Geometry of genus 9 Fano 4-fold

0.25 MB·

by  Han Frederic

Geometry of genus 9 Fano 4-fold
Han Fr´ed´eric
9 Institut de Math´ematiques de Jussieu,
0 Universit´e Paris 7 - Case Postale 7012,
0
2 Place Jussieu
2
75251 Paris Cedex 05 France
n
a email: [email protected]
J
Introduction:
1
1
Let W be a 6 dimensional vector space over the complex number, equiped with a non

] degenerate symplectic form ω. Let Gω be the grassmannian of ω-isotropic 3-dimensional
G
vector subspaces of W. Considering the Plucker embedding, the intersection of G with
ω
A
a generic codimension 2 linear subspace is the Mukai model of a smooth Fano manifold
.
h of dimension 4, genus 9 index 2 and picard number 1. We will also note by P the 13
ω
t
a dimensional projective space spanned by G under it’s plucker embedding.
ω
m
Notations:
[

2 In all the paper, B will be a general double hyperplane section of Gω. For a
v hyperplane H of Pω, we define H¯ = H∩Gω, and for any u ∈ Gω, the corresponding
4 plane of IP(W) will be noted π .
u
5
0 Abstract
1
. On a genus 9 Fano variety, Mukai’s construction gives a natural rank 3 vector
1
0 bundle, but curiously in dimension 4, another phenomena appears. In the first
9 part of this article, we will explain how to construct on a Fano 4-fold of genus 9
0
(named B), a canonical set of four stable vector bundles of rank 2, and prove that
:
v
they are rigid. Those bundles was already known to A. Iliev and K. Ranestad
i
X in [I-R] and the results of this section could be viewed as particular cases of the
r work of A. Kuznetsov (Cf [K]). We’d like here to show their consequences in the
a
geometry of the 4-fold, and study Zak duality in this case.
Indeed, this “four-ality” (Cf [M]) is also present in the geometry of lines in-
cluded in B, and also in the Chow ring ofB. In section 2 we show that the variety
of lines in B, is an hyperplane section of IP ×IP ×IP ×IP . This description is
1 1 1 1
explicit and could also be interesting in terms of Freudenthal geometries. Then in
section 3, we compute the Chow ring of B which appears to have a rich structure
in codimension 2.
The 4 bundles constructed can embed B in a Grassmannian G(2,6), and the
link with the order one congruence discovered by E. Mezzeti and P de Poi in
[M-dP] will be done in section 4. In particular we will prove that the generic fano
variety of genus 9 and dimension 4 can be obtained by their construction, and
explain the choices involved. We will also describe in this part the normalization
of the non quadraticaly normal variety they constructed, and also its variety of
plane cubics.
1
Aknowledgements:
I’d like to thanks L. Gruson for his constant interest on this work, and also F. Zak
and C. Peskine for fruitful discussions. I’d like also to thanks K. Ranestad and A.
Kuznetsov for pointing and explaining their works.
1 Construction of rank 2 vector bundles on B
This part is devoted to the construction of a canonical set of 4 stable and rigid rank 2
vector bundles on B. Many results are done in a universal way in [K], but we detail this
description touseitinnext sections. Let’sfirst recallsomeclassical geometricproperties
of G . ( Cf [I]) The union of the tangent spaces to G is a quartic hypersurface of P , so
ω ω ω
a general line of P has naturally 4 marked points. Dually, as the variety B is given by a
ω
pencil L of hyperplane sections of G , there are in this pencil, 4 hyperplanes H ,...,H
ω 1 4
tangent to G . Denoting by u the contact point of H with G , we will first construct
ω i i ω
a rank 2 sheaf on H ∩G with singular locus u , and it’s restriction to B will be the
i ω i
vector bundle.
