Geometry of the genus 9 Fano 4-folds

Geometry of the genus 9 Fano 4-folds

Han Fr´ed´eric

Institut de Math´ematiques de Jussieu, Universit´e Paris 7 - Case Postale 7012, Batiment Chevaleret

75205 Paris Cedex 13 email: han@math.jussieu.fr

Abstract

Let W be a 6 dimensional vector space over the complex numbers endowed with a non degenerate symplectic form ω. Let Gω be the Grassmannian of ω- isotropic 3-dimensional vector subspaces ofW. We will also denote by P_ω the 13 dimensional projective space spanned by Gω in the Pl¨ucker embedding. Consid- ering this embedding, the intersection of Gω with a generic codimension 2 linear subspace is the Mukai model of a smooth Fano manifold of dimension 4, genus 9, index 2 and Picard number 1. We give an explicit description of its the variety of lines.

Introduction:

On a genus 9 Fano variety with Picard number 1, Mukai’s construction gives a natural rank 3 vector bundle, but in dimension 4, another phenomena appears. In the first part of this article, we will explain how to construct on a general Fano 4-foldB of genus 9, a canonical set of four stable vector bundles of rank 2, and prove that they are rigid. In next parts, we study the consequences on the geometry of the 4-fold.

Indeed, this “four-ality” (cf [M]) is also present in the geometry of the lines included in this Fano 4-fold, and also in its Chow ring. In section 2, we show explicitly that the variety of lines in B is a hyperplane section of IP₁×IP₁×IP₁ ×IP₁. Then in section 3, we compute the Chow ring of B which appears to have a rich structure in codimension 2.

The four bundles can embed B in a Grassmannian G(2,6), and the link with the order one congruence of lines discovered by E. Mezzetti and P. de Poi in [M-dP] is explained in section 4. In particular we prove that the generic Fano variety of genus 9 and dimension 4 can be obtained by their construction, and explain the choices involved.

We also describe in this part the normalization of the non quadratically normal variety they constructed, and also its variety of plane cubics.

Acknowledgements:

I would like to thank L. Gruson for his constant interest in this work, and also F.

Zak and C. Peskine for fruitful discussions. I would like also to thank K. Ranestad and A. Kuznetsov for pointing out to me and explaining their works. I’m also grateful to the referee for careful reading and many useful comments.

Notations:

In all the paper but section 4.1,B will be a general double hyperplane section ofG_ω. For any uin G_ω, the corresponding plane ofIP(W)will be noted π_u. Furthermore, the restriction of a hyperplane H in P_ω to G_ω will be noted H.¯

1 Construction of rank 2 vector bundles on B

This part is devoted to the construction of a canonical set of four stable and rigid rank two vector bundles on B using a classical technique of modification. Those bundles were already known to A. Iliev and K. Ranestad in [I-R] where they are constructed by projection. They described the link between some moduli spaces of vector bundles in terms of linear sections of the dual variety ofG_ω, with main application to the genus 9 Fano threefold case. Note also that many of the results of this section are obtained in a universal way in derived categories by A. Kuznetsov in [Ku], but we detail this short description to use it in the next sections.

Let’s first recall some classical geometric properties of G_ω (cf [I]). The union of the tangent spaces toG_ωis a quartic hypersurface ofP_ω, so a general line ofP_ω has naturally 4 marked points. Dually, as the varietyB is given by a pencil L of hyperplane sections ofG_ω, there are in this pencil, four hyperplanes H₁, . . . , H₄ tangent toG_ω. Denoting by u_i the unique contact point of H_i with G_ω, we will first construct a rank two sheaf on H_i∩G_ω with singular locus u_i, and its restriction to B will be a vector bundle.

1.1 Data associated to a tangent hyperplane section

Let u ∈ G_ω, and H be a general hyperplane tangent to G_ω at u. We consider the following hyperplane section of G_ω:

H¯_u ={v ∈G_ω, π_v ∩π_u 6=∅}.

The following lemma is proved in [I]:

Lemma 1.1 There exists a conic C in π_u such that v ∈H∩H¯_u ⇐⇒ π_v∩C 6=∅. For H general containing the tangent space of G_ω at u, C is smooth. Furthermore, H∩H¯_u contains the tangent cone TuGω∩Gω ={v ∈Gω|dim(πv∩πu)≥1} which is embedded in P_ω as a cone over a Veronese surface.

Consequently, we will assume that H is such that C is smooth. Now, we consider the following incidence variety:

Z_H ={(p, v)∈C×H|p¯ ∈π_v}

and denote by q₁ and q₂ the projections from C×G_ω to C and to G_ω. Corollary 1.2 The incidence variety Z_H is smooth.

The previous lemma implies that for any point p of C the fiber q₁⁻¹(p) consist of all the isotropic planes containing p, so q⁻¹₁ (p) is a smooth 3-dimensional quadric (cf [I]

proposition 3.2). SoZ_H is a fibration in smooth quadratic threefolds over IP₁.

