The Geometric Genus and the Newton Polyhedron
The various notions of genus which we have discussed up to this point do not tell us how to effectively compute the genera of an algebraic surface if we start from a defining equation f(x, y, z) = 0 in ℂ3 (which determines its birational equivalence class
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Patrick Popescu-Pampu
What is the Genus?
Lecture Notes in Mathematics Editors-in-Chief: J.-M. Morel, Cachan B. Teissier, Paris Advisory Board: Camillo De Lellis, Zurich Mario di Bernardo, Bristol Alessio Figalli, Zurich Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gabor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, Paris and New York Catharina Stroppel, Bonn Anna Wienhard, Heidelberg
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Patrick Popescu-Pampu
What is the Genus?
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Patrick Popescu-Pampu UFR de Mathématiques Université Lille 1 Villeneuve d’Ascq, France
Expanded translation by the author of the original French edition: Patrick Popescu-Pampu, Qu’est-ce que le genre?, in: Histoires de Mathématiques, Actes des Journées X-UPS 2011, Ed. Ecole Polytechnique (2012), ISBN 978-2-7302-1595-4, pp. 55-198 ISSN 0075-8434 Lecture Notes in Mathematics ISBN 978-3-319-42311-1 DOI 10.1007/978-3-319-42312-8
ISSN 1617-9692 (electronic) ISBN 978-3-319-42312-8 (eBook)
Library of Congress Control Number: 2016950015 Mathematics Subject Classification (2010): 01A05, 14-03, 30-03, 55-03 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland
To Ghislaine, Fantin and Line
Preface to the English Translation
In France, some students follow special curricula during the first 2 years of their superior formation, in “classes préparatoires.” There, an intensive training is organized for the entrance examinations to teaching institutions in science or engineering, the so-called “grandes écoles.” Every May, one of those “great schools,” École Polytechnique, organizes a 2day mathematical conference with lectures given by professional mathematicians and addressed to mathematics teachers of “classes préparatoires.” Each year, the
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