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Introduction to Mori Theory

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Introduction to Mori Theory INTRODUCTION TO MORI THEORY Cours de M2 { 2010/2011 Universit´eParis Diderot Olivier Debarre March 11, 2016 2 Contents 1 Aim of the course 7 2 Divisors and line bundles 11 2.1 Weil and Cartier divisors . 11 2.2 Invertible sheaves . 12 2.3 Line bundles . 14 2.4 Linear systems and morphisms to projective spaces . 15 2.5 Globally generated sheaves . 16 2.6 Ample divisors . 17 2.7 Very ample divisors . 18 2.8 A cohomological characterization of ample divisors . 20 3 Intersection of curves and divisors 23 3.1 Curves . 23 3.2 Surfaces . 25 3.3 Blow-ups . 26 n 3.3.1 Blow-up of a point in Pk .................................. 26 n 3.3.2 Blow-up of a point in a subvariety of Pk .......................... 26 3.3.3 Blow-up of a point in a smooth surface . 27 3.4 General intersection numbers . 28 3.5 Intersection of divisors over the complex numbers . 32 3.6 Exercises . 32 4 Ampleness criteria and cones of curves 33 4.1 The Nakai-Moishezon ampleness criterion . 33 4.2 Nef divisors . 34 4.3 The cone of curves and the effective cone . 36 4.4 A numerical characterization of ampleness . 37 4.5 Around the Riemann-Roch theorem . 38 4.6 Relative cone of curves . 39 4.7 Elementary properties of cones . 42 3 4 CONTENTS 4.8 Exercises . 43 5 Surfaces 45 5.1 Preliminary results . 45 5.1.1 The adjunction formula . 45 5.1.2 Serre duality . 45 5.1.3 The Riemann-Roch theorem for curves . 46 5.1.4 The Riemann-Roch theorem for surfaces . 46 5.2 Ruled surfaces . 47 5.3 Extremal rays . 50 5.4 The cone theorem for surfaces . 52 5.5 Rational maps between smooth surfaces . 54 5.6 The minimal model program for surfaces . 56 5.7 Exercises . 56 6 Parametrizing morphisms 59 6.1 Parametrizing rational curves . 59 6.2 Parametrizing morphisms . 60 6.3 Parametrizing morphisms with fixed points . 64 6.4 Lines on a subvariety of a projective space . 64 6.5 Exercises . 66 7 \Bend-and-break" lemmas 67 7.1 Producing rational curves . 67 7.2 Rational curves on Fano varieties . 70 7.3 A stronger bend-and-break lemma . 72 7.4 Rational curves on varieties whose canonical divisor is not nef . 74 7.5 Exercise . 76 8 The cone of curves and the minimal model program 77 8.1 The cone theorem . 77 8.2 Contractions of KX -negative extremal rays . 79 8.3 Different types of contractions . 81 8.4 Fiber contractions . 82 8.5 Divisorial contractions . 83 8.6 Small contractions and flips . 85 8.7 The minimal model program . 89 8.8 Minimal models . 90 8.9 Exercises . 93 9 Varieties with many rational curves 95 CONTENTS 5 9.1 Rational varieties . 95 9.2 Unirational and separably unirational varieties . 95 9.3 Uniruled and separably uniruled varieties . 96 9.4 Free rational curves and separably uniruled varieties . 97 9.5 Rationally connected and separably rationally connected varieties . 100 9.6 Very free rational curves and separably rationally connected varieties . 101 9.7 Smoothing trees of rational curves . 104 9.8 Separably rationally connected varieties over nonclosed fields . 108 9.9 R-equivalence . 109 9.10 Rationally chain connected varieties . 110 9.11 Exercises . 