5. Construction of the Hilbert Scheme: We’ll start with a discussion of Grassmannians, then construct Grothendieck’s quot schemes and Hilbert schemes using the results of the previous two sections.

Fix a field k. The Grassmannian G(m, n) over k is a projective k-scheme which we may define as follows. Start with the “wedge” map:

f : Spec(k[x_{i,j}]) = A_{k}^{mn} → A\binom{n}{m}_{k} = Spec(k[x_{J}])

where 1 ≤ i ≤ m, 1 ≤ j ≤ n and J = (j_{1}, ..., j_{m}) for 1 ≤ j_{1} < ... < j_{m} ≤ n (we’ll call J a sorted m-tuple). Then f is defined by f^{∗}(x_{J}) = det(x_{i,j_{l}}) the determinant of the square submatrix of (x_{i,j}) obtained by removing all

columns except those indexed by J. Define ϕ : A_{k}^{mn} − − > P_{k}^{\binom{n}{m}−1} to be the composition of the morphism f with the rational map to projective space. Thus ϕ is a morphism when restructed to the Zariski open subset U := A_{k}^{mn} − V ({det(x_{i,j_{l}})}) of matrices of “full rank.”

If I = (i_{1}, ..., i_{m}) is an (unsorted) m-tuple of integers with 1 ≤ i_{l} ≤ n and if σ ∈ Σ_{m} is a permutation, let σ(I) = (i_{σ(1)}, ..., i_{σ(m)}). Note that there is at most one sorted m-tuple J in each orbit of Σ_{m} and exactly one if the i_{l} are

distinct. It follows that we obtain well-defined x_{I} ∈ H^{0}(P_{k}^{\binom{n}{m}−1}, O(1)) by setting:

x_{I} = \{ \begin{matrix}0 & if the i_{l} are not distinct \\ (−1)^{sgn(σ)}x_{J} & if σ(I) = J, a sorted multi-index \end{matrix}

The Plücker quadric associated to a pair I, J of m-tuples and an integer 1 ≤ s ≤ m is given by:

Q_{I,J,s} := x_{I}x_{J} − ∑_{l=1}^{m} x_{(i_{1},...,i_{s−1},j_{l},i_{s+1},...,i_{m})}x_{(j_{1},...,j_{l−1},i_{s},j_{l+1},...,j_{m})}

The Grassmannian G(m, n) ⊂ P_{k}^{\binom{n}{m}−1} is the zero scheme of the set of all the Plücker quadrics.

Remark: These quadrics are redundant. For example, if m = 2, n = 4, then

G(2, 4) = V (x_{(1,2)}x_{(3,4)} − x_{(1,3)}x_{(2,4)} + x_{(1,4)}x_{(2,3)})

i.e. G(2, 4) ⊂ P_{k}^{5} is a nonsingular quadric hypersurface.

Proposition: ϕ : U → P_{k}^{\binom{n}{m}−1} factors through G(m, n) and:

(a) ϕ : U → G(m, n) is a locally trivial fiber bundle, with fiber GL(m, k). In particular, ϕ determines a bijection from the set of m-dimensional subspaces of k^{n} to the set of k-rational points of G(m, n).

(b) Let U_{x_{J}} = P_{k}^{\binom{n}{m}−1} − V (x_{J}). Then G(m, n) ∩ U_{x_{J}} ≅ A_{k}^{m(n−m)}. Hence G(m, n) is a nonsingular projective variety over k of dimension m(n − m).

Proof: ϕ is equivariant with respect to the “left multiplication” action μ : GL(m, k) × U → U where GL(m, k) ⊂ A_{k}^{m^{2}} is the open subscheme of invertible m × m matrices. That is, the following diagram commutes:

GL(m, k) × U →^{μ}  U
↓ π  ↓ ϕ
U  →^{ϕ} P_{k}^{\binom{n}{m}−1}

This follows from the fact that if we multiply an m × n matrix A on the left by a square matrix M, then the determinants of the m × m submatrices of A are all multiplied by det(M).

Fix J and consider the closed subscheme A_{k}^{m(n−m)} ≅ Z_{J} ⊂ U of m × n matrices whose square submatrix corresponding to J is the identity. Then the following are easily verified:

(i) μ : GL(m, k) × Z_{J} → ϕ^{−1}(U_{x_{J}}) is an isomorphism.

(ii) ϕ^{∗}(Q_{K,L,s}/x_{J}^{2}) ∈ I(Z_{J}) for all the Plücker quadrics.

