THE HIGHER SECANT VARIETIES OF AN ELLIPTIC NORMAL CURVE

TOM FISHER

Abstract. We study the higher secant varieties of an elliptic normal curve, determining the form of their minimal free resolutions and classifying their determinantal presentations.

1. Introduction

We work over an algebraically closed field k of arbitrary characteristic. An elliptic normal curve C ⊂ ℙ^{n−1} is a smooth curve of genus one and degree n that is contained in no hyperplane. The rth higher secant variety Sec^{r}C is the Zariski closure of the locus of all (r − 1)-planes spanned by r points of C. It is shown in [L] that Sec^{r}C is an irreducible variety of codimension max(n − 2r, 0).

Let R = k[x_{1}, …, x_{n}] be the homogeneous co-ordinate ring of ℙ^{n−1}. For M = ⊕M_{d} a graded R-module we write M(c) for the graded Rmodule with M(c)_{d} = M_{c+d}. We give a new proof of the following theorem that was recently proved independently by Graf v. Bothmer and Hulek [vBH].

Theorem 1.1. Let C ⊂ ℙ^{n−1} be an elliptic normal curve of degree n. Let m = n − 2r. If m ≥ 1 then the homogeneous co-ordinate ring of Sec^{r}C 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}

In particular Sec^{r}C is projectively Gorenstein of codimension m.

Our proof is different from that given in [vBH, §8] in that we build the minimal free resolutions by induction on r and n. The induction step is closely related to a technique of Kustin and Miller [KM, Theorem 1.5] for constructing new Gorenstein ideals from old. This approach provides additional information that is essential to our subsequent work on Pfaffian presentations of elliptic normal curves [F2]. It also serves as a prototype for our forthcoming work on the the minimisation of genus one curves. In this application we use the main results of [CS] and [F1] to start the induction.

Date: 30th May 2006.

A great deal of information may be read off from Theorem 1.1. If m ≥ 2 then the theorem is equivalent to the statement that the homogeneous co-ordinate ring of Sec^{r}C is an extremal Gorenstein ring (in the sense of Schenzel [S]) with a-invariant 0. As we recall in §2, the Betti numbers, Hilbert series and Hilbert polynomial of an extremal Gorenstein ring are explicitly known. For instance we deduce that if n − 2r ≥ 1 then Sec^{r}C has degree

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

Alternative proofs are given in [Ro, §9.3] and [vBH, Proposition 8.5]. The numbers β(r, n) are most conveniently thought of as the number of ways of choosing r elements from ℤ/nℤ such that no two elements are adjacent. The expression as a sum of binomial coefficients is found by considering the subsets that do or do not contain a given element.

We call a homogeneous polynomial of degree r an r-ic. Theorem 1.1 has the following consequence.

Theorem 1.2. Let C ⊂ ℙ^{n−1} be an elliptic normal curve of degree n. The space of (r + 1)-ics containing Sec^{r}C has dimension β(r + 1, n). If n − 2r ≥ 2 then these (r + 1)-ics generate the homogeneous ideal of Sec^{r}C.

In fact we prove Theorem 1.2 first, and then use it in the proof of Theorem 1.1. Both proofs are by induction on r and n, and are guided by the recurrence relation

β(r + 1, n) = β(r + 1, n − 1) + β(r, n − 2).

We adapt the proof of Theorem 1.2 to give the following result on determinantal presentations. First we fix some notation. We write L(D) = H^{0}(C, O(D)) for the Riemann-Roch space associated to a divisor D on C. Let H be the divisor of a hyperplane section, and let D_{1}, D_{2} be divisors on C with D_{1} + D_{2} = H. We write Φ(D_{1}, D_{2}) for the matrix of linear forms representing the multiplication map

L(D_{1}) × L(D_{2}) → L(H).

It is clear that Φ(D_{1}, D_{2}) has rank at most 1 on C, and so has rank at most r on Sec^{r}C. We write I(X) for the homogeneous ideal of a projective variety X.

Definition 1.3. A matrix of linear forms is a determinantal presentation of Sec^{r}C if its (r + 1) × (r + 1) minors generate I(Sec^{r}C).

Determinantal presentations for curves of arbitrary genus have been studied in [EKS]. A set-theoretic generalisation to higher secant varieties is given in [Ra]. In §10 we establish the following analogue of the main result of [EKS].

Theorem 1.4. Let n ≥ 2r + 1. Then Φ(D_{1}, D_{2}) is a determinantal presentation of Sec^{r}C if and only if

(i) deg D_{1}, deg D_{2} ≥ r + 2, and (ii) if deg D_{1} = deg D_{2} = r + 2 then D_{1} ≁ D_{2}.

An easy corollary of Theorem 1.4 is that if n ≥ 2r + 1 then Sec^{r}C has singular locus Sec^{r−1}C. It follows that an elliptic normal curve is uniquely determined by any one of its higher secant varieties that is not the whole of projective space. Since the method is closely related to that given in [vBH, Proposition 8.15] we omit the details. The case r = 2 was previously treated in [GP, Proposition 5.1].

If r = 1 then Sec^{r}C = C. Theorems 1.1 and 1.2 are well known in this case. Theorem 1.1 is stated in [GP, Theorem 5.5] and supported by a reference to [E, Exercise A2.22]. It can also be deduced from the fact C has trivial canonical sheaf. Theorem 1.2 asserts that the space of quadrics vanishing on an elliptic normal curve of degree n ≥ 3 has dimension n(n − 3)/2, and that if n ≥ 4 then these quadrics generate the homogeneous ideal. Generalisations are known both for curves of genus g [M, Corollary to Theorem 8], and for abelian varieties [LB, Chapter 7, Section 4]. The statement that an elliptic normal curve of degree n ≥ 4 is defined by quadrics is proved in [H, IV.1.3].

If r = 2 then Sec^{2}C = Sec C is the ordinary secant variety. In this case, it is shown in [GP, Theorem 5.5] that Theorem 1.1 holds for all but finitely many j-invariants. They compute the degree of the secant variety by the following method, of which [vBH, Proposition 8.5] is a generalisation. Projecting C away from a general (n − 4)-plane we obtain a plane curve C′ with d nodes. Then computing the arithmetic genus of C′ in two different ways gives d + 1 = (n − 1)(n − 2)/2. So Sec C has degree d = n(n − 3)/2 = β(2, n) as claimed.

The special cases where Sec^{r}C ⊂ ℙ^{n−1} has small codimension are also of interest.

If m = 1 then Theorem 1.1 asserts that Sec^{r}C is a hypersurface of degree n. This was previously known when the characteristic of k does not divide n. (The proof for n = 7 given in [GP, Example 2.10] generalises immediately.) We remove this restriction on the characteristic.

If m = 2 then Theorem 1.1 asserts that Sec^{r}C is the complete intersection of two (r + 1)-ics. This is a result of Room (see §7).

If m = 3 then Theorem 1.1 asserts that the homogeneous ideal I = I(Sec^{r}C) is Gorenstein of codimension 3, and that I is generated by a space of (r + 1)-ics of dimension n. By the Buchsbaum-Eisenbud structure theorem [BE1], [BE2] we can write these (r + 1)-ics as the submaximal Pfaffians of an n × n alternating matrix of linear forms on ℙ^{n−1}. This application, explained further in [F2], was the motivation for the present work.

Let us also note that our practical algorithm for computing the Jacobian of an elliptic normal curve of degree 5 (see [F3]) relies on a geometric “accident” that we initially checked by a generic calculation over the modular curve X(5), but is explained here in Corollary 7.5. This application is of course an arithmetic one, since over an algebraically closed field a smooth curve of genus one is its own Jacobian.

The plan of the paper is as follows. In §2 we give the necessary background on extremal Gorenstein ideals. (This will only be needed in §§3,9.) The results of Gross and Popescu [GP] in the case r = 2 were obtained by letting an elliptic normal curve degenerate to a Néron polygon. We give details in §3. In §4 we make some general remarks on higher secant varieties. Then in §§5,6 we restrict to the case of an elliptic normal curve and prove the first part of Theorem 1.2. The second part of Theorem 1.2 is proved in §§7,8. Finally in §9 and §10 we present our proofs of Theorem 1.1 and Theorem 1.4.

2. Extremal Gorenstein ideals

Let R = k[x_{1}, …, x_{n}] be the homogeneous co-ordinate ring of ℙ^{n−1}. Let R_{+} = ⊕_{d≥1}R_{d} be the irrelevant ideal and let M be a finitely generated graded R-module.

Definition 2.1. 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 called minimal if ϕ_{i}(F_{i}) ⊂ R_{+}F_{i−1} for all i.

Lemma 2.2. 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 [E, §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 is equal to n − dim R/I.

Proposition 2.3. Let I ⊂ R be a homogeneous ideal. The following are equivalent.

(i) The quotient ring R/I is a Cohen-Macaulay ring. (ii) proj dim R/I = codim I.

Proof: This follows from [BH, Corollary 2.2.15] or the graded analogue of [E, Corollary 19.5]. □

Definition 2.4. If the conditions of Proposition 2.3 are satisfied then we say that I is a perfect ideal.

Proposition 2.5. Let I ⊂ R be a perfect ideal with minimal graded free resolution

F_{•} : 0 → F_{m} → F_{m−1} → … → F_{1} → F_{0} → 0.

The following are equivalent.

(i) The quotient ring R/I is a Gorenstein ring. (ii) The graded free R-module F_{m} has rank 1. (iii) The complex of free R-modules F_{•} is self-dual.

Proof: This is the graded analogue of [E, Corollary 21.16]. □

Definition 2.6. If the conditions of Proposition 2.5 are satisfied then we say that I is a Gorenstein ideal.

