76 VICTORIA HOSKINS

8.7. The Quot . The is afine which generalises the Grass- mannian in the sense that it parametrises quotients of afixed . In this , we will define the moduli problem that the Quot scheme represents and give an overview of the construction of the Quot scheme following [33]. LetY be a projective scheme and beafixed coherent sheaf onX. Then one can consider the moduli problem of classifying quotientsF of . More precisely, we consider surjective sheaf homomorphismsq: up to the equivalence relationF F→E (q: ) (q � : �) kerq = kerq �. F→E ∼ F→E ⇐⇒ Equivalently, there is a sheaf isomorphismφ: such that the following diagram commutes E→E � q � F E Id φ � � � �. F q� E This gives the naive moduli problem and the following definition of families gives the extended moduli problem.

Definition 8.37. Let be a coherent sheaf overY . Then for any schemeS, we let S :=π Y∗ denote the pullback ofF toY S via the projectionπ :Y S Y.A family ofF quotients F F × Y × → of over a schemeS is a surjective Y S -linear homomorphism of sheaves overY S F O × × q : , S F S →E such that isflat overS. Two familiesq : andq : are equivalent if E S F S →E S� F S →E � kerq S = kerq S� . It is easy to check that we can pullback families, asflatness is preserved by base change; therefore, we let uot ( ) : Sch Set Q Y F → denote the associated moduli functor. Remark 8.38. (1) With these definitions, it is clear that we can think of the Quot scheme as instead parametrising coherent subsheaves of up to equality rather than quotients of up to the above equivalence. Indeed this perspectiveF can also be taken with the GrassmannianF (and even projective space). For us, the quotient perspective will be the most useful. (2) For the moduli problem of the , wefix the dimension of the quotient vector spaces. Similarly for the quotient moduli problem, we canfix invariants, as for two quotient sheaves to be equivalent, they must be isomorphic. Thus we can refine the above moduli functor byfixing the invariants of our quotient sheaves. Definition 8.39. For a coherent sheaf over a projective schemeY equipped with afixed ample E , the Hilbert polynomial of with respect to is a polynomialP( , ) Q[t] L E L E L ∈ such that forl N sufficiently large, ∈ l i i l P( , , l)=χ( ⊗ ) := ( 1) dimH (Y, ⊗ ). E L E⊗L − E⊗L i 0 �≥ Serre’s vanishing theorem states that forl sufficiently large (depending on ), all the higher E groups of l vanish (see [14] III Theorem 5.2). Hence, forl sufficiently large, E⊗L ⊗ P( , , l) = dimH 0(Y, l). E L E⊗L ⊗ The proof that there is such a polynomial is given by reducing to the case ofP n (asL is ample, we can use a power ofL to embedX into a projective space) and then the proof proceeds by induction on the dimensiond of the support of the sheaf (where the inductive step is given by restricting to a hypersurface and the base cased = 0 is trivial as the Hilbert polynomial is constant); for a proof, see [16] Lemma 1.2.1. However, for a smooth projective curveX, we can explicitly write down the Hilbert polynomial of a locally free sheaf overX using the Riemann–Roch Theorem. MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 77

Example 8.40. On a smooth projective genusg curveX, wefix a degree 1 = L X (x) =: X (1). For a overX of rankn and degreed, the twist (m) := O (mO) has rankn and degreed+ mnE. The Riemann–Roch formula gives E E⊗O X χ( (m)) =d+ mn+n(1 g). E − Thus has Hilbert polynomialP(t) = nt+d+n(1 g) of degree 1 with leading coefficient givenE by the rankn. − Definition 8.41. For afixed ample line bundleL onY , we have a decomposition uot ( )= uotP,L( ) Q Y F Q Y F P Q[t] ∈� into Hilbert polynomialsP taken with respect toL. IfY=X is a curve, then we have a decomposition of the Quot moduli functor by ranks and degrees of the quotient sheaf: uot ( )= uotn,d( ). Q X F Q X F (�n,d) Example 8.42. The grassmannian moduli functor is a special example of the Quot moduli functor: n d n r(d, n) = uot − (k ). G Q Speck Theorem 8.43 (Grothendieck). LetY be a projective scheme and an ample invertible sheaf L P, onY . Then for any coherent sheaf overY and any polynomialP , the functor uot Y L( ) F P, Q F is represented by a projective scheme Quot L( ). Y F The idea of the construction is very beautiful but also technical; therefore, we will just give an outline of a proof. We split the proof up into the 4 following steps.

