1 Introuction Let X be a special cubic fourfol, h its hyperplane class, an T the class of an algebraic surface not homologous to any multiple of h 2.

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1 Special Cubic Fourfols Brenan Hassett University of Chicago September 18, 1998 Abstract A cubic fourfol is a smooth cubic hypersurface of imension four; it is special if it contains a surface not homologous to a complete intersection. Special cubic fourfols form a countably innite union of irreucible families C, each a ivisor in the mouli space C of cubic fourfols. For an innite number of these families, the Hoge structure on the nonspecial cohomology of the cubic fourfol is essentially the Hoge structure on the primitive cohomology of a K3 surface. We say that this K3 surface is associate to the special cubic fourfol. In these cases, C is relate to the mouli space N of egree K3 surfaces. In particular, C contains innitely many mouli spaces of polarize K3 surfaces as close subvarieties. We can often construct a corresponence of rational curves on the special cubic fourfol parametrize by the K3 surface which inuces the isomorphism of Hoge structures. For innitely many values of, the Fano variety of lines on the generic cubic fourfol of C is isomorphic to the Hilbert scheme of length-two subschemes of an associate K3 surface. Keywors: K3 surfaces, Noether-Lefschetz loci, perio omains AMS Subject Classications: 14C30, 14J28, 14J35 This work was supporte by Harvar University, a National Science Founation Grauate Fellowship, an a Sloan Founation Dissertation Fellowship. This paper was revise while the author was visiting the Institut Mittag-Leer. 1

2 1 Introuction Let X be a special cubic fourfol, h its hyperplane class, an T the class of an algebraic surface not homologous to any multiple of h 2. The iscriminant is ene as the iscriminant of the saturate lattice spanne by h 2 an T. Let C enote the special cubic fourfols of iscriminant (x 3.2). Theorem (Classication of Special Cubic Fourfols) ( Theorems 3.1.2, 3.2.3, an 4.3.1) C C is an irreucible ivisor an is nonempty i > 6 an 0; 2(mo 6). In section four, we give concrete escriptions of special cubic fourfols with small iscriminants an explain how certain Hoge structures at the bounary of the perio omain arise from singular cubic fourfols. The nonspecial cohomology of a special cubic fourfol consists of the mile cohomology orthogonal to the istinguishe classes h 2 an T. In many cases, it is essentially the primitive cohomology of a K3 surface of egree, which is sai to be associate to the special cubic fourfol. Furthermore, the varieties C surfaces. Let C mar are often closely relate to mouli spaces of polarize K3 enote the marke special cubic fourfols of iscriminant (x 5.2). This is the normalization of C if 2 (mo 6) an is a ouble cover of the normalization otherwise. Theorem (Associate K3 Surfaces an Maps of Mouli Spaces) (Theorems an 5.2.4) Special cubic fourfols of iscriminant have associate K3 surfaces i is not ivisible by four, nine, or any o prime p 1(mo 3). In these cases, there is an open immersion of C mar into the mouli space of polarize K3 surfaces of egree. In particular, an innite number of mouli spaces of polarize K3 surfaces may be realize as mouli spaces of special cubic fourfols. We can explain geometrically the existence of certain associate K3 surfaces. The Fano variety of X parametrizes the lines containe in it. For certain special cubic fourfols these Fano varieties are closely relate to K3 surfaces: Theorem (Geometry of Fano Varieties) (Theorem 6.1.4) Assume that = 2(n 2 +n+1) where n is an integer 2, an let X be a generic special cubic fourfol of iscriminant. Then the Fano variety of X is isomorphic to the Hilbert scheme of length-two subschemes of a K3 surface associate to X. 2

3 We shoul point out that the hypothesis on is stronger than necessary, but simplies the proof consierably. Combining this with the results on maps of mouli spaces, we obtain examples of istinct K3 surfaces with isomorphic Hilbert schemes of length-two subschemes (Proposition ) One motivation for this work is the rationality problem for cubic fourfols. The Hoge structures on cubic fourfols an their relevance to rationality questions have previously been stuie by Zarhin [30]. Izai [15] also stuie Hoge structures on cubic hypersurfaces with a view towar rationality questions. All the examples of cubic fourfols known to be rational ([10] [27] [5] [28]) are special an have associate K3 surfaces. Inee, a birational moel of the K3 surface is blown up in the birational map from P 4 to the cubic fourfol. Is this the case for all rational cubic fourfols? In a subsequent paper [14], we shall apply the methos of this paper to give new examples of rational cubic fourfols. We show there is a countably innite union of ivisors in C 8 parametrizing rational cubic fourfols (C 8 correspons to the cubic fourfols containing a plane). Throughout this paper we work over C. We use the term `generic' to mean `in the complement of some Zariski close proper subset.' The term `lattice' will enote a free abelian group equippe with a nonegenerate symmetric bilinear form. I woul like to acknowlege the help I receive in the course of this project. I benette from conversations with Davi Eisenbu an Elham Izai, an from suggestions by Walter Baily an Robert Frieman. Barry Mazur provie important insights an comments, an Johan e Jong mae many helpful observations an pointe out some errors in an early version of this paper. Finally, Joe Harris introuce me to this beautiful subject; his inspiration an avice have been invaluable. 2 Hoge Theory of Cubic Fourfols 2.1 Cohomology an the Abel-Jacobi Map Let X be a smooth cubic fourfol. The Hoge iamon of X has the form:

4 Let L enote the cohomology group H 4 (X; Z), L 0 the primitive cohomology H 4 (X; Z) 0, an h; i the symmetric nonegenerate intersection form on L. If h is the hyperplane class then h 2 2 L an L 0 = (h 2 )?. Our best tool for unerstaning the mile cohomology of X is the Abel- Jacobi mapping. Let F be the Fano variety of lines of X, the subvariety of the Grassmannian G (1; 5) parametrizing lines containe in X. This is a smooth fourfol ([1] x1). Let Z F X be the `universal line', with projections p an q. The Abel-Jacobi map is ene as the map = p q : H 4 (X; Z)! H 2 (F; Z): Let M = H 2 (F; Z), M 0 the primitive cohomology, an g the class of the hyperplane on F (inuce from the Grassmannian). Recall that (h 2 ) correspons to the lines meeting a coimension-two subspace of P 5, so (h 2 ) = g. Following [3] an [5], we ene the Beauville canonical form (; ) on M so that g an M 0 are orthogonal, (g; g) = 6, an (x; y) = 1 6 g2 xy for x; y 2 M 0. Extening by linearity we obtain an integral form on all of M. Proposition (Beauville-Donagi [5] Prop. 6) The Abel-Jacobi map inuces an isomorphism between L 0 an M 0 ; moreover, for x; y 2 L 0 we have ((x); (y)) = hx; yi. Inee, we may interpret is an isomorphism of Hoge structures H 4 (X; C ) 0! H 2 (F; C ) 0 ( 1): The 1 means that the weight is shifte by two; this reverses the sign of the intersection form. Proposition The mile integral cohomology lattice of a cubic fourfol is L = (+1) 21 ( 1) 2 i.e. the intersection form is iagonalizable over Z with entries 1 along the iagonal. The primitive cohomology is L 0 = B H H E8 E where B =, H = is the hyperbolic plane, an E is the positive enite quaratic form associate to the corresponing Dynkin iagram. 4

