Configuration Spaces over Hilbert Schemes and Applications
The main themes of this book are to establish the triple formula without any hypotheses on the genericity of the morphism, and to develop a theory of complete quadruple points, which is a first step towards proving the quadruple point formula under less r
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Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo
Danielle Dias Patrick Le Barz
Configuration Spaces over Hilbert Schemes and Applications
Springer
Authors Danielle Dias Patrick Le Barz Laboratoire de Mathernatiques Universite de Nice - Sophia Antipolis Pare Valrose F-06108 Nice, France e-mail: [email protected] [email protected]
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Dias. Danielle: Configuration spaces over Hilbert schemes and applications / Danielle Dias ; Patrick LeBarz. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan Paris; Santa Clara; Singapore; Tokyo: Springer, 1996 (Lecture notes in mathematics; 1647) ISBN 3-540-62050-8 NE: LeBarz, Patrick:; GT
Mathematics Subject Classification (1991): 14C05, 14C17 ISSN 0075-8434 ISBN 3-540-62050-8 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms 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 liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1996 Printed in Germany The use of general descriptive names, registered names, trademarks, 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. Typesetting: Camera-ready TEX output by the authors SPIN: 10520222 46/3142-543210 - Printed on acid-free paper
Table of Contents Introduction
1
Part one : Double and triple points formula
9
1
Conventions and notation 1.1 Fundamental facts 1.2 Conventions 1.3 Notation . . .
11
11 11 12
2 Double formula 2.1 The class of H2(X) in H2(Z)
13
13
2.2
Definition of the double class.
16
2.3
Computation of the double class .
18
2.3.1
Computation of M 2
18
.
19 20
2.3.2 2.3.3
3
Triple formula 3.1 The class of H3(X) in H3(Z) 3.2
4
The triple formula
.
26
3.2.1
Some notation
.
26
3.2.2
Computation of M3 and ¢.M3
28
3.2.3 3.2.4
Computation of prhWl.Vl .. Computation of {s(U) X cw}rn
31
3.2.5
Computation of prl.Wl.V2, first part.
3.2.6 3.2.7
Computation of prhWl.V2, second part Conclusion .
35 38 42
Intermediate computations 4.1 4.2
22 22
Flatness of
7rl
and
7r2
. . .
32
44 44 44
VI
4.3 4.4 4.5 4.6 4.7 4.8 4.9
Proof of lemma 4.(iv) and of Proof of lemma 4.(iii) .... Proof of lemma 4.(ii) and (v) Proof of lemma 1 . . . . Flatness of P12 and of P 3 Proof of lemma 4.(i) .. Proof of lemma 3 . . . .
= 0
45 46 47 48 49 51 52
4.10 T
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