Nicaise J., Bourqui D., Sebag J. (eds.) Arc Schemes And Singularities

London: World Scientific, 2020. — 311 p.This title introduces the theory of arc schemes in algebraic geometry and singularity theory, with special emphasis on recent developments around the Nash problem for surfaces. The main challenges are to understand the global and local structure of arc schemes, and how they relate to the nature of the singularities on the variety. Since the arc scheme is an infinite dimensional object, new tools need to be developed to give a precise meaning to the notion of a singular point of the arc scheme.Other related topics are also explored, including motivic integration and dual intersection complexes of resolutions of singularities. Written by leading international experts, it offers a broad overview of different applications of arc schemes in algebraic geometry, singularity theory and representation theory.About the EditorsArc Schemes in Geometry and Differential Algebra
The Grinberg–Kazhdan Formal Arc Theorem and the Newton Groupoids
Non-complete Completions
The Local Structure of Arc Schemes
Arc Schemes of Affine Algebraic Plane Curves and Torsion Kähler Differential Forms
Models of Affine Curves and Ga-actions
Théorèmes de Structure sur les Espaces d’Arcs
Partition Identities and Application to Infinite-Dimensional Gröbner Basis
The Algebraic Answer to the Nash Problem for Normal Surfaces According to de Fernex and Docampo
The Nash Problem from Geometric and Topological Perspective
Motivic and Analytic Nearby Fibers at Infinity and Bifurcation Sets
The Néron Multiplicity Sequence of Singularities
The Dual Complex of Singularities After de Fernex, Kollár and Xu
Log-Regular Models for Products of Degenerations
Arc Scheme and Bernstein Operators