Bi rational Geometry of algebraic varieties

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BIRATIONAL GEOMETRY CAUCHER BIRKAR Contents 1. Introduction: overview 2 2. Preliminaries 7 3. Singularities 10 4. Minimal model program 15 4.1. Kodaira dimension 15 4.2. Basics of minimal model program 16 4.3. Cone and contraction, vanishing, nonvanishing, and base point freeness 18 5. Further studies of minimal model program 26 5.1. Finite generation, flips and log canonical models 26 5.2. Termination of flips 29 5.3. Minimal models 31 5.4. Mori fibre spaces 32 6. Appendix 35 References 35 3April2007. ThesearelecturenotesfortheBirational Geometry coursewhich I taught at the University of Cambridge, Winter 2007. 12 CAUCHER BIRKAR 1. INTRODUCTION: OVERVIEW All varieties in this lecture are assumed to be algebraic overC. What classification means? In every branch of mathematics, the problem of classifying the objects arises naturally as the ultimate un- derstanding of the subject. If we have finitely many objects, then clas- sification means trying to see which object is isomorphic to another, whatever the isomorphism means. In this case, the problem shouldn’t be too difficult. However, if we have infinitely many objects, then clas- sification has a bit different meaning. Since each human can live for a finite length of time, we may simply not have enough time to check every object in the theory. On the other hand, objects in a mathe- matical theory are closely related, so one can hope to describe certain properties of all the objects using only finitely many or some of the objects which are nice in some sense. For example let V be a vector space over a field k. If dimV ∞, then we can take a basis v ,...,v for this vector space and then 1 n P every v∈ V can be uniquely written as v = av where a ∈ k. We i i i can say that we have reduced the classification of elements of V to the classification of the basis. Formally speaking, this is what we want to do in any branch of mathematics but the notion of "basis" and the basis "generating" all the objects could be very different. In algebraic geometry, objects are varieties and relations are de- scribed using maps and morphisms. One of the main driving forces of algebraic geometry is the following Problem 1.0.1 (Classification). Classify varieties up to isomorphism. Some of the standard techniques to solve this problem, among other things, include • Defining invariants (e.g. genus, differentials, cohomology) so that one can distinguish nonisomorphic varieties, • Moduli techniques, that is, parametrising varieties by objects which are themselves varieties,BIRATIONAL GEOMETRY 3 • Modifying the variety (e.g. make it smooth) using certain op- erations (eg birational, finite covers). It is understood that this problem is too difficult even for curves so one needs to try to solve a weaker problem. Problem1.0.2 (Birationalclassification). Classify projective varieties up to birational isomorphism. By Hironaka’s resolution theorem, each projective variety is bira- tional toasmoothprojectivevariety. So, wecantrytoclassifysmooth projective varieties up to birational isomorphism. In fact, smooth bi- rational varieties have good common properties such as common pluri- genera, Kodaira dimension and irregularity. In this stage, one difficulty is that in each birational class, except in dimension one, there are too many smooth varieties. So, one needs to look for a more special and subtle representative. Example 1.0.3 (Curves). Projective curves are one dimensional pro- 2 jective varieties (i.e. compact Riemann surfaces). Curves X inP are defined by a single homogeneous polynomial F. A natural and impor- tant "invariant" is the degree defined as deg(X) = degF. The degree is actually not an invariant because a line and a conic have different degrees but they could be isomorphic. However, using the degree we can simply define an invariant: genus, which is defined as 1 g(X)= (deg(X)−1)(deg(X)−2) 2 Topologically, g(X) is the number of handles on X. Moduli spaces. For each g, there is a moduli space M of