We recall from [BH, Definition 4.4.4]:

Definition 2.7. Let I ⊂ R be a homogeneous ideal. The a-invariant a(R/I) is the degree of the Hilbert series of R/I viewed as a rational function.

Lemma 2.8. Let I ⊂ R be a Gorenstein ideal of codimension m ≥ 2 with I_{d} = 0 for all d ≤ r. Then

a(R/I) ≥ m + 2r − n.

In the case of equality I is generated in degree r + 1.

Proof: Let R/I have minimal graded free resolution

F_{•} : 0 → F_{m} → F_{m−1} → … → F_{1} → F_{0} → 0.

Let d_{i} be the smallest integer such that R(−d_{i}) appears as a direct summand of F_{i}. Since F_{•} is a minimal resolution the sequence d_{1}, d_{2}, …, d_{m} is strictly increasing. We are given that d_{1} ≥ r+1. Since F_{•} is self-dual it follows that d_{m} ≥ d_{m−1} + (r + 1). Thus d_{m} ≥ m + 2r. Using this resolution to compute the Hilbert series we find

a(R/I) = d_{m} − n ≥ m + 2r − n.

In the case of equality we have d_{i} + d_{m−i} = m + 2r for all 0 ≤ i ≤ m. Then since F_{•} is self-dual it must also be pure, i.e. each F_{i} is a direct sum of copies of R(−d_{i}). In particular I is generated in degree d_{1} = r + 1. □

Definition 2.9. If equality holds in Lemma 2.8 then we say that I is an extremal Gorenstein ideal, and that R/I is an extremal Gorenstein ring.

It is shown in [BH, Proposition 4.1.12] that the a-invariant is the largest integer for which the Hilbert function and Hilbert polynomial differ. So our definition of an extremal Gorenstein ring is equivalent to that originally made by Schenzel. The numerical properties of an extremal Gorenstein ring were determined by Schenzel [S, Theorem B]. We rewrite some of his expressions using the numbers β(r, n) defined in the introduction.

Proposition 2.10. Let I ⊂ R be an extremal Gorenstein ideal of codimension m, generated in degree r + 1. Let a = a(R/I) = m + 2r − n.

(i) The minimal graded free resolution of R/I takes the form

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

(ii) The Betti numbers b_{i} = b_{i}(r, m) are given by

b_{i}(r, m) = \frac{1}{r!}.\frac{m + 2r}{(i + r)(m − i + r)}.\frac{(m + r − 1)!}{(i − 1)!(m − i − 1)!}.

In particular I_{r+1} has dimension b_{1}(r, m) = β(r + 1, n + a). (iii) The Hilbert series of R/I is

h(t) = (1 − t)^{a} ∑_{ρ=0}^{r} β(ρ, n + a)t^{ρ}/(1 − t)^{2ρ}.

(iv) The Hilbert polynomial of R/I is

H(d) = ∑_{ρ=0}^{r} β(ρ, n + a)\binom{d + ρ − a − 1}{2ρ − a − 1}.

In particular R/I has multiplicity β(r, n + a).

Proof: (i) This is a consequence of equality in Lemma 2.8. (ii) By [BH, Theorem 4.1.15(a)] we have

b_{i}(r, m) = \frac{m + 2r}{m + r − i} ∏_{j=r+1,j≠r+i}^{m+r−1} \frac{j}{|r + i − j|}.

(iii) The Hilbert series of R/I is

h(t) = (1 + ∑_{i=1}^{m−1}(−1)^{i}b_{i}t^{r+i} + (−1)^{m}t^{m+2r})/(1 − t)^{n}.

Since b_{i}(0, m) = \binom{m}{i}, this reduces in the case r = 0 to

h(t) = (1 − t)^{m−n} = (1 − t)^{a}.

The general case follows by induction on r using the identity

b_{i}(r, m) + b_{i+1}(r − 1, m + 2) = β(r, m + 2r)\binom{m}{i}.

(iv) This follows from (iii) and the binomial theorem. □

Remark 2.11. Schenzel [S] takes the proof in a different order, leading to different (but of course equivalent) expressions. By reducing to the zero-dimensional case he computes the Hilbert series as

h(t) = ∑_{ρ=−r}^{r} \binom{m − 1 − r + |ρ|}{m − 1}t^{r+ρ}/(1 − t)^{n−m}.

He then computes the Betti numbers from the Hilbert series, and leaves the answer in terms of binomial coefficients.

If n−2r ≥ 2 then Theorem 1.1 is equivalent to the statement that the homogeneous co-ordinate ring of Sec^{r}C is an extremal Gorenstein ring with a-invariant 0. The Hilbert series, Hilbert function and degree of Sec^{r}C are found by setting a = 0 in the above formulae. In particular Theorem 1.2 may be deduced from Theorem 1.1.

3. Néron polygons and cyclic polytopes

Definition 3.1. Let n ≥ 3 be an integer. A Néron polygon C ⊂ ℙ^{n−1} of degree n is the union of n lines ℓ_{1}, …, ℓ_{n} spanning ℙ^{n−1} and arranged such that ℓ_{i} meets ℓ_{j} if and only if i − j ≡ ±1 (mod n).

It is well known that Néron polygons arise as degenerations of elliptic normal curves. In this section we show, following [GP], that Theorems 1.1 and 1.2 hold when C is replaced by a Néron polygon. This provides a valuable heuristic and explains the appearance of the numbers β(r, n). However our proofs of Theorems 1.1 and 1.2 are independent of the results in this section.

Let ℙ^{n−1} have co-ordinates (x_{1} : …: x_{n}), where the subscripts are read modulo n. Let Γ_{n} be the standard Néron polygon, that is to say, with vertices P_{1} = (1 : 0 : 0 : …: 0), P_{2} = (0 : 1 : 0 : …: 0), …, P_{n} = (0 : 0 : …: 0 : 1) and edges ℓ_{i} = P_{i}P_{i+1} for i ∈ ℤ/nℤ. The following definitions are recalled from [BH, §5.1].

Definition 3.2. A simplicial complex Δ on vertex set V = {v_{1}, …, v_{n}} is a collection of subsets of V, called faces, such that

(i) If F ∈ Δ and G ⊂ F then G ∈ Δ. (ii) All singleton sets {v_{i}} belong to Δ.

Definition 3.3. Let Δ be a simplicial complex on vertex set {1, …, n}. The Stanley-Reisner ring of Δ is k[Δ] = k[x_{1}, …, x_{n}]/I_{Δ} where I_{Δ} is the ideal generated by all monomials ∏_{i∈F} x_{i} with F ∉ Δ.

For r ≥ 1 and n ≥ 2r + 1 we put

Δ_{r,n} = {A ⊂ ℤ/nℤ : A ⊂ B ∪ (1 + B) for some |B| ≤ r}.

Then Δ_{r,n} is a simplicial complex on vertex set ℤ/nℤ and k[Δ_{r,n}] is the homogeneous co-ordinate ring of Sec^{r}Γ_{n}.

Lemma 3.4. Let A ⊂ ℤ/nℤ be a proper subset.

(i) There is a unique subset A^{∗} ⊂ ℤ/nℤ satisfying

A^{∗} ⊂ A ⊂ A^{∗} ∪ (1 + A^{∗}) and A^{∗} ∩ (1 + A^{∗}) = ∅.

(ii) |A^{∗}| = min{|B| : A ⊂ B ∪ (1 + B)}. (iii) |A^{∗}| = max{|B| : B ⊂ A, B ∩ (1 + B) = ∅}.

Proof: (i) We may assume that 0 ∉ A. Initially we take A^{∗} = ∅. Then for i = 1, 2, …, n−1 we add i to A^{∗} if i ∈ A and i−1 ∉ A^{∗}. This construction yields the unique subset A^{∗} with the stated properties. (ii) and (iii). It is clear that the righthand side of (iii) is at most the righthand side of (ii). The existence of A^{∗} establishes equality. □

Lemma 3.5. The maximal faces (facets) of Δ_{r,n} are

{B ∪ (1 + B) : B ⊂ ℤ/nℤ, |B| = r, B ∩ (1 + B) = ∅}.

In particular Sec^{r}Γ_{n} is the union of β(r, n) (2r − 1)-planes.

Proof: Since ℤ/nℤ is not a face we have

Δ_{r,n} = {A ⊂ ℤ/nℤ : |A^{∗}| ≤ r}.

By Lemma 3.4(ii) every maximal face A satisfies |A^{∗}| = r. Since A ⊂ A^{∗} ∪ (1 + A^{∗}) the maximality of A then gives A = A^{∗} ∪ (1 + A^{∗}). □

If n = 2r + 1 then Δ_{r,n} consists of all proper subsets of ℤ/nℤ and I(Sec^{r}Γ_{n}) is generated by ∏_{i=1}^{n} x_{i}.

Lemma 3.6. If n ≥ 2r + 2 then the minimal non-faces of Δ_{r,n} are

{B : B ⊂ ℤ/nℤ, |B| = r + 1, B ∩ (1 + B) = ∅}.

In particular I(Sec^{r}Γ_{n}) is generated by a space of (r + 1)-ics of dimension β(r + 1, n).

Proof: The non-faces of Δ_{r,n} are

{A ⊂ ℤ/nℤ : A = ℤ/nℤ or |A^{∗}| ≥ r + 1}.

Since n ≥ 2r + 2 the non-face ℤ/nℤ is not minimal. By Lemma 3.4(iii) every minimal non-face A satisfies |A^{∗}| = r + 1. Since A ⊃ A^{∗} the minimality of A then gives A = A^{∗}. □

Lemma 3.7. (i) The Hilbert series of k[Δ_{r,n}] is

h(t) = ∑_{ρ=0}^{r} β(ρ, n)t^{ρ}/(1 − t)^{2ρ}.

