Pfaffian bundles on cubic surfaces and configurations of planes

Pfaffian bundles on cubic surfaces and configurations of planes

Han Fr´ ed´ eric March 10, 2014

Institut de Math´ematiques de Jussieu - Paris Rive Gauche, Universit´e Paris 7 - 5 rue Thomas Mann,

Batiment Sophie-Germain

75205 Paris Cedex 13, FRANCE email: [email protected]

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 P5. Applications to the geometry of Palatini threefolds and Debarre-Voisin Hyper-K¨ahler manifolds are also discussed.

1 Introduction

Let V₆ be a complex vector space of dimension six and denote the projective space Proj(S^•(V6)) by P⁵. Let Wn be a complex vector space of dimension n ≥ 2. To help the reader to distinguish the projectivization of W_n^∨ from the other projective spaces considered in this article, let us reserve the bold notation Pn−1 for Proj(S^•(W_n)) and the symbol P for the other ones.

Definition 1.1 For n ≥ 2, a general element of Wn⊗

2

VV6 defines a skew-symmetric element M of Hom_S^•_(Wn)(V₆^∨, V₆). Its Pfaffian defines a cubic hypersurface Pf(M) in Pn−1 which is smooth forn ≤6. We have the exact sequence

0 −−→ V₆^∨⊗ O_Pn−1(−1) −−→^M V₆⊗ O_Pn−1 −−→ F −−→ 0,

where F is a rank-2sheaf over Pf(M). Forn ≤6, the sheaf F is locally free on Pf(M).

We will call it the Pfaffian bundle defined by M.

It is known from classical works on representations of a cubic form by Pfaffians ([Be], [Do]) that forn ≥6 a general cubic is not a Pfaffian and that for 3≤n ≤5 the Pfaffian bundles over a fixed Pfaffian cubic have moduli spaces of positive dimension.

The main result of this article concerns the casen = 4. In this situation we have the following results from [Be]:

. Every smooth cubic surface of P₃ can be defined by a linear Pfaffian.

. Let (W₄ ⊗

2

VV₆)^sm be the open subset of W₄ ⊗

2

VV₆ corresponding to smooth Pfaffian surfaces. For any element M of (W4 ⊗

2

VV6)^sm, the Pfaffian sheaf is a stable rank-2 vector bundle over the Pfaffian surface Pf(M). Moreover, it is an arithmetically Cohen-Macaulay sheaf and every arithmetically Cohen-Macaulay rank-2 vector bundle over a smooth cubic surface S with determinant OS(2) is a Pfaffian bundle.

. The quotient of (W₄⊗

2

VV₆)^sm byGL(V₆) for the following action GL(V₆)×(W₄⊗

2

VV₆)^sm → (W₄⊗

2

VV₆)^sm (P, M) 7→ ^tP ·M·P

is isomorphic to the space of pairs (S, F), whereS is a smooth cubic surface of P₃ andF an isomorphism class¹ of a Pfaffian bundle onS. It is a geometric quotient.

In this article we obtain a geometric interpretation of these orbits.

Definition 1.2 A complete pentahedron inscribed in a cubic divisor S of P₃ is a set {H₀, . . . , H₄} of 5 planes of P₃ such that:

i) (H₀, . . . , H₄) is a projective basis of P^∨₃.

ii) The 10points (Hi∩Hj∩Hk)0≤i<j<k≤4 are on S.

We define the subsetH of|O_P3(3)|×|O_P3(5)|(resp. the subsetH_ord of|O_P3(3)|×(P^∨₃)⁵) to be the set of elements (S,Π) such that S is a smooth cubic surface of P₃ and Π is a complete pentahedron inscribed inS (resp. complete pentahedron inscribed inS with an ordering of the five planes). For a fixed cubic surface S of P₃, 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:

1of vector bundles over the fixed cubic surfaceS

Theorem 1.3 There is a natural birational map from(W₄⊗

2

VV₆)^sm/GL(V₆)toH such that the composition with the projection to |O_P3(3)| is the Pfaffian map. In particular:

. (W₄⊗

2

VV₆)^sm/GL(V₆) is a rational variety of dimension 24.

. Let S be a general cubic surface. The moduli space of Pfaffian bundles on S is birational to HS.

We explain this theorem with two constructions. The first construction is a rational map Φ₁ (Definition 3.11) related to a classical problem of hyperplane restriction of the Pfaffian bundles. So in Section 2, we start with the easy casen = 3 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 3, we construct Φ₁ 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 3-dimensional varieties X inP5 such that h⁰(O_X(2))> h⁰(O_P5(2)) 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^∨₃. Then we describe this linear system in Proposition 3.8. It turns out that its exceptional locus² is 5 points of P^∨₃. This achieves the construction of Φ₁.

