GENUS ONE CURVES DEFINED BY PFAFFIANS

TOM FISHER

Abstract. We show that genus one normal curves of odd degree can be presented by alternating matrices of linear forms, and give an algorithm for computing these matrices.

1. Introduction

Let C be a smooth projective curve of genus one defined over an arbitrary field k. If D is a k-rational divisor on C of degree n ≥ 3 then we may embed C ⊂ ℙ^{n−1} via the complete linear system |D|. The image is called a genus one normal curve. Equivalently, a genus one normal curve is a smooth curve of genus one and degree n that spans ℙ^{n−1}. The genus one normal curves of degrees 3 and 4 are plane cubics and intersections of two quadrics respectively.

It is well known that the homogeneous ideal of a genus one normal curve of degree n ≥ 4 is generated by a vector space of quadrics of dimension n(n − 3)/2. It follows that for n ≥ 5 the homogeneous ideal is not a complete intersection. Moreover a generic choice of n(n − 3)/2 quadrics will not define a curve, let alone a genus one normal curve.

In the case n = 5 the homogeneous ideal is generated by the 4 × 4 Pfaffians of a 5×5 alternating matrix of linear forms. This is a classical fact: see [2, Lemma 4.4(i)] and the references given there to works of Bianchi and Klein. It is also a special case of the structure theorem for Gorenstein ideals of codimension 3, due to Buchsbaum and Eisenbud [5], [6].

We generalise this observation to curves of higher odd degree. Specifically for n ≥ 5 an odd integer we show that the homogeneous ideal of a genus one normal curve of degree n is generated by the 4×4 Pfaffians of an n × n alternating matrix of linear forms. The proof over an algebraically closed field has already appeared in [12]. Here we concentrate on arithmetic and algorithmic aspects. The key definition is Definition 1.1. Let n ≥ 3 be an odd integer. Let C be a genus one normal curve of degree n. A Klein matrix Φ for C is an n×n alternating matrix of linear forms defined over k such that

Date: 1st June 2006.

(i) Φ has rank at most 2 on C, and (ii) the n submaximal Pfaffians of Φ are linearly independent.

This definition is well suited to computation. Indeed condition (i) holds if and only if the 4 × 4 Pfaffians of Φ belong to the homogeneous ideal of C. Condition (ii) may be checked directly. In §§3,5 we recall some alternative characterisations of Klein matrices.

If n ≥ 5 then C is precisely the rank 2 locus of Φ. In fact a much stronger result is true. Let Sec^{r}C be the rth higher secant variety of C, i.e. the Zariski closure of the locus of all (r − 1)-planes spanned by r points on C. As special cases we have Sec^{0}C = ∅, Sec^{1}C = C and Sec^{2}C = Sec C. We write I(X) for the homogeneous ideal of a projective variety X. By convention I(∅) is the irrelevant ideal.

Theorem 1.2. Let n ≥ 3 be an odd integer. Let C be a genus one normal curve of degree n with Klein matrix Φ. If n ≥ 2r + 3 then (i) Sec^{r}C is the locus where Φ has rank at most 2r, and (ii) I(Sec^{r}C) is generated by the (2r + 2) × (2r + 2) Pfaffians of Φ.

Proof: It suffices to prove the theorem over an algebraically closed field, and for this we refer to [12, Corollary 1.8]. □

We show that Klein matrices exist and are unique.

Theorem 1.3. Let n ≥ 3 be an odd integer.

(a) Every genus one normal curve of degree n has a Klein matrix.

(b) If Φ_{1} and Φ_{2} are Klein matrices for C then (i) There exist B ∈ GL_{n}(k) and c ∈ k^{×} such that Φ_{1} = cB^{T} Φ_{2}B. (ii) The matrix B is uniquely determined up to scalars.

If k is algebraically closed then we may extract a square root and so dispense with the constant c in Theorem 1.3(b). With this convention the matrix B is uniquely determined up to sign.

We give two different proofs of Theorem 1.3. The first proof (see §3) uses rank 2 vector bundles and Galois descent. The second proof (see §§4,5) replaces C by a higher secant variety of codimension 3 and applies the Buchsbaum-Eisenbud structure theorem. In both approaches the assumption that n is odd is essential. In §§6,7 we turn the seond proof into an algorithm for computing Klein matrices. We then use our algorithm to give some examples in §8.

In the case n = 5 the existence of Klein matrices has the following converse, pointed out to me by Nick Shepherd-Barron.

Proposition 1.4. A generic 5 × 5 alternating matrix of linear forms on ℙ^{4} is a Klein matrix for a genus one normal quintic.

Proof: For the proof we may assume that k is algebraically closed. Let Gr(2, 5) be the Grassmannian of 2-dimensional vector subspaces of a 5-dimensional vector space V. The image of the Plücker embedding Gr(2, 5) ↪ ℙ(∧^{2}V) = ℙ^{9} is defined by the 4 × 4 Pfaffians of a generic 5 × 5 alternating matrix. Let H be hyperplane section for this embedding. It is known that Gr(2, 5) has dimension 6, degree 5 and canonical divisor K = −5H. We apply Bertini’s theorem [15, II, 8.18] to show that a generic ℙ^{4}-section is a smooth irreducible curve C. Since K = −5H the adjunction formula [15, II, 8.20] gives K_{C} ∼ 0. Thus C is a genus one curve of degree 5 defined by quadrics. Since there are no such curves in ℙ^{3} it follows that C spans ℙ^{4}. To fit our definition of a Klein matrix, it only remains to show that the 4 × 4 Pfaffians of Φ are linearly independent. One possible proof of this is given in [12, Corollary 5.5]. □

Proposition 1.4 makes it possible to develop an invariant theory for genus one curves of degree n = 5 analogous to that classically known in the cases n = 2, 3, 4. We give details in a sequel to this paper. In particular we have found an algorithm for computing the Jacobian of a genus one normal quintic. The first step is to compute a Klein matrix using the algorithm described in this paper.

The analogue of Proposition 1.4 for n ≥ 7 is false, as is already apparent by counting dimensions. Nonetheless it is hoped that Klein matrices will find applications to the study of genus one curves of higher odd degree. We hint at these in §9 where we explore the relationship between Klein matrices and the index of a genus one curve.

Remark 1.5. In the geometric literature the term used is “elliptic normal curve” rather than “genus one normal curve”. We have changed the name since we are interested in curves that do not have or are not known to have a k-rational point. The word “normal” means that the curve is embedded by a complete linear system, equivalently that the homogeneous co-ordinate ring is integrally closed.

2. Pfaffians

We recall some basic facts about Pfaffians.

Definition 2.1. A square matrix over a ring R is alternating if it is skew-symmetric and all its diagonal entries are zero.

We temporarily work over a field k of characteristic 0. Let A = (a_{ij}) be an n × n alternating matrix with entries in k. If n = 2r is even then the Pfaffian of A is

(1)  pf(A) = \frac{1}{2^{r}r!} ∑_{σ∈S_{2r}} sign(σ) ∏_{i=1}^{r} a_{σ(2i−1)σ(2i)}.

Standard calculations, cf. [3, Ch. IX, §5], show that

(2)  pf(P^{T} AP) = det(P) pf(A),
(3)  pf (\begin{matrix}0 & A \\ −A^{T} & 0 \end{matrix}) = ± det(A).

It is well known that any alternating matrix over a field is congruent to a matrix of the form

(\begin{matrix}0 & I_{t} & 0 \\ −I_{t} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) .

In particular alternating matrices have even rank. It also follows, with the help of (2) and (3), that det(A) = pf(A)^{2}.

The denominator in (1) may be avoided by restricting the summation to an appropriate subset of S_{2r}. Alternatively, let k be the field of fractions of the polynomial ring

R_{0} = ℤ[a_{12}, a_{13}, …, a_{n−1 n}].

