On the geometry of certain Fano threefolds
Olivier DEBARRE
Institut de Recherche Math´ematique Avanc´ee (IRMA)
Pisa, June 1, 2007
Fano manifolds
Dimension 1: P1 (1 family).
Fano manifolds
AFano1 manifoldis a complex projective manifoldX with det(TX) ample, i.e., det(TX) has a Hermitian metric with positive
curvature.
Dimension 1: P1 (1 family).
Fano manifolds
AFano1 manifoldis a complex projective manifoldX with det(TX) ample, i.e., det(TX) has a Hermitian metric with positive
curvature.
Dimension 1: P1 (1 family).
Fano manifolds
AFano1 manifoldis a complex projective manifoldX with det(TX) ample, i.e., det(TX) has a Hermitian metric with positive
curvature.
Dimension 1: P1 (1 family).
Dimension 2 (del Pezzo2 surfaces): P2 blown-up in at most 8 points, or P1×P1 (10 families).
Fano manifolds
AFano1 manifoldis a complex projective manifoldX with det(TX) ample, i.e., det(TX) has a Hermitian metric with positive
curvature.
Dimension 1: P1 (1 family).
Dimension 2 (del Pezzo2 surfaces): P2 blown-up in at most 8 points, or P1×P1 (10 families).
Dimension 3: all classified (17 + 88 families).
Fano manifolds
AFano1 manifoldis a complex projective manifoldX with det(TX) ample, i.e., det(TX) has a Hermitian metric with positive
curvature.
Dimension 1: P1 (1 family).
Dimension 2 (del Pezzo2 surfaces): P2 blown-up in at most 8 points, or P1×P1 (10 families).
Dimension 3: all classified (17 + 88 families).
Dimension n: finitely many families.
Examples
The projective space Pn.
Submanifolds in Pn+s defined by (general) homogeneous equations of degrees d1, . . . ,ds >1 with
d1+· · ·+ds ≤n+s, such as a cubic surface X3 ⊂P3; a cubic threefold X3 ⊂P4.
The threefold X10 intersection in P(∧2C5) =P9 of the Grassmannian G(2,5), a quadric, and a P7.
Examples
The projective space Pn.
Submanifolds in Pn+s defined by (general) homogeneous equations of degrees d1, . . . ,ds >1 with
d1+· · ·+ds ≤n+s, such as a cubic surface X3 ⊂P3; a cubic threefold X3 ⊂P4.
The threefold X10 intersection in P(∧2C5) =P9 of the Grassmannian G(2,5), a quadric, and a P7.
Examples
The projective space Pn.
Submanifolds in Pn+s defined by (general) homogeneous equations of degrees d1, . . . ,ds >1 with
d1+· · ·+ds ≤n+s, such as
a cubic surface X3 ⊂P3; a cubic threefold X3 ⊂P4.
The threefold X10 intersection in P(∧2C5) =P9 of the Grassmannian G(2,5), a quadric, and a P7.
Examples
The projective space Pn.
Submanifolds in Pn+s defined by (general) homogeneous equations of degrees d1, . . . ,ds >1 with
d1+· · ·+ds ≤n+s, such as a cubic surface X3 ⊂P3;
a cubic threefold X3 ⊂P4.
The threefold X10 intersection in P(∧2C5) =P9 of the Grassmannian G(2,5), a quadric, and a P7.
Examples
The projective space Pn.
Submanifolds in Pn+s defined by (general) homogeneous equations of degrees d1, . . . ,ds >1 with
d1+· · ·+ds ≤n+s, such as a cubic surface X3 ⊂P3; a cubic threefold X3 ⊂P4.
The threefold X10 intersection in P(∧2C5) =P9 of the Grassmannian G(2,5), a quadric, and a P7.
Examples
The projective space Pn.
Submanifolds in Pn+s defined by (general) homogeneous equations of degrees d1, . . . ,ds >1 with
d1+· · ·+ds ≤n+s, such as a cubic surface X3 ⊂P3; a cubic threefold X3 ⊂P4.
The threefold X10 intersection in P(∧2C5) =P9 of the Grassmannian G(2,5), a quadric, and a P7.
Rationality
X is rationalif a (Zariski) open subset ofX is isomorphic to an open subset of aPn, i.e., there exists a birational isomorphism Pn99K∼ X.
Equivalently, the fieldC(X) of rational functions onX is a purely transcendental extensionC(t1, . . . ,tn) of C.
Or, (a dense open subset of)X can be parametrized in a one-to-one way by rational functions int1, . . . ,tn.
Rationality
X is rationalif a (Zariski) open subset ofX is isomorphic to an open subset of aPn, i.e., there exists a birational isomorphism Pn99K∼ X.
Equivalently, the fieldC(X) of rational functions onX is a purely transcendental extensionC(t1, . . . ,tn) of C.
Or, (a dense open subset of)X can be parametrized in a one-to-one way by rational functions int1, . . . ,tn.
Rationality
X is rationalif a (Zariski) open subset ofX is isomorphic to an open subset of aPn, i.e., there exists a birational isomorphism Pn99K∼ X.