The geometric configuration of these five planes is only explained in Section 4 by the construction of a rational map Φ2 (Corollary 4.3). This time, it is a problem of linear spaces multisecant to G(2, V₆). The key step to construct Φ₂ is the surprising Proposition 4.1, with following summary.

Proposition The projection of the Grassmannian G(2, V₆) ⊂ P¹⁴ from a general 3-dimensional projective space has a unique singular point of order 5.

The claim that Φ₁ and Φ₂ are generically the same and also their birationality are proved in Section 4.2 from the explicit formula of Theorem 4.7. This ends the proof of Theorem 1.3. 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(2, V₆) by the diagonal action ofP GL(V₆). 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. 7.2]. 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

2i.e., the image of the contracted locus

for Palatini threefolds in a six-dimensional variety of P⁹. Those varieties of dimension six were discovered by C. Peskine. They are of independent interest because they are smooth and non quadratically normal in P9 (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 toP2×P2. So we will also explain in this section the consequences on Peskine varieties of the work on the Palatini threefolds done in Section 3.

Acknowledgement:

I would like to thank I. Dolgachev for encouraging discussions and references and J. D´eserti and O. Debarre for their attentive proofreading.

2 Invariants of Pfaffian bundles over plane cubics.

2.1 Ruled surfaces in P

and the case n = 3.

In this section, we work out in detail the case n = 3. The following easy lemma gives an important classification result for the next sections.

Lemma 2.1 For a general element M of W₃⊗

2

VV₆, we consider the exact sequence of Definition 1.1

0 −−→ V₆^∨⊗ O_P2(−1) −−→^M V₆⊗ O_P2 −−→ 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 h⁰(L²) = 0;

b) F is the unique unsplit extension

0→θ(1) →F →θ(1) →0 where θ² =O_C and θ 6=O_C;

c) F =θ(1)⊕θ(1) where θ² =O_C and θ6=O_C.

Proof: To simplify the notation, denote F(−1) by ¯F. First one can remark that h⁰( ¯F) = 0, and that ¯F ' F¯^∨ because M is skew-symmetric. So we have ∧²F¯ = O_C. We choose a point p on C. We will now prove that there is a point r of C such that h⁰( ¯F(p−r))>0.

From the Riemann-Roch theorem, the bundle ¯F(p) has a pencil of sections. It gives, onP1×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 thath⁰( ¯F(p−r))>0. Since h⁰( ¯F) = 0, we obtain thatO_C(p−r) is not trivial, and we are in one of the 3 cases above.

Remark 2.2 With the notation of the previous lemma, we will say that a plane in P⁵ is M-isotropic, if it is isotropic for all the elements of

2

VV₆ in the image ofM :W₃^∨ →

2

VV₆. From sequence (1), the ruled surface Proj(S^•(F)) has a natural embedding of degree 6 in P5, such that

• in case a), it contains two plane cubic curves and the planes spanned by these curves are disjoint in P5,

• in case b), it contains only one plane cubic,

• in case c), it is a divisor of bidegree (0,3) in a Segre variety P1 ×P2 ⊂ P5. So it contains infinitely many plane cubics.

Moreover, the planes in P5 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 3 ofF 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

∧²(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 P5. Con- versely, from any M-isotropic plane, we obtain some element P of GL(V6) such that

tP ·M ·P =

0 −^tA

A B

, where A, B are 3 ×3 matrices with linear entries. So the cokernel of A gives an invertible quotient of F of degree 3 as in Lemma 2.1.

2.2 Universal settings and the SL(V

)-invariant double cover

Definition 2.3 Let G(3, V₆^∨) and G(3,

2

VV6) be the Grassmannians of 3-dimensional vector subspaces of V₆^∨ and

2

VV₆. Denote by K₃ and R₃ their tautological subbundles.

We define the isotropic incidence

Z ⊂G(3, V₆^∨)×G(3,

2

VV₆) −−→^p2 G(3,

2

VV₆)

p1



 y G(3, V₆^∨)

Z ={(A^∨, B)|A^∨ is isotropic for all the elements of B ⊂

2

^V₆}

to be the vanishing locus of the uniqueSL(V₆)-invariant section of

2

VK₃^∨R^∨₃. Denote by U the open subset of G(3,

2

VV₆) made of vector spaces such that the intersection of their projectivizations with the Pfaffian hypersurface ofP(

2

VV₆)is a smooth cubic curve.

The restriction of Z to G(3, V₆^∨)× U will be denoted by Z_U. Let E₁₂ be the rank-12 bundle defined by the exact sequence

0 −−→ E₁₂ −−→

2

VV₆⊗ O_G(3,V^∨

6 ) −−→

2

VK₃^∨ −−→ 0. (2)

I would like to thank A. Kuznetsov for the following description ofZ from the relative Grassmannian.