Since pf(A)^{2} = det(A) ∈ R_{0} it follows by unique factorisation that pf(A) ∈ R_{0}. We use this polynomial to define the Pfaffian of an alternating matrix over an arbitrary ring.

Pfaffians, just like determinants, may be expanded along a row. We write A^{{i,j}} for the matrix obtained from A by deleting the ith and jth rows and columns. Using (1) we find

(4)  pf(A) = ∑_{j=2}^{n}(−1)^{j}a_{1j} pf(A^{{1,j}}).

For example in the 4 × 4 case we have

pf (\begin{matrix}0 & a_{12} & a_{13} & a_{14} \\ & 0 & a_{23} & a_{24} \\ & & 0 & a_{34} \\ & & & 0 \end{matrix}) = a_{12}a_{34} − a_{13}a_{24} + a_{14}a_{23}.

Let A be a matrix over a ring R. We write I_{m}(A) for the ideal generated by the m × m minors of A. Likewise if A is alternating and m is even then we write Pf_{m}(A) for the ideal generated by the m × m Pfaffians of A, where it is understood that these are the Pfaffians of the m × m submatrices obtained by deleting the same rows and columns. The proof of (2) gives Pf_{m}(P^{T} AP) ⊂ Pf_{m}(A). So if P is invertible then Pf_{m}(P^{T} AP) = Pf_{m}(A).

For any fixed value of m the following lemma is equivalent to some identities in R_{0}.

Lemma 2.2. Let A be an n×n alternating matrix over a ring R. Then

I_{m}(A) ⊂ \{ \begin{matrix}Pf_{m}(A)^{2} & if m is even, \\ Pf_{m−1}(A) Pf_{m+1}(A) & if m is odd. \end{matrix}

If 2 ∈ R^{×} or m = n − 1 then the reverse inclusions hold.

Proof: We refer to [16, §3, §4] for the inclusions “⊂” and “⊃”, the latter being conditional on 2 ∈ R^{×}. The case m = n − 1 is covered by the identities of Cayley cited in [16, §2.2]. Let us note that the inclusion “⊃” does not hold in general. Indeed if A is the generic 4 × 4 alternating matrix over R_{0} = ℤ[a_{12}, …, a_{34}] then I_{2}(A) ≠ Pf_{2}(A)^{2}. □

The next lemma is recalled from [6, Corollary 2.6]. If 2 ∈ R^{×} then it is an immediate corollary of Lemma 2.2.

Lemma 2.3. Let A be an alternating matrix over a ring R. If m is even then the ideals I_{m}(A) and Pf_{m}(A) have the same radical.

Proof: By Lemma 2.2 we have I_{m}(A) ⊂ Pf_{m}(A). We recall that Pf_{m}(A) is generated by finitely many Pfaffians, say p_{1}, …, p_{N}. Since each p_{i}^{2} is a m × m minor of A we have:

f ∈ Pf_{m}(A) ⟹ f^{N+1} ∈ I_{m}(A).

Thus I_{m}(A) ⊂ Pf_{m}(A) ⊂ √I_{m}(A). □

Definition 2.4. Let A be an n × n alternating matrix with n odd. The vector of submaximal Pfaffians of A is P = (p_{1}, …, p_{n}) where p_{i} = (−1)^{i} pf(A^{{i}}) and A^{{i}} is the matrix obtained by deleting the ith row and column of A.

We recall from [4, §3.4]:

Lemma 2.5. Let n be an odd integer. Let A be an n × n alternating matrix with vector of submaximal Pfaffians P. Then P A = 0.

Proof: Let A′ be the (n + 1) × (n + 1) matrix obtained from A by repeating the first row and column. We apply (4) to A′. □

Lemma 2.6. Let n be an odd integer. Let A_{1}, A_{2} and B be n × n matrices over a ring R. Suppose that A_{1} and A_{2} are alternating with vectors of submaximal Pfaffians P_{1} and P_{2}. If A_{1} = B^{T} A_{2}B then P_{1}B^{T} = det(B)P_{2}.

Proof: It suffices to give a proof in the case R = k is an algebraically closed field. We may further suppose that A_{2} has rank n − 1. If B is invertible then from P_{1}B^{T} A_{2} = P_{1}A_{1}B^{−1} = 0 and P_{2}A_{2} = 0 we deduce that P_{1}B^{T} = cP_{2} for some c ∈ k. Then fixing A_{2} and varying B we get P_{1}B^{T} = f(B)P_{2} where f is a polynomial in the entries of B and f(B) ≠ 0 whenever B is invertible. Hence f(B) = c(det B)^{r} for some c ∈ k^{×} and r ≥ 0. By considering the case B is a scalar matrix it follows that c = 1 and r = 1. □

3. Vector bundles

Let n ≥ 3 be an odd integer and let C be a genus one normal curve of degree n. Let O(1) be the line bundle of degree n on C associated to the hyperplane section. For E a rank 2 vector bundle on C with det E ≃ O(1) we write Φ(E) for the alternating matrix of linear forms representing the determinant map

∧^{2}H^{0}(C, E) → H^{0}(C, O(1)).

Notice that to make Φ(E) explicit it is necessary both to choose a basis for H^{0}(C, E) and to fix an isomorphism det E ≃ O(1). We recall the characterisation of Klein matrices in terms of vector bundles.

Proposition 3.1. Let Φ be an n×n alternating matrix of linear forms defined over k. Then Φ is a Klein matrix for C if and only if it is of the form Φ(E) for E an indecomposable rank 2 vector bundle on C with det E ≃ O(1).

Proof: See [12, Proposition 5.3]. □

We recall a special case of a theorem of Atiyah.

Proposition 3.2. Let k be an algebraically closed field.

(i) There is a unique indecomposable rank 2 vector bundle E on C with det E ≃ O(1). (ii) If char (k) ≠ 2 then End(E) ≃ k.

Proof. (i) See [1, Corollary to Theorem 7]. (ii) We take global sections in [1, Lemma 22]. □

Proposition 3.1 tells us that to prove the existence of Klein matrices over a non-algebraically closed field it suffices to remove the restriction on k in Proposition 3.2(i). If k is perfect and char (k) ≠ 2 this may be achieved by Galois descent, as we now explain. These restrictions on k will remain in force for the rest of this section.

Lemma 3.3. Let E_{1} and E_{2} be indecomposable k-rational rank 2 vector bundles on C with det E_{1} ≃ det E_{2} ≃ O(1). Then E_{1} ≃ E_{2} over k.

Proof: By Proposition 3.2(i) there is an isomorphism ϕ : E_{1} → E_{2} defined over \overline{k}. By Proposition 3.2(ii) the cocycle σ ↦ ϕ^{σ}ϕ^{−1} takes values in \overline{k}^{×}. We are done by Hilbert’s theorem 90. □

Proposition 3.4. There exists an indecomposable k-rational rank 2 vector bundle E on C with det E ≃ O(1).

Proof: Since n is odd there is a field extension k_{0}/k of odd degree with C(k_{0}) ≠ ∅. Let P, Q ∈ C(k_{0}) with (n − 1)P + Q ∼ H, where H is the hyperplane section. Let E_{1} be the non-trivial extension of O(Q) by O and let E = E_{1} ⊗ O(((n − 1)/2)P). Then E is an indecomposable k_{0}-rational rank 2 vector bundle on C with det E ≃ O(1).

We must show that E descends to k. Let k_{1}/k be the Galois closure of k_{0}/k. We write G = Gal(k_{1}/k) and H = Gal(k_{1}/k_{0}). For each σ ∈ G, Lemma 3.3 gives an isomorphism h_{σ} : E → E^{σ} defined over k_{1}. By Proposition 3.2(ii) the cocycle (σ, τ) ↦ h_{στ}^{−1}(h_{σ})^{τ}h_{τ} takes values in k_{1}^{×} and so defines an element ξ ∈ H^{2}(G, k_{1}^{×}). We write the elements of this group additively. Since det E ≃ O(1) is k-rational, we have 2ξ = 0. Since E is defined over k_{0}, we have res_{H}^{G}ξ = 0. But [k_{0} : k] is odd, so

ξ = [k_{0} : k]ξ = cor_{H}^{G} res_{H}^{G} ξ = 0.