n N Step 1. Reduce to the case whereY=P ,L= n (1) and is a trivial vector bundle n . O P F O P Step 2. Form sufficiently large, construct an injective natural transformation of moduli func- tors P, (1) N uot nO ( n )� r(V,P(m)) Q P OP →G N 0 to the Grassmannian moduli functor ofP(m)-dimensional quotients ofV :=k H ( n (m)). ⊗ OP P, (1) N Step 3. Prove that uot nO ( n ) is represented by a locally closed subscheme of Gr(V,P(m)). Q P OP Step 4. Prove that the Quot scheme is proper using the valuative criterion for properness.

Before we explain the proof of each step, we need the following definition. Definition 8.44. A natural transformation of presheavesη: is a closed (resp. open, M � →M resp. locally closed) immersion ifη S is injective for every schemeS and moreover, for any natural transformationγ:h S from the functor of points of a schemeS, there is a closed (resp. open, resp. locally closed)→M subschemeS S such that � ⊂ hS = � hS � ∼ M ×M where thefibre product is given by

( � hS)(T)= (f:T S h S(T),F �(T)):γ T (f)=η T (F) (T) . M ×M → ∈ ∈M ∈M Sketch of Step 1. First,� we can assume thatL is very ample by replacingL by a suffi�ciently large positive power ofL; this will only change the Hilbert polynomial. ThenL defines a n projective embeddingi:Y� P such thatL is the pullback of n (1). Sincei is a closed → O P 78 VICTORIA HOSKINS immersion,i is coherent and, moreoveri is exact; therefore, we can push-forward quotient ∗F ∗ sheaves onY toP n. Hence, one obtains a natural transformation

uotY ( ) uot Pn (i ), Q F →Q ∗F which is injective asi ∗i = Id. We claim that this natural transformation is a in the sense of the above∗ definition. More precisely, we claim for any schemeS and natural transformationh S uot Pn (i ) there exists a closed subschemeS � S with the following →Q ∗F ⊂ property: a morphismf:T S determines a family in uot Y ( )(T ) if and only if the morphismf factors viaS →. To define the closed subschemeQ associatedF to a mapη:h � S → uotPn (i ), we let (i ) S � denote the family overS of quotients associated toη S(idS) Qand apply∗F (id i) to∗ obtainF aE homomorphism of sheaves overY S S × ∗ × ( ) S = i∗(i ) S i ∗ ; F ∼ ∗F → E then we takeS � S to be the closed subscheme on which this homomorphism is surjective (the fact that this is⊂ closed follows from a semi-continuity argument). Hence, we may assume that (Y,L) = (Pn, (1)). We can tensorO any quotient sheaf by a power of (1) and this induces a natural transformation between O uot n ( ) = uot n ( (r)) Q P F ∼ Q P F⊗O (under this natural transformation the Hilbert polynomial undergoes a explicit transformation corresponding to this tensorisation). Hence, by replacing with (r) := (r), we can assume without loss of generality that has trivial higherF cohomologyF groupsF⊗O and is globally generated; that is, the evaluation mapF N 0 n ⊕n = H (P , ) n OP ∼ F ⊗O P →F is surjective andN=P( , 0). By composition, this surjection induces a natural transformation F N uot n ( ) n ( n ) Q P F →Q P OP which can also be shown to be a closed embedding. In conclusion, we obtain a natural trans- formation P �,L P, (1) N uot ( ) uot nO ( n ) Q Y F →Q P OP which is a closed embedding of moduli functors.

