Pfaffian bundles on cubic surfaces and configurations of planes

Pfaffian bundles on cubic surfaces and configurations of planes Han Frédéric March 10, 014 Institut de Mathématiques de Jussieu - Paris Rive Gauche, Université Paris 7-5 rue Thomas Mann, Batiment Sophie-Germain 7505 Paris Cedex 1, FRANCE email: han@math.jussieu.fr Mathematical Subject Classification: 14J60 Abstract We construct a canonical birational map between the moduli space of Pfaffian vector bundles on a cubic surface and the space of complete pentahedra inscribed in the cubic surface. The universal situation is also considered and we obtain a rationality result. As a by-product, we provide an explicit normal form for five general lines in P 5. Applications to the geometry of Palatini threefolds and Debarre-Voisin Hyper-Kähler manifolds are also discussed. 1 Introduction Let V 6 be a complex vector space of dimension six and denote the projective space Proj(S (V 6 )) by P 5. Let W n be a complex vector space of dimension n. To help the reader to distinguish the projectivization of Wn from the other projective spaces considered in this article, let us reserve the bold notation P n 1 for Proj(S (W n )) and the symbol P for the other ones. Definition 1.1 For n, a general element of W n V6 defines a skew-symmetric element M of Hom S (W n)(v6, V 6 ). Its Pfaffian defines a cubic hypersurface Pf(M) in P n 1 which is smooth for n 6. We have the exact sequence 0 V 6 O Pn 1 ( 1) M V 6 O Pn 1 F 0, where F is a rank- sheaf over Pf(M). For n 6, the sheaf F is locally free on Pf(M). We will call it the Pfaffian bundle defined by M. 1

It is known from classical works on representations of a cubic form by Pfaffians ([Be], [Do]) that for n 6 a general cubic is not a Pfaffian and that for n 5 the Pfaffian bundles over a fixed Pfaffian cubic have moduli spaces of positive dimension. The main result of this article concerns the case n = 4. In this situation we have the following results from [Be]:. Every smooth cubic surface of P can be defined by a linear Pfaffian.. Let (W 4 V6 ) sm be the open subset of W 4 V6 corresponding to smooth Pfaffian surfaces. For any element M of (W 4 V6 ) sm, the Pfaffian sheaf is a stable rank- vector bundle over the Pfaffian surface Pf(M). Moreover, it is an arithmetically Cohen-Macaulay sheaf and every arithmetically Cohen-Macaulay rank- vector bundle over a smooth cubic surface S with determinant O S () is a Pfaffian bundle.. The quotient of (W 4 V6 ) sm by GL(V 6 ) for the following action GL(V 6 ) (W 4 V6 ) sm (W 4 V6 ) sm (P, M) t P M P is isomorphic to the space of pairs (S, F ), where S is a smooth cubic surface of P and F an isomorphism class 1 of a Pfaffian bundle on S. It is a geometric quotient. In this article we obtain a geometric interpretation of these orbits. Definition 1. A complete pentahedron inscribed in a cubic divisor S of P is a set {H 0,..., H 4 } of 5 planes of P such that: i) (H 0,..., H 4 ) is a projective basis of P. ii) The 10 points (H i H j H k ) 0 i<j<k 4 are on S. We define the subset H of O P () O P (5) (resp. the subset H ord of O P () (P ) 5 ) to be the set of elements (S, Π) such that S is a smooth cubic surface of P and Π is a complete pentahedron inscribed in S (resp. complete pentahedron inscribed in S with an ordering of the five planes). For a fixed cubic surface S of P, denote by H S the set of complete pentahedra inscribed in S. The first four sections give two natural methods to construct five hyperplane sections of a cubic surface from a Pfaffian vector bundle. Eventually they are generically identical and we obtain: 1 of vector bundles over the fixed cubic surface S

Theorem 1. There is a natural birational map from (W 4 V6 ) sm /GL(V 6 ) to H such that the composition with the projection to O P () is the Pfaffian map. In particular:. (W 4 V6 ) sm /GL(V 6 ) is a rational variety of dimension 4.. Let S be a general cubic surface. The moduli space of Pfaffian bundles on S is birational to H S. We explain this theorem with two constructions. The first construction is a rational map Φ 1 (Definition.11) related to a classical problem of hyperplane restriction of the Pfaffian bundles. So in Section, we start with the easy case n = and introduce some invariants of these bundles. The universal situation is then described because many geometric interpretations of the later sections are specializations of this construction. In Section, we construct Φ 1 from the case n = 4. The projectivization of a Pfaffian bundle on a cubic surface is called a Palatini threefold. Such varieties are the only known examples of smooth -dimensional varieties X in P 5 such that h 0 (O X ()) > h 0 (O P5 ()) and they often appear in lists of exceptions to some geometrical property ([Fa-Fa], [Fa-Me], [Me-Po], [Ot]). Some of their classical properties are also described in [Do] and [Ok], but here some new results are required. First we give an interpretation of their anticanonical linear system to prove that it is P. Then we describe this linear system in Proposition.8. It turns out that its exceptional locus is 5 points of P. This achieves the construction of Φ 1. The geometric configuration of these five planes is only explained in Section 4 by the construction of a rational map Φ (Corollary 4.). This time, it is a problem of linear spaces multisecant to G(, V 6 ). The key step to construct Φ is the surprising Proposition 4.1, with following summary. Proposition The projection of the Grassmannian G(, V 6 ) P 14 -dimensional projective space has a unique singular point of order 5. from a general The claim that Φ 1 and Φ are generically the same and also their birationality are proved in Section 4. from the explicit formula of Theorem 4.7. This ends the proof of Theorem 1.. As a by-product, we obtain in Corollary 4.11 an explicit generically finite parametrization of the quotient of the product of five copies of G(, V 6 ) by the diagonal action of P GL(V 6 ). I would like to thank the referee for pointing out that the rationality of this quotient was proved in [Zai]. Recently, F. Tanturri ([T]) found an algorithm to obtain a Pfaffian representation from the equation of a cubic surface. His construction relies on finding an arithmetically Gorenstein subscheme as in [Be, Prop. 7.]. We do not know any direct relation with the notion of inscribed pentahedra. In the last section we investigate those properties over a base. We explain how the Debarre-Voisin holomorphic symplectic manifold can be considered as a parameter space i.e., the image of the contracted locus

