Math. Ann. 282, 667~80 (1988) A m �9 Springer-Verlag 1988

General Components of the Noether-Lefschetz Locus and their Density in the Space of all Surfaces

Ciro Ciliberto a, Joe Harris 2, and Rick Miranda 3

1 Dipartimento di Matematica, Universith di Rorna II, Via Orazio Raimondo, 1-00173 Roma, Italy 2 Department of Mathematics, Harvard University, Cambridge, MA 02138, USA 3 Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA

I. Introduction

Let ~3 be the projective space of dimension 3 over an algebraically closed field k of any characteristic. We denote by X(d) the projective space of dimension

whose points correspond to all surfaces of degree d in ~3; and by S(d) the open subset of I2(d) whose points correspond to smooth surfaces. The Noether-Lefschetz theorem says that if d > 4, for a general point s in S(d) the corresponding surface S has Picard group Pic(S) -~Z generated by (9s(1). More precisely one can define for d > 4 the Noether-Lefschetz locus NL(d)C S(d) to be the set of points s correspond- ing to surfaces S such that Pic(S) is not generated by ~)s(1). Then the Noether- Lefschetz theorem asserts that NL(d) is a countable union of proper irreducible closed subvarieties of S(d).

Let now V be an irreducible component of NL(d) whose codimension in S(d) we will denote by c(V). Then, at least over the complex numbers, one has the inequalities

d - 3 < c(V) < pg(d), (~))

where

is the geometric genus of any smooth surface of degree d in p3 (see [CGGH], I-G]). Of the two inequalities contained in (@) the one

c(V) < p~(a) (|

All three authors were partially supported by the NSF during the preparation ofthis manuscript; the first author was supported as well by the C.N.R. and the M.P.I.

668 C. Ciliberto et al.

is very easy to understand: it simply says that it is at most pg(d) conditions on the moduli of a surface for an integral cycle on that surface to be algebraic: we require the vanishing on that cycle of all the integrals of the holomorphic 2-forms of the surface. It is also natural to expect that for most cycles these conditions should be independent, or, in other words, that for most components V of NL(d) the equality should hold in (| For this reason we will call such components of NL(d) the general components, calling other components special. Our purpose in this paper is to sketch a first attempt of description of the configuration in S(d) of the general components of the Noether-Lefschetz locus; in particular we will prove the following density theorem:

(*) Theorem. For any d > 4 the Noether-Lefschetz locus NL(d) contains infinitely many general components; and the union of these components is Zariski dense in S(d).

The proof uses the degeneration techniques employed in [GH] to give an algebro-geometric proof of the Noether-Lefschetz theorem. It is based on an inductive argument (the theorem is known to be true for d = 4, but also in this case we give a new proof of it) both to prove the existence of infinitely many general components of NL(d) and to show that their closures in S(dl intersect the closed set R(t0={points of X(d) corresponding to surfaces containing a plane as a component} in dosed subvarieties of R(d) whose union is dense in R(d). Since R(d) is not a divisor, this would not be enough for the proof of Theorem (*). But we are in fact able to prove also that the directions in which our general components of NL(d) approach R(d) are dense in the set o fall normal directions to R(d) in 2(d). In other words we first blow up X(d) along R(d) (the geometry of this blow up is described in Sect. 2 below) and then verify'that the intersection of the closures of our general components of NL(d) with the exceptional divisor are dense in the exceptional divisor itself (this is worked out in Sects. 3, 4).

Over the complex numbers one can, of course, look for a more refined density theorem, to the effect that NL(d) is dense in S(d) in the natural topology. This result does hold and it is in fact a consequence of Theorem (*), as was pointed out to us by Green. We very briefly report on Green's idea of the proof in Sect. 5.

To conclude this introduction, we note that there are still many interesting (and naive) questions about the Noether-Lefschetz locus for which we have no answer. First about the general components of the Noether-Lefschetz locus: does our inductive construction give, at least in principle, all of them? What is the Picard group of a surface corresponding to a general point of one of our general components? (Of course, one might ask the same question for any component of NL(d).) One may suspect the answer to the last question should be ~E 2 (and in this direction seem to point recent results by Angelo Lopez [L]), but we have not been able to prove it. But the most interesting questions concern, in our opinion, the special components of NL(d). In particular, is their number finite? Is it possible to describe and classify them in some way? It is may be worthwhile to recall, in this circle of ideas, the following:

Cmiimension d - 3 Conjecture (see [CGGH]): I f d > 4, the only component of NL(d) of codimension d - 3 is the set of points of S(d) corresponding to surfaces containing a line.