We may therefore scale the h_{σ} so that h_{στ} = (h_{σ})^{τ}h_{τ} for all σ, τ ∈ G. As explained in [20, Chapter V, §4] this is precisely the descent data required to descend E from k_{1} to k. □

We give our first proof of Theorem 1.3 under the assumptions on k specified above, namely that k is perfect and char (k) ≠ 2.

Proof of Theorem 1.3: (a) The existence of Klein matrices follows immediately from Propositions 3.1 and 3.4.

(b) Let Φ_{1} and Φ_{2} be Klein matrices for C. By Proposition 3.1 we have Φ_{1} = Φ(E_{1}) and Φ_{2} = Φ(E_{2}) where E_{1} and E_{2} are indecomposable rank 2 vector bundles on C with det E_{1} ≃ det E_{2} ≃ O(1). By Lemma 3.3 there is an isomorphism E_{1} ≃ E_{2}. Let B ∈ GL_{n}(k) represent the corresponding map on global sections. The isomorphisms det E_{1} ≃ O(1) and det E_{2} ≃ O(1) used in the construction of Φ(E_{1}) and Φ(E_{2}) now give rise to an automorphism of O(1). Since End(O(1)) ≃ H^{0}(C, O) ≃ k it follows that Φ_{1} = cB^{T} Φ_{2}B for some c ∈ k^{×}.

Finally we must show that if Φ(E) = cB^{T} Φ(E)B then B is a scalar matrix. Such a matrix B describes an endomorphism of H^{0}(C, E). We saw in [12, §4] that E may be recovered from Φ(E). So B extends to an endomorphism of E. Then Proposition 3.2(ii) tells us that B is a scalar matrix. □

The approach via vector bundles was crucial to the proof of Theorem 1.2. We have now seen that it leads rapidly to a proof of the existence and uniqueness of Klein matrices. Unfortunately it does not appear to give any practical method for computing Klein matrices. In §5 we give an alternative proof of Theorem 1.3 that is both constructive and applies over an arbitrary field.

4. Gorenstein Ideals

We prepare for our second proof of the existence and uniqueness of Klein matrices by recalling some basic commutative algebra. Our main references are [4] and [8]. We work over an arbitrary field k.

Let R = k[x_{1}, …, x_{n}] be the homogeneous co-ordinate ring of ℙ^{n−1}, with its usual grading R = ⊕_{d≥0}R_{d}. Let R_{+} = ⊕_{d≥1}R_{d} be the irrelevant ideal and let M be any finitely generated graded R-module. Definition 4.1. A graded free resolution of M is a complex of graded free R-modules

F_{•} : … ⟶ F_{2} ⟶^{ϕ_{2}} F_{1} ⟶^{ϕ_{1}} F_{0} ⟶ 0

that is exact except at F_{0} where the homology is M. The resolution F_{•} is minimal if ϕ_{i}(F_{i}) ⊂ R_{+}F_{i−1} for all i. Lemma 4.2. Let F_{•} be a minimal graded free resolution of M. Then any graded free resolution of M is a direct sum of F_{•} and a trivial complex. In particular minimal resolutions are unique up to isomorphism. Proof: See [8, §20.1]. □

The common length of all minimal resolutions is called the projective dimension of M and denoted proj dim M. Hilbert’s syzygy theorem asserts that proj dim M ≤ n. We write codim I for the codimension, or height, of an ideal I ⊂ R. It follows by the Auslander-Buchsbaum formula that codim I ≤ proj dim R/I. A homogeneous ideal I ⊂ R is perfect if codim I = proj dim R/I.

The dual of M is M^{∗} = Hom_{R}(M, R). The dual of a complex of graded free R-modules is again a complex of graded free R-modules. Definition 4.3. A perfect ideal I ⊂ R is a Gorenstein ideal if the minimal graded free resolution of R/I is isomorphic to its dual (up to a shift in degree).

We recall a theorem of Buchsbaum and Eisenbud [5], [6].

Proposition 4.4. Let I ⊂ R be a Gorenstein ideal of codimension 3. Let R/I have minimal graded free resolution F_{•}. Then there is an isomorphism of complexes

\xymatrix{0 \ar[r] & F_{3} \ar[r]^{ϕ_{3}} \ar[d]^{s_{3}} & F_{2} \ar[r]^{ϕ_{2}} \ar[d]^{s_{2}} & F_{1} \ar[r]^{ϕ_{1}} \ar[d]^{s_{1}} & F_{0} \ar[r] \ar[d]^{s_{0}} & 0 \\ 0 \ar[r] & F_{0}^{∗} \ar[r]^{ϕ_{1}^{∗}} & F_{1}^{∗} \ar[r]^{ϕ_{2}^{∗}} & F_{2}^{∗} \ar[r]^{ϕ_{3}^{∗}} & F_{3}^{∗} \ar[r] & 0}

with s_{1}ϕ_{2} (= ϕ_{2}^{∗}s_{2}) represented by an alternating matrix.

Proof: We know by Definition 4.3 that there is an isomorphism of complexes. The proof works by putting an algebra structure on ⊕_{i=0}^{3}F_{i} and then using this algebra structure to choose an isomorphism with the required additional property. We refer to [4, §3.4] for details. □

From now on we label all maps of graded free R-modules by the matrices that represent them.

Definition 4.5. Let m ≥ 3 be an odd integer. Let r_{1}, …, r_{m} be integers that are either all odd or all even. A Buchsbaum-Eisenbud matrix Φ is an m × m alternating matrix with each entry Φ_{ij} either 0 or a non-constant homogeneous polynomial of degree (r_{i} + r_{j})/2.

Let Φ be a Buchsbaum-Eisenbud matrix, and let P = (p_{1}, …, p_{m}) be the vector of submaximal Pfaffians of Φ as defined in §2. Then each p_{i} is either 0 or a non-constant homogeneous polynomial of degree t_{i} = (s − r_{i})/2 where s = ∑_{j=1}^{m} r_{j}. We write R(d) for the graded free R-module of rank 1 with R(d)_{e} = R_{d+e}.

Lemma 4.6. Let Φ be a Buchsbaum-Eisenbud matrix, and let P be the vector of submaximal Pfaffians of Φ. Then there is a complex of graded free R-modules

F_{•}(Φ) : 0 → R(−s) →^{P^{T}} ⊕_{i=1}^{m}R(−s + t_{i}) →^{Φ} ⊕_{i=1}^{m}R(−t_{i}) →^{P} R → 0.

Proof: This is immediate from Lemma 2.5. □

We recall the structure theorem of Buchsbaum and Eisenbud [5], [6] for Gorenstein ideal of codimension 3.

Theorem 4.7 (Buchsbaum-Eisenbud). (a) Let I ⊂ R be a homogeneous ideal of codimension at least 3 generated by the submaximal Pfaffians of a Buchsbaum-Eisenbud matrix Φ. Then F_{•}(Φ) is a minimal graded free resolution of R/I. In particular I is a Gorenstein ideal of codimension 3.

(b) Let I ⊂ R be a Gorenstein ideal of codimension 3. Then there exists a Buchsbaum-Eisenbud matrix Φ such that F_{•}(Φ) is a minimal graded free resolution of R/I. In particular I is generated by the submaximal Pfaffians of Φ.

Proof: (a) Since the submaximal Pfaffians of Φ are not all zero, rank(Φ) = m − 1 and rank(P) = 1. By Lemma 2.3 the ideals I = Pf_{m−1}(Φ) and I_{m−1}(Φ) have the same radical. So I_{m−1}(Φ) has codimension at least 3. Then F_{•}(Φ) is exact by the Buchsbaum-Eisenbud acyclicity criterion: see [4, Theorem 1.4.13] or [8, Theorem 20.9]. It follows from Definition 4.5 that F_{•}(Φ) is minimal. So F_{•}(Φ) is a minimal graded free resolution of R/I and proj dim R/I = 3. But in general we have codim I ≤ proj dim R/I. So I is a Gorenstein ideal of codimension 3.

