Arc Schemes and Singularities

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.
Contents: Introduction (David Bourqui, Johannes Nicaise and Julien Sebag)Arc Schemes in Geometry and Differential Algebra (David Bourqui, Johannes Nicaise and Julien Sebag)The Grinberg–Kazhdan Formal Arc Theorem and the Newton Groupoids (Vladimir Drinfeld)Non-Complete Completions (Mercedes Haiech)The Local Structure of Arc Scheme (David Bourqui and Julien Sebag)Arc Schemes of Affine Algebraic Plane Curves and Torsion Kähler Differential Forms (David Bourqui and Julien Sebag)Models of Affine Curves and 𝔾ₐ-actions (Kevin Langlois)Théorèmes de Structure sur les Espaces d'Arcs (Alexis Bouthier)Partition Identities and Application to Infinite-Dimensional Gröbner Basis and Vice Versa (Pooneh Afsharijoo and Hussein Mourtada)The Algebraic Answer to the Nash Problem for Normal Surfaces According to de Fernex and Docampo (Monique Lejeune-Jalabert)The Nash Problem from a Geometric and Topological Perspective (J Fernández de Bobadilla and M Pe Pereira)Motivic and Analytic Nearby Fibers at Infinity and Bifurcation Sets (Lorenzo Fantini and Michel Raibaut)The Néron Multiplicity Sequence of Singularities (Beatriz Pascual-Escudero and Julien Sebag)The Dual Complex of Singularities After de Fernex, Kollár and Xu (Mirko Mauri)Log-Regular Models for Products of Degenerations (Morgan V Brown and Enrica Mazzon)Arc Scheme and Bernstein Operators (Michel Gros, Luis Narváez Macarro and Julien Sebag)Index
Readership: Graduate students interested in algebraic geometry and the theory of arc schemes and the solution of the Nash problem for surfaces. Singularities;Arc Schemes;Nash Problem;Resolution of Singularities;Motivic Integration0Key Features:It contains a thorough introduction to arc schemes from different geometric and algebraic viewpointsIt contains a complete account of the solution of the Nash problem for surfaces by some of the most important experts in the field, including both authors of the Annals of Mathematics paper, where the solution was presentedIt includes an updated and extended version of an unpublished article by the leading mathematician, Vladimir Drinfeld (University of Chicago)