General Components of the Noether-Lefsehetz Locus 669

As a matter of fact, no component V of NL(d) of codimension strictly smaller than 2d - 7 is known ifd > 5 except the one corresponding to surfaces containing a line. There is a component of codimension 2 d - 7 , corresponding to surfaces containing a conic; but no one as yet has exhibited any other component of codimension 2 d - 7. So the codimension d - 3 conjecture mentioned above seems to be only the first one of a whole chain of possible conjectures, each one included in the other, of which the second could be the following:

Codimension 2 d - 7 Conjecture. I f d>4, the only components of NL(d) of eodimension at most 2 d - 7 are given either by the set of points of S(d) corresponding to surfaces containing a line (a component of codimension d - 3 ) or by the set of points of S(d) corresponding to surfaces containing a conic (a component of codimension 2 d - 7).

The techniques employed in the proof of Theorem (*), we believe, could prove to be of some use also in the analysis of these conjectures.

2. Blowing ep the Locus of Surfaces Containing a Plane and Pulling back the Universal Family

In this paragraph we will provide some technical tools useful in the proof of the density theorem. The first point is to give a geometric interpretation for the exceptional divisor of the blow up of E(d) along the locus R(d) of surfaces containing a plane. More precisely, we let Rv(d)cR(d) be the locus of points corresponding to surfaces of the type So = TwP, where T is a smooth surface of degree d - 1 and P is a plane such that the curve C = Tc~P is smooth. As above, we denote by S(d) the locus of points corresponding to smooth surfaces and set U(d) = Rv(d)wS(d ) (observe that U(d) is an open neighborhood of Rv(d), i.e., a small deformation of a surface in Rv(d) is either smooth or again in Rv(d)). Clearly Rv(d) is smooth, of codimension

in U(d). Let now

p: 0(d) V(d)

be the blow up of U(d) along Rv(d ). We want to prove that for any point So in Rv(cO corresponding to a reducible surface So = T u P the points in p-1(So) are in a natural way in one to one correspondence with the divisors of the complete linear series [(gc(d)[ on the curve C = TriP. Let us make this statement more precise.

First we notice that there is a smooth family of curves

~CRu(d ) x pa

R d)

670 C. Ciliberto et al.

such that for any point So corresponding to a reducible surface So = T o P the fibre of n over so is just the curve C = TemP. Since c~ is a family of curves in •3, we do have the line bundle 0~(1) on ~ and we can then consider the sheaf 8 = 7~.(p,(d) on Rv(d) which clearly is locally free of rank r(d). On the other hand we can also consider another locally free sheaf of the same rank r(d) on Rtr(d), namely the normal sheaf X~ , .v of Rv(d) in U(d). Now we define a map

tp: XR~. v--,8

of vector bundles in the following way. Let so be, as usual, a point in Rv(d). A normal vector to Rv(d ) at So can be assigned by giving the class, modulo TRotd~, SO, of a tangent vector v to 2~(d) at So. If So is defined by the equation hg = 0, where h is the linear form defining P and g the polynomial of degree d - 1 whose zero locus is T, then v corresponds to a first order deformation of So defined by an equation of the type hg + ef = 0 where e 2 = 0 and f is a homogeneous polynomial of degree d. But now the restriction o f f to the curve C defined by h = g = 0, gives us a section in H~ (Pc(d)) which we define to be the image under cp of the class ofv. The map ~0 is well defined and turns out to be an isomorphism. In fact, with the above notation, v determines the zero section in H~ (Pc(d)) if and only if one has f = ah + bg where a, b are forms of degrees d - 1 and 1 respectively. But in this case v is a tangent vector to Rv(d) corresponding to the first order deformation (h + eb)(g + ea)= O.

Summarizing the above analysis, we can rephrase the statement about the fibres of p in the following way:

Proposition 1. The exceptional divisor E of the blow up p: O(d)~ U(d) is naturally isomorphic to P(~).

In view of this proposition it will be natural to look at points x in E as pairs (S o, X), where So is a reducible surface T u P with C = TriP smooth and X is a divisor in I(Pc(d)l.

The second and final point of this paragraph will be the smoothing, at least over a suitable open subset of 0(d), of the pull back of the universal family of surfaces. We start in fact by considering the universal family of surfaces of degree d in pa

c x

1 x(d) .

