On the Hilbert scheme of curves in 5 higher-dimensional projective space 9 9 1 n Barbara Fantechi∗ — Rita Pardini∗ a J 7 1 1 Abstract v 9 In this paper we prove that, for any n ≥ 3, there exist infinitely many 0 r ∈N and for each of them a smooth, connected curve C in IPr such 0 r 1 thatCr lies onexactly nirreduciblecomponents oftheHilbertscheme 0 Hilb(IPr). This is proven by reducing the problem to an analogous 5 statement for the moduli of surfaces of general type. 9 / m o 1 Introduction e g - It is well-known that the Hilbert scheme parametrizing subschemes of IPr g l can be singular at points corresponding to smooth curves as soon as r ≥ 3; a : actually Mumford [M] gave an example of an everywhere singular irreducible v component. If r = 3, it has been proven in [EHM] that the open subset of i X 3 the Hilbert scheme parametrizing smooth curves in IP with given genus and r a degree can have arbitrarily many components when the genus and the degree grow (in fact, they prove that no polynomial estimate on the number of such components holds). Our main result is the following: Theorem 4.4. Let n ≥ 3 be an integer. Then there exist infinitely many integers r, and for each of them a smooth, irreducible curve C ⊂ IPr such r that C lies exactly on n components of the Hilbert scheme of IPr. r The idea of the proof is very simple. Firstly, we modify a construction of [FP] to obtain a regular surface S of general type which lies on n components ∗ Both authors are members of GNSAGA of CNR. 1 of the moduli space; secondly, we consider a suitable pluricanonical embed- ding of this surface and intersect its image with a high-degree hypersurface F to construct the curve C we are interested in. Finally, we prove that all embedded deformations of C are induced by embedded deformations of F and S. Acknowledgements. We are grateful to Ciro Ciliberto, who told us about this problem and suggested that we might apply to it the results of [FP]. 2 Notation and preliminaries All varieties will be assumed smooth and projective over the complex num- bersunless thecontraryisexplicitly stated. AvarietyY willbecalled regular if H1(Y,O ) = 0. If F is a sheaf on Y, let hi(Y,F) = dimHi(Y,F). If t is Y a real number, we denote its integral part by [t]. Let ζ3 = exp(2πi/3). In this paper we will be concerned with abelian covers of a very special type; we collect here the necessary notational set-up. Let n be an integer ≥ 2, and let G = Zn3, G∗ its dual; let e1,...,en be the canonical basis of G, and χ1,...,χn the dual basis of G∗ (i.e., χj(ei) = 1 if i 6= j and χj(ej) = ζ3). Let e0 = −(e1 +...+en). Let I = {0,...,n}, and to each i ∈ I associate the pair (H ,ψ ) where H is the cyclic subgroup of G i i i generated by ei, and ψi ∈ Hi∗ is the character such that ψi(ei) = ζ3. Let Y be a smooth projective variety, and (G,I) as above: a (G,I)-cover of Y is a normal variety X and a Galois cover f : X → Y with Galois group G and (nonempty) branch divisors D (for i ∈ I) having (H ,ψ ) as inertia i i i group and induced character (see [P] for details). Lemma 2.1 To give a smooth (G,I)-cover of Y is equivalent to giving line bundles L and F , for j = 1,...,n, together with smooth nonempty divisors j Di ∈ |Mi| (where M0 = L and, for i ≥ 1, Mi = L−3Fi) such that the union of the D ’s has normal crossings. i Proof. From [P] we know that the cover is determined by its reduced building data, divisors D for i ∈ I and line bundles L for j = 1,...,n i j satisfying the relation 3Lj ≡ Dj + 2D0. Letting Mi = O(Di), and putting M0 = L, Fj = L−Lj, the equations become precisely Mj = L−3Fj. 2 As the natural map H → G is surjective, the covers we consider Li∈I i will be totally ramified. For χ ∈ G∗, let as usual