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Chapter V. Fano Varieties

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Chapter V. Fano Varieties Chapter V. Fano Varieties A variety X is called Fano if the anticanonical bundle of X is ample. Thus Fano surfaces are the same as Del pezzo surfaces. The importance of Fano varieties in the theory of higher dimensional varieties is similar to the sig­ nificance of Del Pezzo surfaces in the two dimensional theory. The interest in Fano varieties increased recently since Mori's program predicts that every uniruled variety is birational to a fiberspace whose general fiber is a Fano variety (with terminal singularities). From this point of view it is more important to study the general prop­ erties of Fano varieties with terminal singularities than to understand the properties of smooth Fano varieties. At the moment, however, we know much more about smooth Fano varieties, and their theory should serve as a guide to the more subtle questions of singular Fano varieties. Fano varieties also appear naturally as important examples of varieties. In characteristic zero every projective variety which is homogeneous under a linear algebraic group is Fano (1.4), and their study is indispensable for the theory of algebraic groups. Also, Fano varieties have a very rich internal geometry, which makes their study very rewarding. This is one of the reasons for the success of the theory of Fano threefolds. This is a beautiful subject, about which I say essentially nothing. Section 1 is devoted to presenting the basic examples of Fano varieties and to the study of low degree rational curves on them. The largest class of examples are weighted complete intersections (1.2-3); these are probably the most accessible by elementary methods. Homogeneous spaces also provide many examples but their detailed study requires the machinery of algebraic groups (1.4). The most studied examples are the moduli spaces of stable vector bundles with fixed determinant on curves. Their theory deserves a monograph in itself; we mention them for sake of completeness only. The cone of curves of a Fano variety X is generated by rational curves Ci C X such that -Kx ,Ci S dim X +1 (1.6). This and many other examples lead to the following Principle. The geometry of a Fano variety is governed by rational curves of low degree. J. Kollár, Rational Curves on Algebraic Varieties © Springer-Verlag Berlin Heidelberg 1996 V.1 Low Degree Rational Curves on Fano Varieties 239 The rest of Sect. 1 is devoted to various assertions that support this prin­ ciple. The aim of Sect. 2 is to prove two general results about Fano varieties. Any smooth Fano variety is rationally chain connected (2.1, 2.13). For any dimension, there are only finitely many deformation types of Fano varieties, at least in characteristic zero (2.3, 2.14). This result implies that, in principle, it is possible to obtain a complete list of Fano varieties of any given dimension. In dimension three this has been accomplished, but the complexity of the arguments and the length of the list suggests that already in dimension 4 this is not a feasible project. Section 3 discusses Mori's characterization of Ipm as the only algebraic variety whose tangent bundle is ample (3.2). [Mori79] is the article where the bend-and-break technique was first introduced. Mori's arguments are very elegant and many of his ideas found later applications. I give a somewhat shortened version of the proof. This result can also be considered as another example of the validity of the above principle. The canonical line bundle of pn is O( -n - I}, thus lines in pn have anticanonical degree n + 1 and there are no curves with smaller anticanonical degree. Ampleness of the tangent bundle of a variety X easily implies that there are no rational curves of anticanonical degree less than dim X + 1 (3.6.1). Section 4 is more like an overgrown exercise about lines of low degree hypersurfaces. The main technical result says that the family of all lines is connected when dimension count suggests that it might be so. This can be used to prove that the group of I-cycles modulo algebraic equivalence is one dimensional (4.1). In some cases it also implies that the group of 1- cycles modulo rational equivalence is one dimensional (4.2). Even in this very concrete situation there are many interesting open problems. There are many similarities between rational and rationally connected varieties. In fact, it is not easy to show that not every rationally connected variety is rational. This question is studied in