The Hilbert Scheme of Space Curves: Ghost terms, linkage and obstructed curves

Jan O. Kleppe

December 10, 2008

Summary of talks. We consider space curves X with homogeneous ideal I and Rao module M which often satisfy _{0}Ext_{R}^{2}(M, M) = 0. We find necessary and sufficient conditions for unobstructedness, and we compute the dimension of the Hilbert scheme, H(d, g), at (X) under the sufficient conditions. In the diameter one case (M_{v} = 0 for v ≠ some c), the necessary and sufficient conditions coincide, and the obstructedness turns out to be equivalent to the non-vanishing of certain graded Betti numbers, e.g. there are repeated direct free factors ("ghost-terms"), in the minimal free resolution of I. Moreover by taking suitable deformations we show how to kill some of the "ghost-terms" in the free resolution of a curve of arbitrary diameter. For Buchsbaum curves of diameter at most 2, we simplify in this way the minimal resolution further, allowing us to see when a singular point of H(d, g) sits in the intersection of several, or lies in a unique irreducible component of H(d, g). As a consequence we get that any irreducible component of H(d, g) is generically smooth in the diameter 1 case.

1 Introduction.

The main object of these notes is the Hilbert scheme H(d, g) of space curves. As a set the Hilbert scheme H(d, g) is

H(d, g) = {(X) | X ⊆ ℙ^{3} a curve of degree d and arithmetic genus g}

where we by a curve mean an equidimensional, locally Cohen-Macaulay (lCM) one-dimensional subscheme of the projective 3-space ℙ^{3} defined over an algebraically closed field k. The Hilbert scheme H(d, g) is actually the representing object of a correspondingly defined functor of (flat) deformations. Its existence as a scheme was proved in the late fifties by Grothendieck [14].

In these notes we will focus on the structure of the Hilbert scheme H(d, g). Much is still unknown concerning questions related to irreducibility, number of components, dimension and smoothness of H(d, g). For particular classes of space curves, some results are known, e.g. that the open subset of H(d, g) of arithmetically Cohen-Macaulay curves is smooth of known dimension [8]. Further progress to the questions was made in [35], [19], [20], [2], [3], [36], [37], [25], [12], [26], [7], [10], [11], [31] and [22], to mention some paper of particular relevance for these notes. In the talks we will report on the work [22] with a special look to obstructedness of Buchsbaum curves, interpreted by their minimal resolutions, and the set of irreducible components of H(d, g) containing Buchsbaum and arithmetically CM curves.

In the following we will use these Notations and terminologies

• R = k[X_{0}, X_{1}, X_{2}, X_{3}] a polynomial ring over k = \overline{k}.

• 𝔪 = (X_{0}, .., X_{3}) is the irrelevant maximal ideal.

• I_{X} := the ideal sheaf of X in ℙ = ℙ^{3}.

• N_{X} := Hom_{O_{ℙ}}(I_{X}, O_{X}) the normal sheaf of X ⊆ ℙ^{3}.

• H^{i}(F) := H^{i}(ℙ, F), where F is a coherent O_{ℙ}-Module.

• H_{∗}^{i}(F) = ⊕_{v} H^{i}(F(v)).

• h^{i}(F) := dim H^{i}(F), and χ(F) = Σ(−1)^{i}h^{i}(F).

• Γ_{𝔪}(N) := ⊕_{v} ker(N_{v} → Γ(ℙ, Ñ(v))), where N is a graded R module of finite type.

• H_{𝔪}^{i}(N) is the right derived functor of Γ_{𝔪}(N).

• I = I(X) := H_{∗}^{0}(I_{X}) is the homogeneous ideal of X ⊆ ℙ.

• M = M(X) := H_{∗}^{1}(I_{X}) is the Hartshorne-Rao module of X ⊆ ℙ.

• E = E(X) := H_{∗}^{1}(O_{X}).

• The postulation of X ⊆ ℙ is the function γ(v) = h^{0}(I_{X}(v)) defined over the integers.

• ρ(v) = ρ_{X}(v) := h^{1}(I_{X}(v)) is the deficiency function, and

• σ(v) = σ_{X}(v) := h^{1}(O_{X}(v)) is the specialization function.

• _{v}ext(N_{1}, N_{2}) = dim_{v}Ext^{i}(N_{1}, N_{2}) etc., i.e. we use small letters for the k-dimension.

Attached to a curve X ⊆ ℙ^{3} we define the following numbers;

• s(X) := min{n| h^{0}(I_{X}(n)) ≠ 0},

• e(X) = max{n|h^{1}(O_{X}(n)) ≠ 0},

• c(X) = max{n|h^{1}(I_{X}(n)) ≠ 0}, provided M ≠ 0.

• b(X) = min{n|h^{1}(I_{X}(n)) ≠ 0}, provided M ≠ 0.

• Then diam M = c(X) − b(X) + 1 is the diameter of M, or of X (let diam M = 0 if M = 0).

• If M = 0, we say X is arithmetically Cohen-Macaulay (ACM).

• If M ≠ 0 and c(X) < s(X), or if M = 0, we say X has maximal rank.

• If M ≠ 0, c(X) < s(X) and e(X) < b(X) (or if M = 0, and e(X) < s(X)), we say X has seminatural cohomology.

• If 𝔪 · M(X) = 0, then X is a Buchsbaum curve.

For the Hilbert scheme and its strata we say that

• X is unobstructed if the Hilbert scheme, H(d, g), is smooth at the corresponding point (X), otherwise X is obstructed.

• H_{γ,ρ} ⊆ H(d, g) is the stratum consisting of curves X with constant cohomology, i.e. γ_{X} and ρ_{X} do not vary with X [25].

• H_{γ} ⊆ H(d, g) consists of curves with constant postulation γ.

• H(d, g)_{S} ⊆ H(d, g) is the open subscheme consisting of smooth connected curves.

• The curve in a small open irreducible subset of H(d, g) (small enough to satisfy all the openness properties which we want it to have) is called a generic curve of H(d, g).

• If a generic curve has a certain property, then there is an non-empty open irreducible subset of H(d, g) of curves having this property.

• A generization X′ of X in H(d, g) is a generic curve of some irreducible subset of H(d, g) containing (X). Then X is a specialization of X′ in H(d, g).

2 Preliminaries.

In this section we shortly review some results which we will need later on and we include references for a more thorough reading.

2.1 Cohomology groups and deformations.

Let N, N_{i} be graded R-modules of finite type, and recall that the right derived functor vExt_{𝔪}^{i}(N, −) of _{v}Γ_{𝔪}^{0}(Hom_{R}(N, −)) is equipped with a spectral sequence ([13], exp. VI)

E_{2}^{p,q} = _{v}Ext_{R}^{p}(N_{1}, H_{𝔪}^{q}(N_{2})) converging to _{v}Ext_{𝔪}^{p+q}(N_{1}, N_{2})  (1)

and a duality isomorphism ([21], Thm. 1.1, [19] Thm. 2.1.4 and Rem. 2.1.5 for a full proof);

vExt_{𝔪}^{i}(N_{2}, N_{1}) ≅ _{−v−4} Ext_{R}^{4−i}(N_{1}, N_{2})^{∨} where (−)^{∨} = Hom_{k}(−, k),  (2)

which generalizes the usual Gorenstein duality _{v}H_{𝔪}^{i}(N) ≃ _{−v}Ext_{R}^{4−i}(N, R(−4))^{∨}. These groups fit into a long exact sequence ([13], exp. VI)

→ _{v}Ext_{𝔪}^{i}(N_{1}, N_{2}) → _{v}Ext_{R}^{i}(N_{1}, N_{2}) → Ext_{O_{ℙ}}^{i}(Ñ_{1}, Ñ_{2}(v)) → _{v}Ext_{𝔪}^{i+1}(N_{1}, N_{2}) → (3)

which in particular relates the deformation theory of the curve X ⊆ ℙ^{3}, described by

H^{i−1}(N_{X}) ≅ Ext_{O_{ℙ}}^{i}(I_{X}, I_{X}) for i = 1, 2  (4)

(cf. [19], Rem. 2.2.6 for a proof of this isomorphism), to the deformation theory of the quotient A := R/I(X), or equivalently of the homogeneous ideal I = I(X), described by

0Ext_{R}^{i}(I, I) for i = 1, 2

(cf. [19], Thm. 2.2.1 and [36], Thm. 2.3 where Walter manages to get rid of the “generically complete intersection” assumption of [19], § 2.2), in the following exact sequence

0 → _{v}Ext_{R}^{1}(I, I) → H^{0}(N_{X}(v)) → _{v}Hom_{R}(I, M) ⟶^{α} _{v}Ext_{R}^{2}(I, I) → H^{1}(N_{X}(v)) → . (5)

Note that M = H_{𝔪}^{2}(I) and _{v}Ext_{𝔪}^{2}(I, I) ≅ _{v}Hom_{R}(I, M). In this situation Walter also proved that the map α : _{v}Hom_{R}(I, M) → _{v}Ext_{R}^{2}(I, I) of (5) factorizes via _{v}Ext_{R}^{2}(M, M) in a natural way ([36], Thm. 2.3), the factorization is in fact given by a certain edge homomorphism of the spectral sequence (1) with N_{1} = M, N_{2} = I and p + q = 4.

Let Def_{A} be the deformation functor of deforming A = R/I as a graded quotient of R, defined on the category of local artinian k-algebras with residue field k. Let Hilb_{X} be the corresponding deformation functor of X ⊆ ℙ^{3} (i.e the local Hilbert functor at X) defined on the same category. Note that (5) and the assumption _{0}Hom(I, M) = 0 lead to

Def_{A} ≅ Hilb_{X}  (6)

because the assumption leads to an isomorphism of their tangent spaces and an injection of their obstruction spaces. This was first proved in [17] (Thm. 3.6 and Rem. 3.7) in this generality using a cohomological proof, but the case M = 0 was already proved in [8] with a direct proof.

Lemma 1. With M = H_{∗}^{1}(I_{X}) and E = H_{∗}^{1}(O_{X}) we have

vExt_{R}^{i}(M, M) ≅ _{−v−4} Ext_{R}^{4−i}(M, M)^{∨} , i, v ∈ ℤ.

Moreover if _{v}Ext_{R}^{2}(M, M) = 0, then there is an exact sequence

0 → _{−v−4}Ext_{R}^{1}(M, M) → _{v}Ext_{R}^{1}(I, M)^{∨} → _{−v−4}Hom_{R}(M, E) → 0 .

