Liaison invariants and the Hilbert scheme of codimension 2 subschemes in ℙ^{n+2}

Jan O. Kleppe

Abstract

In this paper we study the Hilbert scheme Hilb^{p(v)}(ℙ) of equidimensional locally Cohen-Macaulay codimension 2 subschemes, with a special look to surfaces in ℙ^{4} and 3-folds in ℙ^{5}, and the Hilbert scheme stratification H_{γ,ρ} of constant cohomology. For every (X) ∈ Hilb^{p(v)}(ℙ) we define a number δ_{X} in terms of the graded Betti numbers of the homogeneous ideal of X and we prove that 1 + δ_{X} − dim_{(X)} H_{γ,ρ} and 1 + δ_{X} − dim T_{γ,ρ} are CI-biliaison invariants where T_{γ,ρ} is the tangent space of H_{γ,ρ} at (X). As a corollary we get a formula for the dimension of any generically smooth component of Hilb^{p(v)}(ℙ) in terms of δ_{X} and the CI-biliaison invariant. Both invariants are equal in this case.

Recall that, for space curves C, Martin-Deschamps and Perrin have proved the smoothness of the “morphism” ϕ : H_{γ,ρ} → E_{ρ} := isomorphism classes of graded modules M satisfying dim M_{v} = ρ(v), given by sending C onto its Rao module. For surfaces X in ℙ^{4} we have two Rao modules M_{i} ≃ ⊕H^{i}(I_{X}(v)) of dimension ρ_{i}(v), ρ := (ρ_{1}, ρ_{2}) and an induced extension b ∈ _{0}Ext^{2}(M_{2}, M_{1}) and a result of Horrocks and Rao saying that a triple D := (M_{1}, M_{2}, b) of modules M_{i} of finite length and an extension b as above determine a surface X up to biliaison. We prove that the corresponding “morphism” φ : H_{γ,ρ} → V_{ρ} = isomorphism classes of graded modules M_{i} satisfying dim(M_{i})_{v} = ρ_{i}(v) and commuting with b, is smooth, and we get a smoothness criterion for H_{γ,ρ}, i.e. for the equality of the two biliaison invariants. Moreover we get some smoothness results for Hilb^{p(v)}(ℙ), valid also for 3-folds, and we give examples of obstructed surfaces and 3-folds. The linkage result we prove in this paper turns out to be useful in determining the structure and dimension of H_{γ,ρ}, and for proving the main biliaison theorem above. AMS Subject Classification. 14C05, 14D15, 14M06, 14M07, 14B15, 13D02. Keywords. Hilbert scheme, surfaces in 4-space, 3-folds in 5-space, unobstructedness, graded Betti numbers, liaison, normal sheaf.

Contents

1 Introduction. 2

2 Notations and terminology. 3

3 The dimension of H(d, g) and biliaison invariants. 5

4 The dimension and the smoothness of H(d, p, π). 8

5 The smoothness of the “morphism” φ : H_{γ,ρ} → V_{ρ}. 11

6 The tangent space of H_{γ,ρ}. 15

7 Linkage of surfaces. 17

8 Obstructed surfaces in ℙ^{4}. 21

9 Even liaison of codimension 2 subschemes of ℙ^{n+2}. 24

1 Introduction.

A main object of this paper is to find the dimension of the Hilbert scheme, Hilb^{p(v)}(ℙ), of equidimensional locally Cohen-Macaulay (lCM) codimension 2 subschemes of ℙ := ℙ^{n+2}. As an initial ambitious goal we look for a formula for the dimension of any reduced component V of the Hilbert scheme Hilb^{p(v)}(ℙ) in terms of the graded Betti numbers of the homogeneous ideal I_{X} of a general element (X) of V. Since we expect the matrices in the minimal resolution of I_{X} to play a role, it seems natural to modify our goal by introducing a biliaison invariant in the dimension formula. Indeed in this paper we explicitly define an invariant δ_{X}^{n+1}(−n − 3) in terms of the graded Betti numbers of I_{X} and H_{∗}^{n}(O_{X}) and we prove that

dim V = 1 + δ_{X}^{n+1}(−n − 3) − sumext(X)

where sumext(X) is a CI-biliaison invariant (Corollary 9.4). In the case X is a curve (n = 1) with Hartshorne-Rao module M, we use results of [38] to prove

sumext(X) = ∑_{i=0}^{1} _{0}ext_{R}^{i}(M, M) ,

(Theorem 3.7 and Remark 3.9) and there is a similar, but much more complicated formula in the surface case (which we may deduce from Remark 6.4).

Let H_{γ,ρ} ⊆ Hilb^{p(v)}(ℙ) be the Hilbert scheme whose k-points (X) corresponds to equidimensional lCM codimension 2 subschemes X of ℙ^{n+2} with constant cohomology (see [38] for the curve case). If X is any equidimensional lCM codimension 2 subscheme of ℙ, we define obsumext(X) in the following way,

obsumext(X) = 1 + δ_{X}^{n+1}(−n − 3) − dim_{(X)} H_{γ,ρ} .

We define sumext(X) by the same expression provided we have replaced H_{γ,ρ} by its tangent space, T_{γ,ρ}, at (X). Then we prove that sumext(X) and obsumext(X) are CI-biliaison invariants (Theorem 9.1). Since every arithmetically Cohen-Macaulay codimension 2 subscheme is in the liaison class of a complete intersection (CI) by Gaeta’s theorem, it follows that sumext(X) = obsumext(X) = 0 and that dim_{(X)} Hilb^{p(v)}(ℙ) = 1 + δ_{X}^{n+1}(−n − 3) for n > 0 if X is arithmetically Cohen-Macaulay (Corollary 9.6). Even though we do not prove the explicit expression of sumext(X) in terms the Rao modules of X in general, the theorem is motivated from the fact that the Rao modules are invariant under biliaison up to shift. In fact it seems more effective to compute sumext(X) and obsumext(X) by considering a nice representative X′ in its even liaison class, e.g. the minimal element, and to compute δ_{X′}^{n+1}(−n − 3), dim_{(X′)} H_{γ,ρ}, and dim T_{γ,ρ} for X′.

Since the curve case of the results above is rather well understood ([38], [33]), we will in the present paper mostly concentrate on the study of the Hilbert scheme H(d, p, π) of surfaces of degree d and arithmetic (resp. sectional) genus p (resp. π). Recall that, for space curves C, Martin-Deschamps and Perrin proved the smoothness of the “morphism” ϕ : H_{γ,ρ} → E_{ρ}: = isomorphism classes of graded R-modules M satisfying dim M_{v} = ρ(v), given by sending C onto its Rao module. Earlier Rao proved that any graded R-module M of finite length determines the liaison class of a curve, up to dual and shift in the grading ([47]). Note that Rao’s result is related to the surjectivity of ϕ, while the smoothness of ϕ implies infinitesimal surjectivity. For surfaces in ℙ^{4} there is a result in Bolondi’s paper [6], stating that a triple D := (M_{1}, M_{2}, b) of graded modules M_{i} of finite length and an extension b ∈ _{0}Ext^{2}(M_{2}, M_{1}) determine the biliaison class of a surface X such that M_{i} ≃ ⊕H^{i}(I_{X}(v)) modulo some shift in the grading. The result is a consequence of the main theorem of [48] and Horrocks’ classification of stable vector bundles ([24]), as mentioned by Rao in [48]. Therefore it is natural to consider the stratification H_{γ,ρ} of H(d, p, π) where now ρ := (ρ_{1}, ρ_{2}) and ρ_{i}(v) = dim H^{i}(I_{X}(v)), and to ask for the smoothness of the corresponding “morphism” φ : H_{γ,ρ} → V_{ρ} := isomorphism classes of triples (M_{1}, M_{2}, b) where M_{i} are graded R-modules which satisfy dim(M_{i})_{v} = ρ_{i}(v) and where an isomorhpism between triples is an isomorphism between the corresponding modules which commutes with the extensions. We prove in section 5 that the answer is yes (Theorem 5.3). As a corollary we get a smoothness criterion for H_{γ,ρ} (Corollary 5.4, Remark 6.3), i.e. for the equality sumext(X) = obsumext(X) to hold. Note that we do not prove that φ extends to a morphism of schemes; we only prove that the corresponding morphism of the local deformation functors is formally smooth. This, however, takes fully care of what we want.

In section 6 we determine the tangent space of H_{γ,ρ} at (X), and we prove a local isomorphism H_{γ,ρ} ≃ H(d, p, π) at (X) under some conditions (Proposition 6.1, Remark 6.2). Note, however, that if X has seminatural cohomology, we know that H_{γ,ρ} ≃ H(d, p, π) at (X) by the semicontinuity of dim H^{i}(I_{X}(v)) and this observation mostly suffices for our applications. In section 7 we prove a useful linkage result (Theorem 7.1) which we apply to determine the structure and the dimension of H_{γ,ρ} and to prove our main theorem on the biliaison invariants. In this section we also give conditions for a linked surface to be e.g. non-generic, thus proving the existence of surfaces with “smaller” cohomology in some cases (Proposition 7.4).

Since the technical problems in describing well the stratification of H(d, p, π) and the morphism φ are quite complicated (see [31]), we don’t follow up this trace for equidimensional lCM codimension 2 subschemes X ⊆ ℙ^{n+2} of dimension n ≥ 3. Instead we only use our main theorem on the biliaison invariance of sumext(X) and obsumext(X) together with some new results on the smoothness and the dimension of Hilb^{p(v)}(ℙ) in our study of the Hilbert schemes of e.g. 3-folds in section 9. We also give a vanishing criterion for H^{1}(N_{X}), but unfortunately, as in [33], the results we get require that the Hartshorne-Rao modules are rather “small”. When the conditions of these vanishing criteria do not hold, we give examples of obstructed surfaces and 3-folds.

Acknowledgment. I heartily thank prof. G. Bolondi at Bologna for the discussion with him on this topic. As the reader will see, especially for the results in section 5 and 6, Bolondi’s paper [6] is a main source of ideas for the work presented here. It was prof. G. Bolondi who introduced me to the idea of extending the results of [6], as Martin-Deschamps and Perrin do for space curves, to get a stratified description of the Hilbert scheme H(d, p, π), and who pointed out several interesting things to be proved (see also [31]). Parts of the paper are also a natural continuation of [8] and [9]. Moreover I warmly thank Hirokazu Nasu at Chiba for his valuable comments and useful Macaulay 2 computations to the obstructed surface in Example 8.3, which led me to include examples of smooth obstructed surfaces (Example 8.4).

2 Notations and terminology.

A surface (resp. curve) X is an equidimensional, locally Cohen-Macaulay subscheme (lCM) of ℙ^{4} (resp. ℙ^{3}) of dimension 2 (resp. 1) with sheaf ideal I_{X} and normal sheaf N_{X} = Hom_{O_{ℙ}} (I_{X}, O_{X}). If F is a coherent O_{ℙ} -Module, we let H^{i}(F) = H^{i}(ℙ, F), H_{∗}^{i}(F) = ⊕_{v}H^{i}(F(v)) and h^{i}(F) = dim H^{i}(F), and we denote by χ(F) = Σ(−1)^{i}h^{i}(F) the Euler-Poincaré characteristic. Then p(v) = χ(O_{X}(v)) is the Hilbert polynomial of X. Put n = dim X and

\begin{aligned}s(X) & = min\{v|h^{0}(I_{X}(v)) ≠ 0\}, \\ e(X) & = max\{v|h^{n}(O_{X}(v)) ≠ 0\}. \end{aligned}

Let I = I_{X} = H_{∗}^{0}(I_{X}) be the homogeneous ideal. I is a graded module over the polynomial ring R = k[X_{0}, X_{1}, .., X_{n+2}], where k is supposed to be algebraically closed (and of characteristic zero in section 5, 6 and in all examples since we there may use results and methods of papers relying on this assumption). The postulation γ of X is the function defined over the integers by γ(v) = γ_{X}(v) = h^{0}(I_{X}(v)).

Let Hilb^{p(v)}(ℙ^{n+2}) denote the Hilbert scheme of equidimensional lCM codimension 2 subschemes of ℙ^{n+2} with Hilbert polynomial p (cf. [21]). X is called unobstructed if Hilb^{p(v)}(ℙ^{n+2}) is smooth at the corresponding point (X), otherwise X is obstructed. A subscheme of ℙ^{n+2} belonging to a sufficiently small open irreducible subset U of Hilb^{p(v)}(ℙ^{n+2}) (small enough so that any (X) of U satisfies all the openness properties which we want it to have) is called a generic subscheme of Hilb^{p(v)}(ℙ^{n+2}), and accordingly, if we state that a generic subscheme has a certain property, then there is a non-empty open irreducible subset of Hilb^{p(v)}(ℙ^{n+2}) of subschemes having this property.

In the case of curves we put H(d, g) = Hilb^{p(v)}(ℙ^{n+2}) provided p(v) = dv+1−g. Moreover we let M = M(C) := H_{∗}^{1}(I_{C}) be the deficiency or Hartshorne-Rao module of the curve C. The deficiency function ρ is defined by ρ(v) = h^{1}(I_{C}(v)). Let H(d, g)_{γ,ρ} (resp. H(d, g)_{γ}) denote the subscheme of H(d, g) of curves with constant cohomology given by γ and ρ, (resp. constant postulation γ) where “constant” means flat deformations of the corresponding modules, see [38]. Let Def_{M} be the local deformation functor consisting of graded deformations M_{S} of M to ℙ^{3} × Spec(S) modulo graded isomorphisms of M_{S} over M, where S is a local artinian k-algebra with residue field k, i.e. such that M_{S} is S-flat and M_{S} ⊗ k = M.

For a surface X we define the arithmetic genus p by p = χ(O_{X}) − 1, while the sectional genus π is given by χ(O_{X}(1)) = d − π + 1 + χ(O_{X}). By Riemann-Roch’s theorem we have

p(v) = χ(O_{X}(v)) = \frac{1}{2}dv^{2} − (π − 1 − \frac{1}{2}d)v + χ(O_{X}).  (1)

Put H(d, p, π) = Hilb^{p(v)}(ℙ^{n+2}) in this case. Moreover let M_{i} = M_{i}(X) be the deficiency modules H_{∗}^{i}(I_{X}) for i = 1,2. The deficiency ρ = (ρ_{1}, ρ_{2}) of X is the function defined over the integers by ρ(v) = ρ_{X}(v) = (ρ_{1}(v), ρ_{2}(v)) where ρ_{i}(v) = h^{i}(I_{X}(v)) for i = 1, 2. Let H_{γ,ρ} = H(d, p, π)_{γ,ρ} (resp. H_{γ} = H(d, p, π)_{γ}) denote the subscheme of H(d, p, π) of surfaces with constant cohomology given by γ and ρ, (resp. constant postulation γ) where again “constant” means flat deformations of the corresponding modules.

For the notion of linkage, we refer to [39]. Note that liaison (resp. even liaison or biliaison) is the equivalence relation generated by linkage (resp. direct linkages in an even number of steps).

For any graded R-module N, we have the right derived functors H_{𝔪}^{i}(N) and _{v}Ext_{𝔪}^{i}(N, −) of Γ_{𝔪}(N) := ⊕_{v} ker(N_{v} → Γ(ℙ, Ñ(v))) and Γ_{𝔪}(Hom_{R}(N, −))_{v} respectively (cf. [20], exp. VI or [22]) where 𝔪 = (X_{0}, .., X_{n+2}). We use small letters for the k-dimension and subscript v for the homogeneous part of degree v, e.g. _{v}ext_{𝔪}^{i}(N_{1}, N_{2}) = dim _{v}Ext_{𝔪}^{i}(N_{1}, N_{2}).

Let N_{1} and N_{2} be graded R-modules of finite type. As in [33] we need the spectral sequence

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

([20], exp. VI) and the duality isomorphism

vExt_{𝔪}^{i}(N_{2}, N_{1}) ≅ _{−v−n−3}Ext_{R}^{n+3−i}(N_{1}, N_{2})^{∨}, i, v ∈ ℤ  (3)

where (−)^{∨} = Hom_{k}(−, k) (cf. [30], Thm. 1.1, see [28], Thm. 2.1.4 for a full proof). Moreover there is a long exact sequence

→ _{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}) →  (4)

([20], exp. VI) which at least for equidimensional, lCM subschemes of codimension 2 (with n > 0) relate the deformation theory of X, described by H^{i−1}(N_{X}) ≃ Ext_{O_{ℙ}}^{i} (Ĩ, Ĩ) for i = 1, 2 (cf. [28], Rem. 2.2.6), to the deformation theory of the homogeneous ideal I = I_{X}, described by _{0}Ext_{R}^{i}(I, I), in the following exact sequence

0 → _{v}Ext_{R}^{1}(I, I) → H^{0}(N_{X}(v)) → _{v}Ext_{𝔪}^{2}(I, I)

⟶^{α} _{v}Ext_{R}^{2}(I, I) → H^{1}(N_{X}(v)) → _{v}Ext_{𝔪}^{3}(I, I) →

(5)

see [49] or [17] for related works on such deformation functors.