(iii) ϕ|_{Z_{J}} : Z_{J} → G(m, n) ∩ U_{x_{J}} is an isomorphism.

The proposition now follows quickly. By (i) and (ii), and the fact that ϕ is μ-equivariant, it follows that ϕ^{∗}(Q_{I,J,s}/x_{J}^{2}) = 0 ∈ I(ϕ^{−1}U_{x_{J}}) hence that ϕ factors through the Grassmannian.

Next, (i) and (iii) imply that ϕ : U → G(m, n) is a locally trivial fibration. And again using the action, we see that given a k-rational point x ∈ G(m, n), the closed points in the fiber ϕ^{−1}(x) are just the orbit of an m × n matrix A under the left action by GL(m, k). If we write the rows of A as vectors v_{1}, ..., v_{m}, then this identifies x with the subspace of k^{n} generated by v_{1}, ..., v_{m}, giving the bijection. This completes the proof of the Proposition.

Given a scheme T, consider exact sequences: (∗) 0 → K → O_{T}^{n} → Q → 0 where K, Q are locally free sheaves on T of ranks m and n − m, respectively. There is a natural equivalence relation on these. Namely, we’ll say two such sequences (∗) and (∗′) are equivalent if there is a commuting diagram:

(∗) 0 → K → O_{T}^{n} → Q → 0
↓ ∥ ↓
(∗′) 0 → K′ → O_{T}^{n} → Q′ → 0

where the vertical maps are isomorphisms.

Definition: Let F_{m,n} ∈ Sch_{k}/Sets be the functor defined by:

F_{m,n}(T) := {equivalence classes of exact sequences (∗)} with F_{m,n}(f) = f^{∗}

Theorem: The Grassmannian G(m, n) represents the functor F_{m,n}.

Proof: First, we define a transformation of functors: u : F_{m,n} → h_{G(m,n)} to the functor of points of G(m, n) which we’ll then invert.

Given a sequence (∗) on T, choose an affine covering T = ∪Spec(A_{i}) such that K may be trivialized over each Spec(A_{i}). Then over each such affine, the map K|_{Spec(A_{i})} ≅ O_{Spec(A_{i})}^{m} → O_{Spec(A_{i})}^{n} from (∗) is given by an m×n array of elements of A_{i}, determining a morphism f_{i} : Spec(A_{i}) → A_{k}^{mn}. The fact that the cokernel of K → O_{T}^{n} is also locally free implies that each f_{i} factors through U. These morphisms do not patch together, but the compositions ϕ ∘ f_{i} (for the map ϕ : U → G(m, n) defined earlier) do patch to a morphism f : T → G(m, n), and this morphism does not depend upon the trivializations or the sequence (∗) within its equivalence class.

Thus u(T) : F_{m,n}(T) → h_{G(m,n)}(T) is defined on the level of objects. One easily checks that u is a transformation of functors.

To invert u, we find the universal family on G(m, n). The key is to construct the sub-bundle K ↪ O_{G(m,n)}^{n}. To do this, we use the action μ again. Recall the open cover W_{J} := G(m, n) ∩ U_{x_{J}} ≅ Z_{J} ≅ A_{k}^{m(n−m)} of the Grassmannian from (iii) of the proof of the proposition. Putting this together with (i) of the same proof, we obtain isomorphisms:

ρ_{I,J} : GL(m, k) × (W_{I} ∩ W_{J}) ≅ ϕ^{−1}(W_{I} ∩ W_{J}) ≅ GL(m, k) × (W_{I} ∩ W_{J}) identifying W_{I} ∩ W_{J} with open subsets of Z_{I} on the left side and Z_{J} on the right. The morphism to GL(m, k) determined by ρ_{I,J} descends to a morphism g_{I,J} : W_{I} ∩ W_{J} → GL(m, k) which satisfies the cocycle condition, defining the locally free sheaf K of rank m on G(m, n).

The embeddings W_{J} ≅ Z_{J} ↪ A_{k}^{mn} give maps O_{W_{J}}^{m} → O_{W_{J}}^{n} which glue together via the transition functions to an injective map K ↪ O_{G(m,n)}^{n}. The cokernel is determined up to isomorphism, and is locally free. This is our universal family. Via the pull-back applied to this family, we obtain a transformation v : h_{G(m,n)} → F_{m,n}.

It is immediate from the constructions that u and v are inverses(!)

Corollary: G(m, n) ≅ G(n − m, n).