Proposition 2.4 The isotropic incidenceZ is isomorphic to the relative Grassmannian G(3, E₁₂)of linear subspaces of the bundle E₁₂. The projection ZU → U ⊂G(3,

2

VV₆) is generically finite of degree 2. The fiber of this morphism over an element of type a), b), c) in Lemma 2.1 consists in G(3, V₆^∨) of 2 points, 1 point, and a rational cubic curve.

Proof: Let (µ, ν) be an element ofG(3, V₆^∨)×G(3,

2

VV₆). The fiber of a vector bundle at µ(resp. ν) will be denoted by the name of the bundle with the indexµ (resp. ν). The vector space K_3,µ is isotropic for all the skew-symmetric forms defined by the elements of R_3,ν if and only if (µ, ν)∈Z, but also if and only if the composition

R_3,ν −−→

2

VV₆ −−→

2

VK_3,µ^∨

is the zero map. So (µ, ν) ∈ Z ⇐⇒ R_3,ν ⊂ E_12,µ and we have the equality Z = G(3, E₁₂).

The end of the assertion follows immediately from Lemma 2.1 and Remark 2.2.

Corollary 2.5 The locus U_c in U ⊂G(3,

2

VV₆) of planes of type c) has codimension 3.

Define a relation R on Uc by pRp⁰ if and only if p1(p⁻¹₂ (p)) = p1(p⁻¹₂ (p⁰)). For any elementpof U_c, there is a six-dimensional subspaceL_p of

2

VV₆ such that the equivalence class of p for R is an open subset of G(3, L_p).

Proof: From Proposition 2.4, for any p in U_c, p₁(p⁻¹₂ (p)) is a smooth rational cubic curve C_p in G(3, V₆^∨). So the restriction of E₁₂ to C_p is O^⊕6

P1 ⊕ O^⊕6

P1 (−1) and it has a natural trivial subbundle of rank 6. Let L_p be the six-dimensional vector subspace of

2

VV6 obtained from the image of this subbundle by the injection of sequence (2).

Proposition 2.4 describes p⁻¹₁ (C_p) as the relative Grassmannian G(3, E12|C_p). Let F be a subbundle of rank 3 of

E12|C_p =L_p⊗ O_P1 ⊕ O^⊕6

P1(−1).

Case c) appears when the line bundle ∧³F^∨ contracts the curve C_p. But ∧³F^∨ is not ample if and only if F is a trivial subbundle of L_p⊗ O_P1. So p⁻¹₁ (C_p)∩p⁻¹₂ (U_c) is (U ∩G(3, L_p))×C_p, and the equivalence class ofpforRisU ∩G(3, L_p). So the dimension of U_c is the sum of the dimension of G(3,6) and the dimension of the family of rational cubic curves in G(3, V₆^∨). In conclusion U_c, has dimension 33 and codimension 3 in G(3,

2

VV6).

3 Palatini threefolds

In this section, we study the case n = 4.

3.1 Definition and classical properties

Definition 3.1 A smooth 3-dimensional subvariety X of P5 is called a Palatini three- fold³ if there exists an element of M of W₄⊗

2

VV₆ 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.528]) the singular variety of the linear system |W₄^∨| of linear line complexes in |V₆^∨|.

Notation 3.2 In this section, denote by X a Palatini threefold in P5, by h the class of a hyperplane of P5, by S the Pfaffian cubic surface in P₃, and by s the pullback on X of the class of a hyperplane of P3. The cotangent bundle of P⁵ will be denoted by Ω¹_P5.

So we can immediately obtain the well known resolution of its ideal:

Remark 3.3 The ideal I_X of a Palatini threefold X in P5 has the following resolution 0 −−→ W₄^∨ ⊗ O_P5 −−→^M Ω¹

P5(2h) −−→ I_X(4h) −−→ 0 (3)

and we have the famous equality ([Me-dP] problem 1) h⁰O_X(2h) = h⁰O_P5(2h) + 1.

The following results are classical generalizations of properties of a Veronese surface embedded inP⁴.

Remark 3.4 A Palatini threefold X has a natural embedding in the point/plane inci- dence variety of P5. It is defined as follows.

To a point x of X, we associate the plane π_x of P5 swept out by the lines through x which are quadrisecant toX (i.e., the lines throughx contained in all the line complexes associated to X).