Equivalently, the fieldC(X) of rational functions onX is a purely transcendental extensionC(t1, . . . ,tn) of C.
Or, (a dense open subset of)X can be parametrized in a one-to-one way by rational functions int1, . . . ,tn.
Rationality
X is rationalif a (Zariski) open subset ofX is isomorphic to an open subset of aPn, i.e., there exists a birational isomorphism Pn99K∼ X.
Equivalently, the fieldC(X) of rational functions onX is a purely transcendental extensionC(t1, . . . ,tn) of C.
Or, (a dense open subset of)X can be parametrized in a one-to-one way by rational functions int1, . . . ,tn.
Rationality of cubic surfaces
Rationality of cubic surfaces
A (smooth) cubic surface inP3 is rational.2
Rationality of cubic surfaces
A (smooth) cubic surface inP3 is rational.2
Rationality of cubic surfaces
Proof. A smooth cubic surfaceX ⊂P3 contains skew lines `and
`0.
The map
`×`0 99K X
(x,x0) 7−→ third point of intersection of the linehx,x0i with X is
dominant and `×`0 99K∼ P2;
birational: the only preimage of x00∈X is
x =h`0,x00i ∩` , x0=h`,x00i ∩`0.
The surfaceX is in fact P2 blown-up in 6 points hence rational (as all del Pezzo surfaces).
Rationality of cubic surfaces
Proof. A smooth cubic surfaceX ⊂P3 contains skew lines `and
`0. The map
`×`0 99K X
(x,x0) 7−→ third point of intersection of the linehx,x0i with X is
dominant and `×`0 99K∼ P2;
birational: the only preimage of x00∈X is
x =h`0,x00i ∩` , x0=h`,x00i ∩`0.
The surfaceX is in fact P2 blown-up in 6 points hence rational (as all del Pezzo surfaces).
Rationality of cubic surfaces
Proof. A smooth cubic surfaceX ⊂P3 contains skew lines `and
`0. The map
`×`0 99K X
(x,x0) 7−→ third point of intersection of the linehx,x0i with X
is
dominant and `×`0 99K∼ P2;
birational: the only preimage of x00∈X is
x =h`0,x00i ∩` , x0=h`,x00i ∩`0.
The surfaceX is in fact P2 blown-up in 6 points hence rational (as all del Pezzo surfaces).
Rationality of cubic surfaces
Proof. A smooth cubic surfaceX ⊂P3 contains skew lines `and
`0. The map
`×`0 99K X
(x,x0) 7−→ third point of intersection of the linehx,x0i with X is
dominant and `×`0 99K∼ P2;
birational: the only preimage of x00∈X is
x =h`0,x00i ∩` , x0=h`,x00i ∩`0.
The surfaceX is in fact P2 blown-up in 6 points hence rational (as all del Pezzo surfaces).
Rationality of cubic surfaces
Proof. A smooth cubic surfaceX ⊂P3 contains skew lines `and
`0. The map
`×`0 99K X
(x,x0) 7−→ third point of intersection of the linehx,x0i with X is
dominant and `×`0 99K∼ P2;
birational: the only preimage of x00∈X is
x =h`0,x00i ∩` , x0=h`,x00i ∩`0.
The surfaceX is in fact P2 blown-up in 6 points hence rational (as all del Pezzo surfaces).
Rationality of cubic surfaces
Proof. A smooth cubic surfaceX ⊂P3 contains skew lines `and
`0. The map
`×`0 99K X
(x,x0) 7−→ third point of intersection of the linehx,x0i with X is
dominant and `×`0 99K∼ P2;
birational: the only preimage of x00∈X is
x =h`0,x00i ∩` , x0=h`,x00i ∩`0.
Unirationality
X is unirationalif there exists a dominant rational map Pn99KX. Equivalently, the fieldC(X) is containedin a purely transcendental extensionC(t1, . . . ,tn) ofC.
Or, (a dense open subset of)X can be parametrized in a finite-to-one way by rational functions int1, . . . ,tn.
Unirationality
X is unirationalif there exists a dominant rational map Pn99KX.
Equivalently, the fieldC(X) is containedin a purely transcendental extensionC(t1, . . . ,tn) ofC.
Or, (a dense open subset of)X can be parametrized in a finite-to-one way by rational functions int1, . . . ,tn.
Unirationality
X is unirationalif there exists a dominant rational map Pn99KX. Equivalently, the fieldC(X) is containedin a purely transcendental extensionC(t1, . . . ,tn) ofC.
Or, (a dense open subset of)X can be parametrized in a finite-to-one way by rational functions int1, . . . ,tn.
Unirationality
X is unirationalif there exists a dominant rational map Pn99KX. Equivalently, the fieldC(X) is containedin a purely transcendental extensionC(t1, . . . ,tn) ofC.
Or, (a dense open subset of)X can be parametrized in a finite-to-one way by rational functions int1, . . . ,tn.
Unirationality
X is unirationalif there exists a dominant rational map Pn99KX. Equivalently, the fieldC(X) is containedin a purely transcendental extensionC(t1, . . . ,tn) ofC.
