Singularities in Algebraic and Analytic Geometry

Singularities in Algebraic and Analytic Geometry PDF

Author: Caroline Grant Melles

Publisher: American Mathematical Soc.

Published: 2000

Total Pages: 202

ISBN-13: 0821820052

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This volume contains the proceedings of an AMS special session held at the 1999 Joint Mathematics Meetings in San Antonio. The participants were an international group of researchers studying singularities from algebraic and analytic viewpoints. The contributed papers contain original results as well as some expository and historical material. This volume is dedicated to Oscar Zariski, on the one hundredth anniversary of his birth. Topics include the role of valuation theory in algebraic geometry with recent applications to the structure of morphisms; algorithmic approaches to resolution of equisingular surface singularities and locally toric varieties; weak subintegral closures of ideals and Rees valuations; constructions of universal weakly subintegral extensions of rings; direct-sum decompositions of finitely generated modules; construction and examples of resolution graphs of surface singularities; Jacobians of meromorphic curves; investigation of spectral numbers of curve singularities using Puiseux pairs; Gröbner basis calculations of Hochschild homology for hypersurfaces with isolated singularities; and the theory of characteristic classes of singular spaces - a brief history with conjectures and open problems.

Introduction to Singularities and Deformations

Introduction to Singularities and Deformations PDF

Author: Gert-Martin Greuel

Publisher: Springer Science & Business Media

Published: 2007-02-23

Total Pages: 482

ISBN-13: 3540284192

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Singularity theory is a young, rapidly-growing topic with connections to algebraic geometry, complex analysis, commutative algebra, representations theory, Lie groups theory and topology, and many 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. It includes complete proofs.

Introduction to Singularities

Introduction to Singularities PDF

Author: Shihoko Ishii

Publisher: Springer

Published: 2014-11-19

Total Pages: 227

ISBN-13: 443155081X

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This book is an introduction to singularities for graduate students and researchers. It is said that algebraic geometry originated in the seventeenth century with the famous work Discours de la méthode pour bien conduire sa raison, et chercher la vérité dans les sciences by Descartes. In that book he introduced coordinates to the study of geometry. After its publication, research on algebraic varieties developed steadily. Many beautiful results emerged in mathematicians’ works. Most of them were about non-singular varieties. Singularities were considered “bad” objects that interfered with knowledge of the structure of an algebraic variety. In the past three decades, however, it has become clear that singularities are necessary for us to have a good description of the framework of varieties. For example, it is impossible to formulate minimal model theory for higher-dimensional cases without singularities. Another example is that the moduli spaces of varieties have natural compactification, the boundaries of which correspond to singular varieties. A remarkable fact is that the study of singularities is developing and people are beginning to see that singularities are interesting and can be handled by human beings. This book is a handy introduction to singularities for anyone interested in singularities. The focus is on an isolated singularity in an algebraic variety. After preparation of varieties, sheaves, and homological algebra, some known results about 2-dim ensional isolated singularities are introduced. Then a classification of higher-dimensional isolated singularities is shown according to plurigenera and the behavior of singularities under a deformation is studied.

Singularities of Mappings

Singularities of Mappings PDF

Author: David Mond

Publisher: Springer Nature

Published: 2020-01-23

Total Pages: 567

ISBN-13: 3030344401

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The first monograph on singularities of mappings for many years, this book provides an introduction to the subject and an account of recent developments concerning the local structure of complex analytic mappings. Part I of the book develops the now classical real C∞ and complex analytic theories jointly. Standard topics such as stability, deformation theory and finite determinacy, are covered in this part. In Part II of the book, the authors focus on the complex case. The treatment is centred around the idea of the "nearby stable object" associated to an unstable map-germ, which includes in particular the images and discriminants of stable perturbations of unstable singularities. This part includes recent research results, bringing the reader up to date on the topic. By focusing on singularities of mappings, rather than spaces, this book provides a necessary addition to the literature. Many examples and exercises, as well as appendices on background material, make it an invaluable guide for graduate students and a key reference for researchers. A number of graduate level courses on singularities of mappings could be based on the material it contains.

Lectures on Resolution of Singularities (AM-166)

Lectures on Resolution of Singularities (AM-166) PDF

Author: János Kollár

Publisher: Princeton University Press

Published: 2009-01-10

Total Pages: 215

ISBN-13: 1400827809

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Resolution of singularities is a powerful and frequently used tool in algebraic geometry. In this book, János Kollár provides a comprehensive treatment of the characteristic 0 case. He describes more than a dozen proofs for curves, many based on the original papers of Newton, Riemann, and Noether. Kollár goes back to the original sources and presents them in a modern context. He addresses three methods for surfaces, and gives a self-contained and entirely elementary proof of a strong and functorial resolution in all dimensions. Based on a series of lectures at Princeton University and written in an informal yet lucid style, this book is aimed at readers who are interested in both the historical roots of the modern methods and in a simple and transparent proof of this important theorem.