(b) Granted Proposition 4.4 it suffices to prove

Proposition 4.8. Let I ⊂ R be a Gorenstein ideal of codimension 3. Suppose that R/I has minimal graded free resolution

G_{•} : 0 → R(−s) →^{Q^{T}} ⊕_{i=1}^{m}R(−s + t_{i}) →^{Φ} ⊕_{i=1}^{m}R(−t_{i}) →^{Q} R → 0

where Φ is an alternating matrix. Then Φ is a Buchsbaum-Eisenbud matrix and Q is a scalar multiple of the vector of submaximal Pfaffians of Φ. In particular F_{•}(Φ) is a minimal graded free resolution of R/I.

Proof: Let Q = (q_{1}, …, q_{m}) where q_{i} is a homogeneous polynomial of degree t_{i}. Then Φ is an m × m alternating matrix with each entry Φ_{ij} either 0 or a non-constant homogeneous polynomial of degree s − t_{i} − t_{j} = (r_{i} + r_{j})/2 where r_{i} = s − 2t_{i}. Since G_{•} is exact it is clear that Φ has rank m − 1, and so m is odd. Hence Φ satisfies our definition of a Buchsbaum-Eisenbud matrix.

Let J = Pf_{m−1}(Φ). By Lemma 2.3 it has the same radical as I_{m−1}(Φ). Then [8, Corollary 20.12] applied to G_{•} gives √I ⊂ √J. So J has codimension at least 3. By Theorem 4.7(a), R/J has minimal graded free resolution F_{•}(Φ). Let P be the vector of submaximal Pfaffians of Φ. Since both F_{•}(Φ) and G_{•} are exact we have that im(P^{T}) = ker(Φ) = im(Q^{T}). Hence Q is a scalar multiple of P and I = J. □

This completes the proof of Theorem 4.7. □

The following uniqueness result is sufficient for our purposes. It is indicated on page 471 of [6] that a stronger result is true.

Proposition 4.9. Let I ⊂ R be a Gorenstein ideal of codimension 3. Suppose that R/I has minimal graded free resolutions

0 ⟶ R(−s) ⟶^{Q_{1}^{T}} R(−s + t)^{m} ⟶^{Φ_{1}} R(−t)^{m} ⟶^{Q_{1}} R ⟶ 0

and

0 ⟶ R(−s) ⟶^{Q_{2}^{T}} R(−s + t)^{m} ⟶^{Φ_{2}} R(−t)^{m} ⟶^{Q_{2}} R ⟶ 0

where Φ_{1} and Φ_{2} are alternating matrices. Then Φ_{1} = cB^{T} Φ_{2}B for some B ∈ GL_{m}(k) and c ∈ k^{×}. Moreover B is uniquely determined up to scalars.

Proof: By Lemma 4.2 there is an isomorphism of complexes

[figure: 0; R(−s); Q_{1}^{T}; c^{−1}; R(−s + t)^{m}; Φ_{1}; B; R(−t)^{m}; Q_{1}; A^{−T}; R; 0; 0; R(−s); Q_{2}^{T}; R(−s + t)^{m}; Φ_{2}; R(−t)^{m}; Q_{2}; R; 0]

for some A, B ∈ GL_{m}(k) and c ∈ k^{×}. By [8, Lemma 20.3] this isomorphism is unique up to homotopy. Since we have restricted to the case of pure resolutions, there are in fact no homotopies. Hence A and B are uniquely determined. Comparing the above diagram with its dual we find that A = cB, and so Φ_{1} = cB^{T} Φ_{2}B.

For the final statement we suppose Φ_{1} = cB^{T} Φ_{2}B. Let P_{1} and P_{2} be the vectors of submaximal Pfaffians of Φ_{1} and Φ_{2}. Then Lemma 2.6 shows that P_{1}B^{T} is a scalar multiple of P_{2}. According to Proposition 4.8 the entries of P_{1} and P_{2} are bases for the vector space of t-ics generating I. Hence B is uniquely determined up to scalars. □

5. Klein matrices

Let C ⊂ ℙ^{n−1} be a genus one normal curve of degree n. The rth higher secant variety Sec^{r}C ⊂ ℙ^{n−1} was defined in §1. It has codimension max(n − 2r, 0). So if n is odd then there is a higher secant variety of codimension 3. This leads to a characterisation of Klein matrices in terms of higher secant varieties.

Proposition 5.1. Let n ≥ 3 be an odd integer. Let C be a genus one normal curve of degree n. Let Φ be an n×n alternating matrix of linear forms. Then Φ is a Klein matrix for C if and only if the submaximal Pfaffians of Φ generate I(Sec^{r}C) where r = (n − 3)/2.

Proof: See [12, Proposition 3.1]. □

It is well known that genus one normal curves are projectively Gorenstein. We recall the generalisation of this result to higher secant varieties. Let β(r, n) be the number of ways of choosing r elements of ℤ/nℤ such that no two elements are adjacent. By considering the subsets that do or do not contain a given element, one finds

(5)  β(r, n) = \binom{n − r}{r} + \binom{n − r − 1}{r − 1} = \frac{n(n − r − 1)!}{r!(n − 2r)!}.

We call a homogeneous polynomial of degree r an r-ic.

Proposition 5.2. Let C be a genus one normal curve of degree n. If m = n − 2r ≥ 2 then I = I(Sec^{r}C) is a Gorenstein ideal and R/I has a minimal graded free resolution of the form

\begin{matrix}0 → R(−n) → R(−n + r + 1)^{b_{m−1}} → R(−n + r + 2)^{b_{m−2}} → … \\ … → R(−r − 2)^{b_{2}} → R(−r − 1)^{b_{1}} → R → 0. \end{matrix}

where b_{1} = β(r + 1, n). In particular I is generated by a vector space of (r + 1)-ics of dimension β(r + 1, n).

Proof: It suffices to prove the proposition over an algebraically closed field, and for this we refer to [11] or [22]. □

We put everything together to prove the existence and uniqueness of Klein matrices over an arbitrary field.

Proof of Theorem 1.3: (a) Let C be a genus one normal curve of degree n = 2r + 3 and let I = I(Sec^{r}C). Proposition 5.2 tells us that I is a Gorenstein ideal of codimension 3, and that R/I has a minimal graded free resolution of the form

0 → R(−n) → R(−r − 2)^{n} → R(−r − 1)^{n} → R → 0.

We apply Theorem 4.7(b) to show that I is generated by the submaximal Pfaffians of an n × n alternating matrix of linear forms Φ. Then Φ is a Klein matrix by Proposition 5.1.

(b) Let n = 2r + 3 and let I = I(Sec^{r}C). By Proposition 5.1 and Theorem 4.7(a) the complexes F_{•}(Φ_{1}) and F_{•}(Φ_{2}) are minimal graded free resolutions of R/I. We are done by Proposition 4.9. □

In the next two sections we turn the above proof into an algorithm for computing Klein matrices.

6. Computing the higher secant varieties

Let X ⊂ ℙ^{n−1} be a projective variety. If I(X) has generators f_{1}, …, f_{m} then I(Sec X) may be computed by eliminating y_{1}, …, y_{n} from the ideal

(f_{1}(x + y), …, f_{m}(x + y), f_{1}(y), …, f_{m}(y))

in k[x_{1}, …, x_{n}, y_{1}, …, y_{n}]. This is extremely inefficient. The purpose of this section is to explain a better method. The following definition is not the usual one (see [8, §3.9]) but is convenient for our purposes.

Definition 6.1. The rth symbolic power of I(X) is the set of polynomials in R that vanish to order at least r at every point of X.