Or, (a dense open subset of)X can be parametrized in a finite-to-one way by rational functions int1, . . . ,tn.
For curves and surfaces, rationality is equivalent to unirationality.1
Unirationality
X is unirationalif there exists a dominant rational map Pn99KX. Equivalently, the fieldC(X) is containedin a purely transcendental extensionC(t1, . . . ,tn) ofC.
Or, (a dense open subset of)X can be parametrized in a finite-to-one way by rational functions int1, . . . ,tn.
For curves and surfaces, rationality is equivalent to unirationality.1
A (smooth) cubic hypersurface in Pn+1 is unirational for n ≥2.2
Unirationality of cubic hypersurfaces
Proof. A smooth cubic hypersurfaceX ⊂Pn+1 contains a line`.
A line`x tangent toX at a point x of `meetsX in a third point. The induced map
P(TX|`) 99K X
`x ⊂TX,x 7−→ third point of intersection of`x with X
is
dominant and P(TX|`)99K∼ Pn; of degree 2:
Unirationality of cubic hypersurfaces
Proof. A smooth cubic hypersurfaceX ⊂Pn+1 contains a line`.
A line`x tangent toX at a point x of `meetsX in a third point.
The induced map
P(TX|`) 99K X
`x ⊂TX,x 7−→ third point of intersection of`x with X
is
dominant and P(TX|`)99K∼ Pn; of degree 2:
Unirationality of cubic hypersurfaces
Proof. A smooth cubic hypersurfaceX ⊂Pn+1 contains a line`.
A line`x tangent toX at a point x of `meetsX in a third point.
The induced map
P(TX|`) 99K X
`x ⊂TX,x 7−→ third point of intersection of`x with X
is
dominant and P(TX|`)99K∼ Pn; of degree 2:
Unirationality of cubic hypersurfaces
Proof. A smooth cubic hypersurfaceX ⊂Pn+1 contains a line`.
A line`x tangent toX at a point x of `meetsX in a third point.
The induced map
P(TX|`) 99K X
`x ⊂TX,x 7−→ third point of intersection of`x with X
is
dominant and P(TX|`)99K∼ Pn;
of degree 2:
Unirationality of cubic hypersurfaces
Proof. A smooth cubic hypersurfaceX ⊂Pn+1 contains a line`.
A line`x tangent toX at a point x of `meetsX in a third point.
The induced map
P(TX|`) 99K X
`x ⊂TX,x 7−→ third point of intersection of`x with X
is
dominant and P(TX|`)99K∼ Pn; of degree 2:
Unirationality of cubic hypersurfaces
Proof. A smooth cubic hypersurfaceX ⊂Pn+1 contains a line`.
A line`x tangent toX at a point x of `meetsX in a third point.
The induced map
P(TX|`) 99K X
`x ⊂TX,x 7−→ third point of intersection of`x with X
is
dominant and P(TX|`)99K∼ Pn;
of degree 2: if x0 ∈X, the intersection of X with the plane h`,x0i is the union of`and a conicc.
Unirationality of cubic hypersurfaces
Proof. A smooth cubic hypersurfaceX ⊂Pn+1 contains a line`.
A line`x tangent toX at a point x of `meetsX in a third point.
The induced map
P(TX|`) 99K X
`x ⊂TX,x 7−→ third point of intersection of`x with X
is
dominant and P(TX|`)99K∼ Pn;
of degree 2: if x0 ∈X, the intersection of X with the plane h`,x0i is the union of`and a conicc. The two preimages of
0 ∩
Unirationality of Fano threefolds
Most Fano threefolds are unirational...
...but smooth cubic hypersurfaces inP4 are notrational (Clemens–Griffiths, 1972).
Some (smooth) cubic hypersurfaces inPn,n≥5, such as (x03+· · ·+x2m+13 = 0)⊂P2m+1
are rational (odd-dimensional examples were constructed by M. Mella)...
...but the rationality of general cubics is unknown!
Unirationality of Fano threefolds
Most Fano threefolds are unirational...
...but smooth cubic hypersurfaces inP4 are notrational (Clemens–Griffiths, 1972).
Some (smooth) cubic hypersurfaces inPn,n≥5, such as (x03+· · ·+x2m+13 = 0)⊂P2m+1
are rational (odd-dimensional examples were constructed by M. Mella)...
...but the rationality of general cubics is unknown!
Unirationality of Fano threefolds
Most Fano threefolds are unirational...
...but smooth cubic hypersurfaces inP4 are notrational (Clemens–Griffiths, 1972).
Some (smooth) cubic hypersurfaces inPn,n≥5, such as (x03+· · ·+x2m+13 = 0)⊂P2m+1
are rational (odd-dimensional examples were constructed by M.
Mella)...
...but the rationality of general cubics is unknown!
Unirationality of Fano threefolds
Most Fano threefolds are unirational...
...but smooth cubic hypersurfaces inP4 are notrational (Clemens–Griffiths, 1972).