3 The dimension of H(d, g) and biliaison invariants.

In this section we consider the Hilbert scheme, H(d, g), of curves in ℙ^{3} and results which we would like to generalize to surfaces in ℙ^{4}. We will focus on the dimension of the Hilbert schemes and some biliaison invariants which we naturally detect from this point of view.

Recall that χ(N_{C}(v)) = 2dv + 4d and that χ(N_{C}) = 4d is a lower bound for dim_{(C)} H(d, g). For this reason the number 4d is often called the expected dimension of H(d, g) even though it often does not give the correct dimension of H(d, g) at (C). E.g. at ACM, generically comlete intersection curves the dimension is never 4d if e(C) ≥ s(C).

To give a more reliable estimate for the dimension of the components of H(d, g), we have found it convenient to introduce the following invariant, defined in terms of the numbers n_{j,i} appearing in a minimal resolution of the homogeneous ideal I_{C} of C:

0 → ⨁_{i=1}^{r_{3}} R(−n_{3,i}) → ⨁_{i=1}^{r_{2}} R(−n_{2,i}) → ⨁_{i=1}^{r_{1}} R(−n_{1,i}) → I_{C} → 0 .  (6)

Note that we can define the graded Betti numbers, β_{j,k}, of I_{C} by just putting ⊕_{k=1}^{∞}R(−k)^{β_{j,k}} := ⊕_{i=1}^{r_{j}}R(−n_{j,i}).

Definition 3.1. If C is a curve in ℙ^{3}, we let

δ_{C}^{j}(v) := ∑_{i} h^{j}(I_{C}(n_{1,i} + v)) − ∑_{i} h^{j}(I_{C}(n_{2,i} + v)) + ∑_{i} h^{j}(I_{C}(n_{3,i} + v)).

Put δ^{j}(v) = δ_{C}^{j}(v). In [33] we proved the following result (Lem. 2.2 of [33])

Lemma 3.2. Let C be any curve of degree d in ℙ^{3}. Then the following expressions are equal

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

Remark 3.3. Comparing with the results and notations of [38] we recognize 1 − δ^{0}(0) as δ_{γ} and δ^{1}(−4) as ϵ_{γ,δ} in their terminology. By Lemma 3.2 it follows that the dimension of the Hilbert scheme H_{γ,M} of constant postulation and Rao module, which they show is δ_{γ} + ϵ_{γ,δ} −_{0}hom(M, M) (Thm. 3.8, page 171), is also equal to 1 + δ^{2}(−4) −_{0}hom(M, M).

Note that the difference of the ext-numbers in Lemma 3.2 is a lower bound for dim O_{H(d,g)_{γ},(C)} ([33], proof of Thm. 2.6 (i)). Mainly since H(d, g)_{γ} is a subscheme of H(d, g), we used this lower bound in [35], Thm. 24, to prove the following result

Theorem 3.4. Let C be a curve in ℙ^{3} and let δ^{j}(v) = δ_{C}^{j}(v) for any j and v. Then the dimension of H(d, g) at (C) satisfies

dim_{(C)} H(d, g) ≥ 1 − δ^{0}(0) = 4d + δ^{2}(0) − δ^{1}(0).

Moreover if C is a generic curve of a generically smooth component V of H(d, g) and M = H_{∗}^{1}(I_{C}), then

dim V = 4d + δ^{2}(0) − δ^{1}(0) + _{−4}hom_{R}(I_{C}, M)

where _{−4}Hom_{R}(I_{C}, M) is the kernel of the map

⨁_{i} H^{1}(I_{C}(n_{1,i} − 4)) → ⨁_{i} H^{1}(I_{C}(n_{2,i} − 4))

induced by the corresponding map in (6).

Remark 3.5. Let C be any curve in ℙ^{3} and suppose

−4Hom_{R}(I_{C}, M) = _{0}Hom_{R}(I_{C}, M) = 0.

Then C is unobstructed and the lower bound of the inequality of Theorem 3.4 is equal to dim_{(C)} H(d, g) by Thm. 2.6 of [33].

Remark 3.6. Let C be any curve in ℙ^{3}.

(i) If M = 0, then δ^{1}(0) = 0 and we can use Remark 3.5 to see that C is unobstructed and that the lower bound of Theorem 3.4 is equal to dim_{(C)} H(d, g). This coincides with [13]. (ii) If diam M = 1, dim M = r and C is a generic curve, then C is unobstructed by [33] Cor. 1.6 and the lower bound is equal to 4d + δ^{2}(0) + rβ_{2,c}. Indeed rβ_{1,c} = 0 for a generic curve by [33], Cor. 4.4. Moreover in this case the “correction” number _{−4}hom_{R}(I_{C}, M) is equal to rβ_{1,c+4}. Hence we get

dim V = 4d + δ^{2}(0) + r(β_{2,c} + β_{1,c+4}).

This coincides with the dimension formula of [33], Thm. 3.4.

Theorem 3.4 is a consequence of the inclusion H(d, g)_{γ} ↪ H(d, g) of schemes. One may try the same argument for the inclusion H(d, g)_{γ,ρ} ↪ H(d, g) since we also for these schemes know tangent and obstruction spaces. This leads to

Theorem 3.7. Let C be a curve in ℙ^{3} and M = H_{∗}^{1}(I_{C}). Then the dimension of H(d, g) at (C) satisfies

dim_{(C)} H(d, g) ≥ 1 + δ^{2}(−4) − ∑_{i=0}^{2} _{0}ext_{R}^{i}(M, M).

Moreover if C is a generic curve of a generically smooth component V of H(d, g), then

dim V = 4d + δ^{2}(0) − δ^{1}(0) + δ^{1}(−4) − ∑_{i=0}^{1} _{0}ext_{R}^{i}(M, M)

= 1 + δ^{2}(−4) − ∑_{i=0}^{1} _{0}ext_{R}^{i}(M, M).

Proof. We consider the stratification H(d, g)_{γ,ρ} of the Hilbert scheme H(d, g) and the “morphism” ϕ : H(d, g)_{γ,ρ} → E_{ρ}: = isomorphism classes of R-modules M given by mapping (C) onto M(C). By [38], Thm. 1.5, ϕ is smooth, and H(d, g)_{γ,M} := ϕ^{−1}(M) is a scheme of dimension 1+δ^{2}(−4)−_{0}hom(M, M) (see Remark 3.3). If we ignore the scheme structures, we may still, for each curve C, consider the corresponding local deformation functor, ϕ_{C}, of ϕ at (C), defined on the category of local artinian k-algebras with residue field k. ϕ_{C} is smooth of fiber dimension as above by the results of [38], see also [33], Rem. 2.12 for the curve case and Theorem 5.3 of this paper for the corresponding result for surfaces.

It is well known that _{0}Ext_{R}^{i}(M, M) for i = 1, 2, determine the local graded deformation functor, Def_{M}, of the R-module M := M(C), e.g.

0ext^{1}(M, M) −_{0}ext^{2}(M, M) ≤ dim E_{ρ,M} ≤ _{0}ext^{1}(M, M),

where E_{ρ,M} is the hull of Def_{M} ([37], Thm. 4.2.4). Moreover we have equality to the right if and only if Def_{M} is formally smooth. Combining with the smoothness of ϕ_{C} and its fiber dimension we get

1 + δ^{2}(−4) − ∑_{i=0}^{2}_{0}ext^{i}(M, M) ≤ dim_{(C)} H(d, g)_{γ,ρ}
≤ 1 + δ^{2}(−4) −_{0}hom(M, M) +_{0}ext^{1}(M, M)

(7)

with equality to the right if and only if H(d, g)_{γ,ρ} is smooth at (C). This proves the inequality of the theorem since dim_{(C)} H(d, g) ≥ dim_{(C)} H(d, g)_{γ,ρ}. We also get the final statement because, at a generic curve C with postulation γ and deficiency ρ, H(d, g)_{γ,ρ} ≅ H(d, g) around (C)! Indeed if we have dim_{(C)} H(d, g)_{γ,ρ} < dim_{(C)} H(d, g), then a small neighborhood of (C) in H(d, g)_{γ,ρ} is not open in H(d, g), contradicting the assumption that C is generic in H(d, g). Hence we have equality in dimensions and in fact a local isomorphism (e.g. by generic flatness) since H(d, g) is smooth at (C). It follows that H(d, g)_{γ,ρ} is smooth at (C) and the inequality of (7) to the right is an equality.

Remark 3.8. Let T_{γ,ρ} be the tangent space of H(d, g)_{γ,ρ} at (C). Then we easily see from the proof that the upper bound in (7) is equal to dim T_{γ,ρ}.

If we want to generalize Theorem 3.7 to codimension 2 subschemes in ℙ^{n+2}, the explicit replacements of ∑_{i=0}^{1} _{0}ext^{i}(M, M) in the generalized statements seem to be very complicated. However observing that ∑_{i=0}^{1} _{0}ext^{i}(M, M) is a biliaison invariant (since M is, up to a twist), it seems to be the following weaker form of Theorem 3.7 and (7) which is natural to generalize:

Remark 3.9. If we define sumext(C) and obsumext(C) by sumext(C) = 1 + δ^{2}(−4) − dim T_{γ,ρ} and obsumext(C) = 1+δ^{2}(−4)−dim_{(C)} H(d, g)_{γ,ρ}, then sumext(C) and obsumext(C) are biliaison invariants. We have

sumext(C) ≤ obsumext(C)

and equality holds if and only if H(d, g)_{γ,ρ} is smooth at (C). Furthermore if C is unobstructed and generic in H(d, g), then

dim_{(C)} H(d, g) = 1 + δ^{2}(−4) − sumext(C) .

We have not yet proved that obsumext(C) is a biliaison invariant, but it will follow from later results, or from [38], Thm. 1.5 and Remark 3.3.

For curves we have

sumext(C) = ∑_{i=0}^{1} _{0}ext_{R}^{i}(M, M) , and  (8)

∑_{i=0}^{1} _{0}ext_{R}^{i}(M, M) ≤ obsumext(C) ≤ ∑_{i=0}^{2} _{0}ext_{R}^{i}(M, M)  (9)

which we may use to compute sumext(C) and estimate obsumext(C). We may also compute these invariants somewhere in the even liaison class, e.g. by letting C be the minimal curve and computing dim_{(C)} H(d, g)_{γ,ρ} , dim T_{γ,ρ} and δ^{2}(−4) in this case. If D is in the even liaison class of C, D ∈ H_{γ′,ρ′}, and if we can compute δ_{D}^{2}(−4), then we get the dimensions of H_{γ′,ρ′} and T_{γ′,ρ′}, from the biliaison invariants.

4 The dimension and the smoothness of H(d, p, π).

In this section we consider the Hilbert scheme, H(d, p, π), of surfaces in ℙ^{4}. Our goal is to see how far we can generalize the results of the preceding section to surfaces. We will focus on the dimension and the smoothness of the Hilbert scheme.

To compute the dimension of the components of H(d, p, π), we consider the minimal resolution of I = I_{X}:

0 → ⨁_{i=1}^{r_{4}} R(−n_{4,i}) → ⨁_{i=1}^{r_{3}} R(−n_{3,i}) → ⨁_{i=1}^{r_{2}} R(−n_{2,i}) → ⨁_{i=1}^{r_{1}} R(−n_{1,i}) → I → 0,  (10)

and the invariant δ^{j}(v) = δ_{X}^{j}(v) defined by

\begin{matrix}δ_{X}^{j}(v) = ∑_{i} h^{j}(I_{X}(n_{1,i} + v)) & − ∑_{i} h^{j}(I_{X}(n_{2,i} + v)) \\ + ∑_{i} h^{j} & (I_{X}(n_{3,i} + v)) − ∑_{i} h^{j}(I_{X}(n_{4,i} + v)). \end{matrix}  (11)

Proposition 4.1. Let X be any surface in ℙ^{4} of degree d and sectional genus π. Then the following expressions are equal

0ext_{R}^{1}(I, I) − _{0}ext_{R}^{2}(I, I) + _{0}ext_{R}^{3}(I, I) = 1 − δ^{0}(0) = χ(N_{X}) − δ^{0}(−5)

= χ(N_{X}) − δ^{3}(0) + δ^{2}(0) − δ^{1}(0) = 1 + δ^{3}(−5) − δ^{2}(−5) + δ^{1}(−5). (12)

Moreover

χ(N_{X}(v)) = dv^{2} + 5dv + 5(2d + π − 1) − d^{2} + 2χ(O_{X}).  (13)

Proof. The first upper equality follows easily by applying _{v}Hom_{R}(−, I) (for v = 0) to the resolution (10) because Hom_{R}(I, I) ≃ R and because the alternating sum of the dimension of the terms in a complex equals the alternating sum of the dimension of its homology groups. Similarly we compute δ^{0}(−5) which through the duality (3) leads to the alternating sum of _{0}ext_{𝔪}^{i}(I, I). Combining with (5), recalling Hom_{O_{ℙ}} (I_{X}, I_{X}) ≅ O_{ℙ} and Ext_{O_{ℙ}}^{1} (I_{X}, I_{X}) ≅ N_{X}, we get the next equality in the first line. The other equalities involving δ^{j}(v) follow from (2), (3) and (4) as outlined in [33], Lem 2.2 in the curve case. The surface case is technically more complicated because the spectral sequence of the proof, E_{2}^{p,q} = _{v}Ext_{R}^{p}(I, H_{𝔪}^{q}(I)), contains one more non-vanishing term. The principal parts of the proof are, however, the same, and we leave this part to the reader. Similarly the arguments of [33], Rem 2.4, lead to the formula

χ(N_{X}(v)) = χ(O_{X}(v)) + χ(O_{X}(−v − 5)) − d^{2}  (14)

for any surface X, from which (13) of Proposition 4.1 easily follows provided we combine with (1). Since we do not have a reference of (13) in the generality of an arbitrary surface (i.e. locally Cohen-Macaulay and equidimensional, see Remark below) and since the arguments of [33], Rem 2.4 was only sketched, we will include a proof of (14).

Firstly, we compute χ(O_{X}(v)) = χ(O_{ℙ} (v)) − χ(I_{X}(v)), χ(O_{ℙ} (v)) = \binom{v+4}{4}, directly from (10) as a large sum of binomials. Recalling that χ(O_{X}(v)) is the polynomial (1) of degree 2, we get

∑_{j=1}^{4}(−1)^{j−1}r_{j} = 1 , ∑_{j=1}^{4}(−1)^{j−1} ∑_{i} n_{j,i} = 0 and ∑_{j=1}^{4}(−1)^{j−1} ∑_{i} n_{j,i}^{2} = −2d .  (15)

Now as in the very first part of the proof, we apply _{v}Hom_{R}(−, I) to (10). Since we get _{v}Ext_{R}^{i}(I, I) ≅ H^{i−1}(N_{X}(v)) for v >> 0 and i ≥ 1 directly from (2), (3) and (4) and we have Hom_{R}(I, I) ≃ R, we find

dim R_{v} − χ(N_{X}(v)) = δ^{0}(v) = ∑_{j=1}^{4}(−1)^{j−1} ∑_{i} χ(I_{X}(n_{j,i} + v)) , v >> 0 .  (16)

By (10),

χ(I_{X}(−v − 5)) = ∑_{j=1}^{4}(−1)^{j−1} ∑_{i} χ(O_{ℙ} (−n_{j,i} − v − 5)) = ∑_{j=1}^{4}(−1)^{j−1} ∑_{i} χ(O_{ℙ} (n_{j,i} + v)).

The right hand side of (16) is therefore equal to

χ(I_{X}(−v − 5)) − ∑_{j=1}^{4}(−1)^{j−1} ∑_{i} χ(O_{X}(n_{j,i} + v)).

Then we compute ∑_{j=1}^{4}(−1)^{j−1} ∑_{i} χ(O_{X}(n_{j,i} + v)) by just using (1) and (15). We get exactly

∑_{j=1}^{4}(−1)^{j−1} ∑_{i} χ(O_{X}(n_{j,i} + v)) = χ(O_{X}(v)) − d^{2},

and (16) translates to dim R_{v} − χ(N_{X}(v)) = χ(I_{X}(−v − 5)) − χ(O_{X}(v)) + d^{2} and we get (14).

Remark 4.2. Note that the formula (13) of Proposition 4.1 is certainly straightforward to prove for smooth surfaces by combining the well known formula

χ(N_{X}(v)) = dv^{2} + 5dv + 5(d − π + 1) − 2K^{2} + 14χ(O_{X})

with the double point formula d^{2} − 10d − 5H.K − 2K^{2} + 12χ(O_{X}) = 0.