Proof. The duality follows from (2) since M is artinian. By (2) we also have _{v}Ext_{R}^{1}(I, M)^{∨} ≅ −v−4Ext_{𝔪}^{3}(M, I) and then we conclude by a standard exact sequence associated to (1).

2.2 Minimal resolutions, linkage and deformations.

Let X be a curve in ℙ^{3}. Related to a minimal resolution,

0 → ⊕_{i}R(−i)^{β_{3,i}} → ⊕_{i}R(−i)^{β_{2,i}} → ⊕_{i}R(−i)^{β_{1,i}} → I → 0 ,  (7)

of the homogeneous ideal I of X, we define the following invariant:

Definition 2.

δ^{j}(v) = ∑_{i} β_{1,i} · h^{j}(I_{X}(i + v)) − ∑_{i} β_{2,i} · h^{j}(I_{X}(i + v)) + ∑_{i} β_{3,i} · h^{j}(I_{X}(i + v)) .

We also write β_{j,i}(X) = β_{j,i}. We get (see [22], Lem. 2.2, for a proof).

Lemma 3.

0ext_{R}^{1}(I, I) −_{0}ext_{R}^{2}(I, I) = 1 − δ^{0}(0) = 4d + δ^{2}(0) − δ^{1}(0) = 1 + δ^{2}(−4) − δ^{1}(−4) .

Remark 4. In [25] the numbers 1 − δ^{0}(0) and δ^{1}(−4) were called δ_{γ} and ϵ_{γ,δ} respectively. By Lemma 3 it follows that the dimension of the tangent space to the Hilbert scheme H_{γ,ρ} at (X), which they show is δ_{γ} + ϵ_{γ,δ} −_{0}hom(M, M) +_{0}ext^{1}(M, M) (Thm. 4.2, page 173), is also equal to 1 + δ^{2}(−4) −_{0}hom(M, M) +_{0}ext^{1}(M, M).

In these notes to the talks we need the notion of linkage and the following proposition (see [20] , Prop. 3.12 and Prop. 3.8, for a proof). We consider I_{X/Y} := I_{X}/I_{Y} as the sheaf ideal of X in Y. Definition 5. Two curves X and X′ are said to be (algebraically) CI-linked if there exists a complete intersection curve (a CI) Y such that

I_{X}/I_{Y} ≅ Hom_{O_{ℙ}}(O_{X′}, O_{Y}) and I_{X′}/I_{Y} ≅ Hom_{O_{ℙ}}(O_{X}, O_{Y}) .

Thus if Y is a CI of type (f, g), then we know that the dualizing sheaf satisfies ω_{Y} ≅ O_{Y} (f + g − 4), and hence we get

I_{X/Y} ≅ ω_{X′}(4 − f − g) and I_{X′/Y} ≅ ω_{X}(4 − f − g) .  (8)

Proposition 6. Let X and X′ be curves in ℙ^{3} which are linked (algebraically) by a complete intersection of two surfaces of degrees f and g. If

H^{1}(I_{X}(v)) = 0 for v = f, g, f − 4 and g − 4,

then X is unobstructed (resp. generic) if and only if X′ is unobstructed (resp. generic).

Now recall Rao’s theorem concerning the form of a minimal resolution (7) of I = I(X). Let

0 → L_{4} ⟶^{σ} L_{3} → L_{2} → L_{1} → L_{0} → M → 0  (9)

is the minimal resolution of M = H_{∗}^{1}(I_{X}). Then (7) and

0 → L_{4} ⟶^{σ⊕0} L_{3} ⊕ F_{2} → F_{1} → I → 0  (10)

are isomorphic. Here the composition of L_{4} → L_{3} ⊕F_{2} with the natural projection L_{3} ⊕F_{2} → F_{2} is zero ([34], Thm. 2.5).

It is well known that the Hartshorne-Rao module M is a biliaison invariant, up to twist ([34]). Moreover using (8) and the fact that ω_{X′} ≅ Ext^{2}(O_{X′}, O_{ℙ}(−4)) it is rather straightforward to find a resolution of I_{X′} in terms of the resolution (7) and some part of the resolution of the dual of M, by the mapping cone construction. Using the mapping cone construction twice, we get a nice relationship between the minimal resolution (10) of I and a free resolution of the homogeneous ideal of the bilinked curve Z. Indeed suppose we link X, first using a CI of type (f, g), then a CI of type (f′, g′), and put h = f′ + g′ − f − g. Then

0 → L_{4}(−h) ⟶^{σ(−h)⊕0} L_{3}(−h) ⊕ F_{2Z} → F_{1}(−h) ⊕ R(−f′) ⊕ R(−g′) → I(Z) → 0 (11)

is a free resolution of I(Z) where F_{2Z} := F_{2}(−h) ⊕ R(−f − h) ⊕ R(−g − h) (see [27]). Note that this is not necessarily a minimal resolution. For instance if one of the hypersurfaces of the second linkage is the same as one of the hypersurfaces of the first linkage (e.g. f = f′), the direct factor R(−g′) is redundant. Example 7. (Sernesi [35], [6]) If X is “2 skew lines”, and we link twice via CI’s of type (4, 2) and (4, 6), using a common hypersurface of degree 4 in both linkages, then we get a curve Z of degree 18 and genus 39 and with minimal resolution

0 → R(−8) → R(−8) ⊕ R(−7)^{4} → R(−6)^{4} ⊕ R(−4) → I(Z) → 0 ,

cf. (11). If we compare it to the Rao form (10), we see that F_{2} = R(−8) and that 0 → L_{4} = R(−8) → L_{3} = R(−7)^{4} is the leftmost part in the minimal resolution “the Koszul resolution” of M(Z) where M(Z) = R/𝔪(−4).

3 Sufficient conditions for unobstructedness.

In these notes we will focus on the structure of the Hilbert scheme H(d, g). As mentioned earlier, much is still unknown concerning questions related to irreducibility, number of components, dimension and smoothness of H(d, g). Some results are, however, known. In 1975 Ellingsrud [8] managed to prove that the open subset of H(d, g) of arithmetically Cohen-Macaulay curves (resp. with fixed postulation) is smooth (resp. irreducible), and he computed the dimension of the corresponding components. A generalization of this result in the direction of smoothness and dimension was given in [19] (see Theorem 8 (i) below) while the irreducibility was generalized by Bolondi [2]. Later, Martin-Deschamps and Perrin gave a stratification H_{γ,ρ} of H(d, g) obtained by deforming space curves with constant cohomology [25]. Their results lead rather immediately to (iii) in the result below. In [22] we made some further progress and proved among other things the following result.

Theorem 8. Let X be a curve in ℙ^{3} of degree d and arithmetic genus g, let I = H_{∗}^{0}(I_{X}), M = H_{∗}^{1}(I_{X}) and E = H_{∗}^{1}(O_{X}) and suppose at least one of the following conditions:

(i) _{v}Hom_{R}(I, M) = 0 for v = 0 and v = −4 ,
(ii) _{v}Hom_{R}(M, E) = 0 for v = 0 and v = −4 , or
(iii) _{0}Hom_{R}(I, M) = 0 , _{0}Hom_{R}(M, E) = 0 and _{0}Ext_{R}^{2}(M, M) = 0 .

Then H(d, g) is smooth at (X), i.e. X is unobstructed. Moreover if _{0}Ext_{R}^{i}(M, M) = 0 for i ≥ 2, then the dimension of the Hilbert scheme at (X) is

dim_{(X)} H(d, g) = 4d + δ^{2}(0) +_{−4}hom_{R}(I, M) +_{−4}hom_{R}(M, E) .

The following proof is a simplification of the proof of Thm. 2.6 in [22].

Proof. Suppose (i). To see that X is unobstructed we combine (6) with

0Ext_{R}^{2}(I, I) ≅ _{−4}Ext_{𝔪}^{2}(I, I)^{∨} ≅ _{−4}Hom(I, M)^{∨} ,  (12)

which we deduce from (1) and (2). It follows that Def_{A} is smooth by the assumption −4Hom(I, M) = 0, whence X is unobstructed. The proof is really nothing more than to interpret the exact sequence (5) in terms of deformation theory.

(iii) One may deduce the unobstructedness of X from results in [25] by combining Thm. 1.5, page 135 with their tangent space descriptions, pp. 155-166. We will, however, give a partially independent proof, using an exact sequence which we need later on. Indeed it suffices to prove, and interpret in terms of deformation theory, the exact sequence

0 → T_{γ,ρ} → _{0}Ext_{R}^{1}(I, I) → _{0}Hom_{R}(M, E) → _{0}Ext_{R}^{2}(M, M) → _{0}Ext_{R}^{2}(I, I) →  (13)

where T_{γ,ρ} is the tangent space of the Hilbert scheme of constant cohomology H_{γ,ρ} at (X). To prove it we use the spectral sequence (1) and the duality (2) twice and Walter’s remark on the factorization of α via _{0}Ext_{R}^{2}(M, M) in (5), see [22], Thm. 2.6 (iii) for details.

To see that X is unobstructed, we get by (13) and the vanishing of _{0}Hom_{R}(M, E) an isomorphism between the local Hilbert functor of constant cohomology at X and Def_{A}. The former functor is smooth because _{0}Ext^{2}(M, M) contains in a natural way the obstructions of deforming a curve in H_{γ,ρ} (cf. [25], Thm. 1.5, page 135). Hence we conclude by (6).

(ii) We claim that the unobstructedness of X follows from Proposition 6. Indeed if we take a complete intersection Y ⊇ X of two surfaces of degrees f and g such that the conditions of Proposition 6 hold (such Y exists), then the corresponding linked curve X′ satisfies

vHom_{R}(I(X′), M(X′)) ≅ _{v}Hom_{R}(M(X), E(X)) for v = 0 and v = −4  (14)

because M(X′) (resp. I(X′)/I(Y)) is the dual of M(X)(f +g−4) (resp. E(X)(f +g−4)) and vHom_{R}(I(Y), M(X′)) = 0 for v = 0, −4 by assumption. Hence we conclude by Proposition 6 and the proof of (i) above.

It remains to prove the dimension formula, i.e. to prove

h^{1}(N_{X}) = δ^{2}(0) +_{−4}hom_{R}(I, M) +_{−4}hom(M, E) .

Since this follows from the next lemma, we are done.

