Arc Schemes And Singularities

by David Bourqui, Johannes Nicaise, Julien Sebag

Overview

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.

Product Details

ISBN-13:	9781786347190
Publisher:	World Scientific Publishing Europe Ltd
Publication date:	04/04/2020
Pages:	312
Product dimensions:	6.00(w) x 9.00(h) x 0.75(d)

Table of Contents

Introduction; Arc Schemes in Geometry and Differential Algebra; The Grinberg–Kazhdan Formal Arc Theorem and the Newton Groupoids; On Non-Complete Completions; The Local Structure of Arc Scheme; Arc Schemes of Affine Algebraic Plane Curves and Torsion KäHler Differential Forms; Models of Affine Curves and ???-actions; Théorèmes de Structure Sur Les Espaces D'arcs; Partition Identities and Application to Infinite Dimensional Groebner Basis and Vice Versa; The Algebraic Answer to the Nash Problem for Normal Surfaces According to De Fernex and Docampo; The Nash Problem from a 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; List of Participants