On nodal prime Fano threefolds of degree 10

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Fano threefolds The nodal Fano threefoldX₁₀ ReconstructingX₁₀ Verra threefolds Period maps

On nodal prime Fano threefolds of degree 10

Olivier DEBARRE (joint with Atanas ILIEV and Laurent MANIVEL)

´Ecole Normale Sup´erieure de Paris

Trento, September 2010

Olivier DEBARRE On nodal prime Fano threefolds of degree 10

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AFano varietyis a complex projective varietyX with −K_X ample.

Smooth Fano threefoldsform 105 irreducible families: 88 with Picard numbers >1;

17 with Picard number 1: 7 with indices>1;

10 with index 1: one for each degree in{2,4,6,8, . . . ,18,22}. X₁₀: Fano threefold with Picard number 1, index 1, and degree 10.

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Smooth Fano threefoldsform 105 irreducible families:

88 with Picard numbers >1; 17 with Picard number 1:

7 with indices>1;

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88 with Picard numbers >1;

17 with Picard number 1: 7 with indices>1;

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17 with Picard number 1:

7 with indices>1;

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7 with indices>1;

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7 with indices>1;

10 with index 1:

one for each degree in{2,4,6,8, . . . ,18,22}. X₁₀: Fano threefold with Picard number 1, index 1, and degree 10.

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7 with indices>1;

10 with index 1: one for each degree in{2,4,6,8, . . . ,18,22}.

X₁₀: Fano threefold with Picard number 1, index 1, and degree 10.

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7 with indices>1;

10 with index 1: one for each degree in{2,4,6,8, . . . ,18,22}.

X₁₀: Fano threefold with Picard number 1, index 1, and degree 10.

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The fourfoldW_Oand the threefoldX_Oin P⁶_O The double ´etale coverπ:eΓ₆→Γ₆ A (birational) conic bundle structure onX

A generalX10 is the intersection inP(∧²V5) =P⁹ of

the Grassmannian G(2,V5), a general P⁷,

a general quadric Ω, and

X₁₀=G(2,V₅)∩P⁷∩Ω⊂P⁷ is anticanonically embedded.

Note:

W =G(2,V5)∩P⁷ is a smooth fourfold, independent of P⁷; for Ω general quadric cone with vertex O ∈W general, X =W ∩Ω has a single node at O.

From now on, X ⊂P⁷ will be such a nodal Fano threefold.

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A generalX10 is the intersection inP(∧²V5) =P⁹ of the Grassmannian G(2,V5),

a general P⁷, a general quadric Ω, and

Note:

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a general P⁷,

a general quadric Ω, and

Note:

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a general P⁷, a general quadric Ω,

and

Note:

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Note:

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Note:

W =G(2,V5)∩P⁷ is a smooth fourfold, independent of P⁷;

for Ω general quadric cone with vertex O ∈W general, X =W ∩Ω has a single node at O.

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Note:

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Note:

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IfpO :P⁷ 99KP⁶_O is the projection fromO,

W_O :=p_O(W)⊂P⁶_O is the base-locus of a pencil (Ω_p)_p∈Γ₁ of quadrics of rank 6 (all such pencils are isomorphic);

WO contains P³_W :=pO(TW,O);

Sing(W_O) is the locus of the vertices of the Ω_p, a smooth rational cubic curve in P³_W;

and

X_O :=p_O(X)⊂P⁶_O is the intersection ofW_O with the general quadric ΩO :=pO(Ω);

Sing(X_O) = Sing(W_O)∩Ω_O consists of six points (corresponding to the six lines inX through O).

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WO :=pO(W)⊂P⁶_O is the base-locus of a pencil (Ωp)p∈Γ1 of quadrics of rank 6

(all such pencils are isomorphic); WO contains P³_W :=pO(TW,O);

and

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WO :=pO(W)⊂P⁶_O is the base-locus of a pencil (Ωp)p∈Γ1 of quadrics of rank 6 (all such pencils are isomorphic);

and

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and

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and

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and

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and

Sing(XO) = Sing(WO)∩ΩO consists of six points (corresponding to the six lines inX throughO).

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The threefoldX_O is therefore the base-locus of the net of quadrics Π :=hΩ_O,Γ₁i in P⁶_O. Consider

thediscriminant curveΓ₇ ⊂Π corresponding to singular quadrics (all of rank 6), union of

the line Γ₁ of quadrics containingW_O and

a smooth sextic Γ₆meeting Γ₁transversely at the six points corresponding to the quadrics in Γ₁ with vertices at the six singular points ofXO;

the double ´etale cover

π:eΓ6∪Γ¹₁∪Γ²₁→Γ6∪Γ1

corresponding to the choice of a family of 3-planes contained in a quadric of rank 6 in Π (P³_W defines the component Γ¹₁).