Notations:

Let σ be the class of a point of C, and denote by L be the vector space H⁰O_C(σ) viewed as SL₂-representation. In all the paper, we will identify L with its dual, and denote by S_iL the symmetric power of order i of L.

For any integers a and b, the sheaf q₁^∗O_C(a.σ)⊗q₂^∗(O_Gω(b))on Z_H will be denoted O_ZH(a, b).

Let K and Q be the tautological¹ bundles of rank three on G_ω, such that the fol- lowing sequence is exact:

0→K →W ⊗ O_Gω →Q→0.

Proposition 1.3 Fori >0we haveRⁱq2∗O_ZH(1,0) = 0, and the resolution ofq2∗O_ZH(1,0) as a O_Gω-module is given by the following exact sequence:

0→S₃L⊗ O_Gω(−1)→L⊗

2

^Q^v→L⊗ O_Gω →q2∗O_ZH(1,0)→0 (1) Proof: We consider the injection fromq₁^∗(OC(−2σ)) toW⊗OC×Gω given by the conicC.

The incidence Z_H is the locus where the map from q^∗₁(O_C(−2σ))⊕q^∗₂K to W⊗ OC×G_ω

is not injective, hence Z_H is obtained in C ×G_ω as the zero locus of a section of the bundleOC(2σ)Q.

Let K^. be the Koszul complex ^Vi(O_C(−2σ)Q^v) of this section. We obtain propo- sition 1.3 from the Leray spectral sequence applied toK^. twisted by OC×Gω(σ).

Restricting the above surjection L⊗ O_Gω →q2∗O_ZH(1,0) to the hyperplane section H, we obtain:¯

Proposition 1.4 The sheaf E on H¯ defined by the following exact sequence:

0→ E →L⊗ OH¯ →q2∗O_ZH(1,0)→0 is reflexive of rank 2, has c₁(E) = −1 and is locally free outside u.

Proof: From lemma 1.1 we have q2(ZH) = H∩H¯u. So set theoretically the support of q2∗O_ZH(1,0) is H∩H¯_u, and E has rank 2 on ¯H. The exact sequence (1) in proposition 1.3 proves that the O_Gω-module q2∗O_ZH(1,0) has local projective dimension 2. So we obtain from the Auslander-Buchsbaum formula that its local depth is 4. Therefore, this module is locally Cohen-Macaulay and E is reflexive from [H2] Corollary 1.5. We also deduce from the sequence (1) that the Chern polynomial of q2∗O_ZH(1,0) at order 2 is:

1−2c2(Q). But on Gω we have the relation 2c2(Q) = (c1(Q))², soc1(E) =−1 and the support ofq2∗O_ZH(1,0) is the reduced schemeH∩H¯_u. As the equation ofH annihilates q2∗O_ZH(1,0), we can also consider this sheaf as aOH¯-module with the same local depth.

Remarking that ¯H − {u} is smooth of dimension 5, we obtain from the Auslander- Buchsbaum formula, that the OH,x¯ -module (q_2∗O_ZH(1,0))_x has projective dimension 1 for every closed point xof ¯H− {u}, so E is locally free outside u.

Furthermore, we deduce from Bott’s theorem on G_ω the following:

1Remark that onGωthe bundlesQandK^v are isomorphic

Corollary 1.5 We have the following equality L=H⁰(O_ZH(1,0)) =H⁰(q_2∗O_ZH(1,0)), and for i > 0, all the groups Hⁱ(O_ZH(1,0)) and Hⁱ(q2∗O_ZH(1,0)) are zero. For i ≥ 0 all the groups Hⁱ(q2∗O_ZH(1,−1)) and Hⁱ(q2∗O_ZH(1,−1)) are zero.

Proof: We will prove that, on the isotropic GrassmannianG_ω, all the cohomology groups of the bundles ^Vi Q^v and (^Vi Q^v)(−1) vanish for i ∈ {1,2,3}. Indeed, with the nota- tions of [W] 4.3.3 and 4.3.4, they correspond to the partitions (0,0,−1), (0,−1,−1), (−1,−1,−1), (−1,−1,−2), (−1,−2,−2), (−2,−2,−2). Now recall that the half sum of positive roots is ρ = (3,2,1), so α+ρ either contains a 0 or is (2,1,−1). So in all casesα+ρ is invariant by a signed permutation, and the sheaves have zero cohomology.

The corollary is now a direct consequence of these vanishing and of proposition 1.3 and its proof.

Corollary 1.6 We have: ∀i ≥ 0, hⁱ(E) = hⁱ(E(−1)) = 0. The vector space V = H⁰(E(1)) has dimension 6 and hⁱ(E(1)) = 0 for i > 0. Furthermore, E(1) is generated by its global sections.

Proof: The vanishing of the cohomology of E and E(−1) is a direct consequence of the definition of E and of the previous corollary.