The intersection X∩π_x is a plane curve of degree 3 and a residual point supported onx. 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 P5, F. Zak introduced the following vector bundle:

Definition 3.5 The canonical extension on P⁵ displayed in the second column of the

3or a Palatini scroll

following diagram of exact sequences

0



 y

0 −−→ W₄^∨ ⊗ O_P5(−h) −−→ Ω¹_P5(h) −−→ IX(3h) −−→ 0

∼



 y



 y 0 −−→ W₄^∨ ⊗ O_P5(−h) −−→^M V₆⊗ O_P5



 y O_P5(h)



 y 0

induces on a Palatini threefold X the following extension with middle term a rank 3 vector bundle EX (where N_X^∨ is the conormal bundle of X in P⁵)

0 −−→ N_X^∨(3h) −−→ E_X −−→ O_X(h) −−→ 0.

Moreover, the restriction toX of the second line of the previous diagram gives the exact sequence

0 −−→ OX(−h−s) −−→ W₄^∨⊗ OX(−h) −−→^M V6⊗ OX −−→ EX −−→ 0 (4) and the determinant of E_X is O_X(3h−s).

From the inclusion W₄^∨ ⊂

2

VV6 and the identification W4 =

3

VW₄^∨, we can consider P^∨₃ as a subvariety of G(3,

2

VV₆).

Proposition 3.6 Let Z₄ be the restriction of the isotropic incidence Z ⊂ G(3, V₆^∨)× G(3,

2

VV₆) to G(3, V₆^∨)×P^∨₃. Then Z₄ is isomorphic to X and the projection from Z₄ to G(3, V₆^∨) is the embedding of X given by

3

VE_X. It is the natural map x 7→ π_x of Remark 3.4.

Proof: Let us first recall the classical description of quadrisecant lines to X. Let A^∨ and B be the 3-dimensional vector subspaces of V₆^∨ and W₄^∨ corresponding to a point ofZ₄. Denote by A⁰ the kernel of the surjection from V₆ to A. The restriction of Ω¹_P5(1) toP(A^∨) is A⁰⊗ O_P(A^∨₎⊕Ω¹

P(A^∨)(1).

From the isotropicity of P(A^∨) with respect to all the elements of B ⊂

2

VV₆, we see that the restriction of

M :W₄^∨⊗ O_P5(−h)→Ω¹

P5(h) toP(A^∨) is the direct sum of the following maps

B⊗ O_P(A^∨₎(−1)→A⁰ ⊗ O_P(A^∨₎ and (W₄^∨/B)⊗ O_P(A^∨₎(−1)→Ω¹

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) pointµofP(A^∨)∩X. So we have constructed a morphism

ζ : Z₄ → X (A^∨, B) 7→ µ.

Moreover, the vanishing at µ shows by specialization of sequence (4) that the fiber of E_X^∨ atµ isA^∨. So the first projection p₁ is also the composition

Z₄ −−→^ζ X ^|

3

VEX|

−−−−→ G(3, V₆^∨).

The proof of the statement is now reduced to the proof of the embedding of Z4 in G(3, V₆^∨). But the fiber of this morphism over the point of G(3, V₆^∨) corresponding to A^∨ is a single point because A^∨ is not isotropic for all the elements of W₄^∨. So this projection of Z4 is one-to-one, and it must be an isomorphism because the fibers are given by linear conditions.

3.2 Anticanonical properties

Although it is classical that the canonical class K_X of a Palatini threefold satisfies K_X³ =−2 ([Ok]), the following identification and next proposition seem new.

Lemma 3.7 The canonical invertible sheafω_X of X is isomorphic toO_X(s−2h). With Notation 3.2, we have from the equality W4 =H⁰(OS(1)) a canonical isomorphism

H⁰(ω_X^∨)'W₄^∨.

Proof: The isomorphismω_X ' O_X(s−2h) can be computed directly from Definition 3.1.

We obtain the isomorphism H⁰(ω_X^∨) ' W₄^∨ from the isomorphism between X and Z₄ found in Proposition 3.6 and the fact thatω_Z^∨₄ is the pull-back of O_P^∨

3(1).

Proposition 3.8 The linear system |ω_X^∨| has no base points and gives a morphism of degree 2:

X −−→^2:1 P^∨₃ ⊂G(3,

2

VV₆).

It contracts5 rational curves. These curves are smooth of degree 3for the embedding of X in P5 and also for the embedding of X in G(3, V₆^∨).

Proof: From Proposition 3.6 and Lemma 3.7, the anticanonical map is the restriction of the projectionp₂ (Definition 2.3) to the incidence variety Z₄. So it is base point free of degree 2. The contracted curves of this morphism correspond to case c) of Lemma 2.1.

Such curve parametrizes the planes in a Segre varietyP¹×P². So it is a smooth rational cubic curve in G(3, V₆^∨). By definition, on a contracted curve, the divisors 2h and s are equivalent because ω^∨_X = O_X(2h−s). So those curves have the same degree with respect to h and 3h−s. It remains to prove that the image of those curves in P^∨₃ is 5 points. This will be proved in Lemma 3.10 by the description of the ideal inP^∨₃ of this exceptional locus as a determinantal ideal.