Now we come to the analogue of Theorem 3.4. Also in this case _{0}ext_{R}^{1}(I, I) − _{0}ext_{R}^{2}(I, I) is a lower bound of H(d, p, π)_{γ}. Since the basic part of the proof of the Theorem below is similar to the proof of Theorem 3.4, we will only sketch the proof. Note that in the surface case, we do not succeed so nicely as in the curve case because the lower bound above is not directly given by the first equality of Proposition 4.1, due to the term _{0}ext_{R}^{3}(I, I). Since we have _{0}Ext_{R}^{3}(I, I) ≅ −5Ext_{𝔪}^{2}(I, I)^{∨} ≅ _{−5}Hom_{R}(I, M_{1})^{∨} by (2) and (3) and M_{1} ≅ H_{𝔪}^{2}(I) we get at least

Proposition 4.3. Let X be a surface in ℙ^{4}, let M_{i} = H_{∗}^{i}(I_{X}) for i = 1,2 and put I = I_{X} and δ^{j}(v) = δ_{X}^{j}(v) for any j and v. Then the dimension of H(d, p, π) at (X) satisfies

dim_{(X)} H(d, p, π) ≥ 1 + δ^{3}(−5) − δ^{2}(−5) + δ^{1}(−5) − ∑_{i} h^{1}(I_{X}(n_{1,i} − 5)).

Moreover let X be a generic surface of a generically smooth component V of H(d, p, π) and suppose −5Hom_{R}(I, M_{2}) = 0. Then

dim V = 1 + δ^{3}(−5) − δ^{2}(−5) + δ^{1}(−5) − ∑_{i=0}^{1} _{−5}ext_{R}^{i}(I, M_{1}).

Proof. For the inequality, we remark that

0ext_{R}^{3}(I, I) = _{−5}hom_{R}(I, M_{1}) ≤ ∑_{i} h^{1}(I_{X}(n_{1,i} − 5))

because _{−5}Hom_{R}(I, M_{1}) is the kernel of the map ⊕_{i}H^{1}(I_{X}(n_{1,i} − 5)) ⟶ ⊕_{i}H^{1}(I_{X}(n_{2,i} − 5)) induced by the corresponding map in (10). We conclude by Proposition 4.1.

To find dim V we proceed as in the proof of Theorem 3.4 (see the last part of the proof of Theorem 3.7 for a close idea), and we get dim V = _{0}ext_{R}^{1}(I, I), i.e.

dim V = 1 + δ^{3}(−5) − δ^{2}(−5) + δ^{1}(−5) + _{0}ext_{R}^{2}(I, I) − _{0}ext_{R}^{3}(I, I).

By (3) we have _{0}ext_{R}^{2}(I, I) = _{−5}ext_{𝔪}^{3}(I, I) and we conclude by the exact sequence associated to (2),

0 → _{−5}Ext_{R}^{1}(I, H_{𝔪}^{2}(I)) → _{−5}Ext_{𝔪}^{3}(I, I) → _{−5}Hom_{R}(I, H_{𝔪}^{3}(I)) → _{−5}Ext_{R}^{2}(I, H_{𝔪}^{2}(I)) → .  (17)

Under more specific assumptions we are able to prove,

Proposition 4.4. Let X be any surface in ℙ^{4} and suppose

0Hom_{R}(I, M_{1}) = _{−5}Ext_{R}^{1}(I, M_{1}) = _{−5}Hom_{R}(I, M_{2}) = 0.

Then X is unobstructed and

dim_{(X)} H(d, p, π) = 1 + δ^{3}(−5) − δ^{2}(−5) + δ^{1}(−5) −_{−5}hom_{R}(I, M_{1}).

Proof. Due to [27], Rem. 3.7 (cf. [49], Thm. 2.1), H(d, p, π)_{γ} ≅ H(d, p, π) at (X) provided 0Hom_{R}(I, M_{1}) = 0. Then we see by the arguments of (17) that _{0}Ext_{R}^{2}(I, I) = 0. It follows that H(d, p, π)_{γ} is smooth at (X) of dimension _{0}ext_{R}^{1}(I, I). Then we conclude by Proposition 4.1.

Remark 4.5. (i) Proposition 4.4 is mainly proved in [30], sect. 1. In [30] we moreover use (2) and (3) to prove a vanishing result for H^{1}(N_{X}). Indeed we show that H^{1}(N_{X}) = 0 provided

H^{1}(I_{X}(n_{2,i})) = H^{1}(I_{X}(n_{2,i} − 5)) = 0 and H^{2}(I_{X}(n_{1,i})) = H^{2}(I_{X}(n_{1,i} − 5)) = 0

for every i.

(ii) Let X be an arithmetically Cohen-Macaulay surface in ℙ^{4}. Then M_{1} = M_{2} = 0 and δ^{1}(v) = δ^{2}(v) = 0 for every v and we can use Proposition 4.4 to see that X is unobstructed and dim_{(X)} H(d, p, π) = 1 + δ^{3}(−5) = 1 − δ^{0}(0). This coincides with [13].

We will illustrate the results of this section by an example. If the assumptions of Proposition 4.4 or Remark 4.5 are not satisfied, then the surface may be obstructed, and we refer to section 8 for such examples.

Example 4.6. Let X be the smooth rational surface with invariants d = 11, π = 11 (no 6-secant) and K^{2} = −11 (cf. [43] or [11], B1.17, see also [16]). In this case the graded modules M_{i} ≃ ⊕H^{i}(I_{X}(v)) are supported at two consecutive degrees and satisfy

dim H^{1}(I_{X}(3)) = 2,  dim H^{2}(I_{X}(1)) = 3,
dim H^{1}(I_{X}(4)) = 1,  dim H^{2}(I_{X}(2)) = 1.

Moreover I = I_{X} admits a minimal resolution (cf. [11])

0 → R(−9) → R(−8)^{⊕3} ⊕ R(−7)^{⊕3} → R(−7)^{⊕2} ⊕ R(−6)^{⊕12} → R(−5)^{⊕10} → I → 0.

It follows that _{−5}Hom_{R}(I, M_{2}) = 0 and _{−5}Ext_{R}^{i}(I, M_{1}) = 0 for i = 0, 1. By Proposition 4.4, H(d, p, π) is smooth at (X) and

dim_{(X)} H(d, p, π) = 1 + δ^{3}(−5) − δ^{3}(−5) + δ^{1}(−5)
= 1 + 12h^{2}(I_{X}(1)) − h^{2}(I_{X}(2)) + 3h^{1}(I_{X}(3)) − h^{1}(I_{X}(4)) = 41.

In this example it is, however, easier to use Proposition 4.1 to get

\begin{aligned}1 + δ^{3}(−5) − δ^{2}(−5) + δ^{1}(−5) & = χ(N_{X}) − δ^{3}(0) + δ^{2}(0) − δ^{1}(0) \\ & = 5(2d + π − 1) − d^{2} + 2χ(O_{X}) = 41 \end{aligned}

because δ^{i}(0) for i > 0 is easily seen to be zero. We may also use Remark 4.5 to see H^{1}(N_{X}) = 0. Since any smooth surface satisfies

H^{2}(N_{X}) = 0 provided H^{2}(O_{X}(1)) = 0

(due to the existence of the natural surjection O_{X}(1)^{5} → N_{X}), we may conclude as above directly from dim H^{0}(N_{X}) = χ(N_{X}) = 41.

One may hope that a generalization of Theorem 3.7 to surfaces will contain a more complete result. To do it we need to generalize some of the theorems in [38] to surfaces. This will be done in the next two sections. The biliaison statements of Remark 3.9 will be generalized to any codimension 2 lCM equidimensional subscheme of ℙ^{n+2} and carried out in later sections.

5 The smoothness of the “morphism” φ : H_{γ,ρ} → V_{ρ}.

In this section we prove the local smoothness of the “morphism” φ : H_{γ,ρ} → V_{ρ} := isomorphism classes of graded R-modules M_{1} and M_{2} satisfying dim(M_{i})_{v} = ρ_{i}(v) and commuting with b, given by sending the surface X onto the class of the triple (M_{1}, M_{2}, b) where M_{i} = H_{∗}^{i}(I_{X}) and b ∈ 0Ext_{R}^{2}(M_{2}, M_{1}) is the extension determined by X (cf. Remark 5.2 (ii)). To prove our theorem we first take in Proposition 5.1 a close look to Bolondi’s short exact “resolution” of the homogeneous ideal of a surface X ([6]) and how we can define the extension b given in Horrock’s paper [24]. As in [11] the ideal is the cokernel of some syzygy modules of M_{1} and M_{2}, up to direct free factors. The proposition somehow uses and extends a result of Rao for a curve C, namely that the minimal resolution of I_{C} can be put in the following form

0 → L_{4} ⟶^{σ⊕0} L_{3} ⊕ F_{2} → F_{1} → I_{C} → 0  (18)

where 0 → L_{4} →^{σ} L_{3} → ... → M → 0 is a minimal resolution of M and F_{i} are free modules ([47]). Moreover we use local flatness criteria to generalize Bolondi’s construction in [6] so that it works for flat resolutions over a local ring, rather than over a field. This is also the approach of [23] in the curve case.

Let X be a surface in ℙ^{4} and let

\begin{matrix}0 → P_{5} ⟶^{σ_{5}} P_{4} ⟶^{σ_{4}} P_{3} ⟶^{σ_{3}} … ⟶ P_{0} ⟶^{σ_{0}} M_{1} → 0, \\ 0 → Q_{5} ⟶^{τ_{5}} Q_{4} ⟶^{τ_{4}} Q_{3} ⟶^{τ_{3}} … ⟶ Q_{0} ⟶^{τ_{0}} M_{2} → 0 \end{matrix}  (19)

(for short σ_{•} : P_{•} → M_{1} → 0 and τ_{•} : Q_{•} → M_{2} → 0) be minimal free resolutions over R. Let K_{•} and L_{•} be the syzygies of M_{1} and M_{2} respectively, i.e. K_{i} = ker σ_{i} and L_{i} = ker τ_{i}. Recall that syzygies have nice cohomological properties ([11], [6]), for instance

\begin{matrix}M_{1} = H_{∗}^{1}(K̃_{1}) and H_{∗}^{2}(K̃_{1}) = H_{∗}^{3}(K̃_{1}) = 0, \\ M_{2} = H_{∗}^{3}(L̃_{3}) and H_{∗}^{1}(L̃_{3}) = H_{∗}^{2}(L̃_{3}) = 0. \end{matrix}  (20)

There is a strong connection between the resolutions (19), the minimal resolution (10) of I = I_{X} and the following minimal resolutions of A = H_{∗}^{0}(O_{X});

0 → P′_{3} ⟶^{σ′_{3}} P′_{2} ⟶^{σ′_{2}} P′_{1} ⟶^{σ′_{1}} P_{0} ⊕ R → A → 0  (21)

where the morphism P_{0} ⊕ R → A of (21) is naturally deduced from P_{0} → M_{1} of (19) and the exact sequence R → A → M_{1} → 0 and where σ′_{•} : P′_{•} → ker(P_{0} ⊕ R → A) → 0 is a minimal R-free resolution (cf. [38], p. 46). The connection we have in mind can be formulated and proved for a family of surfaces with constant cohomology, at least locally, e.g. we can replace the field k by a local k-algebra S with residue field k. Now, in [6], Bolondi uses some ideas of Horrocks [24] to define an element b ∈ _{0}Ext_{R}^{2}(M_{2}, M_{1}) and the “Horrocks triple” D =: (M_{1}, M_{2}, b) associated to X such that, conversely given D = (M_{1}, M_{2}, b) where M_{i} are R-modules of finite length, there is a surface X whose homogeneous ideal I is defined in the following way. For some integer h ∈ ℤ there is an exact sequence 0 → L′_{3} → K′_{1} → I(h) → 0 where L′_{3} (resp. K′_{1}) is isomorphic to the syzygy L_{3} (resp. K_{1}) up to some R-free module F_{L} (resp. F_{K}). Up to biliaison this construction is the inverse to the first approach which defines (M_{1}, M_{2}, b) from a given X. To prove the main smoothness theorem of this section, we need to adapt the approach above by determining F_{L} and F_{K} more explicitly and such that it works over (at least an artinian) S. Using also ideas of Rao’s paper [47], we can prove

Proposition 5.1. Let X be a surface in ℙ_{S}^{4}, flat over a local noetherian k-algebra S with residue field k, and suppose that M_{1} = H_{∗}^{1}(I_{X}), M_{2} = H_{∗}^{2}(I_{X}) and I = I_{X} are flat S-modules. Then there exist minimal R-free resolutions of M_{i}, I and A = H_{∗}^{0}(O_{X}) as in (19), (10) and (21), with R = S[X_{0}, X_{1}, .., X_{4}]. Moreover let L′_{3} = ker σ′_{1} and let K′_{1} be the kernel of the composition of σ′_{1} and the natural projection P_{0} ⊕ R → P_{0}, cf. (21). Then there is an exact sequence

0 ⟶ L′_{3} ⟶^{b′} K′_{1} ⟶ I ⟶ 0  (22)

of flat graded S-modules and a surjective morphism d : _{0}Hom_{R}(L′_{3}, K′_{1}) ⟶ _{0}Ext_{R}^{2}(M_{2}, M_{1}), defining a triple (M_{1}, M_{2}, b) where b = d(b′), coinciding with the uniquely defined “Horrocks triple” of [24] or [6]. Moreover L′_{3} (resp. K′_{1}) is the direct sum of a 3rd syzygy of M_{2} (resp. 1st syzygy of M_{1}) up to a direct free factor, i.e. there exist R-free modules F_{L} and F_{K} such that the horizontal exact sequences in the diagram

0 ⟶ K′_{1} ⟶ P′_{1} ⟶ P_{0}

↓  ∘  ↓  ∘  ∥

0 → K_{1} ⊕ F_{K} → P_{1} ⊕ F_{K} ⟶^{σ_{1}⊕0} P_{0}

are isomorphic (i.e., the downarrows are isomorphisms). Similarly, the exact sequences 0 → Q_{5} ⟶^{(τ_{5},0)} Q_{4} ⊕ F_{L} → L_{3} ⊕ F_{L} → 0 and 0 → P′_{3} → P′_{2} → L′_{3} → 0 are isomorphic as well.

Remark 5.2. (i) By a surface X ⊆ ℙ_{S}^{4} in Proposition 5.1 we actually mean that X ×_{Spec(S)} Spec(k) is a surface (i.e. locally Cohen-Macaulay and equidimensional of dimension 2).

(ii) The proposition above, defining the “Horrocks triple” (M_{1}, M_{2}, b) from a given X ⊆ ℙ_{S}^{4}, can be regarded as our definition of the “morphism” φ : H_{γ,ρ} → V_{ρ} = isomorphism classes of graded R-modules M_{1} and M_{2} satisfying dim(M_{i})_{v} = ρ_{i}(v) and commuting with b.

Proof. We obviously have minimal resolutions of M_{i} ⊗_{S} k, I_{X} ⊗_{S} k and A ⊗_{S} k as described above with R = k[X_{0}, X_{1}, .., X_{4}], cf. (19), (10) and (21). These resolutions can easily be lifted to the minimal resolution of the proposition by cutting into short exact sequences and using the flatness of the modules involved.

By the definition of L′_{3} and K′_{1} there is a commutative diagram

0 ⟶ R ⟶ R ⟶ 0
↓ ∘ ↓
0 ⟶ L′_{3} ⟶ P′_{1} ⟶ P_{0} ⊕ R ⟶ A ⟶ 0
∥ ∘ ↓ ∘ ↓
0 ⟶ K′_{1} ⟶ P′_{1} ⟶ P_{0} ⟶ M_{1} ⟶ 0

and we get the exact sequence (22) by the snake lemma. Comparing the lower exact sequence in the last diagram with the following part of the minimal resolution of M_{1}; → P_{1} → P_{0} → M_{1} → 0, we get the commutative diagram of the proposition because K_{1} is the 1st syzygy of M_{1}.

To prove the corresponding commutative diagram for L′_{3} and L_{3}, we sheafify (22), and we get M_{2} ≃ H_{∗}^{3}(L̃′_{3}). Recalling the definition of L′_{3}, we get the exact sequence

H_{∗}^{4}(P̃′_{2})^{∨} → H_{∗}^{4}(P̃′_{3})^{∨} → M_{2}^{∨} ≃ Ext_{R}^{5}(M_{2}, R(−5)) → 0

which we compare to the minimal resolution

Q_{4}^{∨} → Q_{5}^{∨} → Ext_{R}^{5}(M_{2}, R) → 0

obtained by applying Hom_{R}(−, R) to the resolution Q_{•} → M_{2}. Recalling H_{∗}^{4}(P̃′_{i})^{∨}(5) ≃ P_{i}^{′∨}, we get the conclusion, as in the proof of Thm. 2.5 of [47].

Finally to define the morphism d and to see that the defined triple (M_{1}, M_{2}, b) corresponds to the one given by Horrocks’ construction (seen to be unique by [24]), one may consult [6] for the case S = k which, however, generalizes to a local ring S. The important part is as follows. The definition of K′_{1} and K_{0} implies

Ext^{2}(M_{2}, M_{1}) ≃ Ext^{3}(M_{2}, K_{0}) ≃ Ext^{4}(M_{2}, K′_{1}).

