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MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 71

8.4. Vector bundles and locally free sheaves. We will often use the equivalence between the of algebraic vector bundles onX and the category of locally free sheaves. We recall that this equivalence is given by associating to an algebraic vector bundleF X the → F of sections ofF . Under this equivalence, the trivial line bundleX A 1 onX corresponds to × the structure sheaf X . We will use theO notation to mean a sheaf or locally free sheaf andE to mean a vector . We also denote theE stalk of atx by and thefibre ofE atx byE . E E x x For a smooth projective curveX, the local rings X,x are DVRs, which are PIDs. Using this one can show the following. O Exercise 8.18. Prove the following statements for a smooth projective curveX. a) Any torsion free sheaf onX is locally free. b) A subsheaf of a locally free sheaf overX is locally free. c) A non-zero homomorphismf: of locally free sheaves overX with rk = 1 is injective. L→E L One should be careful when going between vector bundles and locally free sheaves, as this correspondence does not preserve subobjects. More precisely, if is a locally free sheaf with F associated vector bundleF and is a subsheaf, then the map on stalks x x is injective for allx X. However, the mapE⊂ onFfibres of the associated vector bundlesE E →F F is not ∈ x → x necessarily injective, asE x is obtained by tensoring x with the residuefieldk(x) ∼= k, which is not exact in general. E

Example 8.19. For an effective divisorD, we have that X ( D)� X is a locally free subsheaf but this does not induce a vector subbundle of theO trivial− line→ bundle,O as a has no non-trivial vector subbundles. However, if we have a subsheaf of a locally free sheaf for which the quotient := / is torsion free (and so locally free,E asX is a curve), thenF the associated vector bundleG EF isE a vector subbundle ofF , because if we the short 0 0 →E x →F x →G x → with the residuefield, then we get a long exact sequence 1 Tor (k, x) E x F x G x 0, ···→ OX,x G → → → → 1 where Tor (k, x) = 0 as x isflat. OX,x G G Definition 8.20. Let be a subsheaf of a locally free sheaf and letE andF denote the corresponding vector bundles.E Then the vector subbundle ofFF generically generated byE is a vector subbundle E ofF which is the associated the locally free sheaf 1 :=π − ( ( / )) E T F E whereπ: / and ( / ) denotes the torsion subsheaf of / (i.e. ( / )/ ( / ) is torsion free).F→F E T F E F E F E T F E Indeed, as / is torsion free (and so locally free), the vector bundle homomorphism as- F E sociated to is injective; that is, E is a vector subbundle ofF . Furthermore, we have that E→F rk = rk and deg . E E E≥E Example 8.21. LetD be an effective divisor and consider the subsheaf := ( D) of L � O X − := ; then the vector subbundle ofL generically generated byL is L =L. L O X � � The category of locally free sheaves is not an and also the category of vector bundles is not abelian. Given a homomorphism of locally free sheavesf: , the quotient /kerf may not be locally free (and similarly for the image). Similarly,E→ theG (and the image)E of a of vector bundles may not be a vector bundle; essentially because the rank can jump. Instead, we can define the vector subbundle that is generically generated by the kernel (and the same for the image) sheaf theoretically. 72 VICTORIA HOSKINS

