On supersingular varieties

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On supersingular varieties

Ichiro Shimada

Hiroshima University

24 September, 2010, Nagoya

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Frobenius supersingular varieties Definition

LetX be a smooth projective variety overF_q. The following are equivalent:

(i) There is a polynomial N(t)∈Z[t] such that

|X(F_q^ν)|=N(q^ν) for all ν ∈Z_>0.

(ii) The eigenvalues of theqth power Frobenius on thel-adic cohomology ring are powers of q by integers.

If these are satisfied, thenb_2i−1(X) = 0 and N(t) =

dimXX

i=0

b_2i(X)tⁱ.

We say thatX isFrobenius supersingularif (i) and (ii) are satisfied.

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Frobenius supersingular varieties An example

If the cohomology ring ofX is generated by the classes of algebraic cycles overF_q, then X is Frobenius supersingular.

The converse is true if the Tate conjecture is assumed.

We have examples of Frobenius supersingular varieties of non-negative Kodaira dimension.

Theorem

The Fermat variety

X :={x₀^q+1+· · ·+x_2m+1^q+1 = 0} ⊂P^2m+1

of dimension 2m and degree q+ 1 regarded as a variety overF_q² is Frobenius supersingular.

This follows from

|X(F_q2)|= 1 +q²+· · ·+q^4m+ (b_2m(X)−1)q^2m.

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Frobenius supersingular varieties Problems

Problems on Frobenius supersingular varieties

Construct non-trivial examples.

Prove (or disprove) the unirationality.

Present explicitly algebraic cycles that generate the cohomology ring.

Investigate the lattice given by the intersection pairing of algebraic cycles.

Produce dense lattices by the intersection pairing in small characteristics.

We discuss these problems

for the classical example ofFermat varieties of degreeq+ 1, and for the new example ofFrobenius incidence varieties.

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Unirationality and Supersingularity

A varietyX is called(purely-inseparably) unirationalif there is a dominant (purely-inseparable) rational map

Pⁿ· · →X.

Theorem (Shioda)

LetS be a smooth projective surface defined overk = ¯k. IfS is unirational, then the Picard numberρ(S) is equal tob₂(S); that is, S issupersingular in the sense of Shioda.

The converse is conjectured to be true forK3 surfaces.

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Artin-Shioda conjecture

Every supersingularK3 surface S (in the sense of Shioda) is conjectured to be (purely-inseparably) unirational.

The discriminant of the N´eron-Severi latticeNS(S) is−p^2σ(S), whereσ(S) is a positive integer≤10, which is called theArtin invariantof S.

The conjecture is confirmed to be true in the following cases:

p odd andσ(S)≤2 (Ogus and Shioda):

p = 2 (Rudakov and Shafarevich, S.-):

p = 3 andσ(S)≤6

(Rudakov and Shafarevich, S.- and De Qi Zhang):

p = 5 andσ(S)≤3 (S.- and Pho Duc Tai).

Method: The structure theorem forNS(S) by Rudakov-Shafarevich.

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Fermat varieties Unirationality

Fermat variety of degree q + 1

Unirationality of the Fermat variety

Theorem (Shioda-Katsura, S.-)

The Fermat varietyX of degreeq+ 1 and dimension n≥2 in characteristicp >0 is purely-inseparably unirational, whereq =p^ν. Indeed,X contains a linear subspace Λ⊂Pⁿ⁺¹ of dimension [n/2].

The unirationality is proved by the projection from the center Λ.

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Fermat varieties

Terminologies about lattices

Lattice

By aquasi-lattice, we mean a free Z-moduleLof finite rank with a symmetric bilinear form

( , ) :L×L→Z.

If the symmetric bilinear form is non-degenerate, we say that Lis alattice.

IfLis a quasi-lattice, thenL/L^⊥ is a lattice, where L^⊥:={x ∈L | (x,y) = 0for ally ∈L}.

