NON-OBSTRUCTED SUBCANONICAL SPACE CURVES

NON-OBSTRUCTED SUBCANONICAL SPACE CURVES


Introduction
In 1960 ( [G]), A. Grothendieck proved that there is a k-projective scheme Hilb71 t) which parametrizes from the functorial point of view all closed subschemes of P' with given Hilbert polynomial p(t) E Q[t] ; though so far very few of these schemes has been studied in detail and there are no general results about these schemms concerning connected components, dirnension, smoothness, rationality, topological invariants, . . .From now on, we will say that a closed subscheme X C P" is non-obstructed if the corresporrding point x of the Hilbert scheme Hilb,(t) is non-singular; otherwise, we will say that X is obstructed.A geometrical characterization of non-obstructedness is not known oven for smooth space curves and several examples of obstructed smooth space curves has been given, for instante, in [M], [S], [EF], [K1], [K2], [K3], [E1] ; and [BKM] .
In this paper, we will prove the non-obstructedness of subcanonical 2-Buchsbaum space curves (Cf.Theorem 2.5) .As a corollary we will get that 2-Buchsbaum quasi-complete space curves (Cf.Definition 2.6) not contairied in a surface of degree < 9 are also non-obstructed (Cf.Proposition 2.7) .
Recall that a curve C C P3 is subcanonical if the canonical sheaf wC of C is isornorphic to Oc(a) for some integer a, and a curve C C P3 is k-Buchsbaum if and only if k = min{t 1 irn,'M(C) = 0} where m = (X0, Xl , X2, X3) and M(C) is the Hartshorne-Rao module of C. Note that a curve C C P3 is arithmetically Cohen-Macaulay (Resp.arithmetically Buchsbaum) if and only if is 0-Buchsbaum (Resp.1-Buchsbaurn) .So, the notion of k-Buchsbaum can be viewed as a natural extension of the notions arithmetically Cohen-Macaulay and arithmetically Buchsbaum .Moreover, every curve C C P3 is k-Buchsbaum for some integer k.
A classical theorem of Gherardelli says that a smooth irreducible subcanonical curve C C P3 is arithmetically Cohen-Macaulay if and only if it is complete intersection (For a weaker characterization of complete intersection space curves see [CV]) .In [EF], Ellia-Fiorentini prove that an integral subcanonical curve C C P3 is arithmetically Buchsbaum if and only if C is the zero scheme of a section of N(t), t >_ 1, where N is the null correlation bundle .Therefore, subcanonical k-Buchsbaum space curves, 0 _< k < 1, are non-obstructed .The aim of this paper is to extend this knowledge to subcanonical, 2-Buchsbaum space curves with the hopo of finding a clue which could facilítate the study of arbitrary subcanonical space curves.
In section 1, we establish some preliminary results.In section 2; we prove the main results of this papen .We see that any subcanonical, 2-Buchsbaum space curve is in the oven liaison chas of three disjoint linos.From this we can show that any subcanonical, 2-Buchsbaum space curve is non-obstructed, it has maximal rank and we give a resolution of its ideal sheaf.In section 3; -,ve conclude by studying some examples and adding some remarks.

Notations
Throughout this papen we work oven an algebraically closed field k of characteristic zero .We set S = k[Xo , . . .; X3] , ni = (Xo , . . .; X3) C S and P3 = Proj (S) .By a curve we mean a closed, locally Cohen-Macaulay, one-dimensional subscheme X C P3.For a coherent sheaf F on X, F(n) as usual will be F ® OX(n) and we let h'F(n) _ dimk H'(X, F(n)) .
Given a curve C C P3 , we denote d = degree of C, p,, = arithmetic genus of C, s = rnin{t, 1 H°IC(t) =,É 0} ; e = max{t j HIOC(t) 0 0} and c = max{t 1 H'Ic(t) 7~0} (c = -oo, if C is arithmetically Cohen-Macaulay) .For a curve C in P3 , the Hartshorne-Rao module M(C) = ®aH'IC(n) is a graded S-module of finite length.Recall that a curve C in P3 is said to be a-subcanonical if the canonical sheaf wc of C is isomorphic to Oc(a) and a curve C in P3 is said te have maximal rank if the restriction map H °(P3, Op3(ra)) --> H°(C, Oc(n)) is of maximal rank, for every integer n.
For a coherent sheaf E en P3 we denote H* (E) the graded S-module ®,,Hl(P3, .E(n)) .A rank 2 vector bundle E en P3 is said to be stable if II °Eorm = 0 where Enonn denotes the twist of E which has first Chern class equal 0 or -1.Our main reference for this subject is [H] .

