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.

Systematically treats the global geometry of equisingular families of algebraic curves on algebraic surfaces Accumulates the material spread over numerous, recent and classical journal publications, and elaborates it into a unified theory which allows one to approach all main problems in the subject and to answer several classical questions in this area Provides a guide to a variety of methods, results and applications of singular algebraic curves and their families Offers a detailed presentation of the background stuff (including the global deformation theory and the original Viro patchworking construction) which leads the reader to the main ideas of the theory

Autorentext

Gert-Martin Greuel: Born 1944, Studies of Mathematics and Physics at Univ. Göttingen and ETH Zürich, Diploma 1971, PhD 1973 (Göttingen) and Habilitation 1980 in Mathematics (Bonn), 1980–1981 Professor at Univ. Osnabrück (C3), 1981–2010 Professor Univ. Kaiserslautern (C4), 2010 – 2015 Distinguished Senior Professor at Univ. Kaiserslautern, 2015 Emeritus. 2002 – 2013 Director of Mathematisches Forschungsinstitut Oberwolfach, 2009 Dr.h.c. from Leibniz Univ. Hannover, 2011 Honorary Member of Real Sociedad Matemática Española, 2012 – 2015 Editor-in-Chief of Zentralblatt MATH (zbMATH), 2004 First Richard D. Jenks Prize for Excellence in Software Engineering to the Singular team, 2013 Media Prize Mathematik by Deutsche Mathematiker Vereinigung. Christoph Lossen: Born in 1967, Study of mathematics and economical sciences at the University of Kaiserslautern, 1994 Diploma in Mathematics, 1998 PhD at the University of Kaiserslautern, 2002 State doctorate (Habilitation), 2002-2006 Assistant Professor (Hochschuldozent) at TU Kaiserslautern, since 2006 Administrative Director of the Department of Mathematics at TU Kaiserslautern (since 2023 University of Kaiserslautern-Landau (RPTU)). Eugenii I. Shustin: Born 1957, Studies of Mathematics at Leningrad State Univ. and Gorky State Univ., Ms. 1979, PhD 1984 (Leningrad). 1984-87 Assistant Prof. at Gorky Civil Eng. Inst., 1987-92 Associate Prof. Kuibyshev State Univ., 1992-96 Associate Prof. Tel Aviv Univ., 1996-now Full Prof. Tel Aviv University. 1990 Invited lecturer at ICM[1]90, Kyoto, 2002 Bessel Research Award from Alexander von Humboldt Foundation, 2018-now The Bauer-Neuman Chair in Real and Complex Geometry.

Inhalt

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