Singular Algebraic Curves: With an Appendix by Oleg Viro

ISBN-13: 9783030033491 / Angielski / Twarda / 2019 / 553 str.

Gert-Martin Greuel; Christoph Lossen; Eugenii Shustin

Recenzje

"The book is very well written and sufficiently self-contained. Certainly, the reader is assumed to be familiar with the basics of algebraic geometry and singularity theory; however, most necessary results, even classical, are carefully stated and either proved in the book or supplied with abundant references. As a nice surprise, the book has an appendix on patchworking, which is the first English publication of the original text by Oleg Viro." (Alex Degtyarev, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 123, 2021)

"The book is written in a very clear style, many topics are illustrated by various nice pictures, visual diagrams, etc. ... this book is comprehensible, interesting and useful for graduate and post-graduate students; it is also very valuable for advanced researchers, lecturers, and practicians working in singularity theory, algebraic geometry, topology, combinatorics, tropical geometry and in other fields of mathematics and its applications." (Aleksandr G. Aleksandrov, zbMATH 1411.14001, 2019)

Spis treści

Zero-Dimensional Schemes for Singularities.- Global Deformation Theory.- H ¹-Vanishing Theorems.- Equisingular Families of Curves.

Blurb

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.