Explicit projective embeddings of standard opens of the Hilbert scheme of points

We describe explicitly how certain standard opens of the Hilbert scheme of points are embedded into Grassmannians. The standard opens of the Hilbert scheme that we consider are given as the intersection of a corresponding basic open affine of the Grassmannian and a closed stratum determined by a Fitting ideal.


Introduction
The main result in this article is an explicit projective embedding of the standard open subschemes of the Hilbert scheme of points in affine space. By taking unions we get an embedding of certain natural subschemes of the Hilbert scheme of points. Apolarity schemes, commutator ideals and the space of non-degenerate families of n + 1 points in projective n-space are examples and applications of our result.
A standard open subscheme Hilb β ⊆ Hilb N X of the Hilbert scheme of N points on an affine scheme X is determined by a sequence β of global sections with |β| = N . The scheme Hilb β is characterized by parameterizing all closed subschemes of the ambient scheme such that β forms a basis for their global sections. Standard open subschemes seem to first have appeared in Haimans article [Hai98] as a tool in his description of the Hilbert scheme of points in the complex affine plane. The definition of these schemes Hilb β were formalized and generalized for Hilbert scheme of points on general affine schemes, independently by several authors [KKR05], [GLS07a] and [Hui06].
Considerable interest have been devoted to the schemes Hilb β , and in particular with the Hilbert scheme of points on affine n-space and with β a sequence of monomials satisfying the order ideal condition, see e.g. [KR08], [KK05] and [Led11].
However, what have been missing from the discussion of the schemes Hilb β is a natural projective embedding. The schemes Hilb β are defined and realized abstractly, and they glue canonically for different choices of β. The information is however local, and it has not been clear how to realize these charts and their unions in a projective space. In the present article we give a precise answer to that question.
Having the sequence of monomials β fixed we let z d β denote the degree d homogenization, and we view z d β as global sections of the degree d-forms on projective n-space. These global sections z d β determine a basic open affine D + (z d β) of the Grassmannian G of locally free, rank N = |β| quotients of the vector space V d of d-forms on projective n-space. We prove the following result.
Theorem 1. Let β be an order ideal sequence of monomials in the polynomial ring A[t 1 , . . . , t n ]. Let d be an integer such that d ≥ d(β) + 1, where d(β) is the highest degree of the monomials in β. Then the standard open subscheme Hilb β of the Hilbert scheme of N = |β| points on affine n-space A n A , is given as the locally closed subscheme Here Fitt N −1 (E) is the closed stratum determined by the (N − 1)'th Fitting ideal of a coherent sheaf E (see Definition 2.6.1) on the Grassmannian G.
Having the length N of the sequence β fixed, the sheaf E and the Grassmannian G depend only on the integer d. Thus by fixing d, and considering unions, we obtain a description of certain natural subschemes of the Hilbert scheme of points in affine n-space, as well as projective n-space. We can vary β in A[t 1 , . . . , t n ] with d(β) + 1 ≤ d, and we can in addition vary which hyperplane in projective n-space whose complement is our affine n-space. The resulting quasi-projective scheme parametrizes closed, length N , subschemes in projective n-space, that fiberwise have global sections given by a sequence β with d(β) + 1 ≤ d. We refer to such subschemes as d-strongly generated of length N (see Section 3). These schemes are particular instances of the bounded regularity schemes introduced in [BBR15]. For the space of d-strongly generated subschemes of length N , our result yields an explicit embedding of these parameter schemes.
We work out a particular example, the 2-strongly generated subschemes of projective n-space, of length n + 1. These subschemes are particularly simple being cut out by quadrics, and we describe how the quasi-projective parameter space is embedded in the Grassmannian of rank n + 1 quotients of the vector space of degree two forms.
An application of our explicit description is within the theory of apolarity schemes. The scheme VPS n+1 Z of length n + 1 subschemes in P n apolar to the annihilator scheme Z of a smooth quadratic surface was introduced in [RS00] (see Section 6.2). We give a direct argument reproving their result that the space VPS n+1 Z is closed in the Grassmannian of rank n+1 quotients of the vector space of two-forms on projective n-space.
