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functor of points very ample

Grassmann and the Functor of Points

In the last chapter, we saw that a scheme can be identified with its functor of points. This observation leads us to two classes of questions:

• Given a scheme, can we describe its functor of points in elementary terms?

• Given a functor from schemes to sets, can we determine if it comes from a scheme?

Example 3.1. The fundamental example related to the first question is the functor of points associated to projective space. Let T be a k-scheme, and let f : T → P_{k}^{n} be any T -valued point. By pulling back the standard very ample invertible sheaf on P^{n}, we get an invertible sheaf L = f^{∗}(O(1)) on T. Moreover, the standard coordinate functions pull back to yield sections s_{0} = f^{∗}(x_{0}), …, s_{n} = f^{∗}(x_{n}) ∈ Γ(T, L). Since the x_{i} do not all vanish at the same time, the s_{i} generate the sheaf L locally. Now the collection of data (L; s_{0}, …, s_{n}) depends not just on the T -point f, but also on a choice of coordinates on projective space. We can remove this ambiguity by identifying two such sets of data if they differ by an isomorphism of invertible sheaves on T. So, let us consider the functor I_{n} that assigns to the scheme T the set

I_{n}(T) = {(L; s_{0}, …, s_{n})}/modulo isomorphism,

where L is any invertible sheaf and the s_{i} are global sections that generate L. We have a natural transformation of functors

ι : h_{P^{n}} → I_{n}.

Proposition 3.2. The natural transformation ι is a natural isomorphism.

Proof: The main point is to show that any collection of data (L; s_{0}, …, s_{n}) determines a unique morphism to projective space. This is clear on the open subset {t ∈ T : s_{i}(t) ≠ 0}, where the ratios s_{j}/ − s_{i} determine a map into A^{n} ⊂ P^{n}. It is straightforward to check that these morphisms glue together.

Remark 3.3. It is worth pointing out that this rather abstract way of looking at projective space actually fits quite closely with the classical definition. Let k be a field. There is only one invertible sheaf on Spec(k), corresponding to a vector space

◯^{c}Coombes 1996  1

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2 Chapter 3: Grassmann and the Functor of Points

Picard group Picard functor Picard functor Picard functor Grassmann functor Grassmann functor

V of dimension one. The elements s_{i} ∈ V generate if and only if at least one of them is nonzero. Changing things by an isomorphism corresponds to choosing a different generator for V, which translates into multiplying all the s_{i} by the same nonzero constant.

Example 3.4. Let’s turn our attention to the second class of problems raised above: Given a functor, how can we hope to recognize it as a scheme? For example, consider the functor that assigns to any scheme T its Picard group Pic(T). Now the Picard group classifies invertible sheaves up to isomorphism. So, given a T -point of projective space, we can pullback O(1) to produce an element of Pic(T); i.e., a T -valued point of the Picard functor. Thus, we have a natural transformation

h_{P^{n}} → Pic.

In this way, we can view projective space as a preliminary attempt to represent the Picard functor. This construction generalizes to other projective schemes: Let R be any graded k-algebra that is generated by R_{1}, the elements of degree 1. Then Proj(R) is also equipped with a natural invertible sheaf O(1), and the same technique with pullbacks gives rise to a natural transformation

h_{Proj(R)} → Pic.

Perhaps we have generalized too far; this seems to say that every projective variety is an approximate representation of the Picard functor. In any event, this construction allows us at least to begin to appreciate the complexity of the Picard functor.

By replacing invertible sheaves with locally free sheaves, we can construct a functor analogous to the elementary functor I_{n} = h_{P^{n}}. The resulting functor will be called the Grassmann functor of r-dimensional linear subspaces of affine n + 1 space, and we will denote it by G_{r,n}. In full, it is defined by

G_{r,n}(T) = {(E; s_{0}, …, s_{n})}/modulo isomorphism,

where E is any locally free sheaf of rank r and the s_{i} are global sections that generate the sheaf E. We would like to answer the second question for the Grassmann functors: Is G_{r,n} represented by a scheme?

Classifying Vector Subspaces

Before attempting to represent the Grassmann functor, let us instead work in a more elementary setting. Let k be an algebraically closed field and let V be a k-vector space of dimension n. Define G(r, V) to be the set of k-linear vector subspaces of V of dimension r. We want to show that G(k, V) is equal to the set of k-points of a certain projective variety. (The only reason to work in an algebraically closed field is that the set of k-points, which equals the set of closed points, determines the variety.)

◯^{c}Coombes 1996 exterior power totally decomposable

First, recall the definition of the exterior power algebra of a module V over an arbitrary ring R. We let ⋀^{r}(V) denote the quotient of the r^{th} tensor product V ⊗_{R} V ⊗_{R} ⋯ ⊗_{R} V by the relations

(⋯ ⊗ x ⊗ y ⊗ ⋯) = −(⋯ ⊗ y ⊗ x ⊗ ⋯)

for all x, y ∈ V.