smooth g projective curves of genus g.  1 0 iff X 'P  g(X)= 1 iff X is elliptic  ≥2 iff X is of general type These correspond to   0 iff X has positive curvature g(X)= 1 iff X has zero curvature  ≥2 iff X has negative curvature 0 On the other hand, g(X) = h (X,ω ) where ω =O (K ) is the X X X X d canonical sheaf, that is, Ω .4 CAUCHER BIRKAR º 0 iff degK 0 X g(X)= ≥1 iff degK ≥0 X Definition 1.0.4. For a smooth projective variety X, define the m-th plurigenus as 0 ⊗m P (X):=h (X,ω ) m X Note that P (X)=g(X). Define the Kodaira dimension of X as 1 logP (X) m κ(X):=limsup logm m→∞ IfdimX =d,thenκ(X)∈−∞,0,1,...,d. Moreover,theKodaira dimension and the plurigenera P (X) are all birational invariants. m Example 1.0.5. If dimX =1, then º −∞ iff degK 0 X κ(X)= ≥0 iff degK ≥0 X In higher dimension, the analogue of canonical with negative degree is the notion of a Mori fibre space defined as a fibre type contraction Y → Z which is a K -negative extremal fibration with connected Y fibres. But the analogue of canonical divisor with nonnegative degree is the notion of a minimal variety defined as Y having K nef, i.e., Y K ·C≥0 for any curve C on Y. Y Conjecture 1.0.6 (Minimal model). Let X be a smooth projective va- riety. Then, • If κ(X) = −∞, then X is birational to a Mori fibre space Y →Z. • If κ(X)≥0, then X is birational to a minimal variety Y. Conjecture 1.0.7 (Abundance). Let Y be a minimal variety. Then, there is an Iitaka fibration φ: Y → S with connected fibres and an ample divisor H on S such that ∗ K =φ H Y In fact,BIRATIONAL GEOMETRY 5 • φ(C)=pt.⇐⇒K ·C=0 for any curve C on Y. Y • dimS =κ(Y). Example1.0.8(Enriques-Kodairaclassificationofsurfaces). Themin- imal model conjecture and the abundance conjecture hold for surfaces. More precisely, 2 1 κ(X) = −∞ =⇒ X is birational to P or a P -bundle over some curve. κ(X) = 0 =⇒ X is birational to a K3 surface, an Enriques sur- face or an ´etale quotient of an abelian surface. The Iitaka fibration φ: Y →S =pt. is trivial. κ(X) = 1 =⇒ X is birational to a minimal elliptic surface Y. The Iitaka fibration φ: Y →S is an elliptic fibration. κ(X) = 2 =⇒ X is birational to a minimal model Y. The Iitaka fibration φ: Y →S is birational. Remark 1.0.9 (Classical MMP). To get the above classification one can use the classical minimal model program (MMP) as follows. If 1 2 there is a −1-curve E (i.e. E ' P and E = −1) on X, then by Castelnuovo theorem we can contract E by a birational morphism f: X → X where X is also smooth. Now replace X with X and 1 1 1 continue the process. In each step, the Picard number ρ(X) drops by 1. Therefore, after finitely many steps, we get a smooth projective variety Y with no−1-curves. Such Y turns out to be among one of the classes in Enriques classification. In higher dimension (ie. dim≥3), we would like to have a program similar to the classical MMP for surfaces. However, this is not with- out difficulty. Despite being a very active area of algebraic geometry and large number of people working on this program for nearly three decades, there are still many fundamental problems to be solved. For theprecisemeaningoftheterminologyseeothersectionsofthelecture. In any case, some of the main problems including the minimal model conjecture and abundance conjecture are: • Minimal model conjecture. Open in dimension≥5. • Abundance conjecture. Open in dimension≥4. • Flip conjecture: Flips exist. Recently solved. • Terminationconjecture: Anysequenceofflipsterminates. Open in dimension≥4.6 CAUCHER BIRKAR • Finite generation conjecture: Foranysmoothprojectivevariety X, the canonical ring ∞ M 0 ⊗m R(X):= H (X,ω ) X m=0 is finitely generated. Recently solved. • Alexeev-Borisov conjecture: Fano varieties of dimension d with bounded singularities, are bounded. Open in dimension≥3. • ACC conjectures on singularities: Certain invariants of singu- larities(i.e. minimallogdiscrepanciesandlogcanonicalthresh- olds) satisfy the ascending chain condition (ACC). Open in di- mension≥3 and≥4 respectively. • Uniruledness: NegativeKodairadimensionisequivalenttounir- uledness. Open in