To prove the second assertion, we restrict the sequence (1) to the hyperplane section H, and we obtain the following monad¯ ²:

0→S₃L⊗ OH¯(−1)→L⊗

2

^Q^vH¯ → E →0

whose cohomology is Tor¹(q2∗(O_ZH(1,0)),OH¯) which is equal to q2∗(O_ZH(1,−1)) be- cause we proved in proposition 1.4 that its support is a subscheme of H, so the multi- plication by the equation ofH fromq2∗(O_ZH(1,0))⊗ O_Gω(−1) to q2∗(O_ZH(1,0))⊗ O_Gω is the zero map. Twisting this monad byOH¯(1) and using corollary 1.5, we obtain that H⁰(E(1)) is the quotient ofL⊗W byS₃L⊕LbecauseW =H⁰(QH¯) andQ= (^V2 Q^v)(1).

Furthermore, the right part of the monad gives a sujection from L⊗QH¯ to E(1). Since L⊗QH¯ is generated by its global sections, so is E(1).

The vanishing ofhⁱ(E(1)) fori >0 is a corollary of the vanishing ofhⁱ(q2∗(O_ZH(1,0)), hⁱ(QH¯) and hⁱ(OH¯) fori >0.

Remark 1.7 The two vector spacesV andW of dimension 6 have different roles. More precisely, the conic C gives a marked subspace of W so that we have the followingSL₂- equivariant sequences:

0→S₂L→W →S₂L→0 and 0→L→V →S₃L→0

2A monad is a complex of vector bundles given by an injection followed by a surjection (cf [OSS])

1.2 The 4 rank 2 vector bundles on B

The pencil of hyperplanes defining B contains the 4 tangent hyperplanes H_i, so we can apply the previous construction to construct a rank 2 sheaf E_i on each of the ¯H_i, and define by E_i the restriction of E_i to B. Because B is smooth, it doesn’t contain the contact pointsu_i, so E_i is locally free on B.

Corollary 1.8 All the cohomology groups of the vector bundlesE_i vanish. In particular, the rank 2 vector bundles E_i are stable. The vector space H⁰(E_i(1)) has dimension 6, and we have H^j(Ei(1)) = 0 for j > 0. The bundles Ei(1) are generated by their global sections.

Proof: It is a direct consequence of corollary 1.6, because B is a hyperplane section of ¯H_i. (Note that the stability condition for E_i is equivalent to h⁰(E_i) = 0, because c₁(E_i) = −1 andE_i has rank 2).

1.3 The restricted incidences

Now, for each of the 4 hyperplanesHi containingB and tangent to Gω at the point ui, letC_i be the conic of the projective planeπ_ui constructed in 1.1. Consider the restriction of the incidence varietyZ_Hi toB. In other words, let Z_i, Z_i⁰ be:

Z_i ={(p, v)∈C_i×B|p∈π_v}, Z_i⁰ ={(p, v)∈Z_i|dim(π_v∩π_ui)>0}

and still denote byq₁ and q₂ the projections from C_i×G_ω to C_i and G_ω.

Lemma 1.9 Let p be a fixed point of C_i. The scheme Z_i,p = q₂(q₁⁻¹(p)∩Z_i) is a 2 dimensional irreducible quadric in Pω. For a general choice of p on Ci, the quadric Zi,p

is smooth. The restriction ofq₂ toZ_i⁰ is a double cover of a Veronese surfaceV_i =q₂(Z_i⁰).

Proof: In fact, we already mentioned that {v ∈ G_ω|p ∈ π_v} is a smooth quadric of dimension 3 (cf [I] prop 3.2), so it doesn’t contain planes. This scheme is included inH_i, so Z_i,p is just a hyperplane section of this smooth quadric, so it is always irreducible, and for a general p it is smooth because B is also general. Consider the cone {v ∈ G_ω|dim(π_ui ∩ π_v) > 0} described in lemma 1.1. As u_i ∈/ B, the surface V_i is the intersection of this cone with a hyperplane which doesn’t contain the vertex u_i, so it’s a Veronese surface.

Notations:

We denote by σ_i the class of a point on C_i, and by h₃ the class of a hyperplane in P_ω (the Pl¨ucker embedding of G_ω).

Proposition 1.10 LetΠbe the following projective bundle, andhbe the class ofO_Π(1).

The incidence Z_i is a divisor of class 2h in

Π =P roj(O_Ci(2σ_i)⊕S₂L⊗ O_Ci).

Furthermore we have h3 ∼h+ 2σi and σi is also the class of the fiber of a point on the base of the fibration Π . The divisor Z_i⁰ of Z_i is equivalent to the restriction to Z_i of h−2σ_i.

Proof: Denote by e_i the vector bundle image of the map from O_Ci(−2σ_i) to W ⊗ O_Ci associated to the embedding of C_i in π_ui, and by e^⊥_i its orthogonal with respect to ω. Choose an element φ⁰ of ^V3 W^v such that kerφ⁰ gives a hyperplane section of G_ω containing B and different from the ¯H_i (i.e φ⁰(u_i) 6= 0). We can remark that the incidence Z_i is given overC_i by the isotropic 2-dimensional subspaces l of ^e_e^⊥i

i such that φ⁰(ei ∧(∧²l)) = 0. Indeed, the condition φi(ei ∧(∧²l)) = 0 is already satisfied by the definition of C_i and lemma 1.1 (here φ_i denotes a trilinear form of kernel H_i).