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Fermat varieties Lattice of algebraic cycles

Lattices associated with the Fermat varieties

The Fermat variety

X :={x₀^q+1+· · ·+x_2m+1^q+1 = 0} ⊂ P^2m+1

of dimension 2mand degree q+ 1 contains manym-dimensional linear subspaces Λ_i. The number is

Ym ν=0

(q^2ν+1+ 1).

Each of them is defined overF_q².

LetNe(X)⊂A^m(X) be theZ-module generated by the rational equivalence classes of Λ_i, whereA(X) is the Chow ring.

By the intersection pairing

Ne(X)×Ne(X) → Z, we can considerNe(X) as a quasi-lattice.

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LetN(X) :=Ne(X)/Ne(X)^⊥ be the associated lattice.

Theorem (Tate, S.-)

(1) The rank ofN(X) is equal to b_2m(X).

(2) The discriminant ofN(X) is a power ofp.

Corollary

The cycle map induces an isomorphismN(X)⊗Q_l ∼=H^2m(X,Q_l).

The assertion (2) is an analogue of the result that the discriminant of the N´eron-Severi latticeNS(S) of a supersinglar K3 surface S is a power ofp.

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Leth∈ N(X) be the numerical equivalence class of a linear plane sectionX ∩P^m+1.

We put

N_prim(X) :={x∈ N(X) | (x,h) = 0}=hhi^⊥. Theorem

The lattice [−1]^mN_prim(X) is positive-definite.

Here [−1]^mN_prim(X) is the lattice obtained from N_prim(X) by changing the sign with (−1)^m.

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Fermat varieties

Definition of dense lattices

Dense lattices

LetLbe a positive-definite lattice of rank m.

Theminimal norm ofL is defined by

N_min(L) := min{x²|x ∈L,x 6= 0}, and thenormalized center density ofL is defined by

δ(L) := (discL)^−1/2·(N_min(L)/4)^m/2.

Minkowski and Hlawka proved in a non-constructive way that, for eachm, there is a positive-definite latticeL of rankmwith

δ(L) > MH(m) := ζ(m) 2^m−1V_m, whereV_m is the volume of them-dimensional unit ball.

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Fermat varieties

Definition of dense lattices

We say that a positive-definite latticeL of rankmis denseif δ(L) > MH(m).

The intersection pairing of algebraic cycles in positive characteristic has been used to construct dense lattices.

For example, Elkies and Shioda constructed many dense lattices as Mordell-Weil lattices of elliptic surfaces in positive characteristics.

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Fermat varieties

Dense lattice in characteristic 2

Dense lattices arising from Fermat varieties

LetX be the Fermatcubic variety of dimension 2m in characteristic 2.

Recall thatX contains manym-dimensional linear subspaces Λ_i. We consider the positive-definite lattice

h[Λ_i]−[Λ_j]i ⊂ [−1]^mN_prim(X)

generated by the classes [Λ_i]−[Λ_j]. Their properties are as follows:

dimX rank N_min log₂δ log₂MH name 2 6 2 −3.792... −7.344... E₆ 4 22 4 −1.792... −13.915... Λ₂₂ 6 86 8 34.207... 19.320... N₈₆

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Frobenius incidence varieties Definition

Frobenius incidence variety

We fix ann-dimensional linear spaceV overF_p with n≥3.

We denote byG_n,l =G_n^n−l the Grassmannian variety of l-dimensional subspaces ofV.

LetF be a field of characteristic p, and consider an F-rational linear subspaceL∈G_n,l(F) of V.

Letφbe thepth power Frobenius morphism ofG_n,l. For a positive integerν, we put

L^(p^ν⁾:=φ^ν(L).

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Frobenius incidence varieties Definition

Letl andc be positive integers such that l+c <n.

We denote byI_n,l^c the incidence subvariety of G_n,l×G_n^c: I_n,l^c (F) ={(L,M)∈G_n,l(F)×G_n^c(F) | L⊂M }.

