THE FANO THREEFOLD X10

OLIVIER DEBARRE

This is joint work in progress with A. Iliev (Sofia) and L. Manivel (Grenoble).

1. Fano threefolds

A smooth projective complex variety X is called a Fano variety if

−K_X is ample.

Fano curves are all isomorphic to P¹.

Fano surfaces are called del Pezzo surfaces. They are: P² blown-up in at most 8 points (in general position), or P¹×P¹. These surfaces are all rational.

It is known that there are only finitely many families of Fano varieties of any fixed dimension, but classification has been achieved only in dimensions up to 3.

We assume here X that has dimension 3 and that its Picard group is Z·h, where h is an ample class (“Fano varieties of the first kind”).

We define the degree as the integer

d= (−K_X)³ >0

thegenusbyg = ¹₂(−K_X)³+ 1 (it is the genus of the intersection of two canonical divisors), and the index as the (positive) integer such that [−K_X] =rh.

For Fano varieties of the first kind of index 1, Iskovskikh proved that the set of possible values for g is {2, . . . ,10,12} and that when g ≥4, the linear system | −K_X| defines an embedding

X ,→P^g+1

with image a smooth subvarietyX2g−2 of degree 2g−2. Moreover,

• for g = 4, X₆ is the complete intersection in P⁵ of a quadric and a cubic;

• forg = 5, X₈ is the complete intersection in P⁶ of 3 quadrics;

• for g = 6, X₁₀ is the complete intersection in P(∧²C⁵) of the Grassmannian G(2,5), a quadric, and a linear space of codi- mension 2 (or a degeneration thereof).

(There are also descriptions for other genera.)

For the case of index 2, I will only mention a couple of examples:

1

• a double cover of P³ ramified along a smooth quartic (called a quartic double solid);

• a cubic in P⁴.

2. Rationality questions

Simple geometric constructions prove that most of these varieties are unirational: all the ones that are mentioned above are, and there are only a few cases that are undecided.¹ The rationality question is more difficult and has been approached in several ways.

• By studying the group of birational automorphisms (if it is not

“big” enough”, the variety cannot be rational). Iskovskikh and Pukhlikov showed with this method (among other things) that noX₆ is rational (although they are all unirational).

• By studying the intermediate Jacobian J(X) =H^2,1(X)^∨/ImH3(X,Z)

which is a principally polarized abelian variety. If X is ratio- nal, the theorem on resolution of indeterminacies of rational maps imply thatJ(X) is the Jacobian of a (possibly reducible) curve. In particular, the singular locus of a theta divisor has codimension ≤4 in J(X).

The second method is particularly efficient whenX has a conic bundle structureX →P²: the intermediate Jacobian is then isomorphic to the Prym variety associated with a double ´etale cover of the discriminant curve C ⊂P², and singularities of theta divisors of Prym varieties are known (Mumford).

It has been used to prove the nonrationality of

• any (smooth) cubic threefold (projection from a line makes them—birationally—into a conic bundle): the theta divisor has dimension 4 and has a unique singular point (Beauville);

• any (smooth) quartic double solid: there is no conic bundle structure, but the singular locus of the theta divisor can still be shown to have codimension 5 (Voisin);

• any (smooth)X₈ (Beauville);

• general X₆ or X₁₀: although they are not conic bundles, they can be degenerated to ones, to which the method applies.

1Among which that of general quartic hypersurfaces in P⁴.

3. The period map

By associating to a Fano threefoldX its intermediate JacobianJ(X), we define aperiod mapfrom the “moduli space”²Mdof Fano threefolds of degreedto the moduli space of principally polarized abelian varieties of the appropriate dimension. The study of the (generic) injectivity of this map is called the Torelli problem.

3.1. Cubic hypersurfaces. As mentioned earlier, the theta divisor has a unique singular point. It is a triple point, and the projectified tangent cone to the theta divisor at that point is the cubic (Beauville)!

Therefore, the Torelli property holds.³

3.2. Intersections of three quadrics. Any X₈ can be explicitly re- covered from the theta divisor (and its singular locus),⁴ hence the map

M8 −→A14

is injective.

4. The Abel–Jacobi map

In the study of the geometry of a Fano threefold X of the first kind, an essential role is played by lines or conics on X (the degree is com- puted with respect to the ample classh).

