Cone of Curves, and Fano 3-Folds

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1 Proceedings of the International Congress of Mathematicians August 6-24, 983, Warszawa SHIGEFUMI MOBI* Cone of Curves, and Fano 3-Folds. Cones of curves, extremal rays, and Fano 3-folds Through the solution of Frankel and Hartshorne conjectures (Mori [0], Sin and Yau [20]), the importance of finding a rational curve on a given algebraic variety is exibited. It is done through a cone of curves and extremal rays. Let X be an ^-dimensional non-singular projective variety defined over an algebraically closed field Jc of characteristic #>0. Let N Z (X) be the set of numerical equivalence classes of -cycles on X and let N(X) = N Z (X) Z R. Then W(X) is a real vector space of dimension Q(X), and in N(X) we consider the smallest closed convex cone NJS(X) (the cone of curves) containing all the classes of irreducible curves which are stable under multiplication by R+, the set of non-negative real numbers. This cone was originally considered by Hironaka [3] and Kleinian [5]. À new notion here is the extremal ray, which is defined as follows. Let JÖL(Z) = {z e NB(X) (*. ~K X ) < 0}. We say that a half line B = R + z c NE(X) is an extremal ray if there exists a closed convex cone B => NJE_(X) such that E < B and NE(X) ^E+B. THDEOKEM (Mori []). NPJ(X) is the smallest closed convex cone containing NB^(X) and all the extremal rays. An arbitrary extremal ray is generated by the class of a (possibly singular) rational curve 0 such that < (0. K x ) <w+l. A divisor D on X is called numerically effective if (D. Ö) > 0 for an arbitrary irreducible curve 0. An obvious corollary is * Partially supported "by the Japan-U.S.A. exchange foundation. [747]

2 748 Section 6: S. Mori COROLLARY. X has an extremal ray if and only if K x is not numerically As a special case of the theorem, one has COROLLARY. If K x is ample, then NE(X) is spanned by a finite number of extremal rays. If dimx = 2, or dimx = 3 and p = 0, extremal rays are geometrically explained by means of "contractions". THEOREM DEFINITION. Assume that dim X = 2, or dim X = 3 and p = 0. For an arbitrary extremal ray E ofx, there exists a morphism f: X->Y to a projective variety such that and (i) /* x Y (ii) an arbitrary irreducible curve Ö is collapsed to a point if and only if the class [0] of G belongs to E. Such an f is imique up to cm isomorphism und (iii) K x is f-ample. We call f the contraction of E. For dimx = 2, the description of the extremal ray is equivalent to the following classical result. THEOREM. Assume that dim X = 2, and X has cm extremal ray E. Let f: X->Y be the contraction of E. Then one of the following hold. (i) f is the contraction of an exceptional curve Ö of the first Mnd, and E =B + [0]. (ii) f is a P l -bundle, and E is generated by a fiber off. (iii) X ~P 2, Y ~Specfc, and E =NE(X). The result for dim JC = 3 and p = 0 turns out to be a natural generalization of the above. THEOREM. Assume that dim X = 3, p = 0, and X has an extremal ray E. Let f: X->Y be the contraction of E. Then Q(X) = Q( Y)+l, and one of the following holds. (i) / contracts an irreducible divisor D to a curve or to a pointy f is then the blow-up of Y along the reduced closed subscheme f(b), and one of the following 5 cases holds: (i.l) Y is non-singular, f(jd) is a non-singular curve, Dis a P^bundle over /(D), and Ö? JD ( D) is the tautological line bundle, (i.2) Y is non-singular, f(jd) is a point, D ~P 2, and (5 D ( D) -Ml),