Sketch of Step 2. By a result of Mumford and Castelnuovo concerning ‘Castelnuovo– N Mumford regularity’ of subsheaves of Pn , there existsM N (depending onP,n andN) such that for allm M, the following holdsO for any short exact∈ sequence of sheaves ≥ N 0 n 0 →K→O P →F→ such that has Hilbert polynomialP: F (1) the groupsH i of (m), (m) vanish fori> 0, (2) (m) and (m) are globally generated.K F K F n 1 The proof of this result is by induction onn, where one restricts to a hyperplaneH ∼= P − in Pn to do the inductive step; for a full proof, see [33] Theorem 2.3. Now if wefixm M, we claim there is a natural transformation ≥ P, (1) N N 0 η: uot nO ( n )� r(k H ( n (m)),P(m)). Q P OP →G ⊗ OP N First, let us define this for families overS = Speck: for a quotientq: n with kernel P⊕ � , we have an associated long in cohomology O F K 0 0 N 0 1 0 H ( (m)) H ( n (m)) H ( (m)) H ( (m)) = 0. → K → OP → F → K Hence, we define N 0 0 η (q: n )= H (q(m)) :W H ( (m)) , Speck O P �F � F 0 N N 0 n 0 whereW :=H ( Pn (m)) =k H ( P �(m)) and we have dimH ( �(m)) =P(m), as all higher cohomologyO of (m) vanishes.⊗ O F F MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 79

N To defineη S for a family of quotientsq S : Pn S over an arbitrary schemeS, we essentially do the above process in a family. MoreO × precisely,→E we letπ :X S S be the S × → projection and push forwardq S(m) byπ S toS to obtain a surjective homomorphism of sheaves overS N S W = πS ( Pn S(m)) π S ( (m)). O ⊗ ∼ ∗ O × → ∗ E By our assumptions onm and the semi-continuity theorem,π S ( (m)) is locally free of rank P(m) (as the higher rank direct images of (m) vanish so the claim∗ E follows by EGA III 7.9.9). Hence, we have a family ofP(m)-dimensionalE quotients ofW overS, which defines the desired S-point in the Grassmannian. Let Gr = Gr(W, P(m)). We claim this natural transformationη is an injection. Let us explain how to reconstructq from the morphismf :S Gr corresponding to the surjection S qs → πS (qs(m)) : S W π S ( (m)). ∗ O ⊗ → ∗ E Over the Grassmannian, we have a universal inclusion (and a corresponding surjection) � W, KGr →O Gr ⊗ whose pullback toS via the morphismS Gr is the homomorphism → N πS∗ πS ( S(m)) V S =π S∗ πS ( Pn S(m)), ∗ K → ⊗O ∗ O × where := kerq . We claim that the homomorphism K S S N N f:π S∗ πS ( S(m)) π S∗ πS ( Pn S(m)) Pn S(m) ∗ K → ∗ O × →O × has cokernelq S(m). To prove the claim, consider the following commutative diagram N 0 �πS∗ πS ( S(m)) �πS∗ πS ( Pn S(m)) �πS∗ πS ( (m)) �0 ∗ K ∗ O × ∗ E

� N � � 0 � S(m) � Pn S(m) � (m) �0 K O × E whose rows are exact and whose columns are surjective by our assumption onm. Finally, we can recoverq fromq (m) by twisting by ( m). S S O − Sketch of Step 3. For any morphismf:T Gr = Gr(W, P(m)), we let denote the → K T,f pullback of the universal subsheaf Gr on the Grassmannian toT viaf. Then consider the induced composition K N N hT,f :π T∗ T,f π T∗ (W T ) = πT∗ πT ( Pn T (m)) T P n (m) K → ⊗O ∼ ∗ O × →O × n N whereπ T :P T T denotes the projection. Letq T,f (m): Pn T (m) T,f (m) denote the × → n O × →F cokernel ofh T,f ; then T,f is a coherent sheaf overP T. We claim that theF natural transformation defined in× Step 2 is a locally closed immersion. To prove the claim, we need to show for any morphismh:S Gr, there is a unique locally closed subschemeS � S with the property that a morphism→f:T S factors viaS � S if � → → � → and only if the sheaf T,h f isflat overT and has Hilbert polynomialP at eacht T . This F ◦ ∈ locally closed subschemeS � S is constructed as the stratum with Hilbert polynomialP in ⊂ n theflattening stratification for the sheaf T,f � overT P . For details, see [33] Theorem 4.3. P N F × Let QuotPn ( Pn ) be the locally closed subscheme of Gr associated to the identity morphism on Gr (which correspondsO to the universal family on the Grassmannian); then it follows from P N the above arguments that this scheme represents the functor uot Pn ( Pn ). The Grassmannian Gr = Gr(W, P(m)) has its P¨ucker embeddingQ intoO projective space P(m) Gr(W, P(m))� P( W ∨). → ∧ Therefore, we have a locally closed embedding N P(m) (6) Quot n ( ⊕n ,P)� P( W ∨). P OP → ∧ In particular, the Quot scheme is quasi-projective; hence, separated and offinite type. 80 VICTORIA HOSKINS