for Palatini threefolds in a six-dimensional variety of P 9. Those varieties of dimension six were discovered by C. Peskine. They are of independent interest because they are smooth and non quadratically normal in P 9 (they are boundary cases in Zak s theory of quadratic normality). However, most of their geometric properties are unknown. In particular, it would be very interesting to understand those varieties from a Palatini threefold in a similar way that a Veronese surface is related to P P. So we will also explain in this section the consequences on Peskine varieties of the work on the Palatini threefolds done in Section. Acknowledgement: I would like to thank I. Dolgachev for encouraging discussions and references and J. Déserti and O. Debarre for their attentive proofreading. Invariants of Pfaffian bundles over plane cubics..1 Ruled surfaces in P 5 and the case n =. In this section, we work out in detail the case n =. The following easy lemma gives an important classification result for the next sections. Lemma.1 For a general element M of W V6, we consider the exact sequence of Definition 1.1 0 V 6 O P ( 1) M V 6 O P F 0. (1) The cokernel F is isomorphic to one of the following bundles over the Pfaffian curve C = Pf(M): a) L(1) L (1), where L is a line bundle of degree 0 on C such that h 0 (L ) = 0; b) F is the unique unsplit extension where θ = O C and θ O C ; 0 θ(1) F θ(1) 0 c) F = θ(1) θ(1) where θ = O C and θ O C. Proof: To simplify the notation, denote F ( 1) by F. First one can remark that h 0 ( F ) = 0, and that F F because M is skew-symmetric. So we have F = OC. We choose a point p on C. We will now prove that there is a point r of C such that h 0 ( F (p r)) > 0. From the Riemann-Roch theorem, the bundle F (p) has a pencil of sections. It gives, on P 1 C, a section of the bundle O P1 (1) F (p). From the computation of the second Chern class of this bundle, this section has a non-empty vanishing locus. So there is a point r of C such that h 0 ( F (p r)) > 0. Since h 0 ( F ) = 0, we obtain that O C (p r) is not trivial, and we are in one of the cases above. 4

Remark. With the notation of the previous lemma, we will say that a plane in P 5 is M-isotropic, if it is isotropic for all the elements of V6 in the image of M : W V6. From sequence (1), the ruled surface Proj(S (F )) has a natural embedding of degree 6 in P 5, such that in case a), it contains two plane cubic curves and the planes spanned by these curves are disjoint in P 5, in case b), it contains only one plane cubic, in case c), it is a divisor of bidegree (0, ) in a Segre variety P 1 P P 5. So it contains infinitely many plane cubics. Moreover, the planes in P 5 containing such a cubic curve are the M-isotropic planes. Proof: For the first part of the remark, just note that any invertible quotient of degree of F embeds the Pfaffian curve in Proj(S (F )) as a plane cubic. The skew-symmetric map M in the resolution (1) of F induces the isomorphism (F ( 1)) O C. So the line bundles L(1), L (1), θ(1) in the above extension or direct sums are isotropic. Taking global sections, they give M-isotropic planes in P 5. Conversely, from( any M-isotropic ) plane, we obtain some element P of GL(V 6 ) such that 0 t P M P = t A, where A, B are matrices with linear entries. So the A B cokernel of A gives an invertible quotient of F of degree as in Lemma.1.. Universal settings and the SL(V 6 )-invariant double cover Definition. Let G(, V 6 ) and G(, V6 ) be the Grassmannians of -dimensional vector subspaces of V 6 and V6. Denote by K and R their tautological subbundles. We define the isotropic incidence Z G(, V6 ) G(, p V6 ) G(, V6 ) p 1 G(, V 6 ) Z = {(A, B) A is isotropic for all the elements of B V6 } to be the vanishing locus of the unique SL(V 6 )-invariant section of K R. Denote by U the open subset of G(, V6 ) made of vector spaces such that the intersection of their projectivizations with the Pfaffian hypersurface of P( V6 ) is a smooth cubic curve. The restriction of Z to G(, V6 ) U will be denoted by Z U. Let E 1 be the rank-1 bundle defined by the exact sequence 0 E 1 V6 O G(,V 6 ) K 0. () 5

I would like to thank A. Kuznetsov for the following description of Z from the relative Grassmannian. Proposition.4 The isotropic incidence Z is isomorphic to the relative Grassmannian G(, E 1 ) of linear subspaces of the bundle E 1. The projection Z U U G(, V6 ) is generically finite of degree. The fiber of this morphism over an element of type a), b), c) in Lemma.1 consists in G(, V6 ) of points, 1 point, and a rational cubic curve. Proof: Let (µ, ν) be an element of G(, V 6 ) G(, V6 ). The fiber of a vector bundle at µ (resp. ν) will be denoted by the name of the bundle with the index µ (resp. ν). The vector space K,µ is isotropic for all the skew-symmetric forms defined by the elements of R,ν if and only if (µ, ν) Z, but also if and only if the composition R,ν V6 K,µ is the zero map. So (µ, ν) Z R,ν E 1,µ and we have the equality Z = G(, E 1 ). The end of the assertion follows immediately from Lemma.1 and Remark.. Corollary.5 The locus U c in U G(, V6 ) of planes of type c) has codimension. Define a relation R on U c by prp if and only if p 1 (p 1 (p)) = p 1 (p 1 (p )). For any element p of U c, there is a six-dimensional subspace L p of V6 such that the equivalence class of p for R is an open subset of G(, L p ). Proof: From Proposition.4, for any p in U c, p 1 (p 1 (p)) is a smooth rational cubic curve C p in G(, V6 ). So the restriction of E 1 to C p is O 6 P 1 O 6 P 1 ( 1) and it has a natural trivial subbundle of rank 6. Let L p be the six-dimensional vector subspace of V6 obtained from the image of this subbundle by the injection of sequence (). Proposition.4 describes p 1 1 (C p ) as the relative Grassmannian G(, E 1 Cp ). Let F be a subbundle of rank of E 1 Cp = L p O P1 O 6 P 1 ( 1). Case c) appears when the line bundle F contracts the curve C p. But F is not ample if and only if F is a trivial subbundle of L p O P1. So p 1 1 (C p ) p 1 (U c ) is (U G(, L p )) C p, and the equivalence class of p for R is U G(, L p ). So the dimension of U c is the sum of the dimension of G(, 6) and the dimension of the family of rational cubic curves in G(, V6 ). In conclusion U c, has dimension and codimension in G(, V6 ). Palatini threefolds In this section, we study the case n = 4. 6

