Curves of degree two and ropes on a line: their ideals and even liaison classes

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Journal of Algebra 265 (2003) 772–793

www.elsevier.com/locate/jalgebr

Curves of degree two and ropes on a line:their ideals and even liaison classes

Uwe Nagel,a,∗ Roberto Notari,b and Maria Luisa Spreaficob

a Fachbereich Mathematik und Informatik, Universität Paderborn, D-33095 Paderborn, Germanyb Dipartimento di Matematica, Politecnico di Torino, 1-10129 Torino, Italy

Received 23 June 2002

Communicated by Craig Huneke

Abstract

In this paper we investigate degree two curves of arbitrary codimension. This requires to sturopes supported on a line. After establishing several characterizations of ropes supported owe describe their homogeneous ideals and Hartshorne–Rao modules; in particular we charthe arithmetically Buchsbaum ropes. Then we describe the even Gorenstein liaison classenon-arithmetically Buchsbaum, non-degenerate ropes generalizing a well-know result of Miglidouble lines of codimension two. The result relies on the explicit description of certain arithmeGorenstein curves with maximal tangent spaces at each point. As a consequence, we can deccurves of degree two belong to the same Gorenstein liaison class. Finally, we show as an apphow ropes can be used in order to construct quasi-extremal curves. 2003 Published by Elsevier Inc.

1. Introduction

The curves of smallest degree in projective space are lines where we mean by aa closed locally Cohen–Macaulay subscheme of pure dimension one. Let us consinext case, i.e., curves of degree two. Such a curve is either an irreducible conic, atwo lines, or a double line, the latter being the most interesting. Double lines inP

3 arecompletely described by Ferrand’s construction (cf. [10]). Their liaison classes have

* Corresponding author. Current address: Department of Mathematics, University of Kentucky, 715 PaOffice Tower, Lexington, KY 40506-0027, USA.

E-mail addresses:[email protected] (U. Nagel), [email protected] (R. Notari), spreafico@polito.(M.L. Spreafico).

0021-8693/03/$ – see front matter 2003 Published by Elsevier Inc.doi:10.1016/S0021-8693(03)00237-0

U. Nagel et al. / Journal of Algebra 265 (2003) 772–793 773

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characterized by Migliore in [17]. The goal of this paper is to extend these results to dlines of arbitrary codimension. In order to achieve this we are forced to consider nodouble lines, but more generally ropes supported on a line.

Let Y be a smooth, irreducible curve. Anα-ropeC on Y is a curve which is a multiplicity α structure onY satisfying(IY )2 ⊂ IC ⊂ IY . Ropes have been studied, for exampin [1,7,19]. In this paper, we are interested in ropes whose support is a line.

For the rest of this introduction a rope always means a rope supported on a line.such(n − k)-ropeC ⊂ P

n is uniquely determined by the following data: the supportline L and an injective morphism

ϕBC :k⊕

j=1

OL(−βj − 1) →On−1L (−1)

which does not drop rank (cf. Theorem 2.4).We use this description in order to study the ropes from the point of view of lia

theory. While classically only complete intersection were used in order to link twoschemes directly, recent results suggest that a more satisfactory liaison theory is oby allowing arithmetically Gorenstein schemes to give the direct links (cf., e.g., [5,120]). We adopt this point of view. Thus, liaison always means Gorenstein liaison inpaper.

Observe, that the degree of a non-degenerate arithmetically Gorenstein subschP

n is at leastn + 1. Thus, the residual scheme of a non-degenerate double lindegree� n − 1. This simple observation shows that in contrast to the situation inP

3,it is not sufficient to consider only double lines for determining the even liaison claof double lines. But we show that there are arithmetically Gorenstein curves wlink ropes to ropes. Combining this with our explicit information on the ropes weprove:

Theorem 1.1. Let C,E ⊂ Pn = ProjR be two non-degenerate ropes of the same de

and assume that the arithmetic genus ofC is < (n − 1)/2. Then the following conditionare equivalent:

(a) C andE belong to the same even liaison class.(b) The Hartshorne–Rao modules ofC andE are isomorphic as gradedR-modules.(c) (1) C andE have the same lineL as support;

(2) C andE have the same arithmetic genus;(3) the corresponding mapsϕBC ,ϕBE :

⊕kj=1OL(−βj − 1) → On−1

L (−1) differ by

an automorphism of the targetOn−1L (−1).

Forn = 3 this statement specializes to the main result in [17].Actually, we show a more general result (cf. Theorem 5.4). It says that two rop

the same degree, which are not arithmetically Buchsbaum, belong to the same evenclass if and only if their Hartshorne–Rao modules are isomorphic. While this resin codimension two, a special case of a general theorem of Rao (cf. [27]), it pro

774 U. Nagel et al. / Journal of Algebra 265 (2003) 772–793

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(in higher codimension) some support to the hoped-for generalization of Rao’s theIt is one of the main open problems of liaison theory to decide if this so-called Rcorrespondence is true for Gorenstein liaison (it is not true for complete interseliaison). Positive results in this direction have been obtained mainly for arithmetCohen–Macaulay curves (cf., e.g., [5,13,15,20]) and for arithmetically Buchsbaum c(cf., e.g., [6,13,16]) by using the theory developed in [15], which allows to view Gorenliaison as a theory about divisors on arithmetically Cohen–Macaulay schemeslocally Gorenstein in codimension one, and the classification of arithmetically Buchsdivisors on rational normal scrolls (cf. [23]). Note that we cannot use this approach beropes are typically not generically Gorenstein.

Our cleanest result from a liaison theoretic point of view is Theorem 5.7 wcompletely describes the even liaison classes of curves of degree two.

Families of ropes will be studied in a subsequent paper [25].The paper is organized as follows. In Section 1, after comparing different definitio

general ropes we restrict ourselves to ropes supported on a line. In particular, wea very precise description of the homogeneous ideal. Furthermore, we present gedescriptions of ropes following ideas of C. Segre and J. Harris.

The Hartshorne–Rao module of a rope is computed in Section 3. This result is theto determine the arithmetically Buchsbaum ropes.

Section 4 is devoted to the construction of a family of non-degenerate arithmeGorenstein curves inPn, supported on a line, having degreen + 1 and a tangent spacedimensionn at each point of the support. In Section 5 we use the curves of this famlink ropes to each other. Here we prove our main results on the even liaison classes

The final section is devoted to the construction of curves with maximal cohomoin non-negative degrees, subject to the condition that the general hyperplane senon-degenerate (cf. [8]). Such curves were called quasi-extremal curves in [26]examples have been constructed algebraically. Here we give a geometric construcquasi-extremal curves as suitable unions of an(n − 1)-rope and a plane curve.