Sketch of Step 4. We will prove the valuative criterion for the Quot scheme using its moduli P N functor. The Quot scheme Quot n ( n ) is proper over Speck if and only if for every discrete P P⊕ valuation ringR overk with quotientOfieldK, the restriction map P N P N uot n ( n )(SpecR) uot n ( n )(SpecK) Q P OP →Q P OP is bijective. Since the Quot scheme is separated, we already know that this map is injective. n n N Letj:P K � P R denote the open immersion. Any quotient sheafq K : n K can be lifted → O PK →F N to a quotient sheafq R : n R where R is the image of the homomorphism O PR →F F N N qR : ⊕n j ( ⊕n ) j K . O PR → ∗ OPK → ∗F The sheaf R is torsion free as it as a subsheaf ofj K , which is torsion free asj ∗ is exact, F ∗F j∗j K = K and K is torsion free (as it isflat overK). Hence, R isflat overR, as over a DVR∗F flatF is equivalentF to torsion free and so this gives a well deFfinedR-valued point of the Quot scheme. It remains to check that the image ofq R under the restriction map isq K . Asj ∗ is left exact, the mapj ∗ R j ∗j K = K is injective and the following commutative diagram F → ∗F F N j∗qR ⊕n � j∗ R OPK F

qK � � � FK implies that the vertical homomorphism must also be surjective; thusj ∗ R ∼= j∗j K = K as required. Hence, the Quot scheme is proper over Speck. F ∗F F N Since Quot n ( ⊕n ;P ) is proper over Speck, the embedding (6) is a closed embedding. P OP Remark 8.45. P, (1) As the Quot schemeQ := Quot Y L( ) is afine moduli space, the identity morphism on Q corresponds to a universal quotientF homomorphism

π∗ Y F�U overQ Y , whereπ Y :Q Y Y denotes the projection toY. (2) The Quot× scheme can also× be→ defined in the relative setting, where we replace ourfieldk by a general base schemeS and look at quotients of afixed coherent sheaf on a scheme X S; the construction in the relative case is carried out in [33]. → The Hilbert schemes are special examples of Quot schemes, which also play an important role in the construction of many moduli spaces. P Definition 8.46.A Hilbert scheme is a Quot scheme of the form Quot Y ( Y ) and represents the moduli functor that sends a schemeS to the set of closed subschemesZOY S that are proper andflat overS with the given Hilbert polynomial. ⊂ × Exercise 8.47. For a natural numberd 1, prove that the Hilbert scheme ≥ d Quot 1 ( 1 ) P OP is isomorphic toP d by showing they both have the same functor of points in the following way. a) Show that any familyZ P 1 over Speck in this Hilbert scheme is a degreed hypersurface ⊂ inP 1. 1 1 b) LetS be a scheme andπ S :P S :=P S S denote the projection. Show that any 1 × → 1 familyZ P S overS in this Hilbert scheme is a Cartier divisor inP S and so there is a ⊂ d line subbundle ofπ S ( 1 (d)) which determines a morphismf Z :S P . In particular, ∗ OPS → this gives a natural transformation d uot 1 ( 1 ) h d . Q P OP → P c) Construct the inverse to the above natural transformation using the tautological family of degreed hypersurfaces inP 1 parametrised byP d. MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 81