Generalities
In the present section we recall the definitions and basic properties needed later . Definition See [MM] for general results en k-Buchsbaum curves.Remark 1.2.1.If E is a rank 2, k-Buchsbaum vector bundle en P3, then the zero set of a section of E(n) is a k-Buchsbaum, subcanonical curve .Conversely, any subcanonical, k-Buchsbaum curve corresponds to a rank 2, k-Buchsbaum vector bundle en P3 .
Proof.I will only proof (3) ; for the other results see [H] .By (2), there is a section s E H°E(1) which gives us an exact sequence : where Y is the disjoint union of there linos.

. Subcanonical, 2-Buchsbaum space curves
In this section, we give a complete description of subcanonical, 2-Buchsbaum space curves.
Theorem 2.1 .Let. C C P3 be an integral,, a-subeanon7teal curve.Tlcen, C is 2-Buchsbaum if and only if C is the zero scheme of a section of a rank 2 stable vector bundle E on P3 with cIE = 0 and c2E = 2 .
Proof.. -Let E be a rank 2 stable vector bundle on P3 with cl E = 0 and c2E = 2.By Proposition 1 .3; E is 2-Buchsbaum .In particular, any curve associated to E is 2-Buchsbaum .
Let -'F(t) be the family of curves zero schemes of sections of E(t), t >_ 1, where E is a rank 2 stable vector bundle on p3 with cl E = 0 and c2 E=2 .
(2) Assume t :7~3,4 .Let C be a curve in .F(t) and let NC be its normal bundle .Since the Zariski tangent space of Hilb P3 at the point corresponding to C is isomorphic to H°NC and h°NC = 4deg(C) + h1 Nc, it is enough to compute h'Nc and to check that dim,F(t) _ h°NC.But, C is the zero scheme of a section of E(t), hence Nc -E(t) ® Oc and we have h'Nc = h1 (E(t) ® Oc) = h2 (E(t) ® Ic) .On the other hand, tensoring by E(t) the exact sequence : 0 -> O(-2t) ----> E(-t) -> IC -3 0, taking cohomology and using lemma 2 .3,we get : Putting altogether we have: for t = 1, 2, dim .F(t) = h°NC = 4deg(C) and h'Nc = 0 and for t > 4, dimF(t) = h°NC ; which gives what we want.
Theorem 2 .5 .Let C C P3 be an irreducible, subcanonical curve.If C is k-Buchsbaum and k <_ 2, then C is non-obstructed (¡.e. the corresponding point of the Hilbert scheme is smooth) .
Remark 2 .6 .1 .This definition is equivalent to saying that there are homogeneous elements fl, f2, f3 E I(X) of degrees al, a2, a3, respectively, such that I(X)/(fj, f2, f3) is a graded S-module of finite length.
Remark 2 .6 .2 .Let X be a curve in P3 .If X is the zero scheme of a section of a rank 2 vector bundle on P3 and Z is linked to X then Z is q.c .i .Conversely, if Z is q.c.i. of three surfaces of degrees al < a2 < a3 and X is linked to Z by means of two surfaces of degrees al and aj, respectively, then X is the zero scheme of a section of a rank 2 vector bundle on P3 .

. Comments and Questions
If we try to generalize the results of section 2 to higher values of k, we immediately encounter difficulties of various kinds, to be pointed out presently.
First of all, note that all subcanonical, 0-Buchsbaum curves are in the liaison class of a line; all subcanonical, 1-Buchsbaum curves are in the liaison class of the disjoint union of two lines ; and all subcanonical, 2-Buchsbaum curves are in the liaison class of the disjoint union of three lines .Thus ; it is natural to ask if al] subcanonical ; k-Buchsbaum curves (k > 2) are in .theliaison class of -y = -y(k) linos or, at least, if all subcanonical ; k-Buchsbaum curves (k > 2) are in the sarne liaison class.The answer, in general, is no.For instante : Example 3 .1.Take Yr the disjoint union of 4 linos; Y? the disjoint union of 5 lines, Y3 a general ; irreducible ; smooth, elliptic curve of degree 7 and Y4 a general, irreducible, smooth, elliptic curve of degree 8 .Y,, Y2, Y3 and Yn are subcanonical and it is not difficult to see that they are 3-Buchsbaum .However ; computing their Hartshorne-Rao modules we easily get that they belong to four different liaison classes.
Remarks 1.1 .1 .(a) C is 0-Buchsbaum (Resp .1-Buchsbaum) if and only if C is arithmetically Cohen-Macaulay (Resp.arithmetically Buchsbaum) .(b) For any curve C C P3 there is an integer k such that C is k-Buchsbaum .(c) Let C, .D C P3 be two curves in the same liaison class.Then, C is k-Buchsbaum if and only if D is k-Buchsbaum .Definition 1 .2. ([E], [MM]) A rank 2 vector bundle E en P3 is said te be k-Buchsbaum if and only if k = min{t 1 mtH; E = 0}.