76 VICTORIA HOSKINS 8.7. The Quot scheme. The Quot scheme is afine moduli space which generalises the Grass- mannian in the sense that it parametrises quotients of afixed sheaf. In this section, 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 Grassmannian, 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 invertible sheaf , 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 cohomology 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 line bundle = L X (x) =: X (1). For a vector bundle 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 closed immersion 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) andQ 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 sheaf cohomology 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 exact sequence in cohomology O F K 0 0 N 0 1 0 H ( (m)) H ( n (m)) H ( (m)) H ( (m)) = 0.