A special configuration of 12 conics and a related K3 surface

A special configuration of 12 conics and a related K3 surface

Xavier Roulleau

Geometria em Lisboa (IST) Zoom Seminar, May 26, 2020

On what is this talk about ?

Joint work with David Kohel and Alessandra Sarti A smooth projective surfaceX is called aK3 surfaceif

KX ' OX and H¹(X,OX) ={0}.

By the adjonction formula, the self-intersection of a smooth rational curveC onX isC²=−2, sothat such a curve is called a(−2)-curve.

ExampleLetAbe an Abelian surface. The quotientA/[−1] ofAby the involution [−1] :z→ −z is a surface with 16 nodal singularitiesA1, which are the image of the 16 2-torsion points ofA. The minimal desingularization of A/[−1] is denotedKm(A) and is called aKummer surface. A Kummer surface is K3, it contains a 16A1-configuration, which means 16 disjoint (−2)-curves.

Theorem(Nikulin) LetX be a K3 surface such that there exists a 16A1-configuration on it. Then there exist an Abelian surfaceAsuch that X'Km(A) and under that isomorphism, the 16A1onX comes from the resolution ofA/[−1].

On what is this talk about ?

Shioda raised the following question:

ifKm(A)'Km(B), is it true thatA'B ?

This is false in general (Gritsenko-Huleck, Hosono-Lian-Oguiso-Yau).

In a previous work with A. Sarti, we constructed the first explicit counter-examples.

LetAbe an Abelian surface andσ:A→Abe an order 3 symplectic automorphism.σfixes 9 3-torsion points andA/σhas 9 cusps. The minimal resolution is a K3 surface, denoted byKm3(A), and called ageneralised Kummer surface. A cusp singularity is resolved by aA2-configuration i.e. two (−2)-curvesC1,C2such thatC1C2= 1. The surfaceKm(A) contains a 9A2-configuration.

Theorem(Bertin) LetX be a K3 surface such that there exists a

9A2-configuration on it. ThenX is a generalised Kummer surface: there exist an Abelian surfaceAand a symplectic order 3 automorphismσsuch that X'Km3(A) and under that isomorphism, the 9A2-configuration onX comes from the resolution ofA/σ.

Generalised Shioda question: ifKm3(A)'Km3(B), is it true thatA'B?

On what is this talk about ?

Generalised Shioda question: ifKm3(A)'Km3(B), is it true thatA'B? Let us fixX =Km3(A). It is not difficult to prove that:

PropositionThe isomorphism classes of Abelian surfacesB such that X'Km3(B) are in one to one correspondance with theAut(X)-orbits of 9A2-configurations onX.

In particular, the generalized Shioda question has affirmative answer for the generalized KummerX =Km3(A) if and only if the number ofAut(X)-orbits of 9A2-configurations onX is 1.

Our initial aim was to construct a counter-example, as we did for the classical Kummer surfaces. In this talk we explain how to construct 9 more

9A2-configurations on certain generalised Kummer surfacesX. We will then construct automorphisms of the K3 surfaceX which send the configurations to each others, so that one stays in the same orbit underAut(X).

We do not answer to the generalised Shioda question, but thanks to that question, we know much more on the geometry of the K3 surface we study, and on its automorphism group.

On what is this talk about ?

We do not answer to the generalised Shioda question, but thanks to that question, we know much more on the rich geometry of the K3 surface we study, and on its automorphism group.

Hesse Pencil, Hesse and dual Hesse configurations An interesting (98,126)-configuration of points and conics

Construction of the generalised Kummer surfaceX and its natural fibration Nine new 9A2-configurations onX

Construction of automorphisms ofX Another geometric model ofX

Abstract configurations

LetP,Lbe two finite sets, of respective orderm,n. Recall that a

(ms,nt)-configurationis the data of a subsetR ⊂ P × Lsuch that for any fixedp∈ P, one has:

|{(p, `)∈ R |`∈ L}|=s, and for any fixed`∈ L, one has:

|{(p, `)∈ R |p∈ P}|=t.

That impliesms =nt. Ifm=n, this is called ams-configuration.

In practiceRis rarely made explicit.

ExampleYou may imagine thatP is a set of points, andLis a set of lines. If on all lines inLthere aret points inP, and through all points ofPthere pass slines, that gives an example of a (ms,nt)-configuration.

The Hesse and dual Hesse configuration

Forλ∈P¹, letEλbe the cubic

Eλ: x³+y³+z³−3λxyz= 0.

That cubic is smooth if and only ifλ6∈1, ω, ω²,∞, forω³= 1. The pencil {x³+y³+z³−3λxyz|λ∈P¹}

is called theHesse pencil.

The base locusT9of the Hesse Pencil is also the set of flex points of the elliptic curvesEλ, this is also the set of 3-torsion points of these curves.

LetL12be the set of lines that contains at least 2 points inT9. Then

|L12|= 12.

Each lineL∈ L12contains 3 points inT9. Each pointq∈ T9is contained on 4 lines.