(ii) The Hilbert function of k[Δ_{r,n}] is

H(d) = \{ \begin{matrix}1 & if d = 0 \\ ∑_{ρ=1}^{r} β(ρ, n)\binom{d+ρ−1}{2ρ−1} & if d > 0. \end{matrix}

Proof: The support of a monomial m = x_{1}^{a_{1}} … x_{n}^{a_{n}} is

supp(m) = {i ∈ ℤ/nℤ : a_{i} ≠ 0}.

We must count the monomials of degree d with supp(m) ∈ Δ_{r,n}. Notice that supp(m)^{∗} = B if and only if m is the product of ∏_{i∈B} x_{i} and a monomial m′ with supp(m′) ⊂ B ∪ (1 + B). So for d ≥ 1,

H(d) = ∑_{|B|≤r,B∩(1+B)=∅} # \{m : \begin{matrix}deg(m) = d, \\ supp(m)^{∗} = B \end{matrix} \}

= ∑_{ρ=1}^{r} ∑_{|B|=ρ,B∩(1+B)=∅} # \{m′ : \begin{matrix}deg(m′) = d − ρ, \\ supp(m′) ⊂ B ∪ (1 + B) \end{matrix} \}

= ∑_{ρ=1}^{r} β(ρ, n)\binom{d + ρ − 1}{2ρ − 1}.

The expression for the Hilbert series h(t) = ∑_{d≥0} H(d)t^{d} follows. □

To show that k[Δ_{r,n}] is Gorenstein, we quote

Proposition 3.8. If Δ is a simplicial complex whose geometric realisation is homeomorphic to a sphere, then k[Δ] is a Gorenstein ring.

Proof: See [BH, Corollary 5.6.5]. □

The moment curve M_{d} is the image of

ϕ_{d} : ℝ → ℝ^{d}; t ↦ (t, t^{2}, …, t^{d}).

Definition 3.9. Let 2 ≤ d ≤ n − 1. The cyclic polytope C(n, d) is the convex hull of any n distinct points on the moment curve M_{d}.

Lemma 3.10. The vertex scheme of C(n, 2r) is isomorphic (as a simplicial complex) to Δ_{r,n}.

Proof: Let C(n, 2r) be the convex hull of the points ϕ_{d}(τ_{i}) with τ_{1} < τ_{2} < … < τ_{n}. By [BH, 5.2.7, 5.2.10] the vertex scheme of C(n, 2r) is a simplicial complex on V = {ϕ_{d}(τ_{i}) : 1 ≤ i ≤ n}. We identify the vertex sets V and ℤ/nℤ via ϕ_{d}(τ_{i}) ↔ i. The lemma now follows from the descriptions of the maximal faces (facets) given in Lemma 3.5 and [BH, Theorem 5.2.11]. □

It follows by Proposition 3.8 and Lemma 3.10 that k[Δ_{r,n}] is a Gorenstein ring. Let m = n − 2r and suppose that m ≥ 2. We saw in Lemmas 3.5 and 3.6 that I(Sec^{r}Γ_{n}) has codimension m and is generated in degree r+1. It is clear from the Hilbert series, computed in Lemma 3.7, that k[Δ_{r,n}] has a-invariant 0. This gives equality in Lemma 2.8, so k[Δ_{r,n}] is an extremal Gorenstein ring. The analogue of Theorem 1.1 for Néron polygons now follows from Proposition 2.10(i).

4. Higher secant varieties

Let C ⊂ ℙ^{n−1} be any variety. The rth higher secant variety Sec^{r}C is the Zariski closure of the locus of all (r −1)-planes spanned by r points of C.

Proposition 4.1 (Lange). Let C ⊂ ℙ^{n−1} be an irreducible curve contained in no hyperplane. Then Sec^{r}C is an irreducible variety of dimension min(2r − 1, n − 1).

Proof: See [L]. □

For future reference, we outline a proof of Proposition 4.1 in the case C is an elliptic normal curve. Let H be the divisor of a hyperplane section. For D an effective divisor on C we write \overline{D} for the linear subspace of ℙ^{n−1} cut out by L(H − D) ⊂ L(H). If D is a sum of distinct points then \overline{D} is simply the linear span of these points.

Lemma 4.2. Let C ⊂ ℙ^{n−1} be an elliptic normal curve of degree n, and let D, D_{1}, D_{2} be effective divisors on C. Then (i) dim \overline{D} = \{ \begin{matrix}deg D − 1 & if deg D < n \\ n − 2 & if D ∼ H \\ n − 1 & otherwise. \end{matrix}

(ii) The linear span of \overline{D_{1}} and \overline{D_{2}} is \overline{lcm(D_{1}, D_{2})}. (iii) \overline{D_{1}} ∩ \overline{D_{2}} = \overline{gcd(D_{1}, D_{2})} if lcm(D_{1}, D_{2}) has degree at most n and is not linearly equivalent to H.

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). □

We identify the set of effective divisors on C of degree r with the rth symmetric power S^{r}C. Let

Z = \{(D, P) ∈ S^{r}C × ℙ^{n−1}|P ∈ \overline{D}\}

and write p_{1}, p_{2} for the first and second projections. If r < n then each fibre of p_{1} is an (r − 1)-plane. So dim Z = 2r − 1, and Sec^{r}C = p_{2}(Z) is an irreducible variety of dimension at most 2r − 1.

Let D^{∘} = \overline{D} \ ⋃_{D′<D} \overline{D′}. If n ≥ 2r + 1 then we claim that the restriction of p_{2} to the open subset

U = {(D, P) ∈ S^{r}C × ℙ^{n−1}|P ∈ D^{∘}}

is injective. Indeed if P ∈ D_{1}^{∘} ∩ D_{2}^{∘} for some D_{1}, D_{2} ∈ S^{r}C then Lemma 4.2(iii) gives P ∈ \overline{gcd(D_{1}, D_{2})} since deg lcm(D_{1}, D_{2}) ≤ 2r < n. It follows that D_{1} = D_{2} and this proves our claim. Hence Sec^{r}C has dimension 2r − 1. For the case n ≤ 2r we still refer to [L].

We make some elementary observations concerning the homogeneous ideal of a higher secant variety.

Definition 4.3. A hypersurface {f = 0} ⊂ ℙ^{n−1} contains a variety C ⊂ ℙ^{n−1} with multiplicity r if (passing to affine co-ordinates) the Taylor expansion of f at each point P ∈ C begins with terms of order greater than or equal to r.

Lemma 4.4. Let P_{1}, …, P_{r} span ℙ^{r−1}. Then there are no hypersurfaces of degree r + 1 containing {P_{1}, …, P_{r}} with multiplicity r.

Proof: We choose co-ordinates (x_{1} : …: x_{r}) on ℙ^{r−1} such that P_{1} = (1 : 0 : …: 0), …, P_{r} = (0 : 0 : …: 1). The lemma reduces to the statement that a monomial in x_{1}, …, x_{r} of degree r + 1 cannot be square-free. □

Lemma 4.5. Let C ⊂ ℙ^{n−1} be a variety contained in no hyperplane.

(i) There are no r-ics containing Sec^{r}C. (ii) An (r + 1)-ic contains Sec^{r}C if and only if it contains C with multiplicity r.

Proof: Let f ∈ I(Sec^{r}C) be a homogeneous polynomial. We choose P_{1}, …, P_{n} ∈ C spanning ℙ^{n−1} and then choose co-ordinates (x_{1} : …: x_{n}) on ℙ^{n−1} with P_{1} = (1 : 0 : …: 0), …, P_{n} = (0 : 0 : …: 1). Then every monomial appearing in f involves at least r + 1 of the x_{i}. This proves (i). If deg(f) = r+1 then f contains P_{1} with multiplicity r. The first implication of (ii) follows since P_{1} ∈ C is arbitrary. Conversely if f is an (r + 1)-ic containing C with multiplicity r, and Π ≃ ℙ^{r−1} is spanned by r points of C, then f vanishes on Π by Lemma 4.4. By definition Sec^{r}C is the Zariski closure of the union of all such (r − 1)- planes Π. Thus f ∈ I(Sec^{r}C) as required. □

Remark 4.6. If f ∈ I(Sec^{r}C) and P = (a_{1} : …: a_{n}) ∈ C then it is easy to show that ∑ a_{i} \frac{∂f}{∂x_{i}} ∈ I(Sec^{r−1}C). Since C is contained in no hyperplane we have:

f ∈ I(Sec^{r}C) ⟹ \frac{∂f}{∂x_{i}} ∈ I(Sec^{r−1}C).

This leads to an alternative proof of Lemma 4.5 in the case char (k) = 0. See [CJ] for recent work in this area.

Corollary 4.7. Let C ⊂ ℙ^{n−1} be an irreducible curve contained in no hyperplane. Then any (r + 1)-ic containing Sec^{r}C is irreducible.

Proof: We know by Proposition 4.1 that I(Sec^{r}C) is a prime ideal, and by Lemma 4.5 that it contains no r-ics. □

5. The upper bound

In this section we prove the following upper bound.

Proposition 5.1. Let C ⊂ ℙ^{n−1} be an elliptic normal curve of degree n. Then the space of (r + 1)-ics containing Sec^{r}C has dimension at most β(r + 1, n).

We state an elementary lemma to fix our notation.