112 6 CONTENTS Chapter 1 Aim of the course Let X be a smooth projective variety (over an algebraically closed field). Let C be a curve in X and let D be a hypersurface in X. When C and D meets transversely, we denote by (D C) the number of their intersection points. This \product" can in fact be defined for any curve and any hypersurface;· it is always an integer (which can be negative when C is contained in D) and does not change when one moves C and D. 2 Example 1.1 If C1 and C2 are curves in Pk, we have (this is B´ezout'stheorem) (C C ) = deg(C ) deg(C ): 1 · 2 1 2 The intersection number is here always positive. More generally, it is possible to define the degree of a curve C in Pn in such a way that, for any hypersurface H, we have (H C) = deg(H) deg(C); (1.1) · where deg(H) is the degree of a homogeneous polynomial that defines H. We will define intersection of curves and hypersurfaces in any smooth projective variety X. Then, we will identify two curves which have the same intersection number with each hypersurface (this defines an equivalence relation on the set of all curves). It is useful to introduce some linear algebra in the picture, as follows. Consider finite formal linear combinations with real coefficients of irreducible curves in X (they are called real 1-cycles); these form a gigantic vector space with basis the set of all irreducible curves in X. Extend by bilinearity the intersection product between 1-cycles and hypersurfaces; it takes real values. Define N (X) = real vector space of all 1-cycles = 1-cycles with intersection 0 with all hypersurfaces : 1 f g f g The fundamental fact is that the real vector space N1(X) is finite-dimensional. In this vector space, we define the effective (convex) cone NE(X) as the set of all linear combinations with nonnegative coefficients of classes of curves in X. It is sometimes not closed, and we consider its closure NE(X) (the geometry of closed convex cones is easier to study). If X is a smooth variety contained in Pn and H is the intersection of X with a general hyperplane in Pn, we have (H C) > 0 for all curves C in X (one can always choose a hyperplane which does not contain C). This means· that NE(X) 0 , and in fact also NE(X) 0 , is contained in an open half-space f g f g in N1(X). Equivalently, NE(X) contains no lines. Examples 1.2 1) By (1.1), there is an isomorphism N (Pn) R 1 −! 7 8 CHAPTER 1. AIM OF THE COURSE X X λ [C ] λ deg(C ) i i 7−! i i and NE(Pn) is R+ (not a very interesting cone). 3 2) If X is a smooth quadric in Pk, and C1 and C2 are lines in X which meet, the relations (C1 C2) = 1 and (C C ) = (C C ) = 0 imply that the classes [C ] and [C ] are independent in N (X). In fact,· 1 · 1 2 · 2 1 2 1 N (X) = R[C ] R[C ] and NE(X) = R+[C ] R+[C ]: 1 1 ⊕ 2 1 ⊕ 2 3 3) If X is a smooth cubic in Pk, it contains 27 lines C1;:::;C27 and one can find 6 of them which are pairwise disjoint, say C1;:::;C6. Let C be the smooth plane cubic obtained by cutting X with a general plane. We have N (X) = R[C] R[C ] R[C ]: 1 ⊕ 1 ⊕ · · · ⊕ 6 P The classes of C7;:::;C27 are the 15 classes [C Ci Cj], for 1 i < j 6, and the 6 classes [2C i=k Ci], for 1 k 6. We have − − ≤ ≤ − 6 ≤ ≤ 27 X + NE(X) = R [Ci]: i=1 2 So the effective cone can be quite complicated. One can show that there exists a regular map X Pk which 2 ! contracts exactly C1;:::;C6. We say that X is the blow-up of Pk at 6 points. 4) Although the cone NE(X) is closed in each of the examples above, this is not always the case (it 2 is not closed for the surface X obtained by blowing up Pk at 9 general points; we will come back to this in Example 5.16). Let now f : X Y be a regular map; we assume that fibers of f are connected, and that Y is normal. We denote by NE(f)! the subcone of NE(X) generated by classes of curves contracted by f. The map f is determined by the curves that it contracts, and these curves are the curves whose class is in NE(f). Fundamental fact. The regular map f is characterized (up to isomorphism) by the subcone NE(f). The subcone NE(f) also has the property that.
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