Since we have the morphism p : O(d)~ U(d) Cr.(d) we can pull this family back via p. Now, while .~(d) is smooth, the total space ~ of this new family will not be; but the singularities of ~(d)* are easy to describe. First we observe that the cartesian square

E P ,Ro(d)

General Components of the Noether-Lefschetz Locus 671

defines a smooth family of curves on E. Clearly there is a divisor ~ in c~ with the property that for each point x in E corresponding to a pair (So, X) where X is a divisor on the curve ~- l(x) = re- l(p(x)), ~ restricts to X on ~- l(x). Moreover we see that there is a commutative diagram

~ , #-(d)*l~

E ~E

with y a closed embedding. Then ~' , the image of ~ via % is exactly the singular locus of ~(d)* (see, for instance, the analysis of Sect. 2 in [-GH]). Now, in order to get rid of the singularities of~(d)* we would like to blow it up along ~' . Before this, and in order to make things simpler, we first consider the maximal open subset A of U(d) such that Ac~E is the open subset E 0 of E over which ~ ' ~ E is &ale (these are just the points such that the divisor X consist of d(d- 1) distinct points). We also denote by ~ the pull back of ~-(d)* over A and by D the pull back of ~ ' over A. Finally now, following again the analysis of [-GH], we can see that:

(i) the singular locus of~. ~ is exactly D, and the tangent cones to ~ at the points of D are rank four quadrics;

(ii) blowing up o~ along D we get a smooth variety ~ and a commutative diagram

A , A ,

(iii) moreover the exceptional divisor/3 in ~ can be realized, via the map /3~D, as a rank 4 quadric bundle over D. Precisely if x is a point of Eo, corresponding to a pair (So, X), where So= TwP and X=Pl+...+Pd~d-1) is a divisor on C = Trip formed by d(d- 1) distinct points in the linear series I~c(d)l, then the fibre of ~ A at x consists of a surface reducible in d ( d - 1 ) + 2 components ~,, P, Q,. . . , Qd~d- 1). Here T,, P are the blow ups of T, P respectively at the points p~ . . . . , Pd~d- al and Q~ .... . Qa~a- 1) are rank 4 quadrics; moreover T and P are glued along the strict transform (on both of them) of the curve C, and each quadric Qi is glued to T and P along the exceptional divisors of the blow ups at Pi which are, on the quadric, two lines of different rulings;

(iv) there exists a blow down ~-~fq of~- along/3 such that ~q is smooth, and a commutative diagram

,fa

1 I A ~A

such that, for any point x of Eo (we use the same notation as above), the fibre Sx of fa~A over x is reducible in the two surfaces T and P, glued along the curve C and

672 C. Ciliberto et al.

its strict transform on P (which, abusing notation, we shall again denote by C); in other words, the effect of the map J~ ~fq is to blow down, in any fibre, the rulings of the quadrics Q~ containing the exceptional divisors of the surface T.. The picture of the fiber of ~ over a point of Eo is thus.

The end result of this process is thus a simultaneous small resolution of the singular points of the original family.

In conclusion we need to say a few words about some line bundles on (q. First notice that A - E o is isomorphic, via the restriction of p to A, to S(d), so we shall actually identify A - E o with S(d). Moreover the restriction of f~ ~ A to A - E o can also be identified with the restriction of the universal family ~ ( d ) ~ 2~(d) over S(d). Therefore the existence of the line bundle (9s~td)(1) and the smoothness off~ yield the existence of a line bundle ~ extending the restriction of g0~d)(1 ) over S(d).

Of course Ar is not unique because it is defined up to elements of the subgroup of Pic(f~) of line bundles with trivial restriction to the fibres of fg~A. But the

construction of fq makes it clear that we can make it unique by imposing the condition that for any point x of Eo whose fibre in f~---, A is the surface S,, = TwP, Ar restricts on T u P to the line bundle Aax which is r on T and 6p(l), the pull back to P of re(l), on P. On the other hand ~ is easy to describe: it contains Pic(A), viewed in a natural way as a subgroup of Pic (~), and is generated by Pic(A) and the line bundle X on f# corresponding to the divisor described by all components T corresponding to surfaces of degree d - 1 in the reducible fibres of fqoA. Equivalently, JV ~ is the dual of the line bundle associated to the divisor formed by the components P corresponding to planes in the reducible fibers of fg~A. Note now that by the first description of Y , for any point x of Eo the restriction JV~ of X to Sx = Tu/~ restricts to the line bundle Cp(C) ~ d~(d- 1)| Cp(- E 1 - . . . - E e l ( d - 1)) (here by Ei we mean the exceptional divisor of P corresponding to the point Pi), and by the second description of X it restricts to the line bundle (gr(- C) = O r ( - 1) on T. Thus, if we twist the restriction Yx of X to Sx = T w P with ~x we get the line bundle which is trivial on T and coincides with (9~d)| (91,(-E~ - . . . -Edna- ~)) on P. So for any x in E 0 the line bundles ~ and JV,, generate a subgroup ~ of Pic(Sx) that, by what we have seen above and the analysis in [GH], consists just of the line bundles on S~ that can be deformed onto nearby smooth surfaces to a multiple of the (hyper)plane bundle.