Lemma 9. Let X be any curve in ℙ^{3} such that _{0}Ext_{R}^{i}(M, M) = 0 for i ≥ 2. Then

h^{1}(N_{X}) = δ^{2}(0) +_{−4}hom_{R}(I, M) +_{−4}hom(M, E) .  (15)

Proof. The main observation for proving the lemma is the fact that_{0}Ext_{R}^{2}(M, M) = 0 implies α = 0 in the sequence (5) for v = 0. Since the cokernel of the rightmost map in (5) is 0Ext_{𝔪}^{3}(I, I), it follows that h^{1}(N_{X}) = _{0}ext_{R}^{2}(I, I) +_{0}ext_{𝔪}^{3}(I, I). Moreover using the spectral sequence (1) which converging to_{0}Ext_{𝔪}^{3}(I, I), we get_{0}ext_{𝔪}^{3}(I, I) = _{0}hom_{R}(I, E)+_{0}ext_{R}^{1}(I, M) because

0Ext_{R}^{2}(I, M) ≅ _{−4}Ext_{𝔪}^{2}(M, I)^{∨} ≅ _{−4}Hom(M, M)^{∨} ≅ _{0}Ext_{R}^{4}(M, M) = 0.

Finally since _{0}ext_{R}^{1}(I, M) = _{−4}hom(M, E) by Lemma 1 and δ^{2}(0) = _{0}hom_{R}(I, E) by Remark 7 of [20], we conclude by (12).

Corollary 10. Let X be any curve in ℙ^{3}, let diam M ≤ 2 and suppose e(X) < s(X). If diam M ≠ 0, suppose also e(X) ≤ b(X) and c(X) ≤ s(X). Then

H^{1}(N_{X}) = 0 .

Proof. This follows from (15). We leave the details to the reader (or see [22], Cor. 2.8).

4 Comparison of deformation functors

In this section we will use cohomological methods to investigate when the immersions

H_{γ,ρ} (1) ; ↪ H_{γ} (2) ; ↪ H(d, g)

are isomorphisms around (X). Indeed by the semicontinuity of h^{i}(I_{X}(v)) it is rather obvious that (2) (resp. the composition of (1) and (2)) is an isomorphism at (X) provided X has maximal rank (resp. seminatural cohomology). We shall, however, show the isomorphisms under weaker assumptions. As a consequence we get some interesting information on the semicontinuity of the graded Betti number for curves X in ℙ^{3}.

Proposition 11. Let M = H_{∗}^{1}(I_{X}) and E = H_{∗}^{1}(O_{X}). Then

(a) _{0}Hom_{R}(I, M) = 0 ⇒ H_{γ} ≅ H(d, g) are isomorphic as schemes at (X).

(b) _{0}Hom_{R}(M, E) = 0 ⇒ H_{γ,ρ} ≅ H_{γ} are isomorphic as schemes at (X).

Proof. (a) By interpreting the exact sequence

0 → _{0}Ext_{R}^{1}(I, I) → H^{0}(N_{X}) → _{0}Hom_{R}(I, M) ⟶^{α} _{0}Ext_{R}^{2}(I, I) → H^{1}(N_{X}) → (16)

in terms of deformation theories, as done in (6), we get the conclusion.

(b) was established in the proof of Theorem 8 (iii) by interpreting the exact sequence

0 → T_{γ,ρ} → _{0}Ext_{R}^{1}(I, I) → _{0}Hom_{R}(M, E) → _{0}Ext_{R}^{2}(M, M) → _{0}Ext_{R}^{2}(I, I) →  (17)

in terms of deformation theories.

To see that the assumption above is by much weaker than claiming X to have seminatural cohomology, we include a result for generic curves:

Proposition 12. Let X be a curve in ℙ^{3}, and suppose X is generic in the Hilbert scheme H(d, g) and satisfies _{0}Ext_{R}^{2}(M, M) = 0. Then X is unobstructed if and only if

0Hom_{R}(I, M) = 0 and _{0}Hom_{R}(M, E) = 0 .  (18)

Proof. One way is clear from Theorem 8. Now suppose X is unobstructed and generic with postulation γ and deficiency ρ. By generic flatness we see that H_{γ,ρ} ≅ H_{γ} ≅ H(d, g) near (X) from which we deduce an isomorphism of tangent spaces T_{γ,ρ} ≅ _{0}Ext_{R}^{1}(I, I) ≅ H^{0}(N_{X}). We therefore conclude by the exact sequences (17) and (16), recalling that the map α of (16) factorizes via _{0}Ext_{R}^{2}(M, M), whence α = 0.

As a surprising consequence of Proposition 11, we get some “new” semicontinuity results which we heavily use in section 6.

Corollary 13. Inside H_{γ} and hence inside H_{γ,ρ} the graded Betti numbers obey semicontinuity, i.e. if X′ is a generization of X in H_{γ}, then

β_{i,j}(X′) ≤ β_{i,j}(X) for any i, j .

In particular if _{0}Hom_{R}(I(X), M(X)) = 0, then β_{i,j}(X′) ≤ β_{i,j}(X) for any i,j and every generization X′ of X in H(d, g).

Proof. Apply Nakayama’s lemma to the syzygy modules of (7) (see Rem. 7 of [23] and [33]). Then combine with Proposition 11 (a).

Example 14. It is known that the curve Z of Example 7 sits in the intersection of two irreducible components of H(18, 39)_{S} and moreover that the generic curve Z̃ of one of the components is ACM with minimal resolution

0 → R(−8) ⊕ R(−6)^{2} → R(−5)^{4} → I(Z̃) → 0 ,

see [35], [6]. Looking to the minimal resolution of I(Z) in Example 7, we get β_{1,5}(Z) = 0 while β_{1,5}(Z̃) = 4, i.e. we don’t have semicontinuity for β_{1,5}. In this example Corollary 13 does not apply to generizations of Z in H(d, g) because _{0}Hom_{R}(I(Z), M(Z)) ≠ 0!

5 Necessary conditions for unobstructedness.

First note that we can easily reformulate a part of Theorem 8 as Proposition 15. Let X be a space curve with Rao module M satisfying _{0}Ext_{R}^{2}(M, M) = 0. If X is obstructed, then either

(a) _{0}Hom_{R}(I, M) ≠ 0 and _{0}Hom_{R}(M, E) ≠ 0, or

(b) _{−4}Hom_{R}(I, M) ≠ 0 and _{0}Hom_{R}(M, E) ≠ 0, or

(c) _{0}Hom_{R}(I, M) ≠ 0 and _{−4}Hom_{R}(M, E) ≠ 0.

There is a partial converse to this result, at least for Buchsbaum curves X, or more generally for curves admitting “a Buchsbaum factor” which we denote as M_{[t]}. Definition 16. M = M(X) admits “a Buchsbaum factor” M_{[t]} in degree t if

M ≃ M′ ⊕ M_{[t]}

as R-modules where M_{[t]} supported in degree t and has diameter 1.

If M ≃ M′ ⊕ M_{[t]}, then M has a minimal resolution of the form

0 → L′_{4} ⊕ R(−t − 4)^{r} ⟶^{σ} L′_{3} ⊕ R(−t − 3)^{4r} → L′_{2} ⊕ R(−t − 2)^{6r}... → L′_{0} ⊕ R(−t)^{r} → M → 0

where

0 → L′_{4} → L′_{3} → L′_{2} → L′_{1} → L′_{0} → M′ → 0

is the minimal resolution of M′ and

0 → R(−t − 4)^{r} ⟶^{σ_{[t]}} R(−t − 3)^{4r} → R(−t − 2)^{6r} → R(−t − 1)^{4r} → R(−t)^{r} → M_{[t]} → 0 (19)

is “r times” the Koszul resolution of the R-module k ≅ R/(X_{0}, X_{1}, X_{2}, X_{3}). Remark 17. Suppose M = M(X) admits “a Buchsbaum factor”, M ≃ M′ ⊕ M_{[t]}.

(a) If M′ = 0 then X is Buchsbaum curve of diameter one.

(b) Every Buchsbaum curve (not ACM) has a Buchsbaum factor. Every curve obtained from Liaison addition where one of curves is Buchsbaum, admits a Buchsbaum factor ([27]).

Combining with Rao’s theorem (10), we get a minimal resolution

0 → L_{4} ⟶^{σ⊕0} L_{3} ⊕ F_{2} → F_{1} → I → 0 ,

of I = I(X) where

L_{4} ≅ L′_{4} ⊕ R(−t − 4)^{r} , L_{3} ≅ L′_{3} ⊕ R(−t − 3)^{4r} .

Moreover to define what we will call the fundamental 5-tuple, we write F_{i} as

F_{2} ≅ P_{2} ⊕ R(−t − 4)^{b_{1}} ⊕ R(−t)^{b_{2}} , F_{1} ≅ P_{1} ⊕ R(−t − 4)^{a_{1}} ⊕ R(−t)^{a_{2}} (20)

where P_{i}, for i = 1, 2 are supposed to contain no direct factor of degree t and t + 4.

Definition 18. The fundamental 5-tuple (with respect to obstructedness) associated to a curve X with Buchsbaum factor M_{[t]} in degree t is

(a_{1}, a_{2}, b_{1}, b_{2}, r)

Note that (a_{1}, a_{2}) = (β_{1,t+4}, β_{1,t}) are 1^{st} graded Betti numbers of I = I(X) while

(β′_{2,t+4}, β′_{2,t}, β′_{3,t+4}) := (b_{1}, b_{2}, r)

are graded "Betti number of I off L′_{3} and L′_{4}".

Remark 19.

(a) If M′ = 0, then β′_{i,j} = β_{i,j} are the usual Betti numbers of I = I(X).

(b) Note that the 5-tuple (a_{1}, a_{2}, b_{1}, b_{2}, r) was written as (r, a_{1}, a_{2}, b_{1}, b_{2}) in [22].

Theorem 20. Let X ⊆ ℙ^{3} be a curve, and suppose _{0}Ext_{R}^{2}(M, M) = 0 and M ≅ M′ ⊕ M_{[t]} as R-modules. Then X is obstructed if either

(a) _{0}Hom_{R}(I, M_{[t]}) ≠ 0 and _{0}Hom_{R}(M_{[t]}, E) ≠ 0, or

(b)_{−4}Hom_{R}(I, M_{[t]}) ≠ 0 and _{0}Hom_{R}(M_{[t]}, E) ≠ 0, or

(c) _{0}Hom_{R}(I, M_{[t]}) ≠ 0 and _{−4}Hom_{R}(M_{[t]}, E) ≠ 0.

The proof of (a) relies on the study of a morphism appearing in the Proposition below. It is straightforward to show Proposition 21 by using [12], Prop. 2.13. We will, however, give the proof of Prop. 3.6 of [22] which relies only on (16) and (17). For the proof recall that H^{0}(N_{X}) ≅ Ext_{O_{ℙ}}^{1}(I_{X}, I_{X}) by (4).