L−1 be the corresponding χ 2 eigensheaf in the direct sum decomposition of f O ; in the above notation, ∗ X we will have (for χ = χα11 ···χαnn): n Lχ = nχL−XαjFj, (2.1.1) j=1 where nχ = −[(−α1 − ... − αn)/3]. In particular note that nχ ≥ 1 when χ 6= 1, and n = 1 if and only if 1 ≤ α ≤ 3. We will write L instead of χ P j j L . χj Recall from [P], proof of proposition 4.2 on page 208, that 3KX = π∗(3KY +2(n+1)L−6XFj). (2.1.2) We now recall some results from [FP] in a simplified form (fit for our situa- tion). For details and proofs see [FP], §5. Remark 2.2 (1) Let Y → B be a smooth projective morphism (with B a smooth, connected quasiprojective variety) together with an isomorphism between Y and Y for some o ∈ B, and assume that Y is regular and that o the morphism Y → B has a section σ. Let L be a line bundle on Y; assume that c1(L) is kept fixed by the monodromy action of π1(B,o) on H2(Y,Z). Then for each b ∈ B there is a canonical induced class c1(Lb) on Yb. If, for all b ∈ B, the class c1(Lb) is of type (1,1), then L can be extended to a line bundle L over Y, flat over B; this extension is unique if we require that its restriction to σ(B) be trivial. This follows by applying the results on p. 20 of [MF], and by noting that the relative Picard scheme of Y over B is ´etale over B since all fibres are smooth and regular (it is surjective as c1(Lb) is always of type (1,1)); the condition on the monodromy action implies then that the component of the relative Picard scheme containing [L] is in fact isomorphic to B. Let L be the restriction of L to Y . b b 0 (2) If h (Y ,L ) is either constant in b, or if it only assumes the values b b 1 (for b ∈ Z) and 0, then there is a (nonunique) quasiprojective variety WL → B such that WL is canonically isomorphic to H0(Y ,L ); WL is b b b smooth and irreducible in the former case, while in the latter it is the union of one component isomorphic to B and another being the total space of a line bundle over Z (compare with [FP], theorem 5.8 and remark 5.11). 3 Assumption 2.3 Let S = {(i,χ) ∈ I × G∗|χ 6= ψ−1}. Let X → Y be |Hi i a smooth (G,I)-cover as in lemma 2.1, and Y → B be a smooth projective morphism (with (B,o) a pointed space, and Y isomorphic to Y), such that o remark 2.2, (1) applies to Y → B, for the line bundle L and for each of the F ’s. Assume moreover that remark 2.2, (2) applies for the line bundles j M −L for (i,χ) ∈ S, yielding varieties Wi,χ: let W be the fibred product i χ of the Wi,χ over B. Finally, assume that the germ of B at o maps smoothly to the base of the Kuranishi family of Y, and that the cohomology groups H1(Y,L−1) and H1(Y,T ⊗L−1) vanish for each χ ∈ G∗ \1. χ Y χ Theorem 2.4 Assume that assumption 2.3 holds, and let w ∈ W be a point over o ∈ B corresponding to sections si,χ such that si,χ = 0 if χ 6= 1, and si,1 defines D for i = 0,...,n. Assume also that X has ample canonical class. i One can construct a family of natural deformations of (G,I)-covers X → W; the induced map from the germ of w in W to the Kuranishi family of X is smooth (and, in particular, surjective). Moreover, the flat, projective mor- phism X → W defines a rational map from W to the moduli of surfaces with ample canonical class, regular at w; this map is dominant on each irreducible component of the moduli containing X. Proof. Let L (resp. F ) be the line bundle induced by L (resp. F ) on j j W; as we define M0 to be L, Lj to be L − Fj and Mj to be L − 3Fj for j = 1,...,n,thereareglobal,canonicalisomorphismsφj : 3Lj → Mj+2M0. By [FP], theorem 5.12, the germ of W at w maps smoothly to the base of the Kuranishi family of X. If M is an irreducible component of the moduli containing [X], by the previous result the image of W contains an open set in M (in the strong topology), hence it cannot be contained in a closed subset (in the Zariski 2 topology) and is therefore dominant. 