Our notation is mainly standard and follows [12]. For definitions, basic resultsextensive background on liaison theory we refer to [18].

2. Characterizations of ropes supported on a line

In this section, we provide some characterizations of anα-rope. Then we specialize tropes supported on a line where we obtain more precise results. Finally, we describas a multiple structure on a scroll following a classical construction by C. Segre.

Throughout this paper we will use the following notation. We putr := n− 2 and denoteby R the polynomial ringK[x0, . . . , xr , t, u] over an arbitrary field. It will be conveniento assume that the supporting lineL ⊂ P

n := Proj(R) is L = Proj(K[t, u]).Now, we begin by recalling the definition of anα-rope following [19]. For a more

general definition applying also to non-embedded curves, we refer to [7, DefinitioLet Y be a smooth and irreducible curve inP

n with saturated homogeneous idealIY (thesupport of the rope), and letC be a subscheme ofP

n with homogeneous idealIC .


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Definition 2.1. A curveC ⊂ Pn is said to be anα-rope supported onY if its homogeneous

ideal IC satisfiesI2Y ⊆ IC ⊆ IY and if C is a locally Cohen–Macaulay multiplicit

α structure onY .

Remark 2.2. It is easy to see that anα-ropeC ⊂ Pn supported onY corresponds naturall

to a rankα − 1 subbundleE of the normal bundleNY of Y via the exact sequence

0 → E∗ →OC → OY → 0.

This also shows that the definitions of anα-rope in [7] and above agree for embeddropes.

Next, we show that the definition above is equivalent to a generalization of the defiof doublinggiven by Boratynski and Greco in [3].

Proposition 2.3. With the notation above, the1-dimensional subschemeC ⊂ Pn is an

α-rope onY if and only if the following conditions are satisfied:

(1) I2Y ⊆ IC ⊆ IY ;

(2) IC is unmixed;(3) l(RIY /ICRIY ) = α.

Proof. It is enough to prove that�(RIY /ICRIY ) · degY is the degree of the ropeC. Butthis follows from the “multiplicity formula” (see, e.g., [21, 23.5]).✷

In [19], the authors describe the idealIC of a ropeC, proving that every(n − k)-rope supported onY is obtained by starting with an idealJ = (I2

Y ,F1, . . . ,Fk) and then,possibly, removing embedded points.

In this paper we are interested in ropes supported on a lineL. The next result shows thain this case embedded points do not occur. Without loss of generality, we can fix thL = V (x0, . . . , xr ) = Proj(S) whereS = K[t, u]. If ϕ :F → G is a graded homomorphisof free modules, we denote byIk(ϕ) the ideal generated by thek-minors of a matrixrepresentingϕ. The same notation is also used ifϕ is a morphism of locally free sheaves

Theorem 2.4. LetC ⊆ Pn be a curve of degree at mostn−1. Then the following condition

are equivalent:

(1) C is an(n − k)-rope supported on the lineL;(2) IC = ((IL)2, [x0, . . . , xr ]B) where the matrixB gives a mapϕB :

⊕kj=1 S(−βj −1) →

Sr+1(−1) with codim(Ik(B)) = 2;(3) IC = ((IL)2,F1, . . . ,Fk) whereV (F1, . . . ,Fk) ⊂ P

n is a scheme of codimensionkwhich containsL and is smooth at the points ofL.


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Proof. (1) ⇒ (2). According to Remark 2.2, the ropeC corresponds to a rankr + 1 − k

subbundleE of the normal bundleNL =Or+1P1 (1) via the exact sequence

0→ E∗ → OC →OL → 0.

Let E = ⊕r−ki=0 OP1(1− αi) whereαi � 0. If we apply Hom(−,O

P1) to the embedding oE into NL, we get the exact sequence

0 →k⊕

j=1

OP1(−βj − 1)

ϕB−−−→ Or+1P1 (−1)

ϕA−−−→r−k⊕i=0

OP1(αi − 1) → 0, (1)

where the ideals of maximal minorsIk(ϕB) and Ir+1−k(ϕA) have codimension 2 (cfe.g., [9, Theorem 20.9, Proposition 18.2]). Moreover, the sequence above providexact sequence ofS-modules

0 →k⊕

j=1

S(−βj − 1)ψ−−→ IL/(IL)2 → IL/IC → 0,

whereψ is the composition⊕k

j=1 S(−βj ,−1)ϕB−−→ Sr+1(−1)

∼=−−→ IL/(IL)2. Thus, Imψ

is generated by the residue classes of the entries of[x0, . . . , xr ]B. SinceIC = (IL)2 + J

for some idealJ ⊂ IL, we can writeJ as([x0, . . . , xr ]B) as claimed.(2) ⇒ (3). Now, we want to verify that[x0, . . . , xr ]B is a regular sequence and defin

a scheme which is smooth at every pointp ∈ L. The Jacobian matrixJ of [x0, . . . , xr ]B is

J =[Bt,

∂Bt

∂t[x0, . . . , xr ]t , ∂Bt

∂u[x0, . . . , xr ]t

].

Since the idealIk(ϕB) of the maximal minors ofB has codimension 2,B has rankk atevery point not inV (t, u), i.e.,V ([x0, . . . , xr ]B) is smooth onL.

(3) ⇒ (1). This follows by [19, Corollary 1.7 and Proposition 1.9].✷Remark 2.5.

(i) If we apply Hom(−,OP1(−2)) to the sequence (1) above, we obtain the sequenc

0 →r−k⊕i=0

OP1(−αi − 1)

ϕ∗A−−→Or+1

P1 (−1)ϕ∗

B−−→k⊕

j=1

OP1(βj − 1) → 0,

i.e., At is just the syzygy matrix ofBt . It follows that, via the sequence (1), theare two matricesA,B associated to the(n − k)-rope onL giving the morphismsϕA :Or+1

P1 (−1) → ⊕r−ki=0 OP1(αi − 1) and ϕB :

⊕kj=1OP1(−βj − 1) → Or+1

P1 (−1)

such that the ideals of the maximal minors ofA,B have codimension 2 and satisA · B = 0 (cf. also Proposition 2.9).


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(ii) Throughout the paper we will use the notation of the last theorem and its proparticular, the matricesA andB will be used to describe properties of the ropeC.Sometimes we will writeAC and BC in order to stress the dependency of thmatrices on the ropeC.