Lemma 5.2. Let C_{n} ⊂ ℙ^{n−1} be an elliptic normal curve of degree n. Let P ∈ C_{n} be any point, and choose co-ordinates (x_{1} : …: x_{n}) on ℙ^{n−1} such that P = (0 : 0 : …: 0 : 1) and T_{P} C_{n} = {(0 : …: 0 : s : t)}. Let

π_{1} : ℙ^{n−1} − → ℙ^{n−2}; (x_{1} : …: x_{n}) ↦ (x_{1} : …: x_{n−1})

and

π_{2} : ℙ^{n−1} − → ℙ^{n−3}; (x_{1} : …: x_{n}) ↦ (x_{1} : …: x_{n−2})

be the projection maps away from P and T_{P} C_{n}. We continue to write π_{1}, π_{2} for the morphisms obtained by restricting to C_{n}.

(i) If n ≥ 4 then C_{n−1} = π_{1}(C_{n}) ⊂ ℙ^{n−2} is an elliptic normal curve of degree n − 1. Moreover π_{1}(P) = (0 : …: 0 : 1) ∈ ℙ^{n−2}. (ii) If n ≥ 5 then C_{n−2} = π_{2}(C_{n}) ⊂ ℙ^{n−3} is an elliptic normal curve of degree n − 2.

We first prove Proposition 5.1 in the case r = 1.

Lemma 5.3. Let C ⊂ ℙ^{n−1} be an elliptic normal curve of degree n. Then the space of quadrics containing C has dimension at most β(2, n) = n(n − 3)/2.

Proof: The proof is by induction on n, the case n = 3 being clear. We may therefore suppose that n ≥ 4. Let C_{n} and C_{n−1} be as in Lemma 5.2. Then every quadric vanishing on C_{n} can be written as

(2)  x_{n}g(x_{1}, …, x_{n−2}) + h(x_{1}, …, x_{n−1})

where g is a linear form and h is a quadric. But it is evident that

(3)  I(C_{n}) ∩ k[x_{1}, …, x_{n−1}] = I(C_{n−1}).

These observations, combined with the induction hypothesis, show that the space of quadrics containing C_{n} has dimension at most

β(2, n − 1) + (n − 2) = β(2, n).
□

We generalise (3) and (2) to higher secant varieties.

Lemma 5.4. Let C_{n} and C_{n−1} be as in Lemma 5.2.

(i) Sec^{r}C_{n−1} is the Zariski closure of π_{1}(Sec^{r}C_{n}). (ii) I(Sec^{r}C_{n}) ∩ k[x_{1}, …, x_{n−1}] = I(Sec^{r}C_{n−1}).

Proof: This is clear. □

Lemma 5.5. Let C_{n} and C_{n−2} be as in Lemma 5.2. Let f ∈ I(Sec^{r}C_{n}) be a homogeneous polynomial and write

(4)  f(x_{1}, …, x_{n}) = ∑_{(i,j)≤(p,q)} x_{n−1}^{i}x_{n}^{j}g_{ij}(x_{1}, …, x_{n−2})

where (i, j) ≤ (p, q) means that either j < q or j = q and i ≤ p.

(i) If r = 1 then g_{pq} belongs to the irrelevant ideal. (ii) If r ≥ 2 then g_{pq} ∈ I(Sec^{r−1}C_{n−2}).

Proof: Let H be the divisor of a hyperplane section on C_{n}. By a standard abuse of notation we identify x_{1}, …, x_{n} as a basis for L(H). Then L(H − iP) has basis x_{1}, …, x_{n−i} for i = 0, 1, 2. In particular we have ord_{P} (x_{n}) < ord_{P} (x_{n−1}) < ord_{P} (x_{i}) for all 1 ≤ i ≤ n − 2.

(i) We must show that if deg(f) = p + q then g_{pq} = 0. Suppose that x^{I} = x_{1}^{i_{1}} … x_{n}^{i_{n}} is a monomial appearing in f with

(5)  ord_{P} (x^{I}) ≤ ord_{P} (x_{n−1}^{p}x_{n}^{q}).

We see from (4) that i_{n} ≤ q, and from (5) that i_{n} ≥ q. So i_{n} = q, and repeating the same arguments gives i_{n−1} = p. Thus x_{n−1}^{p}x_{n}^{q} is the only such monomial. Since f ∈ I(C_{n}) the coefficient of x_{n−1}^{p}x_{n}^{q} must therefore be zero, i.e. g_{pq} = 0 as was to be shown.

(ii) We continue to employ the multi-index notation x^{I} = x_{1}^{i_{1}} … x_{n}^{i_{n}} and write |I| = i_{1} + … + i_{n}. We define homogeneous polynomials f_{I} via

f(x_{1} + y_{1}, …, x_{n} + y_{n}) = ∑_{I} f_{I}(y_{1}, …, y_{n})x^{I}.

Fixing (a_{1} : …: a_{n}) ∈ Sec^{r−1}C_{n} we put

f̃(x_{1}, …, x_{n}) = ∑_{|I|=p+q} f_{I}(a_{1}, …, a_{n})x^{I}.

Then f̃ ∈ I(C_{n}) and it follows by (i) that g_{pq}(a_{1}, …, a_{n}) = 0. But (a_{1} : …: a_{n}) ∈ Sec^{r−1}C_{n} was arbitrary. Thus

g_{pq} ∈ I(Sec^{r−1}C_{n}) ∩ k[x_{1}, …, x_{n−2}]

and we are done by a double application of Lemma 5.4(ii). □

Proof of Proposition 5.1: The proof is by induction on r and n. The case r = 1 was treated in Lemma 5.3. If n ≤ 2r then Proposition 4.1 gives Sec^{r}C = ℙ^{n−1} in which case the result is trivial. We may therefore suppose that r ≥ 2 and n ≥ 2r + 1, and that the proposition is known for all smaller values of r and n.

Let C_{n}, C_{n−1} and C_{n−2} as in Lemma 5.2. Let f ∈ I(Sec^{r}C_{n}) be an (r + 1)-ic. By Lemma 5.5(ii), and then Lemma 4.5(i) applied to Sec^{r−1}C_{n−2}, we have

f(x_{1}, …, x_{n}) = x_{n}g(x_{1}, …, x_{n−2}) + h(x_{1}, …, x_{n−1})

where g ∈ I(Sec^{r−1}C_{n−2}) is an r-ic and h ∈ k[x_{1}, …, x_{n−1}] is an (r+1)- ic. By this observation, Lemma 5.4(ii) and the induction hypothesis, we deduce that the space of (r+1)-ics containing Sec^{r}C_{n} has dimension at most

β(r + 1, n − 1) + β(r, n − 2) = β(r + 1, n).
□

6. The lower bound

We present a simplified proof of the following theorem of Knight [K].

Proposition 6.1 (Knight). Let C ⊂ ℙ^{n−1} be an elliptic normal curve of degree n. Then the space of (r + 1)-ics containing Sec^{r}C has dimension β(r + 1, n).

Remark 6.2. Knight states his theorem in terms of the (r + 1)-ics containing C with multiplicity r. By Lemma 4.5(ii) this is equivalent to our version.

Let C be any smooth projective curve, and let H be a divisor on C such that L(H) has basis x_{1}, …, x_{n}. If D_{1} and D_{2} are divisors on C with D_{1} + D_{2} = H then the multiplication map

L(D_{1}) × L(D_{2}) → L(H)

is represented by a matrix of linear forms in k[x_{1}, …, x_{n}]. As in the introduction, we call this matrix Φ(D_{1}, D_{2}). By the multilinearity of determinants, the linear span of the (r+1)×(r+1) minors of Φ(D_{1}, D_{2}) is independent of our choice of bases for L(D_{1}) and L(D_{2}). The following elementary observation is made in [K], [EKS]:

Lemma 6.3. Let C ⊂ ℙ^{n−1} be a smooth curve embedded by a complete linear system. Let D_{1} and D_{2} be divisors on C with D_{1} + D_{2} = H where H is the hyperplane section. Then the (r + 1) × (r + 1) minors of Φ(D_{1}, D_{2}) vanish on Sec^{r}C.

Proof: The matrix Φ(D_{1}, D_{2}) has rank at most 1 on C, and so has rank at most r on Sec^{r}C. □

Lemma 6.4. Let C be a smooth curve of genus one. Let H be a divisor on C of degree n ≥ 2. Let L(H) have basis x_{1}, …, x_{n}. Let V be the vector space of r-ics in k[x_{1}, …, x_{n}] spanned by the r × r minors of all matrices Φ(D_{1}, D_{2}) as D_{1}, D_{2} run over all divisors on C with D_{1} + D_{2} = H and deg D_{1}, deg D_{2} ≥ 1. Then dim V ≥ β(r, n).

Proof: We begin by treating the case r = 1. Let P_{1} and P_{2} be distinct points on C with P_{1} + P_{2} ≁ H. Then L(H) = L(H − P_{1}) + L(H − P_{2}) is spanned by the entries of Φ(P_{1}, H − P_{1}) and Φ(P_{2}, H − P_{2}). So dim V = n = β(1, n) as required.

The proof is now by induction on r and n. The case n < 2r is trivial since β(r, n) = 0. We may therefore suppose that r ≥ 2 and n ≥ 2r, and that the result is known for all smaller values of r and n.