3 Moduli of surfaces of general type The aim of this section is the proof of theorem 3.9, i.e., the explicit construc- tion of regular surfaces with ample canonical class lying on arbitrarily many components of the moduli. This construction can be carried out in a much 4 more general setting (see remark 3.12); we consider only the case needed for our applications, since it is easier to describe. For S a smooth projective surface and x1,...,xn pairwise distinct points of S, we let Bℓ(S;x1,...,xn) denote the surface obtained by blowing up S at x1,...,xn. Construction 3.1 Let S be a regular surface, x0 ∈ S, n a positive integer; let B = B(S,n) be the variety parametrizing data (x1,...,xn,y1,...,yn) where the x ’s are pairwise distinct points in S (for i = 0,...,n), the y ’s are i i pairwise distinct points inBℓ(S;x1,...,xn), such thatyi isnotinfinitely near to xj for i 6= j ≥ 1 and none of the yi’s lies over x0. B is a smooth quasipro- jectivevariety, whichisnaturallyisomorphictoanopensubset oftheproduct of n copies of S×S blown up along the diagonal. Let Y → B be the smooth projective family such that Y , the fibre of Y over the point b, is isomorphic b to Bℓ(Bℓ(S;x1,...,xn);y1,...,yn) for b = (x1,...,xn,y1,...,yn). Note that the morphism Y → B has a section, given by mapping b ∈ B to the inverse image of x0 in Yb. Lemma 3.2 Assume that S is rigid. Let B0 be the open set in B where 0 Aut(S) acts freely (the action being the natural one). Then if b ∈ B , the natural map from the germ of B in b to the Kuranishi family of Y is smooth b 0 of relative dimension h (S,T ). S Proof. The proof is easy and left to the reader. 2 Remark 3.3 For any b ∈ B, b = (x1,...,xn,y1,...,yn), there is a canonical isomorphism NS(Y ) = NS(S)⊕Ze′ ⊕...⊕Ze′ ⊕Ze′′ ⊕...⊕Ze′′, b 1 n 1 n where e′i is the pullback from Bℓ(S;x1,...,xn) of the class of the exceptional divisor over x , and e′′ is the class of the exceptional divisor over y . We will i i i consider this isomorphism fixed, and denote this group by NS. We also let f denote e′ −e′′. Since S is regular, so are all the Y ’s and we will not need i i i b to distinguish between line bundles and their Chern classes. 5 Definition 3.4 Let L ∈ NS, G = Zn as in §2; for χ ∈ G∗, let L ∈ NS be 3 χ defined by equation (2.1.1), with F = f . Let B be the open subset of B i i L consisting of the b’s such that 1. thecohomologygroupsH1(Y ,L−1), H1(Y ,T ⊗L−1)arezeroforeach b χ b Yb χ χ ∈ G∗ \1; 2. the line bundles L and L−3F are very ample on Y , for j = 1,...,n; j b 3. the line bundles L − K and L − 3F − K are ample on Y , for Yb j Yb b j = 1,...,n; 4. the line bundle 3K +2(n+1)L−6 F is ample on Y . Y P j b Note that the first condition is needed to ensure that assumption 2.3 is sat- isfied; the second allows one to choose smooth divisors in the linear systems |L| and |L− 3F | meeting transversally; the third implies that these linear j systems have constant dimension when b varies; and the fourth ensures, in view of equation (2.1.2), that the cover so obtained has ample canonical class (recall that the pullback of an ample line bundle via a finite map is again ample). Lemma 3.5 Let Y be a smooth surface containing m disjoint irreducible 2 curves C1,...,Cm, such that Ci < 0. Then: 1. for any choice of nonnegative integers a1,...,am, the linear system |a1C1 +...+amCm| contains only the divisor a1C1 +...+amCm; 2. for any choice of nonnegative integers a1,...,am−1, and for any b > 0, the linear system |a1C1 +...