Thus the sets (T9,L12) form a (94,123)-configuration of points and lines. That configuration is called theHesse configuration.

The Hesse and dual Hesse configuration

Each of the 4 singular cubicsEλ,λ∈1, ω, ω²,∞is a union 3 lines.

In fact the 12 lines inL12are the union of these 4 triangles. The lines inL12

meet in 12 = 4·3 other points.

LetW →P²be the blow-up atT9, the 9 base points of the elliptic pencil.

There exists a natural fibrationW →P¹with fiberEλatλ, whose 4 singular fibers are ’triangles’ i.e. of type ˜A2.

RemarkRemoving a point inT9and the 4 lines through it, one gets a 83-configuration of points and lines.

Thedual Hesseconfiguration is obtained by taking as a set of pointsP12the lines inL12viewed as points in the dual plane, and as a setL9the lines in the dual plane containing at least 2 of the points inP12. The setsP12,L9form a (123,94)-configuration, called the dual Hesse configuration.

The Hesse and dual Hesse configurations are unique, up to projective automorphisms.

A (9

, 12

)-configuration of points and conics

Letfλbe the formfλ=x³+y³+z³−3λxyz. Start with the cubic

Eλ={fλ= 0},→P². Thedual curveCλ,→P²is the image of the cubicEλby the dual map∇λ: P²→P², (x:y :z)→(^∂f_∂x^λ :^∂f_∂y^λ : ^∂f_∂z^λ). Geometrically, it maps a pointpto the point [TEλ,p] in the dual projective space representing the tangent line toEλatp. The curveCλis

Cλ: (x⁶+y⁶+z⁶) + 2(2λ³−1)(x³y³+x³z³+y³z³)

−6λ²xyz(x³+y³+z³)−3λ(λ³−4)x²y²z²= 0.

It is singular at 9 points, and the singularities area2singularities (local equationx₁²=x₂³). This setP9=P9(λ)of 9 cusps is the image of the base locusT9by the dual map; it is:

p1= (λ: 1 : 1), p4= (λ:ω:ω²), p7= (λ:ω²:ω) p2= (1 :λ: 1), p5= (ω²:λ:ω), p8= (ω:λ:ω²) p3= (1 : 1 :λ), p6= (ω:ω²:λ), p9= (ω²:ω:λ).

The union of the pointspk=pk(λ), λ∈P¹is a lineLk, and the 9 lines L1, . . . ,L9are the 9 lines of the dual Hesse configuration (P12,L9).

A (9

, 12

)-configuration of points and conics

There is a unique conic passing through 5 points in general position.

Theorem

LetC12=C12(λ)be the set of conics that contains at least6points in P9=P9(λ). Then

|C12|= 12, each conic is irreducible

The sets(P9,C12)form a(98,126)-configuration.

The union∪C,C∈ C12has singular setP9(λ)∪ P12, where we recall that P12is the set of intersection points of the lines inL9, the dual Hesse configuration.

Let C ∈ C12be a conic. The intersections with the other conics C⁰ are transverse. One has C∩C⁰⊂ P9, unless for a unique conic C⁰, for which C∩C⁰ contains a unique point inP12.

For q∈ P9definePq=P9\ {q}and letCqbe the8conics that contains q. The set(Pq,Cq)form a85-configurationA˜q.

I. Dolgachev, A. Laface, U. Persson, G. Urzua have also independently

A pencil of sextics through P

Warning : the setP9=P9(λ) of cusps of the sexticCλdepends on the parameterλ, and so isC12=C12(λ). We take the parameterλgeneric.

PropositionThe linear systemδ of sextic curves passing throughP9with multiplicity at least 2 is a pencil. It contains the sextic curveCλand a unique non-reduced curve, 2Ca(λ), where

Ca(λ) : x³+y³+z³−^(λ3_λ⁺²⁾xyz= 0, (1) is the so-calledCayleyan curveofCλ.

Letf :Z →P²be the blow-up at the 9 points inP9. LetE1, . . . ,E9be the 9 exceptional curves.

The moving part of the pull-back toZ of the pencil of sexics is now base point free and define a morphismϕ⁰:Z →P¹for which the curvesEk are 2-sections, which meansEk·(Fiber) = 2.

The strict transformCλ⁰ 'EλonZ ofCλis a curve that intersect the exceptional divisors tangentially at one point.

The associated K3 surface

PropositionThe exists a double coverXλ→Z ofZ branched overC_λ⁰. The surfaceX =Xλis a K3 surface.The pull-back toX of an exceptional curvesEk

is the unionAk+A⁰_k of two (−2)-curves.

(We recall that a (−2)-curve on a K3 surface is a smooth rational curve).

The elliptic fibrationϕ⁰:Z →P¹pull-backs to an elliptic fibration ϕ:X →P¹.

The curvesAk,A⁰_k, κ= 1, . . . ,9 are sections of the elliptic fibration.

Another view point: If instead we consider the double coverY →P²branched over the sextic curveCλ, we see thatXλis the minimal resolution of the 9 cusps singularities over the points inP9.