Let P ∈ C be any point. We may arrange that x_{1}, …, x_{n−i} is a basis for L(H − iP) for i = 0, 1, 2. Let g(x_{1}, …, x_{n−2}) be an (r − 1) × (r − 1) minor of Φ(D_{1}, D_{2}) where D_{1}+D_{2} = H−2P and deg D_{1}, deg D_{2} ≥ 1. It is determined by (r −1)-dimensional subspaces W_{1} ⊂ L(D_{1}) and W_{2} ⊂ L(D_{2}). Since deg D_{1}, deg D_{2} ≥ 1 we may pick w_{1} ∈ L(D_{1} + P) \ L(D_{1}) and w_{2} ∈ L(D_{2}+P)\L(D_{2}). Then w_{1}w_{2} ∈ L(H)\L(H −P). Rescaling w_{1} if necessary we may assume that

w_{1}w_{2} = x_{n} + ℓ(x_{1}, …, x_{n−1})

where ℓ is a linear form. Let f be the r × r minor of Φ(D_{1} + P, D_{2} + P) determined by W_{1} ⊕ ⟨w_{1}⟩ and W_{2} ⊕ ⟨w_{2}⟩. Then

f(x_{1}, …, x_{n}) = x_{n}g(x_{1}, …, x_{n−2}) + h(x_{1}, …, x_{n−1})

for some h ∈ k[x_{1}, …, x_{n−1}]. This construction of f from g, combined with the induction hypothesis shows that V ∩ k[x_{1}, …, x_{n−1}] is a subspace of V of codimension at least β(r − 1, n − 2). But trivially, if f(x_{1}, …, x_{n−1}) is an r × r minor of Φ(D_{1}, D_{2}) with D_{1} + D_{2} = H − P, then it is also an r × r minor of Φ(D_{1}, D_{2} + P). Thus

dim V ≥ β(r, n − 1) + β(r − 1, n − 2) = β(r, n).
□

Lemmas 6.3 and 6.4 show that the inequality established in Proposition 5.1 is in fact an equality. This completes our proof of Proposition 6.1.

Remark 6.5. In Knight’s analogue of Lemma 6.4 he takes H = nP and only considers divisors D_{1} and D_{2} that are multiples of P. It turns out that not all (r + 1)-ics vanishing on Sec^{r}C arise in this way. This makes his proof of Proposition 6.1 more complicated than the one presented here.

We record two immediate corollaries of our proof.

Corollary 6.6. Let C ⊂ ℙ^{n−1} be an elliptic normal curve of degree n. Then the space of (r + 1)-ics containing Sec^{r}C is generated by the (r + 1) × (r + 1) minors of the matrices Φ(D_{1}, D_{2}) as D_{1}, D_{2} run over all divisors on C with D_{1} + D_{2} = H.

Corollary 6.7. Let C_{n} and C_{n−2} as in Lemma 5.2. Then for every r-ic g ∈ I(Sec^{r−1}C_{n−2}) there exists an (r + 1)-ic h ∈ k[x_{1}, …, x_{n−1}] such that x_{n}g + h ∈ I(Sec^{r}C_{n}).

7. Complete Intersections

In this section we prove Theorem 1.1 in the cases m = 1 and m = 2. Notice that we have not yet determined the degree of Sec^{r}C. Our proof for m = 2 follows [Ro, §§9.22-9.26].

Let C ⊂ ℙ^{n−1} be an elliptic normal curve of degree n, with hyperplane section H. As in §4 we write \overline{D} for the linear span of an effective divisor D. The following lemma is [Ro, 9.26.1].

Lemma 7.1 (Room). Let D_{1}, D_{2} be divisors on C with D_{1} + D_{2} = H and deg D_{1} ≤ deg D_{2}. Then

\{rank Φ(D_{1}, D_{2}) < deg D_{1}\} = ⋃_{D∈|D_{1}|} \overline{D}.

Proof: A point P ∈ ℙ^{n−1} corresponds to a codimension 1 subspace V_{P} ⊂ L(H). If Φ(D_{1}, D_{2}) evaluated at P has rank less than deg D_{1} then there exists f ∈ L(D_{1}) such that fg ∈ V_{P} for all g ∈ L(D_{2}). Say (f) = D − D_{1}. Then L(H − D) ⊂ V_{P}, so P ∈ \overline{D}. The converse is obtained by reversing these steps. □

The next proposition is our version of [Ro, 9.22.1]. If n = 2r + 2 then we already know by Proposition 6.1 that the space of (r + 1)-ics vanishing on Sec^{r}C has dimension β(r + 1, n) = 2.

Proposition 7.2 (Room). Let r ≥ 1 and n = 2r + 2.

(i) If D_{1}, D_{2} are divisors on C of degree r + 1 with D_{1} + D_{2} = H then

\{det Φ(D_{1}, D_{2}) = 0\} = ⋃_{D∈|D_{1}|} \overline{D} = ⋃_{D∈|D_{2}|} \overline{D}.

(ii) If D_{1}, D_{2} and D′_{1}, D′_{2} as in (i) with D_{1} ≁ D′_{1} and D_{1} ≁ D′_{2} then

{det Φ(D_{1}, D_{2}) = det Φ(D′_{1}, D′_{2}) = 0} = Sec^{r}C.

Proof: (i) This is a special case of Lemma 7.1. (ii) By Lemma 6.3 each of these (r +1)-ics contains Sec^{r}C. Conversely, if P belongs to the lefthand side, we know by (i) that P ∈ \overline{D} ∩ \overline{D′} for some D ∈ |D_{1}| and D′ ∈ |D′_{1}|. Then D + D′ ≁ H and Lemma 4.2(iii) gives P ∈ \overline{gcd(D, D′)}. Since D ≠ D′ it follows that P ∈ Sec^{r}C. □

Proposition 7.2 tells us that if Sec^{r}C has codimension m = 2 then it is set-theoretically the intersection of two (r + 1)-ics, say f_{1}, f_{2}. It follows by the Nullstellensatz that

(6)  I(Sec^{r}C) = √(f_{1}, f_{2}).

To prove Theorem 1.1 in the case m = 2 we must show that I(Sec^{r}C) = (f_{1}, f_{2}). In fact we treat the case m = 1 at the same time. The notation C_{n}, C_{n−1}, C_{n−2} is recalled from Lemma 5.2.

Proposition 7.3. Let r ≥ 1 and n = 2r + 2. Then

(i) Sec^{r}C_{n−1} is a hypersurface of degree n − 1. (ii) Sec^{r}C_{n} is the complete intersection of two (r + 1)-ics.

Proof: The proof is by induction on r, the case r = 1 being well known. We have already seen that I(Sec^{r}C_{n}) contains linearly independent (r + 1)-ics f_{1}, f_{2}. As in §5 we write

f_{i}(x_{1}, …, x_{n}) = x_{n}g_{i}(x_{1}, …, x_{n−2}) + h_{i}(x_{1}, …, x_{n−1})

for i = 1, 2 where g_{1}, g_{2} ∈ I(Sec^{r−1}C_{n−2}). Lemma 5.4(ii) asserts that

(7)  I(Sec^{r}C_{n}) ∩ k[x_{1}, …, x_{n−1}] = I(Sec^{r}C_{n−1}).

By Proposition 6.1 this ideal contains no (r + 1)-ics. It follows that g_{1}, g_{2} are linearly independent. By induction hypothesis they generate I(Sec^{r−1}C_{n−2}). Moreover g_{1}, g_{2} are irreducible by Corollary 4.7. Thus

(8)  (f_{1}, f_{2}) ∩ k[x_{1}, …, x_{n−1}] = (g_{1}h_{2} − g_{2}h_{1}).

From (6), (7) and (8) we deduce

I(Sec^{r}C_{n−1}) = √(g_{1}h_{2} − g_{2}h_{1}).

Proposition 4.1 tells us that Sec^{r}C_{n−1} is an irreducible hypersurface, say with equation s(x_{1}, …, x_{n−1}) = 0. Since s is irreducible, g_{1}h_{2}−g_{2}h_{1} is a power of s. But deg(g_{1}h_{2} − g_{2}h_{1}) = 2r + 1 = n − 1 and we know by Lemma 4.5(i) that deg(s) ≥ r + 1. Thus

(9)  I(Sec^{r}C_{n−1}) = (g_{1}h_{2} − g_{2}h_{1})

and this completes the proof of (i).

To prove (ii) we must show that I(Sec^{r}C_{n}) = (f_{1}, f_{2}). We suppose, for a contradiction, that f ∈ I(Sec^{r}C_{n}) is a homogeneous polynomial with f ∉ (f_{1}, f_{2}). By Lemma 5.5 we have

f(x_{1}, …, x_{n}) = ∑_{(i,j)≤(p,q)} x_{n−1}^{i}x_{n}^{j}g_{ij}(x_{1}, …, x_{n−2})

with g_{pq} ∈ I(Sec^{r−1}C_{n−2}). We may suppose that f is chosen with (p, q) minimal. Since I(Sec^{r−1}C_{n−2}) is generated by g_{1}, g_{2}, we can write g_{pq} = ξ_{1}g_{1} + ξ_{2}g_{2}. If q ≥ 1 then f − x_{n−1}^{p}x_{n}^{q−1}(ξ_{1}f_{1} + ξ_{2}f_{2}) contradicts our minimal choice of f. Therefore q = 0. So f ∈ k[x_{1}, …, x_{n−1}] and it follows by (7), (8), (9) that f ∈ (f_{1}, f_{2}). This is the required contradiction. □

Remark 7.4. An alternative proof of Proposition 7.3 uses properties of the Heisenberg group in place of Proposition 7.2. It is necessary to split into cases char (k) ∤ (n − 1) and char (k) ∤ n. We omit the details.

We obtain a strengthening of Remark 4.6 that is of independent interest. For instance it explains why, as found in the proof of [H, VI.3.2], the quintic defining Sec C_{5} belongs to I(C_{5})^{2}.

