Minimal model program

Minimal model program
Name	Minimal model program
Other names	Mori program
Field	Algebraic geometry
Introduced	1980s
Notable people	Shigefumi Mori, Miles Reid, Yuri Manin, Jean-Pierre Serre, Alexander Grothendieck
Institutions	Princeton University, University of Tokyo, Harvard University, Institute for Advanced Study

Minimal model program

The Minimal model program is a scheme in modern algebraic geometry aimed at classifying higher-dimensional algebraic varieties by performing birational transformations to arrive at simpler models; it connects birational geometry, classification theory, and singularity theory. The program builds on the work of figures associated with the Italian school, the development of sheaf theory at École Normale Supérieure, and modern innovations from researchers at Kyoto University and Princeton University, intertwining techniques from the study of surfaces, threefolds, and moduli problems. It has deep links to birational invariants studied at Institut des Hautes Études Scientifiques and to conjectures inspired by the work of David Mumford, Igor Shafarevich, and John Tate.

Introduction

The program aims to produce, for a given projective variety, either a minimal model or a Mori fiber space via sequences of divisorial contractions and flips introduced by Shigefumi Mori, inspired by earlier classifications by Federigo Enriques, Guido Castelnuovo, and later formalized by contributors at University of Cambridge and University of Paris. Central objects include canonical bundles studied by Kunihiko Kodaira and pluricanonical maps used in work by Kunihiko Kodaira and Arakelov-related investigations. The approach generalizes the classification of surfaces completed by Enriques, with techniques later refined by scholars associated with Princeton University and Harvard University.

Historical development

Early classification efforts trace to the Italian school with Federigo Enriques and Guido Castelnuovo, while modern foundations were shaped by the introduction of sheaf cohomology by Jean-Pierre Serre and scheme theory by Alexander Grothendieck at Institut des Hautes Études Scientifiques. The minimal model approach for threefolds was initiated by Shigefumi Mori whose work on extremal rays built on cone theorems related to results by Yuri Manin and later extended in collaborations involving mathematicians at University of Tokyo and University of California, Berkeley. Major milestones include Mori's proof of existence of flips in special cases, contributions by Miles Reid on canonical singularities, and breakthroughs by teams associated with Clay Mathematics Institute projects and workshops at Mathematical Sciences Research Institute.

Technical framework and definitions

Key definitions employ the language of schemes from Alexander Grothendieck, divisors as in the work of Oscar Zariski, and singularity classes introduced by Vladimir Voevodsky-influenced schools and refined by Miles Reid. Central notions: canonical divisor, K-negative extremal rays related to cone theorems attributed to Shigefumi Mori, log pairs influenced by studies at Institut des Hautes Études Scientifiques, and flips and flops whose conceptual origins link to ideas explored at University of Cambridge. Varieties are studied up to birational equivalence as in classical treatments by Federigo Enriques and modern expositions at Princeton University; resolution of singularities from work linked to Heisuke Hironaka is often invoked. Techniques incorporate vanishing theorems related to Kunihiko Kodaira and cohomological methods developed by Jean-Pierre Serre and researchers at Harvard University.

Key results and conjectures

Proven results include the classification of surfaces by Federigo Enriques and the existence of extremal contractions for threefolds proven by Shigefumi Mori, with canonical singularity analysis influenced by Miles Reid and collaborators at University of Cambridge. Important conjectures include the abundance conjecture, the termination of flips, and the Borisov–Alexeev–Borisov boundedness statements reminiscent of questions pursued at Clay Mathematics Institute. These conjectures have motivated work by researchers associated with University of Tokyo, Institute for Advanced Study, and research groups led by figures such as Yuri Manin and David Mumford.

Methods and techniques

Technical tools combine Mori theory, vanishing theorems originating from Kunihiko Kodaira and refined by scholars at Institut des Hautes Études Scientifiques, and the study of singularities developed by Miles Reid and researchers at University of Cambridge. Methods also integrate moduli techniques from work by David Mumford and Igor Shafarevich, deformation theory advanced by Alexander Grothendieck, and complex-analytic inputs influenced by scholars at École Normale Supérieure and Harvard University. Computational birational methods used by teams at Massachusetts Institute of Technology and University of California, Berkeley supplement abstract existence proofs, while boundedness techniques inspired by projects at Clay Mathematics Institute drive effective results.

Applications and examples

Applications appear in the classification of threefolds following the Mori program as developed by Shigefumi Mori and collaborators at University of Tokyo; specific examples include Fano varieties studied by Vladimir Iskovskikh and moduli spaces explored by David Mumford. The program informs work on Calabi–Yau varieties relevant to approaches at Princeton University and physical models discussed at institutions like Institute for Advanced Study. Concrete constructions and counterexamples have been produced by research groups at University of Cambridge, Harvard University, and Max Planck Institute for Mathematics.

Open problems and current research

Active research addresses the termination of flips, the abundance conjecture, and boundedness results with ongoing contributions from scholars at University of Tokyo, Princeton University, Institute for Advanced Study, and collaborative networks funded by European Research Council. Contemporary work links the program to mirror symmetry topics pursued at Institut des Hautes Études Scientifiques and string-theory inspired projects at Institute for Advanced Study. Workshops at Mathematical Sciences Research Institute and programs at Clay Mathematics Institute continue to shape progress, with encouragement from award committees such as those of the Fields Medal and institutions including Harvard University.

Category:Algebraic geometry