Supersingular K3 surfaces and lattice theory

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SupersingularK3 surfaces and lattice theory

Supersingular K 3 surfaces and lattice theory

Ichiro Shimada

Hiroshima University

December 04, 2009, Minami-Ohsawa, Tokyo

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SupersingularK3 surfaces and lattice theory Definition of supersingularK3 surfaces

Letk be an algebraically closed field of characteristicp ≥0.

LetX be a smooth projective surface defined overk. Two divisors D₁ andD₂ on X are numerically equivalentand denoted by D₁ ≡D₂ if

D₁.C =D₂.C

holds for any curveC onX, whereD.C denotes the intersection number ofD andC. The Z-module

NS(X) :={divisors on X}/≡

is then equipped with a structure of the lattice by the intersection pairing. We callNS(X) the N´eron-Severi latticeofX.

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The cycle map

NS(X)→H_´et²(X,Q_l) is injective. In particular, we have

rank(NS(X))≤b₂(X).

Whenk =C, we have

NS(X) =H^1,1(X)∩H²(X,Z), and hence

rank(NS(X))≤h^1,1(X).

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Definition

A smooth projective surfaceX is called aK3surface if

∃ a nowhere vanishing regular 2-formω_X onX, and h¹(X,O_X) = 0.

K3 surfaces are 2-dimensional analogue of elliptic curves.

Examples ofK3 surfaces:

double covers of P² branching along smooth curves of deg 6, smooth complete intersections of degree (4) inP³, degree (2,3) in P⁴, (2,2,2) in P⁵,

minimal resolutions ofA/hιi, whereAare abelian surfaces and ιis the involution x7→ −x (p6= 2),

. . ..

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IfX is aK3 surface, then

b₂(X) = 22 and p_g(X) = 1.

In particular, we have

rank(NS(X))≤22.

Ifk=C, then we have

rank(NS(X))≤20.

A complexK3 surface X is calledsingular ifrank(NS(X)) = 20.

We have a complete classification and an explicit method of constructing all singularK3 surfaces due to Shioda and Inose.

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Definition

AK3 surface X is calledsupersingularif rank(NS(X)) = 22.

Problem

Construct all supersingular K3 surfaces.

More precisely, we want to obtain a theorem of the type:

Every supersingular K3 surface is defined by such and such defining equations or constructed by such and such methods.

We have an answer in characteristic 2, but not yet in characteristic>2.

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SupersingularK3 surfaces and lattice theory Unirationality and supersingularity

Definition

An algebraic surfaceS is unirationalif there is a dominant rational mapP²· · · →S.

Ifp= 0, every unirational surface is rational (birational to P²).

In particular, a complexK3 surface can never be unirational.

Theorem

If a smooth projective surfaceS is unirational, then b₂(S) =rank(NS(S)).

Corollary

A unirationalK3 surface is supersingular.

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SupersingularK3 surfaces and lattice theory Examples

Consider the Fermat surface of degreed in P³ F :w^d+x^d+y^d+z^d = 0 in characteristicp>0.

Ifd =q+ 1 with q =p^ν, thenF is defined by the Hermitian form ww¯ +x¯x+yy¯+z¯z = 0 where¯a=a^q,

overF_q, and hence is similar to the quadric surface.

Theorem (Shioda)

Ifd =q+ 1, thenF is unirational.

Corollary

Ifp≡3 mod 4, then the Fermat quartic surface w⁴+x⁴+y⁴+z⁴ = 0

in characteristicp is a supersingularK3 surface. _{8 / 24}

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SupersingularK3 surfaces and lattice theory Examples

Suppose thatp = 2. Let (x,y,z) be the homogeneous coordinates ofP². Consider the purely inseparable double cover

Y :w²=f(x,y,z)(=X

a_ijkxⁱy^jz^k)

ofP², wheref is a general homogeneous polynomial of deg 6.

ThenY has 21 ordinary nodes, and the minimal resolution X ofY is aK3 surface.

The pull-back ofX →P² by the morphismP²→P² given by (ξ, η, ζ)7→(x,y,z) = (ξ², η², ζ²)

is defined by

(w−f^(1/2)(ξ, η, ζ))² = 0 where f^(1/2)(ξ, η, ζ) =X

a^(1/2)_ijk ξⁱη^jζ^k, and its reduced partw =f^(1/2)(ξ, η, ζ) is rational. Therefore X is unirational and hence is supersingular.

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SupersingularK3 surfaces and lattice theory N´eron-Severi lattice of supersingularK3 surfaces

By alattice, we mean a freeZ-module Lof finite rank with a non-degenerate symmetric bilinear form

L×L→Z.

A latticeL is naturally embedded into the dual lattice L^∨:=Hom(L,Z).

Thediscriminant group ofL is the abelian group D_L:=L^∨/L,

which is finite of order

|discL|=|detM_L|,

whereM_L is a symmetric matrix expressing L×L→Z.

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Definition

A latticeL isevenif x²∈2Zholds for any x ∈L.

Definition

Letp be a prime integer. A lattice isp-elementaryif D_L is isomorphic to direct product of the cyclic group of orderp:

D_L∼= (Z/pZ)^⊕t.

Definition

The signature sgn(L) = (s₊,s₋) of a latticeL is the pair of numbers of positive and negative eigenvalues ofM_L (inR).

We say thatLisindefinite if s₊>0 and s₋>0.

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Theorem (Rudakov-Shafarevich, Artin)

The N´eron-Severi latticeNS(X) of a supersingularK3 surfaceX in characteristicp is

even,

of signature (1,21) and p-elementary.

There exists a positive integerσ_X ≤10 such that D_NS(X₎∼= (Z/pZ)^⊕2σ^X.

