On equisingular families of plane curves of degree 6

On equisingular families of plane curves of degree 6

Ichiro Shimada

Hiroshima University

2009.01.19 at University of Tokyo

§1 Four equivalence relations of simple sextics

B⊂P²: a complex reduced projective plane curve of degree 6.

B is a simple sextic

⇔ B has only simle singularities (ADE-singularities)

⇔ the minimal resolution X_B of the double cover Y_B →P² branching along B is a K3 surface

I µ_B : the total Milnor number ofB.

I R_B : theADE type of SingB.

I E_B : the set of exceptional (−2)-curves for the minimal resolution X_B →Y_B. We have |E_B|=µ_B.

I Σ_B ⊂H²(X_B,Z) : the sublattice generated by the classes [E] of E ∈ E_B and the polarization classh:= [ρ^∗O_P2(1)], whereρ:X_B →Y_B →P². We haverankΣ_B = 1 +µ_B.

I Σ_B ⊂H²(X_B,Z) : the primitive closure of Σ_B inH²(X_B,Z).

Example by Zariski in 1930’s

There exist two irreducible simple sextics with six ordinary cusps (that is,R_B = 6A₂)

B_trs ={f³+g² = 0} (torus type) and B_ntrs (non-torus type) that cannot be connected by an equisingular family.

Their differences are described in a several ways:

I π₁(P²\B_trs)∼=Z/2Z∗Z/3Z, while π₁(P²\B_ntrs)∼=Z/2Z×Z/3Z.

I ∃ a smooth conic Γ ={f = 0} passing through the six cusps of B_trs, which splits into two curves Γ₊ and Γ₋ in X_Btrs, while

@ a conic passing through the six cusps ofB_ntrs.

I the finite abelian group Σ_Btrs/Σ_Btrs is cyclic of order 3 generated by [Γ₊]∈Σ_Btrs, while Σ_Bntrs/Σ_Bntrs = 0.

B∼_eqs B⁰ ⇔ B and B⁰ are connected by an equisingular family.

B ∼_latB⁰ ⇔ ∃ a bijectionE_B ∼=E_B⁰ that induces, with φ(h) =h, an isometry of lattices φ: Σ_B ∼= Σ_B⁰.

B ∼_cfg B⁰ ⇔ ∃ tubular nbdsT ⊂P² of B and T⁰⊂P² of B⁰

∃ a homeoϕ: (T,B) →∼ (T⁰,B⁰) such that

• degB_i = degϕ(B_i) for each irred comp B_i of B,

• ϕinduces a local analytic isomorphism at each singular point of B and B⁰.

B∼_top B⁰ ⇔ ∃ a homeoϕ: (P²,B) →∼ (P²,B⁰) that induces a local analytic isom at each singular point.

B ∼_latB⁰

⇒ ⇒ ^(Yang)

B ∼_eqsB⁰ B ∼_cfg B⁰

⇒ ⇒

B ∼_top B⁰

Example

For the example by Zariski, we have

B_trs ∼_cfgB_ntrs, but

B_trs6∼_eqsB_ntrs, B_trs6∼_latB_ntrs, B_trs6∼_topB_ntrs.

Remark

The torus curvesB_trs ={f³+g² = 0} of Zariski form a connected equisingular family. An explicit defining equation of a non-torus curve B_ntrs of Zariski was first given by Oka (1994). The connect- edness of the equisingular family of non-torus curves was established by Degtyarev (2008).

Aim: to compare these four equivalence relations.

§2 Comparison of ∼

_lat

and ∼

_cfg

Using the surjectivity of the period mapping for complexK3 surfaces, Yang (1996) classified all lattice types (the equivalence classes of∼_lat). He has also established an algorithm to determine the configuration type of a given lattice type.

