Geometric Theory of Algebraic Space Curves
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423 S. S. Abhyankar A. M. Sathaye
Geometric Theory of Algebraic Space Curves
Springer-Verlag Berlin· Heidelberg· NewYork 1974
Prof. Dr. Shreeram Shankar Abhyankar Purdue University Division of Mathematical Sciences West Lafayette, IN 47907/USA Prof. Dr. Avinash Madhav Sathaye University of Kentucky Department of Mathematics Lexington, KY 40506/USA
Library of Congress Cataloging in Publication Data
Abhyankar, Shreeram Shankar. Geometric theory of algebraic space curves. (Lecture notes in mathematics ; 423) Includes bibliographical references and indexes. 1. Curves, Algebraic. 2. Algebraic varieties. I. Sathaye, Avinash Madhav, l948joint author. II. Title. III. Series: Lecture notes in mathematics (Berlin) ; 423. QA3.L28 no. 423 [QA567] 510'.8s [516'.35] 74-20717
AMS Subject Classifications (1970): 14-01,
14H99, 14M10
ISBN 3-540-06969-0 Springer-Verlag Berlin · Heidelberg · New York ISBN 0-387-06969-0 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 1974. Printed in Germany Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
PREFACE The original main part of this book was just a sequel to the Montreal Notes [ 3 ].
The main part was the proof to the Theorem
(36.9), namely that "All irreducible nonsingular space curves of degree at most five and genus at most one over an algebraically closed ground field are complete intersections." completely proved in 1971.
As such, the Theorem.was
TWo versions of the proof have been written
and circulated, but none published.
We intended to give a completely
self-contained treatment of the Theorem, and in the process, the size of the proof, or rather, the preparatory material, enlarged; while the proof continued to become clearer and somewhat sharper.
The present
version was finally started in June 1973, and we finally decided that it had to be a book. During October 1973, however, Murthy rendered our main Theorem obsolete by proving that "All irreducible nonsingular space curves of genus at most one over an algebraically closed ground field are complete intersections.
[12].
In fact he proved the well known "Serre's
Conjecture" that all projective modules over k
(when
is algebraically closed)."
kCX,Y,Z]
are free
However he also illustrated in
his proof that, concrete detailed proofs can sometimes be more useful than the theorem itself, by using a concrete description of a basis of three elements for the ideal of an irreducible nonsingular space curve given in the Montreal Notes [ 3 ] as one of the main steps in his proof. Another important feature of
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