(iii ) We may order the formsFi such that degFi = βi + 1.(iv) The onlyn-rope supported on a lineL is the first infinitesimal neighbourhood ofL.

If we takeB as the zero matrix andA as the unit matrix of orderr + 1, we also get inthis case the exact sequence (1) above and the same description of the homoideal as in the theorem.

Lemma 2.6. We have

r−k∑i=0

αi =k∑

j=1

βj .

Proof. This is a standard consequence of the exact sequence (1) and the additivityChern classes.✷

Now, we determine the Hilbert polynomial of an(n − k)-rope supported onL.

Proposition 2.7. LetIC = ((IL)2,F1, . . . ,Fk) be the homogeneous ideal of an(n−k)-ropeC ⊂ P

n supported onL and letβi + 1 be the degree ofFi . Then the Hilbert polynomial oC is

PC(z) = (n − k)z + 1+k∑

j=1

βj .

Proof. This follows immediately by Theorem 2.4 and [19, Theorem 1.5].✷Corollary 2.8. Let C ⊂ P

n be a rope supported onL having degreed and (arithmetic)genusg. Then we haveg � 0. Furthermore, ifC is non-degenerate theng � d − n.

Proof. C is degenerate if and only ifβj = 0 for somej . But k = n − d > −g = ∑kj=1 βj ,

we must have thatβj = 0 for somej . ✷Now, we investigate when two different ideals define the same rope supported

line L. To this end consider the two idealsJ1 = ((IL)2, xB1) andJ2 = ((IL)2, x ′B ′2), where

the entries ofx andx′ generateIL, respectively, such that the two ideals define(n − k)-ropes supported on the lineL. We may assume thatx = [x0, . . . , xr ]. Furthermore, aftea base change inOr+1

P1 (−1), we can also assumex = x′ while replacingB ′2 by B2 = FB ′

2whereF ∈ GLK(r + 1). Then we have

Proposition 2.9. The idealsJ1 = ((IL)2, xB1) and J2 = ((IL)2, xB2) define the sam(n − k)-ropeC if and only ifB1 = B2Q where the matrixQ gives a graded automorphisof

⊕kj=1OP1(−βj − 1).


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Proof. We have thatJ1 = J2 if and only if B1 andB2 can be obtained from each otherelementary graded operations on the columns, i.e., if we have a graded automorphQ

of⊕k

j=1OP1(−βj − 1). ✷As the last topic of this section, we present geometric descriptions of ropes sup

on a line following ideas of Segre [28] and Harris [14], respectively.

Remark 2.10.(i) (Ropes as a multiple structure on a scroll(cf. [28]).) In Theorem 2.4, we associate

a mapϕA :Or+1P1 (−1) → ⊕r−k

i=0 OP1(αi − 1) to every(n − k)-rope supported on a lin

L ∼= P1. Each rowAi of A can be thought as a mapAi :L → P

r . Thus, these rowdefiner + 1 − k rational (not necessarily normal) curvesγ0, . . . , γr−k in P

r . Accordingto Remark 2.5, we have that the pointsA0(t, u), . . . ,Ar−k(t, u) are linearly independenfor each(t, u) ∈ P

1. Let V ⊆ Pr be the scroll which is the union of the(r − k)-planes

〈A0(t, u), . . . ,Ar−k(t, u)〉 spanned by the pointsA0(t, u), . . . ,Ar−k(t, u) where (t, u)

varies inP1. Note that dimV = r + 1− k. Now, we embedL ∼= P

1 andPr in P

r+2 = Pn in

such a way that they are skew linear spaces. Hence, we can assume thatP1 = Proj(K[t, u]),

Pr = Proj(K[x0, . . . , xr ]) andP

n = Proj(K[x0, . . . , xr , t, u]). Let X ⊂ Pn be the scroll of

dimensionr +2−k (= n−k) which is the union of the linear spaces spanned by the pin L (corresponding to(t, u)) and〈A0(t, u), . . . ,Ar−k(t, u)〉 ⊂ P

r . The homogeneous ideof the scrollX is IX = IV + ([x0, . . . , xr ]B) whereB is the syzygy matrix ofA. Then, an(n − k)-ropeC is the first infinitesimal neighbourhood ofL in X, i.e., IC = IX + (IL)2.But IX + (IL)2 = ((IL)2, [x0, . . . , xr ]B) because of the following result.

Claim. V is degenerate if and only ifβj = 0 for somej .

In fact,V is contained in a hyperplaneH if and only if Aj(t, u) ∈ H for every(t, u) ∈ L,and for eachj = 0, . . . , r − k. If we set H = V (

∑bhxh), the last condition become∑

Ajh(t, u)bh = 0 for everyj = 0, . . . , r − k, i.e.,A has a degree 0 syzygy and the lineform definingH is contained in the ideal([x0, . . . , xr ]B).

(ii) (Ropes as generalized double lines.) In [14], Harris describes a double line inP3 byspecifying the underlying lineL and for each pointP ∈ L a normal direction toL by a mapL → P

3/L. The discussion above shows that an(n − k)-rope inPn is given by specifying

the underlying lineL and for each pointP ∈ L a linear space of normal directions toL

having dimensionk. This is done by giving a morphism

ϕB :k⊕

j=1

OL(−βj − 1) →Or+1L (−1)

which is injective at all points ofL.


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3. The Hartshorne–Rao module of the ropes

The goal of this section is to compute the Hartshorne–Rao module of a ropeC ⊂ Pn

supported on a lineL. Then we single out the arithmetically Buchsbaum ropes.Recall that there are two matricesA,B associated to each such rope whose entries

in S = K[t, u] ∼= R/IL (cf. Remark 2.5). Whereas the matrixB is visible in the definingequations ofC, the matrixA determines its Hartshorne–Rao module.

Proposition 3.1. LetC ⊂ Pn be an(n−k)-rope supported on the lineL with homogeneou

idealIC = ((IL)2, [x0, . . . , xr ]B). Then its Hartshorne–Rao moduleH 1∗ (IC) is isomorphic

asR-module tocoker(Sr+1(−1)ϕA−−→ ⊕r−k

j=0 S(αj − 1)).

Proof. If k = 0 thenC is arithmetically Cohen–Macaulay. Thus, the result is true incase (cf. Remark 2.5(iv)).

