Appendices A Projective Bundles We recall here some basic facts about projective bundles. Definition and construction. Let X be an algebraic variety or scheme, and let E be a vector bundle of rank e on X. We denote by π : P(E) −→ X the projective bundle of one-dimensional quotients of X.ThusapointinP(E) is determined by specifying a point x ∈ X together with a one-dimensional quotient of the fibre E(x)ofE at x. More algebraically, P(E) is realized as the scheme P(E)=Proj Sym(E) , OX where Sym(E)=⊕ SmE denotes the symmetric algebra of E.1 Serre line bundle. The projective bundle P(E) carries a line bundle ∗ OP(E)(1), arising as a “tautological” quotient of π E: ∗ π E −→ O P(E)(1) −→ 0. (A.1) ∼ If L is a line bundle, then P(E) = P(E ⊗ L) via an isomorphism under which ∗ OP(E⊗L)(1) corresponds to OP(E)(1) ⊗ π L. For m ≥ 0onehas m π∗O (m)=S E (A.2a) P(E) e−1 m ∗ ∗ R π∗OP(E)(−e − m)= S E ⊗ det E , (A.2b) and all other direct images vanish. Maps to P(E). Let p : Y −→ X be a variety or scheme mapping to X.Then giving a line bundle quotient p∗E L of the pullback of E is equivalent to specifying a map f : Y −→ P(E)overX: 1 Here and elsewhere we do not distinguish between E and the corresponding locally free sheaf of sections. 316 Appendix A. Projective Bundles f / Y @ P(E) @@ yy @@ yy @@ yy p @ y|y π X. ∗ Under this correspondence L = f OP(E)(1), and the given quotient is the pullback of (A.1). In particular, for each >0 there is a natural Veronese embedding of P(E)intoP(S E): ν P(E) /P(S E) EE w EE ww EE ww p E ww π E" w{w X. ∗ th It is determined by the quotient S π E OP(E)() arising by taking symmetric products in (A.1). N´eron–Severi group. Assume that X is projective. Then the N´eron–Severi group of P(E) is determined by 1 ∗ 1 N (P(E)) = π N (X) ⊕ Z · ξE, O where ξE is the class of a divisor representing P(E)(1). Writing ξE also for 2 ∗ the corresponding class in H X;Z , the integral cohomology H P(E); Z of ∗ P(E) is generated as an H X; Z -algebra by ξE, subject to the Grothendieck relation e − ∗ · e−1 ∗ · e−2 − e ∗ ξE π c1(E) ξE + π c2(E) ξE + ... +( 1) π ce(E)=0. One–dimensional subspaces. On a few occasions it will be preferable to consider the bundle Psub(E) of one–dimensional subspaces of E.Thisisrelated ∗ to the bundles of quotients by the isomorphism Psub(E)=P(E ). B Cohomology and Complexes We collect in this appendix several constructions and facts of a homological nature. The first subsection deals with techniques for analyzing the cohomol- ogy of a sheaf. The second focuses on some complexes associated to vector bundle maps. B.1 Cohomology This section lays out a collection of tools that can be useful in computing the cohomology of a coherent sheaf on a variety or scheme. Cohomology and direct images. Let f : Y −→ X be a morphism of varieties or schemes, and consider a sheaf F on Y .Itis frequently important to compute the cohomology of F on Y in terms of data — specifically, the direct images of F —onX. The basic mechanism to this end is the Leray spectral sequence p,q p q F ⇒ p+q F E2 = H X, R f∗ = H Y, . However, one often needs only a special case, in which the sequence degener- ates. Proposition B.1.1. In the situation just described, assume that all the higher j direct images of F vanish, i.e. suppose that R f∗F =0for every j>0.Then i i H Y,F = H X, f∗F for all i. A direct proof (avoiding spectral sequences) is possible: see [280, Exercise III.8.1]. We frequently use this equality without explicit mention. 318 Appendix B. Cohomology and Complexes Resolutions. Let X be a projective (or complete) algebraic variety or scheme. One sometimes attempts to understand the cohomology of a coherent sheaf F on X in terms of a resolution, or something close to a resolution. Thus consider a complex of coherent sheaves ε F• :0−→ F −→ ...