1.1 Data associated to a tangent hyperplane section
Let u ∈ G , and H be a general hyperplane tangent to G at u. For any v in G , denote
ω ω ω
by π the corresponding projective subspace of IP(W), and consider the hyperplane
v
section of G :
ω
H¯ = {v ∈ G ,π ∩π 6= ∅}
u ω v u
It’s proved in [I] the following:
Lemma 1.1 There is a conic C in π such that v ∈ H ∩ H¯ ⇐⇒ π ∩ C 6= ∅. For
u u v
H general containing the tangent space of G at u, C is smooth. Furthermore, H ∩H¯
ω u
contains the tangent cone T G ∩G = {v ∈ G |dimπ ∩π ≥ 1} which is embedded
u ω ω ω v u
in P as a cone over a veronese surface.
ω
Let Z be the following incidence:
Z = {(p,v) ∈ C ×(H ∩G )|p ∈ π }
H ω v
Identifying C with IP , we denote by q and q the projections from C × G to IP
1 1 2 ω 1
and to G , and by L the SL -representation H0O (1). Restricting the surjection
ω 2 IP1
L⊗O → q O (1,0) to the hyperplane section H¯, we obtain:
H¯ 2∗ ZH
Proposition 1.2 The sheaf E defined by the following exact sequence:
0 → E → L⊗O → q O (1,0) → 0
H¯ 2∗ ZH
is reflexive of rank 2, c (E) = −1 and is locally free outside u.
1
2
Proof: From lemma 1.1, the support of q O (1,0) is H ∩ H¯ so it is an hyperplane
2∗ ZH u
section of H¯, hence E has rank 2 and c (E) = −1. Furthermore, for v in q (Z ), the
1 2 H
fiber of the restriction of q to Z : q−1 (v) has length 1 if v is not in H¯ , it has lenght
2 H 2|ZH u
2 if v ∈ H¯ −{u} and q−1 (u) is the curve C. As Z and H¯ −u are smooth, the sheaf
u 2|ZH H
q O (1,0) has projective dimension 1 outside u, hence E is locally free outside u.
2∗ ZH
Denote by S L the SL -representation H0(O (i)) and by K and Q the tautological
i 2 IP1
bundles on G , such that the following sequence is exact.
ω
0 → K → W ⊗O → Q → 0
Gω
Proposition 1.3 Fori > 0we haveRiq O (1,0) = 0, andthe resolution ofq O (1,0)
2∗ ZH 2∗ ZH
in G is given by the following exact sequence:
ω
2
0 → S L⊗O (−1) → L⊗ Qv → L⊗O → q O (1,0) → 0 (1)
3 Gω Gω 2∗ ZH
^
Proof: Weconsider theinjectionfromq∗(O (−2))toW⊗O givenbytheconicC.
1 IP1 IP1×Gω
So the incidence Z is the locus where the map fromq∗(O (−2))⊕q∗K to W⊗O
H 1 IP1 2 IP1×Gω
is not injective, hence Z is obtained in IP ×G as the zero locus of a section of the
H 1 ω
bundle O (2)⊠Q.
IP1
i
Let K. betheKoszulcomplex (O (−2)⊠Qv)ofthissection. Weobtainthepropo-
IP1
sition 1.3 from the Leray spectral sequence applied to K. twisted by O (1,0).
V IP1×Gω
Furthermore, we deduce from Bott’s theorem on G the following:
ω
Corollary 1.4 We have the following equality L = H0(O (1,0) = H0(q O (1,0)),
ZH 2∗ ZH
and for i > 0, all the groups Hi(O (1,0) and Hi(q O (1,0)) are zero. For i ≥ 0 all
ZH 2∗ ZH
the groups Hi(q O (1,−1)) and Hi(q O (1,−1)) are zero.
2∗ ZH 2∗ ZH
i
Proof: We will prove that on the isotropic Grassmannian G , the bundles Qv and
ω
i
( Qv)(−1)areacyclicfori ∈ {1,2,3}. Indeed, withthenotationsof[W]4.3.3aVnd4.3.4,
they correspond to the partitions (0,0,−1), (0,−1,−1), (−1,−1,−1), (−1,−1,−2),
V
(−1,−2,−2),(−2,−2,−2). Nowrecallthatthehalfsumofpositiverootsisρ = (3,2,1),
so α+ρ either contains a 0 or is (2,1,−1). So in all cases α+ρ is invariant by a signed
permutation, and the sheaves are acyclic. The corollary is now a direct consequence of
this acyclicity.
