X. An Introduction to a Theorem of Strano

Martin Kreuzer Fakultät für Mathematik Universität Regensburg

Postfach 397

D-W8400 Regensburg, Germany

An Introduction to a Theorem of Strano X 1

Introduction

This expository paper is intended to familiarize the readers with the following theorem of R. Strano:

If a space curve does not lie on a quadric surface and if its generic hyperplane section is a complete intersection, then the curve itself is a complete intersection.

Our goal was to present the proof of this theorem in an easily comprehensible manner, while making the meaning of “generic hyperplane section” as explicit as possible and keeping track of the characteristics assumptions involved. To this end we proceed as follows.

In section 1 we describe the basic linear algebra result involved (cf. Theorem 1.1). We follow the presentation and generality given by R. Re in [R2]. The principal way of applying this theorem for our purposes is explained in Corollary 1.2.

Then we use this result in section 2 to prove several divisibility properties of the Rao module of an integral space curve. Those properties are used implicitly in [S1] and [S2], but we think they merit attention in their own right. The prototype is given by the following special case of Theorem 2.2:

Let M be the Rao module of an integral space curve and ℓ a generic linear form. If m ∈ M_{i−1} and ℓm = 0, then ℓ′m ∈ ℓM_{i−1} for all linear forms ℓ′. In other words, the elements ℓ′m are all divisible by ℓ.

X 2 An Introduction to a Theorem of Strano

Further divisibility properties rely in Strano’s work on the assumption that the generic hyperplane section of the curve satisfies certain Torconditions. In the case of space curves we are able to interpret those as saying that the canonical module of a generic hyperplane section has no minimal generators in certain degrees (cf. Proposition 2.4).

This allows us to elucidate the lifting of elements in Strano’s proof (cf. [S2], Lemma 2) by translating it into the conclusion “If a homogeneous element of the canonical module is not in the maximal ideal times the module, and if there are no minimal generators in that degree, then the element is zero”. Thus we obtain a second divisibility property of the Rao module (cf. Theorem 2.6), a special case of which is the following statement:

If the canonical module of a generic hyperplane section has no minimal generator in degree −i+1, and if m ∈ M_{i−1} with ℓm = 0, then m is divisible by ℓ, i.e. m ∈ ℓM_{i−2}.

For the general version of this divisibility property we need for the first time the assumption that the characteristic of the ground field is not too small. More precisely, this hypothesis is needed to conclude that the powers of linear forms generate low powers of the maximal ideal (cf. Lemma 2.5).

Section 3 deals with various versions of Strano’s characterization of complete intersection curves. Besides the one cited at the beginning of this introduction, we also have formulations referring to the type of the complete intersection of the generic hyperplane section (cf. Theorem 3.1), to the symmetry of its Hilbert function (cf. Corollary 3.3), and to the dimensions of certain cohomology groups of C alone (cf. Corollary 3.4). We hope that those formulations make the assumptions on “char k” and “genericity” of hyperplane sections involved in the corresponding statements in [S1] and [S2] very clear. Notice also the treatment of the case deg C = 4 in Theorem 3.2, for which the argument given in the proof of 3.1 does not apply.

Finally we would like to mention some complementary topics not included here: R. Re’s extension of Strano’s Theorem for curves in higher projective spaces and higher dimensional varieties (see [R2]), counterexamples to the validity of the theorem in small characteristics (if they exist), and an explanation (and possibly proof) of the theorem in terms of the cohomology modules of the curve itself (in the spirit of Corollary 3.4).

Acknowledgements. The author is deeply indebted to A.V. Geramita who introduced him to Strano’s beautiful papers [S1] and [S2] and acted as a careful recipient and critic of a preliminary version of this paper. Thanks also to the Departments of Mathematics of Queen’s University and Regensburg University for their hospitality during various stages of this work, and to NSERC for financial support.

§ 1 Some Linear Algebra

In this section k denotes a field with infinitely many elements.

Definition. Let V and W be finite dimensional k-vector spaces. Consider Hom_{k}(V, W) and k as affine spaces over k. A regular family of linear maps φ_{t} : V ⟶ W is the image of a morphism of affine varieties

\begin{matrix}k & ⟶ & Hom_{k}(V, W) \\ t & ⟼ & φ_{t} \end{matrix}

In other words, if we choose bases for V and W, then the matrix of φ_{t} consists of polynomial functions in t.

The following theorem about regular families of linear maps will be our main tool to derive divisibility properties of Rao modules of curves. It was first shown implicitly by R. Strano in [S1]. We follow the presentation and generality given by R. Re in [R2].

Theorem 1.1. Let V and W be finite dimensional k-vector spaces, let ψ : V ⟶ W be a linear map and ψ_{t} : V ⟶ W a regular family of linear maps. Furthermore, let f : k ⟶ k be a nonzero polynomial function with f(0) = 0, and consider the regular family of linear maps φ_{t} := f(t)ψ + ψ_{t}. Suppose that the following conditions are satisfied.

a) There is a subspace W′ ⊆ W such that im ψ_{t} ⊆ W′ for every t ∈ k.

b) We have rk φ_{o} ≥ rk ψ and rk φ_{o} ≥ rk φ_{t} for every t ∈ k.