Next, by Gorenstein duality, we know Ext_{R}^{i}(M_{2}, R) = 0 for i ≠ 5. Hence the definition of the syzygies L_{i} leads to Ext^{4}(M_{2}, K′_{1}) ≃ Ext^{3}(L_{0}, K′_{1}) ≃ Ext^{1}(L_{2}, K′_{1}) and to a diagram

0Hom_{R}(Q_{3}, K′_{1}) → _{0}Hom(L_{3}, K′_{1}) → _{0}Ext^{1}(L_{2}, K′_{1}) → 0
↓ ↓
0Hom(L′_{3}, K′_{1}) _{0}Ext_{R}^{2}(M_{2}, M_{1})

(23)

where the horizontal sequence is exact and the first (resp. second) vertical map is injective and split (resp. an isomorphism). We let d : _{0}Hom_{R}(L′_{3}, K′_{1}) → _{0}Ext_{R}^{2}(M_{2}, M_{1}) be the obvious composition, first using the “inverse” of the split map, and we get the conclusions of the proposition.

Now we will show the smoothness of φ. Indeed using Proposition 5.1 for S artinian, we get a rather easy proof of Theorem 5.3. The “morphism” φ : H_{γ,ρ} → V_{ρ} = isomorphism classes of graded R-modules M_{1} and M_{2} satisfying dim(M_{i})_{v} = ρ_{i}(v) and commuting with b, is smooth (i.e. for any surface X in ℙ_{k}^{4}, the corresponding local deformation functor of φ, given by (X_{S} ⊆ ℙ_{S}^{4}) ↦ class of (M_{1S}, M_{2S}, b_{S}), see right below, is formally smooth).

Proof. Let T → S → k be surjections of local artinian k-algebras with residue fields k such that ker(T → S) is a k-module via T → k. Let X_{S} ⊆ ℙ_{S}^{4} be a deformation of X ⊆ ℙ^{4} to S with constant postulation γ and constant deficiency ρ = (ρ_{1}, ρ_{2}). Let (M_{1S}, M_{2S}, b_{S}) be the “Horrocks triple” defined by X_{S} (cf. Proposition 5.1). Note that M_{iS} for i = 1, 2 are S-flat by the definition of H_{γ,ρ}. Let (M_{1T} , M_{2T} , b_{T}) be a given deformation of (M_{1S}, M_{2S}, b_{S}) to T. To prove the smoothness at (X), we must show the existence of a deformation X_{T} ⊆ ℙ_{T}^{4} of X_{S} ⊆ ℙ_{S}^{4}, whose corresponding “Horrocks triple” is precisely (M_{1T} , M_{2T} , b_{T}), modulo graded isomorphisms of (M_{1T} , M_{2T}) commuting with b_{T}.

We have by Proposition 5.1 minimal resolutions of M_{iS}, I_{X_{S}} and A_{S} over R_{S} := S[X_{0}, X_{1}, .., X_{4}] as in (10), (19)-(21) and flat S-modules L_{iS}, K_{iS}, L′_{3S}, K′_{1S} fitting into the exact sequence (22) and a surjection d defined as the composition (cf. (23))

0Hom_{R_{S}}(L′_{3S}, K′_{1S}) → _{0}Hom_{R_{S}}(L_{3S}, K′_{1S}) → _{0}Ext_{R_{S}}^{1}(L_{2S}, K′_{1S}) ≃ _{0}Ext_{R_{S}}^{2}(M_{2S}, M_{1S})
∪∣∣  ∪∣∣  ∪∣∣  ∪∣∣
b′_{S}  ⟶  β_{S}  ⟶  b_{S}  ⟶  b_{S}

(24)

“on the S-level” (β_{S} is simply the image of b′_{S} via the map of (24)) which lifts the corresponding resolutions/modules/sequences on the “k-level”. Since M_{iT} are given deformations of M_{iS}, we can lift the minimal resolutions σ_{•S} : P_{•S} → M_{1S} and τ_{•S} : Q_{•S} → M_{2S} further to T, thus proving the existence of deformations L_{iT}, K_{iT}, L′_{3T}, K′_{1T} of L_{iS}, K_{iS}, L′_{3S}, K′_{1S} resp. (the free submodules F_{LS} and F_{KS} of L′_{3S} and K′_{1S} are lifted trivially). So we have a diagram (23) and hence a sequence (24) “on the T -level” where the elements b′_{T} and β_{T} are not yet defined. The element b_{T} ∈ _{0}Ext^{1}(L_{2T} , K′_{1T}) ≃ 0Ext_{R_{T}}^{2} (M_{2T} , M_{1T}) is, however, given and if we consider the diagram (cf. (23))

0Hom_{R_{T}} (Q_{3T} , K′_{1T}) → _{0}Hom_{R_{T}} (L_{3T} , K′_{1T}) → _{0}Ext_{R_{T}}^{1} (L_{2T} , K′_{1T}) → 0
↓  ∘  ↓ α  ∘  ↓
0Hom_{R_{S}}(Q_{3S}, K′_{1S}) → _{0}Hom_{R_{S}}(L_{3S}, K′_{1S}) → _{0}Ext_{R_{S}}^{1}(L_{2S}, K′_{1S}) → 0

of exact horizontal sequences and surjective vertical maps deduced from 0 → L_{3T} → Q_{3T} → L_{2T} → 0, we easily get a morphism β_{T} ∈ _{0}Hom(L_{3T} , K′_{1T}) such that α(β_{T}) = β_{S}, i.e., β_{T} ⊗_{T} S = β_{S}. Since L′_{3S} ≃ L_{3S} ⊕ F_{LS} we can decompose the map b′_{S} as (β_{S}, γ_{S}) ∈ _{0}Hom(L′_{3S}, K′_{1S}), and taking any lifting γ_{T} : F_{LT} → K′_{1T} of γ_{S}, we get a map b′_{T} = (β_{T} , γ_{T}) ∈ _{0}Hom(L′_{3T} , K′_{1T}) fitting into a commutative diagram

L_{3T} ⊕ F_{LT} ≃ L′_{3T} ⟶^{b′_{T}} K′_{1T}
↓ ∘ ↓
L_{3S} ⊕ F_{LS} ≃ L′_{3S} ⟶^{b′_{S}} K′_{1S} .

Once having proved the existence of such a commutative diagram, we can define a surface X_{T} of ℙ_{T}^{4} with the desired properties, thus proving the claimed smoothness. Indeed it is straightforward to see that coker b′_{T} is a (flat) deformation of coker b′_{S} = I_{X_{S}} to T. Moreover one knows that an R_{T} := T [X_{0}, X_{1}, .., X_{4}]-module coker b′_{T} which lifts a graded ideal I_{X_{S}} is again a graded ideal I_{T} (we can deduce this information by interpreting the isomorphisms H^{i−1}(N_{X}) ≃ Ext_{O_{ℙ}}^{i} (I_{X}, I_{X}) for

i = 1, 2 in terms of their deformation theories from which we see that { \atop coker} b′_{T} is a sheaf ideal, and we conclude by taking global sections, cf. [49] or [33], Lem. 4.8 for further details). Hence we have proved the existence of a surface X_{T} = Proj(R_{T} /I_{T}), flat over T which via T → S reduces to X_{S}. By the construction above the corresponding “Horrocks triple” is precisely the given triple (M_{1T} , M_{2T} , b_{T}), and we are done.

Corollary 5.4. Let X be a surface in ℙ^{4}. If the local deformation functors Def(M_{i}) of M_{i} are formally smooth (for instance if _{0}Ext_{R}^{2}(M_{i}, M_{i}) = 0) for i = 1, 2, and if

0Ext_{R}^{3}(M_{2}, M_{1}) = 0,

then H_{γ,ρ} is smooth at (X).

Proof. With notations as in the very first part of the proof of Theorem 5.3, it suffices to prove that there always exists a deformation (M_{1T} , M_{2T} , b_{T}) of (M_{1S}, M_{2S}, b_{S}) since then the proof above shows the existence of a deformation X_{T} = Proj(R_{T} /I_{T}) which reduces to X_{S} via T → S. Since Def(M_{i}) are formally smooth, it suffices to show the existence of b_{T} which maps to b_{S} ∈ _{0}Ext_{R_{S}}^{2}(M_{2S}, M_{1S}). Let 𝔞 = ker(T → S). If we apply _{0}Hom_{R_{T}} (M_{2T} , −) to the exact sequence

0 → 𝔞 ⊗_{T} M_{1T} ≅ 𝔞 ⊗_{k} M_{1} → M_{1T} → M_{1S} → 0

and use _{0}Ext_{R}^{3}(M_{2}, M_{1}) = 0, we see that _{0}Ext_{R_{T}}^{2} (M_{2T} , M_{1T}) → _{0}Ext_{R_{T}}^{2} (M_{2T} , M_{1S}) is surjective. Hence we get a surjective map

0Ext_{R_{T}}^{1} (L_{3T} , K′_{1T}) ≃ _{0}Ext_{R_{T}}^{2} (M_{2T} , M_{1T}) → _{0}Ext_{R_{S}}^{1}(L_{2S}, K′_{1S}) ≃ _{0}Ext_{R_{S}}^{2}(M_{2S}, M_{1S})

and we are done.

Remark 5.5. If we, as in [38] for curves, had proven the existence of the “fiber” H_{γ,D}, D = (M_{1}, M_{2}, b), of φ as a scheme, then Theorem 5.3 must imply the smoothness of H_{γ,D} while [8] implies its irreducibility. Indeed [8], cor. 3.2 tells that the family of surfaces in ℙ^{4} belonging to the same shift of the same liaison class, with fixed postulation, form an irreducible family, from which we see that H_{γ,D} is irreducible. Note that we can work with H_{γ,D} as a locally closed subset of H_{γ,ρ} (cf. the arguments of [4], cor. 2.2, and combine with Proposition 5.1), even though we have not proved that φ extends to a morphism of representable functors.

6 The tangent space of H_{γ,ρ}.

In this section we determine the tangent space of H_{γ,ρ} at (X) and we give a criterion for H_{γ,ρ} ≅ H(d, p, π) to be isomorphic as schemes at (X). We end this section by considering an example.

Let X be a surface in ℙ^{4} with graded ideal I = I_{X} and let D = (M_{1}, M_{2}, b), M_{i} = H_{∗}^{i}(Ĩ), be its “Horrocks triple”. Recall that _{0}Ext_{R}^{1}(I, I) is the tangent space of H_{γ} at (X) because a deformation in H_{γ} keeps the postulation constant, i.e. it corresponds precisely to a graded deformation of I [38]. Moreover there exist maps

φ_{i} : _{0}Ext_{R}^{1}(I, I) → _{0}Hom_{R}(H_{∗}^{i}(Ĩ), H_{∗}^{i+1}(Ĩ))

taking an extension 0 → I → E → I → 0 of _{0}Ext_{R}^{1}(I, I) onto the connecting homomorphism δ^{i} in the exact sequence

H_{∗}^{i}(Ẽ) → H_{∗}^{i}(Ĩ) ⟶^{δ^{i}} H_{∗}^{i+1}(Ĩ) → H_{∗}^{i+1}(Ẽ).

For saturated homogeneous ideals we have I = H_{∗}^{0}(Ĩ), and it follows that the composition E → H_{∗}^{0}(Ẽ) → H_{∗}^{0}(Ĩ) is surjective, i.e. we get φ_{0} = 0. Moreover note that if δ^{i−1} and δ^{i} are both zero for some i, then the exact sequence 0 → I → E → I → 0 above defines an extension

0 → H_{∗}^{i}(Ĩ) → H_{∗}^{i}(Ẽ) → H_{∗}^{i}(Ĩ) → 0.

Since M_{i} = H_{∗}^{i}(Ĩ) for i = 1, 2 and E = H_{∗}^{3}(Ĩ), there are well-defined morphisms

ψ_{i} : ker(φ_{1}, φ_{2}) → _{0}Ext_{R}^{1}(M_{i}, M_{i}) for i = 1, 2

where (φ_{1}, φ_{2}) : _{0}Ext_{R}^{1}(I, I) → _{0}Hom(M_{1}, M_{2}) × _{0}Hom(M_{2}, E) and φ_{i} are defined above. Recalling ρ = (ρ_{1}, ρ_{2}) we put

0Ext_{R}^{1}(I, I)_{ρ} := ker(φ_{1}, φ_{2}).  (25)

Using base change theorems, as in [38], we easily show that ker(φ_{1}, φ_{2}) is the tangent space of H_{γ,ρ} at (X), i.e. we get

Proposition 6.1. _{0}Ext_{R}^{1}(I, I)_{ρ} is the tangent space of H_{γ,ρ} at (X). In particular if

0Hom_{R}(I, M_{1}) = 0, _{0}Hom_{R}(M_{1}, M_{2}) = 0 and _{0}Hom(M_{2}, E) = 0,  (26)

then the tangent spaces of H_{γ,ρ}, H_{γ} and H(d, p, π) are isomorphic at (X). Indeed H_{γ} ≅ H(d, p, π) as schemes at (X), and if H_{γ,ρ} is smooth at (X), then H_{γ,ρ} ≅ H_{γ} are isomorphic as schemes at (X) as well.

Proof. As earlier remarked, cf. (2) and (5), _{0}Ext_{R}^{1}(I, I) ≅ Ext^{1}(I_{X}, I_{X}) ≅ H^{0}(N_{X}) provided 0Hom_{R}(I, M_{1}) = 0. Moreover _{0}Ext_{R}^{1}(I, I)_{ρ} ≅ _{0}Ext_{R}^{1}(I, I) since φ_{i} = 0 for i = 1, 2.

For the isomorphism as schemes we remark that H_{γ} ≃ H(d, p, π) follows from [27], Thm. 3.6 and Rem. 3.7 (see [49] and [33], proof of Thm. 2.6 (i) for details). Finally if H_{γ,ρ} is smooth at (X), then the embedding H_{γ,ρ} ↪ H_{γ} is smooth at (X) (since the tangent map is surjective), hence etale, hence an isomorphism at (X) since the embedding is universally injective.

Remark 6.2. If we suppose (26), then H_{γ,ρ} ≅ H_{γ} are isomorphic as schemes at (X) by [31], Thm. 3.7 without requiring the smoothness of H_{γ,ρ} at (X). See also Remark 9.3.

In [31] we also gave almost complete proofs of Remark 6.2 and of the following two non-trivial results (cf. [31], Prop. 3.4 and Prop. 3.6). Note that Remark 6.3 generalizes Corollary 5.4.

Remark 6.3. Let X be a surface in ℙ^{4}. Then for i = 1, 2 there exist morphisms e_{i} : _{0}Ext_{R}^{1}(M_{i}, M_{i}) → 0Ext_{R}^{3}(M_{2}, M_{1}) and an induced morphism

ē_{1} : _{0}Ext_{R}^{1}(M_{1}, M_{1}) → _{0}Ext_{R}^{3}(M_{2}, M_{1})/e_{2}(_{0}Ext_{R}^{1}(M_{2}, M_{2}))

such that if the local deformation functors Def(M_{i}) of M_{i} are formally smooth (for instance if 0Ext_{R}^{2}(M_{i}, M_{i}) = 0) for i = 1, 2, and if the morphism ē_{1} is surjective, then V_{ρ} is smooth at D = (M_{1}, M_{2}, b) (i.e. the local deformation functor of D is formally smooth).

Remark 6.4. Let X be a surface in ℙ^{4} and let ϵ = dim coker ē_{1}. Then

dim _{0}Ext_{R}^{1}(I, I)_{ρ} = 1 + δ^{3}(−5) + ∑_{i=0}^{3}(−1)^{i} _{0}ext_{R}^{i}(M_{2}, M_{1})

− ∑_{i=0}^{1}(−1)^{i} _{0}ext_{R}^{i}(M_{1}, M_{1}) − ∑_{i=0}^{1}(−1)^{i} _{0}ext_{R}^{i}(M_{2}, M_{2}) + ϵ.

To illustrate the results we have proved, we consider an example of a surface X of ℙ^{4} where actually V_{ρ} is smooth and non-trivial at the corresponding (M_{1}, M_{2}, b), cf. Corollary 5.4. Moreover all conditions of Proposition 6.1 are satisfied, and it follows that H_{γ,ρ} and H(d, p, π) are isomorphic and smooth at (X).

Example 6.5. Let X be the smooth elliptic surface with invariants d = 11, π = 12 and K^{2} = −4 (cf. [43] or [11], B7.6). Then the graded modules M_{i} ≃ ⊕H^{i}(I_{X}(v)) for i = 1, 2 vanish for every v except in the following cases

h^{1}(I_{X}(3)) = 1, h^{2}(I_{X}(1)) = 2, h^{2}(I_{X}(2)) = 1.

Moreover I = I_{X} admits a minimal resolution (cf. [11])

0 → R(−8) → R(−7)^{⊕6} → R(−6)^{⊕13} → R(−5)^{⊕8} ⊕ R(−4) → I → 0.