Proposition 21. Let X ⊆ ℙ^{3} be a curve, and suppose _{0}Ext_{R}^{2}(M, M) = 0. If the obvious morphism

0Hom_{R}(I, M) ×_{0}Hom_{R}(M, E) ⟶ _{0}Hom_{R}(I, E)

(given by the composition) is non-zero, then X is obstructed.

Proof. It is well known (cf. [24]) that if the Yoneda pairing (inducing the cup product)

< −, − > : Ext_{O_{ℙ}}^{1}(I_{X}, I_{X}) × Ext_{O_{ℙ}}^{1}(I_{X}, I_{X}) → Ext_{O_{ℙ}}^{2}(I_{X}, I_{X}),  (21)

given by composition of resolving complexes, satisfies < λ, λ >≠ 0 for some λ, then X is obstructed. If we let p_{1} : Ext_{O_{ℙ}}^{1}(I_{X}, I_{X}) → _{0}Hom_{R}(I, M) and p_{2} : Ext_{O_{ℙ}}^{1}(I_{X}, I_{X}) → 0Hom_{R}(M, E) be the maps induced by sending an extension onto the corresponding connecting homomorphisms (cf. the maps appearing in (16) and (17)), then <-,-> fits into a commutative diagram of natural maps

Ext_{O_{ℙ}}^{1}(I_{X}, I_{X}) × Ext_{O_{ℙ}}^{1}(I_{X}, I_{X}) ⟶ Ext_{O_{ℙ}}^{2}(I_{X}, I_{X})
↓ p_{1} ↓ p_{2} ↓
0Hom_{R}(I, M) × _{0}Hom_{R}(M, E) ⟶ _{0}Hom_{R}(I, E)

(22)

By (16), _{0}Ext_{R}^{1}(I, I) = ker p_{1}, and p_{1} is surjective because α = 0. Moreover since the composition _{0}Ext_{R}^{1}(I, I) ↪ Ext^{1}(I_{X}, I_{X}) → _{0}Hom_{R}(M, E) is surjective by (17), there exists (λ_{1}, λ_{2}) ∈ Ext^{1}(I_{X}, I_{X}) ×_{0}Ext_{R}^{1}(I, I) such that the composed map p_{2}(λ_{2})p_{1}(λ_{1}) is non-zero by assumption. Using λ_{2} ∈ _{0}Ext_{R}^{1}(I, I) = ker p_{1}, we get

p_{2}(λ_{1} + λ_{2})p_{1}(λ_{1} + λ_{2}) = p_{2}(λ_{1})p_{1}(λ_{1}) + p_{2}(λ_{2})p_{1}(λ_{1})

i.e. either < λ_{1} + λ_{2} , λ_{1} + λ_{2} > or < λ_{1} , λ_{1} > are non-zero, whence X is obstructed.

Remark 22. Assume _{0}Ext_{R}^{2}(M, M) = 0. From (16) and (17), we see that _{0}Hom_{R}(I, M) ≠ 0 and _{0}Hom_{R}(M, E) ≠ 0 if and only if we have the following strict inclusions of tangent spaces

T_{γ,ρ} ⊊ _{0}Ext_{R}^{1}(I, I) ⊊ H^{0}(N_{X})  (23)

By Proposition 21, X is obstructed if (23) holds, cf. [25], ch. X, Prop. 5.9. for the case M ≅ k.

Before proving Theorem 20, we remark that

Lemma 23. Let X ⊆ ℙ^{3} be a curve, and suppose M ≅ M′ ⊕ M_{[t]} as R-modules. If vExt_{R}^{2}(M_{[t]}, M) = 0 for v = 0 and −4, then

(a)  _{0}hom_{R}(I, M_{[t]}) = ra_{2}  and _{−4}hom_{R}(I, M_{[t]}) = ra_{1} .
(b)  _{0}hom_{R}(M_{[t]}, E) = rb_{1}  and _{−4}hom_{R}(M_{[t]}, E) = rb_{2} .

Proof. (a) If we apply _{v}Hom(−, M_{[t]}) to (10), we get _{v}Hom(I, M_{[t]}) ≃ _{v}Hom(F_{1}, M_{[t]}) because 𝔪 · M_{[t]} = 0 and we conclude by (20).

(b) is the dual result. We claim that _{−v−4}Hom_{R}(F_{2}, M_{[t]})^{∨} ≅ _{v}Hom_{R}(M_{[t]}, E) for v = 0 and −4. Indeed, exactly as in Lemma 1, we get by duality (2) and the spectral sequence (1) (which converges to _{−v−4}Ext_{𝔪}^{3}(M_{[t]}, I)) an exact sequence

0 → _{−v−4}Ext_{R}^{1}(M_{[t]}, M) → _{v}Ext_{R}^{1}(I, M_{[t]})^{∨} → _{−v−4}Hom_{R}(M_{[t]}, E) → 0

for v = 0 and −4. We get _{−v−4}Ext_{R}^{1}(M_{[t]}, M)^{∨} ≅ _{v}Ext_{R}^{3}(M, M_{[t]}) by (2) and (1) and vExt_{R}^{3}(M, M_{[t]}) ≅ _{v}Hom_{R}(L_{3}, M_{[t]}) by (9). Interpreting _{v}Ext_{R}^{1}(I, M_{[t]}) similarly via the minimal resolution (10) of I, we get _{v}Ext_{R}^{1}(I, M_{[t]}) ≅ _{v}Hom_{R}(L_{3} ⊕ F_{2}, M_{[t]}). Hence we get the claim and we conclude by (20).

Proof (of Theorem 20). (a) Since _{0}Hom_{R}(I, M_{[t]}) ≅ _{0}Hom_{R}(R(−t)^{a_{2}}, M_{[t]}) by (10) and (20), we get that the composition

0Hom_{R}(I, M_{[t]}) ×_{0}Hom_{R}(M_{[t]}, E) ⟶ _{0}Hom_{R}(I, E)

is non-zero. Since _{0}Hom(I, M) → _{0}Hom(I, M_{[t]}) is surjective by the existence of the R-split morphism M → M_{[t]}, the corresponding composition of Proposition 21 is also non-zero, and (a) is proved.

(b) In [22], there is a result (Prop. 3.8) with a quite complicated proof which implies (b) and (c) of Theorem 20. Here we just remark that we immediately get (b) from (a) and a result of Walter ([37]) in the case M′ = 0. Indeed using [37], Thm. 0.5 we get the obstructedness of X provided _{0}Hom_{R}(I, M) = 0 and the assumptions of (b) holds. Since (a) takes care of the case _{0}Hom_{R}(I, M) ≠ 0 under the assumptions of (b), we get (a)

(c) The conclusion follows from Proposition 6 and (b) exactly as in the proof of Theorem 8 (ii).

Combining Lemma 23 with Theorem 20 and using Proposition 15 for the final conclusion, we get

Corollary 24. Suppose _{0}Ext_{R}^{2}(M, M) = 0 and M ≅ M′ ⊕ M_{[t]} as R-modules. Then X is obstructed if

a_{2} · b_{1} ≠ 0 or a_{1} · b_{1} ≠ 0 or a_{2} · b_{2} ≠ 0

where (a_{1}, a_{2}, b_{1}, b_{2}) = (β_{1,t+4}, β_{1,t}, β′_{2,t+4}, β′_{2,t}).

Moreover if M′ = 0, i.e. M ≅ M_{[c]}, then X is obstructed if and only if

β_{1,c} · β_{2,c+4} ≠ 0 or β_{1,c+4} · β_{2,c+4} ≠ 0 or β_{1,c} · β_{2,c} ≠ 0 .

Remark 25. Let r ≠ 0. For the fundamental 5-tuple (a_{1}, a_{2}, b_{1}, b_{2}, r), we have that

a_{2} · b_{1} = 0 and a_{1} · b_{1} = 0 and a_{2} · b_{2} = 0

is equivalent to requiring that the 5-tuple is of the form

(0, 0, b_{1}, b_{2}, r) , (a_{1}, 0, 0, b_{2}, r) or (a_{1}, a_{2}, 0, 0, r) .

Hence if X is unobstructed, then there are “two consecutive 0’s in the 1^{st} four coordinates of the 5-tuple”. This is equivalent to unobstructedness if diam M = 1.

Example 26. (a) (Sernesi, [6]). Take the curve bilinked to 2 skew lines, as in Example 7. The minimal resolution is

0 → R(−8) → R(−8) ⊕ R(−7)^{4} → R(−6)^{4} ⊕ R(−4) → I → 0 ,

whence c = 4. The corresponding 5-tuple, (β_{1,c+4}, β_{1,c}, β_{2,c+4}, β_{2,c}, β_{3,c+4}), is

(0, 1, 1, 0, 1) .

It follows that the curve X of H(18, 39)_{S} is obstructed by Corollary 24 or Remark 25.

(b) Start with the generic curve of H(8, 5)_{S}. It has 2-dimensional Rao module M and diam M = 1 ([15]). Link with (4, 6), then with (6, 8) using a common hypersurface of degree 6 in both linkages. The minimal resolution is

0 → R(−10)^{2} → R(−10) ⊕ R(−9)^{8} → R(−8)^{7} ⊕ R(−6) → I → 0 ,

whence c = 6. The corresponding 5-tuple is (β_{1,c+4}, β_{1,c}, β_{2,c+4}, β_{2,c}, β_{3,c+4}) = (0, 1, 1, 0, 2), i.e. the curve X of H(32, 109)_{S} is obstructed by Remark 25.

(c) ([3] and [37]). There is a curve in H(33, 117)_{S} of maximal rank and diam M = 1 with minimal resolution

0 → R(−9) → R(−10)^{2} ⊕ R(−9) ⊕ R(−8)^{4} → R(−9) ⊕ R(−8) ⊕ R(−7)^{5} → I(X) → 0 .

The corresponding 5-tuple is (1, 0, 1, 0, 1), i.e. X is obstructed by Remark 25.

In the next section, we shall see that the curve of Example 26 (a) sits in the intersection of two irreducible components, while the curve of Example 26 (b) belongs to a unique irreducible component, by studying the possible generizations of the curves.

6 On the minimal resolution of a generic curve.

In this section we consider generizations of a space curve, i.e. deformations to a more general curve, by simplifying their minimal resolutions (“killing ghost terms”). We shall prove a general result (Theorem 27) valid for any space curve and a more special result (Proposition 31) which holds for curves with a Buchsbaum factor under some conditions. These results are inspired by the corresponding results in [25] which treat the case M ≅ k.