3 Cone of Curves, and Fano 3-Folds 749 (i.3) /(D) is an ordinary double point of Y such $ s factorial, D c^p xp, and ß D { -D) ~ 0^(, i), (i.4) /(D) is a double point of Y,D is isomorphic to an irreducible singular quadric surface Q of P 3, D ( B) ~Q Q (), and (i"5) /(D) is a quadruple point, D c^p 2, and G D ( I>) ~& P (2). (ii) / is flat and one of the following holds: (ii.l) an arbitrary fiber of fis isomorphic to a conic in P 2 as a scheme {f is called a conic bundle), and (ii.2) an arbitrary fiber off is an irreducible reduced surface D such that cop is ample (f is called a del Pezzo fibring). (iii) Y c^ Spec ft, hence Q(X) = and K x is ample (cf. (), (2) and IslkovshiVs classification in (3), (4) of the next theorem). We say that an extremal ray E = B + z is numerically effective if (z. D) > 0 for every irreducible divisor D on X. Then, E is numerically effective exactly in cases (ii), (iii) of the above theorem. We say that X is a Fano n-fold if K x is ample. The index r of a Fano x 3-fold X is, by definition, the greatest integer ( > 0) such that K x erpicx The following results are known in characteristic 0: THEOREM. (0) One has r =,2, 3, or 4, () if r =4, then X ~P 3 ([8]), (2) if r = 3, then X ~ 3-dimensi'onal quadric ([8]), (3) if r = 2, and g(x) = then there are exactly 5 deformation types of X (IsJcovshih [4] and Fujita [], cf. [2] for char p), (4) if r = and Q(X) = then Ishovslcih [4] classified X*B into 0 families using the results of Shohurov [8], [9] (we refer the reader to Iskovsldh's report in this volume for the precise classification of (3) and (4)) cmd (5) If r = and Q(X) > 2 then there are exactly 87 deformation types, Mori-Mukai [2]: 6() # of del types By a technique similar to the one in [0], Janos Kollar proved > THEOREM. In any characteristic an arbitrary Fano n-fold X is uni-ruled. In other words, given any point œ of X, there exists a rational curve Ö through so such that (Ö. K x )^.n+. 0

4 750 Section 6: S. Mori 2. Terminal singularities To continue the process of contraction of extremal rays, one has to consider projective varieties with "terminal singularities" in the sense of Eeid [5]. DEFINITION (Eeid [5]). Let lc = C. Let y be a point of a normal 3-fold Y. Then we say that Y has only terminal singularity (resp. canonical singularity) at y if there is a natural number r such that the Weil divisor r K Y is actually a Cartier divisor and, for some resolution (or equivalently, for an arbitrary resolution) /: Y'->Y, the sections of 0(r K T ) vanish along every divisor in f~*(y) (resp. are regular in a neighborhood of ^(y)) when considered as meromorphic sections of rk r *. THEOREM (Eeid [5]). If (Y,y) is a 3-dimensional terminal singularity, then (Y,y) is the quotient, by a cyclic group action, of an isolated cbv singularity. Such a singularity is, by definition, an isolated 3-dimensional hypersurface singularity (Z, z) whose general hyperplane section through z has a rational double (or smooth) point at z. DEFINITION. Assume that h = C. We say that a normal algebraic variety X is Q-factorial at x if the divisor class group Gl((9 XtX ) of the local ring & XtX at œ is torsion, and that X is Q-factorial if X is Q-factorial at every point (Eeid [6]). Let /: X->Y be a projective morphism such that X is a Q-factorial 3-fold with only terminal singularities and/*^ = & Y. Let us consider -cycle ^a G G which is a linear combination of curves G such that f(g) is a point. Let W z (XjY) be the set of numerical equivalence classes of such -cycles, N(XjY) = N z (XfY) z R and JSfF(X) be the smallest closed convex cone containing all the classes of irreducible curves G on X such that f(g) is a point. We say that a half line E c NF(XjY) is an extremal ray of XjY (or an f-extremal ray) if E is generated by the class of a curve and there is a closed convex cone B of NE(X)Y) such that B $ B, NË(XfY) =E+B, and B ^~NF (XIY), where WF_(X/Y) ={zenb(xly)\ (z. -K x )^0}. We fix the above notation. One can ask the following QUESTION. IS NF(XjY) the smallest closed convex cone containing NF_(X/Y) and all the /-extremal rays? For an arbitrary open convex cone TJ containing NE (XjY), are there only a finite number of /-extremal rays which are not contained in Uu {0}?