Remark 3.9 The previous proposition has the following geometric interpretation inP⁵. Any linear section of the Pfaffian cubic surfaceS gives a sextic ruled surface inX ⊂P5. There are five particular cases where such a surface is in a Segre variety P1×P2 (the five Segre varieties are different in P⁵). Any of these Segre variety intersects X in the corresponding ruled surface of degree 6 and a residual smooth rational curve of degree 3. 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 P₃. Fortunately, they are explicit (cf Remark 4.8). It was an important tool in the construction of the matrix M₄ in Section 4.2.

Lemma 3.10 Let F¯ be the normalized bundle F(−1). The vector space H =H¹ (S²F¯)(−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 −−→

2

VV₆ '

4

VV₆^∨ −−→ S²W₄ −−→ 0. (5) Moreover the ideal of the exceptional locus in P^∨₃ of the projection X ' Z4 → P^∨₃ is given by the 4×4 Pfaffians of a skew-symmetric map

H⊗ O_P^∨

3(−1) −−→ H^∨⊗ O_P^∨

3.

Proof: Let i be an isomorphism ∧²F¯ −→ O^∼ _S. The restriction of F to a plane P is of type c) in Lemma 2.1 if and only if we have h¹(S²( ¯FP)) = 3. The determinantal structure will be obtained from this criterion.

To globalize this condition, let us consider the complex

C^• : 0 −−→ V₆^∨⊗ O_P3(−2) −−→^M V₆⊗ O_P3(−1) −−→ 0.

It is exact in degree −1 with cohomology ¯F in degree 0. The second exterior power of C^• tensorized byO_P3(2) is

0 −−→ S²V₆^∨ ⊗ OP3(−2) −−→ V₆^∨⊗V6⊗ OP3(−1) −−→

2

VV6⊗ OP3 −−→ 0.

Its cohomology in degree (−2,−1,0) is given by (0,(S²F¯)(−1),(∧²F¯)(2)). So the hyper- cohomology spectral sequence of this complex gives the exact sequence (5), the dimension of H, and the vanishings

h⁰((S²F¯)(−1)) =h²((S²F¯)(−1)) =h⁰(S²F¯) =h²(S²F¯) = 0.

Now consider the point/plane incidence variety I3 ⊂ P^∨₃ ×P3 and denote by p^∨₃ andp₃ the restrictions to I₃ of the first and second projections of this product. We have the exact sequence

0 −−→ O_P^∨

3(−1)S²F¯(−1) −−→ O_P^∨

3 S²F¯ −−→ p^∗₃(S²F¯) −−→ 0.

From the Leray spectral sequence and the above vanishings, we have the exact sequence 0 −−→ p^∨_3∗(p^∗₃(S²F¯)) −−→ H¹((S²F¯)(−1))⊗ O_P^∨

3(−1) −−→^dM H¹(S²F¯)⊗ O_P^∨

3 −−→

R¹p^∨_3∗(p^∗₃(S²F¯)) −−→ 0.

Let us now explain how to consider the map dM as a skew-symmetric map. The isomorphism i gives a symmetric isomorphism i⁰ : S²( ¯F) −→^∼ S²( ¯F^∨) so the following square is commutative

(S²F¯)(−1)⊗S²F¯ ⁱ

0⊗id

−−−→ (S²F¯^∨)(−1)⊗S²F¯

id⊗i⁰



 y



 y^τ (S²F¯)(−1)⊗S²F¯^∨ −−→^τ0 O_S(−1).

The cup-productH¹ (S²F¯)(−1)

⊗H¹ (S²F¯)(−1)

→H² (S²F¯⊗S²F¯)(−2)

is anti- commutative, so for any z ∈W₄ the following square⁴ is also anti-commutative

H¹ (S²F¯)(−1)

⊗H¹ (S²F¯)(−1) ^dM,z⊗id

−−−−−→ H¹ S²F¯

⊗H¹ (S²F¯)(−1)

id⊗d_M,z



 y



 y^∪ H¹ (S²F¯)(−1)

⊗H¹ S²F¯

H² S²F¯⊗(S²F¯)(−1)

∪



 y



 y^τ◦(i

0⊗id)

H² (S²F¯)(−1)⊗S²F¯ τ⁰◦(id⊗i⁰)

−−−−−→ H²(O_S(−1)).

In conclusion, the composition H⊗ O_P^∨

3(−1) −−→^dM H¹(S²F¯)⊗ O_P^∨

3

i¯⁰

−−→ H¹(S²F¯^∨)⊗ O_P^∨

3

Serre duality

−−−−−−−→ H^∨⊗ O_P^∨

3

is skew-symmetric and the lemma is proved. Indeed, the type c) case corresponds to the locus where this map has rank at most 2, so it defines 5 points in P^∨₃. This also achieves the proof of Proposition 3.8.