In our main result the chosen integer d can typically be lower than the regularity of the subschemes of length N . This is in contrast to the assumptions of Gotzmann's persistence theorem [Got78], a result typically used when considering embeddings of the Hilbert scheme (e.g. [IK99], [BBR15], [ABM10]). One advantage with our approach is that we get embeddings of the standard open subschemes Hilb β in a lower dimensional Grassmannian than what would be possible if using Gotzmann's persistence theorem. Moreover, we can apply our methods to get similar descriptions of Quot schemes, and these generalizations are outlined in the Appendix.
We also establish an unexpected connection between commutators and Fitting ideals. Central in the constructions of the local open schemes Hilb β are the commutator relations arising from matrices operating on certain vector spaces. We show that the ideal generated by the commutator relations equals the Fitting ideal arising from the graded, global situation. That connection ties together and explains the two different descriptions of the embeddings of the Hilbert scheme of points that appear in [ABM10] and in [Stå14] and [Skj14].

Order ideals and Grassmannians
2.1. Order ideals. Let β be a finite collection of monomials in the polynomial ring A[t 1 , . . . , t n ] over a commutative, unitary ring A. We assume that β has the property that it contains all its divisors: for any λ ∈ β, it holds that for any i = 1, . . . , n and any d ≥ 1. Such a set β is commonly referred to as an order ideal, see e.g. [KKR05].
Lemma 2.2. Let I ⊆ k[t 1 , . . . , t n ] be an ideal in the polynomial ring over a field k, such that the quotient ring is a finite dimensional vector space. Then there exists a sequence β of monomials satisfying (2.1.1), and such that the images of β form a k-vector basis for the quotient ring.
Proof. This is a consequence of the quotient set having a k[t 1 , . . . , t n ]-module structure.
2.4. Closed subschemes of projective space. We say that a closed subscheme Γ ⊆ P n A is generated in degree d if it is defined by a homogeneous ideal J ⊆ A[z, x 1 , . . . , x n ] generated in degree d.
Proposition 2.5. Let Γ ⊆ A n A be a finite closed subscheme, and let β be a sequence of monomials in A[t 1 , . . . , t n ] that forms a basis for the global sections of Γ. We view Γ as a closed subscheme in P n A by identifying A n A as a hyperplane complement. Then we have that Γ is generated in degree d(β) + 1, where d(β) is the top degree of the monomials (2.3.1).
Proof. By assumption the monomials in β form a basis for the A-module E of global sections of Γ. The algebra structure on E is then completely determined by t i µ, where µ ∈ β and i = 1, . . . , n. We have that for some scalars a i,λ in A. The relation arising from (2.5.1) is a polynomial of degree at most d(β)+1. Therefore the degree of the generators of the ideal I ⊆ A[t 1 , . . . , t n ] defining Γ is bounded by d(β) + 1. The homogenization of I with respect to z is then generated in degree d(β) + 1, and determines the closed subscheme Γ ⊆ P n A . 2.6. Fitting ideals and graded quotients. The A-module of degree mforms in the standard graded polynomial ring A[z, x 1 , . . . , x n ] we will denote by V m . Thus A[z, x 1 , . . . , x n ] = m≥0 V m .
If J ⊆ A[z, x 1 , . . . , x n ] is a graded ideal, we let (2.6.1) We let Theorem 2.10. Let A[t 1 , . . . , t n ] denote the polynomial ring in the variables t 1 , . . . , t n over a ring A. Let β be a sequence of monomials satisfying (2.1.1), let N = |β| and fix an integer d ≥ d(β) + 1. The functor parameterizing finite closed subschemes Γ ⊆ A n A such that β forms a basis for the global sections of Γ is represented by the scheme x 1 , . . . , x n ] is the ideal generated by the module J d obtained from the sequence (2.9.1). The universal family we get by localizing in z and taking out the degree zero part. In particular Hilb β is a locally closed subscheme of the Grassmannian G N (V d ).