Families of curves onXare also useful to study the geometry ofJ(X) via the Abel–Jacobi map, which can be defined as follows. Let (C_t)t∈T

be a (connected) family of curves onX. Pick a base-point 0∈T. Then C_t is algebraically, hence homologically, equivalent to C₀, so C_t−C₀ is the boundary of a (real) 3-chain Γ_t, defined modulo H₃(X,Z). The map

T −→ J(X) = H^2,1(X)^∨/H₃(X,Z) t 7−→ (ω7→

Z

Γt

ω)

is therefore well-defined and is called the Abel–Jacobi map. Note that if T is smooth and proper, it factors through the Albanese variety of T:

T −→Alb(T) =H^1,0(T)^∨/H₁(T,Z)−→J(X)

2Although this space may not be well-defined...

3The corresponding moduli space has dimension 10.

4More precisely, the set {x ∈ J(X₈) | x+ Sing(Θ) ⊂ Θ} is a surface whose tangent cone at 0 is the Prym curve.

4.1. Cubic hypersurfaces. Lines on a smooth cubic threefold X are parametrized by a smooth surfaceF(X) (called the Fano surface ofX).

The Abel–Jacobi map F(X) → J(X) is an embedding. Furthermore, the induced morphism Alb(F(X))→J(X) is an isomorphism.

4.2. Intersection of three quadrics. Conics in a general X₈ are parametrized by a smooth surface F(X₈). The Abel–Jacobi map is an embedding and the induced morphism Alb(F(X₈))→ J(X₈) is an isomorphism.

5. The Fano threefold X₁₀

As we saw earlier, most Fano threefolds of degree 10 are obtained as follows. Let V be a 5-dimensional vector space and consider the GrassmannianG(2, V)⊂P(∧²V) = P⁹ in its Pl¨ucker embedding.

For Λ ⊂ P(∧²V) general codimension-2 linear space, the dual line Λ^∨ ⊂P(∧²V^∨) consists of skew forms on V, all of maximal rank 4 and the intersection

W =G(2, V)∩Λ

is a smooth 4-fold of degree 5 in P(∧²V) (independent, modulo the action of PGL(V), of the choice of Λ). We let X =W∩Ω, where Ω is a (general) quadric in P(∧²V).

5.1. The Fano surface of X. Let F_g(X) be the family of conics on X.

Theorem 5.1 (Logachev). The scheme F_g(X) is a smooth connected surface and the morphism Alb(F_g(X)) → J(X) induced by the Abel–

Jacobi map is an isomorphism.

Let us say more about these conics. If c is a conic contained in G(2, V),

• either there exists a unique hyperplane V₄ ⊂ V such that c ⊂ G(2, V4);

• or the 2-planehci is a β-planecontained in G(2, V), i.e., of the type{V₂ ⊂V |V2 ⊂V3}.

There is a unique β-plane, Π, contained in W: it is dual to the unique 3-dimensional subspace of V totally isotropic for all forms in Λ^∨.

It follows that any coniccinX is contained in someG(2, V₄), where the hyperplaneV4 ⊂V is uniquely determined bycunlesscis the conic cX = Π∩Ω.

It is natural to introduce the incidence variety

F(X) ={(c, V₄)∈F_g(X)×P(V^∨)|c⊂G(2, V₄)}

The first projectionF(X)→Fg(X) is an isomorphism except over the point c_X, where the fiber is a line`: it is the blow-up of c_X.

5.2. The involutionι. Let (c, V₄)∈F(X). The intersectionM(V₄) = P(∧²V₄)∩Λ is a 3-plane andM(V₄)∩Xis the intersection of the quadric surfaces G(2, V₄)∩M(V₄) and Ω∩M(V₄), which is a curve of degree 4 and genus 1. The residual curve is another conic ι(c)⊂ X that meets cin two points. This defines a fixed-point-free involution

ι:F(X)−→F(X)

The curves`andι(`) are exceptional and can be contracted byF(X)→ F_m(X). The surfaceF_m(X) is minimal of general type (Logachev). The induced mapF_g(X)→F_m(X) is the blow-up of the pointι(c_X).

5.3. Elementary transformations. We now introduce important bi- rational transformations of X.

5.3.1. Elementary transformation along a conic. Let c be a general conic contained in X and let π_c : Λ99KP⁴ be the projection from the 2-plane hci. If ε:Y →X is the blow-up of c, with exceptional divisor E, the composition π_c ◦ε : Y → P⁴ induces a birational morphism ϕ : Y → Y¯ ⊂ P⁴, where ¯Y is a normal quartic hypersurface with terminal singularities. The only curves that it contracts are

• the strict transforms of the lines that meet c;

• the strict transform of the conic ι(c).