The bundlee^⊥_i is isomorphic toS₂L⊗ O_Ci⊕L⊗ O_Ci(−σ_i), and the trivial factorS₂L corresponds to the planeπ_ui. So the bundle ^e_e^⊥i

i is isomorphic toL⊗O_Ci(σ_i)⊕L⊗O_Ci(−σ_i) where those factors are isotropic for the symplectic form induced by ω. We can take local basiss₀, s₁ and s₂, s₃ of each factors such that the form induced by ω isp_0,2+p_1,3 wherep_j,k denotes the Pl¨ucker coordinates associated to the s_j.

So the relative isotropic GrassmannianG_ω(2,^e_e^⊥i

i) is the intersection ofG(2,^e_e^⊥i

i) with IP(O_Ci(2σ_i)⊕ O_Ci(−2σ_i)⊕S₂L⊗ O_Ci) in IP(^{V2 e}_e^⊥i

i), and the factor O_Ci(2σ_i) still corre- sponds tos0∧s1.

Now we need to compute the kernel of the mape_i⊗^V2(^e_e^⊥i

i) ^φ

0

→ O_Ci. But the assumption φ⁰(u_i)6= 0 proves that it is O_Ci(−4σ_i)⊕L⊗L⊗ O_Ci(−2σ_i).

So we have an exact sequence:

0→ O_Ci(−2σ_i)⊕S₂L⊗ O_Ci →

2

^(e^⊥_i e_i )

(^ω

φ0)

−→ O_Ci⊕e^v_i →0

andZ_i is a divisor of class 2h inP roj(O_Ci(2σ_i)⊕S₂L⊗ O_Ci). The relation h₃ ∼h+ 2σ_i is given by the map e_i⊗^V2(^e_e^⊥i

i)→^V3 W.

The divisor Z_i⁰ of Z_i is locally given by the vanishing of the exterior product with s₀∧s₁ so it is equivalent to (h−2σ_i)|Z_i.

Corollary 1.11 The two dimensional quadrics (Z_i,p)p∈C_i are smooth except in four cases where they are irreducible cones.

Proof: From the definition of Zi,p in lemma 1.9, for any point p of Ci, the quadric Z_i,p is the image of q⁻¹₁ (p)∩ Z_i by the restriction of the linear system |h₃|. As the restriction of h₃ and h to the fibers ofq₁ :Z_i →C_i are equivalent from the proposition 1.10, we just have to study the section of OΠ(2h) found in the proposition 1.10. This section corresponds to a section of S₂(O_Ci(2σ_i)⊕S₂L⊗ O_Ci), and the determinant of this quadratic form is a section ofO_Ci(4σ_i). Lemma 1.9 now implies that there are only four irreducible cones.

1.4 Rigidity of E

We will now study the relation between the conormal bundle of Zi in Ci×B and the bundleE_i.

Lemma 1.12 We have the following exact sequence:

0→ O_Ci×B(−σ_i−h₃)→q₂^∗E_i → I_Zi(σ_i)→q₂^∗(R¹q_2∗I_Zi)(−σ_i)→0

Proof: Considerq₂ as a fibration in IP₁. The resolution of the diagonal of IP₁×IP₁ gives the relative Beilinson’s spectral sequence (cf [OSS] chap 2 §3):

E_a,b¹ = (

−a

^ω_q2(σ_i))⊗R^bq2∗(I_Zi((1 +a)σ_i)) =⇒ I_Zi(σ_i).

By the definition of E_i (cf prop 1.4) we have E_i = q2∗(I_Zi(σ_i)). Furthermore, the projectionq₂(Z_i) is a hyperplane section of B, soq2∗I_Zi =O_B(−h₃). Now remark that R¹q2∗(I_Zi(σ_i)) = 0, because the restriction ofq₂: Z_i ^q−→^2|Zi q₂(Z_i) has all its fibers of length at most 2. The non zero terms inE¹ are:

q₂^∗(R¹q2∗IZi)(−σi) O_B(−σ_i−h₃) E_i

6E_0,.¹

-E_.,0¹

and the spectral sequence ends atE². The filtration ofI_Zi(σ_i) is given by the sequence:

0→E_0,0² →IZi(σi)→E_−1,−1² →0.

So we obtain the lemma by the definition of E² and the above values of E¹.

NB: The support ofR¹q2∗IZi is the natural scheme structure (cf [G-P]) on the scheme of fibers ofq₂ intersecting Z_i in length 2 or more. It is the Veronese surface V_i =q₂(Z_i⁰).

So the previous lemma can now be translated in the following:

Corollary 1.13 The scheme q⁻¹₂ (V_i)∪Z_i is in C_i×B the zero locus of a section of the bundle q^∗₂E_i(σ_i+h₃).

