Introduction to the Mori Program

Frontmatter

Introduction: The Tale of the Mori Program

Abstract

The purpose of this book is to give an introductory and comprehensible account of what we call the Mori program, a program that emerged in the last two decades as an effective approach toward the biregular and/or birational classification theory of higher-dimensional algebraic varieties.

Kenji Matsuki

Chapter 1. Birational Geometry of Surfaces

Abstract

The purpose of this chapter is to present the main features of birational geometry of surfaces in the framework of the Mori program following the main strategic schemes MP 1 through MP 4 as presented in the introduction.

Kenji Matsuki

Chapter 2. Logarithmic Category

Abstract

The purpose of this chapter is to give an introductory account of the logarithmic category introduced by Iitaka and inspired by the works of Grothendieck [2] and Deligne [l][2][3].

Kenji Matsuki

Chapter 3. Overview of the Mori Program

Abstract

The purpose of this chapter is to outline the key points of the Mori program in dimension 3 (or higher), extending the analogy from dimension 2 to higher dimensions as much as possible. At the same time we try to distinguish the remarkable features, such as flips, that are unique to higher dimensions. We will not give any detailed proofs in this chapter. The emphasis is on presenting the global picture at an early stage by taking the reader on a quick roller-coaster ride of the Mori program.

Kenji Matsuki

Chapter 4. Singularities

Abstract

This chapter is an attempt to give a unified and general account of singularities that are indispensable in the study of the (logarithmic) Mori program in higher dimensions, namely

terminal singularities; canonical singularities; log terminal singularities; log canonical singularities.

Kenji Matsuki

Chapter 5. Vanishing Theorems

Abstract

The purpose of this chapter is to present several vanishing theorems of cohomology groups that form the technical backbone of our approach to the Mori program.

Kenji Matsuki

Chapter 6. Base Point Freeness of Adjoint Linear Systems

Abstract

The purpose of this chapter is to discuss “base point freeness” of linear systems of adjoint type, i.e., linear systems of type \( \left| {K_x + B} \right|, \), where K _X denotes the canonical divisor of a variety X and B is a (boundary) divisor with some specific conditions depending on the situation. As is clear from the formulation, the most natural framework for linear systems of adjoing type is that of the logarithmic category discussed in Chapter 2. Our key tool is the logarithmic version of the Kodaira vanishing theorem, i.e., the Kawamata—Viehweg vanishing theorem. Our viewpoint centering on adjoint linear systems, is in the spirit of Ein—Lazarsfeld [1], which applied the method of Kawamata—Reid—Shokurov to solve Fujita’s conjecture in dimension 3.

Kenji Matsuki

Chapter 7. Cone Theorem

Abstract

The purpose of this chapter is to give a proof of the cone theorem, providing the necessary key ingredients: the rationality theorem and boundedness of the denominator. We prove these two theorems by applying the same cohomological arguments developed for the proofs of the base point freeness theorem and the non-vanishing theorem of the previous chapter. We note that our point of view for discussing the behavior of divisors following Kawamata—Reid—Shokurov—Kollar is “dual” to the original approach of Mori, who discusses the behavior of curves directly via deformation theory. We will study Mori’s argument in Chapter 10.

Kenji Matsuki

Chapter 8. Contraction Theorem

Abstract

This chapter starts with characterizing in Section 8-1 the extremal contractions the key geometric operations in the process of the minimal model program, as what we call the contractions of extremal rays described in terms of the convex geometry of the cone of curves, thanks to the cone theorem of Chapter 7. Their existence is guaranteed by the base point freeness theorem of Chapter 6.

Kenji Matsuki

Chapter 9. Flip

Abstract

The purpose of this chapter is to study the main features of “flip,” the key operation in the center of the minimal model program, surrounding the two major conjectures (existence of flip) and (termination of flips). This chapter, however, reveals very little of the most important operation “flip,” discussing almost nothing deep in the direction toward (existence of flip). Our hope is that this introductory book will expose the reader to the subject without too much technical difficulty and motivate him to venture into the core of the theory afterwards.

Kenji Matsuki

Chapter 10. Cone Theorem Revisited

Abstract

The purpose of this chapter is to present Mori’s original idea (cf. Mori [1][2]) to prove the cone theorem (in the smooth case) through his ingenious method of “bend and break” to produce rational curves of some bounded degree (with respect to an ample divisor or to the canonical divisor). This method leads to the result of Miyaoka—Mori [1] claiming the uniruledness of Mori fiber spaces, yielding the generalization by Kawamata [13] claiming that (every irreducible component of) the exceptional locus of an extremal contraction is also uniruled in general.

Kenji Matsuki

Chapter 11. Logarithmic Mori Program

Abstract

The purpose of this chapter is twofold. The first is to review what we have learned so far by going over the main ingredients of the Mori program in the framework of the logarithmic category. The reader who feels confident of and comfortable with the material so far should have no trouble understanding this part and should even skip this review chapter. Mastery and understanding of the log minimal model program will be rewarded when we utilize it as an essential part of the Sarkisov program in Chapter 13. The second is to present some of the subtleties that inevitably arise as we go from the usual category to the logarithmic category. There are some open conjectures even in dimension 3, though their statements are the natural generalizations in the logarithmic category, according to Iitaka’s philosophy, of the corresponding ones in the usual category.

Kenji Matsuki

Chapter 12. Birational Relation among Minimal Models

Abstract

The purpose of this chapter is to discuss the birational relation among minimal models (cf. the strategic scheme MP 3 in the Introduction). In Section 1–8 we observed that a minimal model in dimension 2 in a fixed birational equivalence class is unique. This is no longer true in dimension 3 or higher, i.e., there may exist many minimal models in general even in a fixed birational equivalence class, and here arises a need to study the birational relation among them.

Kenji Matsuki

Chapter 13. Birational Relation Among Mori Fiber Spaces

Abstract

The purpose of this chapter is to discuss the birational relation among Mori fiber spaces. Here we focus our attention on the most important subject, the Sarkisov program, due to Sarkisov [3], Reid [6], Corti [1], which gives an algorithm for factoring a given birational map between Mori fiber spaces into a sequence of certain elementary transformations called “links.” While it is a higher-dimensional analogue of the Castelnuovo—Noether theorem (cf. Theorem 1-8-8), its true meaning becomes clearer in the framework of the logarithmic category, with the main machinery of the program working under the log MMP discussed in Chapter 11. Our presentation is mostly in dimension 3, where all the necessary ingredients are established (with the most subtle part of showing “termination of Sarkisov program” ingeniously settled by Corti [1], as discussed in Section 13–2), leaving the details of the higher-dimensional case to the reader, where the general mechanism goes almost verbatim but some key ingredients still remain conjectural. (See Section 14–5 for the toric Sarkisov program, where we have all the necessary ingredients established in all dimensions.)

Kenji Matsuki

Chapter 14. Birational Geometry of Toric Varieties

Abstract

This chapter is intended as a coffee break after the previous thirteen chapters of hard work. We will just play around with the tone varieties and see all the ingredients of the Mori program at work in terms of the concrete geometry of convex cones, following the paper Reid [5]. It is more of my personal note to his beautiful paper, only to “draw legs on the picture of a snake”

Kenji Matsuki

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