Numbers of lattice types and configuration types:

µ_B 0 1 2 3 4 5 6 7 8 9 10 11

∼_cfg 1 1 2 3 6 10 18 30 53 89 148 246

∼_lat 1 1 2 3 6 10 18 30 53 89 148 246

µ_B 12 13 14 15 16 17 18 19 total

∼_cfg 415 684 1090 1623 2139 2283 1695 623 11159

∼_lat 416 686 1096 1639 2171 2330 1734 629 11308

Definition

A configuration type consisting ofklattice types withk >1 is called alattice Zariski k-plet.

Aim: Desribe all lattice Zariskik-plets.

There are no lattice Zariskik-plets with k≥4.

Example of a lattice Zariski triple

There are three lattice types Λ₁,Λ₂,Λ₃ in the configuration type of B=C ∪Q, where C is a smooth conic andQ is a quartic with a tacnodeP₁ intersecting C atP₂,P₃ with multiplicity 4, so that R_B =A₃+ 2A₇.

These three lattice types are distinguished as follows:

B ∈Λ₁ ⇔ [Σ_B : Σ_B] = 4, B ∈Λ₂ ⇔ [Σ_B : Σ_B] = 8, B ∈Λ₃ ⇔ [Σ_B : Σ_B] = 2.

Definition

A simple sexticBis said to belattice-genericif Σ_B =NS(X_B) holds.

Remark

For any B, there exists a lattice-genericB⁰ such thatB ∼_eqs B⁰. In particular, every lattice type contains a lattice-generic member.

The three lattice types above are distinguished geometrically.

LetB be a lattice-generic member of the configuration type above.

I B ∈Λ₁ if and only if ∃a smooth conic Γ passing through P₁,P₂,P₃ such thatmult_Pi(B,Γ) = 4 for i = 1,2,3.

I B ∈Λ₂ if and only if ∃a line Γ passing throughP₁,P₂,P₃.

I B ∈Λ₃ if and only if there are no such conics or lines.

Z -splitting curves (Z stands for “Zariski”)

Definition

A reduced irreducible curve Γ ⊂ P² is called splitting for B if the strict transform of Γ by X_B → P² splits into distinct irred com- ponents Γ₊ ⊂ X_B and Γ₋ ⊂ X_B, which are called the lifts of Γ.

A splitting curve Γ ⊂ P² is called Z -splitting if the class [Γ₊] of Γ₊⊂X_B is contained in the primitive closure Σ_B ⊂H²(X_B,Z).

Remark

(1) A splitting curve is Z-splitting if and only if it is stable under a small equisingular deformation of B.

(2) We have a numerical criterion (involving the intersection mul- tiplicities of B and Γ) to determine whether a splitting curve Γ is Z-splitting or not.

Example

Consider the torus curve B_trs ={f³+g² = 0} of Zariski, where f andg are general. Then the conic Γ ={f = 0}is Z-splitting.

If f =f₁f₂ with degf₁ = degf₂ = 1, then the lines Γ₁ = {f₁ = 0}

and Γ₂ = {f₂ = 0} are splitting but not Z-splitting. In this case, the simple sextic

B⁰ :={f₁³f₂³+g² = 0}

with R_B⁰ = 6A₂ is contained in the same lattice type asB_trs, but is notlattice-generic.

We have written an algorithm to determine allZ-splitting curves of degree≤2 for a lattice-generic member of a given lattice type. In particular, ifB andB⁰ are lattice-generic and B∼_latB⁰, then the numbers ofZ-splitting lines (resp. conics) forB and forB⁰ are equal.

By this algorithm, we have obtained the following:

Theorem

The lattice types in any lattice Zariski k-plets (k > 1) are distin- guished by the numbers of Z-splitting lines and Z-splitting conics for their lattice-generic members.

We are going to classify allZ-splitting lines and Z-splitting conics for simple sextics.

§3 Classification of Z -splitting lines and conics

For simplicity, we call a pair (B,Γ) of a lattice-generic simple sextic B and a Z-splitting curve Γ alattice-generic Z -pair.