Proof: We only need to find an isomorphism of functors F_{m,n} ≅ F_{n−m,n} by the Theorem. But such an isomorphism is obtained by dualizing the exact sequences (∗) and fixing an isomorphism k^{n} ≅ (k^{n})^{∗}, which then gives simultaneous isomorphisms O_{T}^{n} ≅ (O_{T}^{n})^{∗} for all the schemes T, hence the desired bijection on equivalence classes of exact sequences.

Note: Of course this isomorphism coincides with the identification of a subspace of k^{n} of dimension m with a subspace of (k^{n})^{∗} of dimension n − m by taking the dual of the cokernel. But without the Theorem we cannot deduce an isomorphsm of schemes from this bijection of k-rational points.

Variations: The proof goes through without modification if we replace “k” by “A” (any commutative ring with 1) giving us Grassmannians over Spec(A). If S is any scheme with affine open cover S = ∪Spec(A_{i}), then the Grassmannians over Spec(A_{i}) patch together to give us a Grassmannian G(m, n) over S. But we can do even better than this. If E is any locally free sheaf of rank n on S with trivializations over the Spec(A_{i}), then the Grassmannians over the affines Spec(A_{i}) patch together to give a “twisted” Grassmannian G(m, E) representing the functor from schemes over S to sets:

F_{m,E}(T) = {equivalence classes of sequences (∗)}

where (∗) : 0 → K → E_{T} → Q → 0 are short exact sequences of locally free sheaves on T with rk(K) = m and E_{T} is the pull-back of E under T → S.

The fibers of G(m, E) are just the Grassmannians G(m, n) over k(s) and the corollary generalizes to an isomorphism G(m, E) ≅ G(n − m, E^{∗}).

Fix a Noetherian scheme S, a projective S-scheme X, a coherent sheaf E on X, and a polynomial P (d). Given a Noetherian S-scheme T, let E_{T} be the pull-back of E to X ×_{S} T. For coherent sheaf quotients (∗) : E_{T} → Q → 0 on X ×_{S} T, we declare that (∗) ∼ (∗′) if there is a commuting diagram:

(∗) E_{T} → Q → 0
∥  ↓
(∗′) E_{T} → Q′ → 0

We now define a functor from (Noetherian) S-schemes to sets:

F_{E,P (d)}(T) = {equivalence classes of quotients (∗) such that Q is flat over T and each Q_{t} has Hilbert polynomial P (d)}

F_{E,P (d)}(f) = f̃^{∗} where f̃ is the “base extension”

X ×_{S} U \underrightarrow{_{f}} X ×_{S} T
↓  ↓
U  →^{f}  T

given a morphism f : U → T of S-schemes.

Example! If X = S, P (d) = n − m and E is locally free of rank n, then F_{E,P (d)} is the functor F_{m,E} which was represented by the Grassmannian! After all, Q is flat over T = X ×_{S} T with Hilbert polynomial n − m if and only if Q is locally free of rank n − m(!)

Theorem (Grothendieck’s Quot Scheme): The functor F_{E,P (d)} is always represented by a projective scheme, which we denote by Quot(E, P (d)).

Proof: As in the earlier variations on the Grassmannian, we may as well assume S = Spec(A), since the quot schemes will patch. We prove first:

A Simple Case: Suppose X = P_{A}^{m} and E = OP_{A}^{m}^{n}(l) for some l, n.

Let K be the kernel of (∗) : E_{T} → Q. Then E_{T} = OP_{T}^{m}^{n}(l) is evidently flat over T, hence K is flat over T as well, with constant Hilbert polynomial P′(d) := n \binom{m+l+d}{l+d} − P (d) on the fibers. Since the K_{t} ↪ OP_{t}^{n}^{n}(l) are each coherent subsheaves of a fixed locally free sheaf for each t, uniform vanishing applies. Moreover, it follows from the proof of uniform vanishing in this special case that d_{0} only depends upon P (d), m, n and l and not upon the field k, hence it yields the following “more uniform” result:

There is an d_{0} independent of (∗) and t such that:

H^{i}(P_{t}^{m}, K_{t}(d)) = 0 for all t ∈ T, i > 0, d ≥ d_{0}

Remark: This holds for finite fields as well. If k(t) = F_{q}, then vanishing under the flat base change Spec(\overline{F}_{q}) → Spec(F_{q}) to the (infinite) algebraic closure implies vanishing for F_{q}.

