castelnuovo curves and unobstructed deformations

Math. Z. 174, 141- 147 (1980) Mathematische Zeitschrift

�9 by Springer-Verlag 1980

Castelnuovo Curves and Unobstructed Deformations

Allen Tannenbaum Forschungsinstitut fiir Mathematik, EidgenSssische Technische Hochschule Ztirich, Ziirieh, Switzer- land and Department of Mathematics, Weizmann Institute Rehovot, Rehovot, Israel

Introduction

This note concerns what we feel to be an interesting example about the deformations of space curves. Let ~ be an algebraic family of non-singular space curves and let the closed point t ~ correspond to the space curve F. Then if T t ~ denotes the Zariski tangent space to ~ at t and N denotes the normal bundle to F in IP a, from I-8] or [11] there is a characteristic map

p: Tt~ ~ H~

Now it is a well-known fact from Kodai ra [8], that if H i ( N ) = 0 , then there exists a non-singular family ~ for which p is surjective. In classical language, this is the "completeness of the characteristic linear series". Alternatively one says that F~---~IP 3 is unobstructed if there exists a family of deformations of F in IP 3 over a non-singular parameter scheme ~ for which the corresponding characteristic linear map is surjective.

Mumford in [10] has shown that if for a smooth space curve F, we have that HI(N)+O (here again N denotes the normal bundle to F), then F may be obstructed. In point of fact, Mumford constructs a specific example of an obstructed space curve.

In our note we show that for generic Castelnuovo space curves (see our discussion in (1.1) below) of degree d > 8 , we have HI(N)=I=O, but yet the curves are unobstructed explicitly showing the converse to Kodaira ' s aforementioned result is false. We accomplish this by computing the dimension of the component of the Hilbert scheme of space curves consisting of Castelnuovo curves of a given degree, and showing this is equal to d imH~ where N is the normal bundle to a generic Castelnuovo space curve. Since H~ represents the Zariski tangent space to the point of the Hilbert scheme corresponding to the given curve (see e.g. [11]), this means the point is smooth and hence we are done.

Finally we make some remarks relating our results tO the classical count ([13]) of parameters on which a given family of space curves depends.

0025-5874/80/0174/0141/$01.40

142 A. Tannenbaum

This work was done while the author was a guest at the Forschungsinstitut ftir Mathematik at the ETH, Ziirich. The author would like to thank its director, Professor Benno Eckmann, for his kind invitation to work at the ETH, and the staff for their hospitality.

Notation and Terminology

(i) All our schemes are defined over a fixed algebraically closed field k, char k = 0. By "po in t " of a scheme, unless stated otherwise, we mean "closed point".

(ii) By surface we mean a non-singular integral projective algebraic 2-dimen- sional scheme. By curve on a surface we mean an effective Cartier divisor. By space curve we mean an equidimensional closed subscheme ofIP 3 of dimension 1.

(iii) Given ~ a coherent sheaf on a scheme X, hi(Q)= dim Hi(x, 6). (iv) For X a projective scheme of dimension n, the arithmetic genus pa(X)

is defined to be p~(X)=(-1)"(X(@x)-1) where X((gx)= ~ (-1)ihi((gx). i=0

(v) By a non-degenerate subscheme of IP", we mean a closed subscheme of lP" not contained in any hyperplane.

(vi) On a surface F, "-= " means "is linearly equivalent to". (vii) We use the term "generic" in the following classical sense (see Griffiths-

Harris [4], pp. 20-21): Given a family of schemes parametrized by a variety (i.e. an integral separated scheme of finite type over k), to say that "a generic member of the family has a certain proper ty" means that " the property holds for all closed points in a dense Zariski open subset".

(viii) For other terminology we use the standard definitions of modern algebraic geometry. See e.g. Hartshorne [7].

Section 1. Some General Facts about Castelnuovo Curves

In this section we will discuss some general properties of integral non-degenerate curves of maximal genus for given degree, i.e. of "Castelnuovo curves". In parti- cular we show that if F is a Castelnuovo curve of degree d > 8 and N is its normal bundle in IP 3, then HI(N)+O. We begin with the following remarks:

Remarks (1.1). Recall that from Castelnuovo [3] we have that for given degree d, an integral non-degenerate curve F _ I P n has arithmetic genus p~(F)<M/2

�9 (2 d - (M + 1) ( n - 1 ) - 2) where M = In -~ ]d - 1 is the greatest integer not exceeding

d - 1 - - . This "Castelnuovo bound" is realizable by smooth curves, and following n - 1 Griffiths-Harris [4] we will call integral curves realizing this bound "Castel- nuovo curves". Moreover for d>2n (n>3) every Castelnuovo curve in IP n lies on some non-degenerate irreducible surface of degree ( n - 1).