Then ψ(ker φ_{o}) ⊆ W′.

Proof. Let δ := dim_{k} V and V′ := ψ^{−1}(W′). Suppose that ψ(ker φ_{o}) ⊈ W′, i.e. that ker φ_{o} ⊈ V′. Since rk φ_{o} ≥ rk ψ, we get that dim_{k} V′ ≥ dim_{k} ker ψ ≥ dim_{k} ker φ_{o}. Thus we also have V′ ⊈ ker φ_{o}, because otherwise both subspaces of V would have to agree in contradiction to our assumption.

Now let T := V′ ∩ ker φ_{o}. From what we just saw it follows that there are nonzero subspaces S′, T′ of V such that V′ = S′ ⊕T and ker φ_{o} = T ⊕T′. Next we choose a subspace S of V such that V = S ⊕S′ ⊕T ⊕T′. Finally, we let α := dim_{k} S, β := dim_{k}(S ⊕ S′), γ := dim_{k}(S ⊕ S′ ⊕ T) and we choose a basis {v_{1}, …, v_{δ}} of V such that {v_{1}, …, v_{α}} is a basis of S, {v_{α+1}, …, v_{β}} is a basis of S′, {v_{β+1}, …, v_{γ}} is a basis of T and {v_{γ+1}, …, v_{δ}} is a basis of T′. Notice that 0 ≤ α < β ≤ γ < δ, because S′, T′ ≠ 0.

From our construction we derive the following observations.

1) The set {v_{β+1}, …, v_{δ}} is a basis of ker φ_{o}.

2) The set {φ_{o}(v_{1}), …, φ_{o}(v_{β})} is a basis of im φ_{o}, because it has the correct cardinality and no linear combination of {v_{1}, …, v_{β}} is in ker φ_{o}.

3) The set {ψ(v_{α+1}), …, ψ(v_{γ})} generates im φ_{o} ∩im ψ, because the set {v_{α+1}, …, v_{γ}} is a basis of V′ = ψ^{−1}(W′) ⊇ ψ^{−1}(im ψ_{o}) = ψ^{−1}(im φ_{o}).

4) The set {ψ(v_{1}), …, ψ(v_{α}), ψ(v_{γ+1}, …, ψ(v_{δ})} is linearly independent, because any nontrivial relation yields a nonzero vector in ker ψ ∩ (S ⊕ T′), but ker ψ ⊆ V′ = S′ ⊕ T.

5) The subspace W′′ := kψ(v_{1}) ⊕ … ⊕ kψ(v_{α}) ⊕ kψ(v_{γ+1}) ⊕ … ⊕ kψ(v_{δ}) satisfies W′′ ∩ W′ = (0), because otherwise a nontrivial linear combination of {v_{1}, …, v_{α}, v_{γ+1}, …, v_{δ}} lies in V′ ⊇ ψ^{−1}(W′) in contradiction to the construction of {v_{1}, …, v_{δ}}.

6) For a generic element t ∈ k the map φ_{t} has image im φ_{t} = kφ_{t}(v_{1}) ⊕ …⊕kφ_{t}(v_{β}), because observation 2) implies that {φ_{o}(v_{1}), …, φ_{o}(v_{β})} is linearly independent, so that for generic t ∈ k also the set {φ_{t}(v_{1}), …, φ_{t}(v_{β})} is linearly independent, and from hypothesis b) it follows that we have rk φ_{t} ≤ rk φ_{o} = β.

7) For a generic element t ∈ k we have ker φ_{t}∩(S⊕S′⊕T) ≠ (0), because dim_{k}(ker φ_{t})+dim_{k}(S⊕S′⊕T) = (δ−β)+(β+δ−γ) = 2δ−γ > δ = dim_{k} V.

Let us choose a generic element t ∈ k such that 6), 7) and f(t) ≠ 0 are satisfied. From 7) we get a nontrivial relation

a_{1}φ_{t}(v_{1}) + … + a_{β}φ_{t}(v_{β}) + a_{γ+1}φ_{t}(v_{γ+1}) + … + a_{δ}φ_{t}(v_{δ}) = 0

with a_{1}, …, a_{β}, a_{γ+1}, …, a_{δ} ∈ k. Thus a) implies that

a_{1}ψ(v_{1}) + … + a_{β}ψ(v_{β}) + a_{γ+1}ψ(v_{γ+1}) + … + a_{δ}ψ(v_{δ}) =

= \frac{1}{f(t)}(−a_{1}ψ_{t}(v_{1}) − … − a_{β}ψ_{t}(v_{β}) − a_{γ+1}ψ_{t}(v_{γ+1}) − … − a_{δ}ψ_{t}(v_{δ}))

∈ W′

and using 3) we even get a_{1}ψ(v_{1}) + … + a_{α}ψ(v_{α}) + a_{γ+1}ψ(v_{γ+1}) + … + a_{δ}ψ(v_{δ}) ∈ W′. Now 5) shows a_{1} = … = a_{α} = a_{γ+1} = … = a + δ = 0, and therefore our original relation simplifies to

a_{α+1}φ(v_{α+1}) + … + a_{β}φ_{t}(v_{β}) = 0.