Definition 3.11 Let Σ5 be the fifth symmetric product of P^∨₃. We define a rational map

Φ₁ : (W₄⊗

2

VV₆)^sm/GL(V₆) 99K P(S³(W₄))×Σ₅ M 7→ (S,{h₀, . . . , h₄}) ,

where S is the Pfaffian cubic surface defined by M, and h0, . . . , h4 are the five linear sections of S defined in Proposition 3.8.

In Section 4, we will study the image of this map.

4An overlined symbol denotes the map induced in cohomology.

3.3 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 surfaceV embedded inP4. For instance, in the Veronese situation, the sequence (3) is just replaced by

0 −−→ W₃^∨⊗ O_P4 −−→^M Ω¹

P4(2h) −−→ I_V(3h) −−→ 0.

But the main difference is that in the theory of Severi varieties the embedding of V by the complete linear system |OV(h)| is understood from an interpretation in terms of matrices of size 3×3 of rank 1. For a Palatini threefold, there are no similar results to describe the embedding by the complete linear system |OX(2h)|. The following remark could be a first step in this direction.

Remark 3.12 The restriction of the line bundle ω_X^∨ O_X(s)to the diagonal of X×X gives the natural inclusion

W₄^∨⊗W₄ ⊂H⁰(O_X(2h)).

In other words, the embedding of a Palatini threefold X by |O_X(2h)| has a canonical projection to P(W₄^∨ ⊗W₄). Moreover, the image of X by this projection consists of endomorphisms of W₄ of rank 1.

Proof: It is straightforward from Lemma 3.7.

4 Geometry in

2

V V

4.1 Projections from linear spaces

The Grassmannian variety G(2,6) in its Pl¨ucker 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 P³ and points of multiplicity 5.

Proposition 4.1 Denote by U₅ the subspace of G(5,

2

VV₆)defined by the5-dimensional vector spaces such that the intersection of their projectivization with G(2, V₆) consists of five linearly independent distinct points. Then, a general 4-dimensional subspaceW₄^∨ of

2

VV6 is contained in a unique element of U5. Proof: First remark that the incidence variety

I_4,5 ={(W₄^∨, W₅^∨)|W₄^∨ ⊂W₅^∨ ⊂

2

^V₆, W₅^∨ ∈ U₅}

has the same dimension asG(4,

2

VV₆). We thus have to prove that the natural projection is birational.

Consider a general elementW₄^∨ in the image of this projection and choose an element W₅^∨ such that (W₄^∨, W₅^∨) ∈ I_4,5. Again denote by P₃,P₄ their projectivizations. The vector spaceH⁰(I_P3∪G(2,V₆)(2)) is the kernel of the mapH⁰(I_G(2,V6)(2)) =

4

VV₆^∨ →S²W₄. So it has dimension 5. Now remark that we also haveh⁰(I_P4∪G(2,V6)(2)) = 5 because the ideal of the 5 points P₄∩G(2, V₆) in P₄ is a 10-dimensional space of quadrics. So we proved that P₄ must be in all the quadrics of H⁰(I_P3∪G(2,V₆)(2)). It gives the following linear conditions satisfied by any W₅^∨ of U₅ containingW₄^∨

W₅^∨ ⊂ \

q∈H⁰(I_P

3∪G(2,V6)(2))

(W₄^∨)^⊥q,

where ⊥_q denotes the orthogonal with respect to the quadratic form q on

2

VV₆. So uniqueness ofW₅^∨ will be a consequence of the existence of a W4 such that

\

q∈H⁰(I_P

3∪G(2,V6)(2))

(W₄^∨)^⊥q

has dimension 5. This is true in the following:

Example 4.2 Let us consider a basis(_i) of V₆, and the 5 elements

u₀ =₀∧₃, u₁ =₁∧₄, u₂ =₂∧₅, u₃ = (₀+₁+₂)∧(₄+₃+₅), u₄ = (₁+₄+₂)∧(₃+₁ +₅).

Denote by W₅^∨ the 5-dimensional vector space spanned by the (u_i)0≤i≤4 and W₄^∨ ={ X

0≤i≤4

λ_i.u_i | X

0≤i≤4

λ_i = 0}.

Then T

q∈H⁰(I_P

3∪G(2,V6)(2))

(W₄^∨)^⊥q has dimension 5.