5 We rst prove the statement on the full cohomology. L is unimoular by Poincare uality an has signature (21; 2) by the Riemann bilinear relations. L is o because hh 2 ; h 2 i = h 4 = 3. Using the theory of inenite quaratic forms (e.g. [25] ch.5 x2.2) we conclue the result. Now we turn to the primitive cohomology L 0. By Proposition it suces to compute M 0 ; we rst compute M. In [5] Prop. 6, it is shown that F is a eformation of a variety S [2], where S is a egree-fourteen K3 surface an S [2] enotes the Hilbert scheme of length-two zero-imensional subschemes of S (also calle the blown-up symmetric square of S). By [3] x6 we have the canonical orthogonal ecomposition H 2 (S [2] ; Z) = H 2 (S; Z)? Z where (; ) = 2 an the restriction of (; ) to H 2 (S; Z) is the intersection form. Geometrically, the ivisor 2 correspons to the nonreuce lengthtwo subschemes of S. The cohomology lattice of a K3 surface is well-known (cf. [16] Prop. 1.2) H 2 (S; Z) = := H 3 ( E 8 ) 2 ; so M = H 3 ( E 8 ) 2 ( 2): Furthermore, the polarization g = 2f 5; where f 2 H 2 (S; Z) satises (f; f) = 14 [5]. The automorphisms of H 2 (S; Z) act transitively on the primitive vectors of a given nonzero length ([16] Theorem 2.4). If v 1 an w 1 are elements of the rst summan H with (v 1 ; v 1 ) = (w 1 ; w 1 ) = 0 an (v 1 ; w 1 ) = 1, then we may take f = v 1 + 7w 1 an g = 2v w 1 5. Using v 1 + 3w 1 2 an 5w 1 as the rst two elements of a basis of M 0, we obtain the result. Remark: Note that our computation shows that L 0 is even. 2.2 Hoge Theory an the Torelli Map We review Hoge theory in the context of cubic fourfols; a general introuction to Hoge theory is [12]. Recall that a complete marking of a polarize cubic fourfol is an isomorphism : H 4 (X; Z)! L mapping the square of the hyperplane class to h 2 2 L. If we are given a complete marking, the complex structure on X etermines a istinguishe subspace F 3 (X) = H 3;1 (X; C ) L 0 C satisfying the following properties: 1. F 3 (X) is isotropic with respect to the intersection form <; >; 5

6 2. the Hermitian form H(u; v) = hu; vi on F 3 (X) is positive. Let Q P(L 0 C ) be the quaric hypersurface ene by (1), an let U Q be the topologically open subset where (2) also hols. U is a homogeneous space for the real Lie group SO(L 0 R ) = SO(20; 2). This group has two components; one of them reverses the orientation on the negative enite part of L 0 R, which coincies with (F 3 F 3 ) \ L 0 R. Changing the orientation correspons to exchanging F 3 an F 3 (see x6 of the appenix to [24] for etails). Hence the two connecte components of U parametrize the subspaces F 3 an F 3 = H 1;3 (X) respectively; we enote them D 0 an D 0. The component D 0 is a twenty-imensional open complex manifol, calle the local perio omain for cubic fourfols. Let enote the group of automorphisms of L preserving the intersection form an the istinguishe class h 2, an + the subgroup stabilizing D 0. This is the inex-two subgroup of which preserves the orientation on the negative enite part of L 0 R. + acts holomorphically on D 0 from the left; for a point in D 0 corresponing to the marke cubic fourfol (X; ) the action is (X; ) = (X; ). The orbit space D = + nd 0 exists as an analytic space an is calle the global perio omain. Two cubic fourfols are isomorphic i they are projectively equivalent. Let C enote the coarse mouli space for smooth cubic fourfols, constructe as a Geometric Invariant Theory quotient [18] ch.4 x2. Each cubic fourfol etermines a point in D, an the corresponing map : C! D is calle the perio map. By general results of Hoge theory, this is a holomorphic map of twenty-imensional analytic spaces. For cubic fourfols we can say much more. First, we have the following result ue to Voisin: Theorem (Torelli Theorem for Cubic Fourfols[29]) : C! D is an open immersion of analytic spaces. In particular, if X 1 an X 2 are cubic fourfols an there exists an isomorphism of Hoge structures : H 4 (X 1 ; C )! H 4 (X 2 ; C ), then X 1 an X 2 are isomorphic. Secon, is not just an analytic map: Proposition D is a quasi-projective variety of imension twenty an : C! D is an algebraic map. In x6 of the appenix to [24], it is shown that the manifol D 0 is a boune symmetric omain of type IV. The group + is arithmetically ene an acts 6

7 holomorphically on D 0. In this situation we may introuce the Borel-Baily compactication ([2] x10): there exists a compactication of D 0, compatible with the action of +, so that the quotient is projective. Moreover, + nd 0 is a Zariski open subvariety of this quotient. To complete the proof, we use the following consequence of A. Borel's extension theorem [6]: Let D 0 be a boune symmetric omain, an G an arithmetically ene torsion-free group of automorphisms. Let D = GnD 0 be the quasi-projective quotient space, an Z an algebraic variety. Then any holomorphic map Z! D is algebraically ene. While + has torsion, some normal subgroup H of nite inex is torsionfree ([24] IV Lemma 7.2). Let + (N) enote the subgroup of + acting trivially on L=NL. For some large N, + =H acts faithfully on L=NL so + (N) H an is torsion-free. Let C(N) enote the mouli space of cubic fourfols with marke Z=N Z cohomology. This is a nite (an perhaps isconnecte) cover of C; we use C 0 (N) to enote a connecte component. Let D(N) = + (N)nD 0, which is also nite over D. The perio map lifts to a map N : C 0 (N)! D(N): By Borel's theorem N is algebraic, an a escent argument implies is also algebraic. Remark: It follows that C is a Zariski open subset of D an its complement is ene by algebraic equations. 3 Special Cubic Fourfols 3.1 Basic Denitions Denition A cubic fourfol X is special if it contains an algebraic surface T not homologous to a complete intersection. Let A(X) = H 2;2 (X) \ H 4 (X; Z), which is positive enite by the Riemann bilinear relations. The Hoge conjecture is true for cubic fourfols [31], so A(X) is generate (over Q) by the classes of algebraic cycles; X is special if an only if the rank of A(X) is at least two. This is equivalent to saying that the rank of L \ F 3 (X)? is at least two, or that L 0 \ F 3 (X)? 6= 0. A Hoge structure x 2 D 0 is special if L 0 \ F 3 (x)? is nonzero. Theorem (Structure of Special Cubic Fourfols) Let K L be a positive enite rank-two saturate sublattice containing h 2, [K] the + 7