This finally contradicts observation 6). □

The main application of Theorem 1.1 for our purposes is provided by the following corollary.

Corollary 1.2. Let R = ⨁_{n≥0} R_{n} be a graded k-algebra with R_{o} = k and dim_{k} R_{n} < ∞ for n > 0, and let M = ⨁_{n∈ℤ} M_{n} be a finitely generated graded R-module. For ℓ ∈ R_{1}, i ∈ ℤ and j ≥ 1 denote by μ_{ij}(ℓ) : M_{i−j} ⟶ M_{i} the multiplication by ℓ^{j}.

a) For i ∈ ℤ and j ≥ 1 there is a nonempty Zariski open subset U_{ij} ⊆ R_{1} such that for every ℓ ∈ U_{ij} the multiplication map μ_{ij}(ℓ) has maximal rank among all multiplication maps μ_{ij}(ℓ′), ℓ′ ∈ R_{1}.

b) Let i ∈ ℤ, j ≥ 1 and ℓ ∈ U_{ij} be given. If m ∈ M_{i−j} such that ℓ^{j}m = 0, then (ℓ′)^{j}m ∈ ℓM_{i−1} for every ℓ′ ∈ R_{1}.

Proof. a) Let d := dim_{k} R_{1} and {ℓ_{1}, …, ℓ_{d}} be a basis of R_{1}. Then for ℓ = a_{1}ℓ_{1}+…+a_{d}ℓ_{d} with a_{1}, …, a_{d} ∈ k the condition that μ_{ij}(ℓ) has less than maximal rank is expressed by the vanishing of certain minors of its matrix with respect to some bases of M_{i−j} and M_{i}. This in turn is equivalent to the vanishing of certain polynomials in a_{1}, …, a_{d} and defines a Zariski closed subset of R_{1}.

b) We want to apply Theorem 1.1. Let ψ := μ_{ij}(ℓ′) and φ_{t} := μ_{ij}(ℓ+tℓ′). Because of (ℓ + tℓ′)^{j} = t^{j}(ℓ′)^{j} + \binom{j}{1}t^{j−1}(ℓ′)^{j−1}ℓ + … + \binom{j}{j−1}tℓ′ℓ^{j−1} + ℓ^{j}, we see that φ_{t} is of the form φ_{t} = f(t)ψ + ψ_{t}, where f(t) = t^{j} and ψ_{t} is the multiplication by \binom{j}{1}t^{j−1}(ℓ′)^{j−1}ℓ + … + ℓ^{j}. Notice that φ_{o} = ψ_{o} = μ_{ij}(ℓ) and that ψ_{t} and φ_{t} are regular families of linear maps. Using W′ := im μ_{i−1 1}(ℓ) ⊆ M_{i}, we have im ψ_{t} ⊆ W′ for all t ∈ k. Thus W′ satisfies condition a) of Theorem 1.1. Also condition b) of 1.1 is satisfied, because ℓ ∈ U_{ij} and φ_{o} = μ_{ij}(ℓ).

Therefore Theorem 1.1 yields ψ(ker φ_{o}) ⊆ W′ which is exactly the claim of b). □

§ 2 Divisibility Properties of Rao Modules

From now on we let k be an algebraically closed field, ℙ^{3} the threedimensional projective space over k, and C ⊆ ℙ^{3} an integral curve. We choose coordinates {X_{o}, X_{1}, X_{2}, X_{3}} of ℙ^{3} and let I_{C} ⊆ k[X_{o}, …, X_{3}] be the homogeneous ideal of C. Then R := k[X_{o}, …, X_{3}]/I_{C} is the projective coordinate ring of C. It is a standard graded k-algebra, i.e. R = ⨁_{n≥0} R_{n} with R_{o} = k, dim_{k} R_{1} < ∞ and R = k[R_{1}]. The maximal homogeneous ideal ⨁_{n>0} R_{n} of R is denoted by 𝔪.

Definition. The graded R-module M := H_{𝔪}^{1}(R) is called the Rao module of C.

Denoting by J_{C} ⊆ O_{ℙ} the sheaf of ideals of C, we can describe M also

by M = ⨁_{n∈ℤ} H^{1}J_{C}(n), cf. [G], 2.1.5.

In order to be able to apply Corollary 1.2, we need some finiteness results for the Rao module.

Proposition 2.1. Let M = ⨁_{n∈ℤ} M_{n} be the Rao module of an integral curve C ⊆ ℙ^{3}.

a) For all n ∈ ℤ we have dim_{k} M_{n} < ∞.

b) For all n >> 0 we have M_{n} = 0.

c) For all n ≤ 0 we have M_{n} = 0.

In particular, M is a finitely generated k-vector space and thus also a finitely generated R-module.

Proof. Claims a) and b) follow immediately from Serre’s Finiteness Theorem, cf. [H2], III 5.2. For the proof of c) we follow the argument given in [M].

