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2023 | Book

A Computational View on Hironaka’s Resolution of Singularities Read chapter Read first chapter

New Techniques in Resolution of Singularities

Authors: Dan Abramovich, Anne Frühbis-Krüger, Michael Temkin, Jarosław Włodarczyk

Publisher: Springer International Publishing

Book Series : Oberwolfach Seminars

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

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.

Table of Contents

Frontmatter
A Computational View on Hironaka’s Resolution of Singularities
Abstract
To put the main topic, algorithmic aspects of resolution of singularities, into context and provide references to literature for the interested reader, we now very briefly discuss some facets of desingularization
Anne Frühbis-Krüger
Stacks for Everyone Who Cares About Varieties and Singularities
Abstract
Algebraic geometry is about the geometry of varieties. Anything else is meant to help in understanding the geometry of varieties, and must be measured by that yardstick.
Dan Abramovich
Introduction to Logarithmic Geometry
Abstract
These chapter is based on a minicourse on logarithmic resolution of singularities given by the author, and it provides an extended version of its first part devoted to introduction to logarithmic geometry with a view towards applications to resolution. I do not aim to build a theory with proofs (and this is impossible in a 3–4 lecture long course). The goal is to make the reader familiar with basic definitions, constructions, techniques and results of logarithmic geometry. I formulate most of the results as “Exercises” and try to keep them at a reasonable level of difficulty. References to the literature are also provided. At the first reading of the material it may be worth just to read the formulations and hints or comments about main ideas of the arguments, without trying to solve them or read proofs in the cited papers.
Michael Temkin
Birational Geometry Using Weighted Blowing Up
Abstract
This is an exposition of ideas appearing in Abramovich et al. (Functorial embedded resolution via weighted blowings up, arXiv:1906.07106, 2019), discussing in addition the extent to which one can address other aspects of birational geometry using weighted blowings up.
Dan Abramovich, Michael Temkin, Jarosław Włodarczyk
Relative and Logarithmic Resolution of Singularities
Abstract
When working on this chapter my original plan was to provide an expanded version of lecture notes of a mini-course on logarithmic resolution and semistable reduction, but while working on this I decided to widen the perspective and discuss the non-logarithmic methods too—both the classical method and the weighted (or dream) algorithm. This produces a certain intersection with the material of the other chapters, but I think that the profit is larger than this inconvenience.
Michael Temkin
Weighted Resolution of Singularities. A Rees Algebra Approach

The purpose of these notes is to review the results and ideas of the joint project with Abramovich and Temkin on resolution in characteristic zero using weighted blow-ups, as well as the recent developments by the author.

Jarosław Włodarczyk
New Techniques in Resolution of Singularities: Open Problems
Abstract
When we proposed the Oberwolfach Seminar the first time, we prepared a number of open problems for people to work on. Eighteen months into the pandemic, we held the seminar, 1 year late, and unfortunately many of the problems we had prepared were already taken up by various people—we were caught unprepared.
Dan Abramovich, Anne Frühbis-Krüger, Michael Temkin, Jarosław Włodarczyk
Metadata
Title
New Techniques in Resolution of Singularities
Authors
Dan Abramovich
Anne Frühbis-Krüger
Michael Temkin
Jarosław Włodarczyk
Copyright Year
2023
Electronic ISBN
978-3-031-32115-3
Print ISBN
978-3-031-32114-6
DOI
https://doi.org/10.1007/978-3-031-32115-3

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