Proof. By definition we have that z d β forms a basis for the universal quotient bundle E d of the Grassmannian, when restricted to the basic open subscheme D + (z d β). When we restrict further to the closed subscheme defined by the Fitting ideal Fitt N −1 (E J d+1 ) we obtain by Proposition 2.7 that the graded sheaf O[z, x 1 , . . . , x n ]/J is free and of rank N in every degree m ≥ d, where O is the structure sheaf on Hilb β . In particular we get a closed subscheme Z ⊆ P n A × A Hilb β , which is flat, finite with relative rank N over Hilb β and fiberwise the sequence β will form a basis for its global sections. Universal property follows.
Remark 2.11. The scheme Hilb β described in the Theorem is the basic open subscheme of the Hilbert scheme Hilb N A n A of N points on A n A , and Hilb β is parameterizing finite closed subschemes where β forms a basis for its global sections. These schemes were introduced in [KKR05], [GLS07a] and [Hui06], and can be defined for any sequence of elements β, not only for an order ideal of monomials as we consider. However, the novelty in our description is the projective embedding of these schemes. We do not describe the schemes Hilb β abstractly, but give these in terms of an explicit embedding into a Grassmannian.
Remark 2.12. The assumption on the integer d in the Theorem is only that d ≥ d(β) + 1. Typically that integer d is lower than the regularity for the closed subschemes in A n A of length N = |β|. In particular Gotzmann's persistence theorem [Got78] is not applicable in our situation.
Remark 2.13. In the Appendix we show that the above results can be naturally generalized to module quotients. That situation, with Quot schemes replacing Hilbert schemes, is more notationally complex, but otherwise quite similar.
Example 2.14. We apply our Theorem 2.10 to describe ideals in the polynomial ring A[t 1 , t 2 ] having β = {1, t 1 , t 2 } as a basis for the quotient ring. We let d = 2 = d(β) + 1, and then z 2 β = {z 2 , zx 1 , zx 2 }. The basic open D + (z 2 β) of the Grassmannian of rank 3 quotients of the A-module of two- Here E 2 is the free B-module with basis z 2 β, and J 2 is the kernel of M . The kernel J 2 of M is generated by the three elements The nine elements we obtain by multiplying a, b and c with the linear forms z, x 1 and x 2 generates the degree three part J 3 of the ideal J = (J 2 ). We order the monomials with the lexicographic ordering, where z < x 1 < x 2 .
Then the columns in the (10 × 9)-matrix below, represents the generators of J 3 .
The cokernel of M is by definition E J 3 , and the Fitting ideal Fitt 2 (E J 3 ) is the ideal generated by the (7 × 7)-minors of M . By computing these minors we get that the ideal cutting out Hilb β in the affine 9-space, with coordinates a 1 , . . . , c 3 , is generated by the five elements Example 2.15. An efficient way to compute the Fitting ideals described in Theorem 2.10 is available in the Macaulay2 [GS] program using the package FiniteFittingIdeals, available as of version 1.8, and constructed by the second author of the present article. The following is an example of how to use that package.
We will describe ideals in A[t 1 , t 2 ] where β = {1, t 1 , t 2 , t 2 1 } forms a basis for the quotient ring. We chose d = d(β) + 1 = 3. The basic open subscheme D + (z 3 β) in the Grassmannian of rank 4 quotients of the A-module of 3-forms The output will be 27 degree two polynomials in the variables a 1 , . . . , f 4 .

Strongly generated subschemes
Definition 3.1. Let Γ ⊆ A n k be a closed, finite subscheme with k a field. We say that Γ is d-strongly generated if a sequence of monomials β satisfying (2.1.1) form a basis for the vector space of global sections of Γ, with d ≥ d(β) + 1.
A family of closed subschemes Γ ⊆ A n A , with A a commutative ring, is dstrongly generated if Γ is finite and flat over A, and fiberwise d-strongly generated.
Remark 3.2. A family Γ ⊆ A n A is d-strongly generated, if locally on Spec(A), we can find a sequence of monomials β satisfying (2.1.1) that forms a basis for the global sections of Γ, and where d ≥ d(β) + 1.
Remark 3.3. By Proposition 2.5 we have that a d-strongly generated subscheme Γ ⊆ A n k ⊂ P n k is, considered as a subscheme in projective n-space, generated in degree d.