Corollary 1.5 The sheaves E and E(−1) are acyclic. The vector space V = H0(E(1))
has dimension 6 and ∀i > 0,hi(E(1)) = 0, and E(1) is generated by its global sections.
Proof: The acyclicity of E and E(−1) is a direct consequence of the definition of E and
of the previous corrolary.
3
To obtain the second assertion, we restrict the sequence 1 to the hyperplane section
H¯, so we obtain the following monad1:
2
0 → S L⊗O (−1) → L⊗ Qv → E → 0
3 H¯ H¯
^
whose cohomology is Tor1(q (O (1,0)),O ) which is equal to q (O (1,−1)) be-
2∗ ZH H¯ 2∗ ZH
cause Z ⊂ q−1(H). Twisting this monad by O (1) we obtain that H0(E(1)) is the
H 2 H¯
quotient of L⊗W by S L⊕L because W = H0(Q ). Furthermore, the right part of
3 H¯
the monad gives a sujection from L⊗Q to E(1). But L⊗Q is globally generated,
H¯ H¯
so E(1) is also generated by its global sections.
Thevanishing ofhi(E(1))fori > 0isacorollaryofthevanishing ofhi(q (O (1,0)),
2∗ ZH
hi(Q ) and hi(O ) for i > 0.
H¯ H¯
We can remark that the two vector spaces V and W of dimension 6 have not the
same role. More precisely, the conic C gives a marked subspace of W so that we have
the following:
Remark 1.6 The tangenthyperplaneH givescanonicallythe SL -equivariantsequences:
2
0 → S L → W → S L → 0 and 0 → L → V → S L → 0
2 2 3
1.2 The 4 rank 2 vector bundles on B
The pencil of hyperplanes defining B contains the 4 tangent hyperplanes H , so we can
i
apply the previous construction to construct a rank 2 sheaf E on each of the H¯ , and
i i
define by E the restriction of E to B. Because B is smooth, it doesn’t contain the
i i
contact points u , so E is locally free on B.
i i
Corollary 1.7 All the cohomolgy groups of the vector bundles E vanish. In particular,
i
the rank 2 vector bundles E are stable. The vector space H0(E (1)) has dimension 6,
i i
and ∀j > 0,hj(E (1)) = 0. The bundles E (1) are generated by their global sections.
i i
Proof: It’s a direct consequence of corollary 1.5, because B is a hyperplane section of
H¯ . (Note that the stablility condition for a E is equivalent to h0E = 0)
i i i
1.3 The restricted incidences
Now, for each of the 4 hyperplanes H containing B and tangent to G at some point u ,
i ω i
let C bethe conic of theprojective plane π constructed in1.1. Consider the restriction
i ui
of the incidences Z to B. In other words, let Z ,Z′ be:
Hi i i
Z = {(p,v) ∈ C ×B|p ∈ π }, Z′ = {(p,v) ∈ Z |dim(π ∩π ) > 0}
i i v i i v ui
where q and q still denote the projections from C ×G to C and G .
1 2 i ω i ω
Remark 1.8 Let p be a fixed point of C . The scheme Z = q (q−1(p) ∩ Z ) is a 2
i i,p 2 1 i
dimensional irreducible quadric in P . The restriction of q to Z′ is a double cover of a
ω 2 i
veronese surface V = q (Z′).
i 2 i
1
A monad is complex exact at all terms different from the middle one.
4
Proof: In fact {b ∈ G |p ∈ π } is a smooth quadric of dimension 3 (Cf [I]), so it doesn’t
ω v
contain planes. This scheme is included in H , so Z is just an hyperplane section of
i i,p
this smoothquadric. It is also proved in [I] that{v ∈ G |dimπ ∩π > 0} is a cˆone over
ω ui v
a veronese surface of vertex u . As u ∈/ B, the surface V is the intersection of this cone
i i i
with an hyperplane which doesn’t contain the vertex u , so it’s a veronese surface.
i
Notations:
We denote by σ the class of a point on C , and h the class of a hyperplane in
i i 3
P . (the plucker embedding of G ).