Proof: We compute with [Macaulay2] that H⁰(I_P3∪G(2,V₆)(2)) is generated by the fol- lowing five quadrics in Pl¨ucker coordinates

• p_(3,4)p_(1,5)−p_(1,4)p_(3,5)+p_(1,3)p_(4,5),

• p_(1,2)p_(0,5)−p_(2,4)p_(0,5)−p_(0,2)p_(1,5)+p_(2,3)p_(1,5)+p_(0,1)p_(2,5)−p_(1,3)p_(2,5)+p_(0,4)p_(2,5)− p_(3,4)p_(2,5)+p_(1,2)p_(3,5)+p_(2,4)p_(3,5)−p_(0,2)p_(4,5)−p_(2,3)p_(4,5),

• p_(2,3)p_(0,4)−p_(0,3)p_(2,4)+p_(0,2)p_(3,4)−p_(1,3)p_(0,5)+p_(2,4)p_(0,5)−p_(3,4)p_(0,5)+p_(0,3)p_(1,5)− p_(2,3)p_(1,5)+p_(1,3)p_(2,5)−p_(0,4)p_(2,5)+p_(3,4)p_(2,5)−p_(0,1)p_(3,5)−p_(1,2)p_(3,5)+p_(0,4)p_(3,5)− p_(2,4)p_(3,5)+p_(0,2)p_(4,5)−p_(0,3)p_(4,5)+p_(2,3)p_(4,5),

• p_(1,2)p_(0,4)−p_(0,2)p_(1,4)+p_(0,1)p_(2,4)−p_(2,4)p_(0,5)+p_(2,3)p_(1,5)−p_(1,3)p_(2,5)+p_(0,4)p_(2,5)− p_(3,4)p_(2,5)+p_(1,2)p_(3,5)+p_(2,4)p_(3,5)−p_(0,2)p_(4,5)−p_(2,3)p_(4,5),

• p_(1,2)p_(0,3)−p_(0,2)p_(1,3)+p_(0,1)p_(2,3)−p_(2,4)p_(0,5)+p_(3,4)p_(0,5)+p_(2,3)p_(1,5)−p_(1,3)p_(2,5)+ p_(0,4)p_(2,5)−p_(3,4)p_(2,5)+p_(1,2)p_(3,5)−p_(0,4)p_(3,5)+p_(2,4)p_(3,5)−p_(0,2)p_(4,5)+p_(0,3)p_(4,5)− p_(2,3)p_(4,5),

and check that the ideal of the orthogonal of P₃ with respect to these 5 quadrics is generated by the 10 independent equations

(p_(3,5), p_(0,5)−p_(1,5)+p_(4,5), p_(3,4)+p_(4,5), p_(2,4)−p_(1,5)+p_(4,5), p(0,4)−p(1,5)+p(4,5), p(2,3)−p(1,5), p(1,3)−p(1,5), p(1,2)+p(4,5), p(0,2), p(0,1)).

So this example completes the proof of the birationality of the projection from I_4,5 to G(4,

2

VV₆) and we have proved Proposition 4.1.

Corollary 4.3 With the notation of Definition 1.2, we can define the rational map Φ₂ : (W₄⊗

2

VV₆)^sm/GL(V₆) 99K H

M 7→ (S,{H₀, . . . , H₄})

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(2, V₆) such that P₃ is in the linear span h(u_i)_0≤i≤4i. Then take

H_i =P₃∩h(u_j)_{0≤j≤4,j6=i}i.

Proof: After Proposition 4.1, we only have to explain why {H₀, . . . , H₄} is inscribed on S. But for {i₀, . . . , i₄}={0, . . . ,4}, the pointH_i0 ∩H_i1∩H_i2 is on the line (u_i3, 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 24.

Proof: Let Σ⁰₅ be the image of H in |O_P3(5)| by the second projection. It is an open subset of the symmetric product Σ₅ defined in Definition 3.11. So it is a rational 15- dimensional variety ([G-K-Z, Th. 2.8 p. 137]).

The partial derivatives of order 2 of any element of Σ⁰₅ are linearly independent cubic forms. So they give a rank-10 subsheafF₂ ofH⁰(O_P3(3))⊗ O_Σ⁰

5 locally free with respect to Zariski’s topology.

Now remark that H is the open subset of P(F₂) corresponding to smooth cubic surfaces. So H is rational of dimension 24.

4.2 An explicit formula and proof of Theorem 1.3

Surprisingly, we are able to give in this section an explicit formula. Recently, an ex- plicit 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 ofS, while here, we first choose some pentahedron, then we need to find the unique Pfaffian bundle onS related to this pentahedron;

- 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≤3 be a basis ofW₄, and letA₉ be the following subset ofC¹⁰×P4

A₉ =

((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 6= 0, and for 0≤i < j < k ≤3, a_i,j,k = 1

.