By a theorem of Nagata [12] all such surfaces are rational scrolls or if n = 5, the Veronese surface. For a modern proof of these facts see [4]. See also [15] and [16] for further discussion of the properties of Castelnuovo curves.

In case n = 3, Castelnuovo curves of degree d have arithmetic genus equal to (d - l) (d - 3)

(d-2)2 for d even, or for d odd. Moreover from our above dis- 4 4

Castelnuovo Curves and Unobstructed Deformations 143

cussion all Castelnuovo space curves of degree d > 7 lie on quadric surfaces. (In point of fact in IP 3, even for d < 6 Castelnuovo curves lie on quadrics. See e.g. Har tshorne [7], pp. 351-352). Now if F is a Castelnuovo space curve of degree d lying on a smooth quadric surface Q, and i fE and f are generators of the Neron- Severi group of Q with E 2 = f 2 = 0 and E . f = 1, then F =- d/2 E + d /2 f for d even, and F = ( d + 1 ) / 2 E + ( d - 1)/2f (or of course ( d - 1) /2E+(d+ 1)/2f) for d odd.

We now have the following two lemmas"

Lemma (1.2). Let ~ be a maximal algebraic family of curves in IP 3 with generic member of degree d > 4 and arithmetic genus p, and lying on a non-singular quadric surface. Then the dimension of ~ is p + 2 d + 8.

Proof We just sketch the proof, since this lemma is proven in [14] (see pp. 292-293). First note that if F is an integral space curve of degree d and arithmetic genus p lying on a smooth quadric surface Q, then an easy calculation using the Riemann- Roch theorem on surfaces gives that dim IFle = p + 2 d - 1. Moreover since d > 4, every F ~ belongs to a unique quadric surface. The lemma now follows from the fact that the dimension of the space of quadric surfaces in IP 3 is 9. Q.E.D.

Lemma (1.3). Let F be any non-singular irreducible space curve of degree d and arithmetic genus p. Then if N denotes the normal bundle to F, we have h ~ =4d.

Proof Let i" F~--~IP 3 be the inclusion morphism. Then if Tx=the tangent sheaf of X where X = F , IP 3, we have an exact sequence

O ---~ Tr -~ i * Tre 3 --~ N ---~ O .

Now applying )~ to this exact sequence we see that x(N)=z( i* T~a)-)~(Tr). But by Riemann-Roch on vector bundles we have

z(i* T~3)= 3 ( 1 - p ) + 4 d and

Z ( T r ) = l - p + 2 - 2 p = 3 - 3 p . Q.E.D.

Hence we have the following result mentioned at the beginning of this section:

Proposition (1.4). A Castelnuovo space curve F of" degree d> 8 has HI(N)+O (where N denotes the normal bundle to F in Ip3).

Proof Since h~ is the dimension of the Zariski tangent space of the Hilbert scheme of space curves at the point corresponding to F (see e.g. Mumford [11]) we have by L e m m a (1.2) that h ~ 8. But then by (1.3) we see that h l ( N ) + 4 d > p a ( F ) + 2 d + 8 . Since from (1.1)

(d - 2) 2 d even

4 pa(C) =

( d - 1 ) ( d - 3) d odd

4

we see that for d > 8 we must have h i (N)>0. Q.E.D.

144 A. Tannenbaum

Section 2. Castelnuovo Curves are Unobstructed

In this section we show that for d > 8, smooth Castelnuovo curves lying on non- singular quadric surfaces are unobstructed. Thus from (1.4) of the previous section, this will show that such curves give examples of unobstructed curves with H 1 (N) 4= 0.