Corollary 7.5. Let r ≥ 2 and n = 2r + 2. Let Sec^{r}C_{n−1} have equation s(x_{1}, …, x_{n−1}) = 0. Then \frac{∂s}{∂x_{i}} ∈ I(Sec^{r−1}C_{n−1})^{2} for all i.

Proof: Keeping the notation of the previous proof we may assume that s = g_{1}h_{2} − g_{2}h_{1}. Then

\frac{∂s}{∂x_{n−1}} = g_{1} \frac{∂f_{2}}{∂x_{n−1}} − g_{2} \frac{∂f_{1}}{∂x_{n−1}} .

Since f_{1}, f_{2} ∈ I(Sec^{r}C_{n}) and g_{1}, g_{2} ∈ I(Sec^{r−1}C_{n−2}), it follows by Remark 4.6 and Lemma 5.4 that \frac{∂s}{∂x_{n−1}} ∈ I(Sec^{r−1}C_{n−1})^{2}. But we can make our choice of co-ordinates in Lemma 5.2 so that π_{1}(P) is any point on C_{n−1}. Since C_{n−1} is contained in no hyperplane, it follows that \frac{∂s}{∂x_{i}} ∈ I(Sec^{r−1}C_{n−1})^{2} for all i. □

8. Proof of Theorem 1.2

Let C ⊂ ℙ^{n−1} be an elliptic normal curve of degree n. When we write Sec^{r}C it has so far been implicit that r ≥ 1. To avoid unnecessary repetition of our arguments we adopt the convention that if r = 0 then Sec^{r}C is the empty set and I(Sec^{r}C) is the irrelevant ideal. To complete the proof of Theorem 1.2 we prove

Proposition 8.1. Let C ⊂ ℙ^{n−1} be an elliptic normal curve of degree n, and let r ≥ 0. If n−2r ≥ 2 then I(Sec^{r}C) is generated by (r+1)-ics.

Proof: In view of the above convention, the case r = 0 is clear. The case n = 2r + 2 was proved in Proposition 7.3(ii). We may therefore suppose that r ≥ 1 and n ≥ 2r + 3, and that the proposition is known for all smaller values of r and n.

Let C_{n}, C_{n−1} and C_{n−2} as in Lemma 5.2. By induction hypothesis I(Sec^{r}C_{n−1}) is generated by (r+1)-ics, say f_{1}, …, f_{s}, and I(Sec^{r−1}C_{n−2}) is generated by r-ics, say g_{1}, …, g_{t}. By Corollary 6.7 there exist (r+1)- ics h_{1}, …, h_{t} ∈ k[x_{1}, …, x_{n−1}] with x_{n}g_{i} + h_{i} ∈ I(Sec^{r}C_{n}). We must show that I(Sec^{r}C_{n}) is equal to

I′ = (f_{1}, …, f_{s}, x_{n}g_{1} + h_{1}, …, x_{n}g_{t} + h_{t}).

Suppose for a contradiction that f ∈ I(Sec^{r}C_{n}) is a homogeneous polynomial with f ∉ I′. By Lemma 5.5 we have

f(x_{1}, …, x_{n}) = ∑_{(i,j)≤(p,q)} x_{n−1}^{i}x_{n}^{j}g_{ij}(x_{1}, …, x_{n−2})

with g_{pq} = ∑ ξ_{i}g_{i}. We suppose that f is chosen with (p, q) minimal. If q ≥ 1 then f − x_{n−1}^{p}x_{n}^{q−1} ∑ ξ_{i}(x_{n}g_{i} + h_{i}) contradicts our minimal choice of f. Therefore q = 0 and

f ∈ I(Sec^{r}C_{n}) ∩ k[x_{1}, …, x_{n−1}] = I(Sec^{r}C_{n−1}) ⊂ I′.

This is the required contradiction. □

Theorem 1.2 is obtained by combining Propositions 6.1 and 8.1.

Definition 8.2. Let I be an ideal in a polynomial ring R = R[x]. The leading coefficient ideal of I is the set of leading coefficients of elements of I. It is an ideal in R.

Corollary 8.3. Let n − 2r ≥ 1 and C_{n}, C_{n−2} as in Lemma 5.2. If R = k[x_{1}, …, x_{n−1}] and R = R[x_{n}] then I(Sec^{r−1}C_{n−2})R is the leading coefficient ideal of I(Sec^{r}C_{n}).

Proof: We have shown that I(Sec^{r−1}C_{n−2}) is generated by r-ics, so by Corollary 6.7 it is contained in the leading coefficient ideal of I(Sec^{r}C_{n}). The inclusion of the leading coefficient ideal in I(Sec^{r−1}C_{n−2})R then follows by Lemma 5.5. □

9. Minimal free resoutions

The induction step in the proof of Theorem 1.1 is given by:

Proposition 9.1. Let r ≥ 1 and m ≥ 3. Let I and J be extremal Gorenstein ideals in R = k[x_{1}, …, x_{n−1}]. Let I be a prime ideal in R = R[x_{n}]. Suppose that

(i) I, J and I are generated in degrees r + 1, r and r + 1, (ii) I, J and I have codimensions m − 1, m and m, (iii) I ∩ R = I, (iv) J is the leading coefficient ideal of I. Then I is an extremal Gorenstein ideal.

Proof: Let R/I have minimal graded free resolution

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

We identify F_{0} = R, and fix a basis a_{1}, …, a_{s} for F_{1}. Then I is generated by the (r + 1)-ics f_{1}, …, f_{s} where f_{i} = ϕ_{1}(a_{i}).

Let R/J have minimal graded free resolution

G_{•} : 0 ⟶ G_{m} ⟶^{ψ_{m}} G_{m−1} ⟶ … ⟶ G_{1} ⟶^{ψ_{1}} G_{0} ⟶ 0.

We identify G_{0} = R, and fix a basis b_{1}, …, b_{t} for G_{1}. Then J is generated by the r-ics g_{1}, …, g_{t} where g_{i} = ψ_{1}(b_{i}).

By (i), (iii) and (iv) there exist (r + 1)-ics h_{1}, …, h_{t} such that

(10)  I = (f_{1}, …, f_{s}, x_{n}g_{1} + h_{1}, …, x_{n}g_{t} + h_{t}).

The proof works by constructing a minimal graded free resolution of R/I. First we must define certain graded R-module maps α_{i}, β_{i} and γ_{i}. Let α_{1} : G_{1}(−1) → F_{0} be given by α_{1}(b_{i}) = h_{i}. We extend α_{1} to a map of complexes

(11) \xymatrix{… \ar[r] & G_{3}(−1) \ar[r]^{ψ_{3}} \ar[d]^{α_{3}} & G_{2}(−1) \ar[r]^{ψ_{2}} \ar[d]^{α_{2}} & G_{1}(−1) \ar[d]^{α_{1}} \ar[r]^{ψ_{1}} & G_{0}(−1) \\ … \ar[r] & F_{2} \ar[r]^{ϕ_{2}} & F_{1} \ar[r]^{ϕ_{1}} & F_{0} \ar[r] & 0}

The construction of α_{2} is as follows. Let e_{1}, …, e_{p} be a basis for G_{2}. Then ψ_{2}(e_{j}) = ∑ ξ_{ij}b_{i} for some ξ_{ij} ∈ R. Since ψ_{1}ψ_{2} = 0 it follows that ∑ ξ_{ij}g_{i} = 0. Then

α_{1}ψ_{2}(e_{j}) = ∑ ξ_{ij}h_{i} = ∑ ξ_{ij}(x_{n}g_{i} + h_{i}) ∈ I ∩ R = I.

Since I = im ϕ_{1} we can choose α_{2} with α_{1}ψ_{2} = ϕ_{1}α_{2}. The remaining α_{i} are constructed by standard diagram chasing. (These maps are not unique.)

Lemma 9.2. The map α_{m} : G_{m}(−1) → F_{m−1} is an isomorphism.

Proof: The graded R-modules G_{m}(−1) and F_{m−1} are both copies of R(−m−2r+1). So it suffices to show that α_{m} is non-zero. We suppose for a contradiction that α_{m} = 0. Since I and J are Gorenstein ideals the following diagram, dual to (11), has exact rows.

\xymatrix{0 \ar[r] & F_{0}^{∗} \ar[r]^{ϕ_{1}^{∗}} \ar[d]^{α_{1}^{∗}} & F_{1}^{∗} \ar[r] \ar[d]^{α_{2}^{∗}} & … \ar[r] & F_{m−2}^{∗} \ar[r]^{ϕ_{m−1}^{∗}} \ar[d]^{α_{m−1}^{∗}} & F_{m−1}^{∗} \ar[d]^{α_{m}^{∗}} \\ G_{0}^{∗}(1) \ar[r]^{ψ_{1}^{∗}} & G_{1}^{∗}(1) \ar[r]^{ψ_{2}^{∗}} & G_{2}^{∗}(1) \ar[r] & … \ar[r] & G_{m−1}^{∗}(1) \ar[r]^{ψ_{m}^{∗}} & G_{m}^{∗}(1)}

Since α_{m}^{∗} = 0, this map of complexes is homotopic to zero. Specifically, we construct maps ρ_{i} : F_{i}^{∗} → G_{i}^{∗}(1) with ρ_{m−1} = 0 and

α_{i}^{∗} = ρ_{i}ϕ_{i}^{∗} + ψ_{i}^{∗}ρ_{i−1}

for all i ≤ m − 1. Then α_{1} = ϕ_{1}ρ_{1}^{∗} + ρ_{0}^{∗}ψ_{1}. Since we have identified F_{0} = G_{0} = R, the map ρ_{0}^{∗} : G_{0}(−1) → F_{0} is multiplication by some linear form ℓ ∈ R. Then

ℓg_{i} − h_{i} = ρ_{0}^{∗}ψ_{1}(b_{i}) − α_{1}(b_{i}) = −ϕ_{1}ρ_{1}^{∗}(b_{i}) ∈ I ⊂ I.