It is worth recalling here that the main point of the proof of the Noether- Lefschetz theorem in [GH] consists of showing that, for a general point x in E0, Y~ coincides with Pic(S~). For any x e Eo we shall call any line bundles on S~ not in the group 6e x extra line bundles on Sx.

General Components of the Noether-Lefschetz Locus 673

3. Construction of General Components of the Noether-Lefschetz Locus (1)

In this paragraph and the next we will as promised construct infinitely many components of NL(d) having codimension pg(d) in S(d) and show that they are Zariski dense. The construction consists of two steps, as follows:

Step t. We describe an infinite collection of subvarieties of E0 of codimension at most pg(d) in E0, each of which is a "good candidate" to be a component of the intersection of E o with a general component of NL(d), in the sense that if V o is one of these subvarieties then the general fibre of ~ A over Vo possesses at least one extra line bundle. Moreover we will see by induction that these subvarieties are dense in Eo.

Step 2, We prove that for each subvariety of codimension pg(d) found in step I the extra line bundle mentioned above can actually be deformed to some nearby smooth fibres of f ~ A .

To begin with step 1, let W be a component of N L ( d - 1) if d > 5, and let W be S(3) if d = 4; let ~ be the restriction of the universal family ~ ( d - 1) over W. Then we may find a finite covering W ' ~ W such that on the pull back ~/C' of ~F to W' there is a line bundle g whose restriction to each fibre of ~/F'---, W' is a line bundle not a power of the hyperplane bundle. Since there is a natural surjective morphism E o ~ S ( d - 1 ) we can take the inverse image Ew of W. Then via the cartesian square

E w, ~ E w

I 1 W' , W

we define the variety Ew, whose points naturally correspond to triples (So, X, go) where, with the usual notation, So = T~P, X=p~ + ...+Pd~d-~) is a divisor on C = Tc~P formed by d ( d - 1) distinct points in I•c(d)l, and go is a line bundle on So not equal to (gso(k) for any k. Finally we will consider the 6tale covering Fw,--*E w, of degree d(d-1)! whose fibres over points like (So, X, go) consist of all triples (So, (Pq . . . . , P~d~- ,), go), with (il, ..., idCd- 1~) any permutation of (1,..., d ( d - 1)).

Now, for any vector 0 t=(~ .. . . . ad~d-~)) in Z dtd-1) such that ct~+0 for any i = l . . . . . d (d - 1), and any integer h define Fw,(~, h) to be the variety consisting of all triples (So,(Pt . . . . . Pard-1)), gO) such that

(9c(~1P~ +. . - + ~dtd - ~)P,~d- ~)) ~ (go)| h | OC;

Let F~v,(a, h) be an irreducible component of Fw,(ot, h). Then pushing F~v,(at, h) down to Ew we find a closed subvariety Ew(at, h) of the same dimension of F~v,(0t, h). Moreover it is clear that for any point x = (So, X, go) in Ew(0t, h), the surface Sx = T ~P has some extra line bundles: precisely, for any point (So, (P~ . . . . , Pata- 1)), go) of F'w,(a, h) mapping to (So, X, go), we have the extra line bundle gx that restricts to (~o) ~h on T and to (Pp(otlEl+...+Otatd-1)Ed(d-l)) on P. We will denote by c(Ew(a, h)) the codimension of Ew(ot, h) in Eo.

Now we come to the main result in this paragraph:

674 C. Cil iberto et ai.

Proposition 2. For infinitely many vectors ~ in ~g d(d- 1) and integers h there exist components F~,,(ot, h) of Fw,(ot, h) with

C(Ew(a, h)) = c(W) + (d - 2) (d - 3)/2. (+)

Moreover the union of the corresponding Ew(a, h) is Zariski dense in Eve.

Proof. To begin with, we notice that (+ ) is equivalent to

c(F'w,(Ot, h)) = ( d - 2) (d - 3)/2, ( + +)

where c(F'w,(ot, h)) is the codimension of F~v,(0t, h) in Fw,. So first of all we have to prove that, for infinitely many a, Fw,(a, h) has a component of codimension ( d - 2 ) ( d - 3)/2 in Fw,. Moreover we observe that there is a natural morphism F w, ~Rw(d), where Rw(d) is the closed subvariety ofRv(d ) given as the inverse image of W via the natural map Rv(d)--*S(d- 1). Then it is sufficient to show that there are infinitely many at and h for which there is an open subset "f, of Rw(d) such that, for any point in f , , the fibre over that poiht of any component of Fw,(O~, h) has codimension (d - 2) (d - 3)/2 in the corresponding fibre of Fw,. But now this is a consequence of the following:

Claim. Let C be a smooth plane curve of degree d - 1. Let ~ E ](gc(d)l be the open subset of divisors consisting of d ( d - 1) distinct points, and let ~ be the (d(d- 1))!- sheeted 6tale coveting of ~ whose points correspond to the orderings of the divisors D e ~ . For any vector a =(a l ..... aa(d- 1)) in 7Z a<a- 1) such that al +... + an(a- it = l let ~9, be the morphism

q~,: ~ / - PiS)(C) defined by

r 1," ' ", P d ( d - 1)) = (9C(alPl + " " q- gd(d - 1 )Pdla - 1))"

Then for infinitely many vectors 0t the map ~o, has maximal rank g=g(C) = ( d - 2) ( d - 3)/2 everywhere in ~t.

Assume for a moment the claim and let us see how the proposition follows. Let in fact So = T u P be a point of Rw(d) and let C = TriP. By the claim we know there is some (and in fact infinitely many) II such that ~0p has maximal rank in Y/. Choose one such p and, for any integer h and line bundle L on C with L | g (Pc, consider the composite map

~r , Pic(0(C) , Pic(*0(C),

where the first map is ~o~ and the second map takes glc| to (glc)| this map is just ~0h, (independently of L, of course). By the density of torsion points on the Jacobian of C, there exist infinitely many h and L such that glc| in the image of ~op, and hence infinitely many h such that (glc) ** is in the image of q~h~. We can thus find infinitely many positive integers h such that q~l((g)| is nonempty of pure codimension g(C) in "/f; and moreover the union, as h varies, of these inverse images is dense in Yr. This procedure can now be repeated for any point of Rw(d) and accordingly we find infinitely many components of Fw,(ct, h) for which ( + +) holds and whose union is dense in Fw,. The corresponding components Ew(O t, h) are therefore dense in Ew.

General Components of the Noether-Lefschetz Locus 675

Finally we prove the claim, and for simplicity, we do this only over the complex numbers, the proof requiring for any algebraically closed field only slight modifications. Let (p 1 . . . . , Pdtd- 1~) be a point in ~r We denote by r = d ( d - 1) - g the dimension of ~r and by x = (xl . . . . , x,) coordinates on q/centered at (Pl . . . . , P~t~- ~). If col . . . . ,cog is a basis of the holomorphic differentials of C then for any 0t = (cq, ..., c~d~ d_ 1~) the local expression of q~ is

~ ( x ) = . . . , y. ~i" coj . . . . , i = 1 p

where we may identify PicOtS(C) with J(C) and p is any given point of C. Let us now denote by A the d ( d - 1) x g matrix (coj(p~)) and by B the r x d ( d - 1) jacobian matrix ~(Pl,..., Pat,~-l~)/d(xl . . . . , x,). Note that B has maximal rank r and A, which is just the Brill-Noether matrix of (P~,'",Pdtd-l~) has maximal rank g as well (no holomorphic differential on C vanishes at all the Pi)- Let D(a) be the d ( d - 1)-square diagonal matrix with entries ~ . . . . ,%ta-~r Then the differential of e~, at (P~,-.., Pata- 1~) is the product B. D(a). A and we want to show that we can chose at in infinitely many ways so to have this be of maximal rank g. In order to do this, let us consider a typical minor of order g of the product matrix M(~) = B. D(a). A, say the minor Mj(a) determined by using rowsjl , ...,jg. This minor may be written as

Mj(a) = E (Bj, i" Ai" ~ i t " " ' " " ~ i , ) , i

where A~ is the minor of A determined by the rows of index i~,..., ig, Bj,~ is the minor B using the rows of index Jl .. . . . jg and columns of index il . . . . . ig, and the summation is taken over all multi-indices i = (i 1 . . . . . ig) of order g of (1 . . . . , d(d - 1 )). Now Mi(a) can be regarded as a homogeneous polynomial of degree g in a. This is certainly not divisible by 1 - Y. ~ unless

B i , i �9 A~ = 0

for any multi-index i=( i l , ...,/9). But this is not the case. In fact, there are multi- indices i~ . . . . , ig for which At ~0, and this means that the points Pi~,..., Pig impose independent conditions on the canonical series. Since 16c(d)l contains the canonical series, they give independent conditions to 16)c(d)l too, which in turn yields Bi, ,~0. So for all a's except those satisfying a nontrivial polynomial equation Mj(at)= 0 the map cp, has rank g at (p ~, ..., Papa-~)); the claim follows by the quasi-compactness of U in the Zariski topology.

