Schiffer variations and the generic Torelli theorem for hypersurfaces
Claire Voisin
Abstract
We show how to recover a general hypersurface inPn of sufficiently large degreed dividingn+ 1, from its finite order variation of Hodge structure. We also analyze the two other series of cases not covered by Donagi’s generic Torelli theorem. Combined with Donagi’s theorem, this shows that the generic Torelli theorem for hypersurfaces holds with finitely many exceptions.
0 Introduction
We will consider in this paper smooth hypersurfaces Xf ⊂ Pn of degree d defined by a homogeneous polynomial equationf. The generic Torelli theorem for such varieties is the following statement:
Let Xf be a very general smooth hypersurface of degree d in Pn. Then any smooth hypersurfaceXf0 of degreedinPn such that there exists an isomorphism of Hodge structures Hn−1(Xf,Q)prim∼=Hn−1(Xf0,Q)prim (1) is isomorphic toXf.
Remark 0.1. The Torelli theorem is usually stated for the polarized period map. However, by Mumford-Tate group considerations, we can see that, except in the case of cubic surfaces, the Hodge structure on the primitive cohomology ofXf has a unique polarization (up to a scalar) for very generalXf. So the polarized and nonpolarized statements are equivalent.
Remark 0.2. The Torelli theorem is usually stated for integral Hodge structures. The fact that we can work with rational Hodge structures is related to the nature of the proof which relies on the study of the (complex!) variation of Hodge structures for hypersurfaces of given degree and dimension and its local invariants.
The reason for the “very general” assumption in this statement is the fact that, by Cattani-Deligne-Kaplan [3], the set of pairs (X, X0) such that an isomorphism as in (1) exists, is a countable union of closed algebraic subsets in Ud,n×Ud,n, where Ud,n is the moduli space of smooth hypersurfaces of degree d in Pn. Th generic Torelli theorem thus says that, among these closed algebraic subsets, only the diagonal dominates Ud,n by the first projection.
Donagi proved in [4] the following beautiful result.
Theorem 0.3. The generic Torelli theorem holds for smooth hypersurfaces of degree d in Pn, with(d, n)6= (3,3) or(4,2) and the following possible exceptions:
1. ddivides n+ 1,
2. d= 4,n= 4m+ 1, with m≥1, 3. d= 6,n= 6m+ 2, with m≥1.
The starting point of Donagi’s proof is the description due to Griffiths and Carlson- Griffiths of the infinitesimal variation of Hodge structure of a smooth hypersurface. Denote byS∗=C[X0, . . . , Xn] the graded polynomial ring ofPn and byR∗f =S∗/Jf∗ the Jacobian ring off, where
Jf∗=S∗−d+1h ∂f
∂Xii ⊂S∗ (2)
is the Jacobian ideal of f, generated by the partial derivatives of f. The infinitesimal variation of Hodge structure on the primitive cohomology of degree n−1 of Xf is given, according to Griffiths [6], see also [10, 6.1.3], by linear maps
Rdf →Hom (Hp,q(Xf)prim, Hp−1,q+1(Xf)prim) (3) forp+q=n−1. Here, the spaceRdf is naturally identified with the first order deformations of Xf in Pn modulo the action of PGl(n+ 1). It also identifies via the Kodaira-Spencer map to the subspace H1(Xf, TXf)0 ⊂ H1(Xf, TXf) of deformations of Xf induced by a deformation off. Griffiths constructs residue isomorphisms
ResXf :R(q+1)d−n−1f →∼= Hn−q−1,q(Xf)prim (4) and the paper [2] in turn describes (3) using the isomorphisms (4) as follows:
Theorem 0.4. Via the isomorphisms (4), the maps (3) identify up to a coefficient to the map
Rdf →Hom(R(q+1)d−n−1f , R(q+2)d−n−1f ). (5) induced by multiplication in Rf∗. In other words, the following diagram is commutative up to a coefficient
Rdf //
∼=
Hom (R(q+1)d−n−1f , Rf(q+2)d−n−1)
∼=
H1(Xf, TXf)0 //Hom (Hn−1−q,q(Xf)prim, Hn−2−q,q+1(Xf)prim)
. (6)
Donagi’s proof starts with the observation that Theorem 0.3 is implied by the following result:
Theorem 0.5. Let X be a smooth hypersurface of degreedin Pn. Assume(d, n)6= (3,3), (4,2) and we are not in the cases 1, 2, 3 listed in Theorem 0.3. Then X is determined by its polarized infinitesimal variation of Hodge structures (5).
Concretely, this says that if Xf and Xf0 are two smooth hypersurfaces of degree dand dimensionn−1 such that there exist isomorphisms
Rdf ∼=Rdf0, Rfkd−n−1∼=Rfkd−n−10 , ∀k,
compatible with the Macaulay pairing betweenR(q+1)d−n−1f andR(n−q)d−n−1f (correspond- ing to the Serre pairing betweenHn−1−q,q(Xf)prim andHq,n−1−q(Xf)prim, see [10, 6.2.2]), and such that the following diagram commutes:
Rdf //
L
kHom (Rkd−n−1f , R(k+1)d−n−1f )
Rdf0 //L
kHom (Rkd−n−1f0 , R(k+1)d−n−1f0 )
, (7)
thenXf is isomorphic toXf0.