dimension≥4. Thefollowingtwotheoremsandtheirgeneralisationsarethebuilding blocks of the techniques used in minimal model program. Theorem1.0.10(Adjunction). LetX beasmoothvarietyandY ⊂X a smooth subvariety. Then, (K +Y) =K X Y Y Theorem 1.0.11 (Kodaira vanishing). Let X be a smooth projective variety and H an ample divisor on X. Then, i H (X,K +H)=0 X for any i0.BIRATIONAL GEOMETRY 7 2. PRELIMINARIES Definition2.0.12. In this lecture, by a variety we mean an irreducible 0 quasi-projective variety over C. Two varieties X,X are called bira- 0 tional if there is a rational map f: X 99K X which has an inverse, 0 i.e., if X,X have isomorphic open subsets. Definition 2.0.13 (Contraction). A contraction is a projective mor- phism f: X →Y such that f O =O (so, it has connected fibres). ∗ X Y Example2.0.14(Zariski’smaintheorem). Letf: X →Y beaprojec- tive birational morphism where Y is normal. Then, f is a contraction. Example 2.0.15. Give a finite map which is not a contraction. Remark 2.0.16 (Stein factorisation). Let f: X → Y be a projective morphism. Then, it can be factored through g: X →Z and h: Z→Y such that g is a contraction and h is finite. Definition 2.0.17. Let X be a normal variety. A divisor (resp. Q- P m divisor,R-divisor) is dD whereD are prime divisors andd ∈Z i i i i i=1 (resp. d ∈Q, d ∈R). A Q-divisor D is called Q-Cartier if mD is i i 0 Cartier for some m∈N. TwoQ-divisors D,D are called Q-linearly 0 0 equivalent, denoted by D∼ D, if mD∼mD for some m∈N. X is Q called Q-factorial if everyQ-divisor isQ-Cartier. Definition 2.0.18. Let X be a normal variety and D a divisor on X. Let U = X−X be the smooth subset of X and i: U → X the sing inclusion. SinceX is normal, dimX ≤dimX−2. So, every divisor sing on X is uniquely determined by its restriction to U. In particular, we define the canonical divisor K of X to be the closure of the canonical X divisor K . U For a divisor D, we associate the sheaf O (D) := i O (D). It is X ∗ U well-known that O (K ) is the same as the dualising sheaf ω in the X X X sense of 3, III.7.2 if X is projective. See 4, Proposition 5.75. n Example2.0.19. The canonical divisor ofP is just−(n+1)H where H is a hyperplane.8 CAUCHER BIRKAR Definition 2.0.20. Let X be a projective variety. AQ-Cartier divisor D on X is called nef if D·C ≥ 0 for any curve C on X. Two Q- 0 Cartier divisors D,D are called numerically equivalent, denoted by 0 0 0 D ≡ D, if D·C = D ·C for any curve C on X. Now let V,V be twoR-1-cycles on X. We call them numerically equivalent, denoted by 0 0 V ≡V if D·V =D·V for anyQ-Cartier divisor D on X. Definition 2.0.21. Let X be a projective variety. We define N (X):=group ofR-1-cycles/≡ 1 NE(X):=the cone in N (X) generated by effectiveR-1-cycles 1 NE(X):=closure of NE(X) inside N (X) 1 1 N (X):=(Pic(X)/≡)⊗ R Z 1 It is well-known that ρ(X) := dimN (X) = dimN (X) ∞ which 1 is called the Picard number of X. n Definition2.0.22. Let C⊂R be a cone with the vertex at the origin. A subcone F ⊆ C is called an extremal face of C if for any x,y∈ C, x+y∈ F implies that x,y∈ F. If dimF = 1, we call it an extremal ray. Theorem2.0.23 (Kleimanamplenesscriterion). LetX be a projective variety and D aQ-Cartier divisor. Then, D is ample iff D is positive on NE(X)−0. Theorem 2.0.24. Let X be a projective surface and C an irreducible 2 curve. (i) If C ≤ 0, then C is in the boundary of NE(X). (ii) 2 Moreover, if C 0, then C generates an extremal ray. Proof. See 4, Lemma 1.22. £ Example 2.0.25. Let X be a smooth projective curve. Then we have 0 a natural exact sequence 0 → Pic (X) → Pic(X) → Z → 0. So, 0 Pic/Pic ' Z. Therefore, N (X) ' R and NE(X) is just R . If 1 ≥0 n X =P , we have a similar story. 1 1 2 Example2.0.26. LetX =P ×P bethequadricsurface. N (X)'R 1 and NE(X) has two extremal rays, each one is generated by fibres of one of the two natural projections. 