.1 Definition and classical properties Definition.1 A smooth -dimensional subvariety X of P 5 is called a Palatini three- fold if there exists an element of M of W 4 V6 such that X = Proj(S (F )), where F is the Pfaffian vector bundle defined by M in Definition 1.1 with n = 4. It is also classically called ([Do, p.58]) the singular variety of the linear system W4 of linear line complexes in V6. Notation. In this section, denote by X a Palatini threefold in P 5, by h the class of a hyperplane of P 5, by S the Pfaffian cubic surface in P, and by s the pullback on X of the class of a hyperplane of P. The cotangent bundle of P 5 will be denoted by Ω 1 P 5. So we can immediately obtain the well known resolution of its ideal: Remark. The ideal I X of a Palatini threefold X in P 5 has the following resolution 0 W 4 O P5 M Ω 1 P 5 (h) I X (4h) 0 () and we have the famous equality ([Me-dP] problem 1) h 0 O X (h) = h 0 O P5 (h) + 1. The following results are classical generalizations of properties of a Veronese surface embedded in P 4. Remark.4 A Palatini threefold X has a natural embedding in the point/plane incidence variety of P 5. It is defined as follows. To a point x of X, we associate the plane π x of P 5 swept out by the lines through x which are quadrisecant to X (i.e., the lines through x contained in all the line complexes associated to X). The intersection X π x is a plane curve of degree and a residual point supported on x. So X contains a three dimensional family of plane cubics parametrized by X itself ([Me-Po]). To explain this natural embedding of X in the point/plane incidence of P 5, F. Zak introduced the following vector bundle: Definition.5 The canonical extension on P 5 displayed in the second column of the or a Palatini scroll 7

following diagram of exact sequences 0 W4 O P5 ( h) Ω 1 P 5 (h) I X (h) 0 0 W 4 O P5 ( h) 0 M V 6 O P5 O P5 (h) induces on a Palatini threefold X the following extension with middle term a rank vector bundle E X (where N X is the conormal bundle of X in P 5) 0 N X (h) E X O X (h) 0. Moreover, the restriction to X of the second line of the previous diagram gives the exact sequence 0 O X ( h s) W 4 O X ( h) and the determinant of E X is O X (h s). 0 M V 6 O X E X 0 (4) From the inclusion W 4 V6 and the identification W 4 = W 4, we can consider P as a subvariety of G(, V6 ). Proposition.6 Let Z 4 be the restriction of the isotropic incidence Z G(, V 6 ) G(, V6 ) to G(, V 6 ) P. Then Z 4 is isomorphic to X and the projection from Z 4 to G(, V 6 ) is the embedding of X given by EX. It is the natural map x π x of Remark.4. Proof: Let us first recall the classical description of quadrisecant lines to X. Let A and B be the -dimensional vector subspaces of V6 and W4 corresponding to a point of Z 4. Denote by A the kernel of the surjection from V 6 to A. The restriction of Ω 1 P 5 (1) to P(A ) is A O P(A ) Ω 1 P(A ) (1). From the isotropicity of P(A ) with respect to all the elements of B V6, we see that the restriction of M : W4 O P5 ( h) Ω 1 P 5 (h) to P(A ) is the direct sum of the following maps 8

B O P(A )( 1) A O P(A ) and (W 4 /B) O P(A )( 1) Ω 1 P(A ) (1). The determinant of the first one gives a cubic curve in P(A ) X, and the second map vanishes on a single (residual) point µ of P(A ) X. So we have constructed a morphism ζ : Z 4 X (A, B) µ. Moreover, the vanishing at µ shows by specialization of sequence (4) that the fiber of E X at µ is A. So the first projection p 1 is also the composition Z 4 ζ X EX G(, V6 ). The proof of the statement is now reduced to the proof of the embedding of Z 4 in G(, V6 ). But the fiber of this morphism over the point of G(, V6 ) corresponding to A is a single point because A is not isotropic for all the elements of W4. So this projection of Z 4 is one-to-one, and it must be an isomorphism because the fibers are given by linear conditions.. Anticanonical properties Although it is classical that the canonical class K X of a Palatini threefold satisfies = ([Ok]), the following identification and next proposition seem new. K X Lemma.7 The canonical invertible sheaf ω X of X is isomorphic to O X (s h). With Notation., we have from the equality W 4 = H 0 (O S (1)) a canonical isomorphism H 0 (ω X) W 4. Proof: The isomorphism ω X O X (s h) can be computed directly from Definition.1. We obtain the isomorphism H 0 (ωx ) W 4 from the isomorphism between X and Z 4 found in Proposition.6 and the fact that ωz 4 is the pull-back of O P (1). Proposition.8 The linear system ωx has no base points and gives a morphism of degree : X :1 P G(, V6 ). It contracts 5 rational curves. These curves are smooth of degree for the embedding of X in P 5 and also for the embedding of X in G(, V 6 ). Proof: From Proposition.6 and Lemma.7, the anticanonical map is the restriction of the projection p (Definition.) to the incidence variety Z 4. So it is base point free of degree. The contracted curves of this morphism correspond to case c) of Lemma.1. Such curve parametrizes the planes in a Segre variety P 1 P. So it is a smooth rational cubic curve in G(, V6 ). By definition, on a contracted curve, the divisors h and s are equivalent because ωx = O X(h s). So those curves have the same degree with respect to h and h s. It remains to prove that the image of those curves in P is 5 points. This will be proved in Lemma.10 by the description of the ideal in P of this exceptional locus as a determinantal ideal. 9