Letr :=p^a ands :=p^b be powers ofp by positive integers. We define theFrobenius incidence variety X_n,l^c by

X_n,l^c := (φ^a×id)^∗I_n,l^c ∩ (id×φ^b)^∗I_n,l^c . ThenX_n,l^c is defined over F_p, and we have

X_n,l^c (F) = {(L,M)∈G_n,l(F)×G_n^c(F) | L^(r⁾⊂M and L⊂M^(s) }

= {(L,M)∈G_n,l(F)×G_n^c(F) | L+L^(rs)⊂M^(s)}

= {(L,M)∈G_n,l(F)×G_n^c(F) | L^(r⁾⊂M∩M^(rs)}.

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Frobenius incidence varieties Frobenius supersingularity

Theorem

(1) The scheme X_n,l^c is smooth and geometrically irreducible of dimension (n−l−c)(l+c).

(2) IfX_n,l^c is regarded as a scheme overF_rs, then X_n,l^c is Frobenius supersingular.

The smoothness ofX_n,l^c is proved by computing the dimension of Zariski tangent spaces.

We prove the second assertion by counting the number of F_(rs₎ν-rational points of X_n,l^c .

We put

q:=rs.

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The main ingredient of the proof is the finite set

T_l,d(q,q^ν) :={L∈G_n,l(F_q^ν) | dim(L∩L^(q)) =d }.

Whenl =d, we haveT_l,l(q,q^ν) =G_n,l(F_q) for any ν.

Ford <l, we calculate the cardinality of the set

P := {(L,M)∈G_n,l(F_q^ν)×G_n,2l−d(F_q^ν) | L+L^(q)⊂M }

= {(L,M)∈G_n,l(F_q^ν)×G_n,2l−d(F_q^ν) | L^(q)⊂M ∩M^(q)}, in two ways using the projections

P →G_n,l(F_q^ν) andP →G_n,2l−d(F_q^ν).

Then we get

|P| = Xl t=d

|T_l,t(q,q^ν)| · |G_n−2l_+t,t−d(F_q^ν)|

=

2l−dX

u=l

|T_2l_−d,u(q,q^ν)| · |G_u,l(F_q^ν)|.

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By this equality, we obtain a recursive formula for|T_l,d(q,q^ν)|.

Using the projectionX_n,l^c (F_q^ν)→G_n,l(F_q^ν), we obtain the following:

|X_n,l^c (F_q^ν)|= Xl d=0

|T_l,d(q,q^ν)| · |G_n−2l^c _+d(F_q^ν)|.

By the recursive formula for|T_l,d(q,q^ν)|, we prove that there is a monic polynomialN_n,l^c (t) of degree (l+c)(n−l−c) such that

|X_n,l^c (F_q^ν)|=N_n,l^c (q^ν).

ThereforeX_n,l^c is Frobenius supersingular.

SinceN_n,l^c (t) is monic,X_n,l^c is geometrically irreducible.

Moreover we obtain the Betti numbers ofX_n,l^c .

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Frobenius incidence varieties Examples

Example

Let (x₁:· · ·:x_n) and (y₁:· · ·:y_n) be homogeneous coordinates of G_n,1=P_∗(V) andG_n¹=P^∗(V) that are dual to each other. Then I_n,1¹ ={P

x_iy_i = 0}, and henceX_n,1¹ is defined by

½ x₁^ry₁+· · ·+x_n^r y_n= 0, x₁y₁^s+· · ·+x_ny_n^s = 0.

The Betti numbers ofX_n,1¹ are as follows:

b_2i =b_{2(n−2)−2i} =

(i+ 1 ifi <n−2, n−2 + (qⁿ−1)/(q−1) ifi =n−2.

Whenr =s = 2 (and hence q= 4), X_3,1¹ is the supersingular K3 surface with Artin invariant 1 (Mukai’s model).

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Frobenius incidence varieties Examples

Example

The Betti numbers ofX_7,2² are calculated as follows:

b₀=b₂₄: 1 b₂=b₂₂: 2 b₄=b₂₀: 5

b₆=b₁₈: q⁶+q⁵+q⁴+q³+q²+q + 8 b₈=b₁₆: 2 (q⁶+q⁵+q⁴+q³+q²+q) + 12 b₁₀=b₁₄: 3 (q⁶+q⁵+q⁴+q³+q²+q) + 14 b₁₂: q¹⁰+q⁹+ 2q⁸+ 2q⁷+ 6q⁶+

+6q⁵+ 6q⁴+ 5q³+ 5q²+ 4q+ 16.