+am−1Cm−1 −bCm| is empty. Proof. (1) We prove the theorem by induction on a1 + ... + am, the case where this sum is zero being trivial. Assume without loss of generality 2 that a1 ≥ 1, and let C ∈ |a1C1 + ... + amCm|; then C · C1 = a1C1 < 0, hence C must have a common component with C1; therefore C = C1 +C′, C′ ∈ |(a1 −1)C1 +...+amCm|, and by induction the proof is complete. (2) Assume that there exists C ∈ |a1C1 + ... + am−1Cm−1 − bCm|. Then C +bCm ∈ |a1C1 +...+am−1Cm−1|, contradicting (1). 2 6 Corollary 3.6 Let b ∈ B, b = (x1,...,xn,y1,...,yn) and let a1,...,am be nonnegative integers. Then the line bundle a1f1+...+amfm on Yb is effective if and only if y is infinitely near to x for every i such that a > 0, and in i i i this case it has only one section. Proof. If y is infinitely near to x , then f is a (−2) curve; otherwise it is i i i the difference of two disjoint (−1) curves. In the former case lemma 3.5, (1) 2 applies and in the latter case 3.5, (2) applies. Notation 3.7 We will denote by E the closed subset of B consisting of the points b such that f is effective on Y for i = 1,...,n. i b Lemma 3.8 Let L ∈ NS be a line bundle and assume that E ∩ B 6= ∅. L Then assumption 2.3 holds for the restriction of Y → B to B ; applying L theorem 2.4 yields a quasiprojective variety W. In this case W is the union of 2n smooth irreducible components W , indexed by subsets A ⊂ {1,...,n}. A The dimension of WA and WA′ are equal if #A = #A′, and in particular one has: 1 3 2 dimW −dimW = (#A +6#A −#A). A ∅ 6 The W ’s have a nonempty intersection. A Proof. The verification that assumption 2.3 holds is easy and we leave it to the reader. For A ⊂ {1,...,n}, let E = {b ∈ B |f is effective for i ∈ A} A L i and let W ⊂ W be defined by A W = {(b,s )|b ∈ E and s = 0 for χ 6= 1 and i ∈/ A}. A i,χ A i,χ 3 It is easy to check that W is smooth over E of dimension 1/6(#A + A A 2 6#A +5#A); on the other hand E is smooth of codimension #A in B . A L Finally, W is the union of the W ’s, which are easily seen to be irreducible A components. Moreover, the intersection of the W ’s is clearly equal to W A ∅ intersected with the inverse image of E ∩B . 2 L Theorem 3.9 Let L ∈ NS and b ∈ E ∩ B ∩ B0. Let f : X → Y = Y L b be a smooth (G,I)-cover with building data (D ,L ); let w ∈ W be a point i χ b corresponding to a choice of equations s ∈ O (D ) defining D , with s = 0 i Y i i i,χ 7 for all χ 6= 1. Then the natural map from the germ of W in w to the Kuranishi family of X is smooth. In particular, the Kuranishi family of X is the union of 2n irreducible components, n+1 of which have pairwise different dimension. Moreover, the surface X lies on exactly n+1 components of the moduli space, having pairwise different dimensions. Proof. The first statement is a straightforward application of theorem 2.4, in view of the previous lemma. To prove the second, note that the map from W to the moduli space factors through the action of the symmetric group on n letters, Σ . The quotient W/Σ has exactly n+ 1 irreducible compo- n n nents of pairwise different dimensions. By the previous result, each of these components dominates a component of the moduli; the n+1 components so 2 obtained must all be distinct, as they have different dimensions. Remark 3.10 If L ∈ NS is sufficiently ample, then the intersection E ∩ 0 B ∩ B is nonempty, hence the theorem applies yielding infinitely many L surfaces with different invariants. Corollary 3.11 Given integers n ≥ 2 and m ≥ 5, for infinitely many values of r there exists a smooth, regular surface X in IPr such that O (1) = mK X X and X lies on exactly n+1 irreducible components of the Hilbert scheme. Proof. Let X be a regular surface, with ample K , lying on exactly n+1 X irreducible components of the moduli; let M be the union of the