8.8. GIT set up for construction of the moduli space. Throughout this section, wefix a connected smooth projective curveX and we assume the genus ofX is greater than or equal to 2 to avoid special cases in low genus. Wefix a rankn and a degreed>n(2g 1) (recall that tensoring with a line bundle does not alter semistability and so we can pick− the degree to be arbitrarily large; in fact, eventually we will choosed to be even larger). It follows from Lemma 8.36 that any locally free semistable sheaf of rankr and degreed is globally generated with H1(X, ) = 0. By the Riemann–Roch Theorem,E E dimH 0(X, )=d+n(1 g) =:N. E − If we choose an identificationH 0(X, ) = kN , then the evaluation map E ∼ H0(X, ) , E ⊗O X →E N which is surjective as is globally generated, determines a quotient sheafq: X � . n,d E N O E LetQ := Quot X ( X ) be the Quot scheme of quotient sheaves of the trivial rankN vector N O (s)s bundle X of rankn and degreed. LetR Q denote the open subscheme consisting O N ⊂ 0 of quotientsq: X such that is a (semi)stable locally free sheaf andH (q) is an isomorphism.O For a proof→F that theseF conditions are open see [16] Proposition 2.3.1. The Quot schemeQ parametrises a universal quotient N qQ : Q X � O × U (s)s N (s)s (s)s and we letq : R(s)s X := R(s)s denote the restriction toR . O × →U U| Lemma 8.48. The universal quotient sheaf (s)s overR (s)s X is a family overR (s)s of (semi)stable locally free sheaves overX with invariantsU (n, d) with× the local universal property. Proof. Let be a family over a schemeS of (semi)stable locally free sheaves overX withfixed F invariants (n, d). Then for eachs S, the locally free semistable sheaf s is globally generated with vanishingfirst cohomology∈ by our assumption ond. Therefore,F by the semi-continuity Theorem,π S is a locally free sheaf overS of rankN=d+n(1 g). For eachs S, we need ∗F − ∈ to show there is an open neighbourhoodU S ofs S and a morphismf:S R (s)s such (s)s ⊂ ∈ → that U f ∗ . Pick an open neighbourhoodU s on whichπ S is trivial; that is, we haveF| an isomorphism∼ U � ∗F N Φ: U = (πS ) U . O ∼ ∗F | Then the surjective homomorphism of sheaves overU X × π Φ N U∗ qU : U X �πU∗ πU ( U ) � U O × ∗ F| F| determines a morphismf:U Q to the quot scheme such that there is a commutative diagram → N qu U X � U O × F|

Id ∼= f q N� ∗ Q � U X �f ∗ O × U In particular = f and, as is a family of (semi)stable vector bundles, the morphism F| U ∼ ∗U F f:U Q factors viaR (s)s. → � These families (s)s overR (s)s are not universal families as the morphisms described above are not unique: ifU we takeS = Speck and to be a (semi)stable locally free sheaf, then different 0 N E (s)s choices of isomorphismH (X, ) ∼= k give rise to different points inR . Any two choices of the aboveE isomorphism are related by an element in the general linear group GLN and so it is natural to mod out by the action of this group. n,d N (s)s Lemma 8.49. There is an action ofGL N onQ := Quot X ( X ) such that the orbits inR are in bijective correspondence with the isomorphism classes ofO (semi)stable locally free sheaves onX with invariants(n, d). 82 VICTORIA HOSKINS

Proof. We claim there is a (left) action σ:GL Q Q N × → which onk-points is given by