This gives also a geometric description of the marked pencil of sections of E_i(h₃) given by the natural inclusion L ⊂V found in remark 1.7. Indeed, if we fix a pointp on C_i, the restriction toq⁻¹₁ (p) of the section obtained in corollary 1.13 gives with the notations of lemma 1.9 the following:

Corollary 1.14 For any pointpon the conicC_i, the vector bundleE_i(h₃)has a section vanishing on Z_i,p∪V_i.

We can now study the restriction ofq₂^∗E_i toZ_i.

Proposition 1.15 The restriction q₂^∗Ei|Zi of the vector bundle q₂^∗E_i to Z_i fits into the following exact sequence:

0→ O_Zi(h₃−3σ_i)→q₂^∗Ei|Z_i(h₃)→ O_Zi(3σ_i)→0

Proof: Fix a pointponC_i, and consider the corresponding section ofE_i(h₃) constructed in corollary 1.14. Its pull back gives a section ofq^∗₂Ei(h3) vanishing onq⁻¹₂ (Zi,p∪Vi), so its restriction toZ_i gives a section ofq^∗₂Ei|Z_i(h₃−σ_i−Z_i⁰). Now, using the computation of the class ofZ_i⁰ inZ_i made in proposition 1.10, namely thatO_Zi(Z_i⁰) is O_Zi(h₃−4σ_i), it gives a section of (q^∗₂Ei)|Zi(3σi). We have to prove that it is a non vanishing section.

To obtain this, we compute the second Chern class of (q₂^∗E_i)|Z_i(3σ_i). We will show that its image in the Chow ring ofC_i×B is zero. Denote bya_i the second Chern class of E_i. From lemma 1.13, we obtain the class of Zi in Ci ×B: [Zi] = ai+h3.σi −[Vi]. So we can compute [Z_i].c₂(q₂^∗E_i(3σ_i)). It is (a_i+h₃.σ_i−[V_i]).(a_i−3h₃σ_i), but we will compute in proposition 3.7 the Chow ring ofB, and this class vanishes.

Corollary 1.16 The vector bundles E_i are rigid, in other words: Ext¹(E_i, E_i) = 0.

Proof: From corollary 1.13 we deduce an exact sequence on Ci×B:

0→q₂^∗E_i(−σ_i)→q^∗₂(E_i)⊗q^∗₂(E_i)(h₃)→q^∗₂E_i(h₃ +σ_i)→(q^∗₂E_i(h₃+σ_i))_|Z

i∪q⁻¹₂ (Vi) →0 All the cohomology groups of the bundle q₂^∗E_i(−σ_i) are zero, and the corollary 1.8 gives H⁰(q₂^∗E_i(h₃+σ_i)) =L⊗V and H¹(q^∗₂E_i(h₃+σ_i)) = 0. The liaison exact sequence:

0→ O_q⁻¹

2 (Vi)(−Z_i⁰)→ O_Z

i∪q⁻¹₂ (Vi) → OZi →0 twisted byq₂^∗(Ei(h3+σi)) is:

0→q^∗₂E_i(h₃)⊗ O_q⁻¹

2 (Vi)(σ_i−Z_i⁰)→q₂^∗E_i(h₃+σ_i))_|Z

i∪q₂⁻¹(Vi)→q₂^∗E_i|Zi(h₃+σ_i)→0 As σ_i −Z_i⁰ has degree −1 along the fibers of q₂ : q₂⁻¹(V_i) → V_i, all the cohomology groups of the bundleq^∗₂Ei(h3)⊗ O_q⁻¹

2 (Vi)(σi−Z_i⁰) vanish, so the cohomology ofq₂^∗Ei(h3+ σi))_|Z

i∪q₂⁻¹(Vi) can be computed from its restriction to Zi. Propositions 1.15 and 1.10 show that H⁰(q₂^∗E_i|Zi(h₃ +σ_i)) = S₂L⊕S₂L⊕S₄L. In conclusion, we have the exact sequence:

0→Hom(Ei, Ei)→L⊗V →S2L⊕S2L⊕S4L→Ext¹(Ei, Ei)→0

By corollary 1.8, the bundle E_i is stable, so it is simple, in other words we have Hom(E_i, E_i) = C, and the above exact sequence gives Ext¹(E_i, E_i) = 0.

2 The variety of lines in B

Remark 2.1 Let δ be an isotropic line in IP(W). The set of isotropic planes of δ^⊥ containing δ form a line in Gω(3, W), and all the lines in Gω(3, W) are of this type for a unique element of G_ω(2, W). In other words, the variety of lines in G_ω(3, W) is naturally isomorphic with G_ω(2, W).

Notations:

A point of G_ω(2, W) will be denoted by a lower case letter, and its corresponding line in G_ω(3, W) by the associated upper case letter. The Pl¨ucker hyperplane sec- tion ofG_ω(j, W)will be notedh_j, the tautological subbundle by K_j, and the variety of lines included inB will be denoted F_B. Let I be the following incidence variety:

I =P roj((K₂^⊥ K₂)

v

(h₂)) ={(δ, p)∈F_B×B|p∈∆} ⊂G_ω(2, W)×G_ω(3, W) The projections from I to F_B and B will be denoted by f₁ and f₂.