Then the map

P GL4× A9 → Hord

(P,((a_i,j,k)0≤i<j<k≤4,(b_i)0≤i≤4)) 7→ (S,(H₀, . . . , H₄)) (6) is birational where

X

0≤i<j<k≤4

a_i,j,k.w_i.w_j.w_k = 0

is an equation ofS, and for all 0≤i≤4, w_i = 0 is an equation of H_i with the following equalities: w₄ =

3

P

i=0 b4.wi

bi ,







 w0

w1

w2

w3









=P·







 x0

x1

x2

x3









.

Proof: Let Π = (H0, . . . , H4) be an ordered pentahedron and let P⁰ be the unique projective transformation that sends the ordered pentahedron (x₀, x₁, x₂, x₃, x₀+x₁ + x₂+x₃) to (H₀, . . . , H₄). Denote byh_ithe equation ofH_i defined by







 h0

h1

h2

h3









=P⁰·







 x0

x1

x2

x3









and h₄ =P3

i=0h_i. Cubic surfaces S such that (S,Π) is inH_ord are the smooth surfaces defined by:

X

0≤i<j<k≤4

A_i,j,k.h_i.h_j.h_k = 0, (A_i,j,k)(0≤i<j<k≤4) ∈P9. Now remark that the map

A₉ → P9

(a, b) 7→ (A_i,j,k =a_i,j,kb_ib_jb_k)0≤i<j<k≤4

is birational because we can compute its inverse with the following formulas b₀

b₃ = A_0,1,2 A_1,2,3,b₁

b₃ = A_0,1,2 A_0,2,3,b₂

b₃ = A_0,1,2 A_0,1,3,b₄

b₃ = A_0,1,4

A_0,1,3, a_i,j,4 = A_i,j,4

A_i,j,3 · A_0,1,3 A_0,1,4.

So we obtain the lemma from the equalities w_i = h_ib_i, with P defined by the product of the diagonal matrix (^b_b⁰

4, . . . , ^b_b³

4) with P⁰.

Definition 4.6 Let A⁰₉ be the set of triples (a, b, u) such that (a, b) is an element of A₉ defining a smooth cubic surface

X

0≤i<j<k≤4

ai,j,kwiwjwk= 0, w4 =

3

X

i=0

b₄w_i b_i ,

and u is a root of the following equation in X:

X²+X·(1 +a_0,2,4−a_0,3,4) +a_0,2,4 = 0.

Denote by v =−(1 +a_0,2,4−a_0,3,4)−u the other root and set

e₁ =a_0,2,4+a_1,2,4−a_2,3,4, e₂ = 1+a_1,2,4−a_1,3,4, e₃ = (−a_1,2,4+a_1,3,4−1)v−a_1,2,4−a_0,2,4+a_2,3,4,

M₄ =

















0 u −1 a_1,2,4 e₁ e₂

−u 0 0 0 a_0,2,4 −u

1 0 0 0 −v 1

−a_1,2,4 0 0 0 a_1,2,4v −a_1,2,4

−e₁ −a_0,2,4 v −a_1,2,4v 0 e₃

−e₂ u −1 a_1,2,4 −e₃ 0

















M₀₁₂₃ =

















0 0 0 w0+w3 w3 w3

0 0 0 w₃ w₁+w₃ w₃

0 0 0 w₃ w₃ w₂+w₃

−w0−w3 −w3 −w3 0 0 0

−w₃ −w₁−w₃ −w₃ 0 0 0

−w₃ −w₃ −w₂−w₃ 0 0 0















 .

Theorem 4.7 For a generic element (P,(a, b, u)) of P GL₄ × A⁰₉, the element M of (W4 ⊗

2

VV6) defined by M = M0123 +w4M4 is such that Φ1(M) = Φ2(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 M₄. It was done by searching for the rational cubic curve in P5 associated to the plane w₄ = 0 in Proposition 3.8. We followed these steps:

• From the equation P

0≤i<j<k≤4ai,j,kwiwjwk of S we obtain a sextic curve of ideal (P

0≤i<j<4a_i,j,4w_iw_j,Pf(M)) in P₃. This curve has geometric genus 0 and is sin- gular at the 4 vertices of the pentahedron that are not in the plane w₄ = 0. In the next steps, we lift this curve to P⁵ to obtain the desired cubic curve.

• The intersection of this sextic curve with the plane w4 = 0 gives 6 points on a conic. Let L ⊗θ(1) denote the Pfaffian bundle defined by M₀₁₂₃ on the plane w₄ = 0 (where L is a 2-dimensional vector space).

• The image of the6points by the linear system|θ(1)|is also on a conic. We choose a parametrization of this conic (cf. the equations foru andv in Definition 4.6) to obtain an identification S²L=H⁰(θ(1)).

• So we have the identification V₆ =L⊗S²L and a rational cubic curve in a Segre variety inP⁵. The matrixM4was obtained from the marked element of

2

V(L⊗S²L).

But now that we have foundM₄, it is much easier to check that M satisfies the required properties.