Singularities in Geometry and Topology

Singularities in Geometry and Topology PDF

Author: Jean-Paul Brasselet

Publisher: World Scientific

Published: 2007

Total Pages: 918

ISBN-13: 981270681X

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Singularity theory appears in numerous branches of mathematics, as well as in many emerging areas such as robotics, control theory, imaging, and various evolving areas in physics. The purpose of this proceedings volume is to cover recent developments in singularity theory and to introduce young researchers from developing countries to singularities in geometry and topology. The contributions discuss singularities in both complex and real geometry. As such, they provide a natural continuation of the previous school on singularities held at ICTP (1991), which is recognized as having had a major influence in the field.

Singularities and Foliations. Geometry, Topology and Applications

Singularities and Foliations. Geometry, Topology and Applications PDF

Author: Raimundo Nonato Araújo dos Santos

Publisher: Springer

Published: 2018-03-21

Total Pages: 553

ISBN-13: 3319736396

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This proceedings book brings selected works from two conferences, the 2nd Brazil-Mexico Meeting on Singularity and the 3rd Northeastern Brazilian Meeting on Singularities, that were hold in Salvador, in July 2015. All contributions were carefully peer-reviewed and revised, and cover topics like Equisingularity, Topology and Geometry of Singularities, Topological Classification of Singularities of Mappings, and more. They were written by mathematicians from several countries, including Brazil, Spain, Mexico, Japan and the USA, on relevant topics on Theory of Singularity, such as studies on deformations, Milnor fibration, foliations, Catastrophe theory, and myriad applications. Open problems are also introduced, making this volume a must-read both for graduate students and active researchers in this field.

Singular Algebraic Curves

Singular Algebraic Curves PDF

Author: Gert-Martin Greuel

Publisher: Springer

Published: 2018-12-30

Total Pages: 553

ISBN-13: 3030033503

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Singular algebraic curves have been in the focus of study in algebraic geometry from the very beginning, and till now remain a subject of an active research related to many modern developments in algebraic geometry, symplectic geometry, and tropical geometry. The monograph suggests a unified approach to the geometry of singular algebraic curves on algebraic surfaces and their families, which applies to arbitrary singularities, allows one to treat all main questions concerning the geometry of equisingular families of curves, and, finally, leads to results which can be viewed as the best possible in a reasonable sense. Various methods of the cohomology vanishing theory as well as the patchworking construction with its modifications will be of a special interest for experts in algebraic geometry and singularity theory. The introductory chapters on zero-dimensional schemes and global deformation theory can well serve as a material for special courses and seminars for graduate and post-graduate students.Geometry in general plays a leading role in modern mathematics, and algebraic geometry is the most advanced area of research in geometry. In turn, algebraic curves for more than one century have been the central subject of algebraic geometry both in fundamental theoretic questions and in applications to other fields of mathematics and mathematical physics. Particularly, the local and global study of singular algebraic curves involves a variety of methods and deep ideas from geometry, analysis, algebra, combinatorics and suggests a number of hard classical and newly appeared problems which inspire further development in this research area.

Complex Analytic Desingularization

Complex Analytic Desingularization PDF

Author: José Manuel Aroca

Publisher: Springer

Published: 2018-11-03

Total Pages: 330

ISBN-13: 4431498222

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[From the foreword by B. Teissier] The main ideas of the proof of resolution of singularities of complex-analytic spaces presented here were developed by Heisuke Hironaka in the late 1960s and early 1970s. Since then, a number of proofs, all inspired by Hironaka's general approach, have appeared, the validity of some of them extending beyond the complex analytic case. The proof has now been so streamlined that, although it was seen 50 years ago as one of the most difficult proofs produced by mathematics, it can now be the subject of an advanced university course. Yet, far from being of historical interest only, this long-awaited book will be very rewarding for any mathematician interested in singularity theory. Rather than a proof of a canonical or algorithmic resolution of singularities, what is presented is in fact a masterly study of the infinitely near “worst” singular points of a complex analytic space obtained by successive “permissible” blowing ups and of the way to tame them using certain subspaces of the ambient space. This taming proves by an induction on the dimension that there exist finite sequences of permissible blowing ups at the end of which the worst infinitely near points have disappeared, and this is essentially enough to obtain resolution of singularities. Hironaka’s ideas for resolution of singularities appear here in a purified and geometric form, in part because of the need to overcome the globalization problems appearing in complex analytic geometry. In addition, the book contains an elegant presentation of all the prerequisites of complex analytic geometry, including basic definitions and theorems needed to follow the development of ideas and proofs. Its epilogue presents the use of similar ideas in the resolution of singularities of complex analytic foliations. This text will be particularly useful and interesting for readers of the younger generation who wish to understand one of the most fundamental results in algebraic and analytic geometry and invent possible extensions and applications of the methods created to prove it.