I Z_i

._{f1 f2}& ._{q2 q1}&

G_ω(2, W)⊃F_B B ⊂G_ω(3, W) C_i 'IP₁

Lemma 2.2 The varietyF_B is obtained inG_ω(2, W)as the zero locus of a section of the bundle (^K_K^2⊥

2)^v(h2)⊕(^K_K^2⊥

2)^v(h2). For a general choice of B, FB is smooth and irreducible with ω_FB =O_FB(−h₂).

Proof: As B is a double hyperplane section of Gω(3, W), we have from the previous definition of I the equality: f1∗(f₂^∗O_Gω(3,W)(h₃)) = (^K_K^2⊥

2)^v(h₂). So F_B is the vanishing locus of a section of (^K_K^2⊥

2)^v(h₂)⊕(^K_K^2⊥

2)^v(h₂) and its dualising sheaf is O_FB(−h₂). The choice of a generic 2-dimensional subspace ofH⁰(O_Gω(3,W)(h₃)) corresponds to the choice of a generic section of (^K_K^⊥2

2)^v(h₂)⊕(^K_K^2⊥

2)^v(h₂). Hence, for a general choice ofB, the variety F_B will be smooth becauseK₂^⊥(h₂) is globally generated, and so is ^K_K^2⊥

2(h₂)'(^K_K^2⊥

2)^v(h₂).

We obtain the vanishing of the first cohomology group of the ideal I_FB of F_B in G_ω(2, W) from the exact sequence:

0→ O_Gω(2,W)(−h2)→(K₂^⊥

K₂)^v⊕(K₂^⊥

K₂)^v → IFB →0.

So we haveh⁰(O_FB) = 1 and F_B is connected and smooth, so it is irreducible.

2.1 A morphism from F

to IP

× IP

× IP

× IP

Each of the four conics C_i will enable us to construct a morphism from F_B to IP₁. We have the following geometric hint to expect at least a rational map: a general elementδ ofF_Bgives an isotropic 2-dimensional subspaceL_δofW. In general, the projectivisation ofL^⊥_δ intersects the plane containg C_i in a point p. There is at least an elementmof ∆ such thatp∈πm, so m is inHi∩H¯ui because it is in ∆⊂B ⊂Hi. Now, the definition of H_i and lemma 1.1 prove that pmust be on C_i.

But to show that this rational map is everywhere defined, we will use the vector bundleEi. We start by constructing line bundles on FB.

Lemma 2.3 A line ∆ included in B intersects the Veronese surface V_i if and only if it is included in the hyperplane section H¯ui = q2(Zi) of B. Moreover, any such line is contained in a quadric Z_i,pi for a unique point p_i of C_i. The set v_i ={δ ∈F_B|∆⊂H¯_ui} is equal to f₁(f₂⁻¹(V_i)) and it is a divisor in F_B.

Proof: Letδ be an element ofF_B such that there is a pointv in the intersection ∆∩V_i. By lemma 1.9, the plane π_v intersects π_ui in a line and it contains the line IP(L_δ). So the intersection IP(Lδ)∩πui is not empty and is included in πv⁰ for any point v⁰ of ∆.

So ∆ is included in ¯H_ui.

Now if the line ∆ is included in ¯H_ui, we have from lemma 1.1 that for any b ∈ ∆, the planeπb intersects the conic Ci. Let’s first prove that the line IP(Lδ) must intersect C_i in some point p_i. Indeed, if it was not the case, the intersection of π_b with C_i would cover C_i as b vary in ∆, so IP(L_δ) would be orthogonal to C_i and it would be in the plane πui, but any line in this plane intersects Ci.

So the line ∆ is in the quadric Z_i,pi. Note that IP(L_δ)∩C_i can’t contain another point because the line ∆ can’t be included in the Veronese surface V_i. Furthermore,

proposition 1.10 implies thatZ_i,p⁰

i is a plane section of the quadricZ_i,pi. As this section is irreducible because V_i doesn’t contain lines, the intersection of ∆ and V_i is a single point. In conclusion we havev_i =f₁(f₂⁻¹(V_i)) and the varietiesv_i andV_i have the same dimension, sov_i is a divisor in F_B.

Lemma 2.4 For any point p_i of C_i, the scheme f₂⁻¹(Z_i,pi) has several irreducible com- ponents of dimension 2, but some of these components are contracted by f₁ to a curve.

Proof: The components of f₂⁻¹(Z_i,pi) corresponding to the lines included in Z_i,pi are contracted to curves.

Notations:

Denote by A_i,pi the 2 dimensional part of f₁(f₂⁻¹(Z_i,pi)).

Proposition 2.5 The sheaf f1∗f₂^∗E_i is a line bundle on F_B. There is a natural map µ_i from L⊗ O_FB to the dual bundle of f1∗f₂^∗E_i. The image of µ_i is also a line bundle on F_B, we will denote it byO_FB(α_i). By construction, for anyp_i ∈C_i, the divisorA_i,pi will be in the linear system |O_FB(α_i)|, and we have f1∗f₂^∗E_i =O_FB(−α_i−v_i).

