Arc Schemes and Singularities

About the Editors
Contents
Chap. 1. Introduction -- David Bourqui, Johannes Nicaise and Julien Sebag
	Bibliography
Chap. 2. Arc Schemes in Geometry and Differential Algebra -- David Bourqui, Johannes Nicaise and Julien Sebag
	1. Weil Restrictions
	2. Jet Schemes
	3. Arc Schemes
	4. Some Brief Reminders on Differential Algebra
	5. Adjunction Formulas in Differential Algebra
	6. Algebro-differential Description of Jet/Arc Schemes
	7. The Universal Algebra of Higher Derivations
	Bibliography
Chap. 3. The Grinberg–Kazhdan Formal Arc Theorem and the Newton Groupoids -- Vladimir Drinfeld
	1. Introduction
	2. The Grinberg–Kazhdan Theorem
	3. Rephrasing the Proof from Section 2
	4. Introduction to the Newton Groupoids
	5. Newton Groupoids (Details)
	Bibliography
Chap. 4. Non-complete Completions -- Mercedes Haiech
	1. Introduction
	2. A Necessary and Sufficient Condition to be Adically Complete
	3. A Not I-adically Complete Completion
	Bibliography
Chap. 5. The Local Structure of Arc Schemes -- David Bourqui and Julien Sebag
	1. Introduction
	2. Conventions and Notations
	3. The Drinfeld–Grinberg–Kazhdan Theorem
	4. A Simplification Lemma in Formal Geometry
	5. The Minimal Formal Model of a Rational Non-degenerate Arc
	6. The Case of Degenerate Arcs
	7. Dependency on the Arc
	8. Nilpotency in Formal Neighborhoods
	Bibliography
Chap. 6. Arc Schemes of Affine Algebraic Plane Curves and Torsion Kähler Differential Forms -- David Bourqui and Julien Sebag
	1. Introduction
	2. Conventions and Notations
	3. Proof of Theorem 1.1
	4. Singular Locus of Torsion Kähler Differential Forms
	5. A Structure Statement on Derivation Module of Plane Curves
	6. A Consequence on the Schematic Structure of Arc Schemes Associated with Plane Curves
	7. A SAGE Code to Compute Nilpotent Kähler Differential Forms of Plane Curves
	Bibliography
Chap. 7. Models of Affine Curves and ?_a-actions -- Kevin Langlois
	1. Introduction
	2. Basics
	3. Proof of the Main Result
	Bibliography
Chap. 8. Théorèmes de Structure sur les Espaces d’Arcs -- Alexis Bouthier
	1. Introduction
	2. Préliminaires
	3. Espace D’arcs
	4. Morphismes Pro-lisses
	5. Sur Certains Espaces Non-noethériens
	6. Énoncés Principaux
	7. Vers une Théorie des Faisceaux
	Bibliographie
Chap. 9. Partition Identities and Application to Infinite-Dimensional Gröbner Basis and Vice Versa -- Pooneh Afsharijoo and Hussein Mourtada
	1. Introduction
	2. Hilbert Series and Integer Partitions
	3. The Lex Gröbner Basis of [x^2_1]
	4. Two Color Partitions and the Node
	Bibliography
Chap. 10. The Algebraic Answer to the Nash Problem for Normal Surfaces According to de Fernex and Docampo -- Monique Lejeune-Jalabert
	1. Introduction
	2. Arcs, the Nash Map and the Nash Problem
	3. Arcs and Wedges
	4. Lifting Wedges
	Bibliography
Chap. 11. The Nash Problem from Geometric and Topological Perspective -- J. Fernández de Bobadilla and M. Pe Pereira
	1. Introduction
	2. The Idea of the Proof for Surfaces
	3. Turning the Problem into a Problem of Convergent Wedges
	4. Reduction to an Euler Characteristic Estimate
	5. The Euler Characteristic Estimates
	6. The Returns of a Wedge and Deformation Theoretic Ideas
	7. The Proof by de Fernex and Docampo for the Higher Dimensional Case
	8. The Generalized Nash Problem and the Classical Adjacency Problem
	9. Holomorphic Arcs
	Bibliography
Chap. 12. Motivic and Analytic Nearby Fibers at Infinity and Bifurcation Sets -- Lorenzo Fantini, and Michel Raibaut
	1. Introduction
	2. Motivic Integration and Nearby Cycles
	3. Motivic Nearby Cycles at Infinity and the Motivic Bifurcation Set
	4. Analytic Nearby Fiber at Infinity and the Serre Bifurcation Set
	Bibliography
Chap. 13. The Néron Multiplicity Sequence of Singularities -- Beatriz Pascual-Escudero and Julien Sebag
	1. Introduction
	2. The Néron Multiplicity Sequence of Singularities
	3. Nash and Néron Multiplicity Sequences
	4. Cuspidal Plane Curve Singularities
	Bibliography
Chap. 14. The Dual Complex of Singularities After de Fernex, Kollár and Xu -- Mirko Mauri
	1. Introduction
	2. Notation: Birational Dictionary
	3. Notation: Simplicial Complexes
	4. Dual Complex of a Log Smooth Pair
	5. Dual Complex of a Resolution of Singularities and of a dlt Pair
	6. Proof of Proposition 5.2 and Main Theorem (1)
	7. Running MMP
	8. Proof of Main Theorem (2)
	9. Proof of Main Theorem (3)
	Bibliography
Chap. 15. Log-Regular Models for Products of Degenerations -- Morgan V. Brown and Enrica Mazzon
	1. Introduction
	2. The Skeleton of a Log-Regular Model
	3. The Essential Skeleton of a Product
	4. Applications
	Bibliography
Chap. 16. Arc Scheme and Bernstein Operators -- Michel Gros, Luis Narváez Macarro and Julien Sebag
	1. Introduction
	2. Recollection on Arc Scheme
	3. Recollection of Differential Algebra
	4. Nilpotent Functions on Arc Scheme and Differential Operators
	5. Bernstein Operators
	6. Examples and Further Comments
	Bibliography
Index
A-E
F-R
S-Z