Only finitely many (20) lines inXmeetchenceϕcontracts only finitely many curves: it is a small contraction.

The divisor −E is ϕ-antiample and can be flopped to get a smooth projective threefold Y⁰ and a rational map

χ:Y −→^ϕ Y¯ ^ϕ

0

←−Y⁰

which is an isomorphism in codimension 1. If ¯His a hyperplane section of ¯Y, we have −K_Y⁰ =ϕ^0∗H¯ and −χ(E) is ϕ⁰-ample.

We have ρ(Y⁰) = 2. Since the extremal ray generated by the class of curves contracted byϕ⁰ has K_Y⁰-degree 0 and K_Y⁰ is not nef, the other extremal ray is K_Y⁰-negative and defines a contraction ε⁰ : Y⁰ → X⁰, where

• X⁰ is again a smooth Fano threefold of degree 10 inP⁷;

• ε⁰ is the blow-up of a smooth conic c⁰ in X⁰, with exceptional divisor E⁰ ≡ −2K_Y⁰ −χ(E).

We have a diagram

(1) Y

ε

ϕ@@@@@

@@

χ _ _ _//

_ _ _

_ Y⁰

ε⁰

ϕ⁰

~~}}}}}}}}

Y¯

X ^{_ _ _ψ_c_ _ _//}

πc

??~

~~

~

X⁰

π_c0

``AAA

A

If H⁰ is a hyperplane section of X⁰, we have

χ^∗ε^0∗H⁰ ≡χ^∗(−K_Y⁰ +E⁰)≡χ^∗(−3K_Y⁰ −χ(E))≡ −3ε^∗K_X −4E hence ψ_c is associated with a linear subsystem of |I_c⁴(3)|.

Note that ε^0∗H⁰−E⁰ ≡ −K_Y⁰ ≡ϕ^0∗H, so the picture is symmetric:¯ the elementary transformation of X⁰ along the conic c⁰ is ψ_c⁻¹ :X⁰ 99K X.

Proposition 5.2. Let cbe a general conic on X. There is a birational isomorphism

ϕ_c :F_g(X)99KF_g(X⁰)

which commutes with the (rational) involutions ι on F_g(X) and ι⁰ on F_g(X⁰).

Proof. We will only defineϕ_c. Let ¯cbe general conic onX. It is disjoint from c, hence the span hc,¯ci is a 5-plane that intersects X ⊂ Λ along a canonically embedded genus-6 curve c+ ¯c+ Γ_c,¯c, where Γ_c,¯c is an irreducible sextic.

The implies that Γ_c,¯c meets cand ¯cin 4 points each, hence must be rational since the total genus is 6. The rational mapψ_cis defined on Γ_c,¯c by a linear system of total degree ≤ 3 deg(Γ_c,¯c)−4c·Γ_c,¯c ≤ 2, hence ψ_c(Γ_c,¯c) is a conic contained in X⁰ (there is only a one-dimensional family of lines on X⁰).

This defines a rational map

ϕ_c :F_g(X)99KF_g(X⁰)

whose inverse is checked to be ϕ_c⁰ :F_g(X⁰)99KF_g(X).

Furthermore, the mapϕ_c sends the conicι(c) to the conicc_X⁰, hence factors as (recall that the surface F_m(X), obtained by contracting the curve of ι(`)-conics on F_g(X), is minimal)

F_g(X)^{blow−up of}−−−−−−−−−→^ι(cX)F_m(X)−→^∼ F_m(X⁰)^{blow−up of}←−−−−−−−^cF_g(X⁰)

In other words, the surface Fg(X⁰) is isomorphic to the surfaceFm(X) blown up at the point c.

We will now write X_c instead of X⁰ to highlight the dependence on c.

Theorem 5.3. As cvaries in F(X), the assignment c7→X_c defines a rational map F(X) 99KM10 generically finite onto its image which is contained in a fiber of the period map M10→A10.

The “moduli space” M10 has dimension 22.

Proof. Logachev has proved a Torelli theorem, to the effect that X is determined up to isomorphism by its Fano surfaceF_g(X). The proof is unfortunately hard to follow, and the conclusion not very clear: F_g(X_c) and F_g(X_ι(c)) are isomorphic (viaι). But are X_c andX_ι(c) isomorphic?

