Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

"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"--

Intersection theory (Mathematics) --- Invariants. --- Deformations of singularities. --- Curves.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

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.

Singularities (Mathematics) --- Deformations of singularities. --- Geometry, Algebraic --- Algebra. --- Geometry, algebraic. --- Algebraic Geometry. --- Algebraic geometry --- Geometry --- Mathematics --- Mathematical analysis --- Algebraic geometry. --- Deformations of singularities

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

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.

Lie algebras. --- Twist mappings (Mathematics) --- Operads. --- Deformations of singularities. --- Àlgebres de Lie --- Teoria dels tuistors --- Singularitats (Matemàtica)

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

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. .

Category theory. Homological algebra --- Ordered algebraic structures --- Algebra --- Differential geometry. Global analysis --- algebra --- differentiaal geometrie --- Deformations of singularities. --- Lie algebras. --- Àlgebres de Lie

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

"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."--

Algebraic geometry --- Deformation --- Deformations of singularities --- Geometry, Algebraic --- Déformations de singularités --- Géométrie algébrique --- Deformations of singularities. --- Geometry, Algebraic. --- 512.72 --- Geometry --- Singularities (Mathematics) --- Structure and variations of algebraic varieties. Fibre bundles. Vector bundles. Fibrations --- 512.72 Structure and variations of algebraic varieties. Fibre bundles. Vector bundles. Fibrations --- Déformations de singularités --- Géométrie algébrique --- EPUB-LIV-FT LIVMATHE LIVSTATI SPRINGER-B