In the case of a genus one normal curve the symbolic powers and higher secant varieties are closely related.

Lemma 6.2. Let C be a genus one normal curve of degree n ≥ 2r + 2. Then I(Sec^{r}C) is generated by the vector space of (r + 1)-ics belonging to the rth symbolic power of I(C).

Proof: Granted Proposition 5.2 this is [11, Lemma 4.5]. □

The following variant of Definition 6.1 is more convenient for computations. We employ the multi-index notation x^{α} = x_{1}^{α_{1}} … x_{n}^{α_{n}} where α = (α_{1}, …, α_{n}) and |α| = α_{1} + … + α_{n}.

Definition 6.3. The rth infinitesimal power of I(X) is the set of polynomials f ∈ R with \frac{∂^{|α|}f}{∂x^{α}} ∈ I(X) for all multi-indices α with |α| < r.

If char (k) = 0 then the infinitesimal and symbolic powers agree. An efficient method for computing infinitesimal powers is given in [21, §2]. In our situation we know in advance the degrees of the polynomials we are looking for. This allows us to reduce the computation to linear algebra. As in §4 we write R_{d} for the vector space of homogeneous polynomials of degree d.

Proposition 6.4. Assume char (k) = 0. Let C be a genus one normal curve of degree n ≥ 2r + 2. Let f_{1}, …, f_{m} be a basis for the vector space of r-ics generating I(Sec^{r−1}C). Then I(Sec^{r}C) is generated by

V = \{g ∈ R_{r+1}|\frac{∂g}{∂x_{j}} ∈ ⟨f_{1}, …, f_{m}⟩ for all 1 ≤ j ≤ n\}
= \{∑ ℓ_{i}f_{i} | ∑ ℓ_{i} \frac{∂f_{i}}{∂x_{j}} ∈ ⟨f_{1}, …, f_{m}⟩ for all 1 ≤ j ≤ n\}

where the ℓ_{i} are linear forms.

Proof: Granted Proposition 5.2 it suffices to show that V is the vector space of (r + 1)-ics in I(Sec^{r}C). If g belongs to V then by Lemma 6.2 its partial derivatives belong to the (r − 1)st symbolic power of I(C). Since char (k) = 0 the infinitesimal and symbolic powers agree. So g belongs to the rth symbolic power of I(C). Applying Lemma 6.2 again shows that g belongs to I(Sec^{r}C). Conversely, we saw in [11, Remark 4.6] that if g belongs to I(Sec^{r}C) then its partial derivatives belong to I(Sec^{r−1}C). Since I(Sec^{r}C) ⊂ I(Sec^{r−1}C) the alternative description of V is clear. □

Remark 6.5. In practice we minimise the number of unknowns by using the first expression to calculate V if \binom{n+r}{r+1} < nβ(r, n), and the second expression otherwise.

We recall from [11, Proposition 7.3(i)] that if n = 2r + 1 then Sec^{r}C is a hypersurface of degree n. Proposition 6.4 leaves open the problem of computing an equation for this hypersurface. Fortunately this is not required for the computation of Klein matrices. The following discussion is included for completeness.

Lemma 6.6. Let C be a genus one normal curve of degree n = 2r + 3. Let p_{1}, …, p_{n} be a basis for the vector space of (r + 1)-ics generating I(Sec^{r}C). If r ≥ 1 then det(∂p_{i}/∂x_{j}) belongs to I(Sec^{r+1}C).

Proof: Let P belong to the linear span Π of a set of r + 1 distinct points on C, but not to the linear span of any proper subset. Then any (r + 1)-ic vanishing on Sec^{r}C and at P must also vanish on Π. It follows that the space of (r + 1)-ics vanishing on Sec^{r}C and singular at P has dimension at least r. In particular det(∂p_{i}/∂x_{j}) vanishes at P. We are done since the points P of the above form are Zariski dense in Sec^{r+1}C. □

If n = 5 then it turns out that det(∂p_{i}/∂x_{j}) is a non-zero quintic, and so a generator for I(Sec C). See [14, Theorem 5.3] or [17, VIII.2.5] for further discussion of this result. We give the following version.

Lemma 6.7. Let C be a genus one normal quintic. Let p_{1}, …, p_{5} be a basis for the vector space of quadrics generating I(C). Then the rank of the Jacobian matrix J = (∂p_{i}/∂x_{j}) evaluated at P ∈ ℙ^{4} is

rank J(P) = \{ \begin{matrix}3 & if P ∈ C, \\ 4 & if P ∈ Sec C \ C, \\ 5 & if P ∉ Sec C. \end{matrix}

In particular Sec C has equation det(J) = 0.

Proof. Let C_{P} ⊂ ℙ^{3} be the projection of C away from P. Then

rank J(P) ≤ 5 − r ⟺ C_{P} lies on r linearly independent quadrics.

If P ∉ Sec C then C_{P} is a smooth curve of degree 5. As explained in the proof of [17, IV.3.4], it cannot lie on any quadric. Thus rank J(P) = 5.

If P ∈ Sec C then Lemma 6.6 gives rank J(P) ≤ 4. If P ∉ C then C_{P} ⊂ ℙ^{3} is a curve of degree 5. It therefore lies on at most one quadric and so rank J(P) = 4. If P ∈ C then the Jacobian criterion for smoothness gives rank J(P) = 3. □

If n = 7 then we find in numerical examples that det(∂p_{i}/∂x_{j}) factorises as a scalar times the square of a 7-ic. It follows by Lemma 6.6 that this 7-ic is an equation for Sec^{3}C. We suggest the following generalisation of Lemma 6.7.

Conjecture 6.8. Assume char (k) = 0. Let C be a genus one normal curve of degree n = 2r +3. Let p_{1}, …, p_{n} be a basis for the vector space of (r + 1)-ics generating I(Sec^{r}C). Then det(∂p_{i}/∂x_{j}) is a generator for I(Sec^{r+1}C)^{r}.

7. Computing Klein matrices

Let C be a genus one normal curve of degree n = 2r + 3. We turn the methods of §§4,5 into a practical algorithm for computing Klein matrices. The first step, already discussed in §6, is to compute generators for I = I(Sec^{r}C).

Our next tasks are to compute the graded minimal free resolution of R/I, and to compute an isomorphism between this complex and its dual. Since we are dealing with pure resolutions such an isomorphism is unique. Proposition 4.4 tells us that, for a suitable choice of bases, the map in the middle of our complex is represented by an alternating matrix Φ. This matrix Φ is a Klein matrix for C.

In practice it is extremely wasteful to call a general procedure for computing minimal resolutions. Roughly speaking, we already know that the minimal resolution is self-dual, so we only need to compute half of it. We also know in advance the degrees of all polynomials that appear. This allows us to reduce the computation to linear algebra.

Proposition 7.1. Let I ⊂ R be a Gorenstein ideal of codimension 3. Suppose that R/I has minimal graded free resolution

0 ⟶ R(−s) ⟶^{Q_{2}^{T}} R(−s + t)^{m} ⟶^{Ψ} R(−t)^{m} ⟶^{Q_{1}} R ⟶ 0.

(i) There exists B ∈ GL_{m}(k) such that ΨB is alternating. (ii) The matrices B for which ΨB is alternating form a 1-dimensional vector subspace of Mat_{m}(k). (iii) If Φ = ΨB is alternating and P is the vector of submaximal Pfaffians of Φ then P = c Q_{1} for some c ∈ k. Moreover if B is non-zero then c is non-zero.