2 2 IfX is the blow up ofP at a pointP, thenN (X)'R andNE(X) 1 has two extremal rays. One is generated by the exceptional curve of the blow up and the other one is generated by the birational transform (see below) of all the lines passing through P.BIRATIONAL GEOMETRY 9 If X is a cubic surface, it is well-known that it contains exactly 27 7 lines, N (X)'R and NE(X) has 27 extremal rays, each one gener- 1 ated by one of the 27 lines. If X is an abelian surface, one can prove that NE(X) has a round shape, that is it does not look like polyhedral. This happens because an abelian variety is in some sense homogeneous. Finally, there are surfaces X which have infinitely many−1-curves. So, they have infinitely many extreml rays. To get such a surface one can blow up the projective plane at nine points which are the base points of a general pencil of cubics. We will mostly be interested in the extremal rays on which K is X negative. Definition 2.0.27 (Exceptional set). Let f: X → Y be a birational −1 morphism of varieties. exc(f) is the set of those x∈ X such that f is not regular at f(x). Definition 2.0.28. Let f: X 99K Y be a birational map of normal varieties and V a prime cycle on X. Let U ⊂ X be the open subset where f is regular. If V ∩U = 6 ∅, then define the birational transform P of V to be the closure of f(U∩V) in Y. If V = aV is a cycle and i i P ∼ U∩V 6=∅, then the birational transform of V is defined to be aV i i i ∼ where V is the birational transform of the prime component V . i i Definition 2.0.29 (Log resolution). Let X be a variety and D a Q- divisor on X. A projective birational morphism f: Y → X is a log resolution of X,D if Y is smooth, exc(f) is a divisor and exc(f)∪ −1 f (SuppD) is a simple normal crossing divisor. Theorem 2.0.30 (Hironaka). Let X be a variety and D a Q-divisor on X. Then, a log resolution exists.10 CAUCHER BIRKAR 3. SINGULARITIES Definition 3.0.31. Let X be a variety and D a divisor on X. We call X,D log smooth if X is smooth and the components of D have simple normal crossings. A pair (X,B) consists of a normal variety X and a Q-divisor B with coefficients in 0,1 such that K +B isQ-Cartier. X Now let f: Y →X be a log resolution of a pair (X,B). Then, we can write ∗ K =f (K +B)+A Y X For a prime divisor E on Y, we define the discrepancy of E with respect to (X,B) denoted by d(E,X,B) to be the coefficient of E in A. Note that if E appears as a divisor on any other resolution, then d(E,X,B) is the same. Remark 3.0.32 (Why pairs?). The main reason for considering pairs is the various kinds of adjunction, that is, relating the canonical divisor of two varieties which are closely related. We have already seen the adjunction formula (K +B) =K where X,S are smooth and S is X S S a prime divisor on X. It is natural to consider (X,S) rather than just X. Now let f: X → Z be a finite morphism. It often happens that ∗ K = f (K + B) for some boundary B. Similarly, when f is a X Z ∗ fibration and K ∼ f D for some Q-Cartier divisor D on Z, then X Q ∗ under good conditions K ∼ f (K +B) for some boundary B on Z. X Q Z Kodaira’s canonical bundle formula for an elliptic fibration of a surface is a clear example. Remark 3.0.33. Let X be a smooth variety and D aQ-divisor on X. Let V be a smooth subvariety of X of codimension ≥ 2 and f: Y → X the blow up of X at V, and E the exceptional divisor. Then, the coefficient of E in A is codimV −1−μ D where V ∗ K =f (K +D)+A Y X and μ stands for multiplicity.BIRATIONAL GEOMETRY 11 Definition3.0.34(Singularities). Let(X,B)beapair. Wecallit ter- minal(resp. canonical)ifB =0andthereisalogresolutionf: Y →X for which d(E,X,B)0 (resp. ≥0) for any exceptional prime divisor E of f. We call the pair Kawamatalogterminal ( resp. logcanonical) if there is a log resolution f for which d(E,X,B) −1 (resp. ≥−1) for any prime divisor E on Y which is exceptional for f or the bira- tional transform of a component of B. The pair is called divisorially log terminal if there is a log resolution f for which d(E,X,B) −1 for any exceptional prime divisor E of f. We usually use abbreviations klt, dlt and lc for Kawamata log terminal, divisorially log terminal and log canonical respectively. Lemma 3.0.35. Definition of all kind of singularities (except dlt) is independent of the choice of the log resolution. Proof. Supposethat(X,B)islcwithrespecttoalogresolution(Y,B ). Y 0 0 00 00 Let(Y ,B )beanotherlogresolutionand(Y ,B )acommonblowup Y Y 0 of Y and Y . The remark above shows that in this way we never get a discrepancy less than−1. Other cases are similar. £ Exercise 3.0.36. Prove that if (X,B) is not lc and dimX 1, then for any integer l there is E such that d(E,X,B)l. Exercise 3.0.37. Prove that a smooth variety is terminal. Exercise 3.0.38. Prove that terminal =⇒ canonical =⇒ klt =⇒ dlt =⇒ lc. 0 Exercise 3.0.39. If (X,B +B ) is terminal (resp. canonical, klt, dlt 0 or lc) then so is (X,B) where B ≥0 isQ-Cartier. Exercise3.0.40. Let (X,B) be a pair and f: Y →X a log resolution. ∗ Let B be the divisor on Y for which K +B =f (K +B). Prove Y Y Y X that (X,B) is • terminal iff B ≤0 and SuppB =exc(f), Y Y • canonical iff B ≤0, Y • klt iff each coefficient of B is 1, Y • lc iff each coefficient of B is≤1. Y Example 3.0.41. Let (X,B) be a pair of dimension 1. Then, it is lc (or dlt) iff each coefficient of B is ≤ 1. It is klt iff each of coefficient of B is 1. It is canonical (or terminal) iff B =0. Example 3.0.42. When (X,B) is log smooth then we have the most simple kind of singularities. It is easy to see what type of singularities this pair has.12 CAUCHER BIRKAR 2 Now let (P ,B) be a pair where B is a nodal curve. This pair is lc 2 but not dlt. However, the pair (P ,B) where B is a cusp curve is not lc. Example 3.0.43. Let’s see what terminal, etc. mean for some of the simplest surface singularities. Let X be a smooth surface containing a 1 2 curve E =P with E =−a, a 0. (Equivalently, the normal bundle to E in X has degree −a.) It’s known that one can contract E to get a singular surface Y, f: X → Y. Explicitly, Y is locally analytically 1 a the cone over the rational normal curve P P ; so for a = 1, Y is 2 2 2 3 smooth, and for a=2, Y is the surface node x +y −z =0 inA (a canonical singularity). Then K = (K +E) , and K has degree −2 on E since E is E X E E 1 isomorphic toP , so K ·E =−2+a. This determines the discrepancy X ∗ ∗ cinK =f (K )+cE, becausef (K )·E =K ·(f (E))=K ·0=0. X Y Y Y ∗ Y Namely, c = (a−2)/(−a). So for a = 1,c = 1 and Y is terminal (of course, since it’s smooth); for a=2,c=0 and Y is canonical (here Y is the node); and for a≥3, c is in (−1,0) and Y is klt. Just for comparison: if you contract a curve of genus 1, you get an lc singularity which is not klt; and if you contract a curve of genus at least 2, it is not even lc. Definition 3.0.44. Let f: X → Z be a projective morphism of va- rieties and D a Q-Cartier divisor on X. D is called nef over Z if D·C ≥ 0 for any curve C ⊆ X contracted by f. D is called numer- ically zero over Z if D·C ≥ 0 for any curve C ⊆ X contracted by f. Lemma 3.0.45 (Negativity lemma). Let f: Y → X be a projective birational morphism of normal varieties. Let D be aQ-Cartier divisor on Y such that−D is nef over X. Then, D is effective iff f D is. ∗ Proof. First by localising the problem and taking hyperplane sections we can assume that X,Y are surfaces and D is contracted by f to a point P ∈ X. Now by replacing Y with a resolution and by tak- ing a Cartier divisor H passing through P, we can find an effective exceptional divisor E which is antinef/X. Let e be the minimal non-negative number for whichD+eE≥0. If D isnoteffective, thenD+eE hascoefficientzeroatsomecomponent. On the other hand, E is connected. This is a contradiction. £ Definition 3.0.46 (Minimal resolution). Let X be a normal surface and f: Y → X a resolution. f or Y is called the minimal resolution 0 0 0 of X if f : Y →X is any other resolution, then f is factored through f.BIRATIONAL GEOMETRY 13 Remark3.0.47. The matrixE·E is negative definite for the resolu- i j tion of a normal surface singularity. So, we can compute discrepancies ∗ by intersecting each exceptional curve with K +B =f K . Y Y X Lemma 3.0.48. Let f: Y →X be the minimal resolution of a normal ∗ surface X. Then, K +B =f K where B ≥0. Y Y X Y Proof. First prove that K is nef over X using the formula (K +C)· Y Y C =2p (C)−2forapropercurveC onY. Then,thenegativitylemma a implies that B ≥0. £ Y Theorem 3.0.49. A surface X is terminal iff it is smooth. Proof. If X is smooth then it is terminal. Now suppose that X is terminal and let Y → X be a minimal resolution and let K +B Y Y be the pullback of K . Since X is terminal, d(E,X,0) 0 for any X exceptional divisor. Thus, B 0 a contradiction. So, Y =X. £ Y Corollary 3.0.50. By taking hyperplane sections, one can show that terminal varieties are smooth in codimension two. Remark 3.0.51. The following are equivalent: X has canonical surface singularities (:= Du Val singularities) 3 Locally analytically X is given by the following equations inA : 2 2 n+1 A: x +y +z =0 2 2 n−1 D: x +y z+z =0 2 3 4 E : x +y +z =0 6 2 3 3 E : x +y +yz =0 7 2 3 5 E : x +y +z =0 8 Lemma 3.0.52. If X is klt, then all exceptional curves of the minimal resolution are smooth rational curves. Proof. Let E be a an exceptional curve appearing on the minimal res- olution Y. Then, (K +eE)·E ≤ 0 for some e 1. So, (K +E + Y Y 2 (e−1)E)·E =2p (E)−2+(e−1)E ≤0 which in turn implies that a p (E)≤0. Therefore, E is a smooth rational curve. £ a Remark 3.0.53. the dual graph of a surface klt singularity: A, D and E type 7. Example 3.0.54. Singularities in higher dimension. Let X be defined 2 2 2 2 4 by x + y + z + u = 0 in A . Then, by blowing up the origin of 4 A we get a resolution Y → X such that we have a single exceptional 1 1 divisor E isomorphic to the quadric surface P ×P . Suppose that ∗ 1 K =f K +eE. Take a fibre C of the projection E→P . Either by Y X14 CAUCHER BIRKAR calculation or more advanced methods, one can show that K ·C 0 Y and E·C 0. Therefore, e0. So, it is a terminal singularity. Remark 3.0.55 (Toric varieties). Suppose that X is the toric variety associated to a cone σ⊂N , R • X is smooth iff σ is regular, that is primitive generators of each face of σ consists of a part of a basis of N, • X isQ-factorial iff σ is simplicial, • X is terminal iff σ is terminal, that is, there is m∈ M such Q that m(P) = 1 for each primitive generator P ∈ σ∩N, and m(P)1 for any other P ∈N∩σ−0, • If K isQ-Cartier, then X is klt. X See 2 and 5 for more information.BIRATIONAL GEOMETRY 15 4. MINIMAL MODEL PROGRAM 4.1. Kodaira dimension. Definition4.1.1. LetD beadivisoronanormalprojectivevarietyX. 0 (n−1) If h (D) = n6= 0, then we define the rational map φ : X 99KP D as φ (x)=(f (x):···:f (x)) D 1 n 0 wheref ,...,f is a basis for H (X,D). 1 n Definition 4.1.2 (Kodaira dimension). For a divisor D on a normal projective variety X, define 0 κ(D)=maxdimφ (X)h (X,mD)6=0 mD 0 if h (X,mD)6=0 for some m∈N, and−∞ otherwise. IfD isaQ-divisor,thendefineκ(D)tobeκ(lD)wherelD isintegral. In particular, by definition κ(D)∈−∞,0,...,dimX. For a pair (X,B), define the Kodaira dimension κ(X,B):=κ(K + X B). Definition 4.1.3 (Big divisor). AQ-divisor on a normal variety X of dimension d is big if κ(D) = d. In particular, if κ(X,B) = d, we call (X,B) of general type. Exercise 4.1.4. Prove that the definition of the Kodaira dimension of aQ-divisor is well-defined. That is, κ(D) = κ(lD) for any l∈N if D is integral. Exercise 4.1.5. Let H be an ample divisor on a normal projective variety X. Prove that κ(H)=dimX. Exercise 4.1.6. Let f: Y → X be a contraction of normal projective ∗ varieties and D aQ-Cartier divisor on X. Prove that κ(D)=κ(f D). Exercise 4.1.7. Let D be a divisor on a normal projective variety X. Prove that, 0 • κ(D)=−∞⇐⇒ h (mD)=0 for any m∈N. 0 • κ(D)=0⇐⇒ h (mD)≤1 for any m∈N with equality for some m. 0 • κ(D)≥1⇐⇒ h (mD)≥2 for some m∈N.16 CAUCHER BIRKAR Exercise 4.1.8. Let D,L beQ-divisors on a normal projective vari- 0 ety X where D is big. Prove that there is m∈N such that h (X,mD− L)= 6 0. Exercise 4.1.9. For a Q-divisor D on a projective normal variety X, define 0 P (D):=h (X,bmDc) m Define the Kodaira dimension of D as logP (D) m κ(D):=limsup logm m→∞ Now prove that this definition is equivalent to the one given above. 