Resolution of Surface Singularities Three Lectures with an Appendix
- PDF / 6,940,407 Bytes
- 138 Pages / 468 x 684 pts Page_size
- 102 Downloads / 180 Views
1101 Vi ncent Cossart Jean Giraud Ulrich Orbanz
Resolution of Surface Singularities Three Lectures with an Appendix by H. Hironaka Edited by U. Orbanz
Spri nger-Verlag Berlin Heidelberg New York Tokyo 1984
Authors
Vincent Cossart Universite Pierre et Marie Curie (Paris VI), Mathematiques 4, Place Jussieu, 75005 Paris, France Jean Giraud Universite de Paris-Sud, Centre d'Orsay, Mathematique BAt. 425, 91405 Orsay Cedex, France Ulrich Orbanz Max-Planck-Institut fur Mathematik Gottfried-Glaren-Str. 26, 5300 Bonn, Federal Republic of Germany
AMS Su bject Classification (1980): 14E15, 14J17 ISBN 3-540-13904-4 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13904-4 Springer-Verlag New York Heidelberg Berlin Tokyo This work IS subject to copyright. All rights are reserved, whether the whole or part of the rnatenal 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.
© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
RESOLUTION OF SURFACE SINGULARITIES - THREE LECTURES
Introduction
The problem of resolution of the singularities of an algebraic surface has a long history reaching back to the last century. After its solution (by R. Walker in 1935), it was qUite reasonable to ask for desingularization of algebraic varieties of arbitrary dimensions. For the general problem the surface case and the different methods of its solution were of special importance, not only as a tool for higher dimension, but also as a testing ground for the general case. To quote D. Hilbert: "Vielleicht in den meisten Fallen, wo wir die Antwort auf eine Frage vergeblich suchen, liegt die Ursache des MiBlingens darin, daB wir einfachere noch unvollkornrnen erledigt haben. Es kornrnt dann alLes dar auf an, diese leichteren Probleme aufzufinden und ihre Losungen mit moglichst vollkornmenen Hilfsmitteln und durch verallgemeinerungsfahige Begriffe zu bewerkstelligen. " Meanwhile the general problem of desingularization in characteristic zero has been solved by Hironaka, but there are some points about his proof which are a challenge for further investigations. First of all, since his proof is very complicated, it is natural to look for simplifications. One such simplification for instance is the notion of idealistic exponents, introduced by Hironaka himself. Then, of course, the case of positive characteristics remains open (in dimension> 3). I want to stress another point which seems unsatisfactory: If Hironaka's proof is specialized for surfaces, there is still no substantial simplification of his procedure. This is one reason to look for different methods to desingularize surfaces, thereby hoping to find some more "natural" method for desingularization in the general cas
Data Loading...