1 q g− q g ( N � )=( N � N � ). · O X E O X OX E N To construct the action morphism it suffices to give a family over GLN Q of quotients of X 1 × O with invariants (n, d). The inverse map on the groupi − : GLN GL N determines a universal inversion which is a sheaf isomorphism → (7) τ:k N k N . ⊗O GLN → ⊗O GLN N Letq Q :k Q X denote the universal quotient homomorphism onQ X. Then the actionσ:GL ⊗O Q× Q→ isU determined by the following family of quotient maps× over GL Q N × → N × p∗ τ GLN (pQ X )∗qQ N N × k GLN Q X �k GLN Q X �pQ∗ X ⊗O × × ⊗O × × × U wherep GLN : GLN Q X GL N andp Q X : GLN Q X Q X denote the projection morphisms. × × → × × × → × From the definition ofR (s)s, we see these subschemes are preserved by the action. Consider N N (s)s quotient sheavesq : X � andq : X � inR . Ifg q E q F , then there is an isomor- E O E E O F · ∼ phism = whichfits into a commutative square, and so and are isomorphic. Conversely, E ∼ F E F if = , then there is an induced isomorphismφ:H 0( ) = H0( ). The composition E ∼ F E ∼ F 0 0 1 H (q ) φ H (q )− kN E �H0( ) �H0( ) F �kN E F is an isomorphism which determines a pointg GL such thatg q q . ∈ N · E ∼ F � Remark 8.50. In particular, any coarse moduli space for (semi)stable vector bundles is con- (s)s structed as a categorical quotient of the GLN -action onR . Furthermore, if there is an orbit (s)s space quotient of the GLN -action onR , then this is a coarse moduli space. In fact, as the diagonalG m GL N acts trivially on the Quot scheme, we do not lose anything by instead ⊂ working with the SLN -action. Finally, we would like to linearise the action to construct a categorical quotient via GIT. There is a natural family of invertible sheaves on the Quot scheme arising from Grothendieck’s embedding of the Quot scheme into the : for sufficiently largem, we have a closed immersion n,d N 0 N M 0 N Q = Quot ( )� Gr(H ( (m)),M)� P :=P( H ( (m))∨) X OX → OX → ∧ OX whereM= mr+d+r(1 g). We let m denote the pull back of P(1) to the Quot scheme via this − L O0 N N 0 closed immersion. There is a natural linear action of SLN onH ( X (m)) = (k H ( X (m)), M 0 N O ⊗ O which induces a linear action of SLN onP( H ( (m))∨); hence, m admits a linearisation ∧ OX L of the SLN -action. We can define the linearisation using the universal quotient sheaf onQ X: we have L m U × m = det(πQ ( π X∗ X (m))) L ∗ U⊗ O whereπ :Q X X andπ :Q X Q are the projection morphisms. Furthermore, the X × → Q × → universal quotient sheaf admits a SL N -linearisation: we have equivalent families of quotients sheaves over SL Q givenU by N × (σ id ) q N × X ∗ Q k SL Q X �(σ id X )∗ ⊗O N × × × U and p∗ τ p q SLN Q∗ X Q N N × k SLN Q X �k SLN Q X �pQ∗ X , ⊗O × × ⊗O × × × U MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 83

N whereq Q :k Q X denotes the universal quotient homomorphism,σ:SL N Q Q ⊗O × →U × → denotes the group action,p Q X andp SLN denote the projections from SLN Q X to the relevant factor andτ is the isomorphism× given in (7). Hence, there is an isomorphism× ×

Φ:(σ id X )∗ (p Q X )∗ × U→ × U satisfying the cocycle condition, which gives a linearisation of the SL -action on . Form N U sufficiently large, m is ample and admits an SLN -linearisation, as the construction of m commutes with baseL change form sufficiently large. Hence, atq: N inQ, thefibreL of O X →F the of the associated line bundleL m is naturally isomorphic to an alternating product of exterior powers of the cohomology groups of (m): F i ( 1)i L = detH ∗(X, (m)) = detH (X, (m)) ⊗ − . m,q ∼ F F i 0 �≥ In fact, by the Castelnuovo–Mumford regularity result explained in the second step of the construction of the quot scheme, form sufficiently large, we haveH i(X, (m)) = 0 for alli>0 for all pointsq: N inQ. Therefore, form sufficiently large, thefibreF atq is O X →F L = detH 0(X, (m)). m,q ∼ F