Geometry of the Plane Cremona Maps
This book provides a self-contained exposition of the theory of plane Cremona maps, reviewing the classical theory. The book updates, correctly proves and generalises a number of classical results by allowing any configuration of singularities for the bas
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1769
Springer Berlin
Heidelberg New York
Barcelona
HongKong London
Milan Paris To ky o
Maria Alberich-Carramifiana
Geometry of the Plane Cremona Maps
4
10,
Springer
Author Maria Alberich-Carramifiana
Departamento d'Algebra i Geometria Catalanes, Barcelona, Spain
Gran Via de leg Corts 08007
585
e-mail: [email protected]
Cataloging-in-Publication Data applied for. Die Deutsche Bibliothek
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CIP-Einheitsaufhahme
Alberich-Caffamiiiana, Maria:
Geometry of, the plane cremona map / Maria Alberich-Carramifiana. Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Tokyo: Springer, 2002 (Lecture notes in mathematics ; 1769) -
ISBN 3-540-42816-X
Mathematics
Subject Classification (2000).:14EO5,14EO7
ISSN 00754434 ISBN 3-540-42816-X
Springer-Verlag Berfin Heidelberg New York
subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in, data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are
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Springer-Verlag Berlin Heidelberg 2002
Printed in
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registered names, trademarks, etc. in this publication does not imply, specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
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Typesetting: Camera-ready TEX output by the author SPIN:10856615
41/3143/LK
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543210
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Printed
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To Albert and Antoni
Preface
The basic
by L. Cremona plane birational [14] (1863), [15] (1865), as plane Cremona maps. Other geometers soon brought
theory of plane birational
maps
maps
are
known
substantial additions. Historical r6sum6s To start
with,
let
was
first stated
and henceforth
in his two memoirs
us
explain
can
linear systems and clusters of points. To ]?21 _-., ]?22 we associate the linear system which is the inverse image
by!P
be found in
a
[34]
XVII and
[12].
Cremona maps,
plane given plane Cremona
the connection between
map 4i
(net) W in p21 without fixed part The net W, determines of the net of lines of P'. 2
there is a projectivitYU : ]?2 So the map 0 up to a projectivity of ]?2: 2 2 that u o it is equal to the map ]?22 --4 V, with V the projective space dual
W, which sends x E ]?2 to the hyperplane in W consisting of the divisors passing through x. Observe that a point x E ]?2 is fundamental for 0 (i.e., x to
belongs to the closed subset where!P cannot be defined as a morphism) if and only if x is a
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