−→ F1 −→ F0 −→ F −→ 0, with ε surjective. We record some simple criteria, in increasing order of gener- ality, that allow one to conclude the vanishing of Hk X, F in terms of data involving F•. Proposition B.1.2. (Chasing through complexes). Given an integer k ≥ 0, assume that k k+1 k+ H X, F0 = H X, F1 = ... = H X, F =0. (B.1) k (i). If F• is exact, then H X, F =0. (ii). The same conclusion holds assuming that F• is exact off a subset of X having dimension ≤ k. (iii). More generally still, for i ≥ 0 denote by Hi the homology sheaf ker Fi −→ Fi−1 Hi = im Fi+1 −→ Fi k (with F−1 = F). Still assuming (B.1), the vanishing H X, F =0 holds provided that k+1 k+2 k+ +1 H X, H0 = H X, H1 = ···= H X, H =0. The hypothesis in (ii) means that all the homology sheaves of F• should be supported on an algebraic set of dimension ≤ k. Proof of Proposition B.1.2. This is most easily checked by chopping F• into short exact sequences in the usual way, and chasing through the resulting diagram. Example B.1.3. (Surjectivity on global sections). One can also ask for criteria to guarantee that the map 0 0 H X, F0 −→ H X, F (*) determined by ε is surjective. Assume that 1 2 H X, F1 = H X, F2 = ... = H X, F =0. Then the homomorphism in (*) is surjective provided either that F• is exact off a set of dimension 0, or that one has the case k = 0 of the vanishings in B.1.2 (iii). B.1 Cohomology 319 Remark B.1.4. (Infinite complexes). Note that the vanishing k+i H X, Fi =0 appearing in (B.1) is automatic if k + i>dim X, and similarly for the hy- potheses in (iii). Therefore one can always replace a complex F• of length 0 by a suitable truncation of length ≤ dim X + 1. In particular, the proposition applies also to possibly infinite resolutions of F. Example B.1.3 extends in a similar fashion. Spectral sequences. It is sometimes most efficient to organize cohomological data into a spectral sequence. Suppose for example that the complex F• above + is exact, and denote by F• the truncated complex + F• :0−→ F −→ ...−→ F1 −→ F0 −→ 0. + Thus F• is quasi-isomorphic to the single sheaf F, and hence the hypercoho- + mology of F• coincides with the cohomology of F itself: k + k H X, F• = H X, F . In this case the cohomology of F is computed via the hypercohomology spec- + tral sequence associated to F• : Lemma B.1.5. Assuming that F• is exact, there is a (second quadrant) spectral sequence p,q q ⇒ p+q F E1 = H X, F−p = H X, abutting to the cohomology of F. Example B.1.6. In the situation of the lemma, suppose that there are pos- i itive integers r and s such that all the groups H X, Fj vanish except when i =0andj ≤ s or when i = r and j ≥ s + r + 1. Then the cohomology groups of F are computed as the cohomology of a complex ...−→ Hr X, F −→ Hr X, F −→ H0 X, F s+r+2 s+r+1 s 0 −→ H X, Fs−1 −→ .... (B.2) Filtrations. Instead of a “resolution” such as F•, one sometimes wishes to infer something about the cohomology of a sheaf from a filtration. Suppose then that F is a coherent sheaf on X, and that one is given a filtration K• of F by coherent subsheaves: 0=Kp+1 ⊆ Kp ⊆ ... ⊆ K1 ⊆ K0 = F. As usual, we set Grj = Kj/Kj+1. In general, the cohomology of F is most effi- ciently analyzed via the spectral sequence of a filtered sheaf (cf. [248, Chapter 3, §5]), but in the simplest situations it is just as easy to work directly with the evident short exact sequences linking these data. 320 Appendix B. Cohomology and Complexes Lemma B.1.7. Suppose that there is an integer k ≥ 0 such that all the Grj have vanishing cohomology in all degrees i = k: Hi X, Grj =0 for all i = k and all 0 ≤ j ≤ p. i F k F Then H X, =0for i = k,and H X, has a filtration with graded pieces isomorphic to Hk X, Grj . Direct images. Everything here generalizes to the setting of a proper mor- phism f : X −→ Y (without X or Y necessarily being complete), with coho- mology groups being replaced by direct image sheaves. We leave the details to the reader. B.2 Complexes We next discuss some useful complexes. Throughout this section, X is an irreducible variety or scheme of dimension n. Koszul complex. Let E be a vector bundle on X of rank e,andlet s ∈ Γ X, E be a section of E. This section gives rise to a zero scheme Z =Zeroes(s) ⊆ X : in terms of a local trivialization of E, s is expressed by a vector s =(s1,...,se) of regular functions, and the ideal sheaf of Z is locally generated by the component functions si.