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thediscriminant curveΓ7 ⊂Π corresponding to singular quadrics (all of rank 6), union of

π:eΓ6∪Γ¹₁∪Γ²₁→Γ6∪Γ1

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π:eΓ6∪Γ¹₁∪Γ²₁→Γ6∪Γ1

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the line Γ₁ of quadrics containingW_O and a smooth sextic Γ₆meeting Γ₁transversely

at the six points corresponding to the quadrics in Γ₁ with vertices at the six singular points ofXO;

π:eΓ6∪Γ¹₁∪Γ²₁→Γ6∪Γ1

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a smooth sextic Γ₆meeting Γ₁transversely at the six points corresponding to the quadrics in Γ₁ with vertices at the six singular points ofX_O;

π:eΓ6∪Γ¹₁∪Γ²₁→Γ6∪Γ1

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a smooth sextic Γ₆meeting Γ₁transversely at the six points corresponding to the quadrics in Γ₁ with vertices at the six singular points ofX_O;

π:eΓ6∪Γ¹₁∪Γ²₁→Γ6∪Γ1

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As the base-locus of the net of quadrics Π, the threefoldX_O has a birational conic bundle structurep`:X 99KΠ:

choose a general line `⊂X_O;

to x∈X_O general, associate the unique quadric in Π containing the 2-planehx, `i.

The discriminant curve is Γ₇ = Γ₆∪Γ₁.

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Definition of the isomorphism and injectivity Surjectivity

Special surfaces

LetX₁₀^nodal be the 21-dim’l moduli stack for our nodalX.

Theorem

There is a birational isomorphism

X10^nodal

99K∼

triples (Γ₆,Γ₁,M)

. isom.

where M is an even invertible theta-characteristic onΓ6∪Γ1.

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Special surfaces

LetX₁₀^nodal be the 21-dim’l moduli stack for our nodalX. Theorem

There is a birational isomorphism

X10^nodal

99K∼

triples (Γ₆,Γ₁,M)

. isom.

where M is an even invertible theta-characteristic onΓ6∪Γ1.

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Special surfaces

Sketch of proof. Given X, the curve Γ₇= Γ₆∪Γ₁ parametrizes singular quadrics in the net Π of quadrics containingXO.

Let v : Γ7 ,→ P⁶_O

p 7−→ Vertex(Ω_p) and define

M_X =v^∗OP⁶_O(1)⊗OΓ7(−1). Then (Beauville),

M_X is a theta-characteristic andH⁰(Γ7,M_X) = 0; the double ´etale coverπ:eΓ7 →Γ7 is defined by the line bundle η=M_X(−2), of order 2;

XO ⊂P⁶_O is determined up to projective isomorphism by the pair (Γ₇,M_X).

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Special surfaces

Sketch of proof. Given X, the curve Γ₇= Γ₆∪Γ₁ parametrizes singular quadrics in the net Π of quadrics containingXO. Let

v : Γ7 ,→ P⁶_O p 7−→ Vertex(Ω_p)

and define

M_X =v^∗OP⁶_O(1)⊗OΓ7(−1). Then (Beauville),

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Special surfaces

v : Γ7 ,→ P⁶_O p 7−→ Vertex(Ω_p) and define

M_X =v^∗OP⁶_O(1)⊗OΓ7(−1).

Then (Beauville),

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Special surfaces

M_X =v^∗OP⁶_O(1)⊗OΓ7(−1).

Then (Beauville),

M_X is a theta-characteristic andH⁰(Γ7,M_X) = 0;

the double ´etale coverπ:eΓ7 →Γ7 is defined by the line bundle η=M_X(−2), of order 2;

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Special surfaces

M_X =v^∗OP⁶_O(1)⊗OΓ7(−1).

Then (Beauville),

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Special surfaces

M_X =v^∗OP⁶_O(1)⊗OΓ7(−1).

Then (Beauville),

X_O ⊂P⁶_O is determined up to projective isomorphism by the

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Special surfaces

Conversely, given

general curves Γ₆ and Γ₁ in Π,

an invertible theta-characteristic M on Γ7 = Γ6∪Γ1 with H⁰(Γ7,M) = 0 (such anM always exists (Catanese)), there is a resolution

0−→OΠ(−2)^⊕7−→^A OΠ(−1)^⊕7 −→M −→0, where (Dixon, Catanese, Beauville)

A is a 7×7 symmetric matrix of linear forms; det(A) is an equation for Γ7;

A defines a net of quadrics inP⁶_O whose base-locusX_O is the intersection of WO with a smooth quadric.

Its inverse image under the birational mapW 99KW_O is a threefoldX10 with a single node atO.

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Special surfaces

Conversely, given

an invertible theta-characteristic M on Γ7 = Γ6∪Γ1 with H⁰(Γ₇,M) = 0

(such anM always exists (Catanese)), there is a resolution

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Special surfaces

Conversely, given

an invertible theta-characteristic M on Γ7 = Γ6∪Γ1 with H⁰(Γ₇,M) = 0 (such anM always exists (Catanese)),

there is a resolution

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Special surfaces

Conversely, given

an invertible theta-characteristic M on Γ7 = Γ6∪Γ1 with H⁰(Γ₇,M) = 0 (such anM always exists (Catanese)), there is a resolution

0−→OΠ(−2)^⊕7 −→^A OΠ(−1)^⊕7 −→M −→0, where (Dixon, Catanese, Beauville)

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Special surfaces

Conversely, given

A is a 7×7 symmetric matrix of linear forms;

det(A) is an equation for Γ7;

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Special surfaces

Conversely, given

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Special surfaces

Conversely, given

A defines a net of quadrics inP⁶_O whose base-locusX_O is the intersection of W_O with a smooth quadric.