that most of the
invariant theoretic results were proven only in char. 0 and that there-
fore the applications to the construction of moduli spaces were either
valid only in char. 0, or else depended on the particular invariant theo-
retic results in Ch. 4 which could be established by elementary methods
in all characteristics. However, a conjecture was made which would
extend all the invariant theory to char. ~. Fortunately, W. HABOUSH
[128] has proved this conjecture, hence the book is now more straight-
forward and instead of giving one construction of the moduli space
M, of curves of genus g over Q by means of invariant theory, and one
construction of @, and #,,, (the moduli space of principally polarized
abelian varieties) over Z by means of the covariant of points of finite
order and Torelli’s theorem, we actually give or sketch the following
constructions of these spaces:
1) 4, and &, are constructed by proving the stability of the Chow
forms of both curves and abelian varieties (Ch. 4, § 6; Appendix 3B)
2) M, and #, are independently constructed by covariants of finite
sets of points (Appendix 7C; Ch. 7, § 3)
3) &, is constructed by an explicit embedding by theta constants
(Appendix 7B).
All of this is valid over Z except that the covariant approach to 4,
uses higher Weierstrass points and is valid only over Q (unless one can
prove their finiteness for high enough multiples |v], in char. p: see
Appendix 7C).
VI Preface to second edition
This preface gives us the opportunity to draw attention to some basic
open questions in invariant theory and moduli theory. 3 questions raised
in the 1% edition have been answered:
a) the geometric reductivity of reductive groups has been proven
by HABOUuSsH, op. cit.,
b) the existence of canonical destabilizing flags for unstable points
has been proven by Kempr [171] and RoussEau [285],
c) the stability of Chow forms and Hilbert points of pluricanonically
embedded surfaces of general type has been proven by GIESEKER [116].
Pursuing the ideas in c), leads one to ask:
d) which polarized elliptic and K3-surfaces have stable Chow forms?
and what is more difficult probably:
e) can one compactify the moduli spaces of smooth surfaces by allow-
ing suitable singular polarized surfaces which are still ,,asymptotically
stable“, ie., have stable Chow forms when embedded by any complete
linear system which is a sufficiently large multiple of the polarization?
In a more classical direction, now that the reasons for the existence of
M, and x, are so well understood, the time seems ripe to try to under-
stand their geometry more deeply, e.g.
f) Can one calculate, or bound, some birational invariants* of @, or
&,? Investigate the cohomology ring and chow ring of 4, or Wz.
and
g) Find explicit Siegel modular forms vanishing on the Jacobian
locus or cutting out this locus, and relate the various known special pro-
perties of Jacobians.
For those who want to learn something of the subject of moduli,
we want to say what they will mot find here and where more background
on these topics may be found. The subject of moduli divides at present
into 3 broad areas: deformation theory, geometric invariant theory, and
the theory of period maps. Deformation theory deals with local questions:
infinitesimal deformations of a variety, or analytic germs of deformations.
Period maps deal with the construction of moduli of Hodge structures
and the construction of moduli spaces of vareties by attachinga family
of Hodge structures to a family of vareties. Both of these subjects are
discussed only briefly in this monograph (see Appendix 5B). Unfortuna-
tely, deformation theory has not received a systematic treatment by
anyone:a general introduction to the theory of moduliasa whole includ-
ing deformation theory is given in SESHADRI [304], deformations of
singularitves are treated in ARTIN [45]. The origins of the algebraic
* For certain g, M, and #W, are not unirational: FREITAG [108], [109],
HaRRIS—MuMFoRD [837].