From the exact sequence of sheaves

0 ⟶ J_{C} ⟶ O_{ℙ} ⟶ O_{C} ⟶ 0

we obtain a long exact sequence of k-vector spaces

0 ⟶ H^{o}J_{C} ⟶ H^{o}O_{ℙ} ⟶ H^{o}O_{C} ⟶ M_{o} ⟶ 0.

Since C is integral, h^{o}O_{C} = 1. Together with h^{o}J_{C} = 0 this shows M_{o} = 0.

Now let n ≤ −1 and let ℓ ∈ k[X_{o}, …, X_{3}] be a linear form such that C ⊈ V(ℓ). Then X := C ∩ V(ℓ) is a 0-dimensional subscheme of C and we have an exact sequence

0 ⟶ J_{C}(n) ⟶^{ℓ} J_{C}(n + 1) ⟶ i_{∗}J_{X/V(ℓ)}(n + 1) ⟶ 0

where i : V(ℓ) ⟶ ℙ^{3} is the inclusion. From this we obtain an exact sequence

H^{o}J_{X/V(ℓ)}(n + 1) ⟶ M_{n} ⟶^{ℓ} M_{n+1}.

Since n ≤ −1, we have H^{o}J_{X/V(ℓ)}(n + 1) = 0 and therefore dim_{k} M_{n} ≤ dim_{k} M_{n+1}. Thus

⋯ ≤ dim_{k} M_{−2} ≤ dim_{k} M_{−1} ≤ dim_{k} M_{o} = 0

implies M_{n} = 0 for n ≤ 0. □

Now we are ready to apply Corollary 1.2 word-for-word and obtain for every i > 0 a nonempty Zariski open subset

U_{i} := U_{i1} ∩ … ∩ U_{i i−1} ⊆ R_{1}

satisfying the conditions of the following theorem, which is our first divisibility property.

Theorem 2.2. For every i ≥ 1 there is a nonempty Zariski open subset U_{i} ⊆ R_{1} such that for every ℓ ∈ U_{i} and every j ∈ {1, …, i} the following statement holds.

If m ∈ M_{i−j} and ℓ^{j}m = 0, then (ℓ′)^{j}m ∈ ℓM_{i−1} for every ℓ′ ∈ R_{1}.

Before we proceed with our second divisibility property, we need to introduce and study the canonical module of a hyperplane section of C. From now on we shall assume that the embedding C ⊆ ℙ^{3} is nondegenerate, i.e. that C is not contained in a hyperplane of ℙ^{3}.

Let ℓ ∈ R_{1} \ {0} and X := V(ℓ) ⊆ C. Then X is a 0-dimensional subscheme of C. Let L ∈ k[X_{o}, …, X_{3}]_{1} be a preimage of ℓ and let {Y_{1}, Y_{2}, Y_{3}} be a k-basis of (k[X_{o}, …, X_{3}]/(L))_{1}. Then X ⊆ V(L) ≅ Proj k[Y], where k[Y] := k[Y_{1}, Y_{2}, Y_{3}]. The projective coordinate ring S of X in V(L) is of the form S = k[Y]/I_{X} with a saturated homogeneous ideal I_{X} ⊆ k[Y]. We choose a minimal homogeneous presentation

⨁_{i=1}^{β_{2}} k[Y](−α_{2i}) ⟶^{φ} ⨁_{i=1}^{β_{1}} k[Y](−α_{1i}) ⟶ I_{X} ⟶ 0.

Since I_{X} is saturated, the map φ is in fact injective.

Definition. Let 𝔫 := ⨁_{n>0} S_{n} be the homogeneous maximal ideal of S. The graded S-module

ω_{S} := Hom_{k}(H_{𝔫}^{1}(S), k)

is called the canonical module of S.

Here the S-module structure of ω_{S} is given by

\begin{matrix}S × ω_{S} & ⟶ & ω_{S} \\ (s, f) & ⟼ & (t ↦ f(st)) \end{matrix}

Proposition 2.3. The canonical module ω_{S} of S has a minimal presentation

⨁_{i=1}^{β_{1}} k[Y](α_{1i} − 3) ⟶^{φ^{∗}(−3)} ⨁_{i=1}^{β_{2}} k[Y](α_{2i} − 3) ⟶ ω_{S} ⟶ 0.

Proof. Using the presentation S = k[Y]/I_{X}, we can compute ω_{S} also from the formula ω_{S} = Ext_{k[Y]}^{2}(S, k[Y])(−3), cf. [GW], 2.2.9. This means that if we dualize the above minimal free resultion of I_{X} with Hom_{k[Y]}(—, k[Y])(−3), then ω_{S} will be the cokernel of φ^{∗}(−3). □

Based on the canonical module of S, we can now give a new interpretation to a certain Tor-condition (cf. [S2], teo. 4) which will play a crucial role in the proof of our second divisibility property.

Proposition 2.4. Let n ≥ 0. The following conditions are equivalent. a) Tor_{1}^{k[Y]}(I_{X}, k)_{n+2} = 0

b) If (f_{o}, f_{1}, f_{2}) ∈ k[Y]_{n}^{3} are such that Y_{i}f_{j} − Y_{j}f_{i} ∈ (I_{X})_{n+1} for i, j = 0, 1, 2, then there is an element f ∈ k[Y]_{n−1} such that f_{i} − Y_{i}f ∈ (I_{X})_{n} for i = 1, 2, 3.

c) We have α_{2i} ≠ n + 2 for i = 1, …, β_{2}.

d) The S-module ω_{S} has no minimal generator in degree −n + 1.