Some (smooth) cubic hypersurfaces inPn,n≥5, such as (x03+· · ·+x2m+13 = 0)⊂P2m+1
are rational (odd-dimensional examples were constructed by M.
Mella)...
Jacobians
Jacobian of a curve C. This is the abelian variety
J(C) =H1,0(C)∨/ImH1(C,Z) = (g-dim’l vector space)/(lattice) Natural target for integrating:
Jacobians
Jacobian of a curve C.
This is the abelian variety
J(C) =H1,0(C)∨/ImH1(C,Z) = (g-dim’l vector space)/(lattice) Natural target for integrating:
Jacobians
Jacobian of a curve C. This is the abelian variety
J(C) =H1,0(C)∨/ImH1(C,Z) = (g-dim’l vector space)/(lattice)
Natural target for integrating:
Jacobians
Jacobian of a curve C. This is the abelian variety
J(C) =H1,0(C)∨/ImH1(C,Z) = (g-dim’l vector space)/(lattice) Natural target for integrating:
Jacobians
Jacobian of a curve C. This is the abelian variety
J(C) =H1,0(C)∨/ImH1(C,Z) = (g-dim’l vector space)/(lattice) Natural target for integrating:
C −→ J(C) Abel–Jacobi1 map
Jacobians
Jacobian of a curve C. This is the abelian variety
J(C) =H1,0(C)∨/ImH1(C,Z) = (g-dim’l vector space)/(lattice) Natural target for integrating:
C −→ J(C) Abel–Jacobi1 map
x 7−→
ω7→
Z x x0
ω
(well-defined modulo H1(C,Z))
Jacobians
Jacobian of a curve C. This is the abelian variety
J(C) =H1,0(C)∨/ImH1(C,Z) = (g-dim’l vector space)/(lattice) Natural target for integrating:
C −→ J(C) Abel–Jacobi1 map
x 7−→
ω7→
Z x
x0
ω
(well-defined modulo H1(C,Z)) (x0 fixed point ofC.)
Intermediate Jacobians
Intermediate Jacobian of a Fano threefold X: J(X) =H2,1(X)∨/ImH3(X,Z)
These areprincipally polarized abelian varieties: they contain a hypersurface Θ uniquely defined up to translation.
The pair (J(X),Θ) carries information aboutX.
Intermediate Jacobians
Intermediate Jacobian of a Fano threefold X:
J(X) =H2,1(X)∨/ImH3(X,Z)
These areprincipally polarized abelian varieties: they contain a hypersurface Θ uniquely defined up to translation.
The pair (J(X),Θ) carries information aboutX.
Intermediate Jacobians
Intermediate Jacobian of a Fano threefold X: J(X) =H2,1(X)∨/ImH3(X,Z)
These areprincipally polarized abelian varieties: they contain a hypersurface Θ uniquely defined up to translation.
The pair (J(X),Θ) carries information aboutX.
Intermediate Jacobians
Intermediate Jacobian of a Fano threefold X: J(X) =H2,1(X)∨/ImH3(X,Z)
These areprincipally polarized abelian varieties: they contain a hypersurface Θ uniquely defined up to translation.
The pair (J(X),Θ) carries information aboutX.
Intermediate Jacobians
Intermediate Jacobian of a Fano threefold X: J(X) =H2,1(X)∨/ImH3(X,Z)
These areprincipally polarized abelian varieties: they contain a hypersurface Θ uniquely defined up to translation.
The pair (J(X),Θ) carries information aboutX.
Using Intermediate Jacobians to prove nonrationality
IfX is rational, we get, by Hironaka’s resolution of indeterminacies, Pf3
composition of blow-ups of points and of smooth curves C1, . . . ,Cr
$$H
HH HH HH HH HH HH
P3 _ _ _ _ _// X J(Pf3)'J(C1)× · · · ×J(Cr);
Using Intermediate Jacobians to prove nonrationality
IfX is rational, we get, by Hironaka’s resolution of indeterminacies,
Pf3 composition of blow-ups of points
and of smooth curves C1, . . . ,Cr
$$H
HH HH HH HH HH HH
P3 _ _ _ _ _// X J(Pf3)'J(C1)× · · · ×J(Cr);
Using Intermediate Jacobians to prove nonrationality
IfX is rational, we get, by Hironaka’s resolution of indeterminacies, Pf3
composition of blow-ups of points and of smooth curves C1, . . . ,Cr
$$H
HH HH HH HH HH HH
P3 _ _ _ _ _// X
J(Pf3)'J(C1)× · · · ×J(Cr);
Using Intermediate Jacobians to prove nonrationality
IfX is rational, we get, by Hironaka’s resolution of indeterminacies, Pf3
composition of blow-ups of points and of smooth curves C1, . . . ,Cr
$$H
HH HH HH HH HH HH
P3 _ _ _ _ _// X J(Pf3)'J(C1)× · · · ×J(Cr);
Using Intermediate Jacobians to prove nonrationality
IfX is rational, we get, by Hironaka’s resolution of indeterminacies, Pf3
composition of blow-ups of points and of smooth curves C1, . . . ,Cr
$$H
HH HH HH HH HH HH
P3 _ _ _ _ _// X J(Pf3)'J(C1)× · · · ×J(Cr);
J(X) is also a product of Jacobians of smooth curves;
Using Intermediate Jacobians to prove nonrationality
IfX is rational, we get, by Hironaka’s resolution of indeterminacies, Pf3
composition of blow-ups of points and of smooth curves C1, . . . ,Cr
$$HHHHHHHHHHHHH
P3 _ _ _ _ _// X J(Pf3)'J(C1)× · · · ×J(Cr);
J(X) is also a product of Jacobians of smooth curves;
hence
codimJ(X)Sing(Θ)≤4
Nonrationality of cubic threefolds
Back to a cubic threefoldX3 ⊂P4.