Assumek > 0. We have the following exact sequences ofR-modules

0 →k⊕

j=1

R/IL(−βj − 1)ϕB−−→ (R/IL)r+1(−1)

ϕA−−→r−k⊕i=0

R/IL(αi − 1) → cokerϕA → 0

and

0 →k⊕

j=1

R/IL(−βj − 1)ϕB−−→ (R/IL)r+1(−1) → IL/IC → 0

(cf. the proof of Theorem 2.4). Note, that the Koszul resolution ofR/IL providesExtn−1

R (R/IL,R) ∼= R/IL(n − 1). Thus, dualizing the second sequence with respeR, we obtain the exact sequence

· · · → (R/IL)r+1(n)ϕ∗

B(n−1)−−−−→k⊕

j=1

R/IL(βj + n) → Extn−1R (IL/IC,R) → 0.

It implies Extn−1R (IL/IC,R) ∼= cokerϕ∗

B(n − 1).Dualizing the first sequence with respect toS ∼= R/IL, we get by local duality:

cokerϕA∼= Ext2S

(cokerϕ∗

B,S) ∼= (

cokerϕ∗B

)∨(2),

whereM∨ denotes the gradedK-dual Hom(M,K) of the gradedR-moduleM. Usinglocal duality again, we obtain

H 1m(IL/IC) ∼= Extn−1

R (IL/IC,R)∨(n + 1) ∼= cokerϕA.

Finally, sinceL is arithmetically Cohen–Macaulay, the exact sequence

0 → IL/IC → R/IC → R/IL → 0


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ofs

Raondanted

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s

implies

H 1∗ (IC) ∼= H 1m(R/IC) ∼= H 1

m(IL/IC).

Now, our claim follows. ✷Corollary 3.2. The Rao functionρC(i) := h1(IC(i)) of an(n− k)-ropeC ⊂ P

n supportedon L is

ρC(i) =r−k∑j=0

(i + αj

1

)+

k∑j=1

(i − βj

1

)− (r + 1)

(i

1

).

Proof. This follows directly from the exact sequence (1) and Proposition 3.1.✷The Hartshorne–Rao module of double lines is particularly simple.

Corollary 3.3. LetC ⊂ Pn be a double line. Then its Hartshorne–Rao module is gener

by one element.

Proof. For a double lineC we haven− k = 2, thusr − k = 0. Hence, the Hartshorne–Rmodule ofC is isomorphic toS(α0−1)/(A) where(A) is the ideal generated by the entriof the matrixA. ✷

Finally, we investigate properties of ropes with a given Hartshorne–Rao module.now on it is understood that the ropes we are studying are always supported on a lin

Corollary 3.4. Let M be a gradedR-module of finite length. Then the arithmetic genusa ropeC ⊂ P

n, whose Hartshorne–Rao module is isomorphic toM asR-module, dependonly onM.

Furthermore, if in additionC is non-degenerate, then also the degree ofC is uniquelydetermined byM.

Proof. According to Proposition 3.1, a minimal free resolution of the Hartshorne–moduleM(C) asS-module can be obtained from the sequence (1) by cancelling redufree summands. It follows that the shifts� −2 in the last free module in the associatsequence (1) toC are uniquely determined by the isomorphism class ofM(C). Hence, thefirst claim follows by Proposition 2.7.

If C is non-degenerate, then all the entries of the matrixBC are positive by Theorem 2.4Thus, the shifts in the last free module in the associated sequence (1) toC agree with theones of the last free module of the minimal free resolution ofM. SinceM(C) ∼= M itfollows that the size of the matrixBC is determined byM. Therefore, Theorem 2.4 showthe second claim. ✷


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Remark 3.5. Even if the ropeC is possibly degenerate there is some information ondegree ofC by knowing its Hartshorne–Rao module. In fact, let

0 → F2 → F1 → F0 → M → 0

be a minimal free resolution ofM asS-module. Then there is the following estimatethe degree ofC:

1+ ranks F0 � degC � n − ranks F2.

Proposition 3.1 shows that the Hartshorne–Rao module of a rope inPn is annihilated by

at leastn − 1 linearly independent linear forms. The next result describes when themore such linear forms.

Proposition 3.6. LetC ⊂ Pn be a rope which is not arithmetically Cohen–Macaulay. Th

the following conditions are equivalent:

(1) There are at leastn linearly independent linear forms annihilating the HartshornRao moduleM(C) of C;

(2) C is arithmetically Buchsbaum andM(C) ∼= Km for some positive integerm � n−12 .

Moreover, in this caseC has genus−m and is cut out by quadrics and possibly lineforms. IfC is non-degenerate thenC has degreen − m.

Proof. (1) ⇒ (2). Let l be a linear form annihilatingM(C), but not being contained inIL.Then, M(C) ∼= cokerϕA is an S/lS-module. Therefore, its minimal free resolutionmodule overT := S/lS has the form

0 →m⊕

j=1

T (−δj ) →m⊕

i=1

T (−γi) → M(C) → 0

for some positive integerm. Thus, the minimal free resolution asS-module has the shap

0→m⊕

j=1

S(−δj − 1) →m⊕

j=1

S(−δj ) ⊕m⊕

i=1

S(−γi − 1) →m⊕

i=1

S(−γi) → M(C) → 0.

The exact sequence (1) provides the following free resolution ofM(C):

0 →k⊕

j=1

S(−βj − 1) → Sr+1(−1) →r−k⊕i=0

S(αi − 1) → M(C) → 0.

The comparison of the two resolutions showsδj = 1, γi = 0 for all i and j , and 2m �r + 1= n − 1. SinceT is a polynomial ring in one variable, it follows thatM(C) ∼= Km asclaimed.


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ne

(2) ⇒ (1) is trivial.In order to show the final claim, note that we may assumeβ1 = · · · = βm = 1 and

βm+1 = · · · = βk = 0. But the genus ofC is −m = −β1 − · · · − βk (cf. Corollary 2.7).Moreover, Theorem 2.4 shows thatIC is generated bym quadrics outside of(IL)2 andk − m linear forms ifC has degreen − k. This completes the proof.✷

Observe that not all non-degenerate ropes inPn, which are cut out by quadrics an

satisfyd − g = n, are arithmetically Buchsbaum. For example, for anyn � 4 there is anon-degenerate rope of degreen − 2 and genus 2 which is cut out by quadrics, but whHartshorne–Rao module is isomorphic toS/(t, u)2.

Corollary 3.7. LetC ⊂ Pn be a rope with genusg < (n−1)/2. ThenC is not arithmetically

Buchsbaum.

Proof. This is an immediate consequence of Proposition 3.6.✷Later on, we will need the following result.