Proof. To see that a) and b) are equivalent, consider the Koszul complex of k[Y] with respect to {Y_{0}, Y_{1}, Y_{2}}, i.e. the exact sequence

0 ⟶ k[Y](−3) ⟶ ⨁_{i=1}^{3} k[Y](−2) ⟶^{ψ} ⨁_{i=1}^{3} k[Y](−1) ⟶^{(Y)} k[Y] ⟶ k ⟶ 0

Tensoring with S = k[Y]/I_{X} yields a complex

0 ⟶ S(−3) ⟶^{d_{3}} ⨁_{i=1}^{3} S(−2) ⟶^{d_{2}} ⨁_{i=1}^{3} S(−1) ⟶^{d_{1}} S ⟶ 0

Now b) is nothing but a reformulation of (ker d_{2})_{n+2} = (im d_{3})_{n+2}, and this is equivalent to Tor_{2}^{k[Y]}(k[Y]/I_{X}, k)_{n+2} = Tor_{1}^{k[Y]}(I_{X}, k)_{n+2} = 0.

Next we show that a) and c) are equivalent. In the above presentation of I_{X} let B := im φ. When we tensor the exact sequence

0 ⟶ B ⟶ ⨁_{i=1}^{β_{1}} k[Y](−α_{1i}) ⟶ I_{X} ⟶ 0

with k = k[Y]/(Y_{0}, Y_{1}, Y_{2}), we obtain an exact sequence

0 ⟶ Tor_{2}^{k[Y]}(I_{X}, k) ⟶ B ⊗_{k[Y]} k ⟶ ⨁_{i=1}^{β_{1}} k(−α_{1i}) ⟶ I_{X} ⊗_{k[Y]} k ⟶ 0

Here we used the fact that Tor_{1}^{k[Y]}(k[Y], k) = 0. Since the entries of the matrix of φ are all contained in the ideal (Y_{0}, Y_{1}, Y_{2}), we find that

Tor_{1}^{k[Y]}(I_{X}, k) ≅ B ⊗_{k[Y]} k ≅ ⨁_{i=1}^{β_{2}} k[Y](−α_{2i}) ⊗_{k[Y]} k ≅ ⨁_{i=1}^{β_{2}} k(−α_{2i})

Thus Tor_{1}^{k[Y]}(I_{X}, k)_{n+2} ≅ ⨁_{i=1}^{β_{2}} k_{n+2−α_{2i}}, implying that a) and c) are equivalent.

The equivalence of c) and d) follows immediately from the minimal presentation of ω_{S} given in Proposition 2.3. □

We need one more lemma before we are ready to prove the main result of this section.

Lemma 2.5. Let j ≥ 1. The following conditions are equivalent. a) We have char k = 0 or char k > j.

b) For every n ∈ {1, …, j} the powers ℓ^{n}, ℓ ∈ k[Y]_{1} generate k[Y]_{n} as a k-vector space.

Proof. First we show that a) implies b). Let Z_{1}, …, Z_{n} ∈ {Y_{o}, Y_{1}, Y_{2}}. We have to show that Z_{1} ⋯ Z_{n} is a k-linear combination of n-th powers of linear forms. From a) it follows that n! is a unit in k. Now consider the following formula (cf. [B], I 8.2.2).

Z_{1} ⋯ Z_{n} = (−1)^{n} \frac{1}{n!} ∑_{H⊆\{0,1,2\}}(−1)^{card (H)}(∑_{i∈H} Y_{i})^{n}

It expresses the product Z_{1} ⋯ Z_{n} in the desired way.

To prove the converse by contradiction, we suppose that j ≥ 2 and char k = p ∈ {2, …, j}. Then

(a_{o}Y_{o} + a_{1}Y_{1} + a_{2}Y_{2})^{p} = a_{o}^{p}Y_{o}^{p} + a_{1}^{p}Y_{1}^{p} + a_{2}^{p}Y_{2}^{p}

for all a_{o}, a_{1}, a_{2} ∈ k, and thus for instance Y_{o}^{p−1}Y_{1} is not a linear combination of p-th powers of linear forms. This contradicts our assumption. □

The next theorem contains our second divisibility property of the Rao module.

Theorem 2.6. Let i ≥ 1, U_{i} ⊆ R_{1} as in 2.2, ℓ ∈ U_{i} and j ∈ {1, …, i}. Also suppose that char k = 0 or char k > j and that X := V(ℓ) satisfies the conditions of Proposition 2.4 for i + 1 − j ≤ n ≤ i.

Then every m ∈ M_{i−j} with ℓ^{j}m = 0 is of the form m = ℓm′ with m′ ∈ M_{i−j−1}.