Definition

Let (B,Γ) and (B⁰,Γ⁰) be lattice-genericZ-pairs. We say that (B,Γ) and (B⁰,Γ⁰) are of the same lattice type and write (B,Γ)∼_lat(B⁰,Γ⁰) if there exists a bijection E_B ∼= E_B⁰ that induces an isometry of lattices φ: Σ_B ∼= Σ_B⁰ such that

I φpreservesh, and

I φmaps the class [Γ₊]∈Σ_B to [Γ⁰₊]∈Σ_B⁰ or [Γ⁰₋]∈Σ_B⁰. The lattice type containing a lattice-genericZ-pair (B,Γ) is denoted byλ(B,Γ).

We have (B,Γ)∼_lat(B⁰,Γ⁰) =⇒B ∼_latB⁰ =⇒B ∼_cfg B⁰.

Definition

Theorderof a lattice typeλ(B,Γ) is the order of the class [Γ₊]∈Σ_B in the finite abelian group Σ_B/Σ_B.

Definition

Let λ and λ₀ be lattice types of lattice-generic Z-pairs. We say that λ₀ is a specialization of λ if there exists an analytic family (B_t,Γ_t)_t∈∆of lattice-genericZ-pairs parametrized by a unit disc ∆ such that (B_t,Γ_t)∈λfort 6= 0, and (B₀,Γ₀)∈λ₀.

We are going to classify the lattice types of lattice-genericZ-pairs (B,Γ) with deg Γ≤2 from which all lattice types are obtained by specializations.

Theorem (Classification ofZ-splitting lines)

Let λ be the lattice type of a lattice-generic Z-pair (B,Γ) with deg Γ = 1. Then the order d of λ is 6,8,10 or 12, and λ is a specialization of the following lattice typeλ_lin,d =λ(B_d,Γ_d):

R_Bd degrees of irreducible components λ_lin,6 3A₅ [3,3] (the cubics are smooth) λ_lin,8 A₃+ 2A₇ [2,4] (the quartic has A₃) λ_lin,10 2A₄+A₉ [1,5] (the quintic has 2A₄) λ_lin,12 A₃+A₅+A₁₁ [2,4] (the quartic has A₅).

For each λ_lin,d =λ(B_d,Γ_d), the Z-splitting line Γ_d passes through the three singular points of B_d. The finite abelian group Σ_Bd/Σ_Bd is cyclic of orderd, and is generated by the class [(Γ_d)₊] of the lift.

Theorem (Classification ofZ-splitting conics)

Let λ be the lattice type of a lattice-generic Z-pair (B,Γ) with deg Γ = 2. Then the order d of λ is 3,4,5,6,7 or 8, and λ is a specialization of the following lattice typeλ_con,d =λ(B_d,Γ_d):

R_Bd degs

λ_con,3 6A₂ [6]

λ_con,4 2A₁+ 4A₃ [2,4] (the quartic has 2A₁)

λ_con,5 4A₄ [6]

λ_con,6 2A₁+ 2A₂+ 2A₅ [2,4] (the quartic has 2A₂)

λ_con,7 3A₆ [6]

λ_con,8 A₁+A₃+ 2A₇ [2,4] (the quartic has A₁+A₃).

The finite abelian group Σ_Bd/Σ_Bd is cyclic of orderd, and is gener- ated by the class [(Γ_d)₊] of the lift.

Each of the simple sextics in these “generating” lattice types λ_lin,d (d = 6,8,10,6= 12) and λ_con,d (d = 3, . . . ,8) is a member of lattice Zariskik-plets (k >1).

Example

The simple sextic inλ_con,3 (R_B = 6A₂,degs= [6]) is the torus curve B_trs of Zariski. It has a non-torus partnerB_ntrs.

Example

The simple sextic in λ_lin,8 (R_B = A₃ + 2A₇,degs = [2,4]) is a member of the lattice Zariski triple presented above.