Projection from a line `⊂X3
yields
Xf3 blow-up of`
zzvvvvvvvvvvvvv
conic bundle
Ce
double
´etale cover
X3
projection from `
//
_ _ _ _ _
_ P2 oo ? _C discriminant curve
J(X3) has dimension 5;
Nonrationality of cubic threefolds
Back to a cubic threefoldX3 ⊂P4. Projection from a line`⊂X3
yields
Xf3 blow-up of`
zzvvvvvvvvvvvvv
conic bundle
Ce
double
´etale cover
X3
projection from `
//
_ _ _ _ _
_ P2 oo ? _C discriminant curve
J(X3) has dimension 5;
Nonrationality of cubic threefolds
Back to a cubic threefoldX3 ⊂P4. Projection from a line`⊂X3
yields
Xf3 blow-up of`
zzvvvvvvvvvvvvv
conic bundle
Ce
double
´etale cover
X3
projection from `
//
_ _ _ _ _
_ P2 oo ? _C discriminant curve
J(X3) has dimension 5;
Nonrationality of cubic threefolds
Back to a cubic threefoldX3 ⊂P4. Projection from a line`⊂X3 yields
Xf3
blow-up of`
zzvvvvvvvvvvvvv
conic bundle
Ce
double
´etale cover
X3
projection from `
//
_ _ _ _ _
_ P2 oo ? _C discriminant curve
J(X3) has dimension 5;
Nonrationality of cubic threefolds
Back to a cubic threefoldX3 ⊂P4. Projection from a line`⊂X3 yields
Xf3
blow-up of`
zzvvvvvvvvvvvvv
conic bundle
Ce
double
´etale cover
X3
projection from `
//
_ _ _ _ _
_ P2 oo ? _C discriminant curve
J(X3) has dimension 5;
J(X3) is isomorphic to the Prym1 variety of the coverCe →C; Θ has a unique singular point o, which is a triple point
Nonrationality of cubic threefolds
Back to a cubic threefoldX3 ⊂P4. Projection from a line`⊂X3 yields
Xf3 blow-up of`
zzvvvvvvvvvvvvv
conic bundle
Ce
double
´etale cover
X3
projection from `
//
_ _ _ _ _
_ P2 oo ? _C discriminant curve
J(X3) has dimension 5;
J(X3) is isomorphic to the Prym1 variety of the coverCe →C; Θ has a unique singular point o, which is a triple point
The Abel–Jacobi map
X Fano threefold.
Geometric information about (J(X),Θ) is in general hard to get, in particular whenX is nota conic bundle as before.
Some information can be obtained from families of curves onX: (Ct)t∈T (connected) family of curves on X, base-point 0∈T; Ct is algebraically, hence homologically, equivalent to C0, so Ct−C0 is the boundary of a (real) 3-chain Γt, defined modulo H3(X,Z);
define the Abel–Jacobi map
T −→ J(X) =H2,1(X)∨/H3(X,Z) t 7−→
ω7→
Z
Γt
ω
The Abel–Jacobi map
X Fano threefold.
Geometric information about (J(X),Θ) is in general hard to get, in particular whenX is nota conic bundle as before.
Some information can be obtained from families of curves onX: (Ct)t∈T (connected) family of curves on X, base-point 0∈T; Ct is algebraically, hence homologically, equivalent to C0, so Ct−C0 is the boundary of a (real) 3-chain Γt, defined modulo H3(X,Z);
define the Abel–Jacobi map
T −→ J(X) =H2,1(X)∨/H3(X,Z) t 7−→
ω7→
Z
Γt
ω
The Abel–Jacobi map
X Fano threefold.
Geometric information about (J(X),Θ) is in general hard to get, in particular whenX is nota conic bundle as before.
Some information can be obtained from families of curves onX:
(Ct)t∈T (connected) family of curves on X, base-point 0∈T; Ct is algebraically, hence homologically, equivalent to C0, so Ct−C0 is the boundary of a (real) 3-chain Γt, defined modulo H3(X,Z);
define the Abel–Jacobi map
T −→ J(X) =H2,1(X)∨/H3(X,Z) t 7−→
ω7→
Z
Γt
ω
The Abel–Jacobi map
X Fano threefold.
Geometric information about (J(X),Θ) is in general hard to get, in particular whenX is nota conic bundle as before.
