Vol. 34, 1980 37

Deformat ions of space curves

By

ALLEN TANNENBAU3I

Introduction. I n this note we discuss the problem of flat embedded smoothings of reduced space curves. I n other words given X c p3 a reduced curve, we want to know when X lies on an irreducible component of the Hilber t scheme of curves in p3 with generic member a smooth curve.

Our main result is t h a t given a curve X ---- X1 ~9 X2 c p3 such tha t each Xi is smooth and irreducible (i = 1, 2), X1 is rational, X2 is of degree d and admits a non-special ve ry ample divisor of degTee d, then if X1 and Xe intersect in a single point with distinct tangents a t this point, we can always find a flat smoothing of X in p3.

I n [5] Peskine-Szpiro give an example of a reduced, reducible ari thmetic Cohen- McCaulay space curve which does no t admit smooth determinantal deformations ( that is it cannot be smoothed if we consider deformations of its tangent cone).

However as a simple corollary of our result (see (2.2) below), we will see tha t the I~eskine-Szpiro curve does admi t smooth embedded deformations in ps.

Final ly we note t h a t in [6] we employ similar techniques as those used here to show every reduced connected space curve of ar i thmetic genus 0 m a y be flatly smoothed.

This work was done while the au thor was a galest a t the Forschungsinst i tut ftir Mathemat ik of the E.T.H. , Ziirieh. The author would like to thank its director Professor Beno E c k m a n n and the staff for their kind hospital i ty and support .

�9 and terminology. (i) All our schemes will be projective algebraic defined over a fixed algebraically closed field k, char(k) ~ 0.

(ii) B y curve we mean 1-dimensional scheme and by space curve we mean a curve embedded in projective three space.

(iii) Given ~ a coherent sheaf on a scheme X,

h~(~) = dim H i ( X , ~) .

(iv) :For X a scheme of dimension n, the arithmetic genus Pa (X) is defined to be:

pa(X) = (-- 1)n(Z(X) -- 1) where z ( X ) = ~ (-- 1)~h~((~x). i = 0

(v) For R a commuta t ive r ing with uni ty, we let P~ ---- Proj B[X0 . . . . , Xn]. I f R = k, then we will set pn = p~.

~8 A. TANNENBAUM ARCH. MATH.

(vi) By fiat smoothing of a curve X c pn, we mean a flat family E c P~ param- etrized by T ---- Spec R, R a discrete valuation ring, with the generic fiber of iE smooth, the special fiber isomorphic to X. This implies tha t X lies on the same irreducible component of the Hilbert scheme of curves in pn as a smooth curve. (See [2] for the definition and construction of the Hilbert scheme.)

(vii) Given X r Pu a closed subscheme, we let deg (X) ----- de~ee of X.

(viii) Given Cartier divisors D1, D2 on a nonsingular surface S, we let D I " D2

be their intersection number on S.

(ix) Let X be a smooth curve. Then a divisor D is said to be non-special if hl((Px(D)) -= O, and special ff hl((~x(D)) ~ O.

Section 1. Smooth deformations of reducible space curves. We begin with a proof of the result announced in the Introduction:

Theorem (1.1). Let X ~- Xz w X2 c F 3 be a curve with irreducible components Xz and X2, X~ smooth (i = 1, 2), pa(X1)= O, pa(X2)~-g, d e g ( X ~ ) = d~ (i-----1, 2), such that Xz and X2 intersect in a single point with distinct tangents at this point. Suppose moreover that the embedding o / X 2 into pa is de/ined by a non-special very ample divisor t ) o] degree d2 (so that h 1 (Ox~(D)) = 0). Then X may be flatly smoothed in p3 to a smooth irreducible curve o/genus g and degree dl q- d2.

P r o o f . We first note tha t by a classical result of Halphen (see [4] page 349) the condition tha t X2 has a very ample non-special divisor of degree d2 is equivalent to the condition tha t d2 ~ g § 3 when g ~ 2.