The reason why Theorem 0.5 implies Theorem 0.3 is the fact that, as an easy conse- quence of Macaulay theorem [10, Theorem 6.19], the considered hypersurfaces satisfy the infinitesimal Torelli theorem. It follows that, ifX is very general andX0has an isomorphic polarized Hodge structure onHprimn−1,X0 is also very general and there is an isomorphism of variations of Hodge structures on respective neighborhoods of [X] and [X0] in their moduli space Ud,n. Taking the differential of this isomorphism provides a commutative diagram where the vertical maps are isomorphisms
Rdf //
L
p+q=n−1Hom (Hp,q(Xf)prim, Hp−1,q+1(Xf)prim)
Rdf0 //L
p+q=n−1Hom (Hp,q(Xf0)prim, Hp−1,q+1(Xf0)prim)
. (8)
By theorem 0.4, we then get a commutative diagram (7) to which Theorem 0.5 applies.
Donagi’s method does not work in the case where (d, n) = (4,3), that is, quartic K3 surfaces because Theorem 0.5 is clearly wrong in this case, while Theorem 0.3 is true by Piateski-Shapiro-Shafarevich [8]. In the case of cubic surfaces, the generic Torelli theorem is clearly wrong, since they have moduli, but their variation of Hodge structure is trivial. The case of plane quartics is also a counterexample to the generic Torelli theorem with rational coefficients, since in genus 3, a general curve is not determined by the isogeny class of its Jacobian.
Donagi’s proof of Theorem 0.5 consists in recovering from the data of the IVHS (3) its polynomial structure(see Section 1) given by Theorem 0.4, and more precisely, reconstructing the whole Jacobian ring off from its partial data appearing in (5). His method, based on the use of the symmetrizer lemma (Proposition 1.1), gives nothing more, when d divides n+ 1, than the subringR∗df ⊂R∗f defined as the sum of the graded pieces of R∗f of degree divisible byd. This is why Donagi’s method fails to give the result in that case. The goal of this paper is to extend Theorem 0.3 to most families of hypersurfaces not covered by Donagi’s theorem.
Theorem 0.6. (1) The generic Torelli theorem holds for smooth hypersurfaces of degreed inPn such that d divides n+ 1, andd large enough. In particular, it holds for Calabi-Yau hypersurfaces of degreedlarge enough.
(2) The generic Torelli theorem holds for smooth hypersurfaces of degree 4 in P4m+1, and for smooth hypersurfaces of degree 6in P6m+2 form sufficiently large.
These results combined with Donagi’s theorem 0.3 imply the following result:
Corollary 0.7. The generic Torelli theorem holds for hypersurfaces of degreedinPn with finitely many exceptions.
The proof of Theorem 0.6 (2) will be given in Section 2. We will give there an effective estimate form, which can probably be improved by refining the method. In that case, the method of proof follows closely Donagi’s ideas, and in particular passes through a proof of Theorem 0.5, at least forX generic.
The case (1) of the theorem had been also proved in [11] in the case of quintic threefolds, the first case which is not covered by Theorem 0.3, by extending Theorem 0.5 to that case. It is quite possible that the same strategy works similarly in all cases not covered by Theorem 0.3 and of sufficiently large degree, but the proof given in [11] is very technical and specific, hence is not encouraging. The proof given in the present paper also rests on the algebraic analysis of the finite order variation of Hodge structure, but it does not pass through a proof of Theorem 0.5. Instead, it introduces a main new ingredient, which is the notion ofSchiffer variation of a hypersurface(see Section 3). These Schiffer variations are of the form
ft=f+txd (9)
and we believe they are interesting for their own. The terminology comes from the notion of Schiffer variations for a smooth curveC. They consist in deforming the complex structure of C in a way that is supported on a point pof C. First order Schiffer variations are the elementsup∈P(H1(C, TC)) given by
[H0(C,2KC(−p))]∈P(H0(C,2KC)∗) =P(H1(C, TC)).
First order Schiffer variations (9) of hypersurfaces Xf are supported on a linear section.
They are parameterized by thed-th Veronese embedding of Pn projected toP(Rdf) via the linear projectionP(Sd)99KP(Rdf), so recovering them intrinsically will allow to reconstruct the polynomial structure of the Jacobian ring, or rather its degree divisible bydpart, which, according to Theorem 0.4, is given by the infinitesimal variation of Hodge structure ofXf. Our strategy consists in characterizing Schiffer variations by the formal properties of the variation of Hodge structure of the considered family of hypersurfaces along them. An obvious but key point (see Lemma 3.8) is the fact that the structure of the Jacobian ring does not change much along them. Compared to Donagi’s method which involves the first order properties of the period map, this is a higher order argument. It would be nice to have a better understanding and a more Hodge-theoretic, less formal, characterization of Schiffer variations.
The paper is organized as follows. In Section 1, we discuss the notion of polynomial structure on the data of an infinitesimal variation of Hodge structure of an hypersurface.
We discuss alternative recipes toward proving uniqueness of the polynomial structure. For example, we exhibit a very simple recipe to show that the natural polynomial structure for most hypersurfaces of degreeddividingn+ 1 is rigid. In Section 2, we prove the case (2), that is degrees 4 and 6, of Theorem 0.6. This proof follows Donagi’s argument but provides a different recipe to prove the uniqueness of the polynomial structure of the infinitesimal variation of Hodge structures in these cases.
