COMPUTING STRUCTURE CONSTANTS FOR RINGS OF FINITE RANK FROM MINIMAL FREE RESOLUTIONS

TOM FISHER AND LAZAR RADIČEVIĆ

Abstract. We show how the minimal free resolution of a set of n points in general position in projective space of dimension n − 2 explicitly determines structure constants for a ring of rank n. This generalises previously known constructions of Levi-Delone-Faddeev and Bhargava in the cases n = 3, 4, 5.

1. Introduction

A classical construction, known as the Levi-Delone-Faddeev correspondence [Lev14, DF64] (see also [GGS02]), shows that a binary cubic form

f(x, y) = ax^{3} + bx^{2}y + cxy^{2} + dy^{3}

naturally determines a ring of rank 3. Explicitly, if ξ is a symbol formally satisfying f(ξ, 1) = 0 then ω = aξ and θ = −dξ^{−1} satisfy the relations

(1)  \begin{aligned}ω^{2} & = −ac − bω + aθ, \\ ωθ & = −ad, \\ θ^{2} & = −bd − dω + cθ. \end{aligned}

These relations may be used to define a commutative and associative multiplication on the free module with basis 1, ω, θ. The construction works over any base ring, and gives a discriminant preserving bijection between equivalence classes of binary cubic forms and isomorphism classes of rings of rank 3. This construction was extended to rings of rank 4 and 5 by Bhargava [Bha04, Bha08], who considered pairs of quadratic forms in 3 variables, and 5 × 5 alternating matrices of linear forms in 4 variables.

We describe an extension to rings of rank n for any integer n ⩾ 3. Our main result (Theorem 2.8) shows how a set of n points X ⊂ ℙ^{n−2} in general position determines, by means of an explicit construction involving the minimal free resolution of X, structure constants for an algebra A of rank n. In particular we show that this algebra A is isomorphic to the coordinate ring of X.

We should say straight away that we are not expecting to fully generalise Bhargava’s work, and count number fields of degree n > 5. Instead our motivation comes from the study of genus one curves. If C ⊂ ℙ^{n−1} is a genus one curve of degree n, embedded by a complete linear system, then a generic hyperplane section of C will be a set of n points in general position. The first author showed in [Fis18] how to associate to such a curve C a matrix of quadratic forms Ω describing the invariant differential. One application of Theorem 2.8 is that the associative law then determines some of the equations defining the space of all such Ω’s. Another application is given by the second author in his PhD thesis [Rad21] where for E/ℚ an elliptic curve and n ⩾ 2 an integer, he gives a simple bound on the least discriminant of a degree n number field over which each element of order n in the Tate Shafarevich group of E capitulates.

Date: 16th September 2021.

Having mentioned these applications to the study of curves, in the rest of this article we only consider finite sets of points in projective space.

For the statement of Theorem 2.8 we work over a field K of characteristic 0. However, examination of the proofs shows that all we need is that the characteristic does not divide 2n. In Section 3 (with the main proof postponed to Section 10) we describe a slightly more complicated variant of our construction that works in all characteristics. It is this construction that reduces (in the cases n = 3, 4, 5) to the earlier work of Levi-Delone-Faddeev and Bhargava. It also gives better bounds in [Rad21, Theorem 1.0.1].

In Section 4 we review the connection between non-degenerate algebras of dimension n and sets of n points in ℙ^{n−2} in general position. Then, as explained in Section 5, the proof of Theorem 2.8 comes down to (i) checking our construction of the structure constants behaves well under all changes of co-ordinates, and (ii) checking that the theorem holds for the standard set of n points:

(1 : 0 : …: 0), (0 : 1 : 0 : …: 0), …, (0 : …: 0 : 1), (1 : 1 : …: 1).

We give the proof of (i) in Sections 6 and 7. We may check (ii) for any given n by computer algebra. We give a proof that works for all n in Sections 8 and 9, using an explicit description of the minimal free resolution due to Wilson [Wil13].

Acknowledgements. This article is based on part of the second author’s PhD thesis. We thank Manjul Bhargava and Melanie Wood for useful conversations, and Jack Thorne for alerting us to an oversight in an earlier version of Section 4.

2. Statement of the main theorem

We recall a few basic notions from commutative algebra that will be needed to state our main theorem. Throughout, we work over a field K with algebraic closure K̄. Let R = K[x_{1}, …, x_{m}] be the polynomial ring with its usual grading. For M = ⊕_{d}M_{d} a graded R-module, we write M(c) = ⊕_{d}M_{c+d} for the graded R-module with grading shifted by c. A direct sum of modules of the form R(c) is a called a graded free R-module.

Definition 2.1. A graded free resolution of a graded R-module M is a chain complex F_{•} of graded free R-modules

F_{r} ⟶^{ϕ_{r}} F_{r−1} ⟶^{ϕ_{r−1}} … ⟶^{ϕ_{2}} F_{1} ⟶^{ϕ_{1}} F_{0},

that is exact in degree > 0, and has H_{0}(F_{•}) = F_{0}/ϕ_{1}(F_{1}) ≅ M. Let 𝔪 = (x_{1}, …, x_{m}) be the maximal homogeneous ideal of R. We say a resolution F_{•} is minimal if we have ϕ_{k}(F_{k}) ⊂ 𝔪F_{k−1} for every k ⩾ 1.

Our interest is in the case M ≅ R/I, where I is a homogeneous ideal in R.

Remark 2.2. The minimal free resolution of a module is unique up to an isomorphism of chain complexes. Any such isomorphism consists of changes of bases for the free R-modules F_{k} in the resolution, see [Eis13, Theorem 20.2].

We require that the maps ϕ_{k} respect the grading of the modules. For example, a homomorphism of modules R(−m) → R is defined by multiplication by an element f ∈ R, and this is a graded homomorphism if and only if f is homogeneous of degree m (or zero). By choosing bases for each module F_{k} in the resolution, we may represent the maps ϕ_{k} as matrices of homogeneous polynomials. By abuse of notation we also write ϕ_{k} for these matrices.

Remark 2.3. The condition that the resolution is minimal means that every non-zero entry of every matrix has positive degree. By Nakayama’s lemma, this is equivalent to requiring that ϕ_{k} takes the basis of F_{k} to a minimal set of generators for the kernel of ϕ_{k−1}, see [Eis05, Corollary 1.5]. This characterisation makes it clear that every finitely generated graded module admits a minimal free resolution.

A minimal free resolution of an ideal contains the data of a set of generators for the ideal, the data of all relations (syzygies) that these generators satisfy, the data of relations that these relations satisfy, and so on iteratively. We illustrate this in the following example.

Example 2.4. Let X be the set of four points (1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1), (1 : 1 : 1) in ℙ^{2}. The homogeneous ideal I of X in R = K[x_{1}, x_{2}, x_{3}] is generated by the quadratic forms A = x_{1}(x_{2} − x_{3}) and B = x_{2}(x_{1} − x_{3}). For the first step of the resolution, we can take F_{0} = R, F_{1} = R(−2)^{2}, and let ϕ_{1} : R(−2)^{2} ⟶ R be the map represented by the row matrix (A, B), so that coker(ϕ_{1}) ≅ R/I. To compute the second step, we observe that A and B satisfy the relation B · A + (−A) · B = 0. Furthermore, any equation f · A + g · B = 0 is obtained by multiplying this relation by some r ∈ R, i.e., we have f = r · B and g = −r · A. We now take F_{2} = R(−4), and let ϕ_{2} : R(−4) ⟶ R(−2)^{2} be the map represented by the column matrix (B, −A)^{T}. Since this map is injective, this is where the resolution stops. We obtain the chain complex

(2)  0 ⟶ R(−4) ⟶^{(B,−A)^{T}} R(−2)^{2} ⟶^{(A,B)} R ⟶ 0

which is exact at the middle term and on the left, and is hence the minimal free resolution of R/I.

More generally we consider sets of points of the following form.

Definition 2.5. A zero dimensional variety X ⊂ ℙ^{n−2} defined over K is a set of n points in general position, if X has degree n and the set of geometric points X(K̄) consists of n points in general position, meaning that no subset of X(K̄) of size n − 1 is contained in a hyperplane.

Theorem 2.6. Let n ⩾ 4, and let R = K[x_{1}, …, x_{n−1}] be the coordinate ring of ℙ^{n−2}. Let X ⊂ ℙ^{n−2} be a set of n points in general position. Let I ⊂ R be the homogeneous ideal of X. Then I is (arithmetically) Gorenstein, and the minimal free resolution F_{•} of R/I takes the form

\begin{aligned}0 ⟶ R(−n) ⟶^{ϕ_{n−2}} R(−n + 2)^{b_{n−3}} & ⟶^{ϕ_{n−3}} R(−n + 3)^{b_{n−4}} ⟶^{ϕ_{n−4}} … \\ & … ⟶^{ϕ_{3}} R(−3)^{b_{2}} ⟶^{ϕ_{2}} R(−2)^{b_{1}} ⟶^{ϕ_{1}} R ⟶ 0, \end{aligned}

where the Betti numbers are given by b_{i} = n\binom{n−2}{i} − \binom{n}{i+1}.

Proof. The minimal free resolution is as described in [Wil13, Theorem 138], and the references cited there. For the statement that I is Gorenstein see [Wil13, Corollary 140]. □

We note that ϕ_{1} and ϕ_{n−2} are represented by matrices of quadratic forms, while the maps ϕ_{i}, for 1 < i < n − 2, are represented by matrices of linear forms.

Definition 2.7. The resolution F_{•} determines the following quadratic forms in x_{1}, …, x_{n−1}. (i) For 1 ⩽ a_{1}, a_{2}, …, a_{n−2} ⩽ n − 1 we define

[a_{1}, a_{2}, …, a_{n−2}]_{F_{•}} = \frac{∂ϕ_{1}}{∂x_{a_{1}}} \frac{∂ϕ_{2}}{∂x_{a_{2}}} ⋯ \frac{∂ϕ_{n−2}}{∂x_{a_{n−2}}} ,

where the partial derivative of a matrix is the matrix of partial derivatives of its entries, and the product is matrix multiplication.