We will deform over parameter scheme T = Spec(R), R a discrete valuation ring over /c. Let C be the smooth irreducible curve of genus g of which X2 is the embedded image under the non-special very ample divisor D of degree d2. Denote the fibers of C • T -+ iv by / '1 ,2"2 where 2"1 is the generic fiber and _F2 is the closed fiber. On the surface C • T let D' = Dz -k D2 be a Cartier divisor such tha t D1 intersects each /'~ in one point yt such tha t the intersection multiplicity at yf is dl ----- deg(X1) for i ----- 1, 2, and such tha t D2 on 2"2 induces the very ample divi- sor D (so tha t D2 �9 1"2 = d2) and on 1"1, D2 induces a very ample divisor of de- gree d2. Thus D ' . 2"i = dl + d2 for i = 1, 2.

Now blow up the surface C • T at the closed point Y2- Let ~ ' be the proper transform of D' on the blown-up surface S. Then S is fibered over T (the projection C • T --> T induces a morphism p: S -+ T) with generic fiber 2"~ isomorphic to 2"z and closed fiber ~ isomorphic to C w pi joined at a single point. Note t h a t / ~ ' = r)l ~ ~)2 (D~ is the proper transform of Di i ---- 1, 2) with D1 intersecting the p1 component o f / ' ~ in one point with intersection multiplicity dz, and on the C com- ponent of 2"~, 1~2 induces a non-special very ample divisor D (which may be identified with D) of degree d2. Moreover

D' .]~- - - -dzq-d2 , i : 1 , 2 .

Next ~)' induces a very ample divisor on each of the f ibe r s /~ , _P~. The very ample divisor which it induces on the generic fiber /'~ defines an embedding of 2"~ as a

Vol. 34, 1980 Deformations of space curves 39

curve of de~ee dl ~- d2 in a projective space of dimension dl + d2 - - g. The very ample divisor which ~)' induces on ~ defines an embedding of ~ in a projective space of dimension dl + d2 - - g with the images of the C and p1 irreducible com- ponents joined at one point with distinct tangents. Moreover the image of the p1 component has degree dl.

We now prove tha t ~)' is very ample on S and defines an embedding of S in p~+d~-g. Let t e R be a local parameter. Then we have an exact sequence

0 ~ Os(D') _t.~. (Pz(D') ~ &s(D')/t(gs(~)') ~ 0

(where by "t" we mean multiplication by t). Note tha t Os(b')/tOs(D') _~ D~ where ~ is the closed fiber o f / ) ' over T. Next

taking the long exact cohomology sequence

o-+HO(D') ~ Ho(b') ---> HO(D;) -+HI(i)') -+ HI(D') -+ H~(b~)

and using the fact tha t H 1 (b~) -~ 0 (since ~)' by construction induces a non-special divisor on each of the irreducible components of the closed fiber /~) we get by Nakayama ' s lemma tha t H 1 ( D ' ) = 0 which means we may extend sections from the closed fiber. Let s~ (i = 0 . . . . . dl + d2 - - g) be the sections of f)~ which define the embedding of F~. Then extend these sections to sections of D' and denote the

dx-'bd2--g extended sections again by si. Let ~i ~ (zero set of s/} and let Z ~ ( '~ a~. Then

~ = 0 Z is a closed subset of S and since the map p : S -~ T is proper, Z cannot intersect the generic fiber so tha t Z ---- 0 (on the closed fiber the si have no common zeroes). Hence the 8~ have no base points and so D' defines a morphism

/: S -> p~+~-'-g d0j y .

