Moduli Theory and Classification Theory of Algebraic Varieties
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620 Herbert Popp
Moduli Theory and Classification Theory of Algebraic Varieties
Springer-Verlag Berlin Heidelberg New York 1977
Author Herbert Popp Universitat Mannheim (WH) Lehrstuhl fur Mathematik VI A5 6800 Mannheim/BRD
AMS Subject Classifications (1970): 14020, 14JlO, 14KlO, 32G13 ISBN 3-540-08522-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08522-X 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 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 the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2140/3140-543210
Preface and Introduction
The rough classification of algebraic varieties is obtained by dividing the smooth, projective, algebraic varieties into classes up to birational equivalence according to the structure of the m-canonical maps and the Albanese maps. This classification leads to the Enriques' classification for surfaces (cf.
or [1:?> ] ) .
For higher dimensional varieties the main known results of this theory are described in Ueno's Lecture Notes and in Lectures
and 11. The
classification of 3-dimensional varieties seems to be possible.
(For
curves, the classification obtained is almost trivial and divides the curves into classes according to the genus.) The fine classification is the explicit study of the varieties of the various classes obtained by the rough classification by fibre space methods or by moduli theory. But the rough classification and the fine classification cannot be separated. Certain results from the rough classification point out for which types of algebraic varieties a moduli theory should exist and what the properties of this moduli theory should be. Conversely, to do the hard part of the rough classification, for example the proof of Conjecture C for fibre spaces f
n,m
from page
10
: V --+11, the fine classification and a good moduli
theory for certain types of algebraic varieties of dimension
n-1
must be available. It is the purr-0se of this note, to provide a systematic treatment of moduli theory and to describe the interplay between the rough classification and the fine classification of algebraic varieties, as far as it is known , In this way, many of the recent developments in moduli theory such as the theory of period maps, the rrojectivity of moduli spaces, the theory of fine moduli spaces and the compactification of moduli spaces,
appear natural and their interaction becomes clear. These notes, consisting of eleven lectures and an appendix on classical invariant theory, are based on lectures which I gave in the fall of 1975 at the University of M
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