The main new ideas and results of the paper appear starting from Section 3 where we introduce Schiffer variations of hypersurfaces and discuss their formal properties. The proof of Case (1) of Theorem 0.6 is given in Section 4.2, where we give a characterization of Schiffer variations based on the local analysis of the infinitesimal variation of Hodge structure of the considered hypersurfaces.
Thanks. I thank Nick Shepherd-Barron for reminding me the exception (4,2) in the Donagi generic Torelli theorem stated with rational coefficients as in 0.3. This work was started at MSRI during the program “Birational Geometry and Moduli Spaces” in the Spring 2019. I thank the organizers for inviting me to stay there and the Clay Institute for its generous support.
1 Polynomial structure
The method used by Donagi to prove Theorem 0.5 consists in applying the “symmetrizer lemma” (Proposition 1.1 below), in order to recover from the data (5) the whole Jacobian ring in degrees divisible byl, wherel is the g.c.d. ofn+ 1 andd. This result proved first in [4] for the Jacobian ring of generic hypersurfaces, and reproved in [5] for any smooth hypersurface (and more generally quotientsRf• of the polynomial ring S =C[X0, . . . , Xn] by a regular sequence f• = (f0, . . . , fn) with degfi = d−1), is the following statement.
Consider the multiplication map
Rkf•⊗Rkf0•→Rk+kf 0
• , (10)
a⊗b7→ab.
Proposition 1.1. LetN = (n+1)(d−2). Then, ifMax (k, N−k0)≥d−1andN−k−k0>0, the multiplication map
Rfk0−k
• ⊗Rkf•→Rkf•0
is determined by the multiplication map (10) as follows Rkf0−k
• ={h∈Hom (Rkf•, Rfk•0), bh(a) =ah(b) in Rk+kf 0
• ,∀a, b∈Rkf•}. (11) Coming back to the case of a Jacobian ringRf, when ddivides n+ 1, the infinitesimal variation of Hodge structure (3) ofXf, translated in the form (5), involves only piecesRkf of the Jacobian ring of degreek divisible byd. Hence the symmetrizer lemma allows at best, starting from the IVHS of the hypersurface, to reconstruct the Jacobian ring in degrees divisible byd. At the opposite, whendandn+ 1 are coprime, repeated applications of the symmetrizer lemma allow to reconstruct the whole Jacobian ring. In degree < d−1, the Jacobian ring coincides with the polynomial ring, hence we directly recover in that case the multiplication map
Symd(S1)→Rdf
and its kernelJfd. The proof of Donagi is then finished by applying Mather-Yau’s theorem [7] (see also Proposition 3.2).
This leads us to the following definition. Suppose that we have two integers d, n and the partial data of a graded ring structure R∗, namely finite dimensional vector spaces Rd, R−(n+1)+id, −(n+ 1) +id≥0 with multiplication maps
µi:Rd⊗R−(n+1)+id→R−(n+1)+(i+1)d. (12) Whenddividesn+1, we get all the upper-indices divisible byd, and an actual ring structure Rd∗, but in general (12) is the sort of data provided by the infinitesimal variation of Hodge structure of a hypersurface of degreedinPn. LetSk be the degreekpart of the polynomial ring inn+ 1 variables.
Definition 1.2. A polynomial structure inn+ 1variables for (Rd, R−(n+1)+id, µi)
is the data of a rankn+ 1base-point free linear subspaceJ⊂Sd−1generating a graded ideal J∗⊂S∗, of a linear isomorphism Sd/Jd∼=Rd and, for alli, of linear isomorphisms
S−(n+1)+id/J−(n+1)+id∼=R−(n+1)+id,
compatible with the multiplication maps, that is, making the following diagrams commutative:
Sd⊗S−(n+1)+id //
∼=
S−(n+1)+(i+1)d
∼=
Rd⊗R−(n+1)+id µi //R−(n+1)+(i+1)d
(13)
The group Gl(n+ 1) acts in the obvious way on the set of polynomial structures. We will say that the polynomial structure of (Rd, R−(n+1)+id, µi) is unique if all its polyno- mial structures are conjugate under Gl(n+ 1). As explained above, Donagi’s Theorem 0.5 has the more precise form that, under some assumptions on (d, n), the polynomial struc- ture of the infinitesimal variation of Hodge structures (Rdf, R−(n+1)+idf , µi) of a smooth hypersurface Xf is unique. This statement applies as well to the polynomial structure of (Rdf
•, R−(n+1)+idf
• , µi) for any regular sequence f• of polynomials of degree d−1. We will prove a similar statement in the case (2) (degreesd= 4 andd= 6) of Theorem 0.6, at least for genericf.
For the main series of cases not covered by Donagi’s theorem, namely when d divides n+ 1, we have not been able to prove the uniqueness of the polynomial structure of Rd∗f (even for genericf), although it is likely to be true (and it is proved in [11] ford= 5,n= 4).
We conclude this section by the proof of a weaker statement that provides evidence for the uniqueness. We will say that a polynomial structure is rigid if its small deformations are given by its orbit underGl(n+ 1). We have the following
Proposition 1.3. Assume n+ 1 ≥ 8 and d ≥ 6. Let f ∈ Sd be a generic homogeneous polynomial of degreedinn+ 1variables andRd∗f be its Jacobian ring in degrees divisible by d. Then the natural polynomial structure
Sd∗→Rd∗f given by the quotient map is rigid.
Remark 1.4. The case where d = 4, resp. d = 6, will be studied in next section. We will prove there, using a different recipe, that the polynomial structure onR2∗, resp.R3∗, is unique fornlarge enough.