Remark.9 The previous proposition has the following geometric interpretation in P 5. Any linear section of the Pfaffian cubic surface S gives a sextic ruled surface in X P 5. There are five particular cases where such a surface is in a Segre variety P 1 P (the five Segre varieties are different in P 5 ). Any of these Segre variety intersects X in the corresponding ruled surface of degree 6 and a residual smooth rational curve of degree. These curves are contracted by the anticanonical linear system of X. We could also remark that the projection of such curve on the Pfaffian surface S has degree 6 in P. Fortunately, they are explicit (cf Remark 4.8). It was an important tool in the construction of the matrix M 4 in Section 4.. Lemma.10 Let F be the normalized bundle F ( 1). The vector space H = H 1 ( (S F )( 1) ) has dimension 5 and it is the kernel of the following map given by the Pfaffians of size 4 4 of M: 0 H V6 4 V 6 S W 4 0. (5) Moreover the ideal of the exceptional locus in P of the projection X Z 4 P is given by the 4 4 Pfaffians of a skew-symmetric map H O P ( 1) H O P. Proof: Let i be an isomorphism F O S. The restriction of F to a plane P is of type c) in Lemma.1 if and only if we have h 1 (S ( F P )) =. The determinantal structure will be obtained from this criterion. To globalize this condition, let us consider the complex C : 0 V 6 O P ( ) M V 6 O P ( 1) 0. It is exact in degree 1 with cohomology F in degree 0. The second exterior power of C tensorized by O P () is 0 S V 6 O P ( ) V 6 V 6 O P ( 1) V6 O P 0. Its cohomology in degree (, 1, 0) is given by (0, (S F )( 1), ( F )()). So the hypercohomology spectral sequence of this complex gives the exact sequence (5), the dimension of H, and the vanishings h 0 ((S F )( 1)) = h ((S F )( 1)) = h 0 (S F ) = h (S F ) = 0. Now consider the point/plane incidence variety I P P and denote by p and p the restrictions to I of the first and second projections of this product. We have the exact sequence 0 O P ( 1) S F ( 1) OP S F p (S F ) 0. 10

From the Leray spectral sequence and the above vanishings, we have the exact sequence 0 p (p (S F )) H 1 ((S F )( 1)) OP ( 1) d M H 1 (S F ) OP R 1 p (p (S F )) 0. Let us now explain how to consider the map d M as a skew-symmetric map. The isomorphism i gives a symmetric isomorphism i : S ( F ) S ( F ) so the following square is commutative (S F )( 1) S F id i i id (S F )( 1) S F τ (S F )( 1) S F τ O S ( 1). The cup-product H 1 ( (S F )( 1) ) H 1 ( (S F )( 1) ) H ( (S F S F )( ) ) is anticommutative, so for any z W 4 the following square 4 is also anti-commutative H 1 ( (S F )( 1) ) H 1 ( (S F )( 1) ) id d M,z H ( ) 1 (S F ( ) )( 1) H 1 S F d M,z id H ( ) 1 S F ( ) H 1 (S F )( 1) H ( ) S F (S F )( 1) τ (i id) H ( (S F )( 1) S F ) τ (id i ) H (O S ( 1)). In conclusion, the composition H O P ( 1) d M H 1 (S F ) OP ī H 1 (S F ) O P Serre duality H O P is skew-symmetric and the lemma is proved. Indeed, the type c) case corresponds to the locus where this map has rank at most, so it defines 5 points in P. This also achieves the proof of Proposition.8. Definition.11 Let Σ 5 be the fifth symmetric product of P. We define a rational map Φ 1 : (W 4 V6 ) sm /GL(V 6 ) P(S (W 4 )) Σ 5 M (S, {h 0,..., h 4 }), where S is the Pfaffian cubic surface defined by M, and h 0,..., h 4 are the five linear sections of S defined in Proposition.8. In Section 4, we will study the image of this map. 4 An overlined symbol denotes the map induced in cohomology. 11

. Palatini threefolds and endomorphisms Although this part is not required for the main theorem, let us briefly describe here some connected remarks. The exceptional geometric properties of a Palatini threefold are classically considered as natural generalizations of the geometry of a Veronese surface V embedded in P 4. For instance, in the Veronese situation, the sequence () is just replaced by 0 W O P4 M Ω 1 P 4 (h) I V (h) 0. But the main difference is that in the theory of Severi varieties the embedding of V by the complete linear system O V (h) is understood from an interpretation in terms of matrices of size of rank 1. For a Palatini threefold, there are no similar results to describe the embedding by the complete linear system O X (h). The following remark could be a first step in this direction. Remark.1 The restriction of the line bundle ω X O X(s) to the diagonal of X X gives the natural inclusion W 4 W 4 H 0 (O X (h)). In other words, the embedding of a Palatini threefold X by O X (h) has a canonical projection to P(W 4 W 4 ). Moreover, the image of X by this projection consists of endomorphisms of W 4 of rank 1. Proof: It is straightforward from Lemma.7. 4 Geometry in V6 4.1 Projections from linear spaces The Grassmannian variety G(, 6) in its Plücker embedding is one of the four Severi varieties. It has the following special property: its projection from a general line has a unique triple point ([I-M], [Z]). Here, we prove a similar result for projections from P and points of multiplicity 5. Proposition 4.1 Denote by U 5 the subspace of G(5, V6 ) defined by the 5-dimensional vector spaces such that the intersection of their projectivization with G(, V 6 ) consists of five linearly independent distinct points. Then, a general 4-dimensional subspace W4 of V6 is contained in a unique element of U 5. Proof: First remark that the incidence variety I 4,5 = {(W 4, W 5 ) W 4 W 5 V6, W 5 U 5 } 1