8 orbit of K, an C [K] the cubic fourfols X such that A(X) K 0 for some K 0 2 [K]. Every special cubic fourfol is containe in some C [K], which is an irreucible algebraic ivisor of C, an is nonempty for all but a nite number of [K]. Given such a lattice K, we set K 0 = K \ L 0. Let DK 0 be the x 2 D 0 such that K 0 x? ; this is a hyperplane section of D 0 P(L 0 C ). Each special Hoge structure is containe in some DK. 0 Let K? enote the orthogonal complement to K in L. We see that DK 0 is a topologically open subset of a quaric hypersurface in P(K C? ), has imension nineteen, an classies Hoge structures structures on the lattice K?. As in the previous section, we can prove that DK 0 is a boune symmetric omain of type IV. Let + K = f 2 + : (K) Kg. As before, the quotient + KnD 0 K is quasi-projective. Furthermore, the inuce holomorphic map + KnD 0 K! + nd 0 = D is algebraically ene, so its image is an irreucible algebraic ivisor. We enumerate the ivisors parametrizing special Hoge structures in D. for some K L as above, but K is not Each one correspons to + KnDK 0 uniquely etermine. K 1 an K 2 give rise to the same ivisor if an only if K 1 = (K 2 ) for some 2 +, i.e. + K 1 an + K 2 are conjugate in +. Let D [K] enote the corresponing irreucible ivisor in D. Since C is Zariski open in D (Proposition 2.2.2), C [K] = C \ D [K] is an irreucible algebraic ivisor in C, an D [K] (D C) for nitely many [K]. Denition Let (K; h; i) be a positive enite rank-two lattice containing a istinguishe element h 2 with hh 2 ; h 2 i = 3. A marke (resp. labelle) special cubic fourfol is a cubic fourfol X with the ata of a primitive imbeing of lattices K,! A(X) preserving h 2 (resp. the image of such an imbeing.) A special cubic fourfol is typical if it has a unique labelling. We write D lab for [K] + KnDK 0. The morphism Dlab [K]! D [K] is birational (inee D lab is the normalization of D [K] [K]), so a general point in D [K] has a unique labelling. The ber prouct D lab [K] D C will be enote C lab. [K] 3.2 Discriminants an Special Cubic Fourfols Denition The iscriminant of a labelle special cubic fourfol (X; K) is the eterminant of the intersection matrix of K. 8

9 Proposition Let (X; K) be a labelle special cubic fourfol of iscriminant an let v be a generator of K > 0 an 0; 1 (mo 3) ; 2. 0 := hv; vi = 3. hv; L 0 i = 4. is even. ( ( 3 if 1 (mo 3) ; if 0 (mo 3) 3 3Z if 1 (mo 3) ; Z if 0 (mo 3) The rst three statements are straightforwar computations, so we omit their proofs. The fourth follows from the remark after Proposition We rene the results of the previous section by classifying the orbits of the rank-two sublattices uner the action of +. The following theorem is a consequence of Theorem an Proposition 3.2.4: Theorem (Irreucibility Theorem) The special cubic fourfols possessing a labelling of iscriminant form an irreucible (possibly empty) algebraic ivisor C C. Elements of C are calle special cubic fourfols of iscriminant ; the corresponing rank-two lattice is enote K. We write D for D [K ], D lab for D lab, C [K ] for C [K ], an C lab for C lab. [K ] Proposition Let K an K 0 be saturate rank-two nonegenerate sublattices of L containing h 2. Then K = (K 0 ) for some 2 + if an only if K an K 0 have the same iscriminant. We claim it suces to prove the result for. We n some g 2 + stabilizing sublattices with every possible iscriminant. Take g to be the ientity except on the secon hyperbolic plane in the orthogonal ecomposition for L 0 ; on this component set g equal to multiplication by 1. (We refer to the computation of L 0 in Proposition ) Now we analyze the action of on our rank-two sublattices, or equivalently, on saturate nonegenerate rank-one sublattices K 0 L 0. We apply the results of Nikulin on iscriminant groups an quaratic forms; see [22] 9

10 or [9] for basic enitions an proofs. The elements of x h 2, so they act trivially on the iscriminant groups (Zh 2 ) an (L 0 ) [22] x1.5. Conversely, any automorphism of L 0 that acts trivially on (L 0 ) extens to an element of [22] Let K 0 enote a lattice generate by an element v with hv; vi = 0, q K 0 the quaratic form on (K 0 ), an q the quaratic form on (L 0 ). The lattice L 0 is the unique even lattice of signature (20; 2) with iscriminant quaratic form q [22] Any saturate coimension-one sublattice K? L 0 is the orthogonal complement in L of a rank-two sublattice K, so there is an inuce isomorphism (K? ) = (K) [22] 1.6.1, an (K? ) is generate by at most two elements. This implies the isomorphism class of K? is etermine by its signature an iscriminant form, an any isomorphism of (K? ) preserving the iscriminant quaratic form is inuce by an automorphism of K? [22] Two primitive imbeings of i : K 0! L 0 iering only by an element of are sai to be congruent. Applying the results of [22] x1.15 in our situation, we n the primitive imbeings i : K 0! L 0 correspon to the following ata: 1. a subgroup H q (L 0 ); 2. a subgroup H K 0 (K 0 ); 3. an isomorphism : H K 0! H q preserving the restrictions of the quaratic forms to these subgroups, with graph = f(h; (h)) : h 2 H K 0g (K 0 ) (L 0 ); 4. an even lattice K? with complementary signature an iscriminant form q K?, an an isomorphism K? : q K?!, where = ((q K 0 q)j?)= (an? is the orthogonal complement to with respect to q K 0 q). Another imbeing i 0 with ata (H 0 q ; H 0 K 0 ; 0 ; (K 0 )? ; (K 0 )?) is congruent to i if an only if H K 0 = H 0 K 0 an = 0. Our proof now ivies into two cases. In the rst case H q = f0g, or equivalently, hi(k 0 ); L 0 )i = Z (i.e. 3j). By the characterization above, all primitive imbeings of K 0 are congruent. In the secon case H q = (L 0 ) = Z=3Z, or equivalently, hi(k 0 ); L 0 )i = 3Z. In this case, (K 0 ) has a subgroup H K 0 of orer three an 3j 0. There are two possible isomorphisms between (L 0 ) an H K 0, thus two congruence classes of imbeings of K 0 into L 0. 10