ω ω
Proposition 1.9 The incidence Z is a divisor of classe 2h in
i
Π = Proj(O (2)⊕S L⊗O )
IP1 2 IP1
Furthermore we have h ∼ h+2σ , where h ∼ O (1) and σ is also the class of a point
3 i Π i
on the base of the fibration Π . The divisor Z′ of Z is equivalent to h−2σ .
i i i
Proof: Denotes by e the image of the map from O (−2) to W ⊗ O associated to
i IP1 IP1
3
C . Choose an element φ′ of Wv such that kerφ′ gives an hyperplane section of G
i ω
containing B and different from the H¯ . (i.e φ′(u ) 6= 0). We can remark that the
V i i
e⊥
incidence Z is given over IP by the isotropic 2-dimensional subsbaces l of i , such that
i 1 ei
φ′(e ⊗ ∧2l) = 0, because the condition φ (e ⊗ ∧2l)) = 0 is already satisfied by the
i i i
definition of C and lemma 1.1. (where φ denotes a trilinear form of kernel H )
i i i
The bundle e⊥ is isomorphic to S L⊗O ⊕L⊗O (−1) where the trivial factor
i 2 IP1 IP1
S L correspond to the plane π . So2 the bundle e⊥i is isomorphic to L(1)⊕L(−1) where
2 ui ei
those factors are isotropic for the symplectic form induced by ω. We can take local basis
s ,s and s ,s of each factors such that the form induced by ω is p +p where p
0 1 2 3 0,2 1,3 i,j
denotes the Plucker coordinates associated to the s .
i
e⊥ e⊥
So the relative isotropic grassmannian G (2, i ) is the intersection of G(2, i ) with
ω ei ei
the subsheaf O (2)⊕O (−2)+S L⊗O where the factor O (2) still correspond
IP1 IP1 2 IP1 IP1
to s ∧s .
0 1
2 e⊥ φ′
Now we need to compute the kernel of the map e ⊗ ( i ) → O . But the assump-
i ei IP1
tion φ′(u ) 6= 0 proves that it is O (−4)⊕L⊗L⊗O .
i IP1 IPV1
So we have an exact sequence:
2 e⊥ (ω )
0 → O (−2)⊕S L⊗O → ( i ) −φ→′ O ⊕ev → 0
IP1 2 IP1 e IP1 i
i
^
and Z is a divisor of class 2h in Proj(O (2)⊕S L⊗O ). The relation h ∼ h+2σ
i IP1 2 IP1 3 i
2 e⊥ 3
is given by the map e ⊗ ( i ) → W.
i ei
The divisor Z′ of Z is locally given by the vanishing of the exterior product with
i i V V
s ∧s so it equivalent to h−2σ .
0 1 i
We will now study the relation between the conormal bundle of Z in IP ×B and
i 1
the bundle E .
i
2As SL2-represention, we will identify L with its dual.
5
1.4 Deformations of E
i
Lemma 1.10 We have the following exact sequence:
0 → O (−σ −h ) → q∗E → I (σ ) → q∗(R1q I )(−σ ) → 0
IP1×B i 3 2 i Zi i 2 2∗ Zi i
(where q and q denotes the projections from IP ×B to IP and B)
1 2 1 1
Proof: From the resolution of the diagonal of IP ×IP , we obtain the relative Beilinson’s
1 1
spectral sequence:
−a
E1 = ( ω (σ ))⊗Rbq (I ((1+a)σ )) =⇒ I (σ )
a,b q2 i 2∗ Zi i Zi i
^
By the definition of E (Cf prop 1.2) we have E = q (I (σ )). Furthermore, the
i i 2∗ Zi i
projection q (Z ) is an hyperplane section of B, so q I = O (−1). We can conclude,
2 i 2∗ Zi B
remarking that R1q (I (σ )) = 0, because the restriction of q : Z −q2→|Zi q (Z ) has all
2∗ Zi i 2 i 2 i
its fibers of length at most 2.
NB: The support of R1q I is the natural scheme structure (Cf [G-P]) on the
2∗ Zi
scheme of fibers of q intersecting Z in length 2 or more. It is the veronese surface
2 i
V = q (Z′).
i 2 i
So the previous lemma can now be translated in the following:
Corollary 1.11 The scheme q−1(V )∪Z is in IP ×B the zero locus of a section of the
2 i i 1
bundle q∗E (1,1).