(ii) Let σ be the (n − 2)-cycle (12 … n − 2) in the symmetric group S_{n−2}. We define

[[a_{1}, a_{2}, …, a_{n−2}]]_{F_{•}} = ∑_{k=1}^{n−2}[a_{σ^{2k}(1)}, a_{σ^{2k}(2)}, …, a_{σ^{2k}(n−2)}]_{F_{•}}.

(iii) For 1 ⩽ j ⩽ n − 1 we define Ω_{j} = (−1)^{j}[[1, 2, …, ĵ, …, n − 1]]_{F_{•}}. The choice of resolution will usually be fixed, and we therefore drop the subscripts F_{•}.

For our main result we work over a field K of characteristic 0.

Theorem 2.8. Let X ⊂ ℙ^{n−2} be a set of n points in general position, and let Ω_{1}, …, Ω_{n−1} be the quadratic forms associated to a minimal free resolution of X. Then there exists a commutative and associative K-algebra A, of dimension n, and a K-basis 1 = α_{0}, α_{1}, …, α_{n−1} for A, such that for each 1 ⩽ i, j ⩽ n − 1 we have

α_{i}α_{j} = c_{ij}^{0} + ∑_{k=1}^{n−1} \frac{∂^{2}Ω_{k}}{∂x_{i}∂x_{j}} α_{k},

for some constant c_{ij}^{0} ∈ K. Moreover A is isomorphic to the affine coordinate ring (i.e., ring of global functions) of X, and the α_{i} for 1 ⩽ i ⩽ n − 1 span the trace zero subspace.

Remark 2.9. Following [Bha08, page 68] we can use the associative law to solve for the c_{ij}^{0}. Explicitly, for any 1 ⩽ i, j, k ⩽ n − 1 with i ≠ k, comparing coefficients of α_{k} in α_{i}(α_{j}α_{k}) = (α_{i}α_{j})α_{k} gives

c_{ij}^{0} = ∑_{r=1}^{n−1} (\frac{∂^{2}Ω_{r}}{∂x_{j}∂x_{k}} \frac{∂^{2}Ω_{k}}{∂x_{r}∂x_{i}} − \frac{∂^{2}Ω_{r}}{∂x_{i}∂x_{j}} \frac{∂^{2}Ω_{k}}{∂x_{r}∂x_{k}} ) .

Remark 2.10. The construction of [a_{1}, a_{2}, …, a_{n−2}], and hence of Ω_{1}, …, Ω_{n−1}, is independent of the choice of basis for the free R-modules in the resolution F_{•}, except for the leftmost module R(−n). The quadratic forms Ω_{1}, …, Ω_{n−1} are therefore uniquely determined up to multiplying through by an overall scalar. It is clear that this gives an isomorphic K-algebra.

3. Constructing orders in number fields

In this section we explain the connection between Theorem 2.8 and the previously known constructions due to Levi-Delone-Faddeev and Bhargava for n = 3, 4, 5. Whereas we work with algebras over a field of characteristic zero, the latter constructions work for rings of rank n, i.e., algebras over ℤ. We discuss to what extent this earlier work generalises to larger n.

Let A be an n-dimensional commutative K-algebra with K-basis 1, α_{1}, …, α_{n−1}. The structure constants c_{ij}^{k} for 1 ⩽ i, j, k ⩽ n − 1 are determined by

α_{i}α_{j} = c_{ij}^{0} + ∑_{k=1}^{n−1} c_{ij}^{k}α_{k}.

As noted in Remark 2.9, the c_{ij}^{0} may be recovered from the other structure constants using the associative law. We say that bases 1, α_{1}, …, α_{n−1} and 1, β_{1}, …, β_{n−1} differ by a shear if β_{i} = α_{i} + λ_{i} · 1 for some λ_{1}, …, λ_{n−1} ∈ K.

When n = 3 the algebra constructed by Levi-Delone-Faddeev (as defined by (1) in the introduction) is uniquely determined, up to shear, by

(3)  c_{11}^{2} = a,  c_{11}^{1} − 2c_{12}^{2} = −b,  c_{22}^{2} − 2c_{12}^{1} = c,  c_{22}^{1} = −d.

To compare with the algebra in Theorem 2.8 we consider the minimal free resolution

0 ⟶ R(−3) ⟶^{f} R ⟶ 0

where f(x_{1}, x_{2}) = ax_{1}^{3} + bx_{1}^{2}x_{2} + cx_{1}x_{2}^{2} + dx_{2}^{3}. Using Definition 2.7, we compute

Ω_{1} = −[[2]] = −[2] = −\frac{∂f}{∂x_{2}} = −bx_{1}^{2} − 2cx_{1}x_{2} − 3dx_{2}^{2}

and

Ω_{2} = [[1]] = [1] = \frac{∂f}{∂x_{1}} = 3ax_{1}^{2} + 2bx_{1}x_{2} + cx_{2}^{2}.

From this it is easy to check that

c_{ij}^{k} = \frac{1}{6} \frac{∂^{2}Ω_{k}}{∂x_{i}∂x_{j}}

is a solution to (3).

When n ⩾ 4 the algebra is uniquely determined, up to shear, by the linear combinations of structure constants in the left hand column of Table 3.1, where i, j, k range over all triples of distinct integers with 1 ⩽ i, j, k ⩽ n − 1. These linear combinations appear, with what we believe is a type error, in [Bha08, Equation (21)]. The remaining columns are explained below.

Table 3.1. Structure constants for rings of rank n (up to shear)

n = 4  n = 5  n ⩾ 4
c_{ij}^{k}  ±{jjii} ±{iiℓjj} ±{i, i, 1, 2, … î, …, ĵ, …, k̂, …, n − 1, j, j}
c_{ii}^{j} ±{iiik} ±{ℓiiik} ±{i, i, 1, …, î, …, ĵ, …, n − 1, i}
c_{ij}^{j} − c_{ik}^{k}  ±{iijk} ±{jkℓii} ±{i, i, 1, …, î, …, ĵ, …, k̂, …, n − 1, j, k}
c_{ii}^{i} − c_{ij}^{j} − c_{ik}^{k} ±{ikij} ±{ijℓki} ±{i, j, 1, …, î, …, ĵ, …, k̂, …, n − 1, k, i}

Let F_{•} be a minimal free resolution of a set of n points in general position, with differentials ϕ_{1}, …, ϕ_{n−2} represented by matrices of linear and quadratic forms. We write

ϕ_{1} = ∑_{i⩽j} P (i, j)x_{i}x_{j} and ϕ_{n−2} = ∑_{i⩽j} Q(i, j)x_{i}x_{j},

where the P (i, j) are row vectors, and the Q(i, j) are column vectors. We then define

\{a_{1}a_{2} … a_{n}\} := P (a_{1}, a_{2})\frac{∂ϕ_{2}}{∂x_{a_{3}}} ⋯ \frac{∂ϕ_{n−3}}{∂x_{a_{n−2}}} Q(a_{n−1}, a_{n}).

When n = 4 the minimal free resolution F_{•} takes the form (2) where A and B are ternary quadratic forms. The symbols {ijkℓ} were denoted λ_{kℓ}^{ij} in [Bha04, Section 3.2]. The structure constants in loc. cit., up to shear, are then as recorded in Table 3.1, where ± denotes the sign of the permutation taking 1, 2, 3 to i, j, k.

When n = 5 the structure theorem of Buchsbaum and Eisenbud [BE82] for Gorenstein ideals of codimension 3 shows that the minimal free resolution takes the form

0 ⟶ R(−5) ⟶^{P^{T}} R(−3)^{5} ⟶^{Φ} R(−2)^{5} ⟶^{P} R ⟶ 0,

where Φ is a 5×5 alternating matrix of linear forms, and P is the (signed) row vector of 4×4 Pfaffians of Φ. Our symbols {ijkℓm} differ only by some factors of 2 from those defined in [Bha08, Section 4]. The structure constants in loc. cit., up to shear, are again as recorded in Table 3.1, where ± denotes the sign of the permutation taking 1, 2, 3, 4 to i, j, k, ℓ.

The expressions we give in the right hand column of Table 3.1 are new.

Theorem 3.1. Let n ⩾ 4 be an integer. Then the structure constants

c_{ij}^{k} = \frac{1}{2n} \frac{∂^{2}Ω_{k}}{∂x_{i}∂x_{j}}

satisfy the system of equations in Table 3.1.

Proof. It is clear from the definition that the symbol {⋯ } does not depend on the order of its first two arguments, or the order of its last two arguments.

If n = 4 then by (2) we have [i, j] = −[j, i] and {ijkℓ} = −{kℓij}. Let i, j, k be an even permutation of 1, 2, 3. By Definition 2.7 we have Ω_{k} = −2[i, j]. Using the product rule we compute

\frac{∂^{2}[j, i]}{∂x_{i}∂x_{j}} = 4\{jjii\} + \{ijij\} = 4\{jjii\},
\frac{∂^{2}[i, k]}{∂x_{i}^{2}} = 2\{iiik\} + 2\{iiik\} = 4\{iiik\},
\frac{∂^{2}[i, k]}{∂x_{i}∂x_{j}} + \frac{∂^{2}[i, j]}{∂x_{i}∂x_{k}} = 2\{iijk\} + \{ijik\} + 2\{iijk\} + \{ikij\} = 4\{iijk\},
\frac{∂^{2}[k, j]}{∂x_{i}^{2}} + \frac{∂^{2}[k, i]}{∂x_{i}∂x_{j}} + \frac{∂^{2}[i, j]}{∂x_{i}∂x_{k}} = 2\{ikij\} + \{ikij\} + 2\{jkii\} + 2\{iijk\} + \{ikij\} = 4\{ikij\}.