Since x_{n}g_{i} + h_{i} ∈ I it follows that (ℓ + x_{n})g_{i} ∈ I. But I is a prime ideal generated by (r + 1)-ics, so ℓ = −x_{n} ∉ R and this is the required contradiction. □

It follows from (iii) and (iv) that I ⊂ J. The natural map R/I → R/J extends to a map of complexes

(12)  \xymatrix{… \ar[r] & F_{2} \ar[r]^{ϕ_{2}} \ar[d]^{β_{3}} & F_{1} \ar[r]^{ϕ_{1}} \ar[d]^{β_{2}} & F_{0} \ar[r] \ar[d]^{β_{1}} & 0 \\ … \ar[r] & G_{2} \ar[r]^{ψ_{2}} & G_{1} \ar[r]^{ψ_{1}} & G_{0} \ar[r] & 0}

where β_{1} is the identity map on F_{0} = G_{0} = R.

Composing the α_{i} and the β_{i} gives a map of complexes that we claim is homotopic to zero.

\xymatrix{… \ar[r] & F_{3}(−1) \ar[r]^{ϕ_{3}} \ar[d]^{α_{3}β_{4}} & F_{2}(−1) \ar[r]^{ϕ_{2}} \ar[d]^{α_{2}β_{3}} & F_{1}(−1) \ar[r]^{ϕ_{1}} \ar[d]^{α_{1}β_{2}} & F_{0}(−1) \\ … \ar[r] & F_{2} \ar[r]_{ϕ_{2}} & F_{1} \ar[r]_{ϕ_{1}} & F_{0} \ar[r] & 0}

Specifically, we construct γ_{i} : F_{i−1}(−1) → F_{i−1} with γ_{1} = 0 and

(13)  α_{i}β_{i+1} = ϕ_{i}γ_{i+1} + γ_{i}ϕ_{i}

for all i ≥ 1. The construction of γ_{2} is as follows. We have β_{2}(a_{j}) = ∑ ξ_{ij}b_{i} for some ξ_{ij} ∈ R. Then

f_{j} = β_{1}ϕ_{1}(a_{j}) = ψ_{1}β_{2}(a_{j}) = ∑ ξ_{ij}g_{i}

and

α_{1}β_{2}(a_{j}) = ∑ ξ_{ij}h_{i} = ∑ ξ_{ij}(x_{n}g_{i} + h_{i}) − x_{n}f_{j} ∈ I ∩ R = I.

Since I = im ϕ_{1} we can choose γ_{2} with α_{1}β_{2} = ϕ_{1}γ_{2}. The remaining γ_{i} are constructed by standard diagram chasing.

We have constructed graded R-module maps α_{i}, β_{i} and γ_{i}. We now pass from working over R to working over R = R[x_{n}]. Let F_{i} = F_{i}⊗_{R}R and G_{i} = G_{i} ⊗_{R} R. We continue to write ϕ_{i}, ψ_{i}, α_{i}, β_{i}, γ_{i} for the extensions of these maps to the category of R-modules. We put

δ_{i} = γ_{i} + (−1)^{i}x_{n} : F_{i−1}(−1) → F_{i−1}

so that (13) becomes

(14)  α_{i}β_{i+1} = ϕ_{i}δ_{i+1} + δ_{i}ϕ_{i}.

We consider the diagram

\xymatrix{ & F_{m−1}(−1) \ar[r] \ar[d]^{β_{m}} \ar[ddl]^{δ_{m}} \ar[r] & … \ar[r] & F_{1}(−1) \ar[r]^{ϕ_{1}} \ar[d]^{β_{2}} & F_{0}(−1) \ar[ddl]^{δ_{1}} \ar[d]^{β_{1}} \\ G_{m}(−1) \ar[r]^{ψ_{m}} \ar[d]_{α_{m}} & G_{m−1}(−1) \ar[r] \ar[d]_{α_{m−1}} & … \ar[r] & G_{1}(−1) \ar[r]^{ψ_{1}} \ar[d]_{α_{1}} & G_{0}(−1) \\ F_{m−1} \ar[r]^{ϕ_{m−1}} & F_{m−2} \ar[r] & … \ar[r] & F_{0}}

and form the total complex

… ⟶ F_{i}⊕G_{i}(−1)⊕F_{i−1}(−1) ⟶^{d_{i}} F_{i−1}⊕G_{i−1}(−1)⊕F_{i−2}(−1) ⟶ …

with differential

d_{i} = (\begin{matrix}ϕ_{i} & α_{i} & −δ_{i} \\ 0 & −ψ_{i} & β_{i} \\ 0 & 0 & ϕ_{i−1} \end{matrix}) .

Since F_{•} and G_{•} are complexes, it follows from (14), and the commutativity of the diagrams (11) and (12), that d_{i}d_{i+1} = 0, i.e. this total complex really is a complex.

Lemma 9.3. The total complex is a graded free resolution of R/I.

Proof: The exactness of F_{•} and G_{•} gives exactness of the total complex at all terms up to and including F_{2} ⊕G_{2}(−1)⊕F_{1}(−1). It remains to consider

F_{2} ⊕ G_{2}(−1) ⊕ F_{1}(−1) →^{d_{2}} F_{1} ⊕ G_{1}(−1) ⊕ F_{0}(−1) →^{d_{1}} F_{0} ⊕ G_{0}(−1)

where d_{1}(u, v, w) = (ϕ_{1}(u) + α_{1}(v) + x_{n}w, −ψ_{1}(v) + w). We must show that (i) im d_{2} = ker d_{1} and (ii) coker d_{1} ≃ R/I.

(i) Let (u, v, w) ∈ ker d_{1}. Then

(15)  w = ψ_{1}(v) and ϕ_{1}(u) + α_{1}(v) + ψ_{1}(v)x_{n} = 0.

We must show that (u, v, w) ∈ im d_{2}. We write

u = u_{p}x_{n}^{p} + … + u_{1}x_{n} + u_{0}

and

v = v_{q}x_{n}^{q} + … + v_{1}x_{n} + v_{0}

with u_{0}, …, u_{p} ∈ F_{1} and v_{0}, …, v_{q} ∈ G_{1}. The proof is by induction on Δ(u, v) := max(2p, 2q + 1). We start the induction by noting that if p = q = 0 then u ∈ F_{1}, v ∈ G_{1}. So (15) gives w = 0, whence (u, v, 0) ∈ im d_{2} by the exactness of F_{•} and G_{•}.

We split the induction step into two cases. If p > q we put (u′, v′, w′) = (u, v, w) + d_{2}(0, 0, u_{p}x_{n}^{p−1}). Then Δ(u′, v′) < Δ(u, v) since

d_{2}(0, 0, u_{p}x_{n}^{p−1}) = (−u_{p}x_{n}^{p} − γ_{2}(u_{p})x_{n}^{p−1}, β_{2}(u_{p})x_{n}^{p−1}, ϕ_{1}(u_{p})x_{n}^{p−1}).

If p ≤ q then (15) gives ψ_{1}(v_{q}) = 0. So v_{q} = ψ_{2}(e) for some e ∈ G_{2}. We put (u′, v′, w′) = (u, v, w) + d_{2}(0, ex_{n}^{q}, 0). Then Δ(u′, v′) < Δ(u, v) since

d_{2}(0, ex_{n}^{q}, 0) = (α_{2}(e)x_{n}^{q}, −v_{q}x_{n}^{q}, 0).

(ii) We identify F_{0} ⊕ G_{0}(−1) = R ⊕ R(−1). Then im d_{1} is generated by the pairs (f_{i}, 0), (h_{i}, −g_{i}) and (x_{n}, 1). It follows by (10) that

coker d_{1} = \frac{R ⊕ R(−1)}{im d_{1}} ≃ R/I.

□

The total complex is not minimal. This is rectified by eliminating the isomorphisms α_{m} and β_{1}. If m ≥ 4 we put

\begin{matrix}d′_{m} = (\begin{matrix}β_{m} − ψ_{m}α_{m}^{−1}δ_{m} \\ ϕ_{m−1} \end{matrix}) , & d′_{m−1} = (\begin{matrix}α_{m−1} & −δ_{m−1} \\ −ψ_{m−1} & β_{m−1} \\ 0 & ϕ_{m−2} \end{matrix}) , \\ d′_{2} = (\begin{matrix}ϕ_{2} & α_{2} & −δ_{2} \\ 0 & −ψ_{2} & β_{2} \end{matrix}) , & d′_{1} = (ϕ_{1} α_{1} − δ_{1}β_{1}^{−1}ψ_{1}) , \end{matrix}

and d′_{i} = d_{i} for 3 ≤ i ≤ m − 2. Then R/I has minimal graded free resolution

\begin{matrix}0 ⟶ F_{m−1}(−1) ⟶^{d′_{m}} G_{m−1}(−1) ⊕ F_{m−2}(−1) ⟶^{d′_{m−1}} … \\ … ⟶ F_{2} ⊕ G_{2}(−1) ⊕ F_{1}(−1) ⟶^{d′_{2}} F_{1} ⊕ G_{1}(−1) ⟶^{d′_{1}} F_{0} ⟶ 0. \end{matrix}

The modifications for m = 3 are similar.

Since F_{m−1}(−1) ≃ R(−m−2r) it follows that I is an extremal Gorenstein ideal. This completes the proof of Proposition 9.1. □

Proposition 9.4. Let C ⊂ ℙ^{n−1} be an elliptic normal curve of degree n, and let r ≥ 0. If n−2r ≥ 2 then I(Sec^{r}C) is an extremal Gorenstein ideal.