11 Using [22] x1.15 an Proposition 3.2.2, we can compute the iscriminant quaratic forms of the lattices K? : Proposition If 0 (mo 6) then (K?) = Z= Z Z=3Z, which 3 is cyclic unless 9j. We may choose this isomorphism so that q K?(0; 1) = 2 (mo 2Z) an q 3 K?(1; 0) = 3 (mo 2Z). If 2 (mo 6) then (K? ) = Z=Z. We may choose a generator u so that q K? (u) = Examples (mo 2Z). 4.1 Special Cubic Fourfols with Small Discriminants The examples here are iscusse in more etail in [13]. If T is a smooth surface containe in a cubic fourfol X then ht; Ti = c 2 (N T =X ) = 6h 2 T + 3h T K T + K 2 T T where T is the topological Euler characteristic an h T is the hyperplane class restricte to T =8: Cubic Fourfols Containing a Plane (see [29]) X contains a plane P, so that hp; Pi = 3 an our marking is K 8 = h 2 P h P 1 3 : The cubic fourfols in C 8 generally contain other surfaces, like quaric surfaces an quartic el Pezzo surfaces =12: Cubic Fourfols Containing a Cubic Scroll X contains a rational normal cubic scroll T, so that ht; Ti = 7 an our marking is h 2 T K 12 = h : T

12 4.1.3 =14: Cubic Fourfols Containing a Quartic Scroll/Pfaan Cubic Fourfols X is a cubic fourfol containing a rational normal quartic scroll T, so that ht; Ti = 10 an our marking is K 14 = h 2 T h : T 4 10 Special cubic fourfols of iscriminant 14 generally also contain quintic el Pezzo surfaces an quintic rational scrolls. One can show that the quartic scrolls, quintic scrolls, an quintic el Pezzos on X form families of imensions two, two, an ve respectively. Note that Morin [17] uses a spurious parameter count to euce that the quartic scrolls form a one imensional family. From this, he conclues incorrectly that every cubic fourfol contains a quartic scroll. Another escription of an open subset of C 14 is the Pfaan construction of Beauville an Donagi [5]. The imension counts above follow easily from their results. They also show that the Pfaan cubic fourfols are rational. Finally, we shoul point out that the cubic fourfols containing two isjoint planes possess a marking with iscriminant 14, an thus are also containe in C 14. (See [10] an [27] for more iscussion of these examples.) =20: Cubic Fourfols Containing a Veronese X contains a Veronese surface V, so that hv; V i = 12 an our marking is K 20 = h 2 V h : V =6: Cubic Fourfols with Double Points A ouble point is orinary if its projectivize tangent cone is smooth. Cubic hypersurfaces in P 5 with an orinary ouble point are stable in the sense of Geometric Invariant Theory. This is prove using Mumfor's numerical criterion for stability ([18] x2.1) an the methos of ([18] x4.2). Let ~C enote the quasi-projective variety parametrizing cubic fourfols with (at worst) a single orinary ouble point. 12

13 Let X 0 be a cubic fourfol with a single orinary ouble point p. Projection from p gives a birational map p : X KP 4 which can be factore X 0 = Bl S (P 4 ) q 2??y q! 1 X 0 P 4 where q 1 is the blow-up of the ouble point p an q 2 is the blow-own of the lines containe in X 0 passing through p. These lines are parametrize by a surface S P 4, which is the complete intersection of a quaric an a cubic. The quaric is nonsingular because p is orinary; the complete intersection is smooth because p is the only singularity of X 0. In particular, S is a sextic K3 surface. The inverse map p 1 is given by the linear system of cubic polynomials through this K3 surface. Conversely, given any sextic K3 surface containe in a smooth quaric, the image of P 4 uner this linear system is a cubic fourfol with an orinary ouble point. Note that the sextic K3 surfaces containe in a singular quaric hypersurface are precisely those containing a cubic plane curve. This construction suggests that we associate a sextic K3 surface to any element of ~ C C: Proposition The Torelli map extens to an open immersion ~ : ~C! D: The close set ~ C 6 := ~ C C is mappe into D 6. In x 5.2 we shall see that D 6 coincies with the perio omain for sextic K3 surfaces. A etaile proof of the proposition is given in x4 of [29], so we merely explain some etails neee for our calculations. (It also follows from the elicate analysis of singular cubic fourfols in x 6.3.) Let X 0 be a cubic fourfol with an orinary ouble point an let S be the associate K3 surface. Smoothings of orinary ouble points of even coimension have monoromy satisfying T 2 = I, so any smoothing of X 0 yiels a pure limiting mixe Hoge structure H 4 lim. The corresponing point of the perio omain is enote ~(X 0 ). The limiting Hoge structure may be compute with the Clemens- Schmi exact sequence [7], which implies there is a natural imbeing of the primitive cohomology H 2 (S; C ) 0 ( 1) into H 4 lim. The orthogonal complement to the image consists of a rank-two lattice of integral (2; 2) classes K 6 = h 2 T h T

14 so ~(X 0 ) 2 D Existence of Special Cubic Fourfols D D is nonempty if an only if is positive an congruent to 0; 2 (mo 6) (Proposition 3.2.2), so we restrict to these values of. Theorem (Existence of Special Cubic Fourfols) Let > 6 be an integer with 0; 2 (mo 6). Then the ivisor C is nonempty. We saw in the last section why there are no smooth cubic fourfols of iscriminant six: D 6 correspons to the limiting Hoge structures arising from cubic fourfols with ouble points. In the next section we shall explain why there are no cubic fourfols of iscriminant two: D 2 correspons to the limiting Hoge structures arising from another class of singular cubic fourfols. Is the complement D C equal to D 2 [ D 6? To prove the theorem, we nee the following lemmas: Lemma Let P be an inenite even rank-two lattice representing six. Assume that P is not isomorphic to any of the following: Then there exists a smooth sextic K3 surface S lying on a smooth quaric with Pic(S) = P. Lemma Let P be a rank-two inenite even lattice, f 2 P a primitive element with := f 2 > 0, an assume there is no E 2 P with E 2 = 2 an fe = 0. Then there exists a K3 surface S with Pic(S) = P an f a polarization on S. Moreover, f is very ample unless there exists an elliptic curve C on S with C 2 = 0 an fc = 1 or 2. Recall that enotes the lattice isomorphic to the mile cohomology of a K3 surface. Using the results of x2 of [16], there exists an imbeing P,!. So for some elements of the perio omain P equals the lattice of (1; 1)-classes. The surjectivity of the perio map for K3 surfaces implies the existence of a K3 surface S with Picar group P so that f containe in the Kahler cone of S (see pp. 127 of [4]). This implies f is a polarization of S. 14 :