Remark 1.5. Proposition 1.3 implies that the natural polynomial structure of Rd∗f
• for a generic rankn+ 1 regular sequencef•of degreed−1 homogeneous polynomials is rigid.
We will use in fact only the multiplication map in degreed µ:Rdf×Rdf →R2df .
Proposition 1.3 will be implied by Proposition 1.8 below. For our original polynomial struc- ture onRfd∗, and for eachx∈S1, we get a pair of vector subspaces
Ixd:=xRd−1f ⊂Rdf, Ix2d:=xR2d−1f ⊂R2df , (14) which form an ideal in the sense that
RdfIxd⊂Ix2d. (15)
It is not hard to see that the multiplication map byx, fromRd−1f to Rfd, is injective for a genericx∈S1 whenf is generic withd≥4 and n≥3 (or d≥3 and n≥5). In fact, we even have (statement (ii) will be used only later on)
Lemma 1.6. (i) The multiplication map by x is injective on R2d−1f when f is generic, x∈S1 is generic and
2(2d−1)<(d−2)(n+ 1) (16)
(for example,n+ 1≥5 andd >8, or n+ 1≥6 andd >4 work).
(ii) The multiplication map by x is injective on R3d−1f when f is generic, x ∈ S1 is generic and
2(3d−1)<(d−2)(n+ 1) (for example,n+ 1≥5 andd >8, or n+ 1≥6 andd >4 work).
(iii) The multiplication map byxl is injective onRkf whenf is generic,x∈S1is generic and
2k+l≤(d−2)(n+ 1).
Proof. Take for f the Fermat polynomial fF ermat = Pn
i=0Xid. Then R∗f
F ermat identifies with the polynomial ringH2∗((Pd−2)n+1,C) and x=P
ixi corresponds to an ample class inH2((Pd−2)n+1,C). By hard Leschetz theorem, the multiplication byxis thus injective on R2d−1f
F ermat if 2(2d−1)<(d−2)(n+ 1), and injective onR3d−1f
F ermat if 2(3d−1)<(d−2)(n+ 1).
More generally, the Lefschetz isomorphism for the powerxlgives the injectivity ofxl onRkf when 2k+l≤(d−2)(n+ 1).
Remark 1.7. This estimate is optimal for dimension reasons. Indeed, the dimensions of the graded pieces Rfk are increasing in the interval k ≤ (d−2)(n+1)2 , and decreasing in the interval (d−2)(n+1)2 ≤k≤(d−2)(n+ 1).
It follows from Lemma 1.6 that, assuming inequality (16), the spaceIxd defined in (14) has dimensionrd−1:= dimRd−1f , whileIx2d has generic dimensionr2d−1:= dimR2d−1f . Proposition 1.8. If f is generic of degreedinn+ 1variables, andd≥6, n≥9, then the set of ideals Zideal ={[Ixd] ∈G(rd−1, Rdf), x∈ S1} is a (reduced) component of the closed algebraic subsetZ ⊂G(rd−1, Rdf)defined as
Z ={[W]∈G(rd−1, Rdf),dimRdf·W ≤r2d−1}. (17) Proof. The tangent space toZideal at the point [Ixd]∈ G(rd−1, Rdf) is the image of S1/hxi in Hom (Rd−1f , Rdf/xRd−1f ) =TG(rd−1,Rd
f),[Ixd] given by multiplication by y ∈ S1/hxi, where we identifyIxd with Rd−1f via multiplication byx. Let us now compute the Zariski tangent space toZ at [Ixd] for f andxgeneric. As dimIx2d =r2d−1 is maximal by the claim above, the condition (17) provides the following infinitesimal conditions:
TZ,[Id
x]={h∈Hom (Rd−1f , Rdf/xRd−1f ), X
i
Aih(Bi) = 0 inR2df /xRd−1f , (18) for any K=X
i
Ai⊗Bi∈Rdf⊗Rd−1f such that X
i
AiBi= 0 inR2d−1f }.
Equation (18) says thath:Rd−1f →Rdf/xRd−1f is a “morphism ofRdf-modules”, the set of which we will denote by MorRd
f(Rd−1f , Rdf/xRd−1f ), in the sense that we have a commutative diagram for someh0∈Hom (R2d−1f , Rf2d/hxi)
Rdf⊗Rd−1f //
Id⊗h
R2d−1f
h0
Rdf⊗Rdf/hxi //R2df /hxi
, (19)
where the horizontal maps are given by multiplication. The equality TZideal = TZ at the point [Ixd] is thus equivalent to the fact that all the “Rdf-modules morphisms”h:Rd−1f → Rdf/xRd−1f , are given by multiplication by somey∈S1, followed by reduction modx. This is the statement of the following
Lemma 1.9. Let f be a generic homogeneous degreed polynomial inn+ 1variables with d ≥ 5, n ≥ 9 (or d ≥ 6 and n ≥ 7), and let x ∈ S1 be generic. Then the natural map S1/hxi →MorRd
f(Rd−1f , Rfd/hxi)is surjective.
Proof. The existence ofh0 as in (19) says that for any tensorP
iAi⊗Bi∈Rdf⊗Rd−1f such thatP
iAiBi= 0 inR2d−1f , P
iAih(Bi) = 0 inR2df /hxi.