Proof. Using Theorem 2.2 we get that (ℓ′)^{j}m ∈ ℓM_{i−1} for every ℓ′ ∈ R_{1}. Consider the canonical exact sequence of sheaves

0 ⟶ J_{C}(i + j − 1) ⟶^{L} J_{C}(i − j) ⟶ i_{∗}J_{X/V(L)}(i − j) ⟶ 0 (∗)

where L ∈ k[X_{o}, …, X_{3}] is a preimage of ℓ and i : V(L) ⟶ ℙ^{3} is the inclusion. From this sequence we obtain a k-linear map ϵ : M_{i−j} ⟶ H^{1}J_{X/V(L)}(i − j). Let S be the projective coordinate ring of X in V(L).

In the graded S-module N := ⨁_{n∈ℤ} H^{1}J_{X/V(L)}(n), the element ϵ(n) ∈ N_{i−j} satisfies (\overline{ℓ})^{j}ϵ(m) = 0 for every \overline{ℓ} ∈ S_{1}, because (ℓ′)^{j}m ∈ ℓM_{i−1} for every preimage ℓ′ of \overline{ℓ} in R_{1}. Using our assumption on char k and Lemma 2.5, this implies that S_{j} · ϵ(m) = 0.

Notice that the canonical module ω_{S} of S is given by ω_{S} ≅ Hom_{k}(N, k), because [G], 2.1.5 yields

H_{𝔫}^{1}(S) ≅ H_{𝔫}^{2}(I_{X}) ≅ ⨁_{n∈ℤ} H^{1}J_{X/V(L)}(n) = N.

Fix a complement V of k·ϵ(m) in N_{i−j}, i.e. let N_{i−j} ≅ V ⊕k·ϵ(m) as k-vector spaces. Let p : N ⟶ k be the k-linear map with p|_{N_{n}} = 0 for n ≠ i − j and p|_{N_{i−j}} : N_{i−j} ⟶ k is the projection to k · ϵ(m) along V.

We claim that the condition S_{j} · ϵ(m) = 0 implies that p = 0 or p ∉ S_{j}(ω_{S})_{−i}, when we consider p as an element of (ω_{S})_{j−i} ≅ Hom_{k}(N, k)_{j−i}. For if p ∈ S_{j}(ω_{S})_{−i} \ {0}, then p is of the form p = sq, where s ∈ S_{j} and q ∈ (ω_{S})_{−i} \{0}. But then 1 = p (ϵ(m)) = (sq)(ϵ(m)) = q(sϵ(m)) = q(0) = 0, a contradiction. From our hypothesis and Proposition 2.4 it follows that ω_{S} has no minimal generators in degrees −i + 1, …, j − i. This yields p = 0, and thus ϵ(m) = 0.

Consider the following part of the long exact cohomology sequence of (∗):

⋯ ⟶ M_{i−j−1} ⟶^{ℓ} M_{i−j} ⟶^{ϵ} H^{1}J_{X/V(L)}(i − j) ⟶ ⋯

Since ϵ(m) = 0, it follows that m is of the form m = ℓm′ with m′ ∈ M_{i−j−1}. □

Repeated application of Theorem 2.6 yields more results about the structure of the Rao module. For every i ≥ 1 we denote by U_{i} ⊆ R_{1} the nonempty Zariski open subset given prior to 2.2, and we denote by Ũ_{i} ⊆ R_{1} the nonempty Zariski open subset

Ũ_{i} := U_{1} ∩ … ∩ U_{i}.

Corollary 2.7. Let i ≥ 1, Ũ_{i} ⊆ R_{1} as above, ℓ ∈ Ũ_{i}, and assume that char k = 0 or char k ≥ i. Also suppose that X := V(ℓ) satisfies the conditions of Proposition 2.4 for 2 ≤ n ≤ i.

Then the multiplication map M_{i−1} ⟶^{ℓ} M_{i} is injective.

Proof. Starting with m ∈ M_{i−1}, ℓm = 0, apply Theorem 2.6 with j = 1, …, i − 1. We find that m is of the form m = ℓ^{i−1}m′ with m′ ∈ M_{o}. But M_{o} = 0 by 2.1, so m = 0. □

Notice that in view of the equivalence in Lemma 2.5, we need the full strength of our assumption on char k here.

§ 3 Characterizations of Complete Intersection Curves

In what follows we keep the notations and conventions of the previous section. In particular, we work over an algebraically closed field k of arbitrary characteristic, and C denotes an integral curve in ℙ^{3} which is not contained in a hyperplane.

R. Strano’s characterization of complete intersection curves (cf. [S1], thm. 6) is expressed by the following theorem.

Theorem 3.1. Suppose that there are numbers 3 ≤ s ≤ t such that the following conditions are satisfied.

1) char k = 0 or char k ≥ t.

2) There is a linear form ℓ ∈ Ũ_{t} ⊆ R_{1} such that the hyperplane section X := V(ℓ) of C is the complete intersection of a curve of degree s and a curve of degree t, i.e. a complete intersection of type (s, t).

Then C is the complete intersection of a surface of degree s and a surface of degree t.