4.2. Basics of minimal model program. Definition 4.2.1 (Minimal model-Mori fibre space). A projective lc pair (Y,B ) is called minimal if K + B is nef. A (K + B )- Y Y Y Y Y negative extremal contraction g: Y →Z is called a Mori fibre space if dimY dimZ. Let (X,B), (Y,B ) be lc pairs and f: X 99K Y a birational map Y whose inverse does not contract any divisors such that B = f B. Y ∗ (Y,B) is called a minimal model for (X,B) if K +B is nef and if Y Y d(E,X,B)d(E,Y,B ) for any prime divisor E on X contracted by Y f. g: Y → Z is a Mori fibre space for (X,B) if it is a (K +B )- Y Y negative extremal contraction such that d(E,X,B) d(E,Y,B ) for Y any prime divisor E on X contracted by f. Conjecture4.2.2 (Minimalmodel). Let(X,B) be a projective lc pair. Then, • If κ(X,B)=−∞, then (X,B) has a Mori fibre space. • If κ(X,B)≥0, then (X,B) has a minimal model. Conjecture 4.2.3 (Abundance). Let (Y,B) be a minimal lc pair. Then, m(K +B) is base point free for some m ∈ N. This, in par- Y ticular, means that there is a contraction h: Y → S called the Iitaka fibration and an ampleQ-divisor H on S such that ∗ K +B =h H Y In fact, • h(C)=pt.⇐⇒(K +B)·C=0 for any curve C on Y. Y • dimS =κ(Y,B).BIRATIONAL GEOMETRY 17 Conjecture 4.2.4 (Iitaka). Let f: X →Z be a contraction of smooth projective varieties with smooth general fibre F. Then, κ(X)≥κ(Z)+κ(F) Definition 4.2.5 (Contraction of an extremal ray). Let R be an ex- tremal ray of a normal projective variety X. A contraction f: X →Z is the contraction of R if f(C)=pt.⇐⇒C∈R for any curve C⊂X. Remark 4.2.6 (Types of contraction). For the contraction of an ex- tremal ray R we have the following possibilities: Divisorial: f is birational and contracts divisors. Flipping: f is birational and does not contract divisors. Fibration: f is not birational. Definition 4.2.7 (Flip). Let (X,B) be a pair where X is projective. A (K +B)-flip is a diagram X + 99K X X C C y C y C y C y C y C y + f C y f y Z such that + • X and Z are normal varieties. + • f and f are small projective birational contractions, where small means that they contract subvarieties of codimension≥2. • f is the contraction of an extremal ray R. + • −(K +B) is ample over Z, and K + +B is ample over Z X X + where B is the birational transform of B. Exercise 4.2.8. Suppose that X isQ-factorial and projective. Prove thatQ-factoriality is preserved after divisorial contractions and flips. Definition 4.2.9 (Minimal model program: MMP). The minimal model program can be described in different level of generality. The following seems to be reasonable. Let (X,B) be a dlt pair where X is Q-factorial and projective. The following process is called the minimal model program if exists: If K +B is not nef, then there is an (K +B)-extremal ray R and X X its contraction f: X →Z. If dimZ dimX, then we get a Mori fibre space and we stop. If f is a divisorial contraction, we replace (X,B) with (Z,f B) and continue. If f is flipping, we replace (X,B) with ∗18 CAUCHER BIRKAR + + the flip (X ,B ) and continue. After finitely many steps, we get a minimal model or a Mori fibre space. Example 4.2.10. Classical MMP for smooth projective surfaces. Conjecture4.2.11(Termination). Let(X,B)beadltpairwhereX is Q-factorial and projective. Any sequence of (K +B)-flips terminates. X 4.3. Cone and contraction, vanishing, nonvanishing, and base point freeness. Theorem4.3.1(Coneandcontraction). Let(X,B)beakltpairwhere X is projetive. Then, there is a set of (K +B)-negative extremal rays X R such that i P • NE(X)=NE(X) + R . K +B≥0 i X i • Each R contains the class of some curve C , that is C ∈R . i i i i • R can be contracted. i •R is discrete in NE(X) . i K +B0 X Remark 4.3.2. Remember that a divisor D on a normal variety X is called free if its base locus \ 0 BsD:= SuppD 0 D∼D≥0 n−1 is empty. So, for a free divisor D the rational map φ : X 99KP D associated to D in Definition 4.1.1 is actually a morphism. The Stein factorisation of φ gives us a contraction ψ : X → Y such that D∼ D D ∗ ψ H for some ample divisor H on Y. D Theorem 4.3.3 (Base point free). Let (X,B) be a klt pair where X is projective. Suppose that for a nef Cartier divisor D, there is some a 0 such that aD−(K +B) is ample. Then, mD is free for some X m∈N. Theorem 4.3.4 (Rationality). Let (X,B) be