What classes generates the finite abelian group Σ

_B

/Σ

_B

?

LetB be a lattice-generic simple sextic.

Σ⁰_B ⊂H²(X_B,Z) : the sublattice generated by Σ_B and the reduced parts of the strict transforms of the irred components ofB.

Σ_B ⊂Σ⁰_B ⊂Σ_B. Theorem (Exceptional simple sextic)

(1) If∃aZ-splitting curve of degree≤2, then Σ_B/Σ⁰_B is generated by the classes of the lifts of theseZ-splitting curves.

(2) There exists a lattice-genericB_exc with R_Bexc = 3A₁+ 4A₃ and degsB_exc = [2,4] (the quartic has 3A₁) such that Σ⁰_Bexc 6= Σ_Bexc but there are no Z-splitting curves of degree ≤2.

(3) If Σ⁰_B 6= Σ_B but there are no Z-splitting curves of degree≤2, then the lattice type ofB is a specialization of that ofB_exc.

The group Σ_Bexc/Σ_Bexc is cyclic of order 4, and is generated by the classes of the lift [Γ₊] of Z-splitting cubiccurves Γ. Hence we have

Corollary

For a lattice-generic B, Σ_B =NS(X_B) is generated over Σ⁰_B by the classes of the lifts ofZ-splitting curves of degree ≤3.

Remark

The exceptional simple sexticB_exc =C ∪Q is a member of lattice Zariski couple. Let B_exc⁰ =C⁰∪Q⁰ be a lattice-generic member of the partner in this lattice Zariski couple. Then Σ_Bexc⁰ /Σ_Bexc⁰ is also cyclic of order 4. There exists a Z-splitting conic Γ for B_exc⁰ and Σ_Bexc⁰ /Σ_Bexc⁰ is generated by [Γ₊].

§4 Comparison of ∼

_lat

and ∼

_top

Recall the implications:

B ∼_latB⁰

⇒ ⇒

B ∼_eqsB⁰ B ∼_cfg B⁰

⇒ ⇒

B ∼_top B⁰ For a simple sexticB, we denote by

T_B ⊂H²(X_B,Z)

the orthogonal complement of Σ_B ⊂H²(X_B,Z). (IfB is lattice-generic, thenT_B is the transcendental lattice of X_B.)

Theorem

IfB ∼_top B⁰, then the latticesT_B andT_B⁰ are isomorphic.

Proof

We consider the open K3 surface U_B := ρ⁻¹(P²\B) ⊂X_B. We put

J_∞(U_B) := T

K Im(H₂(U_B \K,Z)→H₂(U_B,Z)),

where K runs through the set of compact subsets of U_B, and put V₂(U_B) :=H₂(U_B,Z)/J_∞(U_B). Then the intersection pairing

ι_B :H₂(U_B,Z)×H₂(U_B,Z)→Z

induces ¯ι_B :V₂(U_B)×V₂(U_B)→Z. By construction, we have B ∼_topB⁰ =⇒ (V₂(U_B),¯ι_B) ∼= (V₂(U_B⁰),¯ι_B⁰).

Then Theorem follows from

(V₂(U_B),¯ι_B)∼=T_B.

We have obtained following theorem by means of the Shioda-Inose construction of the singularK3 surfaces (the complexK3 surfaces with Picard number 20).

Theorem (S.- and Sch¨utt)

LetX andX⁰ be singularK3 surfaces defined overQsuch that their transcendental lattices are in the same genus. Then X and X⁰ are conjugate under the action of Gal(Q/Q).

Corollary

Let B be a simple sextic with µ_B = 19 defined over Q. If the genus containingT_B contains more than one isomorphism classes of lattices, then∃ σ∈Gal(Q/Q) such thatB ∼_latB^σ andB6∼_topB^σ. (Remark that the lattice type is determined algebraically.)

Can this corollary be generalized to equisingular families of simple sextics withµ_B <19?