Proof: By corollary 1.8, the bundleE_i is a quotient of 6O_B(−1) and by proposition 1.4, it is a subsheaf of 2O_B with c₁(E_i) = −1. So its restriction to any line ∆ included in B must be O_∆⊕ O_∆(−1). As a consequence, we have R¹f_1∗f₂^∗E_i = 0 and f_1∗f₂^∗E_i is a line bundle. Denote this line bundle by O_FB(−α⁰_i). Dualising and twisting the exact sequence defining E_i in proposition 1.4, we obtain the following one:

0→L⊗ O_B(−2h₃)→E_i(−h₃)→ L_i →0 (2) whereL_i is supported on the hyperplane section ¯H_ui, and is singular along the Veronese surface V_i. As the incidence I is P roj((^K_K^⊥2

2)^v(h₂)) (where K₂ is the tautological sub- bundle ofW ⊗ O_Gω(2,W)), the relative dualising sheaf ω_f1 is O_I(2h₂−2h₃). So we have R¹f_1∗(f₂^∗E_i(−h₃)) =O_FB(α⁰_i−2h₂). Therefore the base locus of the map:

L⊗ O_FB(−2h₂)→ O_FB(α⁰_i−2h₂)

is the support ofR¹f_1∗f₂^∗(L_i). We will now prove that this sheaf is a line bundle on the divisor v_i defined in lemma 2.3.

The morphism f₁ defined at the beginning of section 2, is projective of relative dimension 1, and the sheaf f₂^∗E_i(−h₃) is flat over F_B. So by base change (cf [H1] 11.2 and 12.11), for any point δ of F_B, the fiber (R¹f1∗f₂^∗E_i(−h₃))_δ is H¹(E_i(−h₃)⊗ O_∆), where ∆ is the line in B corresponding to δ. The restriction of the sequence (2) to ∆ gives the surjection:

2O_∆(−2)→ O_∆(−1)⊕ O_∆(−2)→ L_i ⊗ O_∆→0 (3) When the line ∆ is not in the hyperplane section ¯Hui, the sheafLi⊗ O∆ is supported by the point ¯H_ui∩∆, so in this case we have h¹(L_i⊗ O_∆) = 0. Now, when the line ∆ is in H¯_ui, the sheafL_i⊗ O_∆ has generic rank 1 because the Veronese surfaceV_i can’t contain

the line ∆. We have proved in lemma 2.3 that ∆ intersects V_i, hence for any element of L, the corresponding section ofE_i(h₃) vanishes on ∆. So the map 2O_∆(−2)→ O_∆(−2) induced by the sequence (3) is zero, and for any δ in v_i, we have h¹(L_i ⊗ O_∆) = 1, thereforeR¹f_1∗f₂^∗(L_i) is a line bundle on v_i.

So our pencil of sections of (f1∗f₂^∗E_i)^v can now be interpreted as a base point free pencil of sections of (f1∗f₂^∗E_i)^v(−v_i). In other words, the image ofµ_i is the line bundle O_FB(α_i) = (f_1∗f₂^∗E_i)^v(−v_i). By definition A_i,pi is the closure of {δ ∈ F_B|length(∆ ∩ Z_i,pi) = 1} which was identified set theoretically with an element of the linear system

|α_i|, so we conclude the proof with the following lemma 2.6:

Lemma 2.6 For a generic choice of a point p_i on C_i, the support of the sheaf R¹f1∗(f₂^∗IZ_i,pi∪Vi(−h3)) represents the class α⁰_i, and all its irreducible components are reduced.

Proof: First notice that the point p_i on C_i gives from the corollary 1.13 a section of E_i(h₃), so the exact sequence:

0→ OB(−2h3)→Ei(−h3)→IZ_i,pi∪Vi(−h3)→0.

Applyingf₂^∗ and f1∗ to the previous sequence, we obtain a section of O_FB(α⁰_i) vanishing on the support of the sheafR¹f1∗(f₂^∗I_Zi,pi∪V_i(−h₃)). But this is the definition in [G-P] of the scheme structure on the set of lines included in B and intersecting Zi,pi∪Vi. So to show that this scheme structure is reduced on each component, we have to prove that Z_i,pi and V_i are not contained in the ramification locus of the morphism: f₂ : I → B. We will do those two cases simultaneously by taking a general point m on Zi,pi ∩Vi. Such a point is the intersection of Z_i,pi and another quadric Z_i,p⁰

i, so there pass four distinct lines through m, and from lemma 2.3 there are no other lines in B through m becausemis only on two element of (Zi,p)p∈Ci. So the pointmis not in the ramification of the morphism f₂ : I →B because it is of degree four (cf lemma 3.1, or remind that the tangent cone of G_ω(3, W) is a cone over a Veronese surface).

2.2 Description of the morphism

Denote by ˜µi the morphisms fromFB toCi given by the surjections µi :H⁰(OCi(σi))^v⊗ O_FB →→ O_FB(α_i) constructed in the previous section. In the following, we prove that the morphism fromF_B toC₁×C₂×C₃×C₄ is an embedding for a generic B, and that its image is a hyperplane³ section.