Our result reposes on the following lemma:

Lemma (2.1). Let F ~_ IP 3 be a smooth irreducible curve o f degree d and arithmetic genus p contained in a smooth quadric surface Q. Le t N denote the normal bundle to F in IP 3, and E, f generators o f the Neron-Severi group o f Q (with E z = f 2 =0, E . f = 1). Then if/"=_ a E + b f we have:

for

14d otherwise.

Proo f We use a method similar to that of Mumford in [10]. (See especially pp. 646-647). Accordingly we let N' be the sheaf of normal vector fields to F and in Q, and N" be the sheaf of normal vector fields to Q and in IP 3 which are defined along F.

Then we have an exact sequence of sheaves

O ~ N ' ~ N - , N " ~ O . (*)

But it is well known that N'~-(gr(/" ) (where (gr(/-):= CQ(F)| Or) and N"-~ Cr(2h ) (where h is the divisor class o n / - induced by plane sections).

Now we compute. First from the adjunetion formula we have K r = F 2 + KQ. F and so by Serre duality we have that hl(N')=h~ But K Q - - 2 H (H is a hyperplane section) so hl(N')=0. Then by Riemann-Roch o n / - applied to N' and N" we have:

h~ ') = 1 - p + deg/-2

= 1 - p + d e g ( K r - K e �9 F) (l)

- - p + 2 d - 1

and h ~ '') = 1 - p + 2d + h i (N"). (2)

Next applying the associated long exact cohomology sequence to (*) and using the fact hl(N') =0, from Eqs. (1) and (2) we see

h~ = h~ ') + h~ '')

= 4 d + h l ( N , , ) . (3)

Consequently we need only compute h I(N') . But again by Serre duality and the adjunction formula we have

h 1 (N") = h~ 2 - 4 h)).

Now we have an exact sequence

0 --* OQ(- 4H) ~ @Q(F--4 H)-* (gr(F 2 - 4 h ) -+0.


But h~ hl((gQ(-4H))=O so we have h~176 We must thus compute h~ Noting that H=E+f, by [14], p. 292 (Lemma (2.3)) if in F=-aE+b.f we have a > 4 and b > 4 , then h~ =Pa(F-4H)+2d-16. (This follows immediately from [14], (Lemma (2.3)) for a, b > 4 and a simple computat ion shows the equality is valid also when a, b > 4.)

But then standard formulae give that

so we have that

Now if a < 4 or b < 4, then

p~(F-4H)=p-4d+ 24,

h~ H))= p - 2d + 8.

h ~ ((gQ (F - 4 H)) = 0.

Hence from Eq. (3) above, we see that the lemma is proven. Q.E.D.

We thus have the following theorem:

Theorem (2.2). Any non-singular Castelnuovo space curve lying on a non-singular quadric surface is unobstructed.

Proof. From our remarks in (1.1) it is clear that the set of all Castelnuovo curves of given degree d in IP 3 forms a maximal algebraic family with generic member non-singular and lying on a smooth quadric surface.

Let ~d be the maximal algebraic family of Castelnuovo curves of degree d, and let F e ~ d be a generic member. Moreover let N be the normal bundle to F in IP 3. Then for d > 8 , by (1.1), (1.2), and (2.1) we must have that dim ~ = h ~ i.e. F is unobstructed. For 3 _<d< 7 (we have d > 3 since by definition Castelnuovo curves are non-degenerate) it is classical and easy to show (see e.g. Halphen [6]) that dim ~ d = 4 d and thus again by (1.1) and (2.1) we are done. Q.E.D.

Section 3. On the Classical Count of the Dimension of the Components of the Chow Scheme of Space Curves

We want to conclude our note by making some general remarks relating our above results to the classical count of the number of parameters on which a family of space curves depends.

Remarks (3.1). Classically (see e.g. Severi [13], pp. 158-162) it was an accepted fact that for families of smooth irreducible space curves of degree d>3/4p+3 (p= arithmetic genus), the dimension of the corresponding component of the Chow Scheme of space curves is 4d.

This was based on the following argument: First if 9Np denotes the moduli space of non-singular curves of genus p, then dim 991p = 3 p - 3 + o- where

0 p > 2

a = 1 p = l

3 p = 0 .

146 A. Tannenbaum

Now for a given smooth irreducible curve F, set

Fa ~ = {set of divisors of degree d, and dim [D[ > r}

and let J~a(F) be the image of Fa r in the Jacobian J(F) of F. If F has genus p, we define the "Bril l-Noether number" to be

r=p-(r+l) (p-d+r) .