This proves the theorem in the case n = 4. We give the proof for n ⩾ 5, and specify the correct choice of signs ±, in Theorem 10.1. It may also be checked, using Lemma 10.2, that the expressions in Table 3.1 for n ⩾ 4 do indeed specialise to those in the previous two columns when n = 4 and n = 5. □

We have now checked that Theorem 2.8 agrees, up to a shear, with the previously known constructions for n = 3, 4, 5. However the structure constants do not agree exactly, since in Theorem 2.8 the basis elements α_{1}, …, α_{n−1} are normalised (up to shear) so that they have trace zero, whereas the algebras in [DF64, Bha04, Bha08] are normalised so that

\begin{matrix}n = 3 & c_{12}^{1} = c_{12}^{2} = 0, \\ n = 4 & c_{12}^{1} = c_{12}^{2} = c_{13}^{1} = 0, \\ n = 5 & c_{12}^{1} = c_{12}^{2} = c_{34}^{3} = c_{34}^{4} = 0. \end{matrix}

In general we could normalise our basis by choosing a convention such as

(4)  c_{12}^{2} = c_{23}^{3} = c_{34}^{4} = … = c_{n−2,n−1}^{n−1} = c_{n−1,1}^{1} = 0,

or when n is odd

(5)  c_{12}^{1} = c_{12}^{2} = c_{34}^{3} = c_{34}^{4} = … = c_{n−2,n−1}^{n−2} = c_{n−2,n−1}^{n−1} = 0.

With either convention, it is clear that there is a unique way to modify our basis by a shear so that it satisfies the convention.

Compared to normalising α_{1}, …, α_{n−1} to have trace zero, these conventions break symmetry, but have the advantage of working in all characteristics. They are also useful for constructing orders in number fields as we now explain.

We take K = ℚ and suppose that the minimal free resolution F_{•} has integer coefficients, i.e. the differentials ϕ_{k} are represented by matrices of polynomials in ℤ[x_{1}, …, x_{n−1}]. By Definition 2.7 the structure constants for the ℚ-algebra A in Theorem 2.8 are integral, and so determine an order B ⊂ A. This ring decomposes (as a ℤ-module) as B = ℤ ⊕ B_{0} where B_{0} is the subset of elements of trace zero. Therefore Tr_{A/ℤ}(B) = nℤ, and so any prime dividing n necessarily ramifies in B. More generally, this order can never be maximal at the primes dividing 2n. Indeed, if we choose our structure constants using Table 3.1 and one of the normalisation conventions (4) or (5), then these define an order B′ with B ⊂ B′ ⊂ A. From the factor 2n in the statement of Theorem 3.1 we see that the index of B in B′ is (2n)^{n−1}. For n = 3, 4, 5, the ring B′ is the one constructed in [DF64, Bha04, Bha08]. For general n, working with B′ rather than B gives a larger order with smaller discriminant, and hence a sharper bound in [Rad21, Theorem 1.0.1].

Remark 3.2. We briefly mention three respects in which the theory for n = 3, 4, 5, as described in [GGS02, Bha04, Bha08], is still more developed than that for general n.

(i) We conjecture than any order in an étale ℚ-algebra of rank n is necessarily of the form B′ for some minimal free resolution F_{•} with integer coefficients. This is known for n = 3, 4, 5. It is also true for the ring ℤ × ℤ × … × ℤ by the calculations in Sections 8 and 9. We hope to investigate this conjecture further in future work. (ii) For n = 3, 4, 5 the binary cubics, pairs of ternary quadratics, and alternating matrices of linear forms, parameterise all rings of rank n, including degenerate rings such as ℤ[x]/(x^{n}). It would be interesting to determine if there is a suitable class of “degenerate” minimal free resolutions for n > 5 that correspond to these rings.

(iii) The results for n = 3, 4, 5 have been used by Davenport and Heilbronn [DH71] and Bhargava [Bha05, Bha10] to give an asymptotic count of number fields of degree n ordered by discriminant. Since we do not have a description of the space of minimal free resolutions that lends itself to the counting arguments used in the geometry of numbers, it remains a difficult problem to extend these results to n > 5.

4. Points in general position and étale algebras

In this section we give a slightly different perspective on the classical fact that there is an equivalence of categories between the category of finite sets of points with a continuous action of the absolute Galois group, and the category of finite dimensional étale algebras. These ideas feature prominently in works of Bhargava, see especially the discussion in [Bha08, page 59], as well as the work of his students Wood [Woo09] and Wilson [Wil13].

We fix an integer n ⩾ 3. Let X = {X ⊂ ℙ^{n−2} : X is a set of n points in general position}. Note that X (K) consists of sets X which are defined over K, viewed as zero-dimensional varieties, but the individual (geometric) points of X need not be defined over K. The group PGL_{n−1}(K) acts on X (K) by changes of coordinates.

An n-dimensional commutative K-algebra A is non-degenerate if the trace form associated to A is non-degenerate. This is equivalent to requiring that A is étale over K, or that there exists an isomorphism A ⊗_{K} K̄ ≅ K̄^{n}. For example, the ring A = Γ(X, O_{X}) of global functions on a set of n points X ∈ X (K) is a non-degenerate K-algebra of dimension n.

The following fact seems to be well-known, but we could not find an adequate reference.

Proposition 4.1. Let A be the set of isomorphism classes of non-degenerate n-dimensional K-algebras. Then the map X ↦ Γ(X, O_{X}) induces a bijection between the set of PGL_{n−1}(K)- orbits of X (K) and the set A.

Any two elements of X (K) that lie in the same PGL_{n−1}(K)-orbit have isomorphic rings of global functions, and so map to the same element of A. Therefore the map in Proposition 4.1 is well defined. We must show it is a bijection. First we need two lemmas.

Lemma 4.2. (i) Let A be a non-degenerate n-dimensional K-algebra. Let M be a locally free A-module of rank 1. Then M is free, i.e., it is isomorphic to A as an A-module.

(ii) Let X ⊂ ℙ^{n−2} be a set of n distinct points, and let A = Γ(X, O_{X}). Then X is the image of a map Spec A → ℙ^{n−2} given by (α_{1} : …: α_{n−1}) for some α_{1}, …, α_{n−1} ∈ A.

Proof. (i) Since A is non-degenerate it is isomorphic as an algebra to a direct product of fields, say, A ≅ A_{1} × ⋯ × A_{k}. For each 1 ⩽ i ⩽ k, let e_{i} be the idempotent corresponding to the factor A_{i}, so that ∑_{i=1}^{k} e_{i} = 1, Ae_{i} ≅ A_{i} as an A-module, e_{i}^{2} = e_{i} and e_{i}e_{j} = 0 for all i ≠ j. Then we have the decomposition M = e_{1}M ⊕ e_{2}M ⊕ ⋯ ⊕ e_{k}M, where each module e_{i}M is an A_{i}-vector space. Since M is locally free of rank 1, each e_{i}M is 1-dimensional, and so we may choose a basis vector f_{i}. Then M = Af ≅ A, where f = ∑_{i=1}^{k} f_{i}. (ii) The embedding of X = Spec A in ℙ^{n−2} is determined by global sections ℓ_{1}, …, ℓ_{n−1} belonging to the A-module M = Γ(X, O_{X}(1)). We see by (i) that M is a free A-module of rank 1, say generated by m ∈ M. We then write ℓ_{i} = α_{i}m for some α_{i} ∈ A. □

Lemma 4.3. Let A be a non-degenerate n-dimensional K-algebra. Let X be the image of the map Spec A → ℙ^{n−2} given by (α_{1} : …: α_{n−1}) for some α_{1}, …, α_{n−1} ∈ A. Then X is a set of n points in general position if and only if there exists a unit λ ∈ A^{×} such that λα_{1}, …, λα_{n−1} is a K-basis for the trace zero subspace of A. Moreover, if such a λ exists then it is unique up to multiplication by an element of K^{×}.

Proof. We may assume that α_{1}, …, α_{n−1} are linearly independent over K, since otherwise it is clear that neither condition is satisfied.

The uniqueness follows from the non-degeneracy of the trace form. For existence, it is clear by linear algebra over K that there exists non-zero λ ∈ A such that Tr_{A/K}(λα_{j}) = 0 for all 1 ⩽ j ⩽ n − 1. It remains to show that λ is a unit.

Since A is isomorphic to a product of fields, we may write any element of A as a unit times an idempotent. It therefore suffices to consider λ an idempotent. Writing σ_{1}, …, σ_{n} for the distinct K-algebra homomorphisms A → K̄ we have

X(K̄) = {(σ_{i}(α_{1}) : …: σ_{i}(α_{n−1})) : 1 ⩽ i ⩽ n} ⊂ ℙ^{n−2}(K̄).

Since λ is an idempotent we may order the σ_{i} such that (σ_{1}(λ), …, σ_{n}(λ)) = (1, …, 1, 0, … 0) where there are (say) m ones and n − m zeros. Then ∑_{i=1}^{m} σ_{i}(α_{j}) = Tr_{A/K}(λα_{j}) = 0 for all 1 ⩽ j ⩽ n − 1. Since X is in general position, this forces m = n, and so λ = 1 as required. □

Proof of Proposition 4.1. First, to prove surjectivity, we suppose that A is a non-degenerate n-dimensional K-algebra. Then we pick α_{1}, …, α_{n−1} a K-basis for the trace zero subspace of A, and let X be the image of the map Spec A → ℙ^{n−2} given by (α_{1} : …: α_{n−1}). Lemma 4.3 shows that X is in general position, and we then have X ↦ A. Next, to prove injectivity, we suppose that X_{1} and X_{2} both map to A. By Lemmas 4.2 and 4.3 both X_{1} and X_{2} are embedded in ℙ^{n−2} using (possibly) different choices of bases for the trace zero subspace of A. Hence there exists an element of PGL_{n−1}(K) taking X_{1} to X_{2}. □

5. Overview of the proof

Let V = ⟨x_{1}, …, x_{n−1}⟩ be the space of linear forms on ℙ^{n−2}, and denote the dual basis of V^{∗} by x_{1}^{∗}, …, x_{n−1}^{∗}. The quadratic forms Ω_{1}, …, Ω_{n−1} determine an element Ω in V^{∗} ⊗ S^{2}V via the formula

(6)  Ω := ∑_{j=1}^{n−1} x_{j}^{∗} ⊗ Ω_{j}.