Remark 3.4. The notion of d-strongly generated is stable under base changes, and forms naturally a subfunctor of the Hilbert functor parametrizing closed subschemes Γ ⊆ A n A that are flat and of relative rank N over the base A. 3.5. Cover. Let d and N be two fixed integers, and let A[t 1 , . . . , t n ] denote the polynomial ring over A. Let B denote the collection of all (ordered) sequences β of monomials in A[t 1 , . . . , t n ] satisfying (2.1.1), having cardinality |β| = N , and top degree d(β) + 1 ≤ d. Define the scheme where Hilb β is the scheme in Theorem 2.10.
Proposition 3.6. The functor that parametrizes closed subschemes Γ ⊆ A n A that are d-strongly generated of relative rank N , is represented by the scheme Gen(d, N ) from (3.5.1). If moreover d ≥ N , then we have an equality Proof. Let β and γ be two sequences of monomials satisfying (2.1.1), with |β| = |γ| = N , and where d ≥ max{d(β), d(γ)} + 1. It follows from the universal property described in Theorem 2.10 that the subscheme parametrizes closed subschemes in A n A where both β and γ form a basis for its global sections. The first statement then follows.
If d ≥ N , then any length N -quotient of k[x 1 , . . . , x n ], where k is any field, has some basis β of monomials of the form (2.1.1) by Lemma 2.2. Thus the schemes Hilb β cover Hilb N A n A , and we have proved the Proposition.
Remark 3.7. When the base field k has infinite cardinality, any finite closed subscheme Γ ⊆ P n k is contained in a hyperplane complement A n k ⊂ P n k . Thus, by varying different hyperplanes one obtains, by taking the corresponding unions of (3.5.1), the scheme Gen + (d, N ) parametrizing finite length N subschemes Γ ⊂ P n A that are flat, and fiberwise d-strongly generated. In particular the scheme Gen + (d, N ) will be a locally closed subscheme of the Grassmannian G N (V d ), explicitly described as the unions of the different Hilb β 's intersected with the closed Fitting ideal stratum. The described scheme Gen + (d, N ) is a special case of the bounded regularity locus introduced and considered in [BBR15]. We have not given the details for such a description in general, but will give such a description for a specific example in the next section.
Remark 3.8. When the base field k is infinite, and when d ≥ N , then dstrongly generated length N subschemes of P n A equals those parametrized by the Hilbert scheme of N -points in P n A . Here A is a k-algebra. In that situation the schemes Gen + (d, N ) we get by varying the hyperplane complement A n A ⊂ P n A will cover the Hilbert scheme Hilb N P n A of N points in projective n-space P n A . In fact, we have that Hilb N where J is the kernel of the universal family on the Grassmannian (2.9.1), see [Skj14], [Stå14] for details.

Non-degenerate families of points
4.1. Non-degenerate families. A closed subscheme Γ ⊆ A n k is called nondegenerate if it is not contained in any hyperplane. A flat family Γ ⊆ A n A is non-degenerate if the fiber over any point in Spec(A) is non-degenerate. We make similar definitions for closed subschemes Γ ⊆ P n A being non-degenerate. Proof. Let k be a field, and let Γ ⊆ A n k be a closed subscheme, of rank n + 1 and non-degenerate. Then we claim that β = {1, t 1 , . . . , t n } in the polynomial ring k[t 1 , . . . , t n ] is a basis for the global sections of Γ. Because, assume that 1, t 1 , . . . , t i−1 are linearly independent, but that t i was not a basis element, for some i > 0. We then have an equality t i = a 0 + i−1 j=1 a j t j . Such an equation would determine a hyperplane containing Γ, hence contradicting the non-degeneracy assumption. We have then shown that β form a basis for the global sections of Γ ⊆ A n A over any fiber, hence β form a basis for the global sections everywhere. The result then follows from Theorem 2.10.
Lemma 4.4. Let Γ be n + 1 distinct k-rational points in projective n-space P n k over a field k. If Γ is non-degenerate there exists a hyperplane H not intersecting Γ, that is Γ ⊆ P n k \ H = A n k .