Step 1 is thus concluded. We remark here that, by the maximality of W in the above construction, the subvarieties E ~ a , h) of Eo we find are in fact components of the locus of x 6 E0 such that the surface S, has extra line bundles. Also, before passing to step 2 we want to observe that, although we have, in the above, taken W to be a component of N L ( d - 1), we could as well have taken W to be equal to S ( d - 1), getting with a few slight changes similar results. Of course, we are going to use in what follows only components for which c(W)=pa(d-1) , and since

p~(d) = p~(d- 1) + (d - 2) ( d - 3)/2

we find p~(d)-codimensional subvarieties Ew(0t, h) of E o.

676 c. Ciliberto et al.

4. Construction of General Components of the Noether-Lefschetz Locus (II) and Proof of the Density Theorem

Step 2 essentially consists in proving a rather simple deformation theoretic result for the surfaces corresponding to points of the subvarieties Ew(~t, h) of E 0 introduced in Sect. 3. As we have seen, for any x in Ew(ot, h), the surface S~ has the extra line bundle o~x. Notice now that the line bundle Jr = (Sa)| 2 | X~ has ample restrictions to both components of S~. For any integer n we will denote by g~ the line bundle 8~ | on Sx. It is then very easy to see that there exists an integer N such that for any n > N and for any x in Ew(a, h), one has:

(i) h2(Sx, ~n) = h t(Sx, 8 n) = 0; and (ii) the general curve F in 18~l is a reduced, local complete intersection on Sx. Moreover if, as usual, Yr, s~ denotes the normal bundle o f f in S~, we have, by

virtue of (i): (iii) hi(r, ~r,s~) = h2(Sx, (-~ = Po(d) �9 Now we recall that we have a family f ~ A , that x ~ Ew(a, h) C A, and that S~ is

the fibre of f r over x. Therefore we can look at F as a closed subscheme o f ~ and we can consider the dimension h(F) of the Hilbert scheme parametrizing closed subschemes offr at the point corresponding to F. What we shall need is an estimate for h(F). Precisely we want to prove the:

Proposition 3. h(F) > N(d) + h~ Yr.sx)-- Po(d) �9

Proof Consider the exact sequence of sheaves

O~ JVr, sx~ Y r , ~ JVs~,~|

and the induced map

f: H~(F, Wr. , )~ HI(F, Wsx,,|

Then, since F is locally a complete intersection, H~(F, ~Vr,,) is known to be an obstruction space for the functor Hilbr of infinitesimal deformations of F in ~ (see [S]); but we can see more, namely that K e r f is an obstruction space for Hilbr and this will yield

h(F) > h~ Jffr, w) - dim K e r f

whence the proposition, by (iii) above, since clearly

,/~S~, ~ | (~r ----- ~ dima

and dimA = N(d). In order to prove that K e r f is an obstruction space for Hilbr we argue as follows. For any Artin ring A o and for any deformation Ao of F in parametrized by Ao there is an obstruction map

~ : Extk(Ao, k )~Ht( F, Jffr,~)

(see IS], prop. (8.4)), and our assertion will follow if the composition of o(A o) with f is zero.

To prove this we first quote a general fact, which says in effect that if we have a fiat family of projective varieties r ~ : ~ B and a connected subvariety YoCXo = ~- ~(bo) of a fiber ofthe family, any flat deformation of Yo in the total space 5f will continue to lie in a fiber of n, even infinitesimally. Specifically, this is the

General Components of the Noether-Lefschetz Locus 677

Lemma. Let n : f ~ B and Yo C Xo be as above. Let ~l : ~I--* B' be a fiat deformation of Yo in ~Y, i.e., a subscheme ~1 C X x B' such that the composition q of the inclusion with the projection to B' is fiat, and such that r I- ~(b~)= Yo. Then there is a unique morphism Q : B'--* B such that the inclusion ~ c~ ~ • B' factors though the inclusion ~g • BB'.

Proof The basic observation here is simply that, since Yo is reduced, projective and connected and ~ is flat, we have

q.(C%) = (gB,.

Now, we may assume that B = SpecA and B' = SpecA' are affine. The composition

~ x B ' ~ B

of the inclusion with projection to s and thence to B induces a map

A = F(B, (fiB)--*F(~, (9,) = r(B', (~B,) = A'

which yields the desired map from B' to B. (We would like to thank Angelo Vistoli for pointing out the simple proof of this

fact).

In the present circumstances, this lemma says that our deformation A o of F in will induce a deformation d~ of S~ in ~ parametrized by A0; and so we get another obstruction map

o(a'o) : Ext~(A o, k)-~ HI(S~, ~Vsx.~).