has the same dimension as G(4, V6 ). We thus have to prove that the natural projection is birational. Consider a general element W4 in the image of this projection and choose an element such that (W4, W5 ) I 4,5. Again denote by P, P 4 their projectivizations. The W 5 vector space H 0 (I P G(,V 6 )()) is the kernel of the map H 0 (I G(,V6 )()) = 4 V 6 S W 4. So it has dimension 5. Now remark that we also have h 0 (I P4 G(,V 6 )()) = 5 because the ideal of the 5 points P 4 G(, V 6 ) in P 4 is a 10-dimensional space of quadrics. So we proved that P 4 must be in all the quadrics of H 0 (I P G(,V 6 )()). It gives the following linear conditions satisfied by any W5 of U 5 containing W4 (W4 ) q, W 5 q H 0 (I P G(,V 6 )()) where q denotes the orthogonal with respect to the quadratic form q on V6. So uniqueness of W5 will be a consequence of the existence of a W 4 such that (W4 ) q q H 0 (I P G(,V 6 )()) has dimension 5. This is true in the following: Example 4. Let us consider a basis (ɛ i ) of V 6, and the 5 elements u 0 = ɛ 0 ɛ, u 1 = ɛ 1 ɛ 4, u = ɛ ɛ 5, u = (ɛ 0 + ɛ 1 + ɛ ) (ɛ 4 + ɛ + ɛ 5 ), u 4 = (ɛ 1 + ɛ 4 + ɛ ) (ɛ + ɛ 1 + ɛ 5 ). Denote by W 5 Then q H 0 (I P G(,V 6 )()) the 5-dimensional vector space spanned by the (u i ) 0 i 4 and = { λ i.u i λ i = 0}. W 4 0 i 4 (W 4 ) q has dimension 5. 0 i 4 Proof: We compute with [Macaulay] that H 0 (I P G(,V 6 )()) is generated by the following five quadrics in Plücker coordinates p (,4) p (1,5) p (1,4) p (,5) + p (1,) p (4,5), p (1,) p (0,5) p (,4) p (0,5) p (0,) p (1,5) +p (,) p (1,5) +p (0,1) p (,5) p (1,) p (,5) +p (0,4) p (,5) p (,4) p (,5) + p (1,) p (,5) + p (,4) p (,5) p (0,) p (4,5) p (,) p (4,5), p (,) p (0,4) p (0,) p (,4) +p (0,) p (,4) p (1,) p (0,5) +p (,4) p (0,5) p (,4) p (0,5) +p (0,) p (1,5) p (,) p (1,5) +p (1,) p (,5) p (0,4) p (,5) +p (,4) p (,5) p (0,1) p (,5) p (1,) p (,5) +p (0,4) p (,5) p (,4) p (,5) + p (0,) p (4,5) p (0,) p (4,5) + p (,) p (4,5), p (1,) p (0,4) p (0,) p (1,4) +p (0,1) p (,4) p (,4) p (0,5) +p (,) p (1,5) p (1,) p (,5) +p (0,4) p (,5) p (,4) p (,5) + p (1,) p (,5) + p (,4) p (,5) p (0,) p (4,5) p (,) p (4,5), 1

p (1,) p (0,) p (0,) p (1,) +p (0,1) p (,) p (,4) p (0,5) +p (,4) p (0,5) +p (,) p (1,5) p (1,) p (,5) + p (0,4) p (,5) p (,4) p (,5) +p (1,) p (,5) p (0,4) p (,5) +p (,4) p (,5) p (0,) p (4,5) +p (0,) p (4,5) p (,) p (4,5), and check that the ideal of the orthogonal of P with respect to these 5 quadrics is generated by the 10 independent equations (p (,5), p (0,5) p (1,5) + p (4,5), p (,4) + p (4,5), p (,4) p (1,5) + p (4,5), p (0,4) p (1,5) + p (4,5), p (,) p (1,5), p (1,) p (1,5), p (1,) + p (4,5), p (0,), p (0,1) ). So this example completes the proof of the birationality of the projection from I 4,5 to G(4, V6 ) and we have proved Proposition 4.1. Corollary 4. With the notation of Definition 1., we can define the rational map Φ : (W 4 V6 ) sm /GL(V 6 ) H M (S, {H 0,..., H 4 }) where S is the Pfaffian cubic surface defined by M, and H i is defined as follows. From Proposition 4.1, consider the five points (u i ) 0 i 4 of G(, V 6 ) such that P is in the linear span (u i ) 0 i 4. Then take H i = P (u j ) 0 j 4,j i. Proof: After Proposition 4.1, we only have to explain why {H 0,..., H 4 } is inscribed on S. But for {i 0,..., i 4 } = {0,..., 4}, the point H i0 H i1 H i is on the line (u i, u i4 ) so it corresponds to a matrix of rank 4 and is on S. Remark 4.4 The variety H is rational of dimension 4. Proof: Let Σ 5 be the image of H in O P (5) by the second projection. It is an open subset of the symmetric product Σ 5 defined in Definition.11. So it is a rational 15- dimensional variety ([G-K-Z, Th..8 p. 17]). The partial derivatives of order of any element of Σ 5 are linearly independent cubic forms. So they give a rank-10 subsheaf F of H 0 (O P ()) O Σ 5 locally free with respect to Zariski s topology. Now remark that H is the open subset of P(F ) corresponding to smooth cubic surfaces. So H is rational of dimension 4. 4. An explicit formula and proof of Theorem 1. Surprisingly, we are able to give in this section an explicit formula. Recently, an explicit result was also found by F. Tanturri in [T]: an algorithm to obtain a Pfaffian representation from a cubic equation. The two main differences, are the following: - first, he only wants to find a Pfaffian representation of S, while here, we first choose some pentahedron, then we need to find the unique Pfaffian bundle on S related to this pentahedron; 14