15 To complete the proof, we apply Saint Donat's results for linear systems on K3 surfaces [23]. Specically, we use Theorems 3.1, 5.2, an 6.1, along with the analysis of xe components in x2.7. To prove Lemma 4.3.2, we note that the image uner jfj is not containe in a singular quaric because P (i.e. S oes not contain a plane cubic). Now we prove the theorem. Let S be one of the K3 surfaces constructe in Lemma an X 0 the corresponing singular cubic fourfol. Let v 2 P be primitive with respect to the sextic polarization. Recall that H 2 (S; C ) 0 ( 1) is naturally imbee into the limiting Hoge structure H 4 lim arising from X 0. The image of v is an integral class of type (2; 2) in H 4, enote lim v0. Relabel H 4 by letting K lim enote the saturation of the lattice Zh 2 + Zv 0. By Proposition 3.2.2, = 1 isc(p ). For each 0; 2 (mo 6) ; > 6, there 2 exist lattices P satisfying the hypotheses of Lemma with iscriminant 2. If = 6n (resp. = 6n + 2) we may take P = n 6 2 (resp. ): 2 2n Set x 0 = ~(X 0 ) so that x 0 2 D 6 \ D : We construct a smoothing : X! 0 where X t is smooth for t 6= 0, an (X t ) 2 D. Let :! D be a holomorphic map such that (0) = x 0 an (u) 2 D D 6 for u 6= 0. The existence of such a curve follows from the construction of D as the quotient + nd 0. Because ~ is an open immersion, we may shrink so that lifts through ~, giving a map :! ~C. Consequently, there exists a ramie base change b : 0! an a family X! 0 so that X t = (b(t)). By construction we have X t 2 C \ 1 (D ) = C for t 6= 0, so C 6= ;: 4.4 =2: The Determinantal Cubic Fourfol The eterminantal cubic fourfol X 0 is ene by the homogeneous equation: R := a b c b e = 0: c e f It is singular where the 2 2 minors of the eterminant are simultaneously zero, i.e. along a Veronese surface V. We shall consier eformations X! 15

16 of X 0 with equations R + tg, where G is the equation of a smooth cubic fourfol, an the curve C V ene by the equation Gj V = 0 is also smooth. Let S be the ouble cover of V branche over C, a egree-two K3 surface. Theorem The limiting mixe Hoge structure arising from X! is pure an special of iscriminant two. The orthogonal complement to K 2 is isomorphic to the primitive Hoge structure H 2 (S; C ) 0 ( 1). This result will not be use elsewhere in this paper. Its proof is essentially a calculation on the semistable reuction for X using the Clemens-Schmi exact sequence [7] (see [13] for etails). Geometrically, X 0 is containe in the ineterminacy locus of the Torelli map, but after blowing up the map is well-ene at the generic point of the exceptional ivisor. Moreover, this exceptional ivisor maps birationally to D 2 D. 5 Associate K3 Surfaces 5.1 Nonspecial Cohomology Denition The nonspecial cohomology lattice of a labelle special cubic fourfol (X; K ) is ene as the orthogonal complement K?. The nonspecial cohomology, enote W X;K, is the polarize Hoge structure inuce on K? by the Hoge structure on H 4 (X; C ) 0. Proposition Let (X; K 14 ) be a generic Pfaan cubic fourfol. Then there exists a egree-fourteen K3 surface S an an isomorphism of Hoge structures W X;K14 = H 2 (S; C ) 0 ( 1): This is a consequence of [5] Prop. 6 (cf. x 2.1) an Proposition However, it is best explaine by observing that the birational map P KX blows up a surface birational to S, which therefore parametrizes a corresponence of rational curves on X. Motivate by this example, we etermine the special cubic fourfols whose nonspecial cohomology is isomorphic to the primitive cohomology of a polarize K3 surface: 16

17 Theorem (Existence of Associate K3 Surfaces) Let (X; K ) be a labelle special cubic fourfol of iscriminant, with nonspecial cohomology W X;K. There exists a polarize K3 surface (S; f) such that W X;K = H 2 (S; C ) 0 ( 1) if an only if the following conitions are satise: j an 96 j ; 2. p6 j if p is an o prime, p 1(mo 3). We say that the pair (S; f) is associate to (X; K ). We rst show the theorem boils own to a computation of lattices (Proposition 5.1.4). Recall that a pseuo-polarization is a ivisor f containe in the closure of the Kahler cone with (f; f) > 0; the primitive cohomology of a pseuo-polarize K3 surface (S; f) is the orthogonal complement to f in H 2 (S; Z). Let 0 be a lattice isomorphic to the primitive mile cohomology of a egree K3 surface. The isomorphism asserte in the theorem implies an isomorphism of lattices K? = 0. On the other han, given a labelle special cubic fourfol (X; K ) an an isomorphism of lattices K? = 0, W X;K (+1) has the form of the primitive cohomology of a pseuo-polarize K3 surface. Inee, since the Torelli map for K3 surfaces is surjective [4] [26], there exists a pseuo-polarize K3 surface (S; f) such that H 2 (S; C ) 0 ( 1) = WX;K. Moreover, X is smooth so H 4 (X; Z) 0 \ H 2;2 (X) oes not contain any classes with self-intersection +2 ([29] x4 Prop. 1). Therefore there are no ( 2)-curves on S orthogonal to f, an f is actually a polarization. Proposition Retain the notation above. K? conitions of Theorem are satise. = 0 if an only if the The automorphisms of = H 2 (S; Z) act transitively on the primitive vectors with (v; v) = 6= 0 ([16] Theorem 2.4), so 0 = ( ) H 2 ( E 8 ) 2 ; let y enote the istinguishe element with (y; y) =. The iscriminant group ( 0) an quaratic form q 0 are equal to Z( y )=Zy, with q 0( y ) = 1( mo 2Z). We etermine when (K?) an ( 0 ) are isomorphic as groups with a Q=2Z-value quaratic form. We rst consier the case 2 (mo 6). Here 17