It follows that _{0}Ext^{i}(M_{j}, M_{j}) = 0 for i ≥ 2 and j = 1, 2 and that _{0}Ext^{3}(M_{2}, M_{1}) = 0. By Corollary 5.4 and Proposition 6.1 we get that H(d, p, π) ≅ H_{γ,ρ} are smooth at (X). If we, however, want to compute the dimension of H(d, p, π) at (X) and will avoid Remark 6.4 which we have not proved, we still have to use the results of section 4. Let us only use the two “most general” results there, Proposition 4.1 and Propositions 4.3, to illustrate the principle of semicontinuity a little extended (to include the semicontinuity of the graded Betti numbers). Let V be the generically smooth component of H(d, p, π) to which (X) belongs. Since H(d, p, π) ≅ H_{γ,ρ} at (X), then a generic surface X̃ of V also belongs to H_{γ,ρ}. Inside H_{γ}, hence inside H_{γ,ρ}, the graded Betti numbers of the homogeneous ideal of the surfaces obey semicontinuity by Remark 7(b) of [34]!! Since we from the minimal resolution of I_{X} can see that, for every i, β_{j,i} ≠ 0 for at most one j and since the Hilbert functions of X and X̃ are the same, they have exactly the same graded Betti numbers. Moreover note that h^{i}(IX̃(v)) = h^{i}(I_{X}(v)) for any i, v since X has seminatural cohomology. It follows that

dim V = 1 + δ^{3}(−5) − δ^{3}(−5) + δ^{1}(−5) =
1 + h^{3}(I_{X}(−1)) + 8h^{3}(I_{X}) + 13h^{2}(I_{X}(1)) − 6h^{2}(I_{X}(2)) − h^{1}(I_{X}(3)) = 50.

Since we have proved dim V = 1 + δ^{3}(−5) − δ^{3}(−5) + δ^{1}(−5) it is easier to use Proposition 4.1 to get

dim V = χ(N_{X}) − δ^{3}(0) + δ^{2}(0) − δ^{1}(0) = 5(2d + π − 1) − d^{2} + 2χ(O_{X}) = 50

because δ^{i}(0) for i > 0 is easily seen to be zero.

7 Linkage of surfaces.

The main result of this section shows how to compute the dimension of H_{γ,ρ} and the dimension of its tangent space at (X) provided we know how to solve the corresponding problem for a linked surface X′ (Theorem 7.1). In another related result (Proposition 7.4 with c > 0) we give conditions on e.g. a generic surface of H(d, p, π) such that corresponding linked surface X′ is non-generic in the sense dim_{(X′)} H_{γ′,ρ′} < dim_{(X′)} H(d′, p′, π′). It follows that a new surface, the generic one with “smaller” cohomology, has to exist! Indeed recall that linkage is a well known method for proving existence of surfaces with certain properties, e.g. see [42], [26], [40], [46], [12], [44], [1] to mention a few papers which use linkage in this way. In these and similar papers we see that the linked surface X′ is usually generic if X is generic. In Remark 7.2 we notice that if certain cohomological assumptions, cf. (30), are satisfied, then X′ is generic if and only if X is generic. Using Proposition 7.4 with c > 0, however, then some of the cohomology groups of (30) are non-zero, and under some assumptions we get the existence of a non-generic surface X′ ∈ H_{γ′,ρ′} and hence a generic one ∉ H_{γ′,ρ′} as well. In proving the results of this section we substantially need the theory of linkage of families developed in [29].

Since the main even liaison result of this paper, which we prove in the final section, requires that the linkage theorem of this section is proven for equidimensional locally Cohen-Macaulay codimension 2 subschemes of ℙ^{n+2}, we prove Theorem 7.1 in this generality. The other results and examples of this section deal, however, with surfaces.

Now, if the surfaces X and X′ are (algebraically) linked by a complete intersection (a CI) Y of type (f, g), then the dualizing sheaf ω_{X′} satisfies ω_{X′} = I_{X/Y} (f+g−5) where I_{X/Y} = ker(O_{Y} → O_{X}) ([45], [39]). Moreover ω_{X} = I_{X′/Y} (f + g − 5) and we get

χ(O_{X}(v)) + χ(O_{X′}(f + g − 5 − v)) = χ(O_{Y} (v))
h^{i}(I_{X′}(v)) = h^{3−i}(I_{X}(f + g − 5 − v)), for i = 1 and 2
h^{i}(I_{X′/Y} (v)) = h^{2−i}(O_{X}(f + g − 5 − v)), for i = 0 and 2
h^{i}(O_{X′}(v)) = h^{2−i}(I_{X/Y} (f + g − 5 − v)), for i = 0 and 2

(27)

from which we deduce d + d′ = fg and π′ − π = (d′ − d)(f + g − 4)/2.

The generalization of (27) to equidimensional lCM codimension 2 subschemes of ℙ^{n+2} is clear, e.g. we have

h^{i}(I_{X′/Y} (v)) = h^{n−i}(O_{X}(f + g − n − 3 − v)), for i = 0 and n.  (28)

Note that we now have n deficiency modules, whose dimensions ρ_{i}(v) = h^{i}(I_{X}(v)), i = 1, 2, ..., n determine the vector function ρ = (ρ_{1}, ..., ρ_{n}). Using this vector function, we easily generalize (25) in such a way that we get the tangent space _{0}Ext_{R}^{1}(I_{X}, I_{X})_{ρ} of the Hilbert scheme H_{γ,ρ} ⊆ Hilb^{p(v)}(ℙ^{n+2}) of constant cohomology in this case. We allow n = 0 in which case there is no ρ and H_{γ,ρ} ⊆ Hilb^{p(v)}(ℙ^{2}) should be taken as the Hilbert scheme of constant postulation (“the postulation Hilbert scheme”) and _{0}Ext_{R}^{1}(I_{X}, I_{X})_{ρ} as _{0}Ext_{R}^{1}(I_{X}, I_{X}). We have (cf. [38] for the curve case of the theorem),

Theorem 7.1. Let X and X′ be two equidimensional locally Cohen-Macaulay codimension 2 subschemes of ℙ^{n+2}, linked by a complete intersection Y ⊆ ℙ^{n+2} of type (f, g), and suppose that (X) (resp. (X′)) belongs to the Hilbert scheme H_{γ,ρ} (resp. H_{γ′,ρ′}) of constant cohomology. Then

i) dim_{(X)} H_{γ,ρ} +h^{0}(I_{X}(f)) + h^{0}(I_{X}(g))
= dim_{(X′)} H_{γ′,ρ′} +h^{0}(I_{X′}(f)) + h^{0}(I_{X′}(g))

or equivalently,

dim_{(X′)} H_{γ′,ρ′} = dim_{(X)} H_{γ,ρ} + h^{0}(I_{X/Y} (f)) + h^{0}(I_{X/Y} (g))
− h^{n}(O_{X}(f − n − 3)) − h^{n}(O_{X}(g − n − 3)).

ii) The dimension formulas of i) remain true if we replace dim_{(X)} H_{γ,ρ} and dim_{(X′)} H_{γ′,ρ′} by the dimensions of their tangent spaces _{0}Ext_{R}^{1}(I_{X}, I_{X})_{ρ} and _{0}Ext_{R}^{1}(I_{X′}, I_{X′})_{ρ′} respectively.

iii) H_{γ,ρ} is smooth at (X) if and only if H_{γ′,ρ′} is smooth at (X′)

Proof. Let D(p(v); f, g) be the Hilbert flag scheme parametrizing pairs (X, Y) of equidimensional lCM codimension 2 subschemes of ℙ^{n+2} such that Y is a CI of type (f, g) containing X. By [29], Thm. 2.6, there is an isomorphism of schemes,

D(p(v); f, g) ≃ D(p′(v); f, g),  (29)

given by sending (X, Y) onto (X′, Y) where X′ is linked to X by Y (cf. [25], Prop. (A.1)). We may suppose n ≥ 1 in Theorem 7.1 since the case n = 0 is completely solved by Prop. 1.7 of [32]. Then the projection morphism p : D(p(v); f, g) → Hilb^{p(v)}(ℙ^{n+2}), given by (X, Y) ↦ (X), is smooth at (X, Y) provided H^{1}(I_{X}(f)) = H^{1}(I_{X}(g)) = 0 ([29], Thm. 1.16 (b)). By [29], Lem. 1.17 and Rem. 1.20, this smoothness holds if we replace the vanishing above with the claim that the set of global sections of the corresponding twisted ideal sheaves over the local ring of Hilb^{p(v)}(ℙ^{n+2}) at (X) are locally free and commute with base change. Hence the following restriction of p to p^{−1}(H_{γ,ρ}), p^{−1}(H_{γ,ρ}) → H_{γ,ρ}, is smooth, (or see [38] for related arguments). Since the fiber dimension of p at (X, Y) is precisely

h^{0}(I_{X/Y} (f)) + h^{0}(I_{X/Y} (g)) = h^{0}(I_{X}(f)) + h^{0}(I_{X}(g)) − h^{0}(I_{Y} (f)) − h^{0}(I_{Y} (g))

by [29], Thm. 1.16 (a), we get any conclusion of the theorem if we combine with (28).

Remark 7.2. Let X and X′ be two surfaces in ℙ^{4}, linked by a CI of type (f, g). Then the arguments of the proof above show that we can, under the assumptions

H^{1}(I_{X}(f)) = H^{1}(I_{X}(g)) = 0 and H^{1}(I_{X′}(f)) = H^{1}(I_{X′}(g)) = 0  (30)

replace H_{γ,ρ} and H_{γ′,ρ′} in Theorem 7.1 (i) (resp. their tangent spaces in Theorem 7.1 (ii) ) by H(d, p, π) and H(d′, p′, π′) (resp. by H^{1}(N_{X}) and H^{1}(N_{X′})) and get valid dimension formulas involving the whole Hilbert schemes (resp. their tangent spaces). Hence assuming (30), it follows that X is unobstructed if and only if X′ is unobstructed, see [29], Prop. 3.12 for a generalization. Note also that we from the proof above (i.e. from [29], Thm. 1.16 (b)) and (30) get that X is generic if and only if X′ is generic, see [29], Prop. 3.8 for a related general result.

Example 7.3. Let X be the smooth rational surface of H(11, 0, 11) of Example 4.6, let Y be a CI of type (5, 5) containing X, and let X′ be the linked surface. Using (27) we deduce χ(O_{X′}(v)) = 7v^{2} − 12v + 9 from χ(O_{X}(v)) = (11v^{2} − 9v + 2)/2, i.e. (X′) belongs to H(d′, p′, π′) = H(14, 8, 20) by (1). Moreover ω_{X′} = I_{X/Y} (5) is globally generated (cf. the resolution of I of Example 4.6) and the graded modules M′_{i} ≃ ⊕H^{i}(I_{X′}(v)) are supported at two consecutive degrees and satisfy

dim H^{1}(I_{X′}(3)) = 1,  dim H^{2}(I_{X′}(1)) = 1,
dim H^{1}(I_{X′}(4)) = 3,  dim H^{2}(I_{X′}(2)) = 2.

From these informations we find the minimal resolution of I′ = I_{X′} to be

0 → R(−9)^{⊕3} → R(−8)^{⊕14} → R(−7)^{⊕23} → R(−6)^{⊕11} ⊕ R(−5)^{⊕2} → I′ → 0.

Combining Example 4.6 and Remark 6.2 we see that H_{γ,ρ} is smooth at (X) and dim_{(X)} H_{γ,ρ} = 41. Thanks to Theorem 7.1, we get that H_{γ′,ρ′} is smooth at (X′) and that

dim_{(X′)} H_{γ′,ρ′} = dim_{(X)} H_{γ,ρ} +2h^{0}(I_{X/Y} (5)) − 2h^{2}(O_{X}(0)) = 57.

Moreover by Remark 7.2 or Proposition 6.1, H(d′, p′, π′) ≃ H_{γ′,ρ′} is smooth at (X′) and dim_{(X′)} H(d′, p′, π′) = 57. Note that in this case we neither have _{0}Ext^{3}(M_{2}, M_{1}) = 0 nor _{−5}Hom_{R}(I, M_{2}) = 0, i.e. we can not use Corollary 5.4 or Proposition 4.4 to conclude that H_{γ′,ρ′} is smooth at (X′). But, as we have seen, the linkage result above takes care of the smoothness and the dimension.

If a surface X of ℙ^{4} is contained in a CI Y of type (f, g), then there is an inclusion map I_{Y} → I_{X} which induces a morphism l_{X/Y}^{i+1} : H^{i}(N_{X}) → H^{i}(O_{X}(f)) ⊕ H^{i}(O_{X}(g)) for every i. We let β_{X/Y} be the composition of l_{X/Y}^{1} with the natural map H^{0}(O_{X}(f))⊕H^{0}(O_{X}(g)) → H^{1}(I_{X}(f))⊕H^{1}(I_{X}(g)).

Proposition 7.4. Let X and X′ be surfaces in ℙ^{4}, geometrically linked by a complete intersection Y ⊆ ℙ^{4} of type (f, g), let (X) ∈ H_{γ,ρ} and (X′) ∈ H_{γ′,ρ′} and suppose dim_{(X)} H_{γ,ρ} = dim_{(X)} H(d, p, π). Let c := dim_{(X′)} H(d′, p′, π′)−dim_{(X′)} H_{γ′,ρ′} and suppose H^{1}(I_{X}(f)) = H^{1}(I_{X}(g)) = 0 and that l_{X/Y}^{2} is injective. Then

h^{1}(I_{X′}(f)) + h^{1}(I_{X′}(g)) − h^{2}(I_{X′}(f)) − h^{2}(I_{X′}(g)) ≤ c ≤ h^{1}(I_{X′}(f)) + h^{1}(I_{X′}(g)) (31)

and we have equality on the right hand side if and only if H(d′, p′, π′) is smooth at (X′). Furthermore, if h^{1}(I_{X′}(v)) · h^{2}(I_{X′}(v)) = 0 for v = f and v = g, then

c = h^{1}(I_{X′}(f)) + h^{1}(I_{X′}(g)).

Proof. Since X and X′ are generically complete intersections (due to geometric linkage) of codimension 2 in ℙ^{4}, it follows that the cotangent sheaves A_{X}^{2} and A_{X′}^{2} are zero (cf. [10]). The vanishing of the obstruction group, A^{2}(X ⊆ Y), of the Hilbert flag scheme D(p(v); f, g) at (X, Y) is therefore equivalent to β_{X/Y} being surjective and l_{X/Y}^{2} being injective by (1.11) of [29], so A^{2}(X ⊆ Y) = 0 by assumption. Moreover since the linkage is geometric, we get A^{2}(X′ ⊆ Y) = 0 by Cor. 2.14 of [29], i.e. β_{X′/Y} is surjective, l_{X′/Y}^{2} is injective and D(p′(v); f, g) is smooth at (X′, Y). Hence [29], Thm. 1.27 applies (to a component V satisfying dim V = dim_{(X′)} H(d′, p′, π′)) to get the bounds of the codimension c above provided we can show that H_{γ′,ρ′}, in a neighborhood of (X′), is dense in an (f, g)-maximal subset of H(d′, p′, π′) (i.e. dense in the image under the first projection of some non-embedded component of D(p′(v); f, g)). By the proof of Theorem 7.1 we see that the restriction of the first projection p′ to p^{′−1}(H_{γ′,ρ′}), p^{′−1}(H_{γ′,ρ′}) → H_{γ′,ρ′}, is smooth. It follows that H_{γ′,ρ′} is, locally at (X′), (f, g)-maximal provided we can show

dim_{(X′,Y)} p^{′−1}(H_{γ′,ρ′}) = dim_{(X′,Y)} D(p′(v); f, g).

Thanks to (29) it suffices to show dim_{(X,Y)} p^{−1}(H_{γ,ρ}) = dim_{(X,Y)} D(p(v); f, g) which readily follows from the assumptions dim_{(X)} H_{γ,ρ} = dim_{(X)} H(d, p, π) and H^{1}(I_{X}(f)) = H^{1}(I_{X}(g)) = 0 because the first projection, p : D(p(v); f, g) → Hilb^{p(v)}(ℙ^{4}), as well as its restriction to p^{−1}(H_{γ,ρ}), are smooth at (X, Y) by Remark 7.2. Then we get the final conclusion from [29], Cor. 1.29, which states that h^{1}(I_{X′}(v)) · h^{2}(I_{X′}(v)) = 0 for v = f and g implies that H(d′, p′, π′) is smooth at (X′) and we are done.

Example 7.5. Let Z be the surface which is linked to the surface (X′) ∈ H(14, 8, 20) of Example 7.3 via a complete intersection of type (5, 6) containing X′. Then (Z) belongs to H(16, 15, 27), ω_{Z} = I_{X′/Y} (6) is globally generated, and M_{i}(Z) = ⊕H^{i}(I_{Z}(v)), i = 1, 2, are supported at two consecutive degrees. Moreover;

\begin{aligned}h^{0}(I_{Z}(5)) & = 1, & h^{1}(I_{Z}(4)) & = 2 & and & h^{1}(I_{Z}(5)) & = 1 \\ h^{2}(O_{Z}(1)) & = 1, & h^{2}(I_{Z}(2)) & = 3 & and & h^{2}(I_{Z}(3)) & = 1 . \end{aligned}  (32)

By Proposition 4.1, we know χ(N_{X′}) = 5(2d′ + π′ − 1) − d^{′2} + 2χ(O_{X′}) = 57 and since we obviously have h^{2}(N_{X′}) = 0 (from h^{2}(O_{X′}(1)) = 0) and we get h^{0}(N_{X′}) = 57 from Example 7.3, we conclude that h^{1}(N_{X′}) = 0. The conditions of Proposition 7.4 are therefore satisfied (replacing X by X′ there).