Some information can be obtained from families of curves onX: (Ct)t∈T (connected) family of curves on X, base-point 0∈T;
Ct is algebraically, hence homologically, equivalent to C0, so Ct−C0 is the boundary of a (real) 3-chain Γt, defined modulo H3(X,Z);
define the Abel–Jacobi map
T −→ J(X) =H2,1(X)∨/H3(X,Z) t 7−→
ω7→
Z
Γt
ω
The Abel–Jacobi map
X Fano threefold.
Geometric information about (J(X),Θ) is in general hard to get, in particular whenX is nota conic bundle as before.
Some information can be obtained from families of curves onX: (Ct)t∈T (connected) family of curves on X, base-point 0∈T; Ct is algebraically, hence homologically, equivalent to C0,
so Ct−C0 is the boundary of a (real) 3-chain Γt, defined modulo H3(X,Z);
define the Abel–Jacobi map
T −→ J(X) =H2,1(X)∨/H3(X,Z) t 7−→
ω7→
Z
Γt
ω
The Abel–Jacobi map
X Fano threefold.
Geometric information about (J(X),Θ) is in general hard to get, in particular whenX is nota conic bundle as before.
Some information can be obtained from families of curves onX: (Ct)t∈T (connected) family of curves on X, base-point 0∈T; Ct is algebraically, hence homologically, equivalent to C0, so Ct−C0 is the boundary of a (real) 3-chain Γt, defined modulo H3(X,Z);
define the Abel–Jacobi map
T −→ J(X) =H2,1(X)∨/H3(X,Z) t 7−→
ω7→
Z
Γt
ω
The Abel–Jacobi map
X Fano threefold.
Geometric information about (J(X),Θ) is in general hard to get, in particular whenX is nota conic bundle as before.
Some information can be obtained from families of curves onX: (Ct)t∈T (connected) family of curves on X, base-point 0∈T; Ct is algebraically, hence homologically, equivalent to C0, so Ct−C0 is the boundary of a (real) 3-chain Γt, defined modulo H3(X,Z);
define the Abel–Jacobi map
T −→ J(X) =H2,1(X)∨/H3(X,Z) t 7−→
ω7→
Z
Γt
ω
The Abel–Jacobi map
X Fano threefold.
Geometric information about (J(X),Θ) is in general hard to get, in particular whenX is nota conic bundle as before.
Some information can be obtained from families of curves onX: (Ct)t∈T (connected) family of curves on X, base-point 0∈T; Ct is algebraically, hence homologically, equivalent to C0, so Ct−C0 is the boundary of a (real) 3-chain Γt, defined modulo H3(X,Z);
define the Abel–Jacobi map
T −→ J(X) =H2,1(X)∨/H3(X,Z) t 7−→
ω7→
Z ω
The Torelli problem
Lines on a cubic threefoldX3 ⊂P4 are parametrized by a smooth surfaceF.
The Abel–Jacobi mapF →J(X3) is an embedding, and Θ =F −F.
We recover Beauville’s result: o is a singular point of Θ.
Moreover, the projectified tangent cone to Θ ato is isomorphic to X3:
X3'P(TCΘ,o)⊂P(TJ(X3),o)'P4
In particular,X3 can be reconstructed from its intermediate Jacobian.
This is theTorelli theorem for cubic threefolds.
The Torelli problem
Lines on a cubic threefoldX3 ⊂P4 are parametrized by a smooth surfaceF.
The Abel–Jacobi mapF →J(X3) is an embedding, and Θ =F −F.
We recover Beauville’s result: o is a singular point of Θ.
Moreover, the projectified tangent cone to Θ ato is isomorphic to X3:
X3'P(TCΘ,o)⊂P(TJ(X3),o)'P4
In particular,X3 can be reconstructed from its intermediate Jacobian.
This is theTorelli theorem for cubic threefolds.
The Torelli problem
Lines on a cubic threefoldX3 ⊂P4 are parametrized by a smooth surfaceF.
The Abel–Jacobi mapF →J(X3) is an embedding, and Θ =F −F.
We recover Beauville’s result: o is a singular point of Θ.
Moreover, the projectified tangent cone to Θ ato is isomorphic to X3:
X3'P(TCΘ,o)⊂P(TJ(X3),o)'P4
In particular,X3 can be reconstructed from its intermediate Jacobian.
This is theTorelli theorem for cubic threefolds.
The Torelli problem
Lines on a cubic threefoldX3 ⊂P4 are parametrized by a smooth surfaceF.
The Abel–Jacobi mapF →J(X3) is an embedding, and Θ =F −F.
We recover Beauville’s result: o is a singular point of Θ.
Moreover, the projectified tangent cone to Θ ato is isomorphic to X3:
X3'P(TCΘ,o)⊂P(TJ(X3),o)'P4
In particular,X3 can be reconstructed from its intermediate Jacobian.
This is theTorelli theorem for cubic threefolds.
The Torelli problem
Lines on a cubic threefoldX3 ⊂P4 are parametrized by a smooth surfaceF.
The Abel–Jacobi mapF →J(X3) is an embedding, and Θ =F −F.