18 both iscriminant groups are isomorphic to Z=Z, so we just nee to check when the quaratic forms are conjugate by an automorphism of Z=Z. Let u an w be generators of (K?) an ( 0) such that q K? 2 1 (u) = (mo 2Z) 3 an q 0 (w) = 1 (mo 2Z) (see Proposition 3.2.5). The quaratic forms are conjugate if an only if the integer 2 1 is a square moulo 2, or equivalently, 3 3 is a square moulo 2. By quaratic reciprocity this is the case if an only if is not ivisible by four an any o prime pj satises p 6 1 (mo 3). A similar argument hols in the case 0 (mo 6). We have seen that the conitions on are necessary for K? to be isomorphic to 0. On the other han, K? is the unique even lattice of signature (19; 2) with iscriminant form ((K?); q K? ) [22] Hence if the iscriminant forms of K? an 0 agree then K? = Isomorphisms of Perio Domains We retain the notation of x 2.1 an x 5.1. Let enote the automorphisms of, an the automorphisms xing some primitive v 2 with (v; v) =, which yiel automorphisms of 0 = v?. As in x 2.2, let N 0 be the local perio omain for egree K3 surfaces, an open nineteen-imensional complex manifol. Let + enote the subgroup stabilizing N 0. As before, N 0 is a boune symmetric omain of type IV, + is an arithmetic group acting holomorphically on N 0, an the quotient N := + =N 0 is therefore a quasi-projective variety, the global perio omain for egree K3 surfaces. We introuce a bit more notation for special cubic fourfols as well. Let G + + be the subgroup acting trivially on K an let D mar enote the marke special Hoge structures of iscriminant, moulo the action of. The ber prouct Dmar D C is written C mar, the marke special cubic G + fourfols of iscriminant. We have natural forgetting maps D mar an C mar! C lab :! D lab Proposition G + = + if 2 (mo 6) an G+ + subgroup if 0 (mo 6). The natural map D mar! D lab if 2 (mo 6) an a ouble cover if 0 (mo 6). Furthermore, D mar G + nd 0 an thus is connecte for all 6= 6. is an inex-two is an isomorphism = We begin with the rst statement. The lattice K has no automorphisms preserving h 2 if 2 (mo 6), so G + = +. If 0 (mo 6) then K has an involution, which acts on K 0 as multiplication by 1. We claim 18

19 it extens to an element 2 +. By Proposition we may assume K 0 = Z(v 1 + w 6 1). We use the notation of x 2.1, so v 1 an w 1 form a basis for a hyperbolic summan H L 0. Choose equal to multiplication by 1 on both hyperbolic summans of L 0 an equal to the ientity elsewhere. We have that 2 + but 62 G+, so G+ is a proper subgroup of +. The secon statement follows immeiately from the rst. As for the thir statement, recall that D lab = + nd 0. Hence for 2 (mo 6) the result is immeiate. For 0 (mo 6), we must check that any 2 + acting nontrivially on K also acts nontrivially on D 0. For 6= 6, if acts nontrivially on K then the inuce action on (K ) is not equal to 1. However, the groups (K ) an (K? ) are isomorphic, so the inuce action on (K?) is not 1. Now D0 is a topologically open subset of a quaric hypersurface in P(K? C ), so only scalar multiplications act trivially on D 0. In particular, necessarily acts nontrivially. Remark: There exists an element G + 6 acting trivially on K? 6. It follows that D6 mar 6= G + 6 nd6 0 but rather that D6 lab = G + 6 nd6. 0 Theorem Let be a positive integer such that there exists an isomorphism j : K?! 0 (see Proposition ) Choose orientations on the negative enite parts of K? an 0 compatible with j, so there is an inuce isomorphism of local perio omains D 0 an N 0. If 6= 6 then there is an inuce isomorphism i : D mar! N ; we also have D lab 6 = N6 : The isomorphism of perio omains epens on the choice of j. Each j inuces an isomorphism of iscriminant groups j 0 : (K?)! ( 0) preserving the Q =2Z-value quaratic forms on these groups [22] x1.3. We enote the set of such isomorphisms Isom((K? ); ( 0 )); the group fn 2 Z=Z : n 2 = 1g acts faithfully an transitively on this set.! N Theorem For 6= 6, the various isomorphisms i : D mar correspon to elements of Isom((K?); ( 0 ))=(1). The isomorphism i 6 : D lab 6! N 6 is unique. These two theorems have the following corollary: Corollary (Immersions into Mouli Spaces of K3 Surfaces) Let 6= 6 be a positive integer such that there exists an isomorphism j : K?! 0. Then there is an imbeing i : C mar,! N ; unique up to the choice of an element of Isom((K?); ( 0 ))=(1): Moreover, there is a unique imbeing i 6 : ~C lab 6,! N 6. 19

20 As we shall see in x6, geometrical consierations will sometimes manate specic choices of i (e.g. in the case = 14). We prove the rst theorem. First, we compare the action of + on 0 to the action of G + on K?. We claim that + is the group of automorphisms of 0 preserving the orientation on the positive enite part of 0 R an acting trivially on the iscriminant group ( 0 ). This follows from the results of [22] x1.4, which imply that any such automorphism extens uniquely to an element of +. Similarly, G+ is the group of automorphisms of K? preserving the orientation on the negative enite part of K? R an acting trivially on the iscriminant group (K?). Now suppose we are given an isomorphism j : K?! 0. This inuces! +, an i : G + nd 0! + nn 0. Applying Proposition 5.2.1, we obtain an isomorphism i : D mar! N for 6= 6. The remark after the proposition also yiels an isomorphism i 6 : D lab 6! N 6. isomorphisms D 0! N 0, G+ We turn to the proof of the secon theorem. We must etermine when inuce the then two ierent isomorphisms j 1 : K?! 0 an j2 : K?! 0 same isomorphism i : G + nd 0! + nn 0. If j2 = j1 for some 2 + j 1 an j2 inuce the same isomorphisms of perio omains. Also, if j1 = j2 then j 1 an j2 inuce the same isomorphism between D0 an N 0, because these manifols lie in the projective spaces P(K? C ) an P( 0 C ). On the other han, assume that j 1 an j 2 inuce the same isomorphism between G + nd 0 an + nn 0. Then there exist 2 G+ an 2 + such that j 1 an j 2 inuce the same isomorphism between D0 an N 0, so j 1 = j 2. We conclue that the isomorphisms between G+ nd 0 an + nn 0 correspon to certain elements of Isom((K? ); ( 0))=(1). It remains to check that each element of Isom((K?); ( 0 ))=(1) actually arises from an isomorphism between K? an 0 respecting the orientations on the negative enite parts. Now K? has an automorphism g reversing the orientation on the negative part an acting trivially on (K ). Take g to be the ientity except on a hyperbolic summan of the orthogonal ecomposition for K? ; on the hyperbolic summan set g equal to multiplication by 1. Hence it suces to show that the automorphisms of K? inuce all the automorphisms of (K? ), which is prove in [22], Theorem an Remark