As we need to allow cases where a point and a line do not generate a plane, we will sometimes work in affine spaces.

Notations:

LetC_B be the affine cone overB. A vector corresponding to a point of a projective space will be denoted by the same letter with a tilde. For a point p_i of the conic Ci, we consider:

F_pi ={δ∈G_ω(2, W)|˜δ∧p˜_i ∈ C_B}

3with respect to|OC₁×C2×C3×C4(σ1+σ2+σ3+σ4)|

Unfortunately, we have to remark that ˜µ_i⁻¹(p_i) is not exactly F_pi∩F_B:

Lemma 2.7 The intersection F_pi ∩F_B is equal to G_ω(2, p^⊥_i )∩F_B. It contains µ˜_i⁻¹(p_i) and residual curves corresponding to the lines included in the quadric Z_i,pi.

Proof: Ifδ be an element ofF_pi, then we have the inclusion< L_δ, p_i >⊂< L_δ, p_i >^⊥. So δ is in G_ω(2, p^⊥_i ) and F_pi ⊂G_ω(2, p^⊥_i ).

Now, let δ be an element of G_ω(2, p^⊥_i )∩F_B. We have from remark 2.1:

f₁(f₂⁻¹(Z_i,pi)) ={δ ∈F_B|∃v ∈B, IP(L_δ)⊂π_v ⊂IP(L^⊥_δ) and p_i ∈π_v}.

Furthermore, this remark also implies that any isotropic plane of IP(W) containing IP(Lδ) is an element of the line ∆, so the vector ˜δ∧p˜i is either 0 or corresponds to an element of B. So we have ˜δ∧p˜_i ∈ C_B. So G_ω(2, p^⊥_i )∩F_B ⊂F_pi ∩F_B.

Now remark that in G_ω(2, W), the scheme G_ω(2, p^⊥_i ) is the zero locus of a section of K₂^v. The restriction of this section to FB vanishes on the divisor Ai,pi defined in proposition 2.5, so F_B∩F_pi contains a divisor of class α_i and the zero locus of a section of (K₂^v)|F_B(−α_i) corresponding to the lines included in Z_i,pi.

Nevertheless, we have the following:

Lemma 2.8 For the generic double hyperplane section B of G_ω(3, W), we can find points p_i (resp. p_j) in the conics C_i (resp. C_j) such that F_pi∩F_pj is a smooth conic in G_ω(2, W). Furthermore, this conic is in F_B and represents the class α_i.α_j.

Proof: We can choosep_i and p_j respectively in C_i and C_j such that, p_i is not inp^⊥_j . So, for any element δof F_pi∩F_pj, the vectors ˜δ∧p˜_i and ˜δ∧p˜_j are different from 0 and give two distinct points of ∆∩B. This implies that ∆ is in B, and that the intersection F_pi∩F_pj is automatically included inF_B. We also deduce from the assumptionp_i ∈/ p^⊥_j that the isotropic Grassmannian G_ω(2, p^⊥_i ∩p^⊥_j ) is a smooth 3-dimensional quadric.

Denote by φk the equation of the hyperplane Hk. Among the four equations (φ_k(˜δ ∧p˜_l) = 0)_k,l∈{i,j} defining the intersection F_pi ∩F_pj in G_ω(2, p^⊥_i ∩p^⊥_j ), the fol- lowing two are automatically satisfied: (φ_k(˜δ∧p˜_k) = 0)k∈{i,j}. Indeed, recall that the vectors ˜δ∧p˜_i and ˜δ ∧p˜_j are in the affine cone over G_ω because d ∈ G_ω(2, p^⊥_i ∩p^⊥_j ).

Then conclude with lemma 1.1 because p_i is in C_i and p_j is in C_j. Consequently, the intersection F_pi ∩F_pj is defined in G_ω(2, p^⊥_i ∩p^⊥_j ) by the two equations φ_i(˜δ∧p˜_j) = 0 and φ_j(˜δ∧p˜_i) = 0.

Let L_δ be the intersection of p^⊥_j and the 3 dimensional vector space U_i represented by the contact pointui ofHi. The corresponding pointδ is an element ofGω(2, p^⊥_i ∩p^⊥_j ) such that φj(˜pi ∧ δ)˜ 6= 0 (because ˜pi ∧ δ˜ corresponds to ui, and B is smooth), but recall from lemma 1.1 that φ_i(˜p_j ∧δ) = 0. So˜ F_pi ∩F_pj is defined by two independent hyperplane sections of a smooth 3 dimensional quadric. So it is a (may be singular) conic.

Note that from the genericity of B we could also assume that IP(p^⊥_i )∩C_j ={a₁, a₂} and IP(p^⊥_j )∩C_i = {b₁, b₂} are 4 distinct points. According to lemma 1.1, the lines (pi, a1), (pi, a2), (pj, b1), (pj, b2) define points of Fpi ∩Fpj, but no three of those lines