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Book
Lectures on deformations of singularities
Authors: --- ---
Year: 1976 Publisher: Bombay: Tata institute of fundamental research,

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Book
Inflectionary Invariants for Isolated Complete Intersection Curve Singularities
Authors: ---
ISBN: 1470473534 Year: 2023 Publisher: Providence, RI : American Mathematical Society,

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"We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let N [greater than or equal to] 2, and consider an isolated complete intersection curve singularity germ f : (CN, 0) [arrow] (CN[minus]1, 0). We define a numerical function m [x in a square] [arrow] ADm(2)(f) that naturally arises when counting mth-order weight-2 inflection points with ramification sequence (0, . . . , 0, 2) in a 1-parameter family of curves acquiring the singularity f [equals] 0, and we compute ADm(2)(f) for several interesting families of pairs (f,m). In particular, for a node defined by f : (x, y) [x in a square] [arrow] xy, we prove that ADm (2)(xy) [equals] [x in a square](m[plus]1)4 [x in a square], and we deduce as a corollary that ADm (2)(f) [greater than or equal to] (mult0 [delta symbol]f ) [x in a square](m[plus]1)4 [x in a square] for any f, where mult0 [delta symbol]f is the multiplicity of the discriminant [delta symbol]f at the origin in the deformation space. Significantly, we prove that the function m [x in a square] [arrow] ADm (2)(f)[minus](mult0 [delta symbol]f ) [x in a square](m[plus]1)4 [x in a square] is an analytic invariant measuring how much the singularity "counts as" an inflection point. We prove similar results for weight-2 inflection points with ramification sequence (0, . . . , 0, 1, 1) and for weight-1 inflection points, and we apply our results to solve a number of related enumerative problems"--

CR-geometry and deformations of isolated singularities
Authors: ---
ISSN: 00659266 ISBN: 082180541X Year: 1997 Publisher: Providence (R.I.): American Mathematical Society

Introduction to singularities and deformations
Authors: --- ---
ISBN: 1280803738 9786610803736 3540284192 3540283803 3642066585 9783540283805 Year: 2007 Publisher: Berlin : Springer,

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Singularity theory is a field of intensive study in modern mathematics with fascinating relations to algebraic geometry, complex analysis, commutative algebra, representation theory, theory of Lie groups, topology, dynamical systems, and many more, and with numerous applications in the natural and technical sciences. This book presents the basic singularity theory of analytic spaces, including local deformation theory, and the theory of plane curve singularities. Plane curve singularities are a classical object of study, rich of ideas and applications, which still is in the center of current research and as such provides an ideal introduction to the general theory. Deformation theory is an important technique in many branches of contemporary algebraic geometry and complex analysis. This introductory text provides the general framework of the theory while still remaining concrete. In the first part of the book the authors develop the relevant techniques, including the Weierstraß preparation theorem, the finite coherence theorem etc., and then treat isolated hypersurface singularities, notably the finite determinacy, classification of simple singularities and topological and analytic invariants. In local deformation theory, emphasis is laid on the issues of versality, obstructions, and equisingular deformations. The book moreover contains a new treatment of equisingular deformations of plane curve singularities including a proof for the smoothness of the mu-constant stratum which is based on deformations of the parameterization. Computational aspects of the theory are discussed as well. Three appendices, including basic facts from sheaf theory, commutative algebra, and formal deformation theory, make the reading self-contained. The material, which can be found partly in other books and partly in research articles, is presented from a unified point of view for the first time. It is given with complete proofs, new in many cases. The book thus can serve as source for special courses in singularity theory and local algebraic and analytic geometry.


Book
Deformation Theory
Author:
ISBN: 1441915958 9786613569165 1441915966 1280391243 Year: 2010 Publisher: New York, NY : Springer New York : Imprint: Springer,

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The basic problem of deformation theory in algebraic geometry involves watching a small deformation of one member of a family of objects, such as varieties, or subschemes in a fixed space, or vector bundles on a fixed scheme. In this new book, Robin Hartshorne studies first what happens over small infinitesimal deformations, and then gradually builds up to more global situations, using methods pioneered by Kodaira and Spencer in the complex analytic case, and adapted and expanded in algebraic geometry by Grothendieck. Topics include: * deformations over the dual numbers; * smoothness and the infinitesimal lifting property; * Zariski tangent space and obstructions to deformation problems; * pro-representable functors of Schlessinger; * infinitesimal study of moduli spaces such as the Hilbert scheme, Picard scheme, moduli of curves, and moduli of stable vector bundles. The author includes numerous exercises, as well as important examples illustrating various aspects of the theory. This text is based on a graduate course taught by the author at the University of California, Berkeley.