21 6 Fano Varieties of Special Cubic Fourfols 6.1 Introuction an Necessary Conitions Here we provie a geometric explanation for the K3 surfaces associate to some special cubic fourfols. The general philosophy unerlying our approach is ue to Mukai [19],[20],[21]. Let S be a polarize K3 surface an let M S be a mouli space of simple sheaves on S. Quite generally, M S is smooth an possesses a natural nonegenerate holomorphic two-form ([19] Theorem 0.1). Furthermore, the Chern classes of the `quasi-universal sheaf' on SM S inuce corresponences between S an M S. If M S is compact of imension two then it is a K3 surface isogenous to S; the Hoge structure of M S can be rea o from the Hoge structure of S an the numerical invariants of the sheaves ([20] Theorem 1.5). Conversely, given a variety F with a nonegenerate holomorphic two-form an an isogeny H 2 (S; Q)! H 2 (F; Q), one can try to interpret F as a mouli space of sheaves on S. In the case where F is a K3 surface, we often have such interpretations ([20] Theorem 1.9). Note that F = S [n] can be interprette as the mouli space of ieal sheaves on S of colength n; such sheaves are simple. Proposition Let X be a cubic fourfol with Fano variety F. Assume there is an isomorphism between F an S [2] for some K3 surface S. Then X has a labelling K such that S is associate to (X; K ); i : C mar,! N may be chosen so that i (X; K ) = S. If (X 1 ; K ) is a generic element of C mar an S 1 = i (X 1 ; K ), then the Fano variety F 1 is isomorphic to S [2] 1. For nongeneric X 1 the isomorphism between F 1 an S [2] 1 can break own. Let X 1 contain two isjoint planes 1 an 2, so that X 1 2 C 14. The proposition hols for = 14, but the (birational) map between F 1 an S [2] 1 acquires ineterminacy at the lines supporte in the i (see [13] for etails). We prove the proposition. As in x2.1, there is an isomorphism H 2 (F; Z) = H 2 (S; Z)? Z an the hyperplane class g = af b where f is some polarization of S with := (f; f). Let K? equal 1 (H 2 (S; Z) 0 ( 1)) where is the Abel-Jacobi map, an set K = (K? )? : Applying Theorem with j = jk?, we obtain a map i with the esire properties. To explain i geometrically, we nee the following result: Theorem (Deformation Spaces of S [2] [3]) Let S be a K3 surface an 2 S [2] be the elements supporte at a single point. The eformation 21

22 space of S [2] is smooth an has imension twenty-one. Deformations of the form S [2] 1 correspon to a ivisor in this space which may be characterize as the eformations for which remains a ivisor. We ene C as the eformations of F for which remains algebraic. Applying Theorem 6.1.2, there is some small analytic neighborhoo in C where the eformations are isomorphic to S [2] 1 for some eformation S 1 of S. This isomorphism hols in an open etale neighborhoo of X in C, so a generic cubic fourfol in C has Fano variety of the form S [2] 1. For which values of are the conclusions of Proposition vali? Theorem gives sucient conitions for the existence of a K3 surface associate to (X; K ), but these o not guarantee that F = S [2] : Proposition Assume that the Fano variety of a generic special cubic fourfol of iscriminant is isomorphic to S [2] for some K3 surface S. Then there exist positive integers n an a such that = 2 n2 +n+1 a 2 : This is equivalent to the existence of a line bunle on S [2] of egree 108, the egree of the Fano variety. For instance, Fano varieties of special cubic fourfols of iscriminant 74 are not generally of the form S [2], because 74a 2 = 2(n 2 + n + 1) has no integral solutions (see [11]). We can prouce innitely many examples of special cubic fourfols with Fano variety isomorphic to the symmetric square of a K3 surface: Theorem Assume that = 2(n 2 + n + 1) where n is an integer 2. Then the Fano variety of a generic special cubic fourfol X of iscriminant is isomorphic to S [2], where S is a K3 surface associate to (X; K ). This is prove in the next two sections. The conition on correspons to setting a = 1 in Proposition The proof of the theorem suggests that the conition of the proposition is the correct sucient conition. 6.2 Ambiguous Symplectic Varieties Denition Let F be an irreucible symplectic Kahler manifol, an assume that there exist K3 surfaces S 1 an S 2 an isomorphisms r 1 : F! S [2] 1 an r 2 : F! S [2] 2 such that r1 1 6= r2 2. Then we say that F is ambiguous. 22

23 Our rst example is a special case of a construction of Beauville an Debarre [8]. Let S be a smooth quartic surface in P 3, p 1 + p 2 a generic point in S [2], an `(p 1 + p 2 ) the line containing p 1 an p 2. By Bezout's theorem `(p 1 + p 2 )\S = p 1 + p 2 + q 1 + q 2 : Setting j(p 1 + p 2 ) = q 1 + q 2 for each p 1 + p 2, we obtain a birational involution j : S [2] 9 9 KS [2] : If S contains no lines then j extens to a biregular morphism. Let f 4 be the egree-four polarization on S an the corresponing class on S [2]. Following [8], one may compute j (x) = x + (x; f 4 ) (f 4 ) on H 2 (S [2] ; Z). Setting r 2 = j r 1, we n that F = S [2] is ambiguous. We igress to give another beautiful example of ambiguous varieties: Proposition Assume that 3j an that the Fano variety F of a generic cubic fourfol in C is isomorphic to S [2] 1 for some K3 surface S 1. Then F is ambiguous. This follows immeiately from Proposition an the results of x5.2, which imply that C lab imbes into a Z=2Z-quotient of N if 3j. 6.3 Construction of the Examples Let X 0 2 ~C 6, F 0 its Fano variety of lines, an S the sextic K3 surface associate to X 0 (see x 4.2). Let : X! be a family in ~C with central ber X 0 an X t smooth for t 6= 0. Let F! be the corresponing family of Fano varieties an X 0! 0 a semistable reuction of X!. For simplicity, we assume that the central ber of the semistable family is of the form X 0 0 = X 0 [ Q where X 0 = Bl S (P 4 ) is the esingularization of X 0, Q is a smooth quaric fourfol, an Q 0 = X 0 \ Q is the smooth quaric in P 4 containing S. This is the case if is a suciently generic smoothing of X 0. Lemma F 0 is singular along the lines through the ouble point, which are parametrize by S. These singularities are orinary coimension-two ouble points an the blow-up : Bl S F 0! F 0 esingularizes F 0. If S 0 oes not contain a line then Bl S F 0 = S [2]. The rst part follows from x 4.2 an [1] For the secon part, we realize by blowing up the Grassmannian G (1; 5) along the locus L(p) of lines containing p. The ber square S! F 0 # # L(p)! G (1; 5) 23

