Resolution of singularities

Resolution of singularities
Name	Resolution of singularities
Field	Algebraic geometry
Introduced	20th century
Notable	Hironaka
Related	Desingularization

Resolution of singularities is a process in algebraic geometry and singularity theory that constructs from a singular algebraic variety or scheme a nonsingular variety related by a proper birational morphism, often achieved by successive blowups. The problem was framed and advanced through contributions by mathematicians and institutions across Europe, North America, and Japan, producing fundamental results influencing work at places such as the Institute for Advanced Study, Princeton University, École Normale Supérieure, University of Tokyo, and Harvard University. Major theorems have connections to the work of Oscar Zariski, Heisuke Hironaka, Janusz Korwin, Serre Prize-level research, and modern programs in birational geometry led at institutes like the Clay Mathematics Institute and Simons Foundation.

Introduction

The aim is to replace a singular algebraic variety with a smooth variety via a sequence of controlled modifications such as blowups centered on subvarieties, respecting birational equivalence and preserving function fields; this idea figures in classical work of Alexander Grothendieck, David Hilbert, Federico Enriques, Kunihiko Kodaira, and Federico Borel. The problem ties to moduli questions studied at California Institute of Technology, cohomological methods developed by Jean-Pierre Serre and Alexander Grothendieck, and deformation theories pushed by researchers at Max Planck Institute for Mathematics and École Polytechnique. For arithmetic applications, links arise to conjectures pursued at Princeton University and Institute for Advanced Study by authors influenced by John Tate and Pierre Deligne.

Historical development

Early ideas trace to work of Bernhard Riemann and the birational classifications by Guido Castelnuovo and Federigo Enriques in Italy, followed by foundational algebraization by Oscar Zariski at Harvard University and later reformulations by Alexander Grothendieck in the language of schemes at the IHÉS. The breakthrough came with Heisuke Hironaka's 1964 theorem resolving singularities for varieties over fields of characteristic zero, an accomplishment recognized broadly across institutions like Kyoto University and Princeton University and influencing subsequent research at Harvard University and ETH Zurich. Later contributions by Abhyankar in positive characteristic at University of Illinois and refinements by Shreeram Abhyankar, Luc Illusie, Ofer Gabber, and colleagues at Institut des Hautes Études Scientifiques and Université Paris-Sud advanced methods and posed new challenges.

Methods and theorems

Core techniques include sequences of blowups along permissible centers, use of valuation theory developed by Oscar Zariski and Shreeram Abhyankar, and invariants inspired by work of Hironaka, Heisuke Hironaka's collaborators, and later algorithmic approaches by teams at Institut des Hautes Études Scientifiques, Universität Bonn, and Universität Münster. Fundamental theorems build on earlier results from David Hilbert's basis theorems and use cohomological tools from Jean-Pierre Serre and Alexander Grothendieck; algorithmic resolution was pursued by researchers at Microsoft Research and CNRS in the computer algebra community. Theorems such as Hironaka's main theorem, refinements by Villamayor and Bierstone-Milman at University of Toronto and McMaster University, and canonical resolution results by teams at Rutgers University and Universidad de Buenos Aires organize the methods.

Resolution in characteristic zero

Hironaka's theorem established existence of resolution for algebraic varieties over fields of characteristic zero, leveraging techniques connected to analytic work in École Normale Supérieure and combinatorial approaches inspired by Oscar Zariski. Subsequent canonical and functorial formulations by Encinas-Villamayor, Bierstone-Milman, and researchers at University of California, Berkeley and University of Toronto produced constructive algorithms used in implementations at Wolfram Research and research groups at Universität Freiburg. Extensions to schemes and complex analytic spaces connect to contributions from Kollár at Princeton University and Mori at Kyoto University within the minimal model program carried out at institutions like Clay Mathematics Institute.

Positive characteristic challenges and progress

Positive characteristic presented deep obstacles highlighted by counterexamples and partial results from Abhyankar at University of Illinois and later work by Hironaka's students and contemporaries; resolution in this setting remains a central open area pursued at University of California, Berkeley, University of Michigan, Université Paris-Sud, and MPI MiS. Progress includes results for surfaces by Zariski and for threefolds by teams involving Cossart and Piltant at École Polytechnique and Université Paris-Saclay, while advances by Kedlaya and Temkin link to nonarchimedean geometry studied at Institute for Advanced Study and University of California, San Diego. Recent breakthroughs by groups at University of Tokyo and IMS address special cases, using alterations introduced by Aise Johan de Jong at Princeton University and techniques refined by researchers at University of California, Los Angeles.

Applications and examples

Resolution techniques apply to moduli problems central to projects at Princeton University and Harvard University, to intersection theory developed by William Fulton and colleagues at Rutgers University, and to arithmetic geometry inspired by Pierre Deligne and John Tate at Institute for Advanced Study. Classic examples include resolving plane curve singularities studied since Bernhard Riemann and Guido Castelnuovo, surface singularities classified by Du Val and explored at University of Cambridge, and singularities arising in minimal model program work by Shigefumi Mori and János Kollár. Computational implementations at SageMath and Magma used by groups at University of Warwick and University of Sydney demonstrate algorithmic applications.

Related concepts and generalizations

Related notions include embedded resolution linked to work at École Normale Supérieure and Université Paris-Sud, toroidal embeddings studied by Kempf and Mumford at Harvard University, and log resolutions integral to log geometry developed by Kazuya Kato and Luc Illusie at University of Tokyo and Institut des Hautes Études Scientifiques. Connections extend to the minimal model program driven by Caucher Birkar and James McKernan at University of Cambridge and Imperial College London, to motivic integration initiated by Maxim Kontsevich at IHÉS and Yuri Manin at Max Planck Institute for Mathematics, and to valuation-theoretic frameworks stemming from Oscar Zariski and refined by Shreeram Abhyankar.

Category:Algebraic geometry