Cubic hypersurfaces over finite fields

Cubic hypersurfaces over finite fields

Olivier DEBARRE

(joint with Antonio LAFACE and Xavier ROULLEAU)

Ecole normale sup´´ erieure, Paris, France

Moscow, HSE, September 2015

Theorem (Chevalley–Warning)

Any subscheme ofP^m_Fq defined by equations of degrees d₁, . . . ,d_s with d₁+· · ·+d_s ≤m has anF_q-point.

→any cubicFq-hypersurface of dimension ≥2 contains an Fq-point.

What about F_q-lines?

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The scheme F (X )

X ⊂Pⁿ⁺¹_k cubic of dimension n.

F(X)⊂Gr(1,Pⁿ⁺¹_k ) projective scheme of lines contained in X.

F(X) connected ifn ≥3.

Sing(X) finite =⇒ F(X) lci of dimension 2n−4 and ω_F(X) =OF(X)(4−n).

X smooth =⇒ F(X) smooth.

High dimensions

Theorem

AnyF_q-cubic of dimension≥5 contains anF_q-line.

Proof. Let x∈X(F_q). The scheme of lines through x contained in X is the intersection inPⁿ of hyperplane, a quadric, and a cubic

→Chevalley–Warning when n≥6.

WhenX is smooth,F(X) is Fano whenn ≥5

→Esnault, and Fakhruddin–Rajan to extend to all X. We now look at cubic surfaces, threefolds, and fourfolds.

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Cubic surfaces

The diagonal cubic surface

x₁³+x₂³+x₃³+ax₄³ = 0 contains noFq-lines when a∈Fq is not a cube.

aexists wheneverq ≡1 (mod 3)

There are smooth cubicF_q-surfaces with no F_q-lines forq arbitrarily large.

The Galkin–Shinder “beautiful formula”

X ⊂Pⁿ⁺¹_F

q F_q-cubic F(X)⊂Gr(1,Pⁿ⁺¹_F

q ) scheme of lines contained in X N_r(X) := Card(X(F_q^r))

Nr(F(X)) = Nr(X)²−2(1 +q^nr)Nr(X) +N2r(X) 2q^2r

+q^(n−2)rN_r(Sing(X)).

This formula comes from a relation between the classes ofX and F(X) in the Grothendieck ring of varieties.

Olivier DEBARRE Cubic hypersurfaces over finite fields 6 of 22

Y smooth projective scheme defined over Fq

P_i(Y,T) := det(Id−TF^∗,Hⁱ(Y,Q_`)) =:

bi(Y)

Y

j=1

(1−Tω_ij)∈Z[T] where|ω_ij|=q^i/2. The trace formula (n:= dim(Y))

N_r(Y) = X

0≤i≤2n

(−1)ⁱTr(F^∗r,Hⁱ(Y,Q_`)) = X

0≤i≤2n

(−1)ⁱ

bi(Y)

X

j=1

ω^r_ij

implies, for the zeta function, Z(Y,T) := expX

r≥1

Nr(Y)T^r r

= Y

0≤i≤2n

Pi(Y,T)^(−1)i+1.

Zeta function of F (X )

X ⊂P⁴_F

q smooth cubic.The trace formula reads Z(X,T) =

Q

1≤j≤10(1−qω_jT)

(1−T)(1−qT)(1−q²T)(1−q³T), withωj algebraic integers and |ω_j|=q^1/2.

The Galkin–Shinder formula implies Z(F(X),T) =

Q

1≤j≤10(1−ωjT)Q

1≤j≤10(1−qωjT) (1−T)(1−q²T)Q

1≤j<k≤10(1−ωjωkT)

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Cohomology of F (X )

This formula gives the Betti numbers ofF(X). Actually, the full Galkin–Shinder relation gives isomorphisms of Gal(Fq/Fq)-modules

H³(X,Q_{`</sub>) <sup>∼ //</sup>H<sup>1</sup>(F(X),Q<sub>`}(1))

V₂

H¹(F(X),Q`) <sup>∼ //</sup>H<sup>2</sup>(F(X),Q`) V₂

H¹(A(F(X)),Q_`) ^{∼ //}

∼OO

H²(A(F(X)),Q_`)

∼OO

The first one can also be obtained using the incidence correspondence.

The second one can also be deduced by smooth and proper base change from the statement in char. 0.

Existence of lines

Theorem

Any smoothFq-cubic threefold contains at least 10Fq-lines if q≥11.

Proof. Write the Frobenius eigenvalues asω₁, . . . , ω₅, ω₁, . . . , ω₅. Setrj :=ωj+ωj ∈[−2√

q,2√

q]. Use the trace formula N₁(F(X)) = 1− X

1≤j≤5

r_j − X

1≤j≤5

qr_j +q²

+ 5q+ X

1≤j<k≤5

(ω_jω_k +ω_jω_k +ω_jω_k +ω_jω_k)

= 1 + 5q+q²−(q+ 1) X

1≤j≤5

r_j + X

1≤j<k≤5

r_jr_k and study the minimum of this real function...

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Examples

We found smooth cubic threefolds overF₂,F₃,F₄, andF₅ with no lines.

The cubic threefold inP⁴_F

5 with equation

x₁³+ 2x₂³+x₂²x3+ 3x1x₃²+x₁²x4+x1x2x4+x1x3x4

+ 3x₂x₃x₄+ 4x₃²x₄+x₂x₄²+ 4x₃x₄²+ 3x₂²x₅+x₁x₃x₅ + 3x2x3x5+ 3x1x4x5+ 3x₄²x5+x2x₅²+ 3x₅³ is smooth and contains noF5-lines and 126F5-points.

RemainsF₇,F₈, andF₉...