We recover Beauville’s result: o is a singular point of Θ.
Moreover, the projectified tangent cone to Θ ato is isomorphic to X3:
X3'P(TCΘ,o)⊂P(TJ(X3),o)'P4
In particular,X3 can be reconstructed from its intermediate Jacobian.
This is theTorelli theorem for cubic threefolds.
The Torelli problem
Lines on a cubic threefoldX3 ⊂P4 are parametrized by a smooth surfaceF.
The Abel–Jacobi mapF →J(X3) is an embedding, and Θ =F −F.
We recover Beauville’s result: o is a singular point of Θ.
Moreover, the projectified tangent cone to Θ ato is isomorphic to X3:
X3'P(TCΘ,o)⊂P(TJ(X3),o)'P4
In particular,X3 can be reconstructed from its intermediate Jacobian.
This is theTorelli theorem for cubic threefolds.
The Torelli problem
Lines on a cubic threefoldX3 ⊂P4 are parametrized by a smooth surfaceF.
The Abel–Jacobi mapF →J(X3) is an embedding, and Θ =F −F.
We recover Beauville’s result: o is a singular point of Θ.
Moreover, the projectified tangent cone to Θ ato is isomorphic to X3:
X3'P(TCΘ,o)⊂P(TJ(X3),o)'P4
In particular,X3 can be reconstructed from its intermediate Jacobian.
This is theTorelli theorem for cubic threefolds.1
The Fano threefold X
10Joint work in progress with A. Iliev (Sofia) and L. Manivel (Grenoble).
Most Fano threefolds of the first kind of degreec13 = 10 are obtained as follows:
V 5-dimensional complex vector space;
The Fano threefold X
10Joint work in progress with A. Iliev (Sofia) and L. Manivel (Grenoble).
Most Fano threefolds of the first kind of degreec13 = 10 are obtained as follows:
V 5-dimensional complex vector space;
The Fano threefold X
10Joint work in progress with A. Iliev (Sofia) and L. Manivel (Grenoble).
Most Fano threefolds of the first kind of degreec13 = 10 are obtained as follows:
V 5-dimensional complex vector space;
The Fano threefold X
10Joint work in progress with A. Iliev (Sofia) and L. Manivel (Grenoble).
Most Fano threefolds of the first kind of degreec13 = 10 are obtained as follows:
V 5-dimensional complex vector space;
G(2,V)⊂P(∧2V) =P9 the Pl¨ucker3 embedding;
The Fano threefold X
10Joint work in progress with A. Iliev (Sofia) and L. Manivel (Grenoble).
Most Fano threefolds of the first kind of degreec13 = 10 are obtained as follows:
V 5-dimensional complex vector space;
G(2,V)⊂P(∧2V) =P9 the Pl¨ucker3 embedding;
P7⊂P(∧2V) general codimension-2 linear space;
The Fano threefold X
10Joint work in progress with A. Iliev (Sofia) and L. Manivel (Grenoble).
Most Fano threefolds of the first kind of degreec13 = 10 are obtained as follows:
V 5-dimensional complex vector space;
G(2,V)⊂P(∧2V) =P9 the Pl¨ucker3 embedding;
P7⊂P(∧2V) general codimension-2 linear space;
G(2,V)∩P7 is a smooth 4-fold of degree 5 (independent, modulo the action of PGL(V), of the choice of P7);
The Fano threefold X
10Joint work in progress with A. Iliev (Sofia) and L. Manivel (Grenoble).
Most Fano threefolds of the first kind of degreec13 = 10 are obtained as follows:
V 5-dimensional complex vector space;
G(2,V)⊂P(∧2V) =P9 the Pl¨ucker3 embedding;
P7⊂P(∧2V) general codimension-2 linear space;
G(2,V)∩P7 is a smooth 4-fold of degree 5 (independent, modulo the action of PGL(V), of the choice of P7);
X10=G(2,V)∩P7∩Ω, where Ω is a (general) quadric.
Rationality and unirationality
AllX10 are unirational.
Some (smooth) degenerationsX0 of X10 are conic bundles for which (by Prym theory) codimJ(X0)Sing(Θ)>4.
This implies the same for a general X10, which is thereforenot rational.
Rationality and unirationality
AllX10 are unirational.
Some (smooth) degenerationsX0 of X10 are conic bundles
for which (by Prym theory) codimJ(X0)Sing(Θ)>4.
This implies the same for a general X10, which is thereforenot rational.
Rationality and unirationality
AllX10 are unirational.
Some (smooth) degenerationsX0 of X10 are conic bundles for which (by Prym theory) codimJ(X0)Sing(Θ)>4.
This implies the same for a general X10, which is thereforenot rational.
Rationality and unirationality
AllX10 are unirational.
Some (smooth) degenerationsX0 of X10 are conic bundles for which (by Prym theory) codimJ(X0)Sing(Θ)>4.
This implies the same for a general X10, which is thereforenot rational.
Conics on X
10For a cubic threefoldX3⊂P4, we used lines to parametrize (via the Abel–Jacobi map) the theta divisor ofJ(X3).