24 gives a natural close imbeing of normal cones C S F 0,! C L(p) G (1; 5)jS: The projectivization P(C L(p) G (1; 5)) correspons to P(C 6 =S), where S is the restriction of the universal subbunle. Note L(p) = P4 an C L(p) G (1; 5)` correspons to the lines such that ` 2 P 4. For ` 2 Sing(F 0 ) the ber of P(C S F 0 )` correspons to those lines such that ` 2 Q 0. These are parametrize by a smooth conic curve, hence F 0 has coimension-two orinary ouble points along S an Bl S F 0 is smooth. This escription implies that we can regar Bl S F 0 as a parameter space for certain curves on X 0. These curves are of the following types: 1. lines on X 0 isjoint from p; 2. unions of proper transforms of lines through p an lines containe in Q 0 X 0. These in turn may be ientie with: 1. two-secants to S P 4 ; 2. three-secants with a istinguishe point s 2 \ S. We emphasize that each line meeting S in more than two points is containe in Q 0 but not in S, an thus is a three-secant to S. We claim elements of S [2] naturally correspon to curves of this type. For each ieal sheaf I of colength two there is a unique line containing the corresponing subscheme. Either is a two-secant, or is a three-secant an s is the support of I=I \S. Lemma Retain the notation an assumptions introuce above. The family of Fano varieties F 0 has orinary coimension-three ouble points along the surface S. The variety F 0 = Bl S (F 0 ) is smooth, an the exceptional ivisor E F 0 0 is a smooth quaric surface bunle over S. The component of F 0 0 ominating F 0 is isomorphic to S [2]. The proof is essentially the same as the rst lemma. Our nest result is: Proposition Retain the notation an assumptions introuce above. Then there is a smooth family F! 0, birational to F 0, such that F u = F u an F 0 = S [2]. 24

25 We start with the family F 0 escribe in the previous lemma. The bers of E! S are all smooth quaric surfaces, so the variety parametrizing rulings of E is an etale ouble cover of S. Since S has no nontrivial etale coverings we may choose a ruling of E. Blowing own E in the irection of this ruling, we obtain a smooth family F. This map inuces an isomorphism from the proper transform of F 0 in F 0 0 to the central ber of F. The proper transform to F 0 in F 0 0 is isomorphic to S [2], so F satises the conitions of the proposition. We now prove Theorem Let S be an algebraic K3 surface with Picar group P = f 6 f 4 f 6 6 n + 5 f 4 n an n 2. By Lemma 4.3.2, such a surface exists an we may assume that jf 6 j imbes it as a smooth sextic surface. The ivisor f 4 is eective because it has positive egree with respect to f 6. We claim that f 4 is very ample. If f 4 were not ample, then there woul exist a ( 2)-curve E with f 4 E 0. This follows from the structure of the Kahler cone of S ([16] x1,x10). Note that f 4 E 6= 0 because P oes not contain a rank-two sublattice of iscriminant 8. Recall that the Picar-Lefschetz reection associate to E is given by the equation r E (x) = x + (E; x)e. Applying this to the class f 4, we n that r E (f 4 ) 2 = 4 an (f 6 ; r E (f 4 )) < (f 6 ; f 4 ). Hence that f 6 an r(f 4 ) span a sublattice with iscriminant smaller than that of P, which is impossible. Finally, applying Lemma we see that the linear system jf 4 j imbes S as a smooth quartic surface. Our hypothesis on P implies that the image of S uner jf 6 j lies on a smooth quaric hypersurface an oes not contain a line, an that the image of S uner jf 4 j also oes not contain a line. In particular, S correspons to a singular cubic fourfol X 0 2 ~C 6. Furthermore S [2] is ambiguous, with an involution j : S [2]! S [2] so that 2 := j = 2f 4 3: Using Proposition an the arguments of x 4.3, X 0 has a smoothing : X! such that (after base change) the corresponing family of smooth symplectic varieties F! 0 is a eformation of S [2] for which 2 remains algebraic. By Theorem the Fano variety F u of Xu 0 is isomorphic to S u [2] : If we choose generally, we may assume that the Xu 0 are typical an that Pic(S u ) is generate by the polarization f 0. Let = Pic(F u ), a lattice (with respect to the canonical form) of iscriminant 2eg (S u ): On the 25

26 other han, is the saturation of Zg + Z 2. Specializing to S [2] we obtain = Z(2f 6 3) + Z(f 6 f 4 ) with iscriminant 4(n 2 + n + 1): In particular, the S u have egree (n) = 2(n 2 +n+1) an the X u are special of iscriminant (n). We have shown that the pure limiting Hoge structures parametrize by D 6 actually arise from smooth symplectic varieties. This may be interprette as a weak surjectivity result for the corresponing Torelli map. It also explains the computation of the limiting mixe Hoge structure H 4 lim in x 4.2. There are a number of ways Theorem might be generalize. We nee not assume that the polarizations f 6 an f 4 actually generate the Picar lattice of S. Another approach is to replace ~C 6 by some other ivisor C parametrizing special cubic fourfols whose Fano varieties are of the form S [2]. To make precise statements one requires explicit escriptions of two complicate close sets: the complement D C an the locus in C where the isomorphism between the Fano varieties an the blown-up symmetric squares breaks own. Finally, Mukai's philosophy suggests that whenever we have an associate K3 surface S, the Fano variety F might be interprette as a suitable mouli space of simple sheaves on S. It woul be interesting to n such interpretations when F cannot be a blown-up symmetric square. References [1] Altman, A. an Kleiman, S., 1977, Founations of the theory of Fano schemes, Compositio Math. 34, [2] Baily, W. L. an Borel, A., 1966, Compactications of arithmetic quotients of boune symmetric omains, Ann. of Math. 84, [3] Beauville, A., 1983, Varietes kahleriennes ont la premiere classe e Chern est nulle, J. Dierential Geom. 18, [4] Beauville, A., 1985, Surjectivite e l'application es perioes, in Geometrie es Surfaces K3: Moules et Perioes-Seminaire Palaiseau, Asterisque 126, Societe mathematique e France, Paris, [5] Beauville, A. an Donagi, R., 1985, La variete es roites 'une hypersurface cubique e imension 4, C.R. Aca. Sc. Paris, Serie I 301,