Theorem 27. Let X be any curve, let

0 → L_{4} ⟶^{σ} L_{3} → L_{2} → L_{1} → L_{0} → M(X) → 0

be a minimal free resolution of M = M(X) and look to the minimal free resolution

0 → L_{4} ⟶^{σ⊕0} L_{3} ⊕ F_{2} → F_{1} → I(X) → 0  (24)

given by Rao’s theorem. If F_{1} and F_{2} has a common free factor;

F_{2} = F′_{2} ⊕ R(−i) , F_{1} = F′_{1} ⊕ R(−i),

then there is a generization X′ of X in H(d, g) with constant postulation and Rao module and with minimal resolution

0 → L_{4} ⟶^{σ⊕0} L_{3} ⊕ F′_{2} → F′_{1} → I(X′) → 0 .

The proof is rather straightforward once we have proven a key lemma. We delay the proof of the lemma and Theorem 27 until the end of this section.

Suppose M = M(X) admits an R-module decomposition M = M′ ⊕ M_{[t]} where the diameter of M_{[t]} is 1 (e.g. X is Buchsbaum). Using Theorem 27 we get immediately

Corollary 28. Let X be a curve and suppose M(X) ≅ M′ ⊕ M_{[t]} as R-modules. Let

(β_{1,t+4}, β_{1,t}, β′_{2,t+4}, β′_{2,t}, β′_{3,t+4}) = (a_{1}, a_{2}, b_{1}, b_{2}, r)

be the corresponding 5-tuple of X.

(a) If a_{1} · b_{1} ≠ 0, then there is a generization X′ of X in H(d, g) with constant postulation and Rao module whose 5-tuple is

(C1) (a_{1} − 1, a_{2}, b_{1} − 1, b_{2}, r) .

(b) If a_{2} · b_{2} ≠ 0, then there is a generization X′ of X in H(d, g) with constant postulation and Rao module whose 5-tuple is

(C2) (a_{1}, a_{2} − 1, b_{1}, b_{2} − 1, r) .

Remark 29. Repeated use of Corollary 28 results in a curve (a generization of X) with Buchsbaum factor M_{[t]} whose corresponding 5-tuple satisfies

a_{1} · b_{1} = 0 and a_{2} · b_{2} = 0 .

Remark 30. In the following we need a slight extension of the notion of a 5-tuple to a special case where r = 0, i.e. we need to consider generizations X′ of a curve X with a Buchsbaum factor in degree t (i.e. M(X) ≅ M′ ⊕ M_{[t]}) for which M(X′) ≅ M′. For such X′ we still have a number t and a minimal resolution of the form (10). Then we use the free summands of (20) appearing in the minimal resolution of I(X′) to define (a_{1}, a_{2}, b_{1}, b_{2}, r) with r = 0.

Proposition 31. Let M(X) ≅ M′ ⊕ M_{[t]} as R-modules and suppose L′_{2} has no factor in degree t and t + 4. Let (a_{1}, a_{2}, b_{1}, b_{2}, r) be the corresponding 5-tuple of X.

(a) If r · b_{1} ≠ 0, then there is a generization X′ of X in H(d, g) with constant postulation and constant M′ whose 5-tuple is

(P1) (a_{1}, a_{2}, b_{1} − 1, b_{2}, r − 1) .

(b) If r·a_{2} ≠ 0, then there is a generization X′ of X in H(d, g) with constant specialization (function) and constant M′ whose 5-tuple is

(P2) (a_{1}, a_{2} − 1, b_{1}, b_{2}, r − 1) .

Remark 32. Repeated use of Proposition 31 results in a generization X′ with 5-tuple satisfying r · a_{2} = 0 and r · b_{1} = 0. Using also Theorem 27, we get a curve X′ whose corresponding 5-tuple is of the form

(a_{1}, 0, 0, b_{2}, r) with r ≠ 0, or (a_{1}, a_{2}, b_{1}, b_{2}, 0) with a_{1} · b_{1} = a_{2} · b_{2} = 0.  (25)

Note that if M′ = 0 then the curve X′ is unobstructed by Corollary 24 (and Theorem 8).

Remark 33. We give details of the proof of Proposition 31 later. For a complete proof, we refer to the proof of Prop. 4.2 of [22]. There is an inaccuracy in the statement of Prop. 4.2 (a) of [22] which states that the resolution for the generization in which r and b_{1} are replaced by r − 1 and b_{1} − 1, is minimal. Looking to the proof of [22] the resolution may be non-minimal at one and only one spot. Indeed if F_{1} contains a direct factor R(−t − 3), this factor may become redundant (we try in the proof of this paper to clarify. I intend to put a revised version, correcting Prop. 4.2, for the corresponding paper on the arXiv). In conclusion the graded Betti numbers of a generization as in (a) do not change except for the change which corresponds to the replacement of r and b_{1} by r − 1 and b_{1} − 1 respectively and for a possible change in which β_{2,t+3} and β_{1,t+3} decrease by the same number i, i ≤ 4. Note that this remark does not affect the conclusion on the 5-tuple in Proposition 31 (a).

Remark 34. Suppose diam M = 1, i.e. M(X) ≅ M_{[t]} and t = c.

(a) By Remark 33 there is a generization as in Proposition 31 (a) whose graded Betti numbers do not change except for β_{3,c+4} and β_{2,c+4}, which both decrease by 1, and for β_{1,c+3} and β_{2,c+3} which may decrease by at most 4, keeping, however, β_{1,c+3} − β_{2,c+3} unchanged. Moreover if we combine with Theorem 27, we can suppose β_{2,c+3} decreases by exactly min{β_{1,c+3}, 4} after a further generization.

(b) Extending some ideas of the proof of Proposition 31 (b) (which we hope to do in a forthcoming paper), we can describe the possible changes of the graded Betti numbers under the generization (P2) in detail. In particular we get that the graded Betti numbers of a generization as in Proposition 31 (b) do not change except β_{3,c+4} and β_{1,c}, which both decrease by 1, and β_{1,v} and β_{2,v} for v ∈ {c + 3, c + 2, c + 1} for which β_{1,c+1} − β_{2,c+1} increases by 4, β_{2,c+2} − β_{1,c+2} increases by 6 and β_{1,c+3} − β_{2,c+3} increases by 4. Moreover combining with Theorem 27, we can take β_{1,c+1} · β_{2,c+1} = 0 and β_{1,c+2} · β_{2,c+2} = 0 after a further generization.

Example 35. (a) ([35]). The 5-tuple of X is

(0, 1, 1, 0, 1) .

Using (P1) and (P2) the curve X admits two generizations to two curves with 5-tuples

(0, 1, 0, 0, 0) and (0, 0, 1, 0, 0) .

(b) The curve X with 5-tuple (0, 1, 1, 0, 2) admits two generizations to two curves with 5-tuples

(0, 1, 0, 0, 1) and (0, 0, 1, 0, 1).

Both these curves, however, admit generizations to a curve with 5-tuple (0, 0, 0, 0, 0), indicating that the curve X, which is obstructed by Corollary 24, may not sit in an intersection of two irreducible components of H(d, g)!!

(c) ([3] and [37]). The 5-tuple of X is (1, 0, 1, 0, 1), i.e. the curve X admits two generizations to two curves with 5-tuples:

(1, 0, 0, 0, 0) and (0, 0, 0, 0, 1)

To see that the 2 directions of generizations in Example 35 (a) and (c) correspond to two different components, while those in Example 35 (b) lead to a unique component, we prove Proposition 37 below which relies on the semicontinuity of the graded Betti numbers (cf. Corollary 13). E.g. we will in Example 35 (c) see that Proposition 37 “separates irreducible components” where the usual semicontinuity of h^{i}(I_{X}(v)) fails! Indeed since both generizations in Example 35 (c) have the same postulation as X, the semicontinuity of h^{i}(I_{X}(v)) can not be used to prove that the two generizations belong to two different irreducible components!

For the rest of this section we suppose M′ = 0, i.e. we consider curves of diameter at most 1 for which we attach a 5-tuple (cf. Remark 30) consisting of the usual graded Betti numbers.

Definition 36. (a) A minimal 5-tuple is a 5-tuple which doesn’t allow further reductions by using either (C1), (C2), (P1) or (P2), i.e. the 5-tuple is of the form (25) or as in the proposition below.

(b) The set of all graded Betti numbers is called minimal provided it doesn’t allow further reductions by using either (P1), (P2) or reductions coming from cancellations of a common direct free factor as in Theorem 27 for some i (in particular β_{1,i} · β_{2,i} = 0 for every i ≠ t + 3 if M = M_{[t]}). We denote the minimal set of all graded Betti numbers by β_{tot}(X), or just β_{tot}.

Proposition 37. Let t be an integer such that M(X)_{v} = 0 for v ≠ t, and let \underline{β} = \underline{β}(X) be the minimal 5-tuple of X, i.e.

\underline{β} = (β_{1,t+4}, 0, 0, β_{2,t}, β_{3,t+4}) with β_{3,t+4} ≠ 0 and t = c,

or

\underline{β} = (β_{1,t+4}, β_{1,t}, β_{2,t+4}, β_{2,t}, 0) with β_{1,i} · β_{2,i} = 0 for i = t, t + 4.

Then X is unobstructed. Let V (\underline{β}) be the unique irreducible component of H(d, g) containing X and let X_{i} be curves as X above with minimal 5-tuples \underline{β}_{i}, i = 1, 2 and with the same t. Then

(i) The generic curve of V (\underline{β}) has 5-tuple = \underline{β}

(ii) If \underline{β}_{1} ≠ \underline{β}_{2}, then V (\underline{β}_{1}) ≠ V (\underline{β}_{2}).

(iii) V (\underline{β}) depends only on β_{tot} or, equivalently, only on the postulation γ_{X} and β_{3,t+4}, not on X.

Proof. (i) X is unobstructed by Corollary 24 (and Theorem 8). Let X′ be the generic curve of V (\underline{β}). We have _{0}Hom_{R}(I(X), M(X)) = 0, cf. Lemma 23, and we get

H_{γ} ≅ H(d, g) at (X)

by Proposition 11. It follows that X′ has the same postulation as X, whence

β_{1,v}(X′) − β_{2,v}(X′) + β_{3,v}(X′) = β_{1,v}(X) − β_{2,v}(X) + β_{3,v}(X)  (26)

for every v, cf. [9], [32]. Moreover \underline{β}(X′) ≤ \underline{β}(X) by Corollary 13. Since \underline{β}(X) is minimal, see the form of \underline{β} of this proposition, we get \underline{β}(X′) = \underline{β}(X), and (i) is proved.

(ii) Suppose V (\underline{β}_{1}) = V (\underline{β}_{2}), i.e. we can suppose their generic curves X′_{1} and X′_{2} coincide. Then (i) leads to a contradiction because

\underline{β_{1}} = \underline{β}(X′_{1}) = \underline{β}(X′_{2}) = \underline{β_{2}} .