5 Cone of Curves, and Fano 3-Folds 75 The answer is affirmative if Y = Specfl; and K x is pseudo-effective (this is the case if H(X) > 0) (Kawamata [7]), or if dinil"^ (done essentially by Tsunoda [2]). If XjY has an extremal ray E, and if E is not numerically effective, one contracts the extremal ray (Kawamata [7]), but if the contracted 3-fold X' is not Q-factorial, we have to blow up X to get a better model (and finally to get a "minimal" model). However, this process is not well understood yet. Possibly one should say that XjY is a minimal model if XjY does not have an extremal ray which is not numerically effective. If dim Y = 0 and K x is pseudo-effective, then this is equivalent to saying that K x is numerically effective (Eeid [5]). Kawamata [6] proved that if dim Y = 0 and X is a minimal model (over Y) of general type then the canonical ring E [X, Q(mK x )) is finitely generated. m Let dim Y == and assume that X\Y has everywhere semi-stable reduction with general fiber X t. It is known that XjY has a minimal model if X t is a K3 surface or an abelian surface (Kulikov [9], Persson and Pinkham [4]) or if X t is an Enriques' surface or a hyperelliptic surface with 2K ^ 0 (Morrison [3] through combinatorial bixational geometry). Tsunoda [2] seems to have proved that X/Y has a minimal model if X t is a mimimal model with n(x) > 0 by using the logarithmic version of cones of curves and extremal rays. If a three-dimensional and Q-factorial Y has only terminal singularities, then the process of finding a minimal model over Y is the same as factoring the birational morphism X-*Y into the product of elementary birational maps. However, this process is not well understood yet. After having written up this paper, the author found that the question marked here was also put as a conjecture in the newly added part of [7]. References [] Fujita T., On the Structure of Polarized Manifolds With Total Deficiency One, I and II, J. Math. Soc. Japan 32 (980), pp , and 33 (98), pp [2] Fujita T., On Polarized Varieties of Small ^-Genera, Tolwlm Math. Journ. 34 (982), pp [3] Hironaka H., On the Theory of Birational Blowing-Up (Thesis), Harvard, 960. [4] Iskovslrih V. A., Fano 3-Folds, I and II, Ism. Ahad. Nauh 88SR Ber. Mat. 4 (977) and 42 (978), English transi.: Math.TJSSB Izv. (977), pp and 2 (978), pp [5] Kleinian S., Toward a Numerical Theory of Ampleness, Ann. of Math. 84 (966), pp [6] Kawamata Y., On the Finiteness of Generators of apluri-oanonieal Bing for a 3-Fold of General Type, to appear.

6 752 Section 6: S. Mori [7] Kawamata Y., Elementary Contractions of Algebraic 3-Folds, to appear. [8] Kobayashi S. and Ochiai T., Characterization of Complex Projective Spacesand Hyperquadrics, J. Math. Kyoto TJ. 3 (973), pp [9] Kulikov V. S., Degenerations of K-3 Surfaces and Enriques' Surfaces, Izv. Ahad. Nauk SSSB Ser. Mat. 4 (977), English transi.: Math. US8B Izv. (977), pp [0] Mori S., Projective Manifolds with Ample Tangent Bundles, Ann. of Math, lid- (979), pp [] Mori S M Threefolds Whose Canonical Bundles are not Numerically Effective, Awn. of Math. 6 (982), pp [2] Mori S., and Mukai S., Manuscripta Math. 36 (98), pp [3] Morrison D., Semistable Degenerations of Enriques 9 and HyperelUptic Surfaces (Thesis), Harvard, 980. [4] Persson U. and Pinkham H., Degeneration of Surfaces with Trivial Canonical Bundle, Ann. of Math. 3 (98), pp [5] Reid M., Minimal Models of Canonical 3-l?olds, Symp. Math., Kinokuniya and North-Holland, 982. [6] Eeid M., Décomposition of Toric Morphisms, to appear. [8] Shokurov V. V., The Smoothness of the General Anticanonical Divisor on a Fano* Variety, lev. Ahad. Nauk SSSB Ser. Mat. 43 (979), English transi.: Math. USSB lev. 4 (980), pp [9] Shokurov V. V., The Existence of Lines on Fano 3-Folds, Izv. Ahad. Nauk SSSB Ser. Mat 43 (979), English transi.: Math. USSB Izv. 5 (980), pp [20] Siu Y. T* and Yau S.-T., Compact Kaehler Manifolds of Positive Bisectional Curvature, Inv. Math. 59 (980), pp [2] Tsunoda S., Degenerations of Minimal Surfaces with Non-Negative Kodaira. pimension (in Japanese), Proc. Symp. Algebraic Geometry, Kinosaki (98). NAGOYA UNIVERSITY NAGOYA, 464, JAPAN