Corollary 3.8. LetC,E ⊂ Pn be(n− k)-ropes. IfC is not arithmetically Buchsbaum, the

the Hartshorne–Rao modules ofC andE are isomorphic asR-modules if and only ifCandE are supported on the same lineL andAE = QACP whereQ is a matrix providinga graded automorphism of

⊕r−ki=0 S(αi − 1) andP ∈ GLK(n − 1).

Proof. Proposition 3.6 guarantees that the linear forms inR annihilatingM(C) ∼= M(E)

generate the ideal of a lineL. Thus, Proposition 3.1 shows thatL must be the support ofCandE.

SinceC andE have the same degree, the matricesAC andAE have the same sizeHence, the second claim follows again by Proposition 3.1.✷

4. Families of arithmetically Gorenstein curves

In this section, we describe a familyFn of arithmetically Gorenstein curves inPn ofdegreen + 1 supported on a fixed line for eachn � 3 with the property that every ropin P

n of degreed supported on the lineL can be linked by means of every curve in tfamily Fn to a rope supported onL having degreen+1−d . The familyFn depends on thechoice of the supporting line. As in the previous sections, we will fix the supporting liL

defined byx0 = · · · = xn−2 = 0, i.e.,L = ProjS whereS = K[t, u]. In order to achievethe goals just described, we have to require that each curveX ∈ Fn satisfies the followingconditions:

(i) X is supported on the lineL;(ii) deg(X) = n + 1;(iii ) the tangent spaceTx,X to X atx has dimensionn for every pointx ∈ L.


re dualgrees of

ve

ofory of

rve

Let

ustg

n

rix

ized in

In fact, the first condition is clear, because the ropes are supported on the lineL. The secondrequirement stems from the fact that the Hartshorne–Rao modules of linked ropes ato each other. Thus, comparing with the sequence (1) we see that the sum of the dethe two ropes should ben + 1. To explain the third condition, we notice that a curveX canbe used to link a curveC to another curveD if for each pointx ∈ C we haveTx,C ⊆ Tx,X.But, we want to use a curveX ∈ Fn to link whatever ropeC we choose. Hence, we hato require that the tangent spaceTx,X has the maximal dimensionn for everyx ∈ L. Asstarting point for the construction of the familyFn, we compute the generic initial idealthe curves in the family with respect to the reverse lexicographic order. For the thegeneric initial ideals we refer to [9,11].

Proposition 4.1. Let X ⊂ Pn be a non-degenerate arithmetically Cohen–Macaulay cu

of degreen + 1. Assume that the characteristic of the ground fieldK is not two. Thenthe generic initial idealgin(IX) of X is generated by the monomialsx3

r and xixj with0 � i, j � r, (i, j) �= (r, r).

Proof. Let Γ be the general hyperplane section ofX. SinceX is arithmetically Cohen–Macaulay, ginIX and ginIΓ are generated by the same monomials. Moreover,Γ ⊂ P

n−1 isnon-degenerate. Thus, its Hilbert function isHΓ = (1, n,n + 1, n + 1, . . .). It follows thatrank(IΓ )2 = (

r2

) − 1. Since ginIΓ is a stable ideal, it must contain the largest(r2

) − 1monomials of degree 2, i.e., gin(IΓ ) contains the monomialsxixj with 0 � i, j � r.(i, j) �= (r, r). (For this argument we need the assumption on the characteristic.)Jdenote the ideal generated by these monomials. Then we get rankK(J )3 = rankK(IΓ )3−1.Therefore, ginIΓ has exactly one minimal generator of degree 3. By stability it mbe x3

r . SinceJ + x3r has the same Hilbert function asIΓ , both ideals are equal provin

the claim. ✷We now use this result to determine all arithmetically Gorenstein curvesX ⊂ P

n

satisfying the conditions (i)–(iii) above. Condition (i) yields thatIX ⊂ IL. The propositionabove shows thatIX is generated by polynomials of the form

fij = xixj − hij x2r + x0l

0ij + · · · + xr l

rij and f = x3

r + q0x0 + · · · + qrxr,

wherehij ∈ K, lhij ∈ (S)1 for every i, j, h, and q0, . . . , qr ∈ (S)2. Then, the Jacobia

matrix of IX contains the formslhi,j and qk as entries. But it must be the zero mat

by condition (iii). It follows thatIX is generated by{xixj − hij x2r , x3

r | 0 � i � j � r,(i, j) �= (r, r)}. The Gorenstein ideals among the ideals of this shape are characterthe next result. It does not depend on the characteristic ofK.

Theorem 4.2. Let IX ⊂ R be the ideal generated byx3r andxixj − hij x2

r wherehij ∈ K

and 0 � i � j � r, (i, j) �= (r, r) and letG = [gij ] be the symmetric(r + 1) × (r + 1)

matrix defined by

gij =

hij if i � j, (i, j) �= (r, r),

hji if i � j, (i, j) �= (r, r),

1 if i = j = r.


e

idealsss, we

the

Then,IX is the ideal of an arithmetically Gorenstein curve if and only if the matrixG isnon-singular.

Furthermore, in this casex3r is not a minimal generator ofIX .

Proof. Put T := K[x0, . . . , xn] ∼= R/(t, u)R. Then the ringR = T/IXT is an artinianreduction of the ringR/IX . Let m be the maximal homogeneous ideal of the ringR. Ofcourse,m is generated byx0, . . . , xr . Sincem3 = 0, the idealIX is Gorenstein if and onlyif the multiplication map:

φ :m/m2 × m/m2 → m2

is non-degenerate (cf. [2]). By the definition ofIX , m/m2 is generated by{x0, . . . , xr} andm2 is generated byx2

r . Hence, we obtain:

φ(xi, xj ) = gij x2r .

This means that the matrix associated to the mapφ is exactly the matrixG. The conclusionfollows. ✷Remark 4.3. Let x denote the vector[x0, . . . , xr ]. The matrixx2

r G can be obtained fromthe productxt · x by reduction moduloIX , where, as usual,xt is the transpose of thmatrixx.

Definition 4.4. We denote byFn the set of arithmetically Gorenstein curvesX ⊂ Pn whose

homogeneous idealIX is generated byxixj − hij x2r wherehij ∈ K and 0� i � j � r,

(i, j) �= (r, r).

We think ofFn as a family of curves depending on the field elementshij . The curvesin F4 are arithmetically Gorenstein of codimension 3. Hence, their homogeneousare generated by the Pfaffians of a skew-symmetric matrix (cf. [4]). For completenecompute this matrix.