The following proposition, which we prove in Section 7, shows that the construction of Ω from the minimal free resolution is invariant under change of basis of V.

Proposition 5.1. Let F_{•} be a minimal free resolution for a set X ⊂ ℙ^{n−2} of n points in general position. We write x_{1}, …, x_{n−1} for our coordinates on ℙ^{n−2}. Let x′_{j} = ∑_{i=1}^{n−1} g_{ij}x_{i} for some g = (g_{ij}) ∈ GL_{n−1}. Writing ϕ_{1}, …, ϕ_{n−2} for the matrices representing the maps in the resolution F_{•}, let F′_{•} be the resolution whose maps are given by matrices ϕ′_{1}, …, ϕ′_{n−2} where

ϕ′_{r}(x_{1}, …, x_{n−1}) = ϕ_{r}(x′_{1}, …, x′_{n−1}).

Let Ω and Ω′ be the elements of V^{∗}⊗S^{2}V associated to the resolutions F_{•} and F′_{•} respectively. Then we have

Ω′ = (det g)(g · Ω),

where the action of g on Ω is the standard action of GL_{n−1} on V^{∗} ⊗ S^{2}V.

Our main theorem (Theorem 2.8) gives an expression for the structure constants of the algebra A in terms of the quadratic forms Ω_{1}, …, Ω_{n−1} determined by a minimal free resolution of X. In fact we prove the following strengthening of that theorem.

Theorem 5.2. Let X ⊂ ℙ^{n−2} be a set of n points in general position. Let A = Γ(X, O_{X}) be the coordinate ring of X and let V be the trace zero subspace of A.

(i) There is a K-basis α_{1}, …, α_{n−1} of V, unique up to multiplication by an overall scalar, such that the embedding of X = Spec A in ℙ^{n−2} is given by (α_{1} : …: α_{n−1}). (ii) Let α_{1}, …, α_{n−1} be as in (i) and let α_{0}^{∗}, α_{1}^{∗}, …, α_{n}^{∗} be the basis for A that is dual to 1, α_{1}, …, α_{n−1} with respect to the trace pairing (x, y) ↦ Tr_{A/K}(xy). If Ω_{1}, …, Ω_{n−1} are the quadratic forms determined by a minimal free resolution of X then there exist constants λ, c_{ij}^{0} ∈ K such that

(7)  α_{i}^{∗}α_{j}^{∗} = c_{ij}^{0} + λ ∑_{k=1}^{n−1} \frac{∂^{2}Ω_{k}}{∂x_{i}∂x_{j}} α_{k}^{∗},

for all 1 ⩽ i, j ⩽ n − 1.

Proof. (i) This follows from Lemmas 4.2 and 4.3. (ii) We claim we are free to make changes of coordinates on ℙ^{n−2}. This is proved by considering the effect of such a change of coordinates on each term in (7). We may organise this calculation as follows. First we use the trace pairing to identify A = (K · 1) ⊕ V^{∗}. Then multiplication in A determines a symmetric bilinear map V^{∗} × V^{∗} ⟶ V^{∗} and hence an element of V^{∗} ⊗S^{2}V. Next we use part (i) of the theorem to identify V with the space of linear forms on ℙ^{n−2}. The quadratic forms Ω_{1}, …, Ω_{n−1} determine an element Ω in V^{∗} ⊗ S^{2}V via (6). The theorem asserts that these two elements of V^{∗} ⊗ S^{2}V are equal, up to multiplication by a scalar. To prove our claim it suffices to show that these two elements transform, under a change of coordinates, according to the natural action of GL(V) on V^{∗} ⊗ S^{2}V. In the first case this is clear from the construction, and in the second case we use Proposition 5.1.

We are also free to extend our base field K, and so may assume by a change of coordinates that X is the standard set of n points in general position given by P_{1} = (1 : 0 …: 0), P_{2} = (0 : 1 : 0 …: 0), …, P_{n−1} = (0 : …: 0 : 1) and P_{n} = (1 : 1 : …: 1). Having reduced to this special case, we identify A ≅ K^{n} via α ↦ (α(P_{1}), …, α(P_{n})). Then the K-basis for the trace zero subspace as determined in part (i) of the theorem is

α_{1} = \frac{1}{n}(1, 0, 0, …, 0, −1),
α_{2} = \frac{1}{n}(0, 1, 0, …, 0, −1),
⋮
α_{n−1} = \frac{1}{n}(0, 0, 0, …, 1, −1),

where at this stage the overall scaling by a factor 1/n is arbitrary, but has been chosen to simplify the calculations that follow. The multiplication on K^{n} is given by multiplication in each component separately, and the trace pairing on K^{n} is given by the standard dot product. Following the statement of part (ii) of the theorem, we compute:

α_{1}^{∗} = (n − 1, −1, −1, …, −1),
α_{2}^{∗} = (−1, n − 1, −1, …, −1),
⋮
α_{n−1}^{∗} = (−1, −1, …, n − 1, −1).

Then for 1 ⩽ i, j ⩽ n − 1, the multiplication is given by

α_{i}^{∗}α_{j}^{∗} = \{ \begin{matrix}−1 − α_{i}^{∗} − α_{j}^{∗} & if i ≠ j, \\ (n − 1) + (n − 2)α_{i}^{∗} & if i = j. \end{matrix}

Equation (7) with λ = 1/2 then follows from the next lemma. □

Lemma 5.3. Let X ⊂ ℙ^{n−2} be the standard set of n points in general position. Then the quadratic forms Ω_{i} are given, up to an overall scalar, by Ω_{i} = nx_{i}^{2} − 2x_{i} ∑_{j=1}^{n−1} x_{j} for all 1 ⩽ i ⩽ n − 1.

We checked this lemma for all n ⩽ 10 by computer algebra, and could easily extend this to some larger values of n. In Sections 8 and 9 we give a proof that is valid for all n.

6. Some symmetries

We prove some symmetries satisfied by the square bracket and double square bracket symbols (see Definition 2.7). For convenience in this section we put m = n − 2, since all our resolutions have length m.

Lemma 6.1. If 2 ⩽ r ⩽ m − 2 then

[a_{1}, …, a_{r}, a_{r+1}, …, a_{m}] = −[a_{1}, …, a_{r+1}, a_{r}, …, a_{m}].

Proof. For this range of r, both ϕ_{r} and ϕ_{r+1} are matrices of linear forms. We differentiate the relation ϕ_{r}ϕ_{r+1} = 0. By the Leibniz rule,

0 = \frac{∂^{2}(ϕ_{r}ϕ_{r+1})}{∂x_{a_{r}}∂x_{a_{r+1}}} = \frac{∂ϕ_{r}}{∂x_{a_{r}}} \frac{∂ϕ_{r+1}}{∂x_{a_{r+1}}} + \frac{∂ϕ_{r}}{∂x_{a_{r+1}}} \frac{∂ϕ_{r+1}}{∂x_{a_{r}}} ,

hence the desired relation. □

Remark 6.2. The lemma holds only for 2 ⩽ r ⩽ m − 2. Indeed, swapping the first two a’s or the last two a’s need not simply result in a sign change. See Lemma 8.1 for an explicit example. We introduce the symbols [[a_{1}, a_{2}, …, a_{m}]] to rectify this; see Lemma 6.5 below.

Lemma 6.3. We have [a_{1}, a_{2}, …, a_{m−1}, a_{m}] = ±[a_{m}, a_{m−1}, …, a_{2}, a_{1}], where the sign is +1 if m ≡ 0, 1 (mod 4) and −1 if m ≡ 2, 3 (mod 4).

Proof. The dual of an R-module M is M^{∗} = Hom_{R}(M, R). If an R-module map M → N between free R-modules is represented by a matrix ϕ (with respect to some bases) then the dual map N^{∗} → M^{∗} is represented by the transpose matrix ϕ^{T} (with respect to the dual bases).

We saw in Theorem 2.6 that X is Gorenstein. This implies that its minimal free resolution F_{•} is self-dual. Explicitly, there is a commutative diagram

[figure: 0; R; ϕ_{m}; ±1; F_{m−1}; …; F_{1}; ϕ_{1}; R; 0; 0; R; ϕ_{1}^{T}; F_{1}^{∗}; …; F_{m−1}^{∗}; ϕ_{m}^{T}; R; 0]

where the vertical maps are isomorphisms, the right most one is the identity map, and the left most one is multiplication by ±1. According to [BH98, page 123], the sign is + if and only if m ≡ 0, 1 (mod 4). The lemma now follows from the definition of the square brackets notation. □

Lemma 6.4. If the terms indicated by … are the same in each case then (i) [a, …, b] + [b, …, a] = 0, and

(ii) [a, b, …, c, d] + [b, a, …, c, d] + [a, b, …, d, c] + [b, a, …, d, c] = 0.