Proof. Since X is a complete intersection of type (s, t), we have ω_{S} ≅ S(s + t − 3) by [GW], 2.2.8 and 2.2.10. Therefore ω_{S} has no minimal generators in degrees ≥ −s − t + 4, and X satisfies the conditions of Proposition 2.4 for n ≤ s + t − 3. Since s ≥ 3, Corollary 2.7 says in particular that the multiplication maps M_{i−1} ⟶^{ℓ} M_{i} are injective for i ≤ t. Using the long exact cohomology sequence of

0 ⟶ J_{C}(n − 1) ⟶^{ℓ} J_{C}(n) ⟶ i_{∗}J_{X/V(L)}(n) ⟶ 0

(cf. proof of 2.6) we find that the canonical homomorphisms H^{o}J_{C}(n) ⟶ H^{o}J_{X/V(L)}(n) are surjective for n ≤ t. Thus we can lift the equations of the curves defining X and get equations of surfaces defining a complete intersection curve of type (s, t) which contains C. Since deg C = deg X = st, the curve C is in fact equal to that complete intersection. □

Remark. Notice that given the equations of C, we can use Corollary 1.2.a to produce explicit equations defining the complement of Ũ_{t}. Obviously Theorem 3.1 applies also, if the generic hyperplane section X = V(ℓ) of C is a complete intersection of type (s, t), i.e. if there is an open subset U ⊆ R_{1} such that for every ℓ ∈ U the hyperplane section X := V(ℓ) of C is a complete intersection of type (s, t).

The dependence of Theorem 3.1 on a concrete pair of numbers (s, t) can be removed as follows.

Theorem 3.2. Suppose that char k = 0 or char k ≥ \frac{1}{3} deg C, that C is not contained in a quadric surface, and that the generic hyperplane section of C is a complete intersection.

Then C is a complete intersection.

Proof. If a hyperplane section X′ of C is a complete intersection of type (s′, t′), then deg C = deg X′ = s′t′. Hence there are only finitely many possibilities for (s′, t′) and there is one pair (s, t) such that the generic hyperplane section of C is a complete intersection of type (s, t). Then we find an element ℓ ∈ Ũ_{t} such that X := V(ℓ) is a complete intersection of type (s, t).

W.l.o.g. we can assume that s ≤ t. We consider the various cases separately. If s = 1, then Proposition 2.1.c and the exact sequence

0 ⟶ J_{C} ⟶^{ℓ} J_{C}(1) ⟶ i_{∗}J_{X/V(L)}(1) ⟶ 0

(cf. proof of 2.6) yield an epimorphism H^{o}J_{C}(1) −↠ H^{o}J_{X/V(L)}(1). This implies that C is contained in a hyperplane and contradicts our nondegeneracy assumption.

If s = 2 and t > 2, then ω_{S} ≅ S(t − 1) and t − 1 ≥ 2 show that ω_{S} has no minimal generator in degree −1. Thus we can apply Corollary 2.7 as in the proof of 3.1, and we get an epimorphism H^{o}J_{C}(2) −↠ H^{o}J_{X/V(L)}(2), contradicting the assumption that C is not contained in a quadric surface.

If s = t = 2, we can use the exact sequences

⋯ → H^{1}J_{X/V(L)}(n) → H^{2}J_{C}(n − 1) ⟶^{ℓ} H^{2}J_{C}(n) → H^{2}J_{X/V(L)}(n) → ⋯

and h^{1}J_{X/V(L)}(n) = h^{2}J_{X/V(L)}(n) = 0 for n ≥ 2 to conclude that h^{2}J_{C}(1) = h^{2}J_{C}(2) = h^{2}J_{C}(3) = ⋯ = 0. In particular, h^{1}O_{C}(2) = h^{2}J_{C}(2) = 0, and by Riemann-Roch (cf. [H2], IV ex. 1.9) we get h^{o}O_{C}(2) = h^{1}O_{C}(2)+2 deg C+ 1 − p_{a}(C) = 9 − p_{a}(C). But then the exact sequence

0 ⟶ H^{o}J_{C}(2) ⟶ H^{o}O_{ℙ}(2) ⟶ H^{o}O_{C}(2) ⟶ H^{1}J_{C}(2) ⟶ 0

shows h^{o}J_{C}(2) = 10 − 9 + p_{a}(C) + h^{1}J_{C}(2) ≥ 1, in contradiction with our assumption that C is not contained in a quardic surface. Notice that we used p_{a}(C) = h^{1}O_{C} ≥ 0 here, which is only true for integral curves C.

Finally, if 3 ≤ s ≤ t, then Theorem 3.1 applies and yields the desired result. □

The following example demonstrates that the assumption “C is not contained in a quadric surface” cannot be removed from Theorem 3.2.

Example. If C is a curve in the linear system of type (a, b) and even degree deg C = a+b = 2t on a nonsingular quadric surface Q ⊆ ℙ^{3}, then the generic hyperplane section of C is a complete intersection of type (2, t). But as long as a ≠ b, the curve C is not a complete intersection.

Proof. Choose L ∈ H^{o}O_{ℙ}(1) such that Q ∩ V(L) is an irreducible conic and X := C ∩ V(L) is a set of 2t distinct points. By dividing X into t pairs of points and drawing the line through each pair, we find a reducible curve of degree t such that X is the complete intersection of that curve and the conic Q ∩ V(L).