Sect. 5. The method first pro­ duces rationally connected varieties in positive characteristic which are not separably uniruled. These can then be lifted to characteristic zero to produce examples of Fano hypersufaces which are not ruled. These techniques produce other interesting examples in positive characteristic as well. V.I Low Degree Rational Curves on Fano Varieties The aim of this section is to present the basic examples of Fano varieties and to prove some simple results about low degree rational curves on them. In this section everything is defined over an algebraically closed field k. 240 Chapter V. Fano Varieties 1.1 Definition. Let X be a smooth projective variety. X is called Fano if -Kx is ample. Let X be a normal projective variety. X is called Q-Fano (or just Fano) if - K X is Q-Cartier and ample. 1.2 Example. Let X c ]pn be a smooth complete intersection of k hyper­ surfaces of degrees d}, ... , dk. Then -Kx = O{n+ 1-L: di ). Thus X is Fano iff L: di < n + 1. More generally, complete intersections in weighted projective spaces also provide lots of examples of Fano varieties. See [Dolgachev82) for a general introduction or [Fletcher89) for a series of examples of Q-Fano threefolds. The basic properties of weighted projective spaces are recalled in the next series of exercises. 1.3 Exercises. Let k be a field and S = k[xo, ... , xn ) the polynomial ring in n + 1 variables. Let ai E N. Define a grading of S by deg Xi = ai. Proj Sis called the weighted projective space of dimension n with weights ai. It is de­ noted by P(ao, ... , an}. We may and do assume that ao, ... , an are relatively prime. The following shortened version of this notation is frequently convenient: P( a~o , ... , a;{' ) denotes ro-times rn.-tinles Because of this convention, one should never use pel, 2, 32) to denote pel, 2, 9)! (1.3.l) Show that P(aO,al, ... ,an) 9:! P(aO,dal, ... ,dan) for every dE N. Thus in working with weighted projective spaces of dimension n we may always assume that any n weights are relatively prime. We say that P(ao, ... ,an) is well formed if this condition is satisfied. For the rest of the section we assume that all weighted projective spaces are well formed. (1.3.2) Show that P(ao, a}, ... ,an) ~ pn / /Lao X ••• X /La" where /Laj denotes the group of a~h roots of unity and it acts on pn via multiplication on the lh coordinate. (If you feel uneasy about group schemes, assume that the characteristic does not divide any of the ai.) (1.3.3) Describe a covering of P{ao, ... ,an) with affine charts. Show that P(ao, ... ,an) has only cyclic quotient sigularities. (1.3.4) Let Oem) denote the coherent sheaf associated to the graded module SCm). Show that Oem) is locally free iff ailm for every i. Show that (1.3.5) Show that the dualizing sheaf Kp of P{ao, ... ,an) is isomorphic to O{ - L: ai). V.l Low Degree Rational Curves on Fano Varieties 241 (1.3.6) Show that lP'(ao, ... ,an) has isolated singularities iff the weights are pairwise relatively prime. (1.3.7) Assume that IP'( ao, ... , an) has isolated singularities and ai Im for every i. Show that the smooth members of IO(m)1 form a dense open set. If H E IO(m)1 is smooth, then KH = Oem - ~ ai)IH. (1.3.8) Let do, . .. ,dr be pairwise relatively prime natural numbers. Let H be a smooth member of O(do'" dr ) on lP'(do, ... , dr ). OH(l) is a locally free sheaf, which is ample and has selfintersection 1. (1.3.9) Let X = Xd1, ... ,dk C lP'(ao, ... ,an) be a smooth (or normal) com­ plete intersection of k hypersurfaces of degrees db'" ,dk' Then -Kx = O(~aj - ~di)' Thus X is Fano iff ~di < ~aj. (1.3.10) Let I: 1P'1 --t lP'(ao, ... , an) be a morphism such that lP'(ao, ... , an) is smooth along im I. Show that I is given by a a collection of sections Ii E HO(lP'l,O(dai)), i = O, ... ,n where d = degf*O(l). d is called the degree of I. (1.3.11) Find further examples of Fano manifolds which are complete intersections in products of weighted projective spaces. Another large class of examples of Fano varieties is provided by homoge­ neous spaces: 1.4 Theorem. Let C be a reduced and connected linear algebraic group and X a proper homogeneous space under C. Pick a point x E X and let H < C be the stabilizer 01 x. Assume that H is reduced (which is always the case in characteristic zero). Then Tx is generated by global sections and K)/ is very ample. Proof As we see in (1.4.2.4), all such homogeneous spaces can be classified in terms of Dynkin diagrams and the theorem can be read off from various assertions about algebraic groups.
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