By flatness, it follows that the sequences: 0 → K_{t} → E_{t} → Q_{t} → 0 are all exact, and uniform vanishing for K_{t} and E_{t} = OP_{t}^{m}^{n}(l) implies uniform vanishing for Q_{t}. It follows from cohomology and base change that:

0 → π_{∗}K(d_{0}) → π_{∗}E_{T} (d_{0}) → π_{∗}Q(d_{0}) → 0

are exact, and moreover that π_{∗}K(d_{0}) and π_{∗}Q(d_{0}) are locally free sheaves of ranks P′(d_{0}) and P (d_{0}) respectively, where π : P_{T}^{n} → T is the projection. Finally, note that π_{∗}E_{T} (d_{0}) is a trivial vector bundle, naturally isomorphic to the pull-back of H^{0}(P_{A}^{m}, OP_{A}^{m}^{n}(l + d_{0})) from Spec(A).

Hence (∗) determines a morphism to the Grassmannian over Spec(A):

T → G = G(P′(d_{0}), H^{0}(P_{A}^{m}, OP_{A}^{m}^{n}(l + d_{0})))

That is, we have defined F_{E,P (d)} → h_{G} on the level of objects. I leave it to you to use flatness again to prove that this is a transformation of functors. Our next task is to find the “image.”

Consider the cokernel:

π^{∗}K → E_{G}(d_{0}) → F → 0

where the first map is the composition π^{∗}K → π^{∗}π_{∗}E_{G}(d_{0}) → E_{G}(d_{0}) and K is the universal subbundle on the Grassmannian. Let Quot(E, P (d)) ⊂ G be the term in the flattening stratification of G with respect to F over which F is flat with Hilbert polynomial P (d_{0} + d). I claim that the transformation of functors factors through Quot(E, P (d)) and that the “universal quotient”: (∗) E_{Quot(E,P (d))} → F(−d_{0})|_{P_{Quot}^{m}} inverts the transformation, proving that this “Quot” scheme represents the functor F_{E,P (d)}. To see this, note that uniform vanishing implies that K_{t}(d_{0}) is generated by its global sections for each t, and by cohomology and base change again, it follows that the natural map: π^{∗}π_{∗}K(d_{0}) → K(d_{0}) is surjective.

Thus if T → G is a morphism associated to a quotient (∗), then the cokernel above pulls back to:

π^{∗}π_{∗}K(d_{0}) → E_{T} (d_{0}) → Q(d_{0}) → 0

where the first map factors through K(d_{0}). This proves not only that the transformation factors through h_{Quot} but it also proves that the universal (∗) inverts this transformation, as desired.

So we have Quot(E, P (d)), given to us as a quasi-projective subscheme of a Grassmannian. To prove it is projective, the valuative criterion tells us that it suffices to prove that every morphism Spec(K) → Quot → S from the fraction field of a discrete valuation ring R that extends to a morphism Spec(R) → S also extends to a morphism Spec(R) → Quot. In terms of the functor, this says that any flat quotient (∗)E_{K} → Q on P_{K}^{n} extends uniquely to a flat quotient (∗)E_{R} → \overline{Q} on P_{R}^{n}. But this is exactly what the properness theorem (or rather its proof) gives us.

This completes the simple case.

Given a coherent sheaf E on a projective scheme i : X ⊂ P_{A}^{m}, we may use Serre’s theorem A to find a surjection O_{X}^{n}(l) → E giving rise to quotients: OP_{T}^{m}^{n}(l) → i_{T}_{∗}E_{T} for each S-scheme T, where i_{T} : X_{T} ↪ P_{T}^{m} is the induced closed immersion. In this way, flat quotients of E_{T} push forward under i_{T} to flat quotients of OP_{T}^{m}^{n}(l) and each (∗) : E_{T} → Q determines a morphism T → Quot(OP_{A}^{m}^{n}(l), P (d)) to the Quot scheme already constructed in the simple case. Let G be the kernel on P_{Quot}^{m}:

0 → G → OP_{Quot}^{m}^{n}(l) → i_{Quot}_{∗}E_{Quot} → 0

and choose d_{1} ≥ d_{0} so that π^{∗}π_{∗}G(d_{1}) → G(d_{1}) is surjective. We have already seen that π_{∗}Q(d_{1}) is locally free on Quot by cohomology and base change. It follows that the zero locus of π_{∗}G(d_{1}) → π_{∗}Q(d_{1}) is a closed subscheme Z ⊂ Quot(OP_{A}^{m}^{n}(l), P (d)). Over this subscheme, the universal quotient lifts to a quotient: i_{Z}_{∗}E_{Z} → Q|_{P_{Z}^{m}} and it follows that Q|_{P_{Z}^{m}} is the push-forward of a sheaf on X_{Z} which is the universal quotient for E_{Z}. In other words, Quot(E, P (d)) := Z represents F_{E,P (d)}.