- technically, his construction starts with five points on S, so it is a problem of extending the 5 5 skew-symmetric matrix in the resolution of the 5 points to a 6 6 one with Pfaffian S, while we start with an inscribed pentahedron. Lemma 4.5 Let (x i ) 0 i be a basis of W 4, and let A 9 be the following subset of C 10 P 4 { A 9 = ((a i,j,k ) 0 i<j<k 4, (b i ) 0 i 4 ) a } 0,1,4 = 1 and for 0 i 4, b i 0,. and for 0 i < j < k, a i,j,k = 1 Then the map P GL 4 A 9 H ord (P, ((a i,j,k ) 0 i<j<k 4, (b i ) 0 i 4 )) (S, (H 0,..., H 4 )) (6) is birational where 0 i<j<k 4 a i,j,k.w i.w j.w k = 0 is an equation of S, and for all 0 i 4, w i = 0 is an equation of H i with the following w 0 x 0 equalities: w 4 = b 4.w i b i, w 1 w i=0 = P x 1 x. w x Proof: Let Π = (H 0,..., H 4 ) be an ordered pentahedron and let P be the unique projective transformation that sends the ordered pentahedron (x 0, x 1, x, x, x 0 + x 1 + h 0 x 0 x +x ) to (H 0,..., H 4 ). Denote by h i the equation of H i defined by h 1 h = P x 1 x h x and h 4 = i=0 h i. Cubic surfaces S such that (S, Π) is in H ord are the smooth surfaces defined by: A i,j,k.h i.h j.h k = 0, (A i,j,k ) (0 i<j<k 4) P 9. 0 i<j<k 4 Now remark that the map A 9 P 9 (a, b) (A i,j,k = a i,j,k b i b j b k ) 0 i<j<k 4 is birational because we can compute its inverse with the following formulas b 0 b = A 0,1, A 1,,, b 1 b = A 0,1, A 0,,, b b = A 0,1, A 0,1,, b 4 b = A 0,1,4 A 0,1,, a i,j,4 = A i,j,4 A i,j, A0,1, A 0,1,4. So we obtain the lemma from the equalities w i = h i b i, with P defined by the product of the diagonal matrix ( b 0 b4,..., b b4 ) with P. Definition 4.6 Let A 9 be the set of triples (a, b, u) such that (a, b) is an element of A 9 defining a smooth cubic surface 0 i<j<k 4 a i,j,k w i w j w k = 0, w 4 = i=0 b 4 w i b i, 15

and u is a root of the following equation in X: X + X (1 + a 0,,4 a 0,,4 ) + a 0,,4 = 0. Denote by v = (1 + a 0,,4 a 0,,4 ) u the other root and set e 1 = a 0,,4 +a 1,,4 a,,4, e = 1+a 1,,4 a 1,,4, e = ( a 1,,4 +a 1,,4 1)v a 1,,4 a 0,,4 +a,,4, 0 u 1 a 1,,4 e 1 e u 0 0 0 a 0,,4 u M 4 = 1 0 0 0 v 1 a 1,,4 0 0 0 a 1,,4 v a 1,,4 e 1 a 0,,4 v a 1,,4 v 0 e e u 1 a 1,,4 e 0 0 0 0 w 0 + w w w 0 0 0 w w 1 + w w M 01 = 0 0 0 w w w + w w 0 w w w 0 0 0. w w 1 w w 0 0 0 w w w w 0 0 0 Theorem 4.7 For a generic element (P, (a, b, u)) of P GL 4 A 9, the element M of (W 4 V6 ) defined by M = M 01 + w 4 M 4 is such that Φ 1 (M) = Φ (M) = (S, Π), where the equation of S and Π are given by the formulas in Lemma 4.5. Remark 4.8 The difficulty was to find M 4. It was done by searching for the rational cubic curve in P 5 associated to the plane w 4 = 0 in Proposition.8. We followed these steps: From the equation 0 i<j<k 4 a i,j,kw i w j w k of S we obtain a sextic curve of ideal ( 0 i<j<4 a i,j,4w i w j, Pf(M)) in P. This curve has geometric genus 0 and is singular at the 4 vertices of the pentahedron that are not in the plane w 4 = 0. In the next steps, we lift this curve to P 5 to obtain the desired cubic curve. The intersection of this sextic curve with the plane w 4 = 0 gives 6 points on a conic. Let L θ(1) denote the Pfaffian bundle defined by M 01 on the plane w 4 = 0 (where L is a -dimensional vector space). The image of the 6 points by the linear system θ(1) is also on a conic. We choose a parametrization of this conic (cf. the equations for u and v in Definition 4.6) to obtain an identification S L = H 0 (θ(1)). So we have the identification V 6 = L S L and a rational cubic curve in a Segre variety in P 5. The matrix M 4 was obtained from the marked element of (L S L). But now that we have found M 4, it is much easier to check that M satisfies the required properties. 16

Proof 5 of Theorem 4.7: - First, one can check that the Pfaffian of M is a 04 w 0 w w 4 +a 04 w 0 w w 4 +a 4 w w w 4 +a 14 w 1 w w 4 +a 14 w 1 w w 4 +w 0 w 1 w 4 + 0 i<j<k - Now to prove that Φ (M) = (S, Π), we just have to remark that M 4 and the four specialization of M 01 at (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1) are matrices of rank. - To obtain that ( Φ 1 (M) = )(S, Π), we need to find 5 elements (P i ) of GL(V 6 ) such 0 that t Ai P i M P i = where A A i 0 i are by symmetric matrices with linear entries. We found the following ones easily 0 0 0 0 0 1 v ( a 0 0 04 a 14 +a 4 ) 0 0 u u P 4 = Id, P = 0 1 0 0 1 + a 14 a 14 0 1 0 0 a 14 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 but the next ones only after understanding that we should use the SL SL SL action that preserves the marked lines in the intersection of the two Segre P 1 P defined by w i = 0 and w 4 = 0 0 0 a 04 a 4 0 0 0 ( u)(a 04 +u) a a 04 1 04 a 04 a 14 ua 04 a 04 +ua 4 a 14 (u+1) a 4 a 04 0 (u+1) u u(a 04 +u) a P 1 = a 04 0 0 04 a 14 +ua 04 +a 04 ua 4 (u+1) a 04 0 0 (u+1) u 0 0 a 4 0 0 1 u 0 0 a 4 1 1 1 0 0 0 1 0 0 1 a 14 0 0 0 0 0 1+v v a 0 0 0 04 +a 4 +va 14 0 u+1 u+1 P = 1 1 v v+a a 14 1 0 04 a 4 va 14 a u+1 u+1 14 1 a 14 0 0 1 0 0 0 0 0 0 1 0 1 a 14 0 0 1 1 1 5 To obtain a more compact presentation, we have suppressed the commas in the indexes of the a i,j,k in the next formulas. w i w j w k. 17