Proof. Part (i) follows from Lemmas 6.1 and 6.3. For the second part, as ϕ_{1}ϕ_{2} = 0 and ϕ_{m−1}ϕ_{m} = 0, we have

0 = \frac{∂^{2}(ϕ_{1}ϕ_{2})}{∂x_{a}∂x_{b}} = \frac{∂ϕ_{1}}{∂x_{a}} \frac{∂ϕ_{2}}{∂x_{b}} + \frac{∂ϕ_{1}}{∂x_{b}} \frac{∂ϕ_{2}}{∂x_{b}} + \frac{∂^{2}ϕ_{1}}{∂x_{a}∂x_{b}} ϕ_{2},

and similarly,

0 = \frac{∂^{2}(ϕ_{m−1}ϕ_{m})}{∂x_{c}∂x_{d}} = \frac{∂ϕ_{m−1}}{∂x_{c}} \frac{∂ϕ_{m}}{∂x_{d}} + \frac{∂ϕ_{m−1}}{∂x_{d}} \frac{∂ϕ_{m}}{∂x_{c}} + ϕ_{m−1} \frac{∂^{2}ϕ_{m}}{∂x_{c}∂x_{d}} .

Thus it suffices to show that

\frac{∂^{2}ϕ_{1}}{∂x_{a}∂x_{b}} ϕ_{2} \frac{∂ϕ_{3}}{∂x_{a_{3}}} … \frac{∂ϕ_{m−2}}{∂x_{a_{m−2}}} ϕ_{m−1} \frac{∂^{2}ϕ_{m}}{∂x_{c}∂x_{d}} = 0.

For 2 ⩽ r ⩽ m − 3 we have

ϕ_{r} \frac{∂ϕ_{r+1}}{∂x_{p}} = −\frac{∂ϕ_{r}}{∂x_{p}} ϕ_{r+1}.

We use this relation to move the undifferentiated term to the right until we get an expression involving ϕ_{m−2}ϕ_{m−1}, which vanishes. □

As in Definition 2.7, we let σ = (123 … m) ∈ S_{m} and define

[[a_{1}, …, a_{m}]] = ∑_{r=0}^{m−1}[a_{σ^{2r}(1)}, …, a_{σ^{2r}(m)}].

Lemma 6.5. For τ ∈ S_{m} we have [[a_{τ(1)}, …, a_{τ(m)}]] = sign(τ)[[a_{1}, …, a_{m}]].

Proof. For m ⩽ 3 the lemma follows easily from Lemma 6.4(i), so we may assume m ⩾ 4. We first prove the lemma in the case τ = (12). We have

[[a_{1}, a_{2}, a_{3}, …, a_{m}]] = [a_{1}, a_{2}, a_{3}, …, a_{m}] + [a_{3}, a_{4}, … a_{m}, a_{1}, a_{2}] + …

We consider the effect of switching a_{1} and a_{2} on each term on the right. For the terms we have not written out, the answer is that they change sign, and indeed this follows from Lemmas 6.1 and 6.4(i), the latter being used for the term [a_{2}, a_{3}, …, a_{m}, a_{1}] which only occurs if n is odd. We are left with the two terms we did write out. We treat them together. By repeatedly using Lemma 6.1 to move a_{3} to the right, we have

[a_{2}, a_{1}, a_{3}, …, a_{m}] = (−1)^{m}[a_{2}, a_{1}, a_{4}, …, a_{m−1}, a_{3}, a_{m}],

and similarly

[a_{3}, a_{4}, …, a_{m}, a_{2}, a_{1}] = −[a_{1}, a_{4}, …, a_{m}, a_{2}, a_{3}] = (−1)^{m}[a_{1}, a_{2}, a_{4}, …, a_{m−1}, a_{m}, a_{3}],

where the first equality is a consequence of Lemma 6.4(i) and the second is a repeated application of Lemma 6.1. By Lemma 6.4(ii) we have

(−1)^{m}([a_{2},a_{1}, a_{4}, …, a_{m−1}, a_{3}, a_{m}] + [a_{1}, a_{2}, a_{4}, …, a_{m−1}, a_{m}, a_{3}])
= (−1)^{m+1}([a_{1}, a_{2}, a_{4}, …, a_{m−1}, a_{3}, a_{m}] + [a_{2}, a_{1}, a_{4}, …, a_{m−1}, a_{m}, a_{3}]).

This is equal, by Lemma 6.4(i) and repeated application of Lemma 6.1, to

−[a_{1}, a_{2}, a_{3}, …, a_{m}] − [a_{3}, a_{4}, …, a_{m}, a_{1}, a_{2}].

This completes the proof in the case τ = (12).

It is immediate from the definition of [[…]] that the lemma holds with τ = σ^{2}. If n is odd then σ^{2} and (12) generate S_{m} and we are done. If n is even then S_{m} is generated by σ^{2}, (12) and (23). So it suffices to prove the lemma for τ = (23). However the proof in this case goes through term by term using Lemmas 6.1 and 6.4(i). □

Remark 6.6. If n is odd then using σ instead of σ^{2} would make no difference in the definition of [[…]]. However if n is even then this would give an expression that is identically zero, as may be seen from Lemma 6.5, noting that σ is an odd permutation.

7. Changes of coordinates

In this section we prove Proposition 5.1. Let V = ⟨x_{1}, …, x_{n−1}⟩ be the space of linear forms on ℙ^{n−2}, and let x_{1}^{∗}, …, x_{n−1}^{∗} be the dual basis for V^{∗}. We identify S^{d}V with the space of degree d homogeneous polynomials F in K[x_{1}, …, x_{n−1}]. The natural left actions of GL_{n−1} on S^{d}V and on V^{∗} are given by

(8)  (g · F)(x_{1}, …, x_{n−1}) = F (x′_{1}, …, x′_{n−1}),

where x′_{j} = ∑_{i=1}^{m} g_{ij}x_{i}, and

(9)  g · x_{j}^{∗} = ∑_{i=1}^{n−1}(g^{−T})_{ij}x_{i}^{∗}.

In Section 2 we defined quadratic forms [[…]] and Ω_{j} associated to a minimal free resolution with differentials ϕ_{r}. Fix any g ∈ GL_{n−1}. We now write [[…]]′ and Ω′_{j} for the quadratic forms associated to the minimal free resolution with differentials ϕ′_{r}(x_{1}, …, x_{m}) = ϕ_{r}(x′_{1}, …, x′_{m}), where x′_{j} = ∑_{i=1}^{m} g_{ij}x_{i}. We must prove that

(10)  ∑_{j=1}^{n−1} x_{j}^{∗} ⊗ Ω′_{j} = (det g) ∑_{j=1}^{n−1} (g · x_{j}^{∗}) ⊗ (g · Ω_{j}) .

It suffices to prove this claim for g running over a set of generators for the group GL_{n−1}, and accordingly we consider g a diagonal matrix, g a permutation matrix, and g a unipotent matrix.

First we suppose that g is diagonal, say with diagonal entries λ_{1}, …, λ_{n−1}. By the chain rule, we have

[[a_{1}, a_{2}, …, a_{n−2}]]′(x_{1}, …, x_{n−1}) = λ_{a_{1}} ⋯ λ_{a_{n−2}}[[a_{1}, a_{2}, …, a_{n−2}]](x′_{1}, …, x′_{n−1}).

Therefore

Ω′_{j}(x_{1}, …, x_{n−1}) = (det g)λ_{j}^{−1}Ω_{j}(x′_{1}, …, x′_{n−1}),

and so (10) follows by (8) and (9).

We next suppose that g is the permutation matrix corresponding to the transposition τ = (ab) for some 1 ⩽ a < b ⩽ n − 1. If j ∉ {a, b} then by Lemma 6.5 we have

[[τ(1), τ(2), …, τ̂(j), …, τ(n − 1)]] = −[[1, 2, …, ĵ, …, n − 1]],

and

[[1, …, â, …, b − 1, a, b + 1, …, n − 1]] = (−1)^{b−a−1}[[1, …, b̂, …, n − 1]],
[[1, …, a − 1, b, a + 1, …, b̂, …, n − 1]] = (−1)^{b−a−1}[[1, …, â, …, n − 1]].

Therefore

Ω′_{j}(x_{1}, …, x_{n−1}) = \{ \begin{matrix}−Ω_{j}(x′_{1}, …, x′_{n−1}) & if j ∉ \{a, b\}, \\ −Ω_{b}(x′_{1}, …, x′_{n−1}) & if j = a, \\ −Ω_{a}(x′_{1}, …, x′_{n−1}) & if j = b, \end{matrix}

and so (10) follows by (8) and (9).

Finally, we consider the case g = I + tE_{21}, where E_{ij} is the (n − 1) × (n − 1) matrix with a 1 in the (i, j)-place, and all other entries 0. The matrices ϕ′_{r} are given by

ϕ′_{r}(x_{1}, …, x_{n−1}) = ϕ_{r}(x_{1} + tx_{2}, x_{2}, …, x_{n−1}).

So by the chain rule

\frac{∂ϕ′_{r}}{∂x_{j}} (x_{1}, …, x_{n−1}) = (\frac{∂ϕ_{r}}{∂x_{j}} + δ_{j2}t\frac{∂ϕ_{r}}{∂x_{1}} ) (x_{1} + tx_{2}, x_{2}, …, x_{n−1}).

Therefore, if 2 ∉ {a_{1}, …, a_{n−2}} then

[[a_{1}, …, a_{n−2}]]′(x_{1}, …, x_{n−1}) = [[a_{1}, …, a_{n−2}]](x′_{1}, …, x′_{n−1}),

whereas if a_{k} = 2 then

[[a_{1}, …, a_{n−2}]]′(x_{1}, …, x_{n−1})

= [[a_{1}, …, a_{n−2}]](x′_{1}, …, x′_{n−1}) + t[[a_{1}, …, a_{k−1}, 1, a_{k+1}, …, a_{n−2}]](x′_{1}, …, x′_{n−1}).

We note that if 1 ∈ {a_{1}, …, a_{n−2}} then the final term [[a_{1}, …, a_{k−1}, 1, a_{k+1}, …, a_{n−2}]] vanishes by Lemma 6.5, since 1 appears twice. Then by definition of the Ω_{j} we have

Ω′_{j}(x_{1}, …, x_{n−1}) = Ω_{j}(x′_{1}, …, x′_{n−1}) − δ_{j1}tΩ_{2}(x′_{1}, …, x′_{n−1}),

and hence

∑_{j=1}^{n−1} x_{j}^{∗} ⊗ Ω′_{j}(x_{1}, …, x_{n−1}) = ∑_{j=1}^{n−1} (x_{j}^{∗} − δ_{j2}tx_{1}^{∗}) ⊗ Ω_{j}(x′_{1}, …, x′_{n−1}).