Example (Arima and S.-)

Consider B_±:z·(G(x,y,z)±√

5·H(x,y,z)) = 0, where G(x,y,z) := −9x⁴z−14x³yz+ 58x³z²−48x²y²z −64x²yz²

+10x²z³+ 108xy³z−20xy²z²−44y⁵+ 10y⁴z, H(x,y,z) := 5x⁴z+ 10x³yz−30x³z²+ 30x²y²z+

+20x²yz²−40xy³z+ 20y⁵.

We have degsB_± = [1,5] with the quintic having A₁₀ and R_B± = A₁₀+A₉. Their transcendental lattices are

T_B+ ∼=

· 2 1 1 28

¸

, T_B− ∼=

· 8 3 3 8

¸ . Hence we haveB₊∼_latB₋ but B₊6∼_topB₋.

Does B ∼

_top

B

⁰

imply B ∼

_lat

B

⁰

?

SinceH²(X_B,Z) is unimodular, we have|discT_B|=|discΣ_B|.

Proposition

Suppose that B∼_cfg B⁰ (and hence Σ_B ∼= ΣB⁰). Then [Σ_B : Σ_B]6= [Σ_B⁰ : Σ_B⁰] =⇒ B6∼_topB⁰. In many cases (but not in all cases), the lattice types in a configuration type are distinguished by [Σ_B : Σ_B].

Proposition

LetB andB⁰ be simple sextics such that

B ∼_cfgB⁰, [Σ_B : Σ_B] = [Σ_B⁰ : Σ_B⁰], B6∼_latB⁰.

Then either the lattice types of [B,B⁰] are specializations of those of [B_exc,B_exc⁰ ], or R_B =A₁+A₃+ 2A₇ anddegsB = [2,4].

§5 Comparison of ∼

_lat

and ∼

_eqs

B ∼_latB⁰

⇒ ⇒

B ∼_eqsB⁰ B ∼_cfg B⁰

⇒ ⇒

B ∼_top B⁰

Using the refined version of the surjectivity of the period mapping for complexK3 surfaces, Degtyarev (2008) has given an algorithm to determine the connected components of the equisingular family (the equivalence classes of∼_eqs) in a given lattice type.

His algorithm involves a calculation of the orthogonal group O(T_B). Since T_B is indefinite forµ_B <19, the complete table of the connected components of the equisingular family has not yet obtainedexcept for the case µ_B = 19.

λ: a given lattice type.

We want to calculate

CES(λ) :=λ/∼_eqs : the set of connected components of the equisingular family inλ.

I Σ : the N´eron-Severi lattice of λ;sgnΣ = (1, µ).

I G ⊂O(Σ) : the subgroup of isometries of Σ preserving the set of exceptional (−2)-curves [E] and the polarizationh.

I Ts: the set of isomorphism classes of even lattices T with sgnT = (2,19−µ) whose discriminant form is (−1) times the discriminant form of Σ (the set of possible transcendental lattices).

We have a natural projection

p:CES(λ)→→ Ts.

ForT ∈ Ts, we put

I L(Σ,T) : the set of even unimodular overlatticesLof Σ⊕T in which Σ andT are primitive (theseL are theK3 lattice).

I cΩ(T) : the set of connected components of the cone {x ∈T ⊗R|x² >0}.

Theorem (Degtyarev) We have

p⁻¹(T) ∼= (G×O(T))\(L(Σ,T)×cΩ(T)).

We say that a connected component isrealif the corresponding (G×O(T))-orbit⊂L(Σ,T)×cΩ(T) is stable under the interchanging of the two elements ofcΩ(T).

Example

The simple sextics with R_B =A₁₈+A₁ form one lattice type. The equisingular family has three connected components; one is real and the other two are non-real. Artal, Carmona and Cogolludo (2002) constructed these simple sextics defined over Q(α), where α is a root of

19x³+ 50x²+ 36x+ 8 = 0,

which has one real root and two non-real roots. Their transