New Techniques in Resolution of Singularities

by Dan Abramovich, Anne Frühbis-Krüger, Michael Temkin, Jaroslaw Wlodarczyk

Overview

Resolution of singularities is notorious as a difficult topic within algebraic geometry. Recent work, aiming at resolution of families and semistable reduction, infused the subject with logarithmic geometry and algebraic stacks, two techniques essential for the current theory of moduli spaces. As a byproduct a short, a simple and efficient functorial resolution procedure in characteristic 0 using just algebraic stacks was produced.

The goals of the book, the result of an Oberwolfach Seminar, are to introduce readers to explicit techniques of resolution of singularities with access to computer implementations, introduce readers to the theories of algebraic stacks and logarithmic structures, and to resolution in families and semistable reduction methods.

Product Details

ISBN-13:	9783031321146
Publisher:	Springer International Publishing
Publication date:	09/15/2023
Series:	Oberwolfach Seminars , #50
Edition description:	1st ed. 2023
Pages:	326
Product dimensions:	6.61(w) x 9.45(h) x 0.00(d)

About the Author

Dan Abramovich studied mathematics at Tel Aviv University and at Harvard, where he received his PhD in 1991. After a postdoctoral fellowship at MIT, he was on faculty at Boston University, where he was Sloan Research Fellow, before moving to Brown University where he is L. Herbert Ballou University Professor. He is Fellow of the American Mathematical Society and the American Association for the Advancement of Science. He works in birational Geometry, Moduli Spaces, and Arithmetic Geometry. His 2018 ICM talk at Rio de Janeiro focused on material described in the present book.

Anne Frühbis-Krüger works in computational algebraic geometry and singularity theory. She received her PhD from Kaiserslautern University in 2000. After a visiting professorship at FU Berlin and more than a decade as apl. Professor at Leibniz Universität Hannover, she moved to her current position at Universität Oldenburg in 2019. She has served as the spokespersonof the 'Fachgruppe Computeralgebra' for two periods and is actively involved in the development of the computer algebra systems Singular and OSCAR.

Michael Temkin is the Maurice and Clara Weil Chair in Mathematics at the Hebrew University of Jerusalem. His main research interests lie within non-archimedean geometry, birational geometry and the interplay between them. In particular, he is interested in resolution of singularities and semistable reduction problems. He graduated from the Weizmann Institute of Science in 2006, and after a postdoc at the University of Pennsylvania and a one-year membership at the IAS joined the Hebrew University of Jerusalem in 2010. Since 2020 he is serving as the editor in chief of the Israel Journal of Mathematics.

Jarosław Włodarczyk works in birational geometry and resolution of singularities. He graduated from Warsaw University in 1993. After a visiting position in Ruhr University Bochum and a scholarship at Grenoble University, he joined Warsaw University where he worked till 2000. After that he moved to Purdue University, where he is now a professor at the Department of Mathematics. Jarosław Włodarczyk was an invited speaker in ICM Madrid 2006, where he presented his work on the weak Factorization Theorem, playing an important role in Algebraic Geometry. He is also a recipient of numerous awards for his research contributions to birational geometry.

Table of Contents

A Computational View on Hironaka’s Resolution of Singularities.- Stacks for Everyone Who Cares About Varieties and Singularities.- Introduction to Logarithmic Geometry.- Birational Geometry Using Weighted Blowing Up.- Relative and Logarithmic Resolution of Singularities.- Weighted Resolution of Singularities. A Rees Algebra Approach.- New Techniques in Resolution of Singularities: Open Problems.

Editorial Reviews

The book under review is a motivated by a series of recent results in the context of the problem of resolution of families and semistable reduction … . the authors have organized the book in seven chapters, where the classical ideas and the main tools of this different framework are presented. … A very comprehensive collection of references is given at the end of each chapter.” (Ana Bravo, zbMATH 1545.14001, 2024)

From the Publisher