But C is not a complete intersection curve, if a ≠ b. W.l.o.g. let a < b. From J_{C/Q} ≅ O_{Q}(−a, −b) and h^{1}O_{Q}(0, a − b) ≠ 0 (cf. [H2], III ex. 5.6) we find that the canonical map H^{o}O_{Q}(a) ⟶ H^{o}O_{C}(a) is not surjective. Hence the canonical map H^{o}O_{ℙ}(a) ⟶ H^{o}O_{C}(a) is not surjective, either, i.e. C is not even arithmetically Cohen-Macaulay. □

For the remainder of this section we shall assume that one of the following conditions is satisfied:

1) char k = 0

2) C is reflexive.

3) C is arithmetically Cohen-Macaulay.

In the first two cases we know by [H1], § 2 resp. [R1], 1.8 and 2.2 that the points of a generic hyperplane section of C are in uniform position.

Definition. Let X be a 0-dimensional reduced subscheme of ℙ^{2}. Denote by H_{X} : ℤ ⟶ ℕ its Hilbert function and let a_{X} := max{n ∈ ℤ : H_{X}(n) < deg X}.

a) We say that X is a Cayley-Bacharach scheme, if H_{Y} (a_{X}) = H_{X}(a_{X}) for every subset Y ⊆ X with deg Y = deg X − 1.

b) We say that X has a symmetric Hilbert function, if for all n ∈ ℤ we have H_{X}(n) + H_{X}(a_{X} − n) = deg X.

Now it follows from [GKR], 2.9 and 5.1 that under any of the above conditions the generic hyperplane section of C is a Cayley-Bacharach scheme.

Corollary 3.3. Suppose that char k = 0 or char k ≥ \frac{1}{3} deg C, that C is not contained in a quadric surface, and that the generic hyperplane section of C has a symmetric Hilbert function.

Then C is a complete intersection curve.

Proof. From [DM], § 4 or [K], § 2 it follows that the generic hyperplane section of C is in fact a complete intersection and we can apply Theorem 3.2. □

Finally we reformulate the condition “The generic hyperplane section of C has a symmetric Hilbert function” purely in terms of the cohomology modules of C.

Corollary 3.4. Suppose that char k = 0 or char k ≥ \frac{1}{3} deg C, and that C is not contained in a quadric surface. We choose ℓ ∈ R_{1} generic, i.e. such that X := V(ℓ) is a Cayley-Bacharach scheme and dim_{k} M/ℓM is maximal.

a) The number a_{X} can be calculated from the cohomology modules of C. Namely, it is the supremum of 1 + max{n ∈ ℤ : h^{1}O_{C}(n) ≠ 0} and sup{n ∈ ℤ : (M/ℓM)_{n} ≠ 0}.

b) For every n ∈ ℤ we have h^{o}O_{C}(n) − h^{o}O_{C}(n − 1) − dim_{k}(M/ℓM)_{n} + h^{o}O_{C}(a_{X} − n) − h^{o}O_{C}(a_{X} − n − 1) − dim_{k}(M/ℓM)_{a_{X}−n} ≤ deg C.

c)The inequalities in b) are equalities for all n ∈ ℤ if and only if C is a complete intersection curve.

Proof. To see a), notice that a_{X} = max{n ∈ ℤ : h^{1}J_{X/V(L)}(n) ≠ 0} and that the exact sequences

0 ⟶ J_{C}(n − 1) ⟶^{ℓ} J_{C}(n) ⟶ i_{∗}J_{X/V(L)}(n) ⟶ 0  (∗)

(cf. proof of 2.6) together with the isomorphisms H^{2}J_{C}(n) ≅ H^{1}O_{C}(n) yield exact sequences

0 ⟶ (M/ℓM)_{n} ⟶ H^{1}J_{X/V(L)}(n) ⟶ H^{1}O_{C}(n − 1) ⟶^{ℓ} H^{1}O_{C}(n) ⟶ 0.

To prove b) we use the inequalities H_{X}(n) + H_{X}(a_{X} − n) ≤ deg X which are true for any Cayley-Bacharach scheme (cf. [GKR], 3.7.a). Let K := ker(M(−1) ⟶^{ℓ} M). From the long exact cohomology sequence of (∗) we calculate

H_{X}(n) = H_{C}(n) − H_{C}(n − 1) − dim_{k} K_{n},

and together with the canonical exact sequence

0 ⟶ R ⟶ ⨁_{n≥0} H^{o}O_{C}(n) ⟶ M ⟶ 0

(cf. [G], 2.1.5) this yields

H_{X}(n) = h^{o}O_{C}(n) − h^{o}O_{C}(n − 1) − dim_{k}(M/ℓM)_{n}.

When we plug this into the above inequalities and use deg C = deg X, we obtain the desired inequalities.

Claim c) follows from the observation that the inequalities H_{X}(n) + H_{X}(a_{X} − n) ≤ deg X are equalities if and only if X has a symmetric Hilbert function, and from Corollary 3.3. □

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