Lemma 3.5. If V is a free R-module of rank n, then ⋀^{r}(V) is free of rank \binom{n}{r}.

Proof: If e_{1}, …, e_{n} is a basis of V, then a basis of the r^{th} exterior power consists of the elements

e_{i_{1}} ∧ e_{i_{2}} ∧ ⋯ ∧ e_{i_{r}}, for all 1 ≤ i_{1} < ⋯ < i_{r} ≤ n.

Corollary 3.6. Let v_{1}, …, v_{r} ∈ V be elements of a finite dimensional vector space V over a field k. Then v_{1} ∧⋯∧v_{r} = 0 ∈ ⋀^{r}(V) if and only if v_{1}, …, v_{r} are linearly dependent.

Proof: Since any linearly independent set can be extended to a basis, the previous lemma shows that their wedge product must be nonzero. On the other hand, if the vectors are linearly dependent, we can write one as a linear combination of the others and use the alternating property to conclude that the wedge product is zero.

So, suppose W ⊂ V is an r-dimensional vector subspace of the finite dimensional vector space V. Choose a basis w_{1}, …, w_{r} of W and consider the element

ω = ω(W) = w_{1} ∧ ⋯ ∧ w_{r} ∈ ⋀^{r}(V).

Choosing a different basis changes ω by multiplying it by a nonzero constant; namely, the determinant of the change of basis matrix. Thus, the class [ω] ∈ P(⋀^{r} V) is well-defined independently of the choice of basis.

Our next task is to investigate the map G(r, V) → P(⋀^{r} V) by W ↦ [ω(W)]. Take an arbitrary multivector ω ∈ ⋀^{r}(V). First, define ϕ(ω) : V → ⋀^{r+1}(V) by ϕ(ω) : v ↦ ω ∧ v. Next, we say that a vector v ∈ V divides ω if there exists an element θ ∈ ⋀^{r−1}(V) such that ω = θ ∧ v. We also say that ω is totally decomposable if we can write ω = v_{1} ∧ … ∧ v_{r} for some vectors v_{i} ∈ V. In this terminology, the image of the Grassmann G(r, V) in P(⋀^{r} V) consists of the classes of totally decomposable vectors.

Lemma 3.7. The map W → [ω(W)] is an injection G(r, V) → P(⋀^{r} V).

Proof: Let W ⊂ V be an r-dimensional vector subspace. We need to show that we can recover W from [ω]; in fact, we claim that W = ker(ϕ(ω)). This will prove the

◯^{c}Coombes 1996 exterior power dual

result, since the kernel is the same independently of which representative we choose for the class [ω] ∈ P(⋀^{r} V). Now fix a basis {w_{1}, …, w_{r}} to define the element ω = ω(W). By the previous corollary, ω ∧ v = 0 if and only if the elements w_{1}, …, w_{r}, v are linearly dependent. But this is clearly true if and only if v ∈ W.

Lemma 3.8. A nonzero vector v ∈ V divides ω ∈ ⋀^{r}(V) if and only if v ∈ ker(ϕ(ω)).

Proof: If v divides ω, then

ϕ(ω)(v) = ω ∧ v = θ ∧ v ∧ v = 0.

On the other hand, suppose ω ∧ v = 0. Extend v = e_{1} to a basis of V, and write ω in terms of the corresponding basis. An explicit computation then shows that e_{1} must appear in every term of ω. By linearity, it follows that v divides ω.

Corollary 3.9. A nonzero element ω ∈ ⋀^{r}(V) is totally decomposable if and only if ker(ϕ(ω)) is r-dimensional.

Proof: Suppose v_{1}, …, v_{s} ∈ V are linearly independent vectors that divide ω. By extending this set to a basis of V, one sees that the wedge product v_{1} ∧ … ∧ v_{s} also divides ω. In particular, we see that the maximum possible dimension of the kernel is r, and that ω is totally decomposable when the maximum is achieved. On the other hand, suppose that ω is totally decomposable, and write ω = v_{1} ∧ … ∧ v_{r}. Since this element is nonzero, the component vectors must be linearly independent. So ker(ϕ(ω)) must have dimension r.

Remark 3.10. Now we can show that G(r, V) is the set of k-points of a closed subset of P(⋀^{r} V). For, we have [ω] ∈ G(r, V) if and only if ker(ϕ(ω)) is r-dimensional, which can only happen if the rank of the matrix ϕ(ω) is n − r. By the proof of the corollary, we only need to have rank(ϕ(ω)) ≥ n−r. But this condition is defined by the vanishing of all n−r+1×n−r+1 minors of the matrix. Since the entries of the matrix are linear forms in the variables on this projective space, this computation shows that the set G(r, V) is cut out by equations of degree n − r + 1.