(iii) By a result of Bolondi (cf. [2]) the subset of H(d, g) of curves of constant postulation and Rao module structure is irreducible. For curves of diameter at most one there is only one structure on the module M(X), namely the trivial structure, in which case “constant Rao module structure” may be replaced by requiring dim M(X) or β_{3,t+4} to be constant. It follows that if the curves X_{1} and X_{2} of the proposition, in addition to having the same \underline{β}, also satisfy γ_{X_{1}} = γ_{X_{2}}, then they define the same irreducible component of H(d, g) and we get the equivalent statement by (26).

Once more we revisit Example 35.

Example 38. In (a) the 5-tuples of the two generizations are minimal, both with t = 4 and both ACM, and they correspond to two different irreducible components of H(18, 39)_{S} containing (X) by Proposition 37. Note that we for this example may separate the two components by the usual semicontinuity because the triple (h^{0}(I_{Z}(4)), h^{1}(I_{Z}(4)), h^{1}(O_{Z}(4))) equals (1, 1, 1) for Z = X, while it is (1, 0, 0) and (0, 0, 1) for the two generizations.

(c) Again the 5-tuples of the two generizations are minimal, both with t = 5, and they correspond to two different irreducible components of H(33, 117)_{S} containing (X) by Proposition 37. In this case one of the generizations is ACM, the other is Buchsbaum of diameter 1. In this case we can not separate the two components by the usual semicontinuity of h^{i}(I_{Z}(v)).

(b) To see that the curve X with 5-tuple (0, 1, 1, 0, 2), which we know is obstructed, belongs to a unique component, we use Proposition 37 (iii). To conclude we need to verify that the postulation of X_{1} and X_{2} coincides where X_{1} and X_{2} are two curves with a common 5-tuple (0, 0, 0, 0, 0) and having X as a common specialization. Since X is Buchsbaum of diameter 1 and t = c = 6, it follows that γ_{X_{1}}(v) = γ_{X_{2}}(v) for v ≠ 6 by the semicontinuity of h^{i}(I_{X}(v)) since h^{0}(I_{X}(v)) − h^{1}(I_{X}(v)) + h^{1}(O_{X}(v)) = χ(I_{X}(v)) is a constant under generization. Moreover we easily see that γ_{X_{1}}(6) = γ_{X_{2}}(6) = 0, and we are done.

Question on possible generizations: If we use (P1), (P2), (C1), (C2) and more generally the removal of “ghost terms” allowed by Theorem 27 (shortly called (Ci) in the following), we find generizations of a diameter one curve X which “allow simplifications”, i.e. except for (P2) which remove “ghost terms”, in the minimal resolution of I(X). The question is: have we found “all” generizations of X in H(d, g), at least from the point of view of its minimal resolution (“all” should be taken modulo those (Ci) which do not change the 5-tuple, cf. Remark 34)? In particular can we find the minimal free resolution of a generic curve of every irreducible component of H(d, g) containing (X) as a combination of (P1),(P2) and (Ci)?

Case (1). To answer, we first suppose the 5-tuple (a_{1}, a_{2}, b_{1}, b_{2}, r) of X satisfies r · a_{2} = 0. As in the proof of Proposition 37 we have _{0}Hom_{R}(I(X), M(X)) = 0 and hence

H_{γ} ≅ H(d, g) at (X).

It follows that every generization of X has postulation γ, i.e. that only consecutive cancellations in the minimal resolution are allowed ([32]). So the answer is YES in this case. Indeed (P1) and (Ci) suffices! Note that, compared with [32], we have by Theorem 27 and Proposition 31 also proved the existence of the generizations!

Case (2). Also in the case r · b_{1} = 0, the answer is YES. Indeed this case follows from the case r · a_{2} = 0 by linkage. Note that if we link X to X_{l} as in Proposition 6, we get, by combining (14) and Lemma 23, that the 5-tuple (a_{1}, a_{2}, b_{1}, b_{2}, r) of X is equal to (b_{2}(X_{l}), b_{1}(X_{l}), a_{2}(X_{l}), a_{1}(X_{l}), r(X_{l})) where (a_{1}(X_{l}), a_{2}(X_{l}), b_{1}(X_{l}), b_{2}(X_{l}), r(X_{l})) is the 5tuple of X_{l}, see the last part of the proof of Proposition 31 for details. In particular we get a_{2}(X_{l}) = 0 in the 5-tuple of X_{l} and since we know “all” generizations of a curve X_{l} with a_{2}(X_{l}) = 0 by the case (1), one may check this gives “all” expected generizations of X. Note that this also nicely explains the somewhat strange generizations of X given by (P2). Indeed a generization given by (P2) corresponds to a generization (P1) for the linked curve, i.e. to the removal of a “ghost term” in the minimal resolution of the linked curve!

Case (3). Finally for the remaining unsolved case a_{2} · b_{1} ≠ 0 the answer seems to be YES, which we hope to treat in a forthcoming paper. Here we only partially give the answer. To simplify, we restrict to curves which satisfy a_{1} · b_{1} = 0 and a_{2} · b_{2} = 0 (letting a_{1} = β_{1,c+4}, a_{2} = β_{1,c}, b_{1} = β_{2,c+4} and b_{2} = β_{2,c}). Let us a little more generally consider the case

\underline{β}(X) := (0, a_{2}, b_{1}, 0, r) and (a_{2} ≠ 0 or b_{1} ≠ 0)  (27)

where proper generizations as in Proposition 31 occur, to give a picture of the existing generizations in H(d, g). Thanks to [2] we remark that any curve D satisfying \underline{β}(D) = \underline{β}(X) and γ_{D}(v) = γ_{X}(v) for v ≠ c, belongs to the same irreducible family as (X), i.e. further generizations of X and D with constant postulation and Rao module structure lead to the "same" generic curve. By Proposition 31, we have

For any pair (i, j) of non-negative integers such that r − i − j ≥ 0, a_{2} − i ≥ 0 and b_{1} − j ≥ 0, there exists a generization X_{ij} of X

in H(d, g) such that \underline{β}(X_{ij}) = (0, a_{2} − i, b_{1} − j, 0, r − i − j).

(28)

Example 39. As an example, let \underline{β}(X) = (0, 3, 2, 0, 4) (such curves exist by Example 41). By (28) we have 10 different generizations X_{ij} among which two curves correspond to the 5-tuples \underline{β}(X_{22}) = (0, 1, 0, 0, 0) and \underline{β}(X_{31}) = (0, 0, 1, 0, 0), i.e. they correspond to two unobstructed ACM curves with different postulation. Hence they belong to two different irreducible components of H(d, g) having (X) in their intersection.

Pushing the arguments of Example 38 (c) and Example 39 a little further, we get at least

Proposition 40. Suppose the Rao module M(X) ≠ 0 is r-dimensional and concentrated in degree c, let a_{1} = β_{1,c+4} and a_{2} = β_{1,c} (resp. b_{1} = β_{2,c+4} and b_{2} = β_{2,c}) be the number of minimal generators (resp. minimal relations) of degree c + 4 and c respectively, and suppose

a_{1} = 0, b_{2} = 0 and a_{2} · b_{1} ≠ 0 .

(a) If r < a_{2}+b_{1}, then the generic curve of every irreducible component containing (X) is ACM. Moreover the number, n(comp, X), of irreducible components containing (X) satisfies

n(comp, X) ≥ min{a_{2}, r} + min{b_{1}, r} − r + 1 ≥ 2 ,

and in the case s(X) = e(X) = c, we have equality to the left.

(b) If r ≥ a_{2} + b_{1} and s(X) = e(X) = c, then X is an obstructed curve which belongs to a unique irreducible component of H(d, g).

Proof. For the details of a proof, see [22], Prop. 4.6.

It is not difficult to find examples of smooth connected curves satisfying even the restrictive condition s(X) = e(X) = c(X) of the proposition (see Example 3.12 of [22] for examples in which a_{2} > 1). See also [1].

Example 41. (chark = 0). We claim that for any pair of positive integers (r, b) there exist smooth connected curves X of diam M(X) = 1 with minimal resolution as in (10) and (20) such that

s(X) = e(X) = c , and

h^{0}(I_{X}(c)) = 1, h^{1}(I_{X}(c)) = r, h^{1}(O_{X}(c)) = b and c = 1 + b + 2r and such that the corresponding 5-tuple is (0, 1, b, 0, r). Note that such curves are obstructed by Corollary 24. To get the examples, we use Chang’s results ([5] or [37], Thm. 4.1) to show the existence of smooth connected curves with Ω-resolution

0 → O_{ℙ}(−2)^{3r−1} ⊕ O_{ℙ}(−4)^{b} → O_{ℙ} ⊕ Ω^{r} ⊕ O_{ℙ}(−3)^{b−1} → I_{X}(c) → 0.

Here Ω is by definition given by the exact sequences

0 → Ω → O_{ℙ}(−1)^{4} → O_{ℙ} → 0 and 0 → O_{ℙ}(−4) → O_{ℙ}(−3)^{4} → O_{ℙ}(−2)^{6} → Ω → 0 (29)

which we deduce from the Koszul resolution of the regular sequence {X_{0}, X_{1}, X_{2}, X_{3}}. Using the mapping cone construction we find the minimal resolution of I = I(X) to be

0 → R(−4)^{r} → R(−4)^{b} ⊕ R(−3)^{4r} → R(−2)^{3r+1} ⊕ R ⊕ R(−3)^{b−1} → I(c) → 0 .

Moreover the degree of X is d = \binom{c+4}{2} − 3r − 7 and the genus is g = (c + 1)d − \binom{c+4}{3} + 5.

The simplest case is (r, b) = (1, 1), which yields curves X with s(X) = 4, d = 18 and g = 39 (Sernesi’s example), while (r, b) = (2, 1) yields curves X with s(X) = 6, d = 32 and g = 109 (Example 35 (b)). More generally, all curves X above satisfy h^{1}(N_{X}) = δ^{2}(0) = b by Lemma 9, and if r > b, then X belongs to a unique irreducible component of H(d, g) by Proposition 40. Other examples of singularities of H(d, g) which belong to a unique irreducible component are known ([18], Rem. 3b) and [11], Thm. 3.10).

To prove Theorem 27 and Proposition 31 we need the following lemma for deforming a module N, which basically tells that if we can lift a (three term) resolution with augmentation N to a complex, then the complex defines a flat deformation of N which in the case N = I(X) has to be a deformation of I(X) as an ideal! Below we mainly follow the arguments of [22], Lem. 4.8.