2 i
This gives also a geometric description of the marked pencil of sections of E (h ) given
i 3
by the natural inclusion L ⊂ V found in remak 1.6. Indeed, if we fixe a point p on C ,
i
the restriction to q−1(p) of the section obtained in corollary 1.11 gives with the notations
1
of lemma 1.8 the following:
Corollary 1.12 For any point p on the conic C , the vector bundle E (h ) has a section
i i 3
vanishing on Z ∪V .
i,p i
We can now study the restriction of E to Z .
i i
Proposition 1.13 The restriction E of the vector bundle q∗E to Z fits into the
i|Zi 2 i i
following exact sequence:
0 → O (h −3σ ) → q∗E (h ) → O (3σ ) → 0
Zi 3 i 2 i|Zi 3 Zi i
Proof: Fix a point p on C , and consider the corresponding section of E (h ) constructed
i i 3
in corollary 1.12. Its pull back gives a section of q∗E (h ) vanishing on q−1(Z ∪V ), so
2 i 3 2 i,p i
its restriction to Z gives a section of q∗E (h −σ −Z′). Now, using the computation
i 2 i|Zi 3 i i
of the class of Z′ in Z made in proposition 1.9, namely that O (Z′) is O (h −4σ ),
i i Zi i Zi 3 i
it gives a section of E (3σ ). We have to prove that it is a non vanishing section. To
i|Zi i
obtain this, we compute the second Chern’s class of E (3σ ). We will show that its
i|Zi i
image in the Chow ring of IP ×B is zero. Denote by a the second Chern class of E .
1 i i
From the lemma 1.11, we obtain the class of Z in IP ×B: [Z ] = a +h .σ −[V ]. So we
i 1 i i 3 i i
can compute [Z ].c (E (3σ )). It is (a +h .σ −[V ]).(a −3h σ ), but we will compute
i 2 i i i 3 i i i 3 i
in proposition 3.7 the Chow ring of B, and this class vanish.
6
Corollary 1.14 The vector bundles E are rigid, in other words we have Ext1(E ,E ) =
i i i
0.
Proof: From the corollary 1.11 we have an exact sequence on IP ×B:
1
0 → q2∗Ei(−σi) → q2∗(Ei)⊗q2∗(Ei)(h3) → q2∗Ei(h3 +σi) → (q2∗Ei(h3 +σi))|Zi∪q2−1(Vi) → 0
The bundle q∗E (−σ ) is acyclic, and the corollary 1.7 gives H0(q∗E (h +σ )) = L⊗V
2 i i 2 i 3 i
and H1(q∗E (h +σ )) = 0. The liaison exact sequence twisted by q∗(E (h +σ)) is:
2 i 3 i 2 i 3
0 → q2∗Ei(h3)⊗Oq2−1(Vi)(σi −Zi′) → q2∗Ei(h3 +σi))|Zi∪q2−1(Vi) → Ei|Zi(h3 +σi) → 0
As σ −Z′ have degree −1 along the fibers of q : q−1(V ) → V , the bundle q∗E (h )⊗
i i 2 2 i i 2 i 3
Oq2−1(Vi)(σi−Zi′)is acyclic, so thecohomologyofq2∗Ei(h3+σi))|Zi∪q2−1(Vi) canbecomputed
with its restriction to Z . The propositions 1.13 and 1.9 show that H0E (h +σ ) =
i i|Zi 3 i
S L⊕S L⊕S L. In conclusion, we have the exact sequence:
2 2 4
0 → Hom(E ,E ) → L⊗V → S L⊕S L⊕S L → Ext1(E ,E ) → 0
i i 2 2 4 i i
By the corollary 1.7, the bundle E is stable, so it is simple, in other words we have
i
Hom(E ,E ) = C, and the above exact sequence gives Ext1(E ,E ) = 0.
i i i i
2 The variety of lines in B
Remark 2.1 Let δ be an isotropic line of IP(W). The set of isotropic planes of δ⊥
containg δ form a line in G (3,6), and all the line in G (3,6) are of this type for a
ω ω
unique element of G (2,6). In other words, the variety of lines in G (3,6) is naturally
ω ω
G (2,6).