Proof. We will do induction on n. The situation with n = 1 is clear. Let Γ = {P 1 , . . . , P n , Q} be n + 1 points in P n k , and let H be a hyperplane containing Γ = {P 1 , . . . , P n }. The non-degeneracy assumption on Γ implies that Q is not in H and that Γ , considered as points in H = P n−1 k , are non-degenerate. By induction hypothesis there exists a hyperplane H ⊂ P n−1 k avoiding the points Γ . The hyperplane H together with any point in R ∈ P n k \ P n−1 k determines a hyperplane H + R. A dimension count shows that not all hyperplanes H + R can contain Q.
Remark 4.5. The non-degeneracy assumption is necessary. In the projective plane over the field with two elements, any three points lying on a line intersect all the seven lines in the plane (since any two lines intersect and there are only three points on a line).
4.6. Cover II. Given a set Γ ⊆ P n k of n + 1 distinct k-rational points, we have, by Lemma 4.4, that Γ is contained in the complement of a hyperplane H. We can choose a finite set B of hyperplanes in P n k such that any Γ is in the complement of one of these hyperplanes. Indeed, this is clear if the field is infinite, and if the field k is finite, then there are only finitely many hyperplanes so the statement is trivial.
For each linear form z determining a hyperplane in B we obtain the polynomial ring A[t 1 , . . . , t n ] after localizing A[x 0 , . . . , x n ] in z and taking out degree zero. In each of these polynomial rings we consider the sequence β z := {1, t 1 , . . . , t n }.
Proposition 4.7. The functor parametrizing closed subschemes P n A that are flat, finite of relative rank n + 1, and non-degenerate, is represented by the scheme Gen (4.6.1). The universal family is the closed subscheme in P n A × A Gen determined by the ideal (J 2 ) ⊆ O[x 0 , . . . , x n ] coming from the universal sequence on the Grassmannian G n+1 (V 2 ) (2.9.1), where O is the structure sheaf on Gen.
Proof. Let z ∈ B be a linear form, and let H ⊆ P n A denote the hyperplane determined by it. By Proposition 4.2 we have that Hilb βz is the scheme representing non-degenerate families not having support in the hyperplane H. It suffices to check that we have a covering over arbitrary field valued points. Let Γ ⊆ P n k be a non-degenerate closed subscheme of length n + 1. It remains to show that there exists a hyperplane H ∈ B that avoids Γ. We may therefore assume that Γ is arranged in the most obtrusive case possible, that is, that Γ consists of n + 1 distinct and k-rational points. The result then follows from the construction of B.

Commutator relations and Fitting ideals
We will in this section show a connection between commutator relations and Fitting ideals. 5.1. Non-commutative monomials. Letβ be a sequence of monomials in the free, non-commutative, algebra A t 1 , . . . , t n in the variables t 1 , . . . , t n over a commutative unital ring A. We assume that the cardinality |β| = N , and that 1 ∈β. Let Eβ denote the free A-module having as a basis the elements ofβ. Assume furthermore that we have (N × N )-matrices T 1 , . . . , T n , where for any λ ∈β we have Lemma 5.2. The matrices T 1 , . . . , T n satisfying (5.1.1) determine an Alinear map ψ : A t 1 , . . . , t n −→ Eβ by sending monomials t a 1 n 1 · · · t a k n k to T a 1 n 1 · · · T a k n k (1). The map ψ is surjective and its kernel is a left ideal. And conversely, a left ideal I ⊆ A t 1 , . . . , t n whereβ form a basis for the quotient module is given by such matrices T 1 , . . . , T n satisfying (5.1.1).