Proof: (i) This is immediate from Proposition 4.4. (ii) Let B ∈ Mat_{m}(k) be a non-zero matrix such that Φ = ΨB is alternating. We construct a commutative diagram

[figure: 0; R(−s); Q_{1}^{T}; R(−s + t)^{m}; Φ; B; R(−t)^{m}; Q_{1}; R; 0; 0; R(−s); Q_{2}^{T}; R(−s + t)^{m}; Ψ; R(−t)^{m}; Q_{1}; R; 0]

The top row is a complex since ΦQ_{1}^{T} = −(Q_{1}Φ)^{T} = 0. The dotted arrow is constructed by a diagram chase. It is multiplication by a constant c ∈ k. The commutativity of the lefthand square gives BQ_{1}^{T} = cQ_{2}^{T}. But the entries of Q_{1} are linearly independent and, recalling that the minimal resolution for R/I is self-dual, the same is true for Q_{2}. So this relation determines B uniquely up to scalars. (iii) If B = 0 then there is nothing to prove. Otherwise it follows by (i) and (ii) that B is invertible. In this case the above diagram is an isomorphism of complexes. The proof is completed by applying Proposition 4.8 to the top row. □

Our algorithm for computing Klein matrices is as follows. We assume char (k) = 0 and write R_{1} for the space of linear forms on ℙ^{n−1}.

Algorithm 7.2. Let n ≥ 5 be an odd integer. Let C ⊂ ℙ^{n−1} be a genus one normal curve of degree n and let r = (n − 3)/2. INPUT: A set of polynomials f_{1}, …, f_{m} defining C. OUTPUT: A Klein matrix Φ for C.

(1) Compute a minimal basis for the radical of (f_{1}, …, f_{m}). This reduces us to the case where f_{1}, …, f_{m} is a basis for the vector space of quadrics generating I(C) = I(Sec^{1}C).

(2) Use Proposition 6.4 to compute a basis p_{1}, …, p_{n} for the vector space of (r + 1)-ics generating I(Sec^{r}C).

(3) Compute an n × n matrix of linear forms Ψ whose columns generate the vector space {(ℓ_{1}, …, ℓ_{n}) ∈ R_{1}^{n}| ∑ ℓ_{i}p_{i} = 0}.

(4) Compute any non-zero solution B ∈ Mat_{n}(k) to ΨB = −B^{T} Ψ^{T}.

(5) Let Φ = ΨB. Check that the vector of submaximal Pfaffians of Φ is a non-zero scalar multiple of (p_{1}, …, p_{n}).

Step 1 may be accomplished using standard Gröbner basis algorithms, as implemented in MAGMA for example. Steps 2, 3 and 4 use only linear algebra. If n = 5 then there is nothing to do in Step 2. Step 5 checks that we have not made a mistake. Since the submaximal Pfaffians are homogeneous polynomials of degree (n − 1)/2, it is not always possible to get rid of the scalar in Step 5 by rescaling Φ.

The proof that Algorithm 7.2 computes a Klein matrix is as follows. Let Q_{1} = (p_{1}, …, p_{n}) and Ψ be the quantities computed in Steps 2 and 3. Then by Proposition 5.2 the hypotheses of Proposition 7.1 are satisfied for I = I(Sec^{r}C) and some (unknown) Q_{2}. The condition of Step 5 is satisfied by Proposition 7.1(iii). Finally Proposition 5.1 shows that Φ is a Klein matrix.

The proof of the uniqueness of Klein matrices given in §5 provides us with the following simple algorithm.

Algorithm 7.3. Let n ≥ 5 be an odd integer. Let C be a genus one normal curve of degree n. INPUT: Klein matrices Φ_{1} and Φ_{2} for C. OUTPUT: Elements B ∈ GL_{m}(k) and c ∈ k^{×} such that Φ_{1} = cB^{T} Φ_{2}B.

(1) Let P_{1}, P_{2} be the vectors of submaximal Pfaffians for Φ_{1}, Φ_{2}.

(2) Compute B ∈ GL_{m}(k) such that BP_{1}^{T} = P_{2}^{T}.

(3) Compute c ∈ k^{×} such that Φ_{1} = cB^{T} Φ_{2}B.

Implementations of Algorithms 7.2 and 7.3 in MAGMA [19] are available from the author’s website [13].

8. Examples

8.1. Elliptic curves. Let E have Weierstrass equation

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}.

We embed E in ℙ^{n−1} via |n.0| to give a genus one normal curve of degree n. If n = 5 then the embedding is given by

(x_{1} : …: x_{5}) = (1 : x : y : x^{2} : xy)

and the image has Klein matrix

Φ_{5} = (\begin{matrix}0 & a_{2}x_{4} + a_{6}x_{1} & x_{5} & x_{4} − a_{1}x_{3} + a_{4}x_{1} & −x_{3} \\ & 0 & −x_{4} & −x_{3} − a_{3}x_{1} & x_{2} \\ & & 0 & −x_{2} & 0 \\ & − & & 0 & −x_{1} \\ & & & & 0 \end{matrix}) .

If n = 7 then the embedding is given by

(x_{1} : …: x_{7}) = (1 : x : y : x^{2} : xy : x^{3} : x^{2}y).

and the image has Klein matrix

Φ_{7} = (\begin{matrix}0 & x_{7} & 0 & −p & x_{3} & x_{5} & 0 \\ & 0 & x_{6} & −q & −x_{5} & 0 & x_{4} \\ & & 0 & x_{5} & −x_{2} & −x_{4} & 0 \\ & & & 0 & r & s & −x_{3} \\ & & − & & 0 & x_{3} & x_{1} \\ & & & & & 0 & x_{2} \\ & & & & & & 0 \end{matrix}) ,

where

\begin{aligned}p & = & x_{6} − a_{1}x_{5} + a_{2}x_{4} − a_{3}x_{3} + a_{4}x_{2} + a_{6}x_{1}, \\ q & = & a_{3}x_{5} − a_{4}x_{4} − a_{6}x_{2}, \\ r & = & x_{4} − a_{1}x_{3} + a_{2}x_{2}, \\ s & = & a_{3}x_{3} − a_{4}x_{2} − a_{6}x_{1}. \end{aligned}

It may be checked directly from Definition 1.1 that Φ_{5} and Φ_{7} are Klein matrices. The above expressions are far from canonical. Different choices are related as described in Theorem 1.3(b).

8.2. Plane cubics. Let C be a smooth curve of genus one, realised as a plane cubic C_{3} ⊂ ℙ^{2} with hyperplane section H. We embed C in ℙ^{8} via |3H| to give a genus one normal curve C_{9}. We explain how to construct a Klein matrix for C_{9} from the ternary cubic U(x, y, z) defining C_{3}.

Let O(1) = O(H) be the line bundle associated to the embedding C_{3} ⊂ ℙ^{2}. It is tautological that C_{3} has Klein matrix

Φ_{3} = (\begin{matrix}0 & z & −y \\ −z & 0 & x \\ y & −x & 0 \end{matrix}) .

By Proposition 3.1, Φ_{3} = Φ(E) where E is an indecomposable rank 2 vector bundle on C with det E ≃ O(1). Since det(E ⊗ O(1)) ≃ O(3) it is natural to consider the matrix of cubic forms

(\begin{matrix}xI_{3} \\ yI_{3} \\ zI_{3} \end{matrix}) (\begin{matrix}0 & z & −y \\ −z & 0 & x \\ y & −x & 0 \end{matrix}) (xI_{3} yI_{3} zI_{3}) .

Let Φ′ be the 9×9 alternating matrix of linear forms on ℙ^{8} obtained by (i) substituting x_{1}, …, x_{10} for x^{3}, y^{3}, z^{3} …, xyz, and (ii) using the equation U = 0 to eliminate one of the x_{i}.

By [12, Lemma 4.2] we have

Φ′ = B^{T} Φ(E ⊗ O(1))B

where B represents the linear map

H^{0}(C, E) ⊗ H^{0}(C, O(1)) → H^{0}(C, E ⊗ O(1)).

It is clear that Φ′ has rank 2 on C_{9}. But since rows 1, 5 and 9 sum to zero, it is not a Klein matrix. However, after step (i), one of the 8 × 8 Pfaffians is a non-zero irreducible quartic. So rank(B) = 8 and Φ′ can be modified to give a Klein matrix by changing the entries in just one row and column.