Claim 1.10. Under the same assumptions as in Lemma 1.9, for a generic q ∈Rd−1f , the multiplication map
q:Rd+1f /hxi →R2df /hxi is injective.
Proof. This is proved again by looking at the Fermat polynomial fF ermat = P
iXid and choosing carefullyxso that multiplication by xis injective onRf2d−1
F ermat, and multiplication byqis injective onRfd+1
F ermat/hxi. We writefF ermat=fF ermat0 +fF ermat00 , where fF ermat0 = P4
i=0Xid and fF ermat00 = Pn
i=5Xid. We take x = P4
i=0Xi and q = (PN
i=5Xi)d−1. We observe that
R∗f
F ermat
∼=R∗f0
F ermat⊗R∗f00 F ermat,
as graded rings, and that xacts by multiplication on the left term Rf∗0
F ermat, while q acts by multiplication on the right term R∗f00
F ermat. So it suffices to show that multiplication by xis injective on Rkf0
F ermat
for k ≤2d−1 and multiplication by q is injective on Rfk00 F ermat
fork ≤d+ 1. The first statement follows from Lemma 1.6 (i) when 2(d+ 1)<5(d−2), hence whend≥5. The second statement holds by Lemma 1.6 (iii) when 2(d+ 1) +d−1≤ (n−4)(d−2), and in particular ifd≥6 andn≥9.
We deduce from Claim 1.10 that for any tensor P
iAi⊗Bi ∈ S1⊗Rd−1f such that P
iAiBi= 0 inRdf, we have X
i
Aih(Bi) = 0 inRfd+1/hxi, (20)
since this becomes true after multiplication by q. It follows now that h vanishes on hxi.
Indeed, letb=xb0. Then for anyy∈S1, we haveyb=xb00 withb00 =yb0. Hence by (20), we getyh(b) =xh(b00) = 0. Henceyh(b) = 0 inRfd+1/hxifor any y∈S1, and it follows, by choosingy such that multiplication by y is injective on Rdf/hxi, thath(b) = 0 in Rdf/hxi.
Thushinduces a morphism
h:Rd−1f /hxi →Rdf/hxi,
which also satisfies (20). Assuming d ≥ 6, n ≥9, we now show by similar arguments as above that for genericz, y∈S1/hxi, the following holds. For anyp, q∈Rdf/hxi,
yp+zq= 0 inRd+1f /hxi ⇒p=zr, q=−yr, (21) for some r ∈ Rfd−1/hxi. Furthermore we already know that the multiplication map by z fromRfd/hxitoRfd+1/hxiis injective. It follows that there exists
h00:Rd−2f /hxi →Rd−1f /hxi inducingh, that is,
h(ap) =ah00(p) (22)
for anyp∈Rd−2f /hxi, and anya∈S1/hxi. Indeed,y andz being as above, we have for any p∈Rd−2f /hxi
y(zp)−z(yp) = 0 inRdf,
hence by (20), we get thatyh(zp)−zh(yp) = 0 inRd+1f /hxi, and by (21), this givesh(zp) = zh00(p), which definesh00. One then shows that the maph00so defined does not depend onz and satisfies (22), which is easy. To finish the proof, we construct similarlyh000:Rd−3f /hxi → Rd−2f /hxi inducingh00 andhiv :Rd−4f /hxi → Rd−3f /hxiinducing h000. As Rif/hxi=Si/hxi fori≤d−2, it is immediate to show thathivis multiplication by some element ofS1, hence alsoh.
The proof of Proposition 1.8 is thus complete.
Proof of Proposition 1.3. Letf be generic of degree d≥6 inn+ 1≥10 variables. We first claim that foranyx∈S1, the multiplication map byx:Rd−1f →Rdf is injective, and that the morphism
Φ :P(S1)→G(rd−1, Rdf), x7→xRd−1f (23) so constructed is an embedding. None of these statements is difficult to prove. The first statement says that iff is a generic homogeneous degreedpolynomials inn+ 1 variables,f
does not satisfy an equation∂u(f)|H = 0 for some hyperplaneH⊂Pn and vector fielduon Pn. The obvious dimension count shows that this holds ifh0(Pn−1,O(d))> n−1 +(n+1)2 2, which holds ifd ≥4, n ≥3. As for the second statement, suppose that xRd−1f =yRd−1f for some non-proportionalx, y∈S1. Then there is a subspace of dimension≥dimSd−1 of pairs (p, q)∈Sd−1×Sd−1 such thatxp=yqin Rdf, that isxp−yq∈Jfd. As the kernel of the mapx−y:Sd−1×Sd−1→Sd is of dimension dimSd−2, this would imply that
dimJfd∩Im (x+y)≥dimSd−1−dimSd−2=h0(Pn−1,O(d−1)). (24) As dimJfd = (n+ 1)2, (24) is impossible if h0(Pn−1,O(d−1)) >(n+ 1)2, which holds if n≥5, d≥4. We thus proved that the map Φ of (23) is injective. That it is an immersion follows in the same way because the differential atxis given by the multiplication map
y7→µy:Rd−1f →Rdf/xRd−1f ,
andµy is zero if and only if yRd−1f ⊂xRfd−1, which has just been excluded. The claim is thus proved.
It follows from the claim and from Proposition 1.8 that, if we have a family of polynomial structures
φt:Sd∗→Rd∗f , withφ0the natural one, then there is an isomorphism
ψt:P(S1)∼=P(S1), such that for anyx∈S1,
φt(xSd−1) =ψt(x)Rd−1f .