Hence, at (Z), we get that H(16, 15, 27)_{γ,ρ} is smooth of codimension 1 in H(16, 15, 27). Moreover H(16, 15, 27) is smooth at (Z), and

dim_{(Z)} H(16, 15, 27)_{γ,ρ} =
dim_{(X′)} H_{γ′,ρ′} +h^{0}(I_{X′/Y} (5)) + h^{0}(I_{X′/Y} (6)) − h^{2}(O_{X′}) − h^{2}(O_{X′}(1)) = 65.

Hence Z belongs to a unique generically smooth component V of H(16, 15, 27) of dimension 66, and since the generic surface Z̃ of V do not have the same cohomology as Z (since Z̃ ∉ H(16, 15, 27)_{γ,ρ}), we must get

dim H^{0}(IZ̃(5)) = dim H^{1}(IZ̃(5)) = 0

while elsewhere the dimension of the cohomology groups is unchanged, i.e. it is as in (32).

8 Obstructed surfaces in ℙ^{4}.

In this section we explicitly prove the existence of obstructed surfaces. Our examples are as close as they can be to the arithmetically Cohen-Macaulay case. Indeed, in the examples, one of the Rao modules in the pair (M_{1}, M_{2}) vanishes, the other is 1-dimensional. Moreover in Proposition 4.4 and Remark 4.5 we gave conditions which imply unobstructedness. Our Example 8.3 is minimal with respect to the mentioned conditions in the sense that only one of the many cohomology groups, claimed in Remark 4.5 (i) to vanish, is non-zero. It also shows that we in Remark 7.2 can not skip the assumption (30) since we in Example 8.3 link an unobstructed surface to an obstructed surface where one of the cohomology groups of (30) is non-zero. Moreover, note that once having constructed one obstructed surface we can find infinitely many by linking under the assumption (30).

In the following proposition we consider a codimension 2 subscheme X of ℙ^{n+2}, containing a CI Y of type (f_{1}, f_{2}), in order to find obstructed codimension 2 subschemes of ℙ^{n+2} for n ≥ 1. In this situation we recall that the inclusion map I_{Y} → I_{X} induces a morphism H^{0}(N_{X}) → ⊕_{i=1}^{2}H^{0}(O_{X}(f_{i})) whose composition with ⊕_{i=1}^{2}H^{0}(O_{X}(f_{i})) → ⊕_{i=1}^{2}H^{1}(I_{X}(f_{i})) we denote β_{X/Y}. Note that we below do not need the cotangent sheaves to vanish since we work only with tangent (and not obstruction) spaces of D(p(v); f_{1}, f_{2}).

Proposition 8.1. Let X be an equidimensional locally Cohen-Macaulay codimension 2 subscheme of ℙ^{n+2}, and let Y and Y_{0} be two complete intersections containing X, both of type (f_{1}, f_{2}) such that

i) β_{X/Y} is surjective and β_{X/Y_{0}} is not surjective,
ii) H^{n}(I_{X}(f_{i} − n − 3)) = 0 for i = 1 and i = 2.

Let X′ (resp. X′_{0}) be linked to X by Y (resp. Y_{0}). Then X_{0} is obstructed. Moreover if X is unobstructed, then so is X′.

Proof. If A^{1}(X ⊆ Y) is the tangent space of the Hilbert flag scheme D(p(v); f_{1}, f_{2}) at (X, Y), then it is shown in [29], (1.11) that there is an exact sequence

0 → ⊕_{i=1}^{2}H^{0}(I_{X/Y} (f_{i})) → A^{1}(X ⊆ Y) → H^{0}(N_{X}) → ⊕_{i=1}^{2}H^{1}(I_{X}(f_{i}))

where the rightmost map is β_{X/Y}. The corresponding exact sequence for (X ⊆ Y_{0}) together with the assumption (i) show that

dim A^{1}(X ⊆ Y) < dim A^{1}(X ⊆ Y_{0})

because it is easy to see h^{0}(I_{X/Y} (v)) = h^{0}(I_{X/Y_{0}}(v)) for every v. We claim that D(p(v); f_{1}, f_{2}) is not smooth at (X, Y_{0}). Suppose the converse. Since it is shown in [29], Thm. 1.16 (a) that the fibers of the first projection p : D(p(v); f_{1}, f_{2}) → Hilb^{p(v)}(ℙ^{n+2}) are irreducible, it follows that there exists an irreducible component W of D(p(v); f_{1}, f_{2}) which contains both points, (X, Y) and (X, Y_{0}). Hence if D(p(v); f_{1}, f_{2}) is smooth at (X, Y_{0}), we get

dim A^{1}(X ⊆ Y_{0}) = dim W ≤ dim_{(X,Y)} D(p(v); f_{1}, f_{2}) ≤ dim A^{1}(X ⊆ Y),

i.e. a contradiction.

Thanks to (29) we get that D(p′(v); f_{1}, f_{2}) is not smooth at (X′_{0}, Y_{0}). Since h^{1}(I_{X′_{0}}(f_{i} −n−3)) = h^{n}(I_{X}(f_{3−i} −n−3)) = 0 for i = 1, 2, cf. (27), and since the vanishing of H^{1}(I_{X′_{0}}(f_{i} −n−3)) implies that the first projection p′ : D(p′(v); f_{1}, f_{2}) → Hilb^{p′(v)}(ℙ^{n+2}) is smooth at (X′_{0}, Y_{0}) by [29], Thm. 1.16 (b), we conclude that X′_{0} is obstructed. Finally, for the last conclusion, if we have the surjectivity of β_{X/Y} and assume the unobstructedness of X, we get that D(p(v); f_{1}, f_{2}) is smooth at (X, Y) by [29], Prop. 3.12. Using (29) and (27) once more we conclude that X′ is unobstructed, and we are done.

We think the surjectivity of β_{X/Y} may often hold, provided the generators of I_{Y} are among the minimal generators of I_{X}, but this is difficult to prove. In the Buchsbaum case, however, it is easy to see the surjectivity, as observed in [7] for curves. Indeed even though the statement of Proposition 8.1 and the remark below generalizes [7], Prop. 2.1 by far, the ideas of the proof are quite close to the idea in Prop. 2.1 of [7].

Remark 8.2. In this remark we consider surfaces in ℙ^{4} with minimal resolution given as in (10). (i) Using (5) and the spectral sequence (2) we get an exact sequence

→ H^{0}(N_{X}) → _{0}Hom_{R}(I_{X}, H_{𝔪}^{2}(I_{X})) ⟶^{α} _{0}Ext_{R}^{2}(I_{X}, I_{X}) →

where _{0}Hom_{R}(I_{X}, H_{𝔪}^{2}(I_{X})) ≃ ⊕_{i}H^{1}(I_{X}(n_{1,i})) provided H^{1}(I_{X}(n_{2,i})) = 0 for any i. The natural map H^{0}(N_{X}) → _{0}Hom_{R}(I_{X}, H_{𝔪}^{2}(I_{X})) ≃ ⊕_{i}H^{1}(I_{X}(n_{1,i})), which we denote β_{X}, is correspondingly defined as β_{X/Y} above, but with the difference that a set of all minimal generators of I_{X} is used. In particular if the generators of I_{Y} are among the minimal generators of I_{X}, then the composition of β_{X} with the projection ⊕_{i}H^{1}(I_{X}(n_{1,i})) → ⊕_{i=1}^{2}H^{1}(I_{X}(f_{i})) is β_{X/Y}. It follows that if

0Ext_{R}^{2}(I_{X}, I_{X}) = 0 and H^{1}(I_{X}(n_{2,i})) = 0 for any i ,

then β_{X/Y} is surjective. Note that, by (3) and (2) (cf. the proof of Proposition 4.4), _{0}Ext_{R}^{2}(I_{X}, I_{X}) = 0 provided _{−5}Ext_{R}^{1}(I, M_{1}) = _{−5}Hom_{R}(I, M_{2}) = 0, i.e. provided

H^{1}(I_{X}(n_{2,i} − 5)) = 0 and H^{2}(I_{X}(n_{1,i} − 5)) = 0 for every i.

(ii) If, however, the minimal generators {F_{1}, F_{2}} of I_{Y} do not belong to a set of minimal generators of I_{X}, say F_{i} = H_{i} · G_{i} for some G_{i} ∈ I_{X}, i = 1, 2, then β_{X/Y} is easily seen to be non-surjective under a manageable assumption. Indeed let g_{i} be the degree of the form G_{i}, let Y_{0} be the CI with homogeneous ideal I_{Y_{0}} = (G_{1}, G_{2}) and suppose the obvious map

h : ⊕_{i=1}^{2}H^{1}(I_{X}(g_{i})) ⟶^{(H_{1},H_{2})} ⊕_{i=1}^{2}H^{1}(I_{X}(f_{i})) is not surjective.

Then β_{X/Y} can not be surjective because it factors via h, i.e. β_{X/Y} = h ∘ β_{X/Y_{0}}!

Example 8.3. If we link the smooth quintic scroll Z of H(5, −1, 1) with Rao modules H_{∗}^{1}(I_{Z}) = 0, H_{∗}^{2}(I_{Z}) ≃ k and minimal resolution (cf. [11], B.2.1),

0 → R(−5) → R(−4)^{⊕5} → R(−3)^{⊕5} → I_{Z} → 0,  (33)

using a CI of type (5, 6) containing Z, then the ideal of the linked surface X has a minimal resolution

0 → R(−11) → R(−10)^{⊕5} → R(−9)^{⊕10} → R(−8)^{⊕5} ⊕ R(−6) ⊕ R(−5) → I_{X} → 0

and Rao modules given by H_{∗}^{2}(I_{X}) = 0, h^{1}(I_{X}(6)) = 1 and H^{1}(I_{X}(v)) = 0 for v ≠ 6. Using (27) we see that (X) belongs to H(d, p, π) = H(25, 99, 71). This surface X has invariants such that Proposition 8.1 and Remark 8.2 apply. Indeed we can link X to two different surfaces X′ and X′_{0} using CI’s Y and Y_{0} containing X, both of type (6, 8), generated in the following way. Let F_{5}, resp. F_{6}, be the minimal generator of I_{X} of degree 5, resp. 6, and let G be a general element of H^{0}(I_{X}(8)). Then we take Y, resp. Y_{0}, to be given by I_{Y} = (F_{6}, G), resp. I_{Y_{0}} = (H · F_{5}, G) where H is a linear form. We may check that all assumptions of Remark 8.2 are satisfied. Hence we get that X′ and X′_{0} belong to a common irreducible component of H(d′, p′, π′) = H(23, 80, 61), that X′_{0} is obstructed with minimal resolution

0 → R(−8) → R(−7)^{⊕5} ⊕ R(−8) ⊕ R(−9) → R(−6)^{⊕6} ⊕ R(−8) → I_{X′_{0}} → 0,

while X′ is unobstructed with minimal resolution

0 → R(−8) → R(−7)^{⊕5} ⊕ R(−9) → R(−6)^{⊕6} → I_{X′} → 0.

Note that it is straightforward to find these resolutions since X′ and X′_{0} are bilinked to Z and we know the minimal resolution of I_{Z}, see [39] or the sequence (39) appearing later in this paper. We observe that common direct free factors (“ghost terms”) are present in the minimal resolution, similar to what happens for obstructed curve with “small Rao module”, cf. [33]. Moreover since the assumptions of Proposition 4.4 are satisfied for X′, we also get the unobstructedness of X′ from that Proposition and the dimension,

dim_{(X′)} H(23, 80, 61) = 1 + δ^{3}(−5) − δ^{2}(−5) + δ^{1}(−5) = 163.

However, since the conditions of Remark 4.5 (i) also hold, we get H^{1}(N_{X′}) = 0 and hence it is easier to compute dim_{(X′)} H(23, 80, 61) by using Proposition 4.1. We get

dim_{(X′)} H(23, 80, 61) = χ(N_{X′}) = 5(2d′ + π′ − 1) − d^{′2} + 2χ(O_{X′}) = 163.

Note that neither the assumptions of Proposition 4.4, nor the assumptions of Remark 4.5 (i), are satisfied for X′_{0}. Indeed Remark 4.5 (i) a little extended will show h^{1}(N_{X′_{0}}) = 1 (i.e. just compute the dimension using (17)). The surface X′_{0} is easily seen to be reducible, as pointed out to me by H. Nasu.

Example 8.4. If we link the surface X′_{0} of Example 8.3 using a general CI of type (9, 9) containing X′_{0}, we get a smooth obstructed surface S of degree 58. Indeed the assumptions of Remark 7.2 are satisfied. So S is obstructed, and we have used Macaulay 2 ([19]) to verify that S is smooth provided the CI’s used in the linkages of Example 8.3 are general enough under the specified restrictions. The surface S is in the biliaison class of the Veronese surface in ℙ^{4}.

Finally if we link S via a general CI of type (9, 12) containing S, we get an obstructed surface S′ of degree 50 by Remark 7.2. We have used Macaulay 2 to verify that the surface is smooth. The surface S′ is in the biliaison class of the quintic elliptic scroll in ℙ^{4}. Since S′ is bilinked to the surface X′_{0} of Example 8.3 we easily find the minimal resolution of I_{S′} to be

0 → R(−11) → R(−10)^{⊕5} ⊕ R(−11) ⊕ R(−12)^{⊕2} → R(−9)^{⊕7} ⊕ R(−11) → I_{S′} → 0.

Note that we again have “ghost terms” in the minimal resolution in degree c+5 where h^{2}(I_{S′}(c)) ≠ 0. This feature seems to be related to obstructedness, as in the curve case, cf. [33].

9 Even liaison of codimension 2 subschemes of ℙ^{n+2}.

In this section we prove the main even liaison theorem of this paper, which holds for any equidimensional lCM codimension 2 subscheme X of ℙ^{n+2}. We also generalize Proposition 4.4 and the vanishing result for h^{1}(N_{X}) of Remark 4.5 to schemes X of dimension n > 2 and we give an example of an obstructed 3-fold.

First we define δ_{X}^{m}(v). Let

0 → ⨁_{i=1}^{r_{n+2}} R(−n_{n+2,i}) → ⨁_{i=1}^{r_{n+1}} R(−n_{n+1,i}) → ... → ⨁_{i=1}^{r_{2}} R(−n_{2,i}) → ⨁_{i=1}^{r_{1}} R(−n_{1,i}) → I → 0  (34)

be a minimal resolution of I = I_{X} and let the invariant δ^{m}(v) = δ_{X}^{m}(v) be defined by

δ_{X}^{m}(v) = ∑_{j=1}^{n+2} ∑_{i=1}^{r_{j}}(−1)^{j+1}h^{m}(I_{X}(n_{j,i} + v)) .  (35)

Since adding common direct free factors in consecutive terms of (34) does not change δ_{X}^{m}(v), the resolution of I does not really need to be minimal in the definition of δ_{X}^{m}(v).

Theorem 9.1. Let X and X′ be two equidimensional locally Cohen-Macaulay codimension 2 subschemes of ℙ^{n+2}, linked to each other in two steps by two complete intersections, and suppose that (X) (resp. (X′)) belongs to the Hilbert scheme H_{γ,ρ} (resp. H_{γ′,ρ′}) of constant cohomology. Then

i) δ_{X}^{n+1}(−n − 3) − dim_{(X)} H_{γ,ρ} = δ_{X′}^{n+1}(−n − 3) − dim_{(X′)} H_{γ′,ρ′} .

In particular obsumext(X) := 1 + δ_{X}^{n+1}(−n − 3) − dim_{(X)} H_{γ,ρ} is a biliaison invariant.

ii) δ_{X}^{n+1}(−n − 3) − dim _{0}Ext_{R}^{1}(I_{X}, I_{X})_{ρ} = δ_{X′}^{n+1}(−n − 3) − dim _{0}Ext_{R}^{1}(I_{X′}, I_{X′})_{ρ′}.

In particular sumext(X) := 1 + δ_{X}^{n+1}(−n − 3) − dim _{0}Ext_{R}^{1}(I_{X}, I_{X})_{ρ} is a biliaison invariant.

iii) We have sumext(X) ≤ obsumext(X), with equality if and only if H_{γ,ρ} is smooth at (X).

Remark 9.2. This result is motivated by Remarks 3.9 and 6.4. Indeed we were quite convinced that Theorem 9.1 was true before starting proving it. Note that the dimension formula of Remark 6.4 was quite involved already for the case n = dim X = 2 and we expect a very complicated formula for n > 2. So Theorem 9.1 may be a good practical approach to the problem of studying H_{γ,ρ} and Hilb^{p(v)}(ℙ^{n+2}) with respect to smoothness and dimension for n > 1. However, except for the other results of this paper, we have no better option for the use of Theorem 9.1 that to first compute sumext(X) and obsumext(X) through a nice representative in the even liaison class, e.g. for the minimal element of the class, before we use it for an arbitrary element in the even liaison class.