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Frobenius incidence varieties Unirationality

Unirationality of X

_n,l^c

Theorem

The Frobenius incidence varietyX_n,l^c is purely-inseparably unirational.

Idea of the proof for the case 2l+c ≤n.

We defineXe ⊂G_n,l ×G_n^c by

Xe(F) ={(L,M) | L⊂M, L^(rs) ⊂M }.

The projectionXe →G_n,l is dominant. Using this projection, we can show thatXe is rational. The map (L,M)7→(L,M^(s)) is a dominant morphism fromXe toX_n,l^c .

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Frobenius incidence varieties Algebraic cycles

Algebraic cycles on X

_n,l^l

Let Λ be anF_rs-rational linear subspace ofV such that l ≤dim Λ≤n−c. We define Σ_Λ⊂G_n,l×G_n^c by

Σ_Λ(F) :={(L,M)∈G_n,l(F)×G_n^c(F) | L⊂Λ and Λ^(r⁾⊂M }.

It follows from Λ^(rs) = Λ that Σ_Λ is contained inX_n,l^c . Whenl =c, we have 2 dim Σ_Λ= dimX_n,l^l .

We can calculate the intersection numbers of these Σ_Λ onX_n,l^l .

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We consider the case wherel =c = 1:

X_n,1¹ ⊂P_∗(V)×P^∗(V).

We put

H:=Im(Aⁿ⁻²(P_∗(V)×P^∗(V))→Aⁿ⁻²(X_n,1¹ ) ).

By the intersection pairing, we can consider the submodule Ne(X_n,1¹ ) :=H+h[Σ_Λ]i ⊂ Aⁿ⁻²(X_n,1¹ ) as a quasi-lattice. Let

N(X_n,1¹ ) :=Ne(X_n,1¹ )/Ne(X_n,1¹ )^⊥ be the associated lattice, and put

N_prim(X_n,1¹ ) :=H^⊥ ⊂ N(X_n,1¹ ).

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Theorem

(1) The rank of N(X_n,1¹ ) isb_2(n−2)(X_n,1¹ ).

(2) The discriminant of N(X_n,1¹ ) is a power ofp.

(3) The lattice [−1]ⁿN_prim(X_n,1¹ ) is positive-definite.

Corollary

The cohomology ring ofX_n,1¹ is generated by the classes of Σ_Λ and the image ofA(P_∗(V)×P^∗(V))→A(X_n,1¹ ).

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Frobenius incidence varieties Dense lattices in characteristic 2

Dense lattices of rank 84 and 85

Theorem

Suppose thatp =r =s = 2. Then N_prim(X_4,1¹ ) is an even positive-definite lattice of rank 84, with discriminant 85·2¹⁶, and with minimal norm 8.

In fact,N_prim(X_4,1¹ ) is a section of a larger latticeM_C of rank 85 =|P³(F₄)|

constructed by the projective geometry overF₄ and a code over R:=Z/8Z.

We put

T :=P³(F₄).

ForS ⊂T, we denote byv_S ∈R^T and ˜v_S ∈Z^T the characteristic functions ofS.

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LetC ⊂R^T be the submodule generated by 2^2−k(v_P −v_P⁰),

whereP andP⁰ areF₄-rational linear subspaces ofP³ of dimension k (k = 0,1,2), and letM_C be the pull-back of C byZ^T →R^T. We define aQ-valued symmetric bilinear form on Z^T by

(˜v_{t},v˜_{t⁰_}) =δ_tt⁰/4 (t,t⁰∈T).

ThenM_C ⊂Z^T is a lattice.

name rank disc N_min log₂δ log₂MH N_prim(X_4,1¹ ) 84 85·2¹⁶ 8 30.795... 17.546...

M_C 85 2²⁰ 8 32.5 18.429...

N₈₆ 86 3·2¹⁶ 8 34.207... 19.320...

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Thank you!

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