ForX10, we will useconics todisprove the Torelli theorem. Conics onX10are parametrized by a smooth connected surface F(X) (Logachev).
The surfaceF(X) is the blow-up at one point of a minimal surface of general typeFm(X).
Conics on X
10For a cubic threefoldX3⊂P4, we used lines to parametrize (via the Abel–Jacobi map) the theta divisor ofJ(X3).
ForX10, we will useconics todisprove the Torelli theorem.
Conics onX10are parametrized by a smooth connected surface F(X) (Logachev).
The surfaceF(X) is the blow-up at one point of a minimal surface of general typeFm(X).
Conics on X
10For a cubic threefoldX3⊂P4, we used lines to parametrize (via the Abel–Jacobi map) the theta divisor ofJ(X3).
ForX10, we will useconics todisprove the Torelli theorem.
Conics onX10are parametrized by a smooth connected surface F(X) (Logachev).
The surfaceF(X) is the blow-up at one point of a minimal surface of general typeFm(X).
Conics on X
10For a cubic threefoldX3⊂P4, we used lines to parametrize (via the Abel–Jacobi map) the theta divisor ofJ(X3).
ForX10, we will useconics todisprove the Torelli theorem.
Conics onX10are parametrized by a smooth connected surface F(X) (Logachev).
The surfaceF(X) is the blow-up at one point of a minimal surface of general typeFm(X).
Elementary transformation along a conic
Letc be a general conic contained in X.
Consider Y
blow-up of c ε
ϕ
++W
WW WW WW WW WW WW WW WW WW WW WW
Y¯ quartic in P4 X
πc
projection fromhci
33g
g g g g g g g g g g g
The only curves contracted byϕare:
the (strict transforms of the 20) lines in X that meetc; (the strict transform of) a conicι(c) that meetsc in 2 points. It is asmall contraction that can be flopped:
χ:Y −→ϕ Y¯ ϕ
0
←−Y0 smooth projective threefold (isomorphism outside curves contracted byϕorϕ0).
Elementary transformation along a conic
Letc be a general conic contained in X. Consider Y
blow-up of c ε
ϕ
++W
WW WW WW WW WW WW WW WW WW WW WW
Y¯ quartic in P4 X
πc
projection fromhci
33g
g g g g g g g g g g g
The only curves contracted byϕare:
the (strict transforms of the 20) lines in X that meetc; (the strict transform of) a conicι(c) that meetsc in 2 points. It is asmall contraction that can be flopped:
χ:Y −→ϕ Y¯ ϕ
0
←−Y0 smooth projective threefold (isomorphism outside curves contracted byϕorϕ0).
Elementary transformation along a conic
Letc be a general conic contained in X. Consider Y
blow-up of c ε
ϕ
++W
WW WW WW WW WW WW WW WW WW WW WW
Y¯ quartic in P4 X
πc
projection fromhci
33g
g g g g g g g g g g g
The only curves contracted byϕare:
the (strict transforms of the 20) lines in X that meetc; (the strict transform of) a conicι(c) that meetsc in 2 points. It is asmall contraction that can be flopped:
χ:Y −→ϕ Y¯ ϕ
0
←−Y0 smooth projective threefold (isomorphism outside curves contracted byϕorϕ0).
Elementary transformation along a conic
Letc be a general conic contained in X. Consider Y
blow-up of c ε
ϕ
++W
WW WW WW WW WW WW WW WW WW WW WW
Y¯ quartic in P4 X
πc
projection fromhci
33g
g g g g g g g g g g g
The only curves contracted byϕare:
the (strict transforms of the 20) lines in X that meetc;
(the strict transform of) a conicι(c) that meetsc in 2 points. It is asmall contraction that can be flopped:
χ:Y −→ϕ Y¯ ϕ
0
←−Y0 smooth projective threefold (isomorphism outside curves contracted byϕorϕ0).
Elementary transformation along a conic
Letc be a general conic contained in X. Consider Y
blow-up of c ε
ϕ
++W
WW WW WW WW WW WW WW WW WW WW WW
Y¯ quartic in P4 X
πc
projection fromhci
33g
g g g g g g g g g g g
The only curves contracted byϕare:
the (strict transforms of the 20) lines in X that meetc; (the strict transform of) a conicι(c) that meetsc in 2 points.
It is asmall contraction that can be flopped: χ:Y −→ϕ Y¯ ϕ
0
←−Y0 smooth projective threefold (isomorphism outside curves contracted byϕorϕ0).
Elementary transformation along a conic
Letc be a general conic contained in X. Consider Y
blow-up of c ε
ϕ
++W
WW WW WW WW WW WW WW WW WW WW WW
Y¯ quartic in P4 X
πc
projection fromhci
33g
g g g g g g g g g g g
The only curves contracted byϕare:
the (strict transforms of the 20) lines in X that meetc; (the strict transform of) a conicι(c) that meetsc in 2 points.
It is asmall contraction that can be flopped:
χ:Y −→ϕ Y¯ ϕ
0
←−Y0 smooth projective threefold (isomorphism outside curves contracted byϕorϕ0).