We must show tha t / is an embedding. Let q: Y--> Spec(R) be the natural projection. Since p and q are proper, / is

proper. Let Y0 be the closed fiber of Y over Spee(R), and let y e Yo- T h e n / - 1 (y) is either empty, or one point x. Since the fibers are finite and / is proper, there exists a neighborhood Uy ~ y such t h a t / : /-1 (Uy) --~ Uy is finite. Then by Nakayama ' s lemma we have giy, y--> Gs,-x is surjective, and hence we conclude tha t

/: /-1 (Uy) -~ Uy

is an embedding. Since ~ J /-1 (Uy )= S, we have ]: S--~ Y is an embedding as required, seYo

Finally note tha t the closed fiber of S ~ p~+d.--g over Spec (R) can be embedded in projective 3-space by generic projection. Hence by Lemma (1.1) [6] page 7, S is embeddable in P~ (via generic projection), i.e. we can always deform in p3. Q.E.D.

Remark (1.2). The above techniques can of course be used to find smooth embedded deformations of more general reducible space curves than those described in (1.1).

40 A. TANNENBAUM ARCH. MATH.

For example one can easily generalize the above proof to show t h a t if

.X = X1 k.J --- '~t X r c p8

is a reduced curve with irreducible components Xi 1 _< i ~ r such t h a t

(i) X1, . . . , Xr-1 are smooth ra t ional curves of degree di 1 --< i _< r - - 1 ;

(ii) Xr is a smooth irreducible curve of genus g and degree dr such t h a t Xr is embedded in p3 via a non-special ve ry ample divisor D of degree dr;

(iii) if Xi and X i intersect , then they intersect a t precisely one point wi th dist inct t angen ts a t this poin t for all 1 ~ i, ] ~ r wi th i ~ ];

(iv) there exists no sequence (il . . . . . i~) (ij a {1 . . . . , r} for 1 -_<= 1" ~ k, ij = ij., if and only if ] = ]') such t h a t

X~ ~ X~ # 0 . . . . , Xi~_~ n X ~ 4= 0, X~ r~ Xi~ # 0 ,

"1OO S" t h a t i s X h a s n o p ;

then X m a y be flat ly smoothed in p3 to a smooth irreducible curve of degree dl ~- "'" -4- dr and genus g. See also [6], Corollary (1.3).

R e m a r k (1.3). Le t X ~ X1 w X2 be as in (i.1) except t h a t we r emove the hypo- thesis t h a t X2 is embedded in p3 via a non-special ve ry ample divisor D. Then such an X m a y not be smoothable in pa.

Indeed let X =- X1 ~ X2 where X1 is a line, X2 is a smooth irreducible plane curve of degree d ~> 4 such t ha t X1 and X2 meet in a single point wi th dist inct tangents (so X1 doesn ' t lie in the plane of X2).

Then note pa(X2) -~ (d - - 1) (d - - 2)/2 :> d - - 3 and pa(X2) ~ 2 for d => 4. There- fore by the result of Ha lphen quoted above in the proof of (1.1), X2 does not have a non-special ve ry ample d i v i s o r / ) of degree d.

Moreover 19a(X)-~pa(X2)-= ( d - 1 ) ( d - 2)/2 and X has degree d- f - 1. Bu t the max ima l ar i thmet ic genus for non-degenera te (i.e. lying in no plane) smooth space curves of degree d -4- 1 is (d - - 1)~/4 for d odd, and d(d -- 2)/4 for d even. (This is a special case of the Castelnuovo bound. See [1], [4], or [7]). Since pa (X) exceeds this m a x i m u m , and ar i thmet ic genus is preserved under flat deformat ions we see X cannot be flat ly smoothed.

Section 2. The example of Peskine-Szpiro.

R e m a r k (2.1). Le t (X0, X1, X2, Xa) be homogeneous coordinates in pa. Then in [5] Peskine-Szpiro consider a space curve X defined by the following ma t r ix :

t x ~ . x~ x0j Taking the 2 • 2 minors of this ma t r ix shows tha t X has defining homogeneous ideal (X0, X1) n (X2, X~ -- XIX~ -4- X~) which means t h a t X consists of a smooth plane

Vol. 34, 1980 Deformations of space curves 41

cubic curve plus a line lying in a different plane intersecting the cubic curve in a single point. Then Peskine-Szpiro prove that X cannot be "smoothed" in p3 in the sense of deforming its tangent cone (and not just the curve) byshowing that such a deformation would smooth X to the complete intersection of two quadric surfaces which contradicts the fact that under such deformations the number of generators of the homogeneous ideals must be preserved. See [5] page 299, for details.