Book
Maurer-Cartan methods in deformation theory : the twisting procedure
Authors: --- ---
ISBN: 9781108963800 9781108965644 1108967027 1108963803 Year: 2024 Publisher: Cambridge ; New York, NY : Cambridge University Press,

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Covering an exceptional range of topics, this text provides a unique overview of the Maurer-Cartan methods in algebra, geometry, topology, and mathematical physics. It offers a new conceptual treatment of the twisting procedure, guiding the reader through various versions with the help of plentiful motivating examples for graduate students as well as researchers. Topics covered include a novel approach to the twisting procedure for operads leading to Kontsevich graph homology and a description of the twisting procedure for (homotopy) associative algebras or (homotopy) Lie algebras using the biggest deformation gauge group ever considered. The book concludes with concise surveys of recent applications in areas including higher category theory and deformation theory.


Book
Deformations of Surface Singularities
Authors: ---
ISBN: 3642391303 3642391311 Year: 2013 Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer,

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The present publication contains a special collection of research and review articles on deformations of surface singularities, that put together serve as an introductory survey of results and methods of the theory, as well as open problems, important examples and connections to other areas of mathematics. The aim is to collect material that will help mathematicians already working or wishing to work in this area to deepen their insight and eliminate the technical barriers in this learning process. This also is supported by review articles providing some global picture and an abundance of examples. Additionally, we introduce some material which emphasizes the newly found relationship with the theory of Stein fillings and symplectic geometry.  This links two main theories of mathematics: low dimensional topology and algebraic geometry. The theory of normal surface singularities is a distinguished part of analytic or algebraic geometry with several important results, its own technical machinery, and several open problems. Recently several connections were established with low dimensional topology, symplectic geometry and theory of Stein fillings. This created an intense mathematical activity with spectacular bridges between the two areas. The theory of deformation of singularities is the key object in these connections. .


Multi
Lie Methods in Deformation Theory
Author:
ISBN: 9789811911859 9789811911842 9789811911866 9789811911873 Year: 2022 Publisher: Singapore Springer Nature

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This book furnishes a comprehensive treatment of differential graded Lie algebras, L-infinity algebras, and their use in deformation theory. We believe it is the first textbook devoted to this subject, although the first chapters are also covered in other sources with a different perspective. Deformation theory is an important subject in algebra and algebraic geometry, with an origin that dates back to Kodaira, Spencer, Kuranishi, Gerstenhaber, and Grothendieck. In the last 30 years, a new approach, based on ideas from rational homotopy theory, has made it possible not only to solve long-standing open problems, but also to clarify the general theory and to relate apparently different features. This approach works over a field of characteristic 0, and the central role is played by the notions of differential graded Lie algebra, L-infinity algebra, and Maurer-Cartan equations. The book is written keeping in mind graduate students with a basic knowledge of homological algebra and complex algebraic geometry as utilized, for instance, in the book by K. Kodaira, Complex Manifolds and Deformation of Complex Structures. Although the main applications in this book concern deformation theory of complex manifolds, vector bundles, and holomorphic maps, the underlying algebraic theory also applies to a wider class of deformation problems, and it is a prerequisite for anyone interested in derived deformation theory. Researchers in algebra, algebraic geometry, algebraic topology, deformation theory, and noncommutative geometry are the major targets for the book. .


Book
Lie Methods in Deformation Theory
Authors: ---
ISBN: 9789811911859 Year: 2022 Publisher: Singapore Springer Nature Singapore :Imprint: Springer


Book
Deformation theory.
Author:
ISBN: 9781441915955 9781441915962 Year: 2010 Publisher: New York Springer

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"The basic problem of deformation theory in algebraic geometry involves watching a small deformation of one member of a family of objects, such as varieties, or subschemes in a fixed space, or vector bundles on a fixed scheme. In this new book, Robin Hartshorne studies first what happens over small infinitesimal deformations, and then gradually builds up to more global situations, using methods pioneered by Kodaira and Spencer in the complex analytic case, and adapted and expanded in algebraic geometry by Grothendieck. Topics include: deformations over the dual numbers; smoothness and the infinitesimal lifting property; Zariski tangent space and obstructions to deformation problems; pro-representable functors of Schlessinger; infinitesimal study of moduli spaces such as the Hilbert scheme, Picard scheme, moduli of curves, and moduli of stable vector bundles. The author includes numerous exercises, as well as important examples illustrating various aspects of the theory. This text is based on a graduate course taught by the author at the University of California, Berkeley."--

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