a klt pair where X is projective. Let H be an ample Cartier divisor on X. Suppose that K +B is not nef. Then, X λ=maxt0t(K +B)+H is nef X a is a rational number. Moreover, one can write λ = where a,b ∈N b and b is bounded depending only on (X,B). Theproofofthisissimilartotheproofofthebasepointfreetheorem and Shokurov nonvanishing theorem. So, we won’t give a proof here. See 4 for a proof.BIRATIONAL GEOMETRY 19 Proof. (ofConeandContractiontheorem)WemayassumethatK +B X is not nef. For any nefQ-Cartier divisor D define F =c∈N (X)D·c=0⊂NE(X) D 1 where the inclusion follows from Kleiman ampleness criterion. Let X C =NE(X) + F K +B≥0 D X D where D runs overQ-Cartier nef divisors for which dimF = 1. Sup- D pose that C = 6 NE(X). Choose a point c ∈ NE(X) which does not belong to C. Now choose a rational linear function α: N (X) → R 1 which is positive onC−0 but negative on c. This linear function is defined by someQ-Cartier divisor G. If t0, then G−t(K +B) is positive on NE(X) and it is X K +B=0 X positive on NE(X) for t½0. Now let K +B≤0 X γ =mint0G−t(K +B) is nef on NE(X) X K +B≤0 X So G−t(K +B) is zero on some point in NE(X) . Then it X K +B≤0 X shouldbepositiveonNE(X) . Therefore,H =G−t(K +B)is K +B≥0 X X ample for some rational numbert. Now the rationality theorem proves that λ=maxt0H +t(K +B) is nef X is a rational number. Put D = H +λ(K +B). Here it may happen X that dimF 1. In that case, let H be an ample divisor which is D 1 linearly independent of K +B on F and let ¯ 0 be sufficiently X D 1 small. For s0 let λ(s,¯ H )=maxt0sD+¯ H +t(K +B) is nef 1 1 1 1 X Obviously, λ(s,¯ H ) is bounded from above where the bound does 1 1 not depend on s. Therefore, if s½0, then F ⊆F sD+¯ H +λ(s,¯ H )(K +B) D 1 1 1 1 X where the inclusion is strict because H and K +B are linearly in- 1 X dependent on F . Putting all together, we get a contradiction. So, D C = NE(X). Note that, by the rationality theorem, the denominator of λ(s,¯ H ) is bounded. 1 1 NowletH ,...,H beampledivisorssuchthattogetherwithK +B 1 n X 1 theyformabasisforN (X). LetT =c∈N (X)(K +B)·c=−1. 1 X Let R be an extremal ray and D a nef Q-Cartier divisor such that R =F . Let c=T ∩R. Then, D ¯ H ·c j j λ(D,¯ H )= =¯ H ·c j j j j −(K +B)·c X20 CAUCHER BIRKAR Therefore, thisispossibleonlyifsuchcdonothaveanaccumulation point inT. This in particular, implies that X NE(X)=NE(X) + R K +B≥0 i X D where R are the (K +B)-negative extremal rays. i X NowletR bea(K +B)-negativeextremalray. Then, thereisanef X CartierdivisorDsuchthatR =F . Thus,fora½0,aD−(K +B)is D X ample. So, by the base point free theorem,mD is free for somem∈N. Thisgivesusacontractionψ : X →Y whichcontractsexactlythose mD curves whose class belong to R. This proves the contractibility of R. On the other hand, the morphism mD is trivial only if mD is ample. By construction, mD is not ample, therefore ψ is not trivial and it mD contracts come curve C. But then the class of C has to be in R. £ Theorem 4.3.5 (Kamawata-Viehweg vanishing). Let (X,B) be a klt pair where X is projective. Let N be an integralQ-Cartier divisor on X such that N ≡K +B+M where M is nef and big. Then, X i H (X,N)=0 for any i0. For a proof of this theorem see 4. Theorem 4.3.6 (Shokurov Nonvanishing). Let (X,B) be a klt pair where X is projective. Let G≥ 0 be a Cartier divisor such that aD+ G−(K +B) is ample for some nef Cartier divisor D. Then, X 0 H (X,mD+G)= 6 0 for m½0. Remark 4.3.7. Let D be a Cartier divisor on a normal variety X. Then, we can write D∼M +F where M is movable and each compo- nent of F is in BsD. M is called the movable part of D and F the ∗ 0 0 fixed part. There is a resolution f: Y → X such that f D∼ M +F 0 such that M is a free divisor. Remark 4.3.8. Every bigQ-divisor D on a normal variety X can be written as D =H +E where H is ample and E≥0. If in addition D is also nef, then 1 1 1 0 D = ((m−1)D+H)+ E =H + E m m m 0 where m ∈N and H is obviously ample. So D can be written as the sum of an ampleQ-divisor and an effectiveQ-divisor with sufficiently small coefficients.

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