Introduction to Singularities and Deformations

In the second edition we do not only correct errors, update references and improve some of the proofs of the text of the first edition, but also add a new chapter on singularities in arbitrary characteristic. We give an overview of several aspects of singularities of algebraic varieties and formal power series defined over a field of arbitrary characteristic (algebraically closed or not). Almost all of the results presented here appeared after the publication of the first edition and some results are new.

In particular, we treat, in arbitrary characteristic, the classical invariants of hypersurface singularities, and we review results on the equisingularity of plane curve singularities, on the classification of parametrizations of plane branches, and on hypersurface and complete intersection singularities with small moduli. Moreover, we discuss and prove determinacy and semicontinuity results of families of ideals and matrices of power series parametrized by an arbitrary Noether base scheme, which are used to prove open loci properties for several singularity invariants. The semicontinuity has surprising applications in the computation of local standard bases of zero dimensional ideals, which are by magnitudes faster than previously known methods.

The chapter contains two appendices. One is by Dmitry Kerner on large submodules within group orbits, which relates to determinacy criteria for singularities in very general contexts. It is focused on methods applicable to a broad class of fields of arbitrary characteristic, while before the theory was mainly restricted to zero characteristic. The second appendix is by Ilya Tyomkin and deals with the geometry of Severi varieties, mainly on toric varieties. It discusses the breakthrough solution to the problem on the irreducibility of Severi varieties of the plane in arbitrary characteristic, with a focus on the characteristic free approach based on tropical geometry.

We try to be self-contained and give proofs whenever possible. However, due to the amount of material, this is not always possible, and we then give precise references to the original sources.

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.