Definition

The integerσ_X is called the Artin invariant of X.

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Theorem

For eachσ, there is only one isomorphism class of lattices Lwith even,

rank(L) = 22, sgn(L) = (1,21) and D_L∼= (Z/pZ)^⊕2σ.

Proof.

Use Eichler’s theorem on isom classes of indefinite lattices.

Corollary

The isomorphism class of the N´eron-Severi latticeNS(X) of a supersingularK3 surface X is determined by the characteristicp and the Artin invariantσ_X.

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The Artin invariant goes down under specialization.

Theorem (Ogus)

IfX andX⁰ are supersingular K3 surfaces in characteristic p with Artin invariant 1, thenX andX⁰ are isomorphic.

Recall that we want to obtain a theorem of the type:

Every supersingular K3 surface in characteristic p with Artin invariant ≤m is defined by such and such defining equations or constructed by such and such methods.

For example:

Theorem (Ogus)

Every supersingularK3 surface in characteristicp 6= 2 with Artin invariant≤2 is a Kummer surface.

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One of the motivations is:

Conjecture (Artin-Shioda)

Every supersingularK3 surface is unirational.

By the theorem of Ogus, we obtain the following:

Corollary

Every supersingularK3 surface with Artin invariant ≤2 is unirational.

To show the unirationality, we need the defining equations.

We can get some defining equations from the latticeNS(X).

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SupersingularK3 surfaces and lattice theory Known results

Theorem (S.-)

Every supersingularK3 surface in characteristic 2 is birational to the purely inseparable double cover

w²=f(x,y,z)

ofP², wheref is a homogeneous polynomial of degree 6.

Corollary (Rudakov-Shafarevich)

Every supersingularK3 surface in characteristic 2 is unirational.

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Theorem (D.Q. Zhang and S.-)

Every supersingularK3 surface X in characteristic 3 with Artin invariant≤6 is birational to the purely inseparable triple cover of a quadratic surfaceP¹×P¹.

Corollary (Rudakov-Shafarevich)

Every supersingularK3 surface in characteristic 3 with Artin invariant≤6 is unirational.

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Theorem (Pho and S.-)

Every supersingularK3 surface X in characteristic 5 with Artin invariant≤3 is birational to the double cover of P² branching along a curve defined by

y⁵z−f(x,z) = 0

wheref(x,z) is a homogeneous polynomial of degree 6.

Corollary

Every supersingularK3 surface X in characteristic 5 with Artin invariant≤3 is unirational.

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Remark

It is difficult to construct supersingularK3 surfaces with big Artin invariantsen masse.

There exist many sporadic examples of supersingularK3 surfaces with big Artin invariants, due to Shioda and Goto.

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How to obtain the defining equations ofXfromNS(X)?

Definition

An effective divisorD on aK3 surfaceX is apolarization ifD²>0 and the complete linear system|D|has no fixed components.

IfD is a polarization, then |D|defines a morphism Φ :X →P^N. LetX →Y →P^N be the Stein factorization of Φ. Then the normal surfaceY, birational toX, has only rational double points.

For a classh∈NS(X), we put

R(X,h) :={x∈NS(X) | (x,h) = 0 and x² =−2}.

ThenR(X,h) is a root system.

Proposition

TheADE-type of the rational double points of Y is equal to the Dynkin type of the root systemR(X,[D]).

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Note that

NS(X)∼=Pic(X) for aK3 surface X.

Proposition

Leth= [D]∈NS(X) be a class with h² >0. Then D is a polarization if and only if the following hold:

D is nef (that is,D.C ≥0 for any curveC), and {x ∈NS(X) | x² = 0 and (x,h) = 1}is empty.

On the other hand, looking at the nef cone ofNS(X)⊗R, we obtain the following:

Proposition

Leth∈NS(X) be a class withh² ≥0. Then there is an isometry ϕofNS(X) such thatϕ(h) is nef.

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Therefore we have:

Theorem

If there exists a vectorh∈NS(X) withd =h² >0 such that {x∈NS(X) | x² = 0 and (x,h) = 1}=∅, thenX has a morphism Φ :X →P^N with the Stein factorization

X →Y →P^N such that

(Φ^∗O(1))² =d, and

the normal surface Y has rational double points of the ADE-type being equal to the Dynkin type of the root system

R(X,h) :={x∈NS(X) | (x,h) = 0 and x²=−2}.

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We give a proof of the following theorem in char 5:

Theorem

Every supersingularK3 surface X in char 5 withσ_X ≤3 is birational to the double cover of a plane branching along a curve defined byy⁵−f(x) = 0 with degf = 6.

We consider the general caseσ_X = 3. ThenNS(X) is

characterized by the properties of being even, sgn = (1,21), and D_NS(X₎∼= (Z/5Z)⁶.

We can construct such a latticeNS(X) lattice-theoretically. For example,

NS(X)∼= (−A₄)⊕(−A₄)⊕(−A₄)⊕(−A₄)⊕(−A₄)⊕

µ 2 1 1 −2

¶ , where (−A₄) denote the negative-definite root lattice of type A₄.

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We have a vectorh∈NS(X) with h² = 2 such that {x ∈NS(X) | x² = 0 and (x,h) = 1}=∅, and

the root system {x∈NS(X) | x²=−2 and (x,h) = 0} is of type 5A₄.

ThereforeX is birational to a double cover ofP² branching along a sextic curveB with 5A₄ singularities. (Such a curve does not exist in char 0.) Then the Gauss map

B→B^∨

is inseparable of degree 5. We can show thatB is defined by an equation of the formy⁵−f(x) = 0.

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