Average numbers of lines

Average numbers of lines computed on random samples of 10⁵ F₂-cubic threefolds

all cubics; average∼9.651 smooth cubics; average∼6.963.

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Singular threefolds

LetX ⊂P⁴_Fq Fq-cubic with a single singular point, of typeA1 or A2.

The curveC of lines in X through the singular point is smooth of genus 4 and canonically embedded inP³_F

q. It has two pencilsg₃¹ andh¹₃, used to embedC in C⁽²⁾ by

x7→g₃¹−x

Clemens–Griffiths, Kouvidakis–van der Geer

F(X) is the non-normal surface obtained by gluing in C⁽²⁾ these two copies ofC.

Lines on singular threefolds

Theorem

For any r≥1, set n_r := Card(C(F_q^r)). We have

Card(F(X)(Fq)) =



















1

2(n₁²+n2)−n1 if g₃¹ 6=h₃¹ are defined overF_q;

1

2(n₁²+n2) +n1 if g₃¹ 6=h₃¹ are not defined overF_q;

1

2(n₁²+n₂) if g₃¹ =h₃¹.

Again, this can also be obtained with the Galkin–Shinder method.

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Lines on singular threefolds

Corollary

When q≥4, any cubic threefold X ⊂P⁴_F

q defined over F_q with a single singular point, of type A1 or A2, contains an Fq-line.

Proof. We need to exclude

n₁=n₂ = 1 andg₃¹6=h¹₃ defined overF_q. Write C(Fq) =C(F_q²) ={x}, andg₃¹≡x+x⁰+x⁰⁰.

g₃¹ is defined over F_q ⇒ x⁰+x⁰⁰ defined overF_q

⇒ x⁰,x⁰⁰ are defined over F_q2

⇒ x⁰ =x⁰⁰=x

⇒ g₃¹≡3x≡h¹₃ Contradiction.

Lines on singular threefolds

n1=n2 = 0. Thenq≤7 (Howe–Lauter–Top).

Weil conjectures forC:

Frobenius rootsω1, . . . , ω4,ω¯1, . . . ,ω¯4, with|ωj|=√ q;

H monic with (real) rootsrj:=ωj+ ¯ωj with|rj| ≤2√ q has integral coefficients;

P

1≤j≤4rj =P

1≤j≤8ωj =q+ 1−n1=q+ 1;

P

1≤j≤4r_j²=P

1≤j≤8(ω_j²+2q) =q²+8q+1−n2=q²+8q+1.

Hence

H(T) =T⁴−(q+ 1)T³−3qT²+aT+b, with |b|=|r₁r₂r₃r₄| ≤16q² and|a|=|P4

j=1b/r_j| ≤32q^3/2 integral.

Computer search: such polynomials with 4 such real roots and q ∈ {2,3,4,5,7}only exist for q ≤3.

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Examples

We found nodal cubics threefolds overF2 and F3 with no lines.

OverF₂:

x₂³+x₂²x₃+x₃³+x₁x₂x₄+x₃²x₄+x₄³+x₁²x₅+x₁x₃x₅+x₂x₄x₅ contains noF₂-lines and

H(T) =T⁴−3T³−6T²+ 24T −15.

OverF₃:

2x₁³+ 2x₁²x2+x1x₂²+ 2x2x₃²+ 2x1x2x4+x2x3x4

+x₁x₄²+ 2x₄³+x₂x₃x₅+ 2x₃²x₅+x₂x₅²+x₅³ contains noF3-lines and

H(T) =T⁴−4T³−9T²+ 47T −32.

Zeta function of F (X )

X ⊂P⁵_F

q smooth cubic.The trace formula reads

Z(X,T) = ¹

(1−T)(1−qT)(1−q³T)(1−q⁴T)Q23

j=1(1−qω_jT),

withωj algebraic integers,|ω_j|=q, andω23=q.

The Galkin–Shinder formula implies Z(F(X),T) = _{(1−T)(1−q}4T)Q ¹

j((1−ωjT)(1−q²ωjT))Q

j≤k(1−ωjωkT).

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Cohomology of F (X )

Again, the Galkin–Shinder relation implies that there are isomorphisms of Gal(F_q/F_q)-modules

H⁴(X,Q_{`</sub>) <sup>∼ //</sup>H<sup>2</sup>(F(X),Q<sub>`}(1))

Sym²H²(F(X),Q_{`</sub>) <sup>∼ //</sup>H<sup>4</sup>(F(X),Q<sub>`}) The first also follows using the incidence correspondence.

The second one can also be deduced by smooth and proper base change from statement in char. 0 (Beauville–Donagi, Bogomolov).

Existence of lines

Theorem

Any smoothF_q-cubic fourfold contains at least 26F_q-lines if q≥5.

One can use another trace formula (Katz). If Sing(X) finite, the cohomology ofOF(X) is very simple (Altman–Kleiman):

dimFqH^j(F(X),OF(X)) = 1

forj ∈ {0,2,4}, the others are 0, and the multiplication H²(F(X),OF(X))⊗H²(F(X),OF(X))→H⁴(F(X),OF(X)) is an isomorphism of Galois modules (Serre duality).

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The Katz trace formula

N₁(F(X)) ≡

4

X

j=0

(−1)^jTr(F,H^j(F(X),OF(X))) (mod p)

≡ 1 +t+t² (modp) Corollary

Assume q≡2 (mod 3). AnyF_q-cubic fourfold with finite singular set contains anFq-line.

This applies toq = 2 and leaves only the casesq ∈ {3,4} open for the existence of a line on a smooth cubic fourfold.

Final question

The computer found a smooth cubic fourfold overF2 with a single line.

Question

We found no cubic fourfolds without lines. Do they exist?

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