P 0 = a 14 (ua 04 +a 04 ua 4 ) ua 04 +a 04 ua 4 0 a 04 a 14 ua 04 a 04 +ua 4 ua 04 +a 04 ua 4 u a 4 a 14 a 14 a 04 0 (ua 04 +a 04 ua 4 ) u a 4 0 0 0 0 0 0 a 14 0 0 (u+1)(ua 1 04 +a 04 ua 4 ) u(a a 4 0 04 a 4 ) u ua 04 +a 04 ua 4 1 1 ( u 1)(ua 0 04 +a 04 ua 4 ) a 4 u 0 0 1 0 0 0 0 0 0 1 and we have proved Theorem 4.7. a 04 a 14 +ua 04 +a 04 ua 4 ua 04 +a 04 ua 4 We are now able to obtain a more explicit version of Theorem 1.. Corollary 4.9 The maps Φ 1 and Φ coincide on an open set and give birational maps (W 4 V6 ) sm /GL(V 6 ) H. Proof: First remark that both spaces are irreducible of dimension 4. Now consider with the notation of Definition 4.6 the following map P GL 4 A 9 W 4 w 0 x 0 V6, where w 4 = b 4.w i b (P, a, b, u) M 01 + w 4.M i and w 1 w 4 i=0 = P. x 1 x. w x Denote by f its composition with the canonical projection from (W 4 V6 ) sm (W 4 V6 ) sm /GL(V 6 ). The map P GL 4 A 9 P GL 4 A 9 has degree because of the permutation of u and v. From the choice of the order of the pentahedron and Lemma 4.5, the rational map from P GL 4 A 9 to H has degree 5!. So we have from Theorem 4.7 the commutative diagram of rational maps to P GL 4 A 9 :1 P GL 4 A 9 1:1 H ord (W 4 V6 ) sm /GL(V 6 ) f Φ 1 Φ H. So Φ 1 and Φ are dominant and coincide on an open set, and we just have to prove that f has degree 5! also. We will do this by providing an example of (S, Π) H such that the permutation of u and v and the permutations of the elements of Π can be obtained by the action of GL(V 6 ). It is more convenient to take an example where all the elements in the preimage of (S, Π) in P GL 4 A 9 have all the same values for (a) and (b). So we end the proof with the following invariant example. Example 4.10 (Clebsch-Sylvester) Consider the values u = e iπ, v = e iπ and for 0 i < j < k 4, a i,j,k = 1. The permutation of u with v and also the permutations of the (w i ) 0 i 4 can be obtained from the action of GL(V 6 ) on M = M 01 + w 4.M 4. Note that if we add the conditions b i = b 4 for 0 i, this is the case of the Clebsch cubic with its Sylvester pentahedron. 5!:1 18

( t0 t Proof: Denote by P T = I 1 t t ) the matrix t 0 0 0 t 1 0 0 0 t 0 0 0 t 1 0 0 0 t 0 0 0 t 1 t 0 0 t 0 0 0 t 0 0 t 0 0 0 t 0 0 t and remark that t P T M 01 P T = M 01 when t 0 t 1 t t = 1. For a square matrix T, let D T be the ( ) T 0 block diagonal matrix. We will first use matrices like D 0 T T to obtain the desired form in the plane w 4 = 0, then correct the last matrix with P T. We found the following matrices: ). Permutation of u and v: if P uv = I. T 01 = 0 1 0 1 0 0 0 0 1 permutes w 0 and w 1. 0 0 1. T 0 = 0 1 0 1 0 0 permutes w 0 and w. 1 1 1. T 0 = 0 1 0 0 0 1 permutes w 0 and w. 0 1 0 v. T 4 = 0 0 1 1 0 0 and P 01 = I, P 0 = I, P 0 = I, P 4 = I ( u 1 ( i.u 6 6 v+ u e iπ i 6 e iπ v ( i 6 u ) v i 6 ( v i 6 i.v then t P uv M 4 P uv = M 4., the conjugation t (D T01 P 01 ) M(D T01 P 01 ), the conjugation t (D T0 P 0 ) M(D T0 P 0 ) ) i 6 u, the conjugation t (D T0 P 0 ) M(D T0 P 0 ) ), with the matrix P defined in Theorem 4.7, the conjugation t (P D T4 P 4 ) M(P D T4 P 4 ) permutes w 4 and w. This completes the proof because we have provided a set of generators of the group of permutations of {w 0, w 1, w, w, w 4 }. So Corollary 4.9 is proved and it implies Theorem 1. from Remark 4.4. 4. A normal form for 5 general lines in P 5 The explicit formulas in Definition 4.6 and Theorem 4.7 have the following straightforward translation. We hope that it should help to handle 5 lines in P 5. Corollary 4.11 Five lines in general position in P 5 can be put in the following form: ɛ 0 ɛ, ɛ 1 ɛ 4, ɛ ɛ 5, (ɛ 0 + ɛ 1 + ɛ ) (ɛ + ɛ 4 + ɛ 5 ) 19