Proof. Let E = Eβ. The map ψ is the composition of the evaluation map ev : End A (E) −→ E at the element 1 ∈ E, and the A-algebra homomorphism T : A t 1 , . . . , t n −→ End A (E) determined by the matrices T 1 , . . . , T n . The kernel of the composite map A t 1 , . . . , t n T / / End A (E) ev / / E is a left ideal. Conversely, assume I is a left ideal such thatβ form a basis for the quotient module A t 1 , . . . , t n /I = E. The multiplication action of the free algebra on the quotient E determines the matrices T 1 , . . . , T n satisfying (5.1.1). The elements inβ determine an A-module homomorphism iβ : E −→ A t 1 , . . . , t n . The conditions (5.1.1) assure that the composition ev •T • iβ is the identity. 5.3. Commutators. By above, we have that a left ideal I ⊆ A t 1 , . . . , t n and monomialsβ that form a basis for the quotient module are equivalent with having matrices T 1 , . . . , T n satisfying (5.1.1). In particular such an ideal I and such a sequenceβ determine the commutator ideal in A, which is the ideal generated by the entries of the matrices A[t 1 , . . . , t n ] means a monomial wordλ in A t 1 , . . . , t n such that c(λ) = λ, where

Monomial lift. A lift of a monomial λ in
denotes the canonical quotient map to the polynomial ring. If β is a sequence of monomials, thenβ denotes a lifted sequence of monomial words in A t 1 , . . . , t n .
Remark 5.5. Letβ be a lift of β, and assume that we have matrices T 1 , . . . , T n satisfying (5.1.1). Then the commutator ideal (5.3.1) is zero precisely when the matrices commute and the corresponding left ideal I ⊆ A t 1 , . . . , t n is such that the canonical map We can localize A[z] x 1 , . . . , x n as an A[z]-module. When we localize with respect to z and take out the degree zero part, we obtain the free algebra A t 1 , . . . , t n where t i = x i /z for i = 1, . . . , n. , and let I ⊆ A t 1 , . . . , t n denote the left ideal we obtain from J by localization in z and taking out degree zero elements. We assume the following.
(1) The graded ideal J ⊆ A[z, x 1 , . . . , x n ] is generated by elements of degree d.  Proof. Both the Fitting ideals and commutator ideals commute with base change A −→ A . We also have that the ideals arising from J ⊗ A A will also satisfy (1), (2) and (3). Thus to show the proposition we can pass to appropriate quotient rings of A.
We first assume that [T i , T j ] = 0 for 1 ≤ i, j ≤ n. We then have that the left ideal I ⊆ A t 1 , . . . , t n is such that (5.7.1) A t 1 , . . . , t n /I = A[t 1 , . . . , t n ]/ c(I).
It follows that the homogeneous left ideal J ⊆ A[z] x 1 , . . . , x n also contains all commutators, that is One verifies that J is the homogenization of c(I). The ideal c(I) determines a closed, finite, subscheme Γ ⊆ A n A , and the homogeneous ideal J determines Γ as a closed subscheme in P n A . By assumption β is an A-module basis of the coordinate ring of Γ, and then we also have that z m β is an A-module basis for the global sections of Γ ⊆ P n A , for m 0, which is the degree m part of the quotient ring (5.7.2). By the defining properties of Fitting ideals, we then have that Fitt N −1 (E J m ) = 0 and that Fitt N (E J m ) = A for m 0. By Proposition 2.5 we have that Γ is generated in degree d, and by assumption z d β is a basis for the degree d part of the quotient ring (5.7.2). It then follows by Proposition 2.7 that the Fitting ideals F N −1 (E J d+m ) = 0 for m ≥ 1. We have therefore proved that the Fitting ideals in question are included in the commutator ideal generated by the entries of the matrices [T i , T j ] (for 1 ≤ i, j ≤ n).
To prove the converse, assume that Fitt N −1 (E J d+1 ) = 0. Proposition 2.7 gives that Fitt N −1 (E J d+m ) = 0, and that z d+m β is a basis for the degree m part of the graded quotient (5.7.2), for m ≥ 1. We need to see that the commutators are zero. The closed subscheme Γ ⊆ P n A determined by the homogeneous ideal J ⊆ A[z, x 1 , . . . , x n ] is then affine, and of relative rank N = |β|. We then get that β is a basis for the coordinate ring of Γ, which we obtain by localizing (5.7.2) in z and taking out the degree zero part. Let I be the ideal in A[t 1 , . . . , t n ] corresponding to the closed immersion Γ ⊆ A n A . As a consequence of exactness of localizations we get a surjection A t 1 , . . . , t n /I −→ A[t 1 , . . . , t n ]/I . By assumptionβ is a basis for the leftmost module, and we have that β is a basis for the module on the right. It follows that the rings are isomorphic. It then follows by Remark 5.5 that [T i , T j ] = 0 for all 1 ≤ i, j ≤ n.