ω
Notations:
A point of G (2,6) will be denoted by a minuscule letter, and the corresponding
ω
line in G (3,6) by the majuscule letter. The variety of lines included in B will be
ω
noted F . Denote by I the incidence point/line in B. In other words:
B
K⊥ v
I = Proj(( 2 ) (h )) = {(δ,p) ∈ F ×B|p ∈ ∆} ⊂ G (2,6)×G (3,6)
2 B ω ω
K
2
The projections from I to F and B will be denoted by p and p .
B 1 2
2.1 A morphism from F to IP × IP × IP × IP
B 1 1 1 1
Each of the 4 conics C will enable us to construct a morphism from F to IP . We have
i B 1
the following geometric hint to expect at least a rational map: A general element δ of
F gives an isotropic 2-dimensional subspace L of W. In general, the projectivisation
B δ
of L⊥ meets the plane containg C in a point p. There is at least an element m of ∆
δ i
such that p ∈ π , so m ∈ ∆ ⊂ B ⊂ H . Now, the definition of H and lemma 1.1 prove
m i i
that p must be on C .
i
But to show that it is everywhere defined, we will use the vector bundle E . We start
i
by constructing line bundles on F .
B
7
Remark 2.2 Any line ∆ included in the hyperplane section H¯ = q (Z ) of B intersect
ui 2 i
the veronese surface V . Furthermore, any such line is in a quadric Z for a unique
i i,pi
point p of C . So the set v = {δ ∈ F |∆ ⊂ H¯ } is a divisor in F .
i i i B ui B
Proof: As the line ∆ is in H¯ , we have from lemma 1.1 that for any b ∈ ∆, the plane
ui
π intersect the conic C . The line IP(L ) must intersect C in some point p . Indeed, if
b i δ i i
it was not the case, this line would be orthogonal to C , so it would be in the plane π ,
i ui
but any line in this plane intersect C .
i
So the line ∆ is in the quadric Z . Note that IP(L ) ∩ C can’t contain another
i,pi δ i
point, because it would imply that ∆ ⊂ V . Furthermore, the proposition 1.9 implies
i
that Z ∩V is a plane section of the quadric Z , so ∆ intersect V in a single point.
i,pi i i,pi i
Remark 2.3 For any point p of C , the scheme p−1(Z ) has several irreducible com-
i i 2 i,pi
ponents of dimension 2, but some of these component are contracted by p to a curve.
1
Denote by A the 2 dimensional part of p (p−1(Z )).
i,pi 1 2 i,pi
Proof: The components of p−1(Z ) corresponding to the lines included in Z are
2 i,pi i,pi
contracted to curves.
Proposition 2.4 The sheaf p p∗E is a line bundle on F . There is a natural map
1∗ 2 i B
f from H0(O (σ ))v ⊗O to the dual bundle of (p p∗E ). The image of f is also a
i Ci i FB 1∗ 2 i i
line bundle on F , we will denote it by O (α ). By construction, for any p ∈ C , the
B FB i i i
divisor A will be in the linear system |O (α )|.
i,pi FB i
Proof: By the corollary 1.7, the bundle E is a quotient of 6O (−1) and by definition
i B
1.2, it is a subsheaf of 2O . So its restriction to any line ∆ included in B must be
B
O ⊕O (−1). So R1p p∗E = 0 and p p∗E is a line bundle. Denote this line bundle
∆ ∆ 1∗ 2 i 1∗ 2 i
by O (−α′). Dualising and twisting the exact sequence defining E , we have the
FB i i
following exact sequence:
0 → L⊗O (−2h ) → E (−h ) → L → 0 (2)
B 3 i 3 i
where L is supported on the hyperplane section H¯ , and is singular along the veronese
i ui
K⊥ v
surface V . As the incidence I is Proj(( 2 ) (h )) (where K is the tautological sub-
i K2 2 2
bundle of W ⊗ O ), the relative dualising sheaf ω is O (2h − 2h ). So we
Gω(2,W) p1 I 2 3
have R1p (p∗E (−h )) = O (α′ − 2h ). So the base locus of this pencil of sections
1∗ 2 i 3 FB i 2
of O (α′) is the support of R1p p∗(L ). We will now prove that this sheaf is a line
FB i 1∗ 2 i
bundle on v .