( ɛ 0 + vɛ 4 ɛ 5 ) (u.ɛ 1 ɛ + a 1,,4 ɛ + e 1.ɛ 4 + e.ɛ 5 ) for some basis (ɛ i ) 0 i 5 of V 6, and some complex parameters u, v, a 1,,4, e 1, e. Proof: Let us use again the notation of Proposition 4.1. From five general lines in P 5, we obtain a 5-dimensional subspace W5 of V6 containing the corresponding decomposable elements. So choose a general 4-dimensional vector subspace W4 of W5. Then, (W4, W5 ) is a general element of the incidence variety I 4,5. Thus from Theorem 4.7 and Corollary 4.9 the corresponding element of W 5 V6 can be written M 01 + w 4.M 4 with the notation of Definition 4.6. So we obtain the corollary. 4.4 Questions on the magic square Remark 4.1 Let X be a non-degenerate subvariety of P n 1. Then the projection of X from a general linear space of dimension d is expected to have a finite number n d,x of points of multiplicity d when d + d(dim(x) n 1) + n = 0. Varieties related to the magic square are famous solutions of this problem for d = or d = with n d,x = 1. For these varieties, what is the number n n d,x? For the Veronese surface we have n,x n,x, but for P P, v (P 1 ), P 1 P 1 P 1 ([H]) we have n d,x = n n d,x = 1. Now, according to Proposition 4.1, this equality is also true for G(, 6). 5 Applications Let V 10 be a 10-dimensional complex vector space. In this section, we will first explain the relationship between two known constructions associated to the choice of a general element of V10. Then we will discuss how the results of the previous section should be related to the symplectic form of the varieties constructed in [D-V]. 5.1 Peskine s example of a 6-fold in P 9 This example was constructed by C. Peskine to obtain a smooth non-quadratically normal variety of codimension. Let P 9 be a 9-dimensional projective space over the complex numbers and denote by V 10 the vector space V 10 = H 0 (O P9 (1)). Let α be a general element of V10 and denote by Ω i P 9 the i-th exterior power of the cotangent sheaf of P 9. From the identification V10 = H 0 (Ω P 9 ()), we obtain a skew-symmetric map M α from (Ω 1 P 9 ) ( 1) to Ω 1 P 9 () and an exact sequence 0 O P9 ( ) (Ω 1 P 9 ) ( 1) M α Ω 1 P9 () I Yα (4) 0, where I Yα is the ideal of the smooth variety of dimension 6 defined by the 8 8 Pfaffians of M α. The following statement is directly deduced from the previous exact sequence. 0

Proposition 5.1 The variety Y α is such that h 1 (I Yα ()) = 1 and its canonical sheaf is ω Yα = O Yα ( ). 5. The Debarre-Voisin manifold as a parameter space Denote by G(6, V10) the Grassmannian of 6-dimensional subspaces of V10. Let K 6 (resp. Q 4 ) be the tautological subbundle (resp. quotient bundle). For any p G(6, V10), the corresponding 5-dimensional projective subspace of P 9 will be denoted by κ p. Debarre and Voisin proved in [D-V] the following. Theorem 5. ([D-V] Th 1.1). Let α be a general element of V10 = H 0 ( K 6 ). The subvariety Z α of G(6, V 10) defined by the vanishing locus of the section α of K 6 is an irreducible hyper-kähler manifold of dimension 4 and second Betti number. We can now remark the following relation between Y α, Z α, and Palatini threefolds. Proposition 5. Let p be a general element of Z α. The scheme defined by the intersection Y α κ p is a Palatini threefold. Proof: The restriction of Ω 1 P 9 (1) to κ p is Ω 1 κ p (1) Oκ 4 p. The vanishing of the restriction of α to κ p implies that the restriction of M α to κ p is ( ) 0 αp (Ω 1 κ p ) ( 1) O 4 κ p t α p β (Ω 1 κ p )() Oκ 4 p (1). So the ideal generated by the Pfaffians of size 8 of this map is also the ideal generated by the maximal minors of α p : Oκ 4 p (Ω 1 κ p )(). In conclusion, the scheme defined by the intersection Y α κ p is a Palatini threefold as in Remark.. Moreover, the following construction globalizes Definition.1 and the Pfaffian cubic surface over Z α. Remark 5.4 The restriction of the bundle K 6 Q 4 to Z α has a non-trivial section. It gives an injective map (Q 4 ) Zα ( K 6 ) Zα. Proof: The section α of ( K 6 ) gives a map from K 6 to ( K 6 ). But the restriction of this map to Z α is zero, so it induces a map from the quotient (Q 4 ) Zα to ( K 6 ) Zα. The injectivity of this map of O Zα -modules follows from the assumption that α is general. 1

5. Conjectures on the symplectic form on Z α Remark 5.5 Let p be a general element of Z α. The tangent space T (Zα,p) to Z α at p contains a canonical set of 5 vector spaces of dimension. Proof: Let p be a general point of Z α. From Remark 5.4, the fiber Q 4,p is a 4-dimensional subspace of ( K 6,p ). From Proposition 4.1, we obtain in K6,p, a canonical set of five vector subspaces (L i ) 0 i 4 of dimension such that Li contains Q 4,p. So the restriction of the map m 1 : K 6,p K 6,p 0 i 4 gives the following commutative diagram of exact sequences K 6,p (7) 0 T (Zα,p) Q 4,p K6,p K 6,p 0 ( 0 i 4 Li ) K6,p K 6,p 0, where the vertical maps are injective and the first row is the normal sequence of Z α in G(6, V10) at the point p. Now remark that m 1 vanishes on each Li L i because L i has dimension. So we can identify the kernel of the second row of the previous diagram with the 10-dimensional vector space Li L i, and we obtain an injection 0 i 4 T (Zα,p) 0 i 4 Li L i. So in general, the kernel of each projection T (Zα,p) Li L i gives a -dimensional vector subspace of T (Zα,p). Now we can remark that five points of G(, T Zα,p) should define a hyperplane γ in TZα,p. From tests with random examples with [Macaulay], we can expect that the ideal of these five lines (l i ) in P(T Zα,p) is generated by the maximal minors of the map K 6,p O P(TZα,p ) Q 4,p O P(TZα,p )(1). Note that this map is obtained from the inclusion of the tangent space to Z α into the tangent space to G(6, V10). But if the skew-symmetric form γ was degenerate, its kernel would give a line in P(T Zα,p) intersecting each l i. But a variety defined by quartic hypersurfaces cannot have a 5-secant line, so we can expect the following Conjecture 5.6 The five vector spaces of dimension canonically defined in Remark 5.5 are maximal isotropic subspaces for the symplectic form on T Zα constructed by Debarre and Voisin.

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[Z] F. Zak. Tangents and secants of algebraic varieties Translations of Mathematical Monographs, 17:viii+164, 199. American Mathematical Society, Providence, RI. Translated from the Russian manuscript by the author. 4