Remark 9.3. For the application of Theorem 9.1 there is one natural situation where H_{γ,ρ} is isomorphic to Hilb^{p(v)}(ℙ^{n+2}) at (X), namely in the case X has seminatural cohomology. We say a subscheme X ⊆ ℙ^{n+2} has seminatural cohomology if for every v ∈ ℤ, at most one of groups H^{0}(I_{X}(v)), H^{1}(I_{X}(v)), ..., H^{n+1}(I_{X}(v)) are non-zero. In this case a generization (i.e. a deformation to more general element in Hilb^{p(v)}(ℙ^{n+2})) of X is forced to have the same cohomology as X by the semicontinuity of h^{i}(I_{X}(v)), i.e. H_{γ,ρ} ≅ Hilb^{p(v)}(ℙ^{n+2}) as schemes at (X).

Proof. Let X be linked to X_{1} by a CI Y ⊆ ℙ^{n+2} of type (f, g) and let X_{1} be linked to X′ by some CI Y′ ⊆ ℙ^{n+2} of type (f′, g′). If (X_{1}) belongs to the Hilbert scheme H_{1} := H_{γ_{1},ρ_{1}} of constant cohomology, then by Theorem 7.1,

dim_{(X_{1})} H_{1} = dim_{(X)} H_{γ,ρ} + h^{0}(I_{X/Y} (f)) + h^{0}(I_{X/Y} (g))
− h^{n}(O_{X}(f − n − 3)) − h^{n}(O_{X}(g − n − 3)),

dim_{(X_{1})} H_{1} = dim_{(X′)} H_{γ′,ρ′} + h^{0}(I_{X′/Y′}(f′)) + h^{0}(I_{X′/Y′}(g′))
− h^{n}(O_{X′}(f′ − n − 3)) − h^{n}(O_{X′}(g′ − n − 3)).

Let h = f′ + g′ − f − g. Using (28) twice we get h^{0}(I_{X′/Y′}(v)) = h^{0}(I_{X/Y} (v − h)). Hence

dim_{(X′)} H_{γ′,ρ′} = dim_{(X)} H_{γ,ρ} + h^{0}(I_{X/Y} (f)) + h^{0}(I_{X/Y} (g))

− h^{0}(I_{X/Y} (f′ − h)) + h^{0}(I_{X/Y} (g′ − h)) + η (36)

where η is defined by

η := h^{n}(O_{X′}(f′ − n − 3)) + h^{n}(O_{X′}(g′ − n − 3)) − h^{n}(O_{X}(f − n − 3)) − h^{n}(O_{X}(g − n − 3)).  (37)

Next we need to find a free resolution of I′ = I_{X′} in terms of the minimal resolution of I = I_{X} in (34). If we define E by the exact sequence

0 → ⊕_{i=1}^{r_{n+2}}R(−n_{n+2,i}) → ... → ⊕_{i=1}^{r_{3}}R(−n_{3,i}) → ⊕_{i=1}^{r_{2}}R(−n_{2,i}) → E → 0,  (38)

we may put (34) in the form 0 → E → ⊕_{i=1}^{r_{1}}R(−n_{1,i}) → I → 0. Then it is well known that there is an exact sequence

0 → E(−h) ⊕ R(−f − h) ⊕ R(−g − h) → ⊕_{i=1}^{r_{1}}R(−n_{1,i} − h) ⊕ R(−f′) ⊕ R(−g′) → I′ → 0  (39)

which combined with (38) yields a free resolution of I′ (see [39]).

We will use this resolution of I′ and (34) to see the connection between δ_{X}^{n+1}(−n − 3) and δ_{X′}^{n+1}(−n − 3). First we need to compute β defined by

β := ∑_{j=1}^{n+2} ∑_{i=1}^{r_{j}}(−1)^{j+1}α(n_{j,i} − n − 3) where α(v) := h^{n}(O_{X′}(v + h)) − h^{n}(O_{X}(v)) .

We claim that

β = h^{0}(I_{X}(f)) + h^{0}(I_{X}(g)) − h^{0}(I_{X}(f′ − h)) − h^{0}(I_{X}(g′ − h)) + h^{0}(I_{X}(−h)).  (40)

Indeed by (28),

α(v) = h^{0}(I_{X_{1}/Y′}(f′ + g′ − n − 3 − v − h)) − h^{0}(I_{X_{1}/Y} (f + g − n − 3 − v)).

Moreover since 0 → I_{Y′} → I_{X_{1}} → I_{X_{1}/Y′} → 0 and 0 → I_{Y} → I_{X_{1}} → I_{X_{1}/Y} → 0 are exact, we get

α(v) = h^{0}(I_{Y} (f + g − n − 3 − v)) − h^{0}(I_{Y′}(f + g − n − 3 − v)).  (41)

Let r(v) := dim R_{(−n−3+v)}. Combining with the minimal resolutions of I_{Y} and I′_{Y}, we get

α(v) := r(f − v) + r(g − v) − r(−v) − r(f′ − h − v) − r(g′ − h − v) + r(−h − v).

Then we get the claim since (34) implies

h^{0}(I_{X}(v)) = ∑_{j=1}^{n+2} ∑_{i=1}^{r_{j}}(−1)^{j+1}r(v − n_{j,i} + n + 3)

for any v and since h^{0}(I_{X}(0)) = 0.

Using the resolution of I′ deduced from (39) and the definition (35) we get

δ_{X′}^{n+1}(−n − 3) = ∑_{j=1}^{n+2} ∑_{i=1}^{r_{j}}(−1)^{j+1}h^{n}(O_{X′}(n_{j,i} + h − n − 3)) + ϵ

where ϵ is defined by

ϵ := h^{n}(O_{X′}(f′ − n − 3)) + h^{n}(O_{X′}(g′ − n − 3))
− h^{n}(O_{X′}(f + h − n − 3)) − h^{n}(O_{X′}(g + h − n − 3)).

Comparing ϵ with η in (37) and recalling the definition of α, we have ϵ = η−α(f −n−3)−α(g−n−3). Moreover the definition of α, the proven claim and (35) lead to δ_{X′}^{n+1}(−n−3) = δ_{X}^{n+1}(−n−3)+β +ϵ. Combining we get

δ_{X′}^{n+1}(−n − 3) = δ_{X}^{n+1}(−n − 3) + β + η − α(f − n − 3) − α(g − n − 3).

Comparing with (36) we get (i) of the Theorem provided we can show that

h^{0}(I_{X/Y} (f)) + h^{0}(I_{X/Y} (g)) − h^{0}(I_{X/Y} (f′ − h)) − h^{0}(I_{X/Y} (g′ − h))
= β − α(f − n − 3) − α(g − n − 3).

Suppose h ≥ 0. Looking at (40), we see it suffices to show

−h^{0}(I_{Y} (f)) − h^{0}(I_{Y} (g)) + h^{0}(I_{Y} (f′ − h)) + h^{0}(I_{Y} (g′ − h)) = −α(f − n − 3) − α(g − n − 3).

Thanks to (41) it remains to show

h^{0}(I_{Y} (f′ − h)) + h^{0}(I_{Y} (g′ − h)) = h^{0}(I_{Y′}(f)) + h^{0}(I_{Y′}(g)).

Using the minimal resolutions of I_{Y} and I_{Y′} and that h = f′ + g′ − f − g ≥ 0, we easily show that both sides of the last equation is equal to dim R_{(f−f′)} + dim R_{(f−g′)} + dim R_{(g−f′)} + dim R_{(g−g′)} and we get what we want, i.e.

δ_{X}^{n+1}(−n − 3) − dim_{(X)} H_{γ,ρ} = δ_{X′}^{n+1}(−n − 3) − dim_{(X′)} H_{γ′,ρ′}  (42)

provided h ≥ 0. Suppose h < 0. Then we can start with X′ and link in two steps back to X, i.e. we get an even liaison with h′ = f + g − f′ − g′ ≥ 0 in which case we know that (42) holds. Hence (42) is proved in general.

To show (ii) of the Theorem we only need to remark that, due to Theorem 7.1, (36) holds if we replace dim_{(X)} H_{γ,ρ} and dim_{(X′)} H_{γ′,ρ′} by the dimension of their tangent spaces _{0}Ext_{R}^{1}(I_{X}, I_{X})_{ρ} and 0Ext_{R}^{1}(I_{X′}, I_{X′})_{ρ′} respectively. With the proof of Theorem 9.1 (i) above, we therefore get (42) with the mentioned replacements, i.e. we get Theorem 9.1 (ii).

Finally Theorem 9.1 (iii) follows by combining (i) and (ii) since e.g. the smoothness of H_{γ,ρ} at (X) is equivalent to dim_{(X)} H_{γ,ρ} = dim _{0}Ext_{R}^{1}(I_{X}, I_{X})_{ρ}.

Corollary 9.4. Let X be an equidimensional lCM codimension 2 subschemes of ℙ^{n+2}, and suppose (X) be a generic point of a generically smooth component V of Hilb^{p(v)}(ℙ^{n+2}). Then sumext(X) = obsumext(X) and

dim V = 1 + δ_{X}^{n+1}(−n − 3) − sumext(X).

Proof. Arguing as the last part of the proof of Theorem 3.7, we get that H_{γ,ρ} is isomorphic to Hilb^{p(v)}(ℙ^{n+2}) at (X). Hence H_{γ,ρ} is smooth at (X). Then we conclude by Theorem 9.1.

Corollary 9.5. Let X be a surface in ℙ^{4}. If the local deformation functors Def(M_{i}) of M_{i} are formally smooth (for instance if _{0}Ext_{R}^{2}(M_{i}, M_{i}) = 0) for i = 1, 2, and if _{0}Ext_{R}^{3}(M_{2}, M_{1}) = 0, then

sumext(X) = obsumext(X).

Proof. By Corollary 5.4 we get that H_{γ,ρ} is smooth at (X) and we conclude by Theorem 9.1 (iii).

Corollary 9.6. Let X be an arithmetically Cohen-Macaulay codimension 2 subschemes of ℙ^{n+2}. Then sumext(X) = obsumext(X) = 0. Moreover,

(i) if n > 0, then X is unobstructed and

dim_{(X)} Hilb^{p(v)}(ℙ^{n+2}) = 1 + δ_{X}^{n+1}(−n − 3) = 1 − δ_{X}^{0}(0) = χ(N_{X}) + (−1)^{n}δ_{X}^{0}(−n − 3),

(ii) if n = 0, then H_{γ} is smooth at (X) and

dim_{(X)} H_{γ} = 1 + δ_{X}^{1}(−3) = 1 − δ_{X}^{0}(0) = h^{0}(N_{X}) + δ_{X}^{0}(−3).

Proof. By Gaeta’s theorem ([14], [15], cf. [2], [3]) X is in the liaison class of a complete intersection Y. Suppose n > 0. Then H_{γ,ρ} ≅ H_{γ} ≅ Hilb^{p(v)}(ℙ^{n+2}) at (X) by [13] or [27], Rem. 3.7, (cf. [49], Thm. 2.1). Thanks to Theorem 9.1 it suffices to show that sumext(Y) = 0, or equivalently that dim _{0}Ext_{R}^{1}(I_{Y} , I_{Y})_{ρ} = 1 + δ_{Y}^{n+1}(−n − 3). By definition, cf. (25), and (5), _{0}Ext_{R}^{1}(I_{Y} , I_{Y})_{ρ} = 0Ext_{R}^{1}(I_{Y} , I_{Y}) = h^{0}(N_{Y}) and it is trivial to show h^{0}(N_{Y}) = 1 + δ_{Y}^{n+1}(−n − 3) by using duality and the minimal resolution of I_{Y}.

Moreover note that for any equidimensional lCM codimension 2 subschemes X of ℙ^{n+2}, we easily show

∑_{i=1}^{n+1} _{0}ext_{R}^{i}(I_{X}, I_{X}) = 1 − δ_{X}^{0}(0) = χ(N_{X}) + (−1)^{n}δ_{X}^{0}(−n − 3).  (43)

as in Proposition 4.1 (see the first sentence of the proof for the left equality and second and third sentence of the proof for the right equality). Hence if X is arithmetically Cohen-Macaulay we get 0ext_{R}^{i}(I_{X}, I_{X}) = 0 for i ≥ 2 and we are done in the case n > 0. The case n = 0 is similar and easier.

Remark 9.7. Corollary 9.6 coincides with [13] if n > 0, and with [18] and [36], Rem. 4.6 if n = 0.

Example 9.8. Let X be the smooth rational surface of H(11, 0, 11) of Example 4.6. Note that X has seminatural cohomology and hence we have H_{γ,ρ} ≅ H(d, p, π) at (X) by Remark 9.3. Moreover I = I_{X} admits a minimal resolution

0 → R(−9) → R(−8)^{⊕3} ⊕ R(−7)^{⊕3} → R(−7)^{⊕2} ⊕ R(−6)^{⊕12} → R(−5)^{⊕10} → I → 0.  (44)

By Example 4.6 we conclude that H_{γ,ρ} ≅ H(d, p, π) is smooth at (X) and that dim_{(X)} H(d, p, π) = 41. However, since X is rational we obviously get 1 + δ_{X}^{3}(−5) = 1 from (44). By Theorem 9.1 we find sumext(X) = obsumext(X) = −40. Now we link twice to get X′, first using a CI of type (5, 5), then a CI of type (5, 6), both times using a common hypersurface of degree 5. Looking at (39) we find a free resolution of I′ = I_{X′} of the form

\begin{matrix}0 → R(−10) → R(−9)^{⊕3} ⊕ R(−8)^{⊕3} → R(−8)^{⊕2} ⊕ R(−7)^{⊕12} ⊕ R(−6) \\ → R(−6)^{⊕10} ⊕ R(−5) → I′ → 0. \end{matrix}  (45)

By (28), h^{2}(O_{X′}) = 15 and h^{2}(O_{X′}(1)) = 1 and we get 1 + δ_{X′}^{3}(−5) = 25. It follows from Theorem 9.1 and Proposition 6.1 that H_{γ′,ρ′} ≅ H(d′, p′, π′) is smooth at (X′) of dimension 1 + δ_{X′}^{3}(−5) − sumext(X) = 65. Compare with Examples 7.3 and 7.5.

Before considering examples of 3-folds, we want to generalize some of the results of section 4. For recent papers on the Hilbert scheme of 3-folds, see [5] and its references. See also [12] for a long list of examples of 3-folds of non general type.

Proposition 9.9. Let X be an equidimensional lCM codimension 2 subschemes of ℙ^{n+2}, let M_{i} = H_{∗}^{i}(I_{X}) for i = 1,...,n and I = I_{X} and suppose

0Hom_{R}(I, M_{1}) = 0 and _{−n−3}Ext_{R}^{n−j}(I, M_{j}) = 0 for every j, 1 ≤ j ≤ n.

Then _{0}Ext_{R}^{2}(I, I) = 0, X is unobstructed and

dim_{(X)} Hilb^{p(v)}(ℙ^{n+2}) = _{0}ext_{R}^{1}(I, I).

E.g. let dim X = 3. Then X is unobstructed and dim_{(X)} Hilb^{p(v)}(ℙ^{5}) = _{0}ext_{R}^{1}(I, I) if, for every i,

H^{1}(I_{X}(n_{1,i})) = H^{3}(I_{X}(n_{1,i} − 6)) = H^{2}(I_{X}(n_{2,i} − 6)) = H^{1}(I_{X}(n_{3,i} − 6)) = 0.  (46)

If in addition

H^{2}(I_{X}(n_{1,i} − 6)) = 0, H^{1}(I_{X}(n_{2,i} − 6)) = 0 and H^{1}(I_{X}(n_{1,i} − 6)) = 0,

for every i, then dim_{(X)} Hilb^{p(v)}(ℙ^{5}) = 1 − δ_{X}^{0}(0) = χ(N_{X}) − δ_{X}^{0}(−6).

Proof. Thanks to [27], Rem. 3.7 (cf. [49], Thm. 2.1), the Hilbert scheme H_{γ} of constant postulation is isomorphic to Hilb^{p(v)}(ℙ^{n+2}) at (X) provided _{0}Hom_{R}(I, M_{1}) = 0. By (3) we get _{0}Ext_{R}^{2}(I, I) = 0 provided_{−n−3}Ext_{𝔪}^{n+1}(I, I) = 0. By (2) and M_{j} ≅ H_{𝔪}^{j+1}(I) we deduce the vanishing of the latter from the assumptions of the proposition. It follows that H_{γ} is smooth at (X) of dimension _{0}ext_{R}^{1}(I, I).

Suppose n = 3. By the definition of _{v}Ext_{R}^{•}(I, −) and (34) we easily prove the vanishing of all Ext_{R}^{•}(I, −)-groups of the first part of the proposition from the explicit vanishings in (46). Moreover due (43), to get the final formula it suffices to show _{0}Ext_{R}^{j}(I, I) = 0 for j = 3, 4. By (3) we must prove _{−n−3}Ext_{𝔪}^{n−j}(I, I) = 0 for j = 0, 1. This is shown in exactly the same way as we did for −n−3Ext_{𝔪}^{n+1}(I, I) = 0, i.e. by using (2) and (34) and we are done.