irreducible components of the moduli space of surfaces with ample canonical class which 0 contain[X]. Letr = h (X,mK )−1; infinitely manysuch X’s(withdistinct X values of r) can be constructed by applying theorem 3.9, in view of remark 3.10. Fix an m-canonical embedding of X in IPr. Every small embedded de- formation of X in IPr is again a smooth surface, m-canonically embedded as X is regular. Let H be the union of the irreducible components of the Hilbert scheme of IPr containing [X], and H0 be the open dense subset of H parametrizing smooth, m-canonically embedded surfaces. 0 The natural map H → M is dominant, and each fibre is irreducible of dimension (r + 1)2 − 1; in fact, the fibre over [X′] is the set of bases of H0(X′,mKX′) modulo the action of the finite group Aut(X′) (and modulo the obvious C∗-action). 8 Inparticularthereisaninducedbijectionbetweenirreduciblecomponents of M and of H, which increases the dimension by (r+1)2 −1. 2 Remark 3.12 The constructions in this section generalize easily to the case where Y is neither regular nor rigid and G is any abelian group. In fact, they also work if the dimension of Y is bigger than 2 (using a suitable, modified form of lemma 3.5). 4 The Hilbert scheme of curves in IPr In this section we apply the results on the Hilbert scheme of surfaces to the Hilbert scheme of curves. We first introduce some notation. If Z ⊂ IPr is a subscheme, wewilldenotebyHilb(Z)theunionoftheirreduciblecomponents of the Hilbert scheme of IPr containing [Z]; we let H(Z) be the germ of Hilb(Z) at [Z]. Note that if F is a hypersurface of degree l, Hilb(F) is naturally isomorphic to IP(H0(IPr,O(l))). Lemma 4.1 Let X ⊂ IPr = IP be a smooth surface, F ⊂ IP be a smooth hypersurface of degree l transversal to X, and let C = X ∩F. Then there is ∼ a natural isomorphism N | = N | ⊕O (l). C/IP C X/IP C C Proof. We have a natural diagram: 0 ↓ N C/F ↓ ց 0 −→ N −→ N −→ N | −→ 0 C/X C/IP X/IP C ց ↓ N | F/IP C ↓ 0 9 However N | = N = O (l), and it is easy to check that the morphism F/IP C C/X C N → N | in the diagram is the identity. This implies that the natural C/X F/IP C map N → N | is also an isomorphism, hence the claimed splitting. C/F X/IP C 2 Proposition 4.2 Let X ⊂ IPr = IP be a smooth, regular, projectively nor- mal surface. Let F be a smooth hypersurface of degree l in IP cutting X transversally along a curve C, and let U ⊂ Hilb(X) × Hilb(F) be the open set of pairs (X′,F′) such that X′ and F′ are smooth and transversal and X′ is projectively normal. If l >> 0, then for every (X′,F′) ∈ U the map H(X′) × H(F′) → H(C′) induced by intersection is smooth, where C′ = X′ ∩F′. Proof. The germ of the Hilbert scheme H(Z) represents the functor of embedded deformations of Z in IP; when Z is smooth, this functor has 0 1 tangent (resp. obstruction) space H (Z,N ) (resp. H (Z,N )). Let Z/IP Z/IP (X′,F′) ∈ U, and C′ = X′ ∩ F′. The map H(X′) × H(F′) → H(C′) induces natural maps on tangent and obstruction spaces; to prove the re- quired smoothness it is enough to prove that the induced maps are sur- jective on tangent spaces and injective on obstruction spaces. Note that Hi(C′,NC′/IP) = Hi(C′,NX′/IP|C′) ⊕ Hi(C′,NC′/IP|C′) by lemma 4.1. Via this isomorphism, the maps we are interested in are induced by the long exact sequences associated to: 0 → NX′/IP ⊗IC′⊂X′ −→ NX′/IP −→ NX′/IP|C′ → 0 0 → NF′/IP ⊗IC′⊂F′ −→ NF′/IP −→ NF′/IP|C′ → 0. Therefore it is enough to prove that H1(X′,NX′/IP ⊗IC′⊂X′) = 0 andthatH0(F′,NF′/IP) → H0(C,NF′/IP|C′)issurjective(remarkthatNF′/IP = OF′(l), hence H1(F′,NF′/IP) = 0 by Kodaira vanishing). For the claimed surjectivity, note that there is a commutative diagram H0(IP,OIP(l)) −→ H0(X′,OX′(l)) ↓ ↓ H0(F′,OF′(l)) −→ H0(C′,OC′(l)) 10