On the other hand, blow-ups of rational curves and rational maps that are isomorphisms in codimension one, such as flops, do not change intermediate Jacobians, hence J(X) and J(X_c) are isomorphic.

5.3.2. Elementary transformation along a line. Let ` be a line con- tained in a general X. We can similarly define an elementary trans- formation along `. If ε : Y → X is the blow-up of `, with ex- ceptional divisor E, projection from ` induces a birational morphism ϕ : Y → Y¯ ⊂ P⁵, whose image has degree 10 and is Gorenstein with terminal singularities. The only curves contracted by ϕ|−K_Y| are the strict transforms of the (11) lines that meet `. Performing a flop, we obtain a diagram

(2) Y

ε

ϕ@@@@@

@@

χ _ _ _//

_ _ _

_ Y⁰

ε⁰

ϕ⁰

~~}}}}}}}}

Y¯

X <sup>_ _ _ψ_`_ _ _//</sup>X`

where

• X` is again a smooth Fano threefold of degree 10 inP⁷;

• ε⁰ is the blow-up of a line `<sup>0</sup> in X`, with exceptional divisor E⁰ ≡ −K_Y⁰ −χ(E).

As before, the elementary transformation of X<sub>`</sub> along the conic `⁰ is ψ_{`</sub><sup>−1</sup> :X<sub>`} 99KX.

IfH⁰ is a hyperplane section of X_`, we have

χ^∗ε^0∗H⁰ ≡χ^∗(−K_Y⁰ +E⁰)≡χ^∗(−2K_Y⁰ −χ(E))≡ −2ε^∗K_X −3E

hence ψ` is associated with a linear subsystem of |I`³(2)|. A general conic con X is therefore sent to a quartic inX_`. Since

ψc(c)·E⁰ =c·χ^∗E⁰ =c·(−KY −E) = 2

the quartic C = ψc(c) meets the line `<sup>0</sup> in 2 points. The curve C∪`⁰ has trivial canonical sheaf, and we will apply the Serre construction to it.

5.3.3. The Serre construction. LetE be a 1-dimensional local intersec- tion projectively normal subscheme of X with trivial canonical sheaf.

There is a rank-2 vector bundleE on X with a sections that vanishes scheme-theoretically onE that fits into an exact sequence

(3) 0−→OX

−→s E −→IE(1)−→0

In particular H⁰(X,E^∨) = H⁰(X,E(−1)) = 0, so E is stable since Pic(X) = Z[OX(1)]. Its Chern classes arec₁(E) = [OX(1)] andc₂(E) = [E].

5.3.4. The moduli space MX(2; 1,5). Let E be a semistable globally generated vector bundle of rank 2 onXwith Chern numbersc₁(E) = 1 and c₂(E) = 5. A general section s of E yields an exact sequence (3) where E is a smooth, projectively normal elliptic quintic on X. From this sequence, one obtains

(4) h¹(X,End(E)) = 2 , h²(X,End(E)) = 0

It follows thatE represents a smooth point of the moduli spaceMX(2; 1,5) of semistable torsion-free sheaves of rank 2 on X with Chern classes c₁ = 1, c₂ = 5, and c₃ = 0.

It turns out that (4) still holds ifE is not locally free, or not globally generated.

Theorem 5.4. The moduli space MX(2; 1,5) is a smooth irreducible surface. The nonlocally free sheaves (resp. the nonglobally generated vector bundles) are parametrized by the curve of lines contained in X.

To prove irreducibility, we proceed as follows.

Let ` be a general line contained in X. Then H<sup>0</sup>(X,I` ⊗E) is 1- dimensional and the corresponding sections vanish on the union of ` and a bisecant rational quartic, whose image by ψ<sub>`</sub> is a conic on X_`. In this way, we obtain a rational map

ϕ_{`</sub> :MX(2; 1,5)99KF(X<sub>`})

Conversely, the inverse image byψ_{`</sub>of a general conic onX<sub>`}is a quartic Cthat meets`in 2 points. Applying the Serre construction toC∪`, we

obtain a vector bundle of rank 2 onX with Chern numbers 1 and 5 and a section that vanishes on C∪`. The map ϕ<sub>`</sub> is therefore birational.

Institut de recherche Math´ematique Avanc´ee, Universit´e Louis Pas- teur et CNRS, 7 rue Ren´e Descartes, 67000 Strasbourg, France

E-mail address: [email protected]