Remark 5.8. In [Stå14] and in [ABM10] two different descriptions of the embedding of the Hilbert scheme Hilb N P n A into the Grassmannian G N (V d ) were given. The first describes the embedding via Fitting ideals, and the second describes the embedding via commutator ideals. Our result above identifying the Fitting ideal with the commutator ideal connects these two different descriptions of the Hilbert scheme.
Example 5.9. Let J ⊆ A[z] x 1 , x 2 be the homogeneous left ideal generated by 1, 2, 3) are elements in A. Note that under the canonical map c : On the other hand, the left ideal I ⊆ A t 1 , t 2 , obtained by localizing J in z and taking out the degree zero part, is generated by The quotient A t 1 , t 2 /I = E is the free A-module with basis {1, t 1 , t 2 }.
We are therefore in a situation where Proposition 5.7 applies. We have β = {1, t 1 , t 2 } =β, and use degree d = 2 = d(β) + 1. The left ideal x 2 is such that the assumptions (1), (2) and (3) are satisfied. Now we want to describe the commutator ideal and the Fitting ideal arising in this particular situation. The matrices, corresponding to the action of A t 1 , t 2 on the quotient module E are The commutator ideal in A is generated by the coefficients of T 1 T 2 −T 2 T 1 , and that matrix is The commutator ideal is generated by five elements as the two lower diagonal elements are equal up to a sign. On the other hand we want to compute the 2nd Fitting ideals of the graded components of E J = A[z, x 1 , x 2 ]/J, in degrees ≥ 3. The Fitting ideal of interest is Fitt 2 (E J 3 ), the one arising from the degree 3 component. That ideal we computed in Example 2.14, and it is readily verified that those five generators listed are, up to signs, the five generators of the commutator ideal. In general the number of generators of the Fitting ideal that we obtain by taking all the prescribed minors, will be far more than the number of generators of the commutator ideal.
Remark 5.10. A technical comment. If one starts with a free A-module E with basis {1, t 1 , t 2 }, and require E to be a quotient module of A t 1 , t 2 , one obtains the matrices The second column in T 2 is here not the same as the third column of T 1 , as was the case in the above example. Backtracking gives now a graded left ideal J = (F 1 , F 2 , F 3 , F 4 ) ⊆ A[z] x 1 , x 2 generated by four elements where F 1 , F 2 and F 3 are as above, but where In general we will not have c(F 3 ) equal to c(F 4 ) via the canonical map c : And then we will not have that the quotient module E = A[z, x 1 , x 2 ]/J = c(F 1 ), c(F 2 ), c(F 3 ), c(F 3 ) will be free in degree 2. In particular we see that the condition (3) of Proposition 5.7 does not imply condition (2).
6. Apolarity schemes 6.1. Cones. Let A[z, x 1 , . . . , x n ] be the homogeneous coordinate ring of projective n-space P n A over A. Let Γ ⊆ P n A be a closed subscheme, and let I denote its corresponding ideal sheaf. The affine cone is the closed subscheme C Γ ⊆ A n+1 A defined by the ideal m≥0 H 0 (I (m)) ⊆ A[z, x 1 , . . . , x n ].
6.2. Apolarity. Let Z ⊆ A n+1 A be a closed subscheme. Inspired by [RS00] we say that a finite subscheme Γ ⊆ P n A is apolar to Z if the affine cone of Γ contains Z as a subscheme, that is A . We let VPS N Z (A) denote the set of closed subschemes Γ ⊆ P n A that are finite, flat, and of relative rank N , and such that Γ is apolar to Z. Then VPS N Z naturally becomes a subfunctor of the Hilbert functor Hilb N P n A parametrizing closed subschemes in P n A that are flat, finite, and of relative rank N .