However we do have the following:

Corollary (2.2). Let X be as in (2.1). Then X may be flatly smoothed (in the sense o/ Theorem (1.1)) to the complete intersection o/ two quadrie our/aces in ~z3.

P r o o f . From (1.1) we have immediately that X may be flatly smoothed in F 3 to an irreducible curve of degree 4 and genus 1. But it is easy to show that all such curves are complete intersections of quadric surfaces (see e.g. [4]). Q.E.D.

Example (2.3). Finally we would like to give an example of a reducible space curve of degree 4 and arithmetic genus 1 very similar to that of (2.1) and show how such a curve may be flatly smoothed to a complete intersection of quadric surfaces in p3. Consider the following matrix (again p3 has homogeneous coordinates X0, X1, X2, X~):

[ XOX3x2 1 I Ixlx3_x~ x~ ~ Xo/.

Then the 2 x 2 minors define a curve Y c F 3 with homogeneous ideal

(X0, X1) c~ (X2, 2 o XoX3 -- X~ X3 + X~).

Clearly Y is the union of a plane irreducible cubic curve with a node and a line not in the same plane passing through the node. Thus Y has degree 4 and arithmetic genus 1.

We now W e a specific flat smoothing of Y to a complete intersection of two quadries. Let H2 c ~z3 be a non-singular quadric surface and let E, / be generators of the Neron-Severi group of H~ with E 2 ~ / 2 ~ 0, E �9 / ~ 1. Let " ~ " stand for "is linearly equivalent to". Then if C is a complete intersection of two quadrics on H2, C - 2E + 2 / a n d note that C is smooth, irreducible (if we take the quadrics to be generic). We may clearly degenerate C on H2 to C 'w 1 where C' --= 2 E + ] is a smooth rational cubic, and l =- / is a line. Note that l �9 C ' - - 2.

Next project C' to p2 from a point on l not contained in l n C'. Then taking the corresponding family (see Hartshorne [4] page 259) we get a flat family of curves with generic fiber isomorphic to C ' w 1 and special fiber an irreducible plane nodal cubic and a line going through the node not contained in the same plane, that is a curve isomorphic to Y. Note this family must be flat since the degree and arith- metic genus are preserved. Finally since C'W 1 is a flat degeneration of C, we see that :Y may be flatly smoothed to C as required.

42 A. TANNENBAUM ARCH. MATH.

References

[i] P. G~FIT~S and J. HARRIS, Principles of Algebraic Geometry. New York 1978. [2] A. GRO~ZCDI~CX, Techniques de construction et th6or~mes d'existence en g~om~trie alg~-

brique. IV, les schemas de Hilbert. S~minaire Bourbaki No. 221, 1--28 (1961). [3] C. H. HxLPH~r Classification des courbes gauches alg~briques. Oeuvres III, Paris 1921. [4] R.H.~TS~O~E, Algebraic Geometry. Graduate Texts in Mathematics, :New York 1977. [5] C. PES~ZINE and L. SzPmo, Liaison des vari~t~s alg~briques I. Invent. Math. 26, 271--302

(1974). [6] A. TA.~'N~NBAU~, Degenerations of Curves in ]p3. Proc. Amer. Math. Soc. 68, 6--10 (1978). [7] A. T.~-~ZNBAU~, On the Geometric Genera of Projective Curves. Math. Ann. 240, 213--221

(1979).

Eingegangen am 10. 9. 1979

Anschrift des Autors"

Allen Tannenbaum Forschungsinstitut fiir Mathematik ETH-Zentrum CH-8092 Ztirich

and

Weizmann Institute of Science Rehovot, Israel