Definition 8.22. Letf:E F be a morphism of vector bundles; then we can define → (1) the vector subbundleK ofE generically generated by the kernel Kerf, which satisfies rkK = rk Kerf degK deg Kerf; ≥ (2) the vector subbundleI ofF generically generated by the image Imagef, which satisfies rkI = rk Imagef degI deg Imagef. ≥ Exercise 8.23. Let be a locally free sheaf of rankr overX. In this exercise, we will prove that there exists a shortE exact sequence of locally free sheaves overX 0 0 →L→E→F→ such that is an invertible sheaf and has rankr 1. L F − a) Show that for any effective divisorD withr degD>h 1( ), the vector bundle (D) admits a by considering the long exact sequence in cohomologyE associatedE to the short exact sequence 0 (D) k 0 →O X →O X → D → tensored by (herek D denotes the skyscraper sheaf with supportD). Deduce that has an invertibleE subsheaf. E b) For an invertible sheaf with deg >2g 2, prove thath 1( ) = 0 using . c) Show that the degreeL of an invertibleL − subsheaf of is boundedL above, using the Riemann–Roch formula for invertible sheaves andL partE b). d) Let to be an invertible subsheaf of of maximal degree; then verify that the quotient ofL is locally free. E F L⊂E Exercise 8.24. In this exercise, we will prove for locally free sheaves and overX that E F deg( ) = rk deg + rk deg E⊗F E F F E by induction on the rank of . E a) Prove the base case where = X (D) by splitting into two cases. IfD is effective, use the short exact sequence E O 0 (D) 0 →O X →O X →O D → and the Riemann–Roch Theorem to prove the result. IfD is not effective, writeD as D1 D 2 for effective divisorsD i and modify by twisting by a line bundle. b) For− the inductive step, use Exercise 8.23. F Example 8.25. Let be a locally free sheaf of rankr and degreed over a genusg smooth projective curveX; thenE for any line bundle , we have that L m χ( ⊗ ) =d+rm deg +r(1 g) E⊗L L − is a degree 1 polynomial inm. 8.5. Semistability. In order to construct moduli spaces of algebraic vector bundles over a smooth projective curve, Mumford introduced a notion of semistability for algebraic vector bundles. One advantage to restricting to semistable bundles offixed rank and degree is that the moduli problem is then bounded (without adding the semistability hypothesis, the mod- uli problem is unbounded; see Example 2.22). A second advantage, which explains the term semistable, is that the notion of semistability for vector bundles corresponds to the notion of semistability coming from an associated GIT problem (which we will describe later on). Definition 8.26. The slope of a non-zero vector bundleE onX is the ratio degE µ(E) := . rkE Remark 8.27. Since the degree and rank are both additive on short exact sequences of vector bundles 0 E F G 0, → → → → it follows that MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 73

(1) If two out of the three bundles have the same slopeµ, the third also has slopeµ; (2)µ(E)<µ(F ) (resp.µ(E)>µ(F )) if and only ifµ(F)<µ(G) (resp.µ(F)>µ(G)). Definition 8.28. A vector bundleE is stable (resp. semistable) if every proper non-zero vector subbundleS E satisfies ⊂ µ(S)<µ(E) (resp. µ(S) µ(E) for semistability). ≤ A vector bundleE is polystable if it is a of stable bundles of the same slope. Remark 8.29. If wefix a rankn and degreed such thatn andd are coprime, then the notion of semistability for vector bundles with invariants (n, d) coincides with the notion of stability. Lemma 8.30. LetL be a line bundle andE a vector bundle overX; then i)L is stable. ii) IfE is stable (resp. semistable), thenE L is stable (resp. semistable). ⊗ Proof. Exercise. � Lemma 8.31. Letf:E F be a non-zero homomorphism of vector bundles overX; then → i) IfE andF are semistable,µ(E) µ(F). ii) IfE andF are stable of the same≤ slope, thenf is an isomorphism. iii) Every stable vector bundleE is simple i.e. EndE=k. Proof. Exercise. �

IfE is a vector bundle which is not semistable, then there exists a subbundleE � E with larger slope thatE, by taking the sum of all vector subbundles ofE with maximal⊂ slope, one obtains a unique maximal destabilising vector subbundle ofE, which is semistable by construction. By iterating this process, one obtains a unique maximal destabilisingfiltration of E known as the Harder–Narasimhanfiltration ofE [13]. Definition 8.32. LetE be a vector bundle; thenE has a Harder–Narasimhanfiltration (0) (1) (s) 0 =E �E � �E =E ··· whereE := (i)/E(i 1) are semistable with slopes i E − µ(E )>µ(E )> >µ(E ). 1 2 ··· s As we have already mentioned, the moduli problem of vector bundles onX withfixed rank n and degreed is unbounded. Therefore, we restrict to the moduli (s)s(n, d) of (semi)stable locally free sheaves. Let us refine our notion of families to familiesM of semistable vector bundles. Definition 8.33. A family over a schemeS of (semi)stable vector bundles onX with invariants (n, d) is a overX S which isflat overS and such that for eachs S, the E × ∈ sheaf s is a (semi) onX with invariants (n, d). WeE say two families and overS are equivalent if there exists an invertible sheaf over S and an isomorphismE F L = π ∗ E ∼ F⊗ SL whereπ :X S S denotes the projection. S × → Lemma 8.34. If there exists a semistable vector bundle overX with invariants(n, d) which is not polystable, then the moduli problem of semistable vector bundles ss(n, d) does not admit a coarse moduli . M Proof. If there exists a semistable sheaf onX which is not polystable, then there is a non-split short exact sequence F 0 � �� 0, →F →F→F → where � and �� are semistable vector bundles with the same slope as . The above short exactF sequenceF corresponds to a non-zero point in Ext( , ) and if weF take the affine line F �� F � through this extension class then we obtain a family of extensions overA 1 = Speck[t]. More 74 VICTORIA HOSKINS precisely, let be the coherent sheaf overX A 1 given by this extension class. Then isflat E × E overA 1 and = fort= 0, and = �� �. Et ∼ F � E 0 ∼ F ⊕F Since is a family of semistable locally free sheaves with thefixed invariants (n, d) which E exhibits the jump phenomenon, there is no coarse by Lemma 2.27.� However, this cannot happen if the notions of semistability and stability coincide, which happens whenn andd are coprime. 8.6. Boundedness of semistable vector bundles. To construct a moduli space of vector bundles onX using GIT, we would like tofind a schemeR that parametrises a family of semistable vector bundles onX offixed rankn and degreed such that any vector bundleF of the given invariants is isomorphic to p for somep R. In this section, we prove an important boundedness result for the family ofF semistable vector∈ bundles onX offixed rank and degree which will enable us to construct such a . In fact, we will show that we can construct a schemeR which parametrises a family with the local universal property. First, we note that we can assume, without loss of generality, that the degree of our vector bundle is sufficiently large: for, if we take a line bundle of degreee, then tensoring with pre- serves (semi)stability and so induces an isomorphism betweenL the moduli of (semi)stableL vector bundles with rank and degree (n, d) and those with rank and degree (n, d+ ne) : ss(n, d) = ss(n, d+ ne). − ⊗L M ∼ M Hence, we can assume thatd>n(2g 1) whereg is the genus ofX. This assumption will be used to prove the boundedness result− for semistable vector bundles. However,first we need to recall the definition of a sheaf being generated by global sections. Definition 8.35. A sheaf is generated by its global sections if the natural evaluation map F 0 ev :H (X, ) X F F ⊗O →F is a surjection. Lemma 8.36. Let be a locally free sheaf overX of rankn and degreed>n(2g 1). If the associated vector bundleF F is semistable, then the following statements hold: − i)H 1(X, )=0; ii) is generatedF by its global sections. F Proof. For i), we argue by contradiction using Serre duality: ifH 1(X, )= 0, then dually there F � would be a non-zero homomorphismf: ω X . We letK F be the vector subbundle generically generated by the kernel off Fwhich→ is a vector subbundle⊂ of rankn 1 with − degK deg kerf deg degω =d (2g 2). ≥ ≥ F− X − − In this case, by semistability ofF , we have d (2g 2) d − − µ(K) µ(F)= ; n 1 ≤ ≤ n − this givesd n(2g 2), which contradicts our assumption on the degree ofF. For ii), we≤ letF− denote thefibre of the vector bundle at a pointx X. If we consider the x ∈ fibreF x as a torsion sheaf overX, then we have a short exact sequence 0 ( x) := ( x) F = k 0 →F − O X − ⊗F→F→ x F⊗ x → which gives rise to an associated long exact sequence in 0 H 0(X, ( x)) H 0(X, ) H 0(F ) H 1(X, ( x)) . → F − → F → x → F − →··· 0 0 Then we need to prove that, for eachx X, the mapH (X, ) H (X,Fx) =F x is surjective. We prove this map is surjective by showing∈ thatH 1(X, ( Fx→)) = 0 using the same argument as in part i) above, where we use the fact that twistingF − by a line bundle does not change semistability: ( x) := X ( x) is semistable with degreed n>n(2g 2) and thus H1(X, ( xF) − ) = 0.O − ⊗F − − O X − ⊗F � MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 75

As mentioned above, these two properties are important for showing boundedness. In fact, we will see that a strictly larger family of vector bundles offixed rank and degree are bounded; namely those that are generated by their global sections and have vanishing 1st cohomology. Given a locally free sheaf of rankn and degreed that is generated by its global sections, we can consider the evaluationF map 0 ev :H (X, ) X F F ⊗O →F which is, by assumption, surjective. If alsoH 1(X, ) = 0, then by the Riemann–Roch formula F χ( )=d+n(1 g) = dimH 0(X, ) dimH 1(X, ) = dimH 0(X, ); F − F − F F that is, the of the 0th cohomology isfixed and equal toN :=d+n(1 g). Therefore, 0 N − we can choose an isomorphismH (X, ) ∼= k and combine this with the evaluation map for , to produce a surjection F F N N q : X =k X F O ⊗O →F from afixed trivial vector bundle. Such surjective homomorphisms from afixed coherent sheaf are parametrised by a projective scheme known as a Quot scheme, which generalises the Grass- mannians.