Arc Schemes And Singularities

311 Pages·2020·16.163 MB·English

by  Johannes Nicaise, David Bourqui, Julien Sebag

Arc Schemes
And
SingulArities
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Arc Schemes
And
SingulArities
Editors
David Bourqui
Université de Rennes 1, France
Johannes Nicaise
Imperial College London, UK
University of Leuven, Belgium
Julien Sebag
Université de Rennes 1, France
World Scientific
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Library of Congress Cataloging-in-Publication Data
Names: Bourqui, David, editor. | Nicaise, Johannes, editor. | Sebag, Julien, editor.
Title: Arc schemes and singularities / edited by David Bourqui (Université de Rennes 1, France),
Johannes Nicaise (Imperial College London, UK & University of Leuven, Belgium),
Julien Sebag (Université de Rennes 1, France).
Description: New Jersey : World Scientific, 2019. | Includes bibliographical references.
Identifiers: LCCN 2019013758 | ISBN 9781786347190 (hc)
Subjects: LCSH: Curves, Algebraic. | Geometry, Algebraic. | Algebraic spaces. |
Geometrical constructions. | Singularities (Mathematics)
Classification: LCC QA567 .A5725 2019 | DDC 516.3/5--dc23
LC record available at https://lccn.loc.gov/2019013758
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2020 by World Scientific Publishing Europe Ltd.
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Printed in Singapore
About the Editors
DavidBourquihasbeenMaîtredeconférencesattheUniversityofRennes
1 since 2004. He works in algebraic geometry and is specifically interested
in the geometry of moduli spaces of curves and arc schemes.
Johannes Nicaise is a Professor of Mathematics at the Imperial College
London and the University of Leuven. He works on the interactions among
non-Archimedean geometry, birational geometry and mirror symmetry. He
is the author of over 40 research articles in leading international journals
and has authored two books.
Julien Sebag is Professor at the University of Rennes 1 since 2009. His
current work deals with algebraic geometry in connection with arithmetic
anddifferentialaspectsofalgebraandgeometry.Heparticularlyfocuseshis
mathematical activity on the study of arc schemes and related topics.
Together with their co-author Antoine Chambert-Loir, Johannes Nicaise
andJulienSebaghavereceivedthe2017FerranSunyeriBalaguerPrizefor
their book Motivic Integration.
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Contents
About the Editors v
Chap. 1. Introduction 1
David Bourqui, Johannes Nicaise and Julien Sebag
Bibliography 4
Chap. 2. Arc Schemes in Geometry
and Differential Algebra 7
David Bourqui, Johannes Nicaise and Julien Sebag
1. Weil Restrictions 7
2. Jet Schemes 11
3. Arc Schemes 14
4. Some Brief Reminders on Differential Algebra 16
5. Adjunction Formulas in Differential Algebra 21
6. Algebro-differential Description of Jet/Arc Schemes 28
7. The Universal Algebra of Higher Derivations 32
Bibliography 35
Chap. 3. The Grinberg–Kazhdan Formal Arc Theorem
and the Newton Groupoids 37
Vladimir Drinfeld
1. Introduction 37
2. The Grinberg–Kazhdan Theorem 38
3. Rephrasing the Proof from Section 2 41
4. Introduction to the Newton Groupoids 43
5. Newton Groupoids (Details) 48
Acknowledgment 56
Bibliography 56
Chap. 4. Non-complete Completions 57
Mercedes Haiech
1. Introduction 57
2. A Necessary and Sufficient Condition to be
Adically Complete 59
3. A Not I-adically Complete Completion 63
Acknowledgments 67
Bibliography 68
viii Contents
Chap. 5. The Local Structure of Arc Schemes 69
David Bourqui and Julien Sebag
1. Introduction 69
2. Conventions and Notations 70
3. The Drinfeld–Grinberg–Kazhdan Theorem 72
4. A Simplification Lemma in Formal Geometry 84
5. The Minimal Formal Model of a Rational
Non-degenerate Arc 85
6. The Case of Degenerate Arcs 87
7. Dependency on the Arc 89
8. Nilpotency in Formal Neighborhoods 93
Bibliography 96
Chap. 6. Arc Schemes of Affine Algebraic Plane Curves and
Torsion Kähler Differential Forms 99
David Bourqui and Julien Sebag
1. Introduction 99
2. Conventions and Notations 101
3. Proof of Theorem 1.1 102
4. Singular Locus of Torsion Kähler Differential Forms 105
5. A Structure Statement on Derivation Module
of Plane Curves 106
6. A Consequence on the Schematic Structure of Arc
Schemes Associated with Plane Curves 108
7. A SAGE Code to Compute Nilpotent Kähler
Differential Forms of Plane Curves 109
Bibliography 110
Chap. 7. Models of Affine Curves and G -actions 113
a
Kevin Langlois
1. Introduction 113
2. Basics 114
3. Proof of the Main Result 116
Acknowledgments 119
Bibliography 119
Chap. 8. Théorèmes de Structure sur les Espaces d’Arcs 121
Alexis Bouthier
1. Introduction 121
2. Préliminaires 122
3. Espace D’arcs 124
4. Morphismes Pro-lisses 131
5. Sur Certains Espaces Non-noethériens 134
Contents ix
6. Énoncés Principaux 138
7. Vers une Théorie des Faisceaux 140
Bibliographie 143
Chap. 9. Partition Identities and Application to Infinite-
Dimensional Gröbner Basis
and Vice Versa 145
Pooneh Afsharijoo and Hussein Mourtada
1. Introduction 145
2. Hilbert Series and Integer Partitions 146
3. The Lex Gröbner Basis of [x2] 152
1
4. Two Color Partitions and the Node 156
Acknowledgments 160
Bibliography 160
Chap. 10. The Algebraic Answer to the Nash Problem
for Normal Surfaces According to
de Fernex and Docampo 163
Monique Lejeune-Jalabert
1. Introduction 163
2. Arcs, the Nash Map and the Nash Problem 164
3. Arcs and Wedges 165
4. Lifting Wedges 166
Bibliography 172
Chap. 11. The Nash Problem from Geometric and Topological
Perspective 173
J. Fernández de Bobadilla and M. Pe Pereira
1. Introduction 173
2. The Idea of the Proof for Surfaces 175
3. Turning the Problem into a Problem
of Convergent Wedges 177
4. Reduction to an Euler Characteristic Estimate 178
5. The Euler Characteristic Estimates 181
6. The Returns of a Wedge and Deformation
Theoretic Ideas 186
7. The Proof by de Fernex and Docampo for the
Higher Dimensional Case 187
8. The Generalized Nash Problem and the Classical
Adjacency Problem 191
9. Holomorphic Arcs 192
Acknowledgments 194
Bibliography 194