Advanced studies in pure mathematics· 2018Q3
Introduction to the Minimal Model Problem
- 1,043citations
- Q3SCImago
- 2018year
Short summary
This paper introduces the foundational concepts and conjectures within the Minimal Model Program (MMP), a major area of algebraic geometry focused on classifying algebraic varieties.
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Key points
- Defines essential concepts like Kleiman's criterion, terminal/canonical singularities, and canonical varieties.
- Introduces major conjectures in the field: Minimal Model, Non-Vanishing, Base Point Free, Flip, and Abundance.
- Covers related theorems such as vanishing theorems (Kawamata-Viehweg, Elkik-Fujita) and the Cone Theorem.
- Provides a structural overview for the classification of algebraic varieties.
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Abstract
Chapter O. Notation and preliminaries § 0-1.Kleiman's criterion for ampleness § 0-2.Definitions of terminal, canonical and (weak) log-terminal singularities § 0-3.Canonical varieties § 0-4.The minimal model conjecture Chapter 1. Vanishing theorems § 1-1.Covering Lemma § 1-2.Vanishing theorem of Kawamata and Viehweg § 1-3.Vanishing theorem of Elkik and Fujita Chapter 2. Non-Vanishing Theorem § 2-1.Non-Vanishing Theorem Chapter 3. Base Point Free Theorem § 3-1.Base Point Free Theorem § 3-2.Contractions of extremal faces § 3-3.Canonical rings of varieties of general type Chapter 4. Cone Theorem § 4-1.Rationality Theorem § 4-2.The proof of the Cone Theorem Chapter 5. Flip Conjecture § 5-1.Types of contractions of extremal rays § 5-2.Flips of toric morphisms Chapter 6. Abundance Conjecture § 6-1.Nef and abundant divisors
The authors' abstract, as published at the source. Advanced studies in pure mathematics, 2018 · DOI ↗
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Field: Geometry and Topology
Geometry and TopologyMathematics