Proposition 4.5. Up to a change of coordinates, the minimal free resolution ofhomogeneous idealIX of a curveX ∈F4 is

0 → R(−5) → R5(−3)E→ R5(−2) → IX → 0,

where the skew-symmetric matrixE is

E =

0 0 cy2 y1 by20 0 y1 y2 y0

−cy2 −y1 0 y0 ay2−y1 −y2 −y0 0 0

−by2 −y0 −ay2 0 0


xis

im

om a

ve.

and y0 = x0 − h02x2, y1 = x1 − h12x2, y2 = x2, anda, b, c, are suitable elements inKdepending onhij ’s.

Proof. First, we explicitly write the generators of the idealIX as in Proposition 4.2:

f1 = x20 − h00x

22, f2 = x0x1 − h01x

22, f3 = x2

1 − h11x22,

f4 = x0x2 − h02x22, f5 = x1x2 − h12x

22, f6 = x3

2.

Second, after performing the following change of coordinates

y0 = x0 − h02x2, y1 = x1 − h12x2, y2 = x2, t ′ = t, u′ = u,

we can choose as new set of generators of the idealIX

f1 − 2h02f4 = y20 − ay2

2, f2 − h12f4 − h02f5 = y0y1 − by22,

f3 − 2h12f5 = y21 − cy2

2, f4 = y0y2, f5 = y1y2, f6 = y32,

wherea = h00 − h202, b = h01 − h02h12, c = h11 − h2

12 The determinant of the matriassociated to the new set of generators ofIX (in the sense of Proposition 4.2)ac − b2 �= 0. Moreover,f6 is not a minimal generator forIX , becausey2f1 −y0f4 +af6 =0 andy2f2 − y1f4 + bf6 = 0, buta andb cannot be both equal to zero. Now, the clafollows from the computation of the Pfaffians of the matrixE. ✷

5. Even Gorenstein liaison classes of ropes

In this section we will combine all the previous results in order to study ropes frliaison theoretic point of view.

Let C ⊂ Pn be an(n − k)-rope supported on the lineL and defined by the idealIC =

((IL)2,F1, . . . ,Fk) = ((IL)2, xBC) and letX ∈ Fn be an arithmetically Gorenstein cursupported onL with homogeneous idealIX and associated matrixG (cf. Proposition 4.2)We have thatIX ⊆ (IL)2 ⊆ IC . Thus, we can Gorenstein linkC via X. We want todetermine the residual curveD. The first step is.

Lemma 5.1. Let D ⊂ Pn be the curve linked to the(n − k)-ropeC via X ∈ Fn. ThenD is

a (k + 1)-rope.

Proof. By definition we have thatID = IX : IC . Notice that(IL)2 ⊆ ID since(IL)3 ⊆ IX

andFi · (IL)2 ⊆ (IL)3, for everyI = 1, . . . , k. It follows by Proposition 2.3 thatD is a(k + 1)-rope. ✷

According to Theorem 2.4, we can write the homogeneous ideal ofD as ID =((IL)2, xBD) for a suitable matrixBD . Now, we want to relate the matrixBD to the matrixG associated toX and to the matrixAC (cf. Remark 2.5).


of

,

mnsnsded

o

e

m.

Proposition 5.2. LetC (respectivelyD) be an(n−k)-rope(respectively(k +1)-rope) withideal IC = ((IL)2, xBC) (respectivelyID = ((IL)2, xBD)) supported onL. ThenC andD

are directly linked by a Gorenstein curveX of the familyFn if and only ifBD = G−1AtCP ,

whereG is the matrix associated toX, the matrixP provides a graded automorphism⊕r−ki=0 S(αi − 1) andS = K[t, u].

Proof. Let us denote thei-th column of a matrixM by (M)i . Then, using Theorem 2.4we obtain from the definition of linked schemes:

(x(BC)i

)(x(BD)j

) = 0 (modIX) for all i, j.

We can write all these equations as

(xBC)t (xBD) = 0 (modIX).

It follows by Remark 4.3 that

(BC)t ( x )t xBD = (BC)tx2r GBD = 0 (modIX).

Since the entries ofBC , BD , G are inS we get

(BC)tGBD = 0.

This shows that the columns ofGBD are syzygies of the module generated by the coluof (BC)t . Since the ideal of maximal minors ofGBD has codimension 2, the columof GBD and (AC)t generate isomorphicS-modules. This means, that there is a graautomorphism of

⊕r−ki=0 S(αi −1) given by a matrixP such that(GBD)t = P tAC , proving

our claim. ✷As a consequence, we get a relation between two ropesC andE which are bilinked.

Corollary 5.3. Two (n − k)-ropes C and E in Pn are linked in two steps by tw

arithmetically Gorenstein curvesX1 and X2 of the familyFn if and only if AE =QACG−1

1 G2, whereGi is the matrix associated toXi , i = 1,2, andQ provides a graded

automorphism of⊕r−k

i=0 S(αi − 1).

Proof. By assumption, there is a(k + 1)-ropeD such thatC andD are directly linked byX1 ∈ Fn andE andD are directly linked byX2 ∈ Fn. According to Proposition 5.2, whave thatBD = G−1

1 (AC)tP1 = G−12 (AE)tP2. The claim follows. ✷

Now, we can prove our main result.

Theorem 5.4. Let C,E ⊂ Pn be (n − k)-ropes which are not arithmetically Buchsbau

Then, the following conditions are equivalent:

(1) C andE are in the same even Gorenstein liaison class;


laryes to

d3

ngmatrixlatter

.

esults

nsion

ao’s

at two

(2) the Hartshorne–Rao modulesM(C) and M(E) of C and E are isomorphic asR-modules.

Proof. It is a general fact of liaison theory that (1) implies (2) (cf., e.g., [22, Corol5.13]). Note, that in our special situation no degree shift is needed. Thus, it sufficshow the converse.

Our assumption ensures that Corollary 3.8 applies. It shows thatC and E aresupported on the same lineL andAE = QACP whereQ is a matrix providing a gradeautomorphism of

⊕r−ki=0 S(αi − 1) and P ∈ GLK(n − 1). Hence, using Corollary 5.

successively our assertion follows from the following fact.

Claim. GLK(n − 1) is equal to its subgroupH generated by{G−11 G2 | Gi is associated to

an arithmetically Gorenstein curveXi ∈Fn (cf. Theorem4.2)}.