Proof: The proof is by induction on r and n. When r = 0 it is understood (see §8) that I(Sec^{r}C) is the irrelevant ideal. By inspection of the Koszul complex, this is an extremal Gorenstein ideal. When n = 2r+2 we saw in Proposition 7.3(ii) that Sec^{r}C is the complete intersection of two (r + 1)-ics. So again I(Sec^{r}C) is an extremal Gorenstein ideal. We may therefore suppose that r ≥ 1 and n ≥ 2r + 3, and that the proposition is known for all smaller values of r and n.

Let C_{n}, C_{n−1} and C_{n−2} as in Lemma 5.2. Let R = k[x_{1}, …, x_{n−1}] and R = R[x_{n}]. We put

I = I(Sec^{r}C_{n−1}), J = I(Sec^{r−1}C_{n−2})R, I = I(Sec^{r}C_{n}).

By induction hypothesis, I and J are extremal Gorenstein ideals. By Proposition 4.1, I is a prime ideal. Let m = n−2r. Hypotheses (i)-(iv) of Proposition 9.1 follow from Theorem 1.2, Proposition 4.1, Lemma 5.4 and Corollary 8.3. Hence I is an extremal Gorenstein ideal. □

Theorem 1.1 follows from Proposition 7.3(i) in the case m = 1, and from Propositions 9.4 and 2.10 in the case m ≥ 2.

10. Determinantal presentations

Let C ⊂ ℙ^{n−1} be an elliptic normal curve of degree n. In the light of Theorem 1.2 we may restate Corollary 6.6 as

Lemma 10.1. If n ≥ 2r + 2 then I(Sec^{r}C) is generated by the (r + 1) × (r + 1) minors of the matrices Φ(D_{1}, D_{2}) as D_{1}, D_{2} run over all divisors on C with D_{1} + D_{2} = H.

The aim of this section is to prove Theorem 1.4. This theorem is a variant of Lemma 10.1 giving necessary and sufficient conditions for I(Sec^{r}C) to be generated by the (r + 1) × (r + 1) minors of a single matrix Φ(D_{1}, D_{2}).

Let C_{P} ⊂ ℙ^{n−2} be the elliptic normal curve of degree n − 1 obtained by projecting away from a point P ∈ C. There is a natural inclusion I(Sec^{r}C_{P}) ⊂ I(Sec^{r}C) where these ideals belong to different rings.

Lemma 10.2. If n ≥ 2r + 3 then I(Sec^{r}C) is generated by the ideals I(Sec^{r}C_{P}) as P runs over any n distinct points on C.

Proof: Let X be a subset of C with |X| ≥ n. Let I be the ideal in k[x_{1}, …, x_{n}] generated by the I(Sec^{r}C_{P}) for P ∈ X. By Lemma 10.1 it suffices to show that if D_{1} + D_{2} = H then all (r + 1) × (r + 1) minors of Φ(D_{1}, D_{2}) belong to I. Swapping D_{1} and D_{2} if necessary we may assume that deg D_{1} ≥ r + 2. Let d = deg D_{1}. We pick distinct points P_{1}, …, P_{d} ∈ X with D_{1} ≁ P_{1} + … + P_{d}. Then

⋂_{i=1}^{d} L(D_{1} − P_{i}) = L(D_{1} − (P_{1} + … + P_{d})) = 0.

So there exists a basis v_{1}, …, v_{d} for L(D_{1}) such that L(D_{1} − P_{i}) has basis v_{1}, …, v̂_{i}, …, v_{d}. Then each (r + 1) × (r + 1) minor of Φ(D_{1}, D_{2}) is an (r + 1) × (r + 1) minor of Φ(D_{1} − P_{i}, D_{2}) for some 1 ≤ i ≤ d. We are done since the latter belong to I(Sec^{r}C_{P_{i}}). □

Remark 10.3. The proof of Lemma 10.2 shows that it would be sufficient for P to run over any n − r distinct points. This improvement is irrelevant for our applications.

We obtain an alternative proof of [H, IV.1.3].

Corollary 10.4. If char (k) ≠ 2 then the homogeneous ideal of an elliptic normal curve of degree n ≥ 4 is generated by rank 3 quadrics.

Proof: By Lemma 10.2 it suffices to prove the case n = 4. It is well known (cf. Lemma 10.6 in the case r = 1) that there is a bijection between the singular fibres of the pencil of quadrics containing an elliptic normal quartic, and the 2-torsion of its Jacobian. So provided char (k) ≠ 2 the pencil is spanned by rank 3 quadrics. □

We make a temporary definition.

Definition 10.5. A divisor pair (D_{1}, D_{2}) consists of divisors D_{1}, D_{2} on C with D_{1} + D_{2} = H and deg D_{1}, deg D_{2} ≥ r + 1. We say that divisor pairs (D_{1}, D_{2}) and (D′_{1}, D′_{2}) are equivalent if D_{1} ∼ D′_{1} or D_{1} ∼ D′_{2}.

If n = 2r + 2 then Theorem 1.2 asserts that Sec^{r}C is the complete intersection of two (r + 1)-ics. We make some further observations.

Lemma 10.6. Let n = 2r + 2. Let V be the 2-dimensional vector space of (r + 1)-ics generating I(Sec^{r}C). Then there is a bijection between the set of equivalence classes of divisor pairs and ℙ(V) given by (D_{1}, D_{2}) ↦ det Φ(D_{1}, D_{2}).

Proof: The injectivity was shown in Proposition 7.2. For the surjectivity we may assume that C is the image of an elliptic curve (E, 0) embedded by |n.0|. We put D_{1} = r.(0) + (P) and D_{2} = r.(0) + (−P). Then P ↦ det Φ(D_{1}, D_{2}) is a non-constant morphism E → ℙ(V) = ℙ^{1} and is therefore surjective. □

We strengthen Lemma 10.1.

Lemma 10.7. If n ≥ 2r + 2 then I(Sec^{r}C) is generated by the (r + 1) × (r + 1) minors of Φ(D_{1}, D_{2}) and Φ(D′_{1}, D′_{2}) where (D_{1}, D_{2}) and (D′_{1}, D′_{2}) are any two inequivalent divisor pairs.

Proof: The proof is by induction on n. The case n = 2r + 2 was treated in Lemma 10.6. We may therefore suppose that n ≥ 2r + 3 and deg D_{1}, deg D′_{1} ≥ r + 2. Let P run over any n distinct points on C with D_{1} − D′_{2} ≁ P. Then (D_{1} − P, D_{2}) and (D′_{1} − P, D′_{2}) are inequivalent divisor pairs on C_{P} ⊂ ℙ^{n−2}. By induction hypothesis the (r + 1) × (r + 1) minors of Φ(D_{1} − P, D_{2}) and Φ(D′_{1} − P, D′_{2}) generate I(Sec^{r}C_{P}). Since these are submatrices of Φ(D_{1}, D_{2}) and Φ(D′_{1}, D′_{2}) we are done by Lemma 10.2. □

Proof of Theorem 1.4: We assume that deg D_{1}, deg D_{2} ≥ r + 1 since otherwise there are no (r + 1) × (r + 1) minors to consider. The proof is divided into 4 cases.

(i) Suppose that deg D_{1}, deg D_{2} ≥ r+2 and D_{1} ≁ D_{2}. Let P run over any n distinct points on C. Then (D_{1} − P, D_{2}) and (D_{1}, D_{2} − P) are inequivalent divisor pairs on C_{P} ⊂ ℙ^{n−2}. We know by Lemma 10.7 that I(Sec^{r}C_{P}) is generated by the (r +1)×(r +1) minors of Φ(D_{1} −P, D_{2}) and Φ(D_{1}, D_{2} − P). Since these are submatrices of Φ(D_{1}, D_{2}) we are done by Lemma 10.2.

(ii) Suppose that deg D_{1}, deg D_{2} ≥ r + 3 and D_{1} ∼ D_{2}. Let P run over any n distinct points on C. We know by case (i) that I(Sec^{r}C_{P}) is generated by the (r + 1) × (r + 1) minors of Φ(D_{1}, D_{2} − P). Since this is a submatrix of Φ(D_{1}, D_{2}) we are done by Lemma 10.2.

(iii) Suppose that deg D_{1} = r+1 and deg D_{2} ≥ r+1. Then Φ(D_{1}, D_{2}) has at most \binom{r+t+1}{t} linearly independent minors where

t = deg D_{2} − deg D_{1} = n − 2r − 2.

By Theorem 1.2 the vector space of (r +1)-ics generating I(Sec^{r}C) has dimension

β(r + 1, n) = \binom{r + t + 1}{t} + \binom{r + t}{t} > \binom{r + t + 1}{t}.

Hence Φ(D_{1}, D_{2}) is not a determinantal presentation of Sec^{r}C.

(iv) Suppose that deg D_{1} = deg D_{2} = r + 2 and D_{1} ∼ D_{2}. Choosing suitable bases for L(D_{1}) and L(D_{2}) we may arrange that Φ(D_{1}, D_{2}) is symmetric. Then Φ(D_{1}, D_{2}) has at most (r + 2)(r + 3)/2 linearly independent minors. By Theorem 1.2 the vector space of (r + 1)-ics generating I(Sec^{r}C) has dimension

β(r + 1, n) = (r + 2)^{2} > (r + 2)(r + 3)/2.

Hence Φ(D_{1}, D_{2}) is not a determinantal presentation of Sec^{r}C. This completes the proof of Theorem 1.4. □

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

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