Now (10) follows by (8) and (9). This completes the proof of Proposition 5.1.

8. Reduction to the key lemma

In Section 5 we reduced the proof of Theorem 2.8 to a verification in the case of the standard set of n points: P_{1} = (1 : 0 …: 0), P_{2} = (0 : 1 : 0 …: 0), …, P_{n−1} = (0 : …: 0 : 1) and P_{n} = (1 : 1 : …: 1). This verification depends on the following “key lemma”, the proof of which we postpone to the next section.

Lemma 8.1. Let X ⊂ ℙ^{n−2} be the standard set of n points in general position. Fix a choice of minimal free resolution for X. Then there exists a scalar λ ∈ K such that whenever a_{1}, …, a_{n−2}, b is a permutation of 1, 2, …, n − 1 we have

[a_{1}, …, a_{n−2}] = ±λ(x_{b} − x_{a_{1}} − x_{a_{n−2}})x_{b},

where ± is the sign of the permutation, and the symbol [⋯] was defined in Section 2.

We use this lemma to compute the quadratic forms Ω_{1}, …, Ω_{n−1} associated to X.

Proof of Lemma 5.3. Recall that we wrote σ for the (n − 2)-cycle in S_{n−2} with σ(1) = 2, σ(2) = 3, …, σ(n − 2) = 1. Let a_{1}, …, a_{n−2}, b be as in the statement of Lemma 8.1. By the definition of [[⋯]], and the fact σ^{2} is an even permutation, we compute

[[a_{1}, a_{2}, …, a_{n−2}]] = ∑_{k=1}^{n−2}[a_{σ^{2k}(1)}, a_{σ^{2k}(2)}, …, a_{σ^{2k}(n−2)}]

= λ ∑_{k=1}^{n−2}(x_{b} − x_{a_{σ^{2k}(1)}} − x_{a_{σ^{2k}(n−2)}})x_{b}

= λ ((n − 2)x_{b}^{2} − 2x_{b} ∑_{j≠b} x_{j})

= λ (nx_{b}^{2} − 2x_{b} ∑_{j=1}^{n−1} x_{j}) .

It follows by the definition of the Ω_{i} and Lemma 6.5 that

Ω_{i} = (−1)^{i}[[1, …, î, …, n − 1]] = (−1)^{n−1}λ (nx_{i}^{2} − 2x_{i} ∑_{j=1}^{n−1} x_{j}) .

Since we are only computing the Ω_{i} up to an overall scalar, the factor (−1)^{n−1}λ may be ignored. This completes the proof of Lemma 5.3. □

9. Proof of the key lemma (=Lemma 8.1)

As before, let X ⊂ ℙ^{n−2} be the standard set of n points in general position. In this section we prove Lemma 8.1. Our approach is inspired by an explicit description of the minimal free resolution of the set X, due to Wilson [Wil13, Chapter 5]. In [Rad21, Section 4.5] we gave a different proof of Lemma 8.1, based on the method of unprojection.

Lemma 9.1. If n ⩾ 4 then the ideal I := I(X) ⊂ K[x_{1}, …, x_{n−1}] = R is generated by the quadratic forms x_{i}(x_{j} −x_{k}) for i, j, k ∈ {1, 2, …, n−1} distinct. If n = 3 then I is generated by x_{1}x_{2}(x_{1} − x_{2}).

Proof. The case n = 3 is obvious. The case n ⩾ 4 is [Wil13, Lemma 146]. The proof is a simple computation, since by Theorem 2.6 we already know that I is generated by quadratic forms. □

For the rest of this section we assume that n ⩾ 5, since the somewhat degenerate cases n = 3, 4 are easy to handle by a direct computation.

We fix a minimal graded free resolution (F_{•}, ϕ) of I. The idea is to describe F_{•} by splicing together Koszul complexes. For each pair J = (j, k), with j, k ∈ {1, 2, …, n − 1} distinct, consider the ideal I^{J} ⊂ I generated by the set of quadratic forms

{x_{i}(x_{j} − x_{k}) : i = 1, 2, …, ĵ, …, k̂, …, n − 1}.

As a graded K[x_{1}, …, x_{n−1}]-module, I^{J} is isomorphic to the ideal generated by the linear forms x_{1}, …, x̂_{j}, …, x̂_{k}, …, x_{n−1}, and so is resolved by a Koszul complex. We write this complex as

K_{•}^{J} : 0 ⟶ ∧^{n−3}E^{J} ⟶^{d_{n−3}} ∧^{n−4}E^{J} ⟶ … ⟶ ∧^{2}E^{J} ⟶^{d_{2}} E^{J} ⟶^{d_{1}} R.

where E^{J} is a free R-module of rank n − 3 with basis e_{1}, …, ê_{j}, …, ê_{k}, …, e_{n−1}, and the differentials d_{m} are given for m > 1 by

(11)  d_{m}(e_{i_{1}} ∧ … ∧ e_{i_{m}}) = ∑_{ℓ=1}^{m}(−1)^{ℓ}x_{i_{ℓ}} · (e_{i_{1}} ∧ … ∧ ê_{i_{ℓ}} ∧ … ∧ e_{i_{m}})

and for m = 1 by d_{1}(e_{i}) = x_{i}(x_{j} − x_{k}).

The inclusion I^{J} ⊂ I induces a map of chain complexes K_{•}^{J} ⟶ F_{•}, i.e. a commutative diagram

[figure: 0; ∧^{n−3}E^{J}; …; ∧^{2}E_{J}; d_{2}; E^{J}; d_{1}; R; 0; F_{n−2}; ϕ_{n−2}; F_{n−3}; …; F_{2}; ϕ_{2}; F_{1}; ϕ_{1}; R]

With notation as in Section 2, we may equally write this as

[figure: 0; R(−n + 2); …; R(−3)^{a_{2}}; d_{2}; R(−2)^{a_{1}}; d_{1}; R; 0; R(−n); ϕ_{n−2}; R(−n + 2)^{b_{n−3}}; …; R(−3)^{b_{2}}; ϕ_{2}; R(−2)^{b_{1}}; ϕ_{1}; R]

where a_{i} = \binom{n−3}{i} and the b_{i} are specified in Theorem 2.6. In particular, all the differentials d_{i} and ϕ_{i} are represented by matrices of linear forms, except for d_{1}, ϕ_{1} and ϕ_{n−2} which are represented by matrices of quadratic forms. The map of chain complexes K_{•}^{J} ⟶ F_{•}, which by construction is unique up to chain homotopy, is actually uniquely determined. This is because any such chain homotopy respects the grading of the modules in the resolutions, and hence must be zero.

We denote the image of e_{i_{1}} ∧…∧e_{i_{m}} in F_{m} by the symbol (i_{1} ∧…∧i_{m})⊗(j, k). It follows from equation (11) that for 2 ⩽ m ⩽ n − 3 we have

(12) ϕ_{m}((i_{1} ∧ … ∧ i_{m}) ⊗ (j, k)) = ∑_{ℓ=1}^{m}(−1)^{ℓ}x_{i_{ℓ}} · (i_{1} ∧ … ∧ î_{ℓ} ∧ … ∧ i_{m}) ⊗ (j, k).

Lemma 9.2. (i) For any i_{1}, …, i_{m}, j, k ∈ {1, 2, …, n − 1} distinct we have

(i_{1} ∧ ⋯ ∧ i_{m}) ⊗ (j, k) + (i_{1} ∧ ⋯ ∧ i_{m}) ⊗ (k, j) = 0.

(ii) For any i_{1}, …, i_{m}, j, k, ℓ ∈ {1, 2, …, n − 1} distinct we have

(i_{1} ∧ ⋯ ∧ i_{m}) ⊗ (j, k) + (i_{1} ∧ ⋯ ∧ i_{m}) ⊗ (k, ℓ) + (i_{1} ∧ ⋯ ∧ i_{m}) ⊗ (ℓ, j) = 0.

(iii) The individual expressions (i_{1} ∧ ⋯ ∧ i_{m}) ⊗ (j, k) are non-zero.

Proof. (i) and (ii). The left hand side of each equation has degree m+1 in F_{m} ≅ R(−m−1)^{b_{m}}. Since ϕ_{m} in injective in this degree, it suffices to check the image under ϕ_{m} is zero. The proof is now by induction on m. If m = 1 then

ϕ_{1}(i ⊗ (j, k) + i ⊗ (k, j)) = x_{i}(x_{j} − x_{k}) + x_{i}(x_{k} − x_{j}) = 0,
ϕ_{1}(i ⊗ (j, k) + i ⊗ (k, ℓ) + i ⊗ (ℓ, j)) = x_{i}(x_{j} − x_{k}) + x_{i}(x_{k} − x_{ℓ}) + x_{i}(x_{ℓ} − x_{j}) = 0.

If m > 1 then we instead use (12) to give a linear combination of x_{1}, …, x_{n−1} where each coefficient vanishes by the induction hypothesis. (iii) This is proved by a similar, but easier, induction. □

We now give a formula for the differential ϕ_{n−2} : F_{n−2} ⟶ F_{n−3}.

Lemma 9.3. The image of ϕ_{n−2} is generated as an R-module by

(13)  t := ∑_{j<k} x_{j}x_{k} · (i_{1} ∧ … ∧ i_{n−3}) ⊗ (j, k),

where for each j < k we pick i_{1}, …, i_{n−3}, j, k an even permutation of 1, 2, …, n − 1.