Every matrix P ∈ GLK(n − 1) is the product of invertible matrices describielementary column operations. Thus, it is enough to prove that each such elementaryis in H . By considering block diagonal matrices we see that it suffices to show thestatement for 2× 2 matrices.

There are the following three types of 2× 2 elementary matrices:

M1 =(

0 11 0

), M2 =

(1 00 x

), M3 =

(1 0x 1

),

where we may assume thatx �= 0. Moreover, it is clear that the identity matrix is inH .Hence, all matrices of the form( x 0

0 1) wherex �= 0 belong toH as well (cf. Theorem 4.2)Therefore, our claim follows from the following identities:

M1 =(

a b

b 1

)−1 (a a + b − b2

a + b − b2 1

)(a 1− b

1− b 1

)−1

,

M2 = M1

(x 00 1

)M1,

M3 =(

a b

b 1

)−1 (b(a + xb)/(b + x) b

b 1

)(b/(b + x) 0

0 1

)−1

. ✷Remark 5.5.

(i) Let C be a rope which is not arithmetically Buchsbaum. Then the previous rcharacterizes all the ropes having degree degC which are in the even liaison clasof C.

(ii) It is shown in [22, Theorem 3.10], that each even liaison class of codimec subschemes determines a stable equivalence class of certain(c + 1)-syzygies.In codimension two this map is in fact injective and provides the so-called Rcorrespondence. It is not known if the same result is true in codimension� 3. Forcurves the hoped—for Rao’s correspondence is equivalent to the statement th


rphic.

equenton in[24,sional

les arebaumto be

re notlence

.

2.9

1)

.

Rao

, thus

of of

3.6ume

inked

curves are evenly linked if and only if their Hartshorne–Rao modules are isomoOur Theorem 5.4 points towards that direction.

Remark 5.6. Some of the methods developed here have been extended in our subspaper [24] in order to extend Theorem 5.4 in two directions. First, the assumptiTheorem 5.4 that the curvesC and E have the same degree can be dropped (cf.Theorem 5.6]). Second, we can allow arithmetically Buchsbaum curves with 1-dimenHartshorne–Rao module (cf. [24, Proposition 4.7]). As a consequence, two ropes inP

4 arein the same even Gorenstein liaison class if and only if their Hartshorne–Rao moduisomorphic. It remains open if this statement is also true for arithmetically Buchscurves inP

n with n � 5 though [24, Proposition 4.10] reduces the number of caseschecked.

We are now ready for the proof of the theorem stated in the introduction.

Proof of Theorem 1.1. The assumption on the genus guarantees that the ropes aarithmetically Buchsbaum by Corollary 3.7. Thus, it only remains to show the equivaof the conditions (b) and (c).

Let us first assume thatM(C) ∼= M(E). Then condition (2) follows by Corollary 3.4Furthermore, Corollary 3.8 shows condition (1) andAE = QACP whereQ is a matrixproviding a graded automorphism of

⊕r−ki=0 S(αi − 1) andP ∈ GLK(n − 1). SinceBC and

BE are syzygy matrices ofAC andAE , respectively, we can assume by PropositionthatBE = P−1 · BC , i.e., condition (3) is satisfied.

Second, we show that conditions (1)–(3) implyM(C) ∼= M(E). By Proposition 3.1, thelatter is equivalent to cokerϕAC

∼= cokerϕAE . But this follows from the exact sequence ((cf. also Remark 2.5). ✷

For double lines we can improve Theorem 5.4.

Theorem 5.7. Let C,E ⊂ Pn be two curves of degree two. ThenC and E belong to the

same even liaison class if and only if their Hartshorne–Rao modules are isomorphic

Proof. Again, it suffices to show the sufficiency of the condition on the Hartshorne–modules for being evenly linked.

If C andE are arithmetically Cohen–Macaulay they are complete intersectionsthe result is known.

If C andE are not arithmetically Buchsbaum, Corollary 3.8 applies. Thus, the proTheorem 5.4 shows thatC andE belong to the same even liaison class.

Finally, assume thatC and E are arithmetically Buchsbaum. Then Propositionimplies thatM(C) ∼= M(E) ∼= K. Thus, after a change of coordinates we may assthat the homogeneous ideal ofC is IC = (x2

0, x0x1, x21, x2, . . . , xr , x0t − x1u). Using the

scheme defined by(x2, . . . , xr, x0x1, x0t − x1u) we can linkC to the pair of skew linesdefined by(x0, x2, . . . , xr, u) ∩ (x1, x2, . . . , xr, t). Similarly, we can linkE to a pair ofskew lines. Our claim follows because any two pairs of two skew lines are evenly lby [27] for n = 3 and by [6,13] or [16] ifn � 4. ✷


iaison

neralgative

sectione

e

es, it

andslope,

]:

e

ducedicatesart.

,

l

Together with Proposition 3.1 this result gives a complete description of the even lclasses containing a curve of degree two.

6. Construction of quasi-extremal curves

In [8] an optimal bound for the Rao function of a curve with non-degenerate gehyperplane section was established. The curves attaining this bound in non-nedegrees were called quasi-extremal curves and investigated in [26]. The goal of thisis to show that such curves can be constructed as the union of an(n−1)-rope and a suitablplane curve.

We begin by recalling the precise definition and some results.

Definition 6.1. Let C ⊂ Pn be a curve of degreed � n + 1 and genusg, and put

a = (d−n+1

2

) − g. ThenC is said to be aquasi-extremal curveif its general hyperplansection is non degenerate and

h1(IC(j)) =

a if 0 � j � d − n + 1,(d−n+2

2

) − g − j + 1 if d − n + 1 � j �(d−n+2

2

) − g,

0 if j �(d−n+2

2

) − g.

While the Rao function of a quasi-extremal curve is fixed in non-negative degrecan vary in negative degrees. In fact, in [26] it is shown:

Theorem 6.2. A numerical function is the Rao function of a quasi-extremal curve ifonly if its graph in negative degrees is a polygonal whose edges have an increasingnot exceedingn − 2 and passing through the point(0, a).

Some insight in the geometry of such curves is given by the following result in [26

Proposition 6.3. Let C be a quasi-extremal curve of degreed . Then C contains a plancurve of degreed − n + 2.

It follows that a quasi-extremal curve with negative genus must have a non-restructure on its part which does not lie in the plane. Nevertheless, the next result indthat we have to expect that this plane must contain the support of the non-reduced p

Lemma 6.4. LetC ⊂ Pn be the union of a ropeD supported on a lineL and a plane curve

P of degree� 3 contained in the planeπ . Assume that the quadrics containingD cut outthe first infinitesimal neighbourhood ofL. Then we have: if C is a quasi-extremal curvethenL is contained inπ .