This is not an optimum set of equations; we need to work harder to get the equations that generate the homogeneous ideal of all functions that vanish on this closed set. Toward this end, it is useful to have a concrete characterization of the dual of the exterior power. We will denote the dual of any R-module V by V^{∗} = Hom_{R}(V, R). Notice that if V is free of rank n, then V^{∗} is also free of rank n. Thus V and V^{∗} are isomorphic, but the isomorphism is not natural, since it depends on choosing bases. Now (⋀^{r}(V))^{∗} = Hom_{R}(⋀^{r}(V), R) is the set of r-multilinear alternating forms on V; i.e., the set of maps f : V^{r} → R that are linear in each variable and satisfy

f(…, x_{i−i}, x_{i}, …) = f(…, x_{i}, x_{i−1}, …).

◯^{c}Coombes 1996 totally decomposable annihilator

If V is free of rank n, then both ⋀^{r}(V^{∗}) and (⋀^{r}(V))^{∗} are free of rank \binom{n}{r}. That tells us that they are abstractly isomorphic, but we actually have a natural isomorphism between them as follows. Let ϕ_{1}, …, ϕ_{r} ∈ V^{∗} and let v_{1}, …, v_{r} ∈ V. Define a homomorphism ϕ_{1} ∧ ⋯ ∧ ϕ_{r} : V^{r} → R by the formula

(ϕ_{1} ∧ ⋯ ∧ ϕ_{r})(v_{1}, …, v_{r}) = det(ϕ_{i}(v_{j})).

Since the determinant is an alternating map, this assignment determines a function ⋀^{r}(V^{∗}) → (⋀^{r}(V))^{∗} that gives the desired isomorphism.

In this situation, we also have an isomorphism between ⋀^{r}(V) and (⋀^{n−r}(V))^{∗} induced by the wedge product pairing

⋀^{r}(V) × ⋀^{n−r}(V) → ⋀^{n}(V) ≈ R.

In particular, an element ω ∈ ⋀^{r}(V) determines in a natural way an element ω^{∗} ∈ ⋀^{n−r}(V^{∗}). Let

ψ(ω) : V^{∗} → ⋀^{n−r+1}(V^{∗})

denote the homomorphism given by ψ(ω)(v^{∗}) = v^{∗} ∧ ω^{∗}. By explicit computation, one can check that ω is totally decomposable if and only if dim(ker(ψ(ω))) ≥ n − r if and only if rank(ψ(ω)) ≤ r. Thus, we can also realize G(r, V) as the set of closed points of a scheme defined by the vanishing of a collection of equations of degree r + 1.

We still haven’t found the best set of equations for the Grassmann. We need to combine the two approaches. First, consider the dual of the exact sequence

0 → ker(ϕ(ω)) → V ⟶^{ϕ(ω)} ⋀^{r+1}(V),

which is just

0 ← (ker(ϕ(ω)))^{∗} ← V^{∗} ⟵^{ϕ(ω)^{t}} ⋀^{r+1}(V^{∗}).

The image of the transpose (since it is the kernel of the next map) is equal to Ann(ker(ϕ(ω))), the annihilator of the kernel of ϕ(ω). An equivalent translation of the above analyses is that ω is totally decomposable if and only if

ker(ψ(ω)) = Ann(ker(ϕ(ω))) = Im(ϕ(ω)^{t}).

Therefore, the Grassmann can be realized as the closed set defined by the equations

< ϕ(ω)^{t}(α), ψ(ω)^{t}(β) >= 0

for all α ∈ ⋀^{r+1}(V^{∗}) and all β ∈ ⋀^{n−r+1}(V). Since both ϕ and ψ have entries that are linear in the coordinates of the ambient projective space P(⋀^{r} V), this is a collection of quadratic equations. By choosing consistent bases for all the

◯^{c}Coombes 1996 Plücker equations Grassmann variety Grassmann variety Grassmann functor functor of points

duals and exterior powers, one can write down the equations explicitly. Writing the coordinates as X_{i_{1},...,i_{r}}, one has

(3.11)  ∑_{m=1}^{r}(−1)^{m}X_{i_{1},...,i_{r−1},j_{m}} ⊗ X_{j_{1},...,j_{m−1},j_{m+1},...,j_{r+1}} = 0

for all collections of indices {i_{1}, …, i_{r−1}} and {j_{1}, …, j_{r+1}}. The equations (3.11) are known as the Plücker equations of the Grassmann variety.