Lemma 42. Let X be a curve in ℙ^{3} whose homogeneous ideal I(X) has a minimal resolution of the following form

(L^{•}) 0 → ⨁_{i} R(−i)^{β_{3,i}} ⟶^{φ} ⨁_{i} R(−i)^{β_{2,i}} ⟶^{ψ} ⨁_{i} R(−i)^{β_{1,i}} → I(X) → 0 .

Let A be a finitely generated k-algebra, B the localization of A in a maximal ideal ℘, and suppose there exists a complex

(L_{B}^{•})  ⨁_{i} R_{B}(−i)^{β_{3,i}} ⟶^{φ_{B}} ⨁_{i} R_{B}(−i)^{β_{2,i}} ⟶^{ψ_{B}} ⨁_{i} R_{B}(−i)^{β_{1,i}} , R_{B} = R ⊗_{k} B ,

such that L_{B}^{•} ⊗_{B} (B/℘) ≅ L^{•}. Then (L_{B}^{•}) is acyclic, φ_{B} is injective and the cokernel of ψ_{B} is a flat deformation of I(X) as an ideal (so coker(ψ_{B}) ⊆ R_{B} defines a flat deformation of X ⊆ ℙ^{3} with constant postulation). Moreover for some a ∈ A−℘, we can extend this conclusion to A_{a} via Spec(B) ↪ Spec(A_{a}), i.e. there exists a flat family of curves X_{Spec(A_{a})} ⊆ ℙ^{3} × Spec(A_{a}) whose homogeneous ideal I(X_{A_{a}}) has a resolution (not necessarily minimal) of the form

(L_{A_{a}}^{•}) 0 → ⨁ R_{A_{a}}(−i)^{β_{3,i}} → ⨁ R_{A_{a}}(−i)^{β_{2,i}} → ⨁ R_{A_{a}}(−i)^{β_{1,i}} → I(X_{A_{a}}) → 0 .

Proof (sketch). If E = coker φ and E_{B} = coker φ_{B}, then one proves that E_{B} ⊗_{B} (B/℘) = E, Tor_{1}(E_{B}, B/℘) = 0 and that φ_{B} is injective. By the local criterion of flatness, E_{B} is a flat deformation of E. Letting Q_{B} = coker(E_{B} → ⊕_{i}R_{B}(−i)^{β_{1,i}}), we can argue as we did for E_{B} to see that Q_{B} is a flat deformation of I(X) and that L_{B}^{•} augmented by Q_{B} is exact.

To prove that Q_{B} is an ideal in R_{B}, we can use (4) to see that a deformation of the O_{ℙ}-Module I_{X} (such as Q̃_{B}) corresponds to a deformation of the curve X in the usual way, i.e. via the cokernel of ĩ: Q̃_{B} → R̃_{B}. We get in particular a morphism H_{∗}^{0}(ĩ): Q_{B} → R_{B} which proves what we want (or use [37]). One may give a direct proof using Hilbert-Burch theorem (cf. [25], page 37-38).

Finally we easily extend the morphism i and any morphism of the resolution L_{B}^{•} to be defined over A_{a′}, for some a′ ∈ A − ℘ (such that L_{A_{a′}}^{•} is a complex). By shrinking Spec A_{a′} to Spec A_{a}, a ∈ A − ℘, we get the exactness of the complex and the flatness of I(X_{A_{a}}) because these properties are open.

Proof (of Theorem 27). In the resolution

0 → L_{4} ⟶^{σ⊕0⊕0} L_{3} ⊕ F′_{2} ⊕ R(−i) ⟶^{ψ} F′_{1} ⊕ R(−i) → I(X) → 0

the matrix of ψ has a zero in the entry which corresponds to R(−i) → R(−i). As in the "Lemma de générisation simplifiantes" ([25], page 189), we can replace the mentioned zero by some indeterminate λ of degree zero. Keeping σ ⊕ 0 ⊕ 0 unchanged, we still have a complex which by Lemma 42 implies the existence a flat family of curves over Spec(A_{a}), A = k[λ], for some a ∈ A−(λ). Since any curve X′ of the family given by Spec(A_{λa}) has a resolution where R(−i) is redundant (R(−i), and only R(−i), is missing in its minimal resolution), and since we may still interpret the Rao module M(X′) as ker H_{∗}^{3}(σ̃ ⊕ 0 ⊕ 0) with σ ⊕ 0 ⊕ 0 as above (so the whole family given by Spec(A_{a}) has constant Rao modules), we get the theorem.

Proof (of Proposition 31). (a) By the assumption M(X) ≅ M′⊕M_{[t]}, the minimal resolution (9) of M is given as the direct sum of the resolution of M′ and the one of M_{[t]}. Let \underline{X} = (X_{0}, X_{1}, X_{2}, X_{3})^{T} and recall (10) and (20). Let η: R(−t − 4) → L′_{4} ⊕ R(−t − 4)^{r} be the map into, say, the last direct factor of R(−t − 4)^{r}, and let

π : L′_{3} ⊕ R(−t − 3)^{4r} ⊕ P′_{2} ⊕ R(−t − 4)^{b_{1}} → R(−t − 4)

be the projection onto the last factor of R(−t − 4)^{b_{1}}. As observed by Martin Deschamps and Perrin in the case M ≅ k ([25], page 189) we can change the 0 component in the matrix of σ ⊕ 0 which corresponds to π(σ ⊕ 0)η: R(−t − 4) → R(−t − 4), to some indeterminate of degree zero. To use Lemma 42 we must change four columns of the matrix A associated to L_{3} ⊕ F_{2} → F_{1}, to get a complex. Indeed let r_{1} := rank F_{1} and look to the last column (a_{k}), 1 ≤ k ≤ r_{1}, of A. Put a_{k} = ∑_{n=0}^{3} γ_{k,n}^{i}X_{n} for every 1 ≤ k ≤ r_{1}. Since the resolution is minimal, such γ_{k,n}^{i} exist. Since the last column of the matrix of σ ⊕ 0 consists of only 0’s and one \underline{X} there are precisely four columns [H_{k,0}^{j}, H_{k,1}^{j}, H_{k,2}^{j}, H_{k,3}^{j}], 1 ≤ k ≤ r_{1}, of A satisfying ∑_{n=0}^{3} H_{k,n}^{j}X_{n} = 0 for every k, which may contribute when we take the product of A with the last column of σ ⊕ 0. Now if we change the trivial map π(σ ⊕ 0)η to the multiplication by an indeterminate λ and simultaneously change the four columns [H_{k,0}^{1}, H_{k,1}^{1}, H_{k,2}^{1}, H_{k,3}^{1}] of A to [H_{k,0}^{1} − γ_{k,0}^{1}λ, H_{k,1}^{1} − γ_{k,1}^{1}λ, H_{k,2}^{1} − γ_{k,2}^{1}λ, H_{k,3}^{1} − γ_{k,3}^{1}λ], leaving the rest of A unchanged, we get the desired complex. By Lemma 42 we get a flat irreducible family of curves having the same (not necessarily minimal) resolution of the homogeneous ideal, hence the same postulation, as X. Since we can remove redundant factors of the resolution in an open set, we have a generization X′ with properties as claimed in Proposition 31. For more details, we refer to the proof of Prop. 4.2 of [22].

Note that in the above proof the Rao module M(X′) will satisfy M(X′) ≅ M′ ⊕ M_{[t−1]}. In particular the module F_{2} (resp. L_{3}) in the minimal resolution (10) of I(X′) increases (resp. decreases) by R(−t − 3)^{4}. If F_{1} contains a direct factor R(−t − 3)^{i}, then some of its summands may become redundant, because when we concretely make the row and column equivalent operations to get the isomorphic resolution in which R(−t − 4) directly may be deleted, the submatrix of A which corresponds to the map R(−t − 3)^{4} → R(−t − 3)^{i}, may have positive rank, say equal to p. Then we remove these p redundant factors as in the proof of Theorem 27. This takes care of the correction in Remark 33.

(b) We will prove (b) by linking X to an X_{l} via a CI of type (f, g) satisfying H^{1}(I_{X}(v)) = 0 for v = f, g, f −4 and g−4, and then apply (a) to X_{l}. To see that X_{l} satisfies the assumptions of (a), let M′(X_{l}) := Ext_{R}^{4}(M′, R)(−f − g). Then M(X_{l}) admits a decomposition M(X_{l}) ≅ M′(X_{l}) ⊕ M_{[f+g−4−t]} as R-modules because of the duality

M(X_{l}) ≅ Ext_{R}^{4}(M, R)(−f − g) ≅ Hom_{k}(M, k)(−f − g + 4) ,  (30)

and the self-duality of M_{[t]}(t). Moreover with L^{∗} := Hom_{R}(L, R) we have an exact sequence → (L′_{2})^{∗} → (L′_{3})^{∗} → (L′_{4})^{∗} → Ext_{R}^{4}(M′, R) ≅ M′(X_{l})(f + g) → 0. Since L′_{2} has no direct free factor of degree t and t + 4, it follows that (L′_{2})^{∗}(−f − g) has no direct free factor of degree f + g − t and f + g − t − 4, and visa versa, see the proof of Prop. 4.2 of [22] for details.

Now since X_{l} satisfies the assumptions of (a), we need to see that the direct free part F_{1} generated in degree t in the resolution of I(X), is equal (at least dimensionally) to the corresponding part of F_{2}(X_{l})(4) in the minimal resolution of I(X_{l}) of the linked curve X_{l}. Indeed since the isomorphism of (14) is given by the duality used in (30), it must commute with their decomposition as R-modules, i.e. we have

0Hom_{R}(I(X), M_{[t]}) ≅ _{0}Hom_{R}(M_{[f+g−4−t]}, E(X_{l})) .  (31)

Then we conclude by Lemma 23 because

vExt_{R}^{2}(M_{[t]}, M)^{∨} ≅ _{−v−4} Ext_{R}^{2}(M, M_{[t]}) ≅ _{−v−4}Hom_{R}(L′_{2}, M_{[t]}) = 0

for v = 0 and 4 by the assumption on L′_{2}. Note that _{v}Ext_{R}^{2}(M_{[f+g−4−t]}, M(X_{l})) = 0 for v = 0 and 4 by the same reason, i.e. because _{−v−4}Hom_{R}((L′_{2})^{∗}(−f − g), M_{[f+g−4−t]}) = 0.