The following recipe was found by studying numerical examples using Algorithm 7.2. We write our ternary cubic in the form U = xα+yβ+zγ where α, β, γ are quadrics in x, y, z. We then consider the matrix of cubic forms P^{T} MP where

M = (\begin{matrix}0 & z & −y & α \\ −z & 0 & x & β \\ y & −x & 0 & γ \\ −α & −β & −γ & 0 \end{matrix}) and P = (\begin{matrix}xI_{3} & yI_{3} & zI_{3} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) .

Let Φ be the 10 × 10 alternating matrix of linear forms on ℙ^{8} obtained by following steps (i) and (ii) above. Since pf(M) = xα + yβ + zγ it is clear that Φ has rank 2 on C_{9}. Deleting the last row and column gives the matrix Φ′ considered above. We claim that deleting the first row and column instead gives a Klein matrix for C_{9}.

To prove our claim it suffices to show that the last row of Φ cannot be written as a linear combination of the other rows. We suppose for a contradiction that this is possible. Reading off from the first 3 columns we find that there are linear forms p, q, r in x, y, z such that

(\begin{matrix}0 & z & −y \\ −z & 0 & x \\ y & −x & 0 \end{matrix}) (\begin{matrix}p \\ q \\ r \end{matrix}) = (\begin{matrix}α \\ β \\ γ \end{matrix}) .

This argument survives step (ii) since U is not divisible by x. Premultiplying by (x y z) it follows that U = xα + yβ + zγ is identically zero. This is the required contradiction.

8.3. Curves with a diagonal action of μ_{n}. In this subsection we assume that char (k)∤n. We take co-ordinates x_{0}, …, x_{n−1} on ℙ^{n−1} and agree to read all subscripts mod n. There is a diagonal action of μ_{n} on ℙ^{n−1} given by ζ : x_{i} ↦ ζ^{i}x_{i}, and an action of ℤ/nℤ given by x_{i} ↦ x_{i+1}.

Let n ≥ 5 be an odd integer, and let C be a genus one normal curve of degree n, invariant under the actions of μ_{n} and ℤ/nℤ. It is shown in [12, Proposition 2.7] that C has a rational point of the form

P_{0} = (0 : a_{1} : a_{2} : …: −a_{2} : −a_{1})

and that C has Klein matrix Φ = (a_{i−j}x_{i+j}). The 4 × 4 Pfaffians of this matrix appear in the works of Klein [18, §11] as relations between theta functions. This is our reason for calling Φ a Klein matrix.

Next we abandon the action of ℤ/nℤ.

Lemma 8.1. Let n = 5 or n = 7. Let C be a genus normal curve of degree n invariant under the diagonal action of μ_{n}. Then I(C) is generated by the quadrics

n = 5  τ_{0}x_{0}^{2} + x_{1}x_{4} − τ_{2}τ_{3}x_{2}x_{3} & cyclic permutes
n = 7 \{ \begin{matrix}τ_{0}x_{0}^{2} + x_{1}x_{6} − (1/λ^{2})τ_{2}τ_{3}τ_{4}τ_{5}x_{2}x_{5} \\ τ_{0}x_{0}^{2} + λx_{1}x_{6} − (1/λ^{3})τ_{2}τ_{3}^{2}τ_{4}^{2}τ_{5}x_{3}x_{4} \end{matrix} & cyclic permutes

for some non-zero λ, τ_{0}, …, τ_{n−1} ∈ k satisfying

∏_{i=0}^{n−1} τ_{i} = \{ \begin{matrix}λ & if n = 5, \\ λ^{4}(λ − 1) & if n = 7. \end{matrix}

Proof: See [10, Proposition 2.8]. □

The variable λ in Lemma 8.1 should be viewed as a co-ordinate on X_{1}(n) ≃ ℙ^{1}. Away from a cusp the above equations define a genus one normal curve. The μ_{5}-invariant curves have Klein matrix

(\begin{matrix}0 & τ_{1}x_{1} & x_{2} & −x_{3} & −τ_{4}x_{4} \\ & 0 & τ_{3}x_{3} & x_{4} & −x_{0} \\ & & 0 & τ_{0}x_{0} & x_{1} \\ & − & & 0 & τ_{2}x_{2} \\ & & & & 0 \end{matrix}) .

The μ_{7}-invariant curves have Klein matrix

( \begin{matrix}0 & τ_{0}τ_{1}τ_{2}x_{1} & −λτ_{2}x_{2} & −λ^{2}x_{3} & λ^{2}x_{4} & λτ_{5}x_{5} & −τ_{5}τ_{6}τ_{0}x_{6} \\ & 0 & τ_{2}τ_{3}τ_{4}x_{3} & −λτ_{4}x_{4} & −λ^{2}x_{5} & λ^{2}x_{6} & λτ_{0}x_{0} \\ & & 0 & τ_{4}τ_{5}τ_{6}x_{5} & −λτ_{6}x_{6} & −λ^{2}x_{0} & λ^{2}x_{1} \\ & & & 0 & τ_{6}τ_{0}τ_{1}x_{0} & −λτ_{1}x_{1} & −λ^{2}x_{2} \\ & & − & & 0 & τ_{1}τ_{2}τ_{3}x_{2} & −λτ_{3}x_{3} \\ & & & & & 0 & τ_{3}τ_{4}τ_{5}x_{4} \\ & & & & & & 0 \end{matrix}) .

In [10] we took k = ℚ and exhibited some counterexamples to the Hasse principle. For example

\begin{aligned}n = 5 & (λ; τ_{0}, …, τ_{4}) & = & (30; 1, 1, 2, 3, 5), \\ n = 7 & (λ; τ_{0}, …, τ_{6}) & = & (6; 1, 1, 1, 1, 2^{4}, 3^{4}, 5). \end{aligned}

8.4. Numerical examples. Let k = ℚ. In Cremona’s tables [7] the first elliptic curve with non-trivial 5-torsion in its Tate-Shafarevich group is labelled 275B3. We saw in [9, Appendix B] that its Tate-Shafarevich group contains a non-zero element with equations

\begin{aligned}0 & = & 11x_{1}^{2} − x_{2}^{2} + 5x_{3}^{2} + x_{4}^{2} − 5x_{5}^{2} \\ 0 & = & x_{2}^{2} + 10x_{2}x_{3} + 5x_{3}^{2} + 2x_{1}x_{4} − x_{2}x_{4} − 5x_{3}x_{4} − 5x_{2}x_{5} − 5x_{3}x_{5} \\ 0 & = & x_{2}^{2} + 2x_{2}x_{3} + 5x_{3}^{2} + x_{2}x_{4} + x_{3}x_{4} + 2x_{1}x_{5} + x_{2}x_{5} + 5x_{3}x_{5} \\ 0 & = & 11x_{1}x_{2} + 55x_{1}x_{3} + 2x_{2}x_{4} + 2x_{4}^{2} − 10x_{3}x_{5} + 10x_{5}^{2} \\ 0 & = & 11x_{1}x_{2} + 11x_{1}x_{3} − 2x_{3}x_{4} + 2x_{2}x_{5} − 4x_{4}x_{5}. \end{aligned}

Algorithm 7.2 computes the Klein matrix

( \begin{matrix}0 & −2x_{5} & 2x_{4} & −x_{2} − x_{3} & x_{2} + 5x_{3} \\ & 0 & −11x_{1} & x_{3} + x_{5} & x_{2} + x_{4} \\ & & 0 & −x_{2} + x_{4} & −5x_{3} + 5x_{5} \\ & − & & 0 & −x_{1} \\ & & & & 0 \end{matrix}) .