Such a projective isomorphism is induced by a linear isomorphism ψ˜t:S1∼=S1,
and composingφtwith the automorphism ofSd∗induced by ˜ψ−1t , we conclude that we may assume that for anyx∈S1,
φt(xSd−1) =xRfd−1. (25) We claim that this implies φt: Sd →Rdf is the natural map of reduction mod Jf. To see this, choose a generalx, so that the multiplication map by x is injective onR2d−1f . The polynomial structure given byφtand satisfying (25) provides two linear maps
φ0t:Sd−1→Rd−1f , φ00t :S2d−1→R2d−1f ,
such thatxφ0t=φt◦x:Sd−1→Rdf,xφ00t =φt◦x:S2d−1→R2df , and the injectivity of the map of multiplication byxonR2d−1f implies that the following diagram commutes, since it commutes after multiplying the maps byx.
Sd−1⊗Sd //
φ0t⊗φt
S2d−1
φ00t
Rd−1f ⊗Rdf //R2d−1f
. (26)
The horizontal maps in the diagram above are the multiplication maps. Following Donagi [4], the multiplication map on the bottom line determines the polynomial structure ofR∗f, because it determines (for d≥3) S1 and the multiplication map S1⊗Rd−1f →Rdf by the symmetrizer lemma 1.1. The diagram (26) then says that up to the action of an automor- phismgofS∗, the polynomial structure given by (φ0t, φt) is the standard one. Finally, asg must act trivially on the spaceZideal of ideals by (25),gis proportional to the identity.
2 The cases of degree 4 or 6
We explain in this section how to recover the polynomial structure of a generic hypersurface of degree 4 or 6 so as to prove Theorem 0.6 (2), namely the cases whered= 4, n= 4m+ 1, or d= 6, n= 6m+ 2, withm large. Note that the methods of Schiffer variations that we will develop later would presumably also apply to this case, but it is much more difficult and does not prove Theorem 0.5 (saying that one can recover a hypersurface from its IVHS).
The congruence conditions are equivalent in both cases to the fact that we haved= 2d0 andgcd(d, n+ 1) =d0, with d= 4 or 6. The infinitesimal variation of Hodge structure
Rdf → ⊕lHom(Rld−n−1f , R(l+1)d−n−1f ), (27) has for smallest degree term the multiplication map
Rdf⊗Rdf0 →Rd+df 0
and the symmetrizer lemma (see Proposition 1.1) allows to reconstruct in these cases the whole ringRdf0∗, and in particular the multiplication map
Rdf0⊗Rdf0 →Rfd. (28) (Note that Rdf0 = Sd0.) We thus only have to explain in both cases how to recover the polynomial structure of (27) from (28), at least for a generic polynomial f. We use the notationSqf2l⊂R2lf for the set of squares
Sqf2l={A2, A∈Rlf} ⊂R2lf.
This is a closed algebraic subset which is a cone inR2lf and we will denote by P(Sq2l) the corresponding closed algebraic subset ofP(R2lf). When d= 4, d0 = 2, (28) determinesSqf4 and we observe thatSqf2 ⊂Rfd0 = R2f =S2 determines the desired polynomial structure, since, passing to the projectivization of these affine cones, P(Sq2f) is the second Veronese embedding of P(S1) in P(S2). Thus the positive generator H of Pic (P(Sqf2)) satisfies the property that H0(P(Sq2f), H) =:V has dimensionn+ 1 and the restriction map (S2)∗ → Sym2V is an isomorphism. The dual isomorphism gives the desired isomorphism Sym2S1∼= S2, withS1:=V∗.
Whend= 6, d0 = 3, (28) determines Sqf6. Next, for anyl and any fixed 06=K ∈S1, denote byKSq2lf ⊂R2l+1f the set of polynomials of the form KA2, for someA∈ Rlf. We have dimSq2=n+ 1, dimKSq2=n+ 1 and ford= 6, the data of the subspacesKSq2⊂ R3f = S3 determines the isomorphism S3 ∼= Sym3(S1), hence the polynomial structure.
Indeed, the singular locus of the varietyS
KKSq2 is the variety of cubesCu3 ⊂ S3, and the same Veronese argument as above shows that it determines the polynomial structure S3=R3f ∼= Sym3S1, withS1=V∗.
We observe now that the spaces Sqf2 ⊂ R2f, resp. KSq2f ⊂ R3f, have the following property
∀A, B∈ Sq2f, AB∈ Sqf4, (29) resp.
∀A, B∈KSq2f, AB∈ Sq6f, (30) We prove now the following result, which concludes the proof of Theorem 0.6 (2).
Proposition 2.1. (1) Let f be a generic homogeneous polynomial of degree 4 in n+ 1 variables, with n ≥ 599. Then the only subvariety T ⊂ Rf2 = S2 of dimension ≥ n+ 1 satisfying the condition
AB∈ Sq4f for anyA, B∈T
isSq2f.
(2) Let f be a generic homogeneous polynomial of degree 6 in n+ 1 variables, with n ≥ 159. Then the only subvarieties T ⊂ R3f = S3 of dimension ≥ n+ 1 satisfying the condition
AB∈ Sq6f for anyA, B∈T are the varietiesKSq2f for06=K∈S1.
Note that in this statement, we can clearly assume thatT is a cone, since the conditions are homogeneous.