Now applying (a) to the linked curve X_{l}, we get a generization of X′_{l} with constant postulation where R(−f − g + 4) is "removed" in its minimal resolution. A further linkage, using a complete intersection of the same type as in the linkage above (such a complete intersection exists by [20], Cor. 3.7) and formula (31) combined with Lemma 23 (replacing X and X_{l} by X′ and X′_{l} in (31), in the case r − 1 = 0 one explicitly constructs the resolution of I(X′) by the mapping cone construction), gives the desired generization X′, and we are done.

Remark 43. Note that the proof of Proposition 31 (b) simplifies in an obvious way in the case of most interest to us, i.e. where the diameter is 1 (M ≅ M_{[c]}).

7 Consequences, remarks and questions.

An interesting consequence of Theorem 27 and Proposition 31 is the following

Corollary 44. Let X be Buchsbaum (or ACM) of diam M(X) ≤ 2. Then there exists a generization X′ of X in H(d, g) such that X′ is Buchsbaum (or ACM) and such that the modules of the three sets

{F_{2}, F_{1}} , {L_{4}, F_{2}} and {L_{4}, F_{1}(−4)}

in the minimal resolution (10) of I(X′) are without common direct free factors. In particular

0Hom_{R}(I(X′), M(X′)) = _{0}Hom_{R}(M(X′), E(X′)) = 0

and H(d, g) is smooth at (X′).

Proof. Since the module structure of M = M(X) is trivial, we get

0Hom_{R}(I, M) ≅ _{0}Hom_{R}(F_{1}, M).

Since M ≅ M_{[c]} ⊕ M_{[c−1]} as R-modules and M_{[t]} is given by (19), it follows that the latter group vanishes if and only if L_{4} and F_{1}(−4) are without common free factors. Moreover, as in the proof of Lemma 23, we have

−4Hom_{R}(F_{2}, M)^{∨} ≅ _{0}Hom_{R}(M, E)

where the former group vanishes if and only if L_{4} and F_{2} are without common free factors. Now, using M ≅ M_{[c]} ⊕ M_{[c−1]}, we can successively apply Proposition 31 to M_{[c]} and M_{[c−1]} to see that {L_{4}, F_{2}} and {L_{4}, F_{1}(−4)} are without common direct free factors. Finally since {F_{2}, F_{1}} are without common direct free factors after performing a suitable generization (Theorem 27), we are done.

All general curves of diam M(X) = 1 are Buchsbaum. Hence

Corollary 45. Every irreducible component of H(d, g) whose generic curve X satisfies diam M = 1 is generically smooth.

Question. Is any irreducible component of H(d, g) whose Rao module of its generic curve is concentrated in at most two consecutive degrees, generically smooth?

We believe the answer is YES. Indeed if we as in Corollary 44 can show the existence of a generization X′ of X such that

0Hom_{R}(I(X′), M(X′)) = _{0}Hom_{R}(M(X′), E(X′)) = 0

for every curve X of diam M(X) = 2, then we can use Theorem 8 to see that H(d, g) is smooth at (X′), whence we get an affirmative answer to the question!

Remark 46. If we simply call an irreducible component V of H(d, g) reduced if H(d, g) is generically smooth along V, then we know that V is reduced if the Rao module M of its generic curve satisfies diam M ≤ 1. Moreover recall that in Mumford’s well known example ([30]) the diameter of M of the obstructed generic curve is 3. Using [18] we find a large number of nonreduced components among which there are examples of any diameter diam M ≥ 3. Indeed, as is well known, a smooth cubic surface X ⊂ ℙ^{3} satisfies Pic(X) ≃ ℤ^{⊕7}. It follows from the main theorem of [18] (or directly from [31]) that the general curve which comes from the linear system (3α, α^{5}, 2) ∈ ℤ^{⊕7} is the generic curve of a non-reduced component of H(d, g) for every α ≥ 4 (and α = 4 is Mumford’s example). Its diameter is 2α − 5. In the same way the general curve which corresponds to (3α+1, α^{5}, 2) ∈ ℤ^{⊕7} is the generic curve of a non-reduced component of H(d, g) of diam M = 2α − 4 for every α ≥ 4. So the question above represents the only open case concerning reducedness of irreducible components of H(d, g) with respect to the diameter of the Rao module of its generic curve.

References

[1] M. Amasaki. Examples of Nonsingular Irreducible Curves Which Give Reducible Singular Points of red(H_{d,g}). Publ. RIMS, Kyoto Univ. 21 (1985), 761-786.

[2] G. Bolondi. Irreducible families of curves with fixed cohomology Arch. der Math., 53 (1989), 300-305

[3] G. Bolondi, J.O. Kleppe and R.M. Miro-Roig. Maximal rank curves and singular points of the Hilbert scheme Compositio Math., 77 (1991), 269-291.

[4] G. Bolondi, J. Migliore. On curves with natural cohomology and their deficiency modules Annales de l’institut Fourier 43 (2) (1993), 325-357.

[5] M-C. Chang. A Filtered Bertini-type Theorem. J. reine angew. Math. 397, (1989), 214-219.

[6] Ph. Ellia, M. Fiorentini. Défaut de postulation et singularités du Schéma de Hilbert. Annali Univ. di Ferrara 30 (1984), 185-198.

[7] Ph. Ellia, A. Hirschowitz, E. Mezzetti On the number of irreducible components of the Hilbert scheme of smooth space curves International J. of Math. Vol.3, No. 6,(1992) 799-807.

[8] G. Ellingsrud. Sur le schéma de Hilbert des variétés de codimension 2 dans ℙ^{e} a cône de Cohen-Macaulay Ann. Scient. Éc. Norm. Sup. 8 (1975), 423-432.

[9] D. Eisenbud. Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Math., volume 150, Springer–Verlag, New York, 1995.

[10] S. Ginouillac. Sur le nombre de composant du schéma de Hilbert des courbes ACM de ℙ_{k}^{3}. C. R. Acad. Sci. Paris Sér. I, Math. 329 (1999), 857-862.

[11] S. Guffroy. Sur l’incomplétude de la série linéaire caractéristique d’une famille de courbes planes a noeuds et a cusps. Nagoya Math. J. 171 (2003), 51-83.

[12] G. Fløystad. Determining obstructions for space curves, with application to non-reduced components of the Hilbert scheme. J. reine angew. Math. 439 (1993), 11-44.

[13] A. Grothendieck. Cohomologie Locale des Faisceaux Cohérents et Théorèmes de Lefschetz Locaux et Globaux. Augmenté d’un exposé par M. Raynaud. (SGA 2). Advanced Studies in Pure Mathematics Vol. 2. North-Holland, Amsterdam (1968).

[14] A. Grothendieck. Les schémas de Hilbert. Séminaire Bourbaki, exp. 221 (1960).

[15] L. Gruson, Chr. Peskine. Genre des courbes de l’espace projectif, in: Proc. Tromsø 1977, Lectures Notes in Math. Vol. 687, Springer–Verlag, New York, (1978).

[16] R. Hartshorne. Algebraic Geometry. Graduate Texts in Math. Vol. 52 Springer–Verlag, New York, 1983.

[17] J. O. Kleppe. Deformations of graded algebras. Math. Scand. 45 (1979), 205–231.

[18] J. O. Kleppe. Non-reduced components of the Hilbert scheme of smooth space curves, in Proc. Rocca di Papa 1985 Lectures Notes in Math. Vol. 1266 Springer–Verlag, New York, (1987).

[19] J. O. Kleppe. The Hilbert-flag scheme, its properties and its connection with the Hilbert scheme. Applications to curves in 3-space. Preprint (part of thesis), March 1981, Univ. of Oslo. http://www.iu.hio.no/˜ jank/papers.htm.

[20] J. O. Kleppe. Liaison of families of subschemes in ℙ^{n}, in “Algebraic Curves and Projective Geometry, Proceedings (Trento, 1988),” Lectures Notes in Math. Vol. 1389 Springer–Verlag, New York, (1989).

[21] J. O. Kleppe. Concerning the existence of nice components in the Hilbert scheme of curves in ℙ^{n} for n = 4 and 5, J. reine angew. Math. 475 (1996), 77-102.

[22] J.O. Kleppe. The Hilbert Scheme of Space Curves of small diameter. Annales de l’institut Fourier 56 no. 5 (2006), p. 1297-1335.

[23] J.O. Kleppe. Families of Artinian and one-dimensional algebras. J. Algebra 311 (2007), 665-701.

[24] A. Laudal. Formal Moduli of Algebraic Structures. Lectures Notes in Math. Vol. 754, Springer–

Verlag, New York, (1979).

[25] M. Martin-Deschamps, D. Perrin. Sur la classification des courbes gauches, Asterisque, 184-185 (1990).

[26] M. Martin-Deschamps, D. Perrin Courbes Gauches et Modules de Rao, J. reine angew. Math. 439 (1993), 103-145.

[27] J. Migliore. Introduction to liaison theory and deficiency modules. Progress in Math., Vol. 165, Birkhäuser Boston, Inc., Boston, MA, 1998.

[28] J. Migliore, U. Nagel. Tetrahedral Curves. Int. Math. Res. Not. (2005), no. 15, p. 899-939.

[29] R. M. Miró-Roig. Unobstructed Arithmetically Buchsbaum Curves. in “Algebraic Curves and Projective Geometry, Proceedings (Trento, 1988),” Lectures Notes in Math. Vol. 1389 Springer– Verlag, New York, (1989).

[30] D. Mumford. Further pathologies in algebraic geometry, Amer. J. Math., 84, 1962, 642-648.

[31] H. Nasu. Obstructions to deforming space curves and non-reduced components of the Hilbert scheme. Publ. RIMS, Kyoto Univ. (2006), no 42, p. 117-141.

[32] I. Peeva. Consecutive cancellations in Betti Numbers. Proc. Amer. Math. Soc. 132 no. 12 (2004), 3503-3507.

[33] Ragusa and Zappala. On the reducibility of the postulation Hilbert scheme. Rend. Circ. Mat. Palermo, Serie II, no LIII (2004), 401-406.

[34] A. P. Rao. Liaison Among Curves. Invent. Math. 50 (1979), 205-217.

[35] E. Sernesi. Un esempio di curva ostruita in ℙ^{3}. Sem. di variabili Complesse, Bologna (1981), 223-231.

[36] C. Walter. Horrocks theory and algebraic space curves. Preprint (1990).

[37] C. Walter. Some examples of obstructed curves in ℙ^{3} In: Complex Projective Geometry. London Math. Soc. Lecture Note Ser. 179 (1992).

OSLO UNIVERSITY COLLEGE, FACULTY OF ENGINEERING, PB. 4 ST. OLAVS PLASS, N-0130 OSLO, NORWAY. E-mail address: JanOddvar.Kleppe@iu.hio.no