It is useful to have a local picture of the Grassmann variety. We’ll start with a coordinate-free description. Let Γ ⊂ V be a vector subspace of dimension n − r, which corresponds to an element γ ∈ ⋀^{n−r}(V) = ⋀^{r}(V)^{∗}. We can interpret γ as a homogeneous linear form on P(⋀^{r} V). So, we can look at the open affine U ⊂ P(⋀^{r} V) where γ ≠ 0. On this affine, one has

G(r, V) ∩ U = {r-spaces W ⊂ V complementary to Γ}.

The complementary subspaces can be identified with the graphs of maps V/Γ → Γ, and so

G(r, V) ∩ U = Hom(V/Γ, Γ) = A^{r(n−r)}.

We can see this identification explicitly if we take Γ to be spanned by the standard vectors e_{r+1}, …, e_{n}. Writing a basis of a subspace W as row vectors, the condition that it be complementary to Γ simply says that the first k ×k minor is nonzero. Up to a change of basis in W, then, a complementary subspace is given by arbitrarily choosing the entries in the matrix

[ \begin{matrix}1 & 0 & … & 0 & a_{1,1} & … & a_{1,n−r} \\ & & … & & & … \\ 0 & 0 & … & 1 & a_{r,1} & … & a_{r,n−r} \end{matrix} ]

Representing the Grassmann Functor

After this long interlude, let us return to the Grassmann functors G_{r,n}. We introduced the Grassmann functors as analogues of the elementary description of the functor of points on projective space, and we wanted to know if we could realize them as points on some scheme.

Theorem 3.12. The Grassmann functor G_{r,n} is the functor of points associated to the projective variety defined by the quadratic equations (3.11).

Proof: Before proceeding, we need to point out that G_{r,n}(k) = G(r, k^{n+1}) in the notation of the preceeding section. Now let T be any scheme, and choose a T - valued point of the Grassmann; i.e., an element ϕ ∈ G_{r,n}(T). The data determining ϕ consists of a locally free sheaf E of rank r on T, and a collection of global sections s_{0}, …, s_{n} that generate the sheaf E. Let L = ⋀^{r}(E) denote the top exterior power of E, so that L is an invertible sheaf on T. We get sections of L by taking

X_{i_{1},...,i_{r}} = s_{i_{1}} ∧ … ∧ s_{i_{r}}

◯^{c}Coombes 1996 Plücker equations Grassmann functor

for all 0 ≤ i_{1} < … < i_{r} ≤ n. The collection (L; {X_{i_{1},...,i_{r}}}) defines a T -valued point of the projective space P^{N} where N = \binom{n+1}{r} − 1. Thus, we have a well defined natural transformation of functors

α : G_{r,n} → h_{P^{N}} .

It is this transformation that we will use to realize the Grassmann as a projective variety.

Now the data defining ϕ also determines a surjective sheaf homomorphism

ϕ : O_{T}^{n+1} → E.

Up to isomorphism, this data is determined by ker(ϕ). Since O_{T}^{n+1} is a fixed sheaf, it is enough to compute ker(ϕ) locally in order to know it globally. Thus, it is enough to show that if X is a T -valued point of P^{N} that satisfies the Plücker equations (3.11), then there is an open cover of T on which X lifts uniquely to a point of the Grassmann functor. So, take the affine open where X_{i_{1},...,i_{r}} ≠ 0. On this affine, we can solve the equations (3.11) and recognize that they define a subvariety isomorphic to A^{r(n+1−r)}, where the free variables can be identified with the X_{i_{1},...,i_{m−1},i_{m+1},...,i_{r},j}. However, to build a point of the Grassmann over this open set means that the section s_{i_{1}}, …, s_{i_{r}} generate the sheaf E. So, the kernel of ϕ has a unique basis of the form

e_{j} − ∑_{m=1}^{r} a_{jm}e_{i_{m}},

where 0 ≤ j ≤ n is distinct from the i_{m}. Computing the Plücker coordinates, one sees that

a_{jm} = (−1)^{r−m} \frac{X_{i_{1},...,i_{m−1},i_{m+1},...,i_{r},j}}{X_{i_{1},...,i_{r}}} .

So, there is one and only one choice of a_{jm} ∈ Γ(T, O_{T}) corresponding to the given Plücker coordinates.

Remark 3.13. It’s probably a good idea, once again, to take stock of how this chapter contributes toward our goal of studying curves on a surface. At a superficial level, this chapter is just an existence proof that our plan to realize some functor as a scheme is not totally hopeless. After all, we just showed how to take the Grassmann functors and represent them as schemes. However, these particular functor-schemes were not chosen randomly. The Grassmann varieties will eventually provide us with a concrete starting place in our quest to represent the functor of curves on an algebraic surface.

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