Definition (The Hilbert Scheme): The Hilbert scheme H_{X,P (d)} is the special case of the quot scheme representing the functor F_{O_{X},P (d)}.

Alternatively, the Hilbert scheme represents the functor:

F_{X,P (d)}(T) = {closed subschemes Z ⊂ X ×_{S} T that are flat over T with Hilbert polynomial P (d)}

via the identification of i : Z ↪ X_{T} with the quotient O_{X_{T}} → i_{∗}O_{Z}.

Example (Mori): Suppose X is a projective scheme over Spec(Z) and X_{F_{p}} has a rational curve C (i.e. χ(C, O_{C}) = 1) of degree d for infinitely many primes p and fixed d. Then X_{Q} has a rational curve of degree d, as well. Indeed, the Hilbert polynomial of C is P (d) = d + 1, and since the Hilbert scheme H_{X,d+1} is projective over Spec(Z), it must have only finitely many components. Thus it has a component whose image in Spec(Z) contains infinitely many primes. Such a component must dominate Spec(Z), and must therefore surject onto Spec(Z) by properness. A point in the fiber of such a component over Spec(Q) yields the desired rational curve.

Observation: Quot schemes “commute with base chage”. That is, if F is a coherent sheaf on a T -scheme Y and we are given a morphism S → T, then the Quot schemes fit into a fiber square:

Quot(F_{S}, P (d)) → Quot(F, P (d))
↓  ↓
S  →  T

This follows from the theorem since the fiber product Quot(F, P (d)) ×_{T} S represents the functor F_{F_{S},P (d)}.

Example (Maps to the Grassmannian): Suppose X is a projective kscheme for some field k and suppose P (d) is the Hilbert polynomial of a locally free sheaf Q of rank n − m which is a quotient of a trivial bundle via (∗) : O_{X}^{n} → Q. Then (∗) determines a k-rational point q ∈ Quot(O_{X}^{n}, P (d)) and also a morphism f_{q} : X → G(m, n). Moreover, the universal quotient O_{X×Quot}^{n} → Q is locally free in a neighborhood of X ×{q} and it follows from properness that there is an open neighborhood q ∈ U in the Quot scheme such that Q|_{X×U} is locally free of rank n−m, hence all the points in U parametrize morphisms X → G(m, n). Thus we may think of the component(s) of the quot scheme containing U as a compactification of a space of maps from X to the Grassmannian.

Example (Another Look at the Grassmannian): Suppose P (d) is the Hilbert polynomial of a linear subspace V ⊂ P_{k}^{n−1} of (projective) dimension m − 1. Then H_{P_{k}^{n−1},P (d)} ≅ G(m, n). Indeed, out of the universal subbundle K ↪ O_{G}^{n}, one obtains an inclusion of projective bundles P (K) ⊂ P_{k}^{n−1} × G which is a flat family over G. This gives a morphism G(m, n) → H_{P_{k}^{n−1},P (d)}. On the other hand, one can show that the only projective subschemes of P_{k}^{n−1} with Hilbert polynomial P (d) are the linear subspaces, and that this morphism is an isomorphism.

Last Example (Hypersurfaces in Projective Space): The complete

linear series P_{k}^{\binom{n+d}{n}−1} of hypersurfaces of degree d in P_{d}^{n} is a Hilbert scheme. Indeed, let x_{0}, ..., x_{n} be the coordinates on P_{k}^{n} and let a_{J} for J = (j_{0}, j_{1}, ..., j_{n})

satisfying ∑ j_{i} = d be the coordinates on P_{k}^{\binom{n+d}{n}−1}. Then:

V (∑_{J} a_{J}x_{0}^{j_{0}}x_{1}^{j_{1}}...x_{n}^{j_{n}}) ⊂ P_{k}^{n} × P_{k}^{\binom{n+d}{n}−1}

is the universal divisor, exhibiting P_{k}^{\binom{n+d}{n}−1} as the Hilbert scheme for hypersurfaces of degree d. Again, one checks that the only projective subscheme of P_{k}^{n} with the Hilbert polynomial of a hypersurface is in fact a hypersurface.