Classically (see Brill-Noether [-2]) it was claimed that dim dS(F)>z. A modern proof of this is given by Laksov-Kleiman in [-9], who moreover show that if r > 0, then J~(F) is non-empty. But the classical geometers claimed even more! It was taken that for generic F, we have dim J~(F) = z and moreover that JS(F) is reduced. For r = 1 or r = 2 this is not too difficult to prove (see e.g. [-5]) and for the case r = 3 (the space curve case of interest to us) a proof may be found in [-1]. (Griffiths- Harris claim to have a proof in the general case and we hope their proof appears soon . )

So for a generic c u r v e FE~CJ~p (we identify a closed point in the moduli space with the curve it represents), dim j3(F)=z=4d-3p -12 when d>3/4p+3. Now noting that d i m P G L ( 3 ) = 15, d i m g ) l p = 3 p - 3 + ~ , and that d i m A u t ( F ) = a, we see that the dimension of the irreducible component of the Chow scheme of space curves containing F should be

= 3 p - 3 + ~r + 4 d - 3 p - 1 2 - Ant(F) + 15

= 4 d .

Remarks (3.2). Now we relate the above classical count to our results in Sects. 1 and 2. First from Proposition (1.4) and Theorem (2.2) we see that if ~a denotes the maximal algebraic family of Castelnuovo curves of degree d, then dim ~a = 4 d for 3<d<7, and d i m ~ a > 4 d for d>8 . This checks exactly with the count of parameters in (3.1). Indeed if F is a Castlenuovo space curve then d > 3/4pa(F)+ 3 if and only if d<8 . The classical tables (e.g. see Halphen [-6]) show that for 3_<d< 7 all irreducible components of the Chow scheme of space curves have d imens ion=4d , while at d = 8 for the first t ime the irreducible component containing the Castelnuovo curves has dimension exceeding 4 d (the dimension = 33 by (1.2)), while the other components all have dimension 32.

References

1. Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: The Brill-Noether problem for space curves. Preprint

2. Brill, A., Noether, M.: Uber die algebraischen Funktionen und ihre Anwendungen in der Geo- metric. Math. Ann. 7, 269-310 (1874)

3. Castelnuovo, G.: Sui multipli di una serie di gruppi di punti appartemente ad una curva alge- brica. Rend. Circ. Mat. Palermo 7, (1893)

4. Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: John Wiley 1968 5. Gunning, R.: Jacobi varieties. Princeton, New Jersey: Princeton University Press 1972 6. Halphen, C.FI.: Classification des courbes gauches alg6briques. Oeuvres III. Paris: Gauthier-

Vitlars 1921


7. Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics 52. Berlin-Heidelberg- New York: Springer 1977

8. Kodaira, K.: A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds. Ann. of Math. 75, 146-162 (1962)

9. Kleiman, S,, Laksov, D.: Another proof of the existence of special divisors. Acta Math. 132, 163-176 (1974)

10. Mumford, D.: Further pathologies in algebraic geometry. Amer. J. Math. 84, 642-648 (1962) 11. Mumford, D. : Lectures on curves on an algebraic surface. Princeton, New Jersey: Princeton Uni-

versity Press 1966 12. Nagata, M.: On rational surfaces I. Mem. Collog. Sci. Univ. Kyoto. Ser. A Math. 32, 351-370

(1960) 13. Severi, F.: Vorlesungen iiber algebraische Geometrie. Leipzig-Berlin: Teubner 1921 14. Tannenbaum, A.: Irreducible components of the Chow scheme of space curves. Math. Z. 162,

287-294 (1978) 15. Tannenbaum, A.: On the geometric genera of projective curves. Math. Ann. 240, 213-221 (i979) 16. Tannenbaum, A.: Families of algebraic curves with nodes. Compositio Math. 41, 107-126

(1980)

Received December 20, 1979

Note Added in Proof. The result that for generic F dimJ~(F)=z and that J~(F) is reduced mentioned in (3.1) has been proven in full generality by Griffiths-Harris. Their proof may be found in their paper: "On the variety of special linear systems on a general algebraic curve", Duke Mathematical Journal 47, 233-272 (1980).

castelnuovo curves and unobstructed deformations

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