Proof of Proposition 2.1. We observe that by a proper specialization argument, the schematic version of the statement (namely thatT satisfying condition (29 or (30) must be reduced) is an open condition on the set of polynomialsf for which Rdf, or equivalentlyJfd, has the right dimension. This will happen in the case d = 4, where we will prove the schematic version of the statements above for one specificf, for whichJfd has the right dimension. In the cased= 6, the schematic statement is not true anymore but it is neither true for the genericf, so the schematic analysis will also allow to conclude by specialization.
Let us first explain the specific polynomials we will use. In the case of degree 4, we will first choose general linear sections P∩P f4 of the Pfaffian quarticPf4 ⊂P(V2
V8), where P⊂P(V2
V8) is a linear subspace of dimension 23, or 24. We get this way polynomials fi
of degree 4 in 24 or 25 variables. In higher dimension, we will then consider polynomials of the form
f =f1(X1,1, . . . , X1,i1) +. . .+fl(Xl,1, . . . , Xl,il),
withi1, . . . , il∈ {24,25}, which allows to construct degree 4 polynomials with any number n+ 1 of variables starting from 600.
In degree 6, we will first choose, for 39≤n≤62, a general linear sectionP∩P f6, where P ⊂ P(V2
V12) is a linear subspace of dimension n and Pf6 ⊂ P(V2
V12) is the Pfaffian sextic hypersurface. We get this way polynomialsfi of degree 6 in n+ 1 variables, where 40≤n+ 1≤63. In higher dimension, we will then consider polynomials of the form
f =f1(X1,1, . . . , X1,i1) +. . .+fl(Xl,1, . . . , Xl,il),
with i1, . . . , il ∈ {40, . . . ,63}, which allows to construct degree 6 polynomials with any numbern+ 1 of variables starting from 160.
Lemma 2.2. For a polynomialf of the form above, Jfd has the right dimension(n+ 1)2. Proof. One hasf =P
jfj, where eachfj involves variablesXj,1, . . . , Xj,ij. It is immediate to check that the statement for each fj implies the statement for f. Turning to the fj, they are either general linear sections of the quartic Pfaffian hypersurface in P27 by a Pn, n= 23 or 24, or of the sextic Pfaffian hypersurface in P65 by a Pn, for 39≤n≤62. Let us show that each of them has no infinitesimal automorphism. The automorphism group of the general Pfaffian hypersurfacePfk⊂P(V2
V2k) is the groupP Gl(2k). We claim that the automorphism group of a general linear section of dimension>2(2k−2) = dimG(2, V2k) is also contained inP Gl(2k). This follows from the fact that after blowing-up inPf2k its singular locus, which parameterizes forms of rank<2k−2, we get a dominant morphism P f]2k →G(2, V2k), which to a denerate form associates its kernel. If we consider a general linear section Xl of Pf2k of dimension >dimG(2, V2k) defined by a r-dimensional vector subspaceW ⊂V2
V2k∗, the same remains true and we get a morphismXel→G(2, V2k) which is dominant with connected fiber of positive dimension. Thus the automorphism group ofXl has to act onG(2, V2k) and it has to identify with the group of automorphisms ofG(2, V2k), or automorphisms ofP(V2k) preserving the spaceW ⊂V2
V2k∗. It is easy to check that this space is zero oncer≥3. Coming back to our situation where k=d= 4 or 6, our choices of r arer = 3 or 4 for k= 4, and 3 ≤r ≤26 fork = 6. In all cases, the variety Xl has dimension>2(2k−2) so the analysis above applies.
We now prove Proposition 2.1 forf as above. Let us first assumed= 4.
Lemma 2.3. Let f =P
jfj be a polynomial of degree 4 in n+ 1 variables as constructed above. Then ifT ⊂S2 is an algebraic subvariety of dimension≥n+ 1 such that
AB∈ Sqf4⊂R4f for anyA, B∈T,T =Sq2.
Proof. Let as abovef =P
lfj. The singular locusZfofXf =V(f) is the join of the singular lociZj of V(fj) in Pij. This means that, introducing the natural rational projection map π:P
P
jij−1
99KQ
lPij−1, one has Zf =π−1(Q
jZj).
Claim 2.4. The varieties Zf are not contained in any quadric.
Proof. Consider first the case of the Pfaffian linear sectionsZj. The claim follows in this case because they are general linear sections of the singular locusZ of the quartic universal Pfaffian Pf4 in P(V2
V8), which is defined by the equations ω3 = 0, that is, by cubics, and is not contained in any quadric. The last point can be seen by looking at the singular locus of Z, which consists of forms of rank 2, that is the Grassmannian G(2, V8∗). Along this locus, the Zariski tangent space of Z is the full Zariski tangent space of P(V2
V8). A quadric containing Z should thus be singular along SingZ. But SingZ =G(2, V8∗) is not contained in any proper linear subspace of P(V2
V8). It follows thatZ is not contained in any quadric. It remains to conclude that the same statement is true for the general linear section Pil−1 ∩P f4, with il = 24,25. Its singular locus Zl is the general linear section Pil−1∩Z, and we show inductively that any quadric containing Zl is the restriction of a quadric containing Z. This statement only needs that Zl is non-empty (it has dimension
≥18 in our case) and that all the successive linear sectionsPj∩Z, withj≥il, are linearly normal in Pj, which is not hard to prove. Finally we have to show that the same is true for a general f =Pfj. In that case, Zf is a join Z1∗. . .∗Zl and a join of varieties not contained in any quadrics is not contained in any quadric.