From the exact sequence

o-~ ~s~(- r)-, Xsx.~-~ Xsx.~| we also obtain the map

g: n~(S~, Xs~.~)-* Ht(r, Jffs~,~|

and it is a matter of local computation (see IS], prop. (4.8)) to check that the composition of o(Ao) with f is equal to the composition of o(A'o) with g. But since Ys~, ~ ~ Os~, one has

H t(S~, .4rs~ ' ~) ~- H t(S~, (gsx) ~ (0)

and this proves the assertion. As an immediate consequence of Proposition 3 we have the

Corollary 4. I f c(W)= pa(d- 1) then there exists some component V of NL(d) such that c(V) = pg(d) and ~'c~E o contains Ew(tX, h) as a component (the closure of V is, of course, taken in A).

Proof Let x be a general point of Eve(a, h), let F be as above and let oct' be a component of maximal dimension of the Hilbert scheme of curves in f# containing the point corresponding to F. Clearly there is a morphism F: o~ ~ A that is proper; since the surfaces in question are regular, the fibres of F consist of a disjoint union of a finite number of projective spaces corresponding to linear systems on the fibres of ~ A . (Indeed, by (i) above, these linear systems can be all assumed to be of the

678 C. Ciliberto et al.

same dimension h~ Jffr, s~): by upper semi-continuity, for nearby curves F t lying on nearby surfaces St we have ht(S~, O(F~))=h2(St,r and it follows that h~ d~(Ft) ) = h~ 6(F)). Then F(W)is a closed subset of A and, by Proposition 3, we have

dim F(Jr ~) _>_ N(d) - pg(d). (,)

Notice now that F(W) contains Ew(a, h) and

dim Ew(ot, h) = N(d) - pg(d)- 1. (**)

Now, all the points ofF(W) correspond to surfaces that support extra line bundles. On the other hand, as we have already remarked at the end of Sect. 3, Ew(a, h) is a component of the locus in E o of points corresponding to such surfaces; se clearly Er~(0t, h) is a component of F(W)c~E o. From before, we have that the codimension of Ew(o~, h) in Eo is equal to pg(d). It follows from this that the codimension ofF(W) in A is at least equal to this; since by (*) the codimension of F(W) is at most pg(d) we may conclude that the codimension of F(Yt ~) in A is exactly equal to pg(d).

It remains to see that F(A~) is indeed a component of the Noether-Lefschetz locus NL(t0. But if it were not, the component V containing it would have to intersect Eo in a subvariety including a component that contained Ew(a, h) and had larger dimension than Eve(a, h), but was still contained in the locus corresponding to surfaces Sx with extra line bundles; and this contradicts the fact that Ew(~t, h) is a component of this locus.

We are now in position to complete the:

Proof of the Density Theorem. For d = 4 the theorem directly follows from Corollary 4 and the last assertion of Proposition 2, since in this case W = S(3) and c(W) = pg(3) = 0. For d > 4 one can use induction, and the theorem is proved in the same manner applying Corollary 4 and Proposition 2.

Before concluding this section it is may be worth remarking the role played by the condition c(lV)= pg(d- 1), which essentially comes into the picture only at the very end of the argument, namely in Corollary 4, in order to deform the extra line bundle off the locus of reducible surfaces. The question remains (and an answer would be interesting in order to understand the special components of the Noether-Lefschetz locus) for which W's with c(W)< pg(d- 1) and which (suitably defined!) Ew(0t, h) a similar deformation theoretic result may be proved.

5. The Density in the Natural Topology over the Complex Numbers (Following Green)

As we said in the introduction, Green pointed out how the existence of just one component of codimension pg(d) of NL(d) yields, over the complex numbers, the density of NL(d) in S(d) in the natural topology. Now we will briefly sketch Green's argument.

First we recall the existence of a real vector bundle W = W2(d) on S(d) whose fibre at a point s corresponding to a surface S is the real vector space H2(S, ~), and

General Components of the Noether-Lefschetz Locus 679

likewise of a complex vector bundle ~r = ~ | with fiber H2(S, ill) at s (o~r and Jfe are the real and complex vector bundles associated to the local system R2n.Z, where n:~(cO~S(d) is the universal surface as above). Note that the bundles and ~ r are flat, so that in particular g r has the structure of holomorphic vector bundle, jig has a subbundle 9f '~' ~(d) of corank 2pg(d) whose fibre at a point s corresponding to a surface S is H L I(S)c~H2(S, IR) and Yfr has subbundles 9r 'i for each i+j = 2, with fibers H ~' J(S); in terms of the holomorphic structure on g c the subbundles ,vg~,J are not in general holomorphic, but the subbundles ~r 2 and o~1 = jgo,2@~f~, 1 are.