Proof. The assumptions imply that(IC)2 = (Iπ )2 ∩ ((IL)2)2. Let J denote the ideagenerated by(IC)2 and consider the following exact sequence

0 → J → Iπ ⊕ (IL)2 → Iπ + (IL)2 → 0.


ve to

fnd

.

According to [26, Lemma 1.5], the Hilbert polynomial ofJ is

pJ (z) =(

z + n

n

)− z2 + (2n − 1)z + 2

2.

Thus, the Hilbert polynomial of the ideal(IL)2 + Iπ is p(z) = (z+n

n

) − (2z + 1), i.e., theideal defines a curve. Hence,L ⊆ π . ✷

Thus, if we want to construct a quasi-extremal curve as in the lemma, we hachooseD as an(n − 1)-rope with supportL in π andP as a plane curve inπ of degreed − n + 1. Thus, we may writeIP = (x0, . . . , xr−1,G), where deg(G) = d − n + 1. Theideal ofD has the formID = ((IL)2,F ) whereF = ∑r

i=0 xifi has degreeβ . It followsthat ID + IP = (x0, . . . , xr−1, x2

r , xrfr ,G). Hence, the intersection ofC and D is a 0-dimensional scheme of degreed − n + 1 if and only if x0, . . . , xr , fr ,G is a regularsequence.

Using this notation we can state the main result of this section.

Theorem 6.5. If the ropeD and the plane curveP meet in a0-dimensional scheme odegreed − n + 1, then the union ofD and P is a quasi-extremal curve of degree d agenus

(d−n+1

2

) − β where−β is the genus ofD.

Proof. PutC = D ∪ P and consider the exact sequence

0 → IC → ID ⊕ IP → ID + IP → 0.

It implies for the Hilbert polynomial ofC

pC = dz + 1−(

d − n + 1

2

)+ β

and the associated cohomology sequence

0 → (ID + IP )sat

ID + IP

→ H 1∗ (IC) → H 1∗ (ID) → H 1∗ (ID∩P ) → ·· · , (∗)

It follows that C is a locally Cohen–Macaulay curve of degreed and genusg =(d−n+1

2

) − β .According to our assumption, we haveID∩P = (ID + IP )sat= (x0, . . . , xr ,G). Thus,

the natural inclusion(IL)2 ↪→ ID∩P factors throughIL. Hence, the mapH 1∗ (ID) →H 1∗ (ID∩P ) in the sequence (∗) factors throughH 1∗ (IL) = 0, i.e., it must be the zero map

Therefore, we obtain the short exact sequence

0 → ID∩P → H 1∗ (IC) → H 1∗ (ID) → 0. (2)
ID + IP


ert

d

theositionon can

e

Using the assumption onD ∩ P again, straightforward calculations give the Hilbfunction ofID + IP . It follows that

dimK

(ID∩P

ID + IP

)j

=

0 if j � 0,j if 0 � j � d − n + 1,d − n + 1 if d − n + 1 � j � β,d + β − n + 1− j if β � j � d + β − n + 1,0 if j � d + β − n + 1.

Applying Corollary 3.2 to the(n − 1)-ropeD we, get from the sequence (2):

h1(IC(j)) =

h1(ID(j)

)if j � 0,

β if 0 � j � d − n + 1,d + β − n + 1− j if d − n + 1 � j � d + β − n + 1,0 if j � d + β − n + 1.

It is easy to see that the general hyperplane section ofC is non-degenerate. Hence,C isa quasi-extremal curve.✷Remark 6.6. The homogeneous ideal of the quasi-extremal curveC in the theoremis IC = ID ∩ IP = (x2

0, x0x1, . . . , xr−1xr, x2r G,FG). Ideals of this form were studie

algebraically in [26, Lemma 3.3].

The genusg of a quasi-extremal curveC of degreed satisfiesg �(d−n+1

2

). Moreover,

if g = (d−n+1

2

)thenC is arithmetically Cohen–Macaulay (cf. [8, Corollary 2.2]), i.e.,

Rao function vanishes. Such curves of maximal genus were constructed in [8, Prop2.4]. Examples of quasi-extremal curves of every possible non-vanishing Rao functibe constructed by means of our theorem.

Corollary 6.7. If ρ is the Rao function of a quasi-extremal curve inPn having degreed

and genusg <(d−n+1

2

)then there is a quasi-extremal curveC ⊂ P

n as in Theorem6.5ofdegreed and genusg satisfying for all integersj

h1(IC(j)) = ρ(j).

Proof. According to the end of the proof of Theorem 6.5 it suffices to chooseD as an(n − 1)-rope of genusg − (

d−n+12

)such that

h1(ID(j)) = ρ(j) for all j < 0.

Comparing with Corollary 3.2 we see that such a ropeD always exist by choosing thmatrixA suitably. ✷


ltipleider

of

cnicoThey

5) 719–

(1981)

Ital. B

ems for

th. 46

. Z. 229

ath.,

.tures

21–48.tion of

ection

165,

Of course, it is possible to construct quasi-extremal curves starting from mustructures of degreen − 1 on a line, which are not ropes. For example, if we consthe triple lineD ⊂ P

4 defined by the ideal

ID = (x2

0, x0x1, x21, x0x2, x1x2, x2

2 + x1t, x0ta + x1u

a)

we can construct a quasi-extremal curve inP4 just by joining it with a plane curveP of

degree at least 2, subject to the condition thatP andD meet in a 0-dimensional schemedegree degP .

Acknowledgments

The authors thank Silvio Greco for useful comments.During the preparation of this paper the first author profited of stays at the Polite

di Torino and the second and third author of a visit at the University of Paderborn.thank these institutions for the warm hospitality.

References

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[3] M. Boratynski, S. Greco, When does an ideal arise from the Ferrand construction?, Boll. Un. Mat.(7) 1 (1987) 247–258.

[4] D. Buchsbaum, D. Eisenbud, Algrebra structures for finite free resolutions, and some structure theorideals of codimension 3, Amer. J. Math. 99 (1977) 447–485.

[5] M. Casanellas, R.M. Miró-Roig, Gorenstein liaison of curves inP4, J. Algebra 230 (2000) 656–664.

[6] M. Casanellas, R.M. Miró-Roig, Gorenstein liaison and special linear configurations, Illinois J. Ma(2002) 129–143.

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Curves of degree two and ropes on a line: their ideals and even liaison classes

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