Proof. We first note that t has degree n in F_{n−3} ≅ R(−n + 2)^{b_{n−3}} and is non-zero by Lemma 9.2(iii). Since F_{n−2} ≅ R(−n) and F_{•} is exact, it suffices to show that t belongs to the kernel of ϕ_{n−3}. We find using (12) that the coefficient of x_{1}x_{2}x_{3} in ϕ_{n−3}(t) is

−(4 ∧ … ∧ (n − 1)) ⊗ (1, 2) − (4 ∧ … ∧ (n − 1)) ⊗ (2, 3) − (4 ∧ … ∧ (n − 1)) ⊗ (3, 1),

which vanishes by Lemma 9.2(ii). The same argument applies to the other coefficients. □

We now prove Lemma 8.1. The symbol […] was defined in terms of the partial derivatives of ϕ_{1}, …, ϕ_{n−2}, so we start by computing these. As ϕ_{1}(i ⊗ (j, k)) = x_{i}(x_{j} − x_{k}), we see that

(14)  \frac{∂ϕ_{1}}{∂x_{i}} (i ⊗ (j, k)) = x_{j} − x_{k}, \frac{∂ϕ_{1}}{∂x_{j}} (i ⊗ (j, k)) = x_{i}, \frac{∂ϕ_{1}}{∂x_{k}} (i ⊗ (j, k)) = −x_{i}.

It is immediate from (12) that for 2 ⩽ m ⩽ n − 3 we have (15)

\frac{∂ϕ_{m}}{∂x_{i}} ((i_{1} ∧ … ∧ i_{m}) ⊗ (j, k)) = \{ \begin{matrix}(−1)^{ℓ}(i_{1} ∧ … ∧ î_{ℓ} ∧ … ∧ i_{m}) ⊗ (j, k) & if i = i_{ℓ} \\ 0 & if i ∉ \{i_{1}, …, i_{m}\} \end{matrix}

Since the statement of Lemma 8.1 allows for an overall scalar λ ∈ K, we may re-scale ϕ_{n−2} so that ϕ_{n−2}(1) = t where t is given by (13). Then by Lemma 9.2(i) we have

(16)  \frac{∂ϕ_{n−2}}{∂x_{k}} (1) = ∑_{\begin{matrix}j=1 \\ j≠k \end{matrix}}^{n−1} x_{j} · (i_{1} ∧ … ∧ i_{n−3}) ⊗ (j, k)

where for each j we pick i_{1}, …, i_{n−3}, j, k an even permutation of 1, 2, …, n − 1. Now let a_{1}, …, a_{n−2}, b be a permutation of 1, 2, …, n − 1. We seek to compute

[a_{1}, a_{2}, …, a_{n−2}] = (\frac{∂ϕ_{1}}{∂x_{a_{1}}} ∘ \frac{∂ϕ_{2}}{∂x_{a_{2}}} ⋯ ∘ \frac{∂ϕ_{n−2}}{∂x_{a_{n−2}}} ) (1).

By (16) we have

[a_{1}, a_{2}, …, a_{n−2}] = (\frac{∂ϕ_{1}}{∂x_{a_{1}}} ∘ \frac{∂ϕ_{2}}{∂x_{a_{2}}} ∘ ⋯ ∘ \frac{∂ϕ_{n−3}}{∂x_{a_{n−3}}} ) ( ∑_{\begin{matrix}j=1 \\ j≠a_{n−2} \end{matrix}}^{n−1} x_{j} · (i_{1} ∧ … ∧ i_{n−3}) ⊗ (j, a_{n−2})) .

where for each j we pick i_{1}, …, i_{n−3}, j, a_{n−2} an even permutation of 1, 2, …, n−1. It is clear by (15) that for a non-zero contribution we need {a_{2}, …, a_{n−3}} ⊂ {i_{1}, …, i_{n−3}}, equivalently {a_{1}, a_{n−2}, b} ⊃ {j, a_{n−2}}. So the only terms to contribute to the sum are those with j = a_{1} and j = b. Using (14) and (15) we compute

[a_{1}, a_{2}, …, a_{n−2}] = ±\frac{∂ϕ_{1}}{∂x_{a_{1}}} (x_{b} · a_{1} ⊗ (b, a_{n−2}) − x_{a_{1}} · b ⊗ (a_{1}, a_{n−2}))
= ±x_{b}(x_{b} − x_{a_{1}} − x_{a_{n−2}}).

Finally, it may be checked that the sign ± only depends on n and the sign of the permutation sending 1, 2, …, n − 1 to a_{1}, a_{2}, …, a_{n−2}, b.

This completes the proof of Lemma 8.1, and hence of Theorem 2.8.

10. Proof of Theorem 3.1

In this section we prove Theorem 3.1 for general n ⩾ 4. This extends the proof for n = 4 in Section 3, and is based on the proof for n odd in [Rad21, Section 3.8].

We write 1, …, îjk, …, n−1 for the sequence of integers 1, 2, …, n−1, with i, j, k deleted (in whatever order they occur). Let ε_{ij} and ε_{ijk} be the signs of the permutations taking 1, 2, …, n − 1 to i, j, 1, …, îj, …, n − 1 and i, j, k, 1, …, îjk, … n − 1, respectively.

With notation as in Sections 2 and 3, we prove the following theorem. It is a refinement of Theorem 3.1, in that we now specify the signs.

Theorem 10.1. Let 1 ⩽ i, j, k ⩽ n − 1 distinct. Then

(17)  \frac{∂^{2}Ω_{k}}{∂x_{i}∂x_{j}} = (−1)^{n+1}ε_{ijk}(2n)\{i, i, 1, …, îjk, …, n − 1, j, j\},
(18)  \frac{∂^{2}Ω_{j}}{∂x_{i}^{2}} = ε_{ij}(2n)\{i, i, 1, …, îj, …, n − 1, i\},
(19)  \frac{∂^{2}Ω_{j}}{∂x_{i}∂x_{j}} − \frac{∂^{2}Ω_{k}}{∂x_{i}∂x_{k}} = (−1)^{n}ε_{ijk}(2n)\{i, i, 1, …, îjk, …, n − 1, j, k\},
(20) \frac{∂^{2}Ω_{i}}{∂x_{i}^{2}} − \frac{∂^{2}Ω_{j}}{∂x_{i}∂x_{j}} − \frac{∂^{2}Ω_{k}}{∂x_{i}∂x_{k}} = (−1)^{n+1}ε_{ijk}(2n)\{i, j, 1, …, îjk, …, n − 1, k, i\}.

For the proof we need some properties of the symbols {⋯ }. First, it is immediate from the definition that the symbol does not depend on the order of the first two terms, or on the order of the last two terms. We have the following additional symmetry properties.

Lemma 10.2. (i) For any τ ∈ S_{n−4} we have

{i, j, a_{τ(1)}, …, a_{τ(n−4)}, k, ℓ} = sign(τ){i, j, a_{1}, …, a_{n−4}, k, ℓ}.

(ii) If i ∈ {a_{1}, …, a_{n−3}} then for any τ ∈ S_{n−3} we have

{i, a_{τ(1)}, …, a_{τ(n−3)}, k, ℓ} = sign(τ){i, a_{1}, …, a_{n−3}, k, ℓ}.

(iii) We have {i, j, a_{1}, …, a_{n−4}, k, ℓ} = −{k, ℓ, a_{1}, …, a_{n−4}, i, j}.

Proof. (i) In the case where τ is a transposition of consecutive elements this is proved exactly as in Lemma 6.1. The general case follows. (ii) Differentiating ϕ_{1}ϕ_{2} = 0 gives {i, i, j, …} + {i, j, i, …} = 0. We are done by (i). (iii) Exactly as in Lemma 6.3, we have {i, j, a_{1}, …, a_{n−4}, k, ℓ} = ±{k, ℓ, a_{n−4}, …, a_{1}, i, j} where the sign is + if and only if n ≡ 2, 3 (mod 4). We are done by (i). □

Lemma 10.3. If 1 ⩽ i, j, k ⩽ n − 1 distinct then

{i, j, k, a_{1}, …, a_{n−3}} + {j, k, i, a_{1}, …, a_{n−3}} + {k, i, j, a_{1}, …, a_{n−3}} = 0.

Proof. This is proved by differentiating ϕ_{1}ϕ_{2} = 0. □

We also have the analogues of Lemmas 10.2(ii) and 10.3 where each symbol is reversed. We write 1_{A} for the indicator function of the event A.

We prove (17) by taking a_{1}, …, a_{n−2} = 1, …, k̂, …, n − 1 in the following lemma.

Lemma 10.4. Let 1 ⩽ i, j, k ⩽ n − 1 distinct. Let a_{1}, …, a_{n−2}, k be a permutation of 1, 2, …, n − 1. Then

(21) \frac{∂^{2}[a_{1}, …, a_{n−2}]}{∂x_{i}∂x_{j}} = ± (2 + 1_{i∈\{a_{1},a_{n−2}\}} + 1_{j∈\{a_{1},a_{n−2}\}}) \{i, i, 1, …, îjk, …, n − 1, j, j\}

and

(22)  \frac{∂^{2}[[a_{1}, …, a_{n−2}]]}{∂x_{i}∂x_{j}} = ±2n\{i, i, 1, …, îjk, …, n − 1, j, j\},

where ± is the sign of the permutation taking a_{1}, …, a_{n−2} to i, 1, …, îjk, …, n − 1, j.

Proof. We first prove (21) when {a_{1}, a_{n−2}} ∩ {i, j} = ∅. Using Lemma 10.2 we compute

\frac{∂^{2}[a_{1}, …, a_{n−2}]}{∂x_{i}∂x_{j}} = \{i, a_{1}, …, a_{n−2}, j\} + \{j, a_{1}, …, a_{n−2}, i\}

= ± ({i, i, 1, …, îjk, …, n − 1, j, j} − {j, j, 1, …, îjk, …, n − 1, i, i})

= ±2{i, i, 1, …, îjk, …, n − 1, j, j}.