i
The morphism p is projective of relative dimension 1. So, for any point δ of F and
1 B
any coherent sheaf F on B, the fiber (R1p p∗F) is H1(F ⊗O ), where ∆ is the line
1∗ 2 δ ∆
in B corresponding to δ. The restriction of the sequence 2 to ∆ gives the surjection:
2O (−2) → O (−1)⊕O (−2) → L ⊗O → 0 (3)
∆ ∆ ∆ i ∆
When the line ∆ is not in the hyperplane section H¯ , the sheaf L ⊗O is supported by
ui i ∆
the point H¯ ∩∆, so in this case we have h1(L ⊗O ) = 0. Now, when the line ∆ is in
ui i ∆
H¯ , the sheaf L ⊗O has generic rank 1 because the veronese surface V can’t contain
ui i ∆ i
8
the line ∆. We have proved in lemma 2.2 that ∆ intersect V , hence for any element of
i
L, the corresponding section of E (h ) vanishes on ∆, so the map 2O (−2) → O (−2)
i 3 ∆ ∆
induced by the sequence (3) is zero, and for any δ in v , we have h1(L ⊗O ) = 1, and
i i ∆
R1p p∗(L ) is a line bundle on v .
1∗ 2 i i
So we have proved that the base locus of |(p (p∗E ))v| is the divisor v . In other
1∗ 2 i i
words, the image of f is the line bundle O (α ) = (p (p∗E ))v(−v ), which is by con-
i FB i 1∗ 2 i i
struction a quotient of L⊗O . By definition A is the closure of {δ ∈ F |lenght(∆∩
FB i,pi B
Z ) = 1} which was identified set theoretically with an element of the linear system
i,pi
|α |, so we conclude the proof with lemma :
i
Lemma 2.5 For a generic choice of a point p on C , the support of the sheaf
i i
R1p (p∗I (−h )) represent the class α′, and all its irreducible components are
1∗ 2 Zi,pi∪Vi 3 i
reduced.
Proof: Firstnoticethatthepointp onC ,givesasectionofE (h ),soanexactsequence:
i i i 3
0 → O (−2h ) → E (−h ) → I (−h ) → 0
B 2 i 3 Zi,pi∪Vi 3
which gives a section of O (α′) vanishing on the support of the sheaf
FB i
R1p (p∗I (−h )). But this is the definition in [G-P] of the scheme structure on
1∗ 2 Zi,pi∪Vi 3
the set of lines included in B and intersecting Z ∪ V . So to show that this scheme
i,pi i
structure is reduced on each component, we have to prove that Z and V are not in
i,pi i
the ramification of the morphism: p : I → B. But a general point m on Z ∩ V is
2 i,pi i
the intersection of Z and another quadric Z , so there pass 4 distinct lines through
i,pi i,p′i
m, and from remark 2.2 there are no other lines in B through m. So m is not in the
ramification of the morphism p : I → B because it is of degree 4. (Cf lemma 3.1).
2
2.2 Description of the morphism
In the previous section, we have contructed 4 morphism f from F to IP . We will now
i B 1
prove that the morphism from F to IP ×IP ×IP ×IP is an embedding for a generic
B 1 1 1 1
B, and that its image is an hyperplane section.
Notations:
For a point p in the conic C , we denote by C the affine cone over B. We
i i B
consider:
F = {δ ∈ G (2,W)|δ∧p ∈ C }
pi ω i B
Unfortunately, we have to remark that f−1(p ) is not exactly F ∩F :
i i pi B
Remark 2.6 The intersection F ∩F is equal to G (2,p⊥)∩F . It contains f−1(p )
pi B ω i B i i
and residual curves corresponding to the lines included in the quadric Z .