Since the K3 surfaceXλcontains 9 disjointA2-configurations, it is a generalised Kummer surface. In our case Birkenhake and Lange proved that X=Km3(A) withA=Eλ×Eλ.

6561 8A

-configurations

Recall that we say that two (−2)-curvesC,C⁰ on a K3 surfaces such that CC⁰= 1 form aA2-configuration. The K3 surfaceX =Xλcontains 9 disjoint A2-configurations, thus it is a generalized Kummer Surface.

We want to know more 9A2-configurations onXλ. For that aim, we use the pull-back toXλof the special setC12of 12 conics. Let us denote byCijklmn the conic that pass through pointsps,s∈ {i,j,k,l,m,n}(when it exists). The following 4 sextic curves are element of the pencilδof sextics with multiplicity

≥2 at thepk’s:

C123456+C123789+C456789,C124578+C134679+C235689, C124689+C135678+C234579,C125679+C134589+C234678.

The strict transform of these curves by the blow-upZ →P²at pointspk are 4 singular fibers of type ˜A2. The pull-back toX of these 4 fibers via the double coverXλ→Z are8 fibers of type ˜A2. These are all the singular fibers of ϕ:Xλ→P¹.

That way, we obtain 3⁸= 6561 8A2-configurations. For eachj∈ {1, ...,9}, the pull-backA⁰j toXλof the setCp_j of 8 conics containing a fixed pointpj∈ P9is among these 8A2-configurations.

9 new 9A

-configurations

Theorem

For j∈ {1, , . . . ,9}, there exists a unique rational quartic curve Qj containing P9with a unique singularity at the point pj. That curve singularity has typed5. The strict transform on Xλof the curve Qjbyη:X →P²is the union of two (−2)-curvesγj, γ_j⁰ which form anA2-configuration.

The curvesγj, γ_j⁰and the16curves inA⁰_jform a9A2-configurationAj.

Xλ

P²

Qj

Cλ −2−3

−1

2 : 1 cover

−1 −2

−2 γ_j⁰

Automorphisms of the generalized Kummer surface X

Theorem

A model of XλinP¹×P²with coordinates t∈P¹,(x:y:z)∈P²is Xλ: x³+y³+z³+λ³(t²+ 3)−4t²

λ²(t²−1) xyz= 0. (2)

The Mordell-Weil group of sections of the natural fibrationϕ:Xλ→P¹is isomorphic toZ×(Z/3Z)²; it is generated by the18curves Ak,A⁰_k’s.

The torsion part acts transitively on the9 9A2-configurations we found.

The equation ofXλis obtained using Magma software (non trivial).

Shimada computed all possible Mordell-weil group of elliptic K3 surfaces. For the assertion on the generators, we use Shioda’s theory of Mordell-Weil lattice, for that aim we compute a base of the N´eron-Severi groupNS(Xλ) ofXλ, and its discriminant.

For the action of the torsion part on the 9 9A2-configurations, we use the explicit addition law on the model (2), (one knows explicit equations of the (−2)-curves in the fibers).

Recall that we denoted byA1,A⁰₁, . . . ,A9,A⁰₉the 18 (−2)-curves of the natural 9A2-configurationC (these are sections ofϕ).

LetC⁰=B1,B₁⁰, . . . ,B9,B₉⁰ be one of the 9A2-configurations we constructed.

Theorem

There exists an automorphismφof Xλsuch thatφ(C) =C⁰.

Suppose that suchφexists.The orthogonal complement ofC is generated by D2, the pull-back of a line inP². One computes that the orthogonal

complements ofC⁰ is also generated by divisorD2⁰ of square 2. Sinceφ^∗must preserve the N´eron-Severi lattice, we haveφ^∗(D2) =D₂⁰.

There are 2⁹9! = 185.794.560 bijective maps

µ:{A1,A⁰1, . . . ,A9,A⁰9} → {B1,B1⁰, . . . ,B9,B9⁰}

which preserves the incidence relations inC andC⁰. Each map extends uniquely to an isometry ˜φµofNS(Xλ)⊗Q. We compute that among these maps, only 864 are isometries ofNS(Xλ) (we know a base of it).

Among the 864 maps, only 36 are effective Hodge isometries, and thus by the Torelli Theorem these 36 maps lift to automorphismsφof the K3Xλsuch that

0

Another construction of X

TheoremA model ofXλin P¹×P²is

X_λ: x³+y³+z³+λ³(t²+ 3)−4t² λ²(t²−1) xyz= 0.

Letπ2:Xλ→P²be the projection on the second factor, the mapπ2contracts the curvesA1, . . . ,A9. By computing the branch locus ofπ2, one gets

Theorem

The surface Xλis the minimal desingularization of the double cover of the plane branched over the union of Eλ: x³+y³+z³−3λxyz= 0and the Hessian of Eλ:

He(λ) : x³+y³+z³+^(λ3−4)

λ² xyz= 0.

The images inP²byπ2of the24irreducible components of the8singular fibers of the fibrationϕ:Xλ→P¹are the12lines of the Hesse configuration.

Thank you !