Remark 6.3. We do not impose the reduced structure on VPS N Z , a condition that is assumed in [RS00], [RS13]. Note also that our definition of apolarity is slightly more general than the one given in [RS00].
Remark 6.4. In [RS00], the authors define VPS as the variety of apolar schemes and VSP as the variety of sums of powers. In [RS13] the notation appears to have changed with VPS for the variety of sums of powers and VAPS for the variety of apolar schemes. We use the notation from the first paper.
Proposition 6.5. Let I ⊆ A[z, x 1 , . . . , x n ] be an ideal generated by homogeneous elements, and let Z ⊆ A n+1 Proof. The scheme Z(F ) ⊆ A n+1 k defined by the ideal F ⊥ ⊆ k[z, x 1 , . . . , x n ] is finite, and flat since the base is a field. By definition the ideal F ⊥ is generated by homogeneous elements of degree two, and it follows by Proposition 6.5 that VPS n+1 Z(F ) is representable by a scheme. Let I denote the ideal sheaf of a flat family Γ ⊆ P n A of relative rank n + 1 over a k-algebra A. Assume that the A-valued point of the Hilbert scheme Hilb n+1 P n is an A-valued point of VPS n+1 Z(F ) . Then we have an inclusion of ideals In particular we have that H 0 (I (0)) = H 0 (I (1)) = 0. Moreover, since the quadratic hyperplane in P n determined by F is non-singular, it follows that H 0 (I (2)) contains no "fake" hyperplanes either. That is, the ideal does not contain degree 2 elements of the form lz, lx 1 , . . . , lx n , for some linear form l in k[z, x 1 , . . . , x n ]. That means that for any point in Spec(A) the fiber of the universal family is non-degenerate, hence the whole family over Spec(A) is a non-degenerate family. Thus, Spec(A) is an A-valued point of the scheme Gen from Proposition 4.7. The scheme Gen is a locally closed subscheme of the Hilbert scheme Hilb n+1 P n and Proposition 4.7 tells us that Gen is embedded in the Grassmannian G n+1 (V 2 ) of rank n + 1 quotients of the A-module of two-forms. As VPS n+1 Z(F ) is closed in the Hilbert scheme, but also closed in the space Gen, it follows that it is closed in the Grassmannian.
Remark 6.8. Proposition 6.7 is stated in [RS13, Corollary 2.2]. Their result is, in parts, based on the computation of the graded Betti numbers of nondegenerate length n + 1 subschemes in P n k . Our proof is perhaps more transparent. In any case our result concerns an embedding of the scheme VPS n+1 Z(F ) with its possibly non-reduced structure.
Appendix A. Strongly generated quotient sheaves We restate several of our previous results in a more general setting with modules and Quot schemes replacing ideals and Hilbert schemes. Standard basic opens of the Quot scheme were introduced in [GLS07b].
A.1. Monomial bases. Let S = A[t 1 , . . . , t n ] be a polynomial ring over a commutative unitary ring A and let F = m i=1 Se i be a free S-module. By a monomial in F , we mean an element of the form λe i , with λ a monomial in the polynomial ring S. Let β be a finite collection of monomials in F . We assume that β has the property that it contains all its divisors: for any λe i ∈ β, it holds that (A.1.1) λe i = t d i λ e i ⇒ t d−1 i λ e i ∈ β, for any i = 1, . . . , n and any d ≥ 1.
Lemma A.2. Let R ⊆ F = m i=1 k[t 1 , . . . , t n ] be a submodule where k is a field, such that the quotient F/R is a finite dimensional k-vector space. Then there exists a sequence β of monomials satisfying (A.1.1) such that the images of the elements of β form a k-vector basis for the quotient module.
Proof. This is a consequence of F/R being a k[t 1 , . . . , t n ]-module. Proposition A.3. Let S = A[t 1 , . . . , t n ] be a polynomial ring over a commutative unitary ring A and let F = m i=1 Se i be a free S-module. Let R ⊆ F be a submodule such that F/R is free of finite rank as an A-module, and let β be a sequence of monomials in F , satisfying (A.1.1), that form an A-basis for F/R. Then we have that R is generated in degree d(β) + 1, where d(β) is the top degree of the monomials.