i,pi
Proof: If δ is already an element of F , then any isotropic plane of IP(W) containing
B
the line IP(L ) is an element of B (Cf remark 2.1). For δ ∈ G (2,p⊥), the plane π is
δ ω i δ∧pi
isotropic, so we have δ ∧p ∈ C and G (2,p⊥)∩F = F ∩F . Now, remark that in
i B ω i B pi B
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G (2,W)theschemeG (2,p⊥)isthezerolocusofasectionofKv. Therestrictionofthis
ω ω i 2
section to F vanishes on the divisor A defined in proposition 2.4, so F ∩F contains
B i,pi B pi
a divisor of class α and the zero locus of a section of (Kv) (−α ) corresponding to
i 2 |FB i
the lines includes in Z .
i,pi
Nevertheless, we have the following:
Lemma 2.7 For the generic double hyperplane section B of G (3,W), we can find
ω
points p (resp p ) in the conics C (resp C ) such that F ∩ F is a smooth conic in
i j i j pi pj
G (2,W). Furthermore, this conic is in F and represent the class α .α .
ω B i j
Proof: We can choose p and p respectively in C and C such that, p is not in p⊥.
i j i j i j
Remark first that this implies that the intersection F ∩F is automatically included
pi pj
in F because p and p are never in the same isotropic plane. This also implies that
B i j
the isotropic grassmannian G (2,p⊥ ∩ p⊥) is a smooth 3-dimensional quadric. The
ω i j
intersection F ∩F is given in this grassmannian by the 2 conditions: d∧p ∈ H and
pi pj i j
d∧p ∈ H for an element d of G (2,p⊥ ∩p⊥). Indeed, as d∧p and d∧p represent
j i ω i j i j
isotropic planes, the conditions d∧p ∈ H and d∧p ∈ H are automatically satisfied
i i j j
because p ∈ C and p ∈ C (Cf lemma 1.1). Let δ be the intersection of p⊥ and the
i i j j j
3 dimensional vector space U represented by the contact point u of H . The space δ
i i i
is an element of G (2,p⊥ ∩p⊥) such that p ∧δ is not in H (because it is u , and B is
ω i j i j i
smooth) but p ∧ δ ∈ H according to lemma 1.1. So the two hyperplane sections are
j i
independent and we have proved that F ∩F is a (may be singular) conic.
pi pj
Note that from the genericity of B we could assume also that IP(p⊥)∩C = {a ,a }
i j 1 2
and IP(p⊥) ∩ C = {b ,b } are 4 distinct points. According to lemma 1.1, the lines
j i 1 2
(p ,a ), (p ,a ), (p ,b ), (p ,b ) are in F ∩ F , but no three of those lines are in the
i 1 i 2 j 1 j 2 pi pj
same plane, so the conic F ∩F contains 4 points with no trisecant line. Hence the
pi pj
conic must be smooth. To conclude that this conic represent the class α .α , we just
i j
have to prove that the residual curve of F ∩ F don’t intersect F . But if δ is such
pi B pj
that ∆ ⊂ Z , the line IP(L ) contains the point p which is not orthogonal to p , so
i,pi δ i j
δ ∈/ F .
pj
Lemma 2.8 When i,j,k are distinct, the morphism from F to C ×C ×C is domi-
B i j k
nant.
Proof: According to lemma 2.7, fora generic choice ofp ∈ C andp ∈ C the subvariety
i i j j
F ∩F of F is a smooth conic, and we can also assume that the line (p p ) doesn’t
pi pj B i j
intersect C . Assume that the induced map from F ∩F to C is not dominant, then
k pi pj k
there is a point p ∈ C such that F ∩F ⊂ F . So for any element d of F ∩ F
k k pi pj pk pi pj
the corresponding line D is in the plane IP(< p ,p ,p >⊥). As p ∈/ (p p ), the vector
i j k k i j
space < p ,p ,p >⊥ has dimension 3, and this contradicts the fact that F ∩F is a
i j k pi pj
smooth conic.
At this point, we need more details about the embedding of F in G (2,W):
B ω
Notations:
Still denote by K the tautological rank 2 subsheaf of W ⊗O , and by −h
2 Gω(2,W) 2
and c its first and second Chern classes. As G (2,W) is an hyperplane section
2 ω
of G(2,W), we will do the computations in G(2,W).
10