If a_{1} = i and a_{n−2} ≠ j then the first term picks up a factor of 2, and the second term is unchanged. If a_{1} = i and a_{n−2} = j then the first term picks up a factor of 4 and the second term vanishes by Lemma 10.2(iii). The other cases are similar.

We deduce (22) from (21) by summing over the n − 2 terms in Definition 2.7(ii). Since σ^{2} is an even permutation, all the terms have the same sign. There are two terms starting or ending in i, and two terms starting or ending in j. This gives an overall numerical factor of 2(n − 2) + 2 + 2 = 2n. □

We prove (18) by taking a_{1}, …, a_{n−2} = 1, …, ĵ, …, n − 1 in the following lemma.

Lemma 10.5. Let 1 ⩽ i, j ⩽ n − 1 distinct. Let a_{1}, …, a_{n−2}, j be a permutation of 1, 2, …, n − 1. Then

\frac{∂^{2}[a_{1}, …, a_{n−2}]}{∂x_{i}^{2}} = ±2 (1 + 1_{i∈\{a_{1},a_{n−2}\}}) \{i, i, 1, …, îj, …, n − 1, i\}

and

\frac{∂^{2}[[a_{1}, …, a_{n−2}]]}{∂x_{i}^{2}} = ±2n\{i, i, 1, …, îj, …, n − 1, i\},

where ± is the sign of the permutation taking a_{1}, …, a_{n−2} to i, 1, …, îj, …, n − 1.

Proof. If i ∉ {a_{1}, a_{n−2}} then

\frac{∂^{2}[a_{1}, …, a_{n−2}]}{∂x_{i}^{2}} = 2\{i, a_{1}, …, a_{n−2}, i\} = ±\{i, i, 1, …, îj, …, n − 1, i\}.

If i ∈ {a_{1}, a_{n−2}} then we pick up an extra factor of 2. This proves the result for [⋯]. We deduce the result for [[⋯]] exactly as before. □

We prove (19) by taking r = 0 and b_{1}, …, b_{s} = 1, …, ĵk, …, n−1 in the following lemma, and also using Lemma 6.5.

Lemma 10.6. Let 1 ⩽ i, j, k ⩽ n − 1 distinct. Let a_{1}, …, a_{r}, b_{1}, …, b_{s}, j, k be a permutation of 1, 2, …, n − 1. Then

\frac{∂^{2}[a_{1}, …, a_{r}, k, b_{1}, …, b_{s}]}{∂x_{i}∂x_{j}} + \frac{∂^{2}[a_{1}, …, a_{r}, j, b_{1}, …, b_{s}]}{∂x_{i}∂x_{k}}

= ±(−1)^{s} (2 + 1_{i∈{a_{1},b_{s}}} + 1_{rs=0}) {i, i, 1, …, îjk, …, n − 1, j, k},

and

\frac{∂^{2}[[a_{1}, …, a_{r}, k, b_{1}, …, b_{s}]]}{∂x_{i}∂x_{j}} + \frac{∂^{2}[[a_{1}, …, a_{r}, j, b_{1}, …, b_{s}]]}{∂x_{i}∂x_{k}}

= ±(−1)^{s}(2n){i, i, 1, …, îjk, …, n − 1, j, k},

where ± is the sign of the permutation taking a_{1}, …, a_{r}, b_{1}, …, b_{s} to i, 1, …, îjk, …, n − 1.

Proof. We first suppose that r, s ⩾ 1. Using Lemmas 10.2 and 10.3, we compute

{i, a_{1}, …, a_{r}, k, b_{1}, …, b_{s}, j} + {i, a_{1}, …, a_{r}, j, b_{1}, …, b_{s}, k}
= (−1)^{s−1} ({i, a_{1}, …, a_{r}, b_{1}, …, b_{s−1}, k, b_{s}, j} + {i, a_{1}, …, a_{r}, b_{1}, …, b_{s−1}, j, b_{s}, k})
= (−1)^{s}{i, a_{1}, …, a_{r}, b_{1}, …, b_{s}, j, k}
= ±(−1)^{s}{i, i, 1, …, îjk, …, n − 1, j, k},

and

{j, a_{1}, …, a_{r}, k, b_{1}, …, b_{s}, i} + {k, a_{1}, …, a_{r}, j, b_{1}, …, b_{s}, i}
= (−1)^{r−1} ({j, a_{1}, k, a_{2}, …, a_{r}, b_{1}, …, b_{s}, i} + {k, a_{1}, j, a_{2}, …, a_{r}, b_{1}, …, b_{s}, i})
= (−1)^{r}{j, k, a_{1}, …, a_{r}, b_{1}, …, b_{s}, i}
= ±(−1)^{s−1}{j, k, 1, …, îjk, …, n − 1, i, i}
= ±(−1)^{s}{i, i, 1, …, îjk, …, n − 1, j, k}.

If i ∉ {a_{1}, b_{s}} then we simply add these two expressions, giving a factor of 2. If i ∈ {a_{1}, b_{s}} then we take twice one expression plus the other, giving a factor of 3.

We next suppose s = 0. The first calculation in the last paragraph is modified by deleting the second line, and introducing a factor of 2 thereafter. If a_{1} ≠ i then this gives an overall factor of 3. If a_{1} = i then we must take twice the first expression, but the second expression vanishes by Lemma 10.2(iii). This gives an overall factor of 4. The case r = 0 is similar.

We deduce the result for [[⋯]] from that for [⋯] as before. □

The following lemma prepares for the proof of (20).

Lemma 10.7. Let 1 ⩽ i, j ⩽ n − 1 distinct. Let a_{1}, …, a_{n−3} be a permutation of 1, 2, …, îj, …, n − 1 with sign ν. Then

(23)  A(i, j) := ν ({i, j, a_{1}, …, a_{n−3}, i} + {i, i, a_{1}, …, a_{n−3}, j})

does not depend on the choice of a_{1}, …, a_{n−3}.

Proof. If ⋯ denotes the same in each case, then by Lemma 10.3 we have

{ij ⋯ rsi} + {ij ⋯ sri} + {ij ⋯ irs} = 0,
{ii ⋯ rsj} + {ii ⋯ srj} + {ii ⋯ jrs} = 0.

Lemma 10.2(ii) shows that the final terms in these two sums differ by a sign. The right hand side of (23) is therefore invariant under switching the last two a’s. The lemma now follows by Lemma 10.2(i). □

Lemma 10.8. Let 1 ⩽ i, j ⩽ n − 1 distinct. Let a_{1}, …, a_{r}, b_{1}, …, b_{s}, i, j be a permutation of 1, 2, …, n − 1. Then

\frac{∂^{2}[a_{1}, …, a_{r}, j, b_{1}, …, b_{s}]}{∂x_{i}^{2}} + 2\frac{∂^{2}[a_{1}, …, a_{r}, i, b_{1}, …, b_{s}]}{∂x_{i}∂x_{j}} = ±(−1)^{r}2 (1 + 1_{rs=0}) A(i, j)

and

\frac{∂^{2}[[a_{1}, …, a_{r}, j, b_{1}, …, b_{s}]]}{∂x_{i}^{2}} + 2\frac{∂^{2}[[a_{1}, …, a_{r}, i, b_{1}, …, b_{s}]]}{∂x_{i}∂x_{j}} = ±(−1)^{r}(2n)A(i, j),

where ± is the sign of the permutation taking a_{1}, …, a_{r}, b_{1}, …, b_{s} to 1, …, îj, …, n − 1.

Proof. If r = 0 then the left hand side equals

(24)  2{i, j, b_{1}, …, b_{s}, i} + 4{i, i, b_{1}, …, b_{s}, j} + 2{j, i, b_{1}, …, b_{s}, i} = ±4A(i, j).

If r, s ⩾ 1 then we instead obtain

2{i, a_{1}, …, a_{r}, j, b_{1}, …, b_{s}, i} + 2{i, a_{1}, …, a_{r}, i, b_{1}, …, b_{s}, j} + 2{j, a_{1}, …, a_{r}, i, b_{1}, …, b_{s}, i}.

Cancelling a factor (−1)^{r}2 and applying Lemma 10.2 term by term gives

−{i, a_{1}, j, …, a_{r}, b_{1}, …, b_{s}, i} + {i, i, a_{1}, …, a_{r}, b_{1}, …, b_{s}, j} − {j, a_{1}, i, …, a_{r}, b_{1}, …, b_{s}, i}.

Applying Lemma 10.3 to the first and third terms shows that this equals ±A(i, j). Finally, when s = 0 the left hand side is

2{i, a_{1}, …, a_{r}, j, i} + 2{i, a_{1}, …, a_{r}, i, j} + 4{j, a_{1}, …, a_{r}, i, i}.

By Lemma 10.2(iii) this is minus the expression we get by replacing b_{1}, …, b_{s} by a_{2}, …, a_{r}, a_{1} in (24). This gives the factor (−1)^{r}.

We deduce the result for [[⋯]] from that for [⋯] as before. □

Taking r = 0 and b_{1}, …, b_{s} = 1, …, îj, …, n − 1 in Lemma 10.8, and appealing to Lemma 6.5, gives

\frac{∂^{2}Ω_{i}}{∂x_{i}^{2}} − 2\frac{∂^{2}Ω_{j}}{∂x_{i}∂x_{j}} = −ε_{ij}(2n)A(i, j).

Taking a_{1}, …, a_{n−3} = 1, …, îjk, …, n − 1, k in Lemma 10.7 shows that this equals

(−1)^{n+1}ε_{ijk}(2n) ({i, j, 1, …, îjk, …, n − 1, k, i} + {i, i, 1, …, îjk, …, n − 1, j, k}) .

Adding (19) gives (20). This completes the proof of Theorem 10.1, and hence of Theorem 3.1.

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University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK

  Email address: T.A.Fisher@dpmms.cam.ac.uk

University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK

  Email address: lazaradicevic@gmail.com