Remark 9.10. (i) We can also generalize Remark 4.5 to equidimensional lCM codimension 2 subschemes X ⊆ ℙ^{n+2} of higher dimension. Indeed using (5), (2) and (3), see the proof above, we get H^{1}(N_{X}) = 0 provided _{0}Ext_{𝔪}^{3}(I, I) = 0 and _{−n−3}Ext_{𝔪}^{n+1}(I, I) = 0, e.g. provided

0Ext_{R}^{j}(I, M_{2−j}) = 0 for 0 ≤ j ≤ 1 and _{−n−3}Ext_{R}^{n−j}(I, M_{j}) = 0 for 1 ≤ j ≤ n.

Similarly H^{2}(N_{X}) = 0 provided _{0}Ext_{𝔪}^{4}(I, I) = 0 and _{−n−3}Ext_{𝔪}^{n}(I, I) = 0, e.g. provided

0Ext_{R}^{j}(I, M_{3−j}) = 0 for 0 ≤ j ≤ 2 and _{−n−3}Ext_{R}^{n−j}(I, M_{j−1}) = 0 for 2 ≤ j ≤ n.

We can in this way easily get a vanishing criteria for H^{q}(N_{X}) = 0 for every q ≥ 1. (ii) Suppose for instance n = dim X = 3. Then H^{1}(N_{X}) = 0 if, for every i,

H^{1}(I_{X}(n_{2,i})) = H^{2}(I_{X}(n_{2,i} − 6)) = 0,
H^{2}(I_{X}(n_{1,i})) = H^{3}(I_{X}(n_{1,i} − 6)) = 0 and H^{1}(I_{X}(n_{3,i} − 6)) = 0.

Moreover H^{2}(N_{X}) = 0 if, for every i,

H^{1}(I_{X}(n_{3,i})) = 0, H^{2}(I_{X}(n_{2,i})) = H^{1}(I_{X}(n_{2,i} − 6)) = 0 and
H^{3}(I_{X}(n_{1,i})) = H^{2}(I_{X}(n_{1,i} − 6)) = 0.

As in the surface case, if some of the assumptions of Proposition 9.9 or Remark 9.10 are not satisfied, we can find examples of obstructed 3-folds (e.g. X′_{0} in the example below). Note that all assumptions of Proposition 9.9 and Remark 9.10 (ii) are satisfied for X′_{0}, except H^{3}(I_{X′_{0}}(n_{1,i}−6)) = 0 for one i.

Example 9.11. We start with the smooth 3-fold Z ⊆ ℙ := ℙ^{5} of [41] of degree 7 with Ω-resolution

0 → O_{ℙ}^{⊕4} → Ω_{ℙ} (2) → I_{Z}(4) → 0,

where Ω_{ℙ} is the kernel of the map O_{ℙ} (−1)^{6} → O_{ℙ} induced by the multiplication with (X_{0}, .., X_{5}). Note that h^{1}(I_{Z}(2)) = 1. If we link Z, first using a CI of type (4, 4) to get a 3-fold Z′, then a CI of type (6, 7) to link Z′ to X, then X is a 3-fold with properties such that Proposition 8.1 applies. Indeed the ideas of Remark 8.2 also apply except for how we proved _{0}Ext_{R}^{2}(I, I) = 0. By the proof of Proposition 9.9, however, we have _{0}Ext_{R}^{2}(I, I) = 0 for 3-folds provided H^{3}(I_{X}(n_{1,i} − 6)) = H^{2}(I_{X}(n_{2,i} −6)) = H^{1}(I_{X}(n_{3,i} −6)) = 0 for all i. To see that all these H^{i}(I_{X}(j))-groups vanish, we first find the minimal resolution of I_{Z′}. Combining the exact sequence 0 → O_{ℙ} → O_{ℙ} (1)^{6} → Ω_{ℙ}^{∨} → 0 with the mapping cone construction for how we get the resolution of I_{Z′} from the resolution of I_{Z}, we find the minimal resolution

0 → R(−6) → R(−5)^{⊕6} → R(−4)^{⊕6} → I_{Z′} → 0.

Hence H_{∗}^{1}(I_{Z′}) = 0, H_{∗}^{2}(I_{Z′}) = 0 and we get H_{∗}^{3}(I_{X}) = 0, H_{∗}^{2}(I_{X}) = 0 and H_{∗}^{1}(I_{X}) ≃ H^{1}(I_{X}(7)) ≃ k, cf. (27). Now since the Koszul resolution induced by the regular sequence {X_{0}, .., X_{5}} implies that

0 → O_{ℙ} (−6) → O_{ℙ} (−5)^{⊕6} → O_{ℙ} (−4)^{⊕15} → O_{ℙ} (−3)^{⊕20} → O_{ℙ} (−2)^{⊕15} → Ω_{ℙ} → 0

is exact, we can use the mapping cone construction to find the following Ω-resolution,

0 → O_{ℙ} (−9)^{⊕6} → Ω_{ℙ} (−7) ⊕ O_{ℙ} (−7) ⊕ O_{ℙ} (−6) → I_{X} → 0

of I_{X}, leading to the minimal resolution

0 → R(−13) → R(−12)^{⊕6} → R(−11)^{⊕15} → ... → I_{X} → 0.

It follows that all n_{3,i} = 11 in the minimal resolution of I_{X} and hence we see that _{0}Ext_{R}^{2}(I, I) = 0.

Then we proceed exactly as in Example 8.3. Indeed we link X to two different 3-folds X′ and X′_{0} using CI’s Y and Y_{0} containing X, both of type (7, 9), as follows. Let F_{6}, resp. F_{7}, be the minimal generator of I_{X} of degree 6, resp. 7, and let G be a general element of H^{0}(I_{X}(9)). Then we take Y, resp. Y_{0}, to be given by I_{Y} = (F_{7}, G), resp. I_{Y_{0}} = (H · F_{6}, G) where H is a linear form. We may check that all assumptions of Proposition 8.1 are satisfied. Hence we get that X′ and X′_{0} belong to a common irreducible component of Hilb^{p(v)}(ℙ^{5}), that X′_{0} is obstructed with minimal resolution

0 → R(−9) → R(−8)^{⊕6} ⊕ R(−9) ⊕ R(−10) → R(−7)^{⊕7} ⊕ R(−9) → I_{X′_{0}} → 0,

cf. (39), while X′ is unobstructed with minimal resolution

0 → R(−9) → R(−8)^{⊕6} ⊕ R(−10) → R(−7)^{⊕7} → I_{X′} → 0.

Again we have “ghost terms” in the minimal resolution of I_{X′_{0}}. From the resolution we find X′_{0} to be of degree 30 and with Hilbert polynomial

p(v) = 5v^{3} − \frac{67}{2}v^{2} + \frac{247}{2}v − 153.

The 3-fold X′_{0} is reducible. Moreover since the assumptions of Proposition 9.9 are satisfied for X′, we also get the unobstructedness of X′ from that Proposition and the dimension, dim_{(X′)} Hilb^{p(v)}(ℙ^{5}) = 1 − δ_{X′}^{0}(0) = 327. Note that the assumptions of Proposition 9.9 are not satisfied for X′_{0}, due to the existence of a minimal generator of degree 9 of I_{X′_{0}} and the fact h^{3}(I_{X′_{0}}(3)) = 1.

Finally since Remark 7.2 generalizes to 3-folds by [29], Prop. 3.12, one may by linkage obtain infinitely many obstructed 3-folds in the liaison class of X′_{0}.

We will finish this section by finding the Hilbert polynomials of O_{X} and N_{X} for any equidimensional lCM 3-fold in ℙ^{5} of degree d and sectional genus π. If S is a general hyperplane section, we have an exact sequence

0 → O_{X}(v − 1) → O_{X}(v) → O_{S}(v) → 0,

and we easily deduce

p(v) := χ(O_{X}(v)) = \frac{1}{6}dv^{3} + \frac{1}{2}(d + 1 − π)v^{2} + (χ(O_{S}) + \frac{d}{3} + \frac{1 − π}{2} )v + χ(O_{X})  (47)

from (1). Moreover

Proposition 9.12. Let X be an equidimensional lCM 3-fold in ℙ^{5} of degree d and sectional genus π and let S be a general hyperplane section. Then

χ(N_{X}(v)) = \frac{1}{3}dv^{3} + 3dv^{2} + (2χ(O_{S}) + 5(π − 1) + \frac{38}{3}d − d^{2})v + (6χ(O_{S}) + 15(π − 1) + 20d − 3d^{2}).

Proof. Since we have no reference for this formula in this generality we sketch a proof. Indeed we claim that

χ(N_{X}(v)) = χ(O_{X}(v)) − χ(O_{X}(−v − 6)) − d^{2}(v + 3).  (48)

Note that, using (48), we get Proposition 9.12 by combining with (47). To show (48), we follow the proof of Proposition 4.1. In addition to the formulas in (15) (where we only replace ∑_{j=1}^{4} by ∑_{j=1}^{5}) we get

∑_{j=1}^{5}(−1)^{j−1} ∑_{i} n_{j,i}^{3} = 6(1 − π − 2d).

Then we proceed as in (16). We get δ^{0}(v) = −χ(I_{X}(−v − 6)) − χ(O_{X}(v)) + (3 + v)d^{2} for v >> 0 and then the claim.

Example 9.13. Let X be the smooth Calabi-Yau 3-fold of [12], sect. 6, with invariants d = 17, π = 32, χ(O_{X}) = 0 and χ(O_{S}) = 24, and deficiency modules M_{1} = 0, M_{2} = 0 and M_{3} given by

h^{3}(I_{X}(1)) = 4, h^{3}(I_{X}(2)) = 2, h^{3}(I_{X}(v)) = 0 for v ∉ {1, 2}.

Following [12] we find that I = I_{X} has the following minimal resolution

0 → R(−8)^{⊕2} → R(−7)^{⊕8} → R(−6)^{⊕5} ⊕ R(−5)^{⊕2} → I → 0.  (49)

All assumptions of Proposition 9.9 are satisfied and we get that Hilb^{p(v)}(ℙ^{5}) is smooth at (X) of dimension

dim_{(X)} Hilb^{p(v)}(ℙ^{5}) = 1 − δ_{X}^{0}(0) = 82.

Let us compute obsumext(X). Note that X has seminatural cohomology and hence we have H_{γ,ρ} ≅ Hilb^{p(v)}(ℙ^{5}) at (X) by Remark 9.3. Since h^{3}(O_{X}) = 1 and h^{3}(O_{X}(−1)) = 24, it follows that obsumext(X) = 1 + δ_{X}^{4}(−6) − 82 = −28 by Theorem 9.1. Now we link twice to get X′, first using a CI of type (5, 6), then a CI of type (5, 5), both times using a common hypersurface of degree 5. This is possible, cf. [12]. Thanks to (39) we find a free resolution of I′ = I_{X′} of the form

0 → R(−7)^{⊕2} → R(−6)^{⊕8} → R(−5)^{⊕6} ⊕ R(−4) → I′ → 0.  (50)

By (28) h^{1}(O_{X′}(−2)) = 19 and h^{1}(O_{X′}(−1)) = 0 and we get 1 + δ_{X′}^{4}(−6) = 20. It follows from Theorem 9.1 and Proposition 6.1 that H_{γ′,ρ′} ≅ Hilb^{p′(v)}(ℙ^{5}) is smooth at (X′) of dimension

1 + δ_{X′}^{4}(−6) − sumext(X) = 48.

We can also use Proposition 9.9 and check that 1 − δ_{X′}^{0}(0) = 48.

References

[1] H. Abo, K. Ranestad. Construction of rational surfaces of degree 12 in projective fourspace J. Alg. Geom. 15 (2006), 323-338.

[2] R. Apéry, Sur certain caractères numériques d’un idéal sans composant impropre, C.R.A.S. 220 (1945), 234–236.

[3] R. Apéry, Sur les courbes de première espèce de l’espace à trois dimensions, C.R.A.S. 220 (1945), 271–272.

[4] E. Ballico, G. Bolondi. The variety of module structures, Arch. der Math. 54 (1990), 397-408.

[5] G. Besana, M.L. Fania. The dimension of the Hilbert scheme of special threefolds. Communications in Algebra, 33 (2005), 3811-3829.

[6] G. Bolondi. Surfaces in ℙ^{4} and deficiency modules. In: Proc. l’Aquila 1992, Contemporary Math. Vol. 162 (1994), 49-63.

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

[8] G. Bolondi, J. Migliore. The Structure of an Even Liaison Class, Trans. AMS. 316 no.1 (1989), 1-37.

[9] G. Bolondi, J. Migliore. The Lazarsfeld-Rao property on an arithmetically Gorenstein variety. Manuscripta Math. 78 (1993), 347-368.

[10] R.O. Buchweitz, “Contributions a la theorie des singularites,” Thesis l’Université Paris VII (1981).

[11] W. Decker, L. Ein, F.O. Schreyer. Construction of surfaces in ℙ^{4} J. Alg. Geom. 2 (1993), 185-237.

[12] W. Decker, S. Popescu. On Surfaces in ℙ^{4} and 3-folds in ℙ^{5} In: Vector bundles in algebraic geometry. (Durham, 1993) London Math. Soc. Lecture Note Ser., 208, 69-100 Cambridge Univ. Press, Cambridge, 1995.

[13] 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.

[14] F. Gaeta, Sulle curve sghembe algebriche di residuale finito, Annali di Matematica s. IV, t. XXVII (1948), 177–241.

[15] F. Gaeta, Nuove ricerche sulle curve sghembe algebriche di residuale finito e sui gruppi di punti del piano, Ann. di Mat. Pura et Appl., ser. 4, 31 (1950), 1–64.

[16] H.-C. Graf van Bothmer, K. Ranestad. Classification of rational surfaces of degree 11 and sectional genus 11 in ℙ^{4}. arXiv:math/0603567v2 [math.AG] 28 Feb 2007.

[17] 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.

[18] G. Gotzmann. Eine Bedingung für die Flachheit und das Hilbertpolynomeines graduierten Ringes. Math. Z., 158 no. 1 (1978), 61-70.

[19] D. Grayson and M. Stillman. Macaulay 2-a software system for algebraic geometry and commutative algebra, available at http://www.math.uiuc.edu/Macaulay2/ .

[20] 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).

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

[22] R. Hartshorne (notes). Local Cohomology. Lecture Notes in Math. Vol 41, Springer–Verlag, New York, 1967.

[23] R. Hartshorne, M. Martin-Deschamps, D. Perrin. Un theoreme de Rao pour les familles de courbes gauches. Journal of Pure and Applied Algebra 155, no. 1, (2001), 53-76.

[24] G. Horrocks. Vector bundles on the punctured spectrum of a local ring. Proc. London Math. Soc. (3) 14 (1964), 689-713.

[25] T. de Jong and D. van Straten, Deformations of normalization of hypersurfaces, Math. Ann. 288 (1990), 527–547.

[26] S. Katz. Hodge numbers of linked surfaces in ℙ^{4}. Duke Math. J. 55 no. 1 (1987), 89-95.

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

[28] 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.

[29] 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 (1989).

[30] 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.

[31] J.O. Kleppe. The Hilbert scheme of surfaces in ℙ^{4}. Preprint. http://www.iu.hio.no/˜ jank/papers.htm.

[32] J.O. Kleppe. The smoothness and the dimension of PGor(H) and of other strata of the punctual Hilbert scheme. J. Algebra 200 no. 2 (1998), 606–628.

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

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

[35] J.O. Kleppe. Moduli spaces of reflexive sheaves of rank 2. ArXiv:math/0803.1077 [math.AG] 7 Mar 2008. To appear in Canadian J. of Math.

[36] J.O. Kleppe and R.M. Miró-Roig, Dimension of families of determinantal schemes, Trans AMS. 357, (2005), 2871-2907.

[37] A. Laudal. Formal Moduli of Algebraic Structures. Lectures Notes in Math. Vol. 754, Springer–Verlag, New York, 1979.

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

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

[40] R. M. Miró-Roig. On the Degree of Smooth non-Arithmetically Cohen-Macaulay Threefolds in ℙ^{5}, Proc. Amer. Math. Soc. 110 (1990), 311-313.

[41] C. Okonek. Über 2-codimensionale Untermannigfaltigkeiten vom Grad 7 in ℙ^{4} und ℙ^{5}. Math. Z. 187 (1984), 209-219.

[42] C. Okonek. Flächen vom Grad 8 in ℙ^{4}. Math. Z. 191 (1986), 207-223.

[43] S. Popescu. On Smooth Surfaces of Degree ≥ 11 in the Projective Fourspace. Thesis, Univ. des Saar-

landes (1993)

[44] S. Popescu, K. Ranestad: Surfaces of Degree 10 in the Projective Fourspace via Linear systems and Linkage. J. Alg. Geom. 5 (1996), 13-76.

[45] Ch. Peskine, L. Szpiro Liaison des varietes algebrique. Invent. Math. 26 (1974), 271-302.

[46] K. Ranestad: On smooth surfaces of degree ten in the projective fourspace. Thesis, Univ. of Oslo (1988)

[47] A.P. Rao. Liaison Among Curves in ℙ^{3}. Invent. Math. 50 (1979), 205-217.

[48] A.P. Rao. Liaison Equivalence Classes. Math. Ann. 258 (1981), 169-173.

[49] 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