Now choose an open subset (in the natural topology) U of S(d) over which out ~ and o~fq: may be trivialized so that F(~lv ) = U x IP and ~'(o~elv)~ U x Pc, where F and Pc are a real and complex projective space of dimension hZ(S,N)-I respectively. Moreover we have maps Gv" ~(~1~' ~ ) ~ and Go,~z:F(o~b)~P~: that are linear and injective on each fibre; and by the above remark Gv,e is in fact holomorphic. Suppose that U intersects some general component of NL(d). We then make the

Basic Claim. The image of G v contains some nonempty open subset ql of ~.

Proof We prove first the corresponding statement for Gv, e, namely that the image of Gv, e contains a nonempty open subset q/e of Pe meeting FcIPr To see this, observe that the hypothesis that U meets a general component of NL(t0 means that there are rational points ~ in ~'e such that the fibre of Gv,e over them has complex codimension equal to dimPe. Since Gv,c is holomorphic it follows that the image of G~,r contains a neighborhood of V.

The claim now follows if we observe that a point V e P = ~(H2(S, Fx))Q Pc ---- ~(H2(S, ~])) that lies in the image of the map Gv, C- i.e., that lies in the subspace H 1" I(S)@H~ z(S)C H2(S, ff~) for some surface S corresponding to a point in U - must also lie in the conjugate subspace H l' I(S)~H2'~ C) and hence in H 1' ~(S), that is, in the image of Gv. The image of Gv thus contains the intersection og of://e with FCPe.

Given the Claim, we see that the inverse image Qv in P ( g ~ ' l(d)tv) via Gv of the rational points of F (not corresponding to the class of the hyperplane section) is dense in the natural topology, and so is the image of Qv in U under the map P(o~ 1' ~(d) I v)-~ U. But this image is nothing else than NL(d), so NL(d)n U is dense in U in the natural topology.

In order to complete the argument it is sufficient to show that for any point x in S(d) there is an open neighborhood U ofx in the natural topology such that g2(d) can be trivialized over U and the (consequently defined) map Gv: ~( jgL l(d)lts ) ~ F is of maximal rank somewhere. In order to see the sufficiency, we simply observe that if we take an open cover { Ui}~, of S(d) such that g2(d) can be trivialized over U~ for any i e J , and such that the transition functions are constant non degenerate h2(S, R ) x h2(S, N) square matrices with coefficients in Z (the structure group of ~2(d) can be always reduced to GL(h2(S, R), Z)), the analytic subsets G(Ui) over which Gv, fails to be of maximal rank patch together defining an analytic subset G of p ( ~ l , t(d))" This is proper because, as we have seen, G(Ui) is whenever Ui intersects a general component of NL(d) and if any one G(Ui) is proper they must all be.

680 C. Ciliberto et al.

Acknowledgements. This paper was written while the first named author was visiting the Department of Mathematics of the Brown University in the Academic year 1986/87. The period spent at Brown has been for him invaluable under many respects and he wants therefore to express his gratitude to all, colleagues and institutions, who made possible for him to enjoy this experience. We would also like to thank David Cox for several useful conversations.

References

I-CGGH] Carlson, J., Green, M., Griffiths, P., Harris, J.: Infinititesimal variations of Hodge structures. Compos. Math. 50, 109-205(1983)

[G] [GH]

ELI Es]

Green, M.: A new proof of the explicit Noether-Lefschetz theorem. Preprint Griffiths, P., Harris, J.: On the Noether-Lefschetz theorem and some remarks on codimension two cycles. Math. Ann. 271, 31-51 (1985) Lopez, A.: Brown University Ph. D. thesis, 1988 Sernesi, E.: Topics on families of projective schemes. Queen's Pap. Pure Appl. Math. 73 (1986)

Received April 1, 1988

Note added in proof. The conjectures stated in the introduction of this paper have both been solved. The Codimension d - 3 Conjecture has been proved by C. Voisin, Une pr6cision concernant le th6or6me de Noether, Math. Ann. 2110, 605-611 (1988), and independently by M. Green, Components of maximal dimension in the Noether-Lefschetz locus, to appear in J. Differ. Geom. The Codimension 2 d - 7 Conjecture was again proved by C. Voisin, first in the case d=5 (Composantes de liet~ de Noether-Lefschetz en degr6 cinq, preprint) and then the general ease (Composante de petite codimension du liefi de Noether-Lefschetz, preprint). Further developments on this subject have been recently announced to the first author of the present paper by Ch. Peskine (joint work with G. Ellingsrud).