We also prove the following
Claim 2.5. (a) The restriction map S1→H0(Zf,OZf(1)) is an isomorphism.
(b) The only n-dimensional family {DA} of divisors on Zf such that for some fixed effective divisor D0,
2DA+D0∈ |OZf(2)|
is the family of hyperplane sections ofZf.
Proof. We recall for this thatZf is the join of theZj⊂Pij, and that each Zj is a smooth linear section of the singular locusZ ofPf4 by either aP24 or aP23. Let us first conclude whenf is one of thefj, so Zf is one of theZj. We observe thatZ ⊂P(V2
V8) is the set of 2-forms of rank≤4 (the generic element ofZ being of rank exactly 4), and has the natural resolution
Ze⊂G(4, V8∗)×P(
2
^V8), Ze={([W4], ω), W4⊂Kerω}.
The statement (a) easily follows from the above description ofZeand the fact that theZjare general linear sections of codimension 3 or 4 ofZ. Let us prove (b). The varietyZeis smooth and, being a projective bundle fibration overG(4, V8∗), has Picard rank 2. Its effective cone is very easy to compute: indeed, the line bundle l which is pulled-back from the Pl¨ucker line bundle on the Grassmannian via the first projectionpr1 is clearly one extremal ray of the effective cone since the corresponding morphism has positive dimensional fibers. There is a second extremal ray of the effective cone, which is the class of the divisorD contracted by the birational mapZe→Z (induced by the second projection pr2). One easily computes that this class is 2h−l, where h is the pull-back of hyperplane class on P(V2
V8) by pr2.
We observe that the fibers ofpr1 are of dimension 5, so that, when we take a general linear section ofZ by a codimension 3 or 4 projective subspace, getting the singular locus Zj of Xj, all the properties above remain satisfied, and thus PicZej =Zhj+Zlj, with effective cone generated by lj and 2hj−lj. We now finish the argument forZj: we lift our data {DA}, D0 toZej. We then have
2hj=De0+ 2DeA
in PicZej, withDe0effective and dim|DeA| ≥n, wherenis dim|h|Zj|by (a). Let us write De0=αhj+βlj= α
2(2hj−lj) + (β+α
2)lj in PicZej, withα, β∈2Z. Then the analysis above shows that
α≥0, β+α 2 ≥0.
As 2hj−De0 is effective, we also have
2−α≥0, −β+ 1−α 2 ≥0.
As α is even, we only get the possibilities α = 0,2. If α = 2, we get β = 0 and thus 2hj−De0 = 0, contradicting the assumption dim|2h−De0| ≥n. If α= 0, we get from the inequalities aboveβ = 0 sinceβ has to be even, and 2h= 2DeA, which proves statement (b) (using the fact that PicZj has no 2-torsion and statement (a)).
We now have to prove the same result for the joinZf =Z1∗Z2. . .∗Zl, which is done inductively on the numberl, assuming, as this is satisfied in our situation, that theZi’s are simply connected. This way we are reduced to consider only the join Z1∗Z2 ⊂ Pn, with n =n1+n2+ 1, of two linearly normal varietiesZ1 ⊂ Pn1, Z2 ⊂ Pn2, which satisfy the properties (a) and (b). We observe thatZ1∗Z2is dominated by aP1-bundle overZ1×Z2, namely
Z^1∗Z2:=P(OZ1(1)⊕ OZ2(1))→π Z1×Z2, (31) the two sections being contracted toZ1, resp. Z2, by the natural morphism toZ1∗Z2⊂Pn. The description (31) of the join immediately proves (a) for Z1∗Z2 once we have it forZ1
andZ2. We now turn to (b). Let h=OP(OZ
1(1)⊕OZ2(1)) on Z^1∗Z2 and let D0 be a fixed effective divisor and{DA}be a mobile family of divisors onZ^1∗Z2 such that
D0+ 2DA= 2h, (32)
dim{DA} ≥n. (33)
Then either D0 or DA is vertical for π. Indeed, they otherwise both restrict to a degree
≥1 divisor on the fibers ofπcontradicting (32). Assume D0is vertical forπ, that is,D0= π−1(D00). The equality 2h−D00= 2DAsays thatD00 = 2D000 as divisors onZ1×Z2and, asZ1
andZ2are simply connected,D000 ∈ |pr∗1D0,100 +pr∗2D0,200 |and both 2D000,1,2D000,2 are effective.
The divisorsD000,ionZi have the property that the linear system|h−pr∗1D0,100 −pr∗2D0,200 |on P(OZ1(1)⊕ OZ2(1)) has dimension ≥n, which says that
dim|OZ1(1)(−D0,100 )|+ dim|OZ2(1)(−D000,2)| ≥n1+n2.
As 2D0,100 is effective onZ1 and 2D0,200 is effective onZ2, we conclude thatD0,i00 = 0 and that theDA’s belong to|OZ1∗Z2(1)|, so (b) is proved in this case.
In the case whereD0is not vertical, then restricting again to the fibers ofπ,D0= 2h−D00 whereD00 is effective and comes fromZ1×Z2, andDA is vertical,DA =π−1(D0A). Hence we have againD00 = 2D000, and dim|h−D00−D0A| ≥n, so the proof concludes as before that D00 = 0 andDA= 0 which contradicts (33).