In the case n = 7 the only known explicit counterexamples to the Hasse principle have a diagonal action of μ_{7}, so have already been treated in §8.3. To give another numerical example we embed the elliptic curve y^{2} + y = x^{3} − x in ℙ^{6} via |6.0 + P | where P = (0, 0). The embedding is given by

(x_{1} : …: x_{7}) = (1 : x : y : x^{2} : xy : x^{3} : (y + 1)/x)

and the image has Klein matrix

( \begin{matrix}0 & x_{1} & −x_{2} & 0 & x_{2} & x_{4} & −x_{1} + x_{3} \\ & 0 & x_{7} & x_{2} & −2x_{7} & x_{1} − x_{3} & −x_{2} \\ & & 0 & −x_{4} & x_{1} + x_{3} & −2x_{2} & x_{1} + x_{7} \\ & & & 0 & −x_{4} & −x_{6} & x_{2} − x_{5} \\ & & − & & 0 & 3x_{2} + x_{5} & −2x_{1} + x_{4} − 2x_{7} \\ & & & & & 0 & x_{1} − x_{2} − 3x_{3} \\ & & & & & & 0 \end{matrix}) .

See [13] for further numerical examples, up to degree n = 11.

9. Relationship with the index

In this section we work over a perfect field k.

Definition 9.1. Let C be a smooth projective curve defined over k. The index of C is the least positive degree of a k-rational divisor on C.

It is clear that the index of a genus one normal curve divides its degree. We give a criterion for these numbers to be equal.

Proposition 9.2. Let n ≥ 5 be an odd integer. Let C be a genus one normal curve of degree n with Klein matrix Φ. Then C has index n if and only if rank Φ(P) = n − 1 for every P ∈ ℙ^{n−1}(k).

Before giving the proof we need a little more notation. Let H be the divisor of a hyperplane section. For D an effective divisor on C we write \overline{D} for the linear subspace cut out by the Riemann-Roch space L(H −D) ⊂ L(H). So if D is a sum of distinct points then \overline{D} is simply the linear span of these points. In this notation

(6)  Sec^{r}C = ⋃_{deg D=r} \overline{D}.

Lemma 9.3. Let C be a genus one normal curve of degree n, and let D, D_{1}, D_{2} be effective divisors on C.

(i) If deg D < n then dim \overline{D} = deg D − 1. (ii) The linear span of \overline{D_{1}} and \overline{D_{2}} is \overline{lcm(D_{1}, D_{2})}. (iii) If deg lcm(D_{1}, D_{2}) < n then \overline{D_{1}} ∩ \overline{D_{2}} = \overline{gcd(D_{1}, D_{2})}.

Proof: (i) This is immediate from Riemann-Roch. (ii) We have L(H − D_{1}) ∩ L(H − D_{2}) = L(H − lcm(D_{1}, D_{2})). (iii) The inclusion “⊃” is clear. Equality follows by counting dimensions using (i) and (ii). □

Proof of Proposition 9.2: Let P ∈ ℙ^{n−1} with rank Φ(P) < n − 1. Theorem 1.2 tells us that P ∈ Sec^{r}C for some r ≤ (n − 3)/2. We take r minimal so that P ∉ Sec^{r−1}C. Then (6) gives P ∈ \overline{D} where D is an effective divisor on C of degree r. If P is k-rational then P^{σ} = P for all σ ∈ Gal(\overline{k}/k). So by Lemma 9.3(iii),

P ∈ \overline{D} ∩ \overline{D^{σ}} = \overline{gcd(D, D^{σ})}.

Since P ∉ Sec^{r−1}C it follows that D^{σ} = D. So D is a k-rational divisor of degree r and this shows that the index of C is less than n.

Conversely, suppose that C has index r with r < n. Since r | n and n ≥ 5 is odd, it follows that r ≤ (n − 3)/2. Let D be a k-rational divisor on C of degree r. By Riemann-Roch we may assume that D is effective. Then \overline{D} is a k-rational linear subspace of ℙ^{n−1}. Let P ∈ \overline{D} be any k-rational point. Since Φ has rank at most 2 on C it follows that rank Φ(P) ≤ 2r < n − 1. □

Remark 9.4. Let E be an elliptic curve curve defined over a number field k. It is conjectured that the Tate-Shafarevich group of E is finite, equivalently there is a bound on the index of an everywhere locally soluble genus one curve with Jacobian E. It is hoped that Proposition 9.2 might be used to show that genus one curves of large index are “arithmetically complicated” in some precise sense (perhaps in terms of a discriminant or a height).

A related result is the following.

Corollary 9.5. Let n ≥ 5 be an odd prime power. Let C be a genus one normal curve of degree n with Klein matrix Φ. If C has index n then every entry of Φ, not on the diagonal, is non-zero.

Proof: We suppose for a contradiction that Φ_{12} = 0. Then the linear forms in the first row of Φ vanish on a line ℓ ⊂ ℙ^{n−1}. Restricting the submaximal Pfaffians of Φ to ℓ we are left with a single homogeneous polynomial of degree d = (n − 1)/2. Since (d, n) = 1 and n is a prime power, this polynomial has a root over a field k_{1} with [k_{1} : k] prime to n. So given that C has index n over k, it still has index n over k_{1}. This is a contradiction to Proposition 9.2. □

References

[1] M.F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3), 7 (1957) 414–452.

[2] A. Aure, W. Decker, K. Hulek, S. Popescu, K. Ranestad, Syzygies of abelian and bielliptic surfaces in ℙ^{4}, Internat. J. Math. 8 (1997), no. 7, 849–919.

[3] N. Bourbaki, Algèbre, Chap. I-X. Hermann, Masson, 1970-1980.

[4] W. Bruns, J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, 1993.

[5] D.A. Buchsbaum and D. Eisenbud, Gorenstein ideals of height 3, Seminar D. Eisenbud/B. Singh/W. Vogel, Vol. 2, pp. 30–48, Teubner-Texte zur Math., 48, Teubner, Leipzig, 1982.

[6] D.A. Buchsbaum and D. Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math. 99 (1977) 447-485.

[7] J.E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1997.

[8] D. Eisenbud, Commutative algebra with a view toward algebraic geometry, GTM 150, Springer-Verlag, 1995.

[9] T.A. Fisher, On 5 and 7 descents for elliptic curves, PhD thesis, University of Cambridge, 2000.

[10] T.A. Fisher, Some examples of 5 and 7 descent for elliptic curves over ℚ, J. Eur. Math. Soc. 3 (2001), 169–201.

[11] T.A. Fisher, The higher secant varieties of an elliptic normal curve, preprint.

[12] T.A. Fisher, Pfaffian presentations of elliptic normal curves, preprint.

[13] T. A. Fisher, website: http://www.dpmms.cam.ac.uk/∼taf1000

[14] M. Gross, S. Popescu, Equations of (1, d)-polarized abelian surfaces. Math. Ann. 310 (1998), no. 2, 333–377.

[15] R. Hartshorne, Algebraic geometry, GTM 52, Springer-Verlag, 1977.

[16] P. Heymans, Pfaffians and skew-symmetric matrices, Proc. London Math. Soc. (3) 19 (1969) 730–768.

[17] K. Hulek, Projective geometry of elliptic curves, Soc. Math. de France, Astérisque 137 (1986).

[18] F. Klein, Über die elliptischen Normalkurven der n-ten Ordnung (1885), in Gesammelte Mathematische Abhandlungen, 3: Elliptische Funktionen etc., R. Fricke et al (eds.) Springer (1923).

[19] MAGMA is described in W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symb. Comb. 24, 235-265 (1997). (See

    also the Magma home page at http://magma.maths.usyd.edu.au/magma/.)

[20] J.-P. Serre, Algebraic groups and class fields, GTM 117, Springer-Verlag, 1988.

[21] A. Simis, Effective computation of symbolic powers by Jacobian matrices, Comm. Algebra 24 (1996), no. 11, 3561–3565.

[22] H.-Chr. Graf v. Bothmer, K. Hulek, Geometric syzygies of elliptic normal curves and their secant varieties, Manuscripta Math. 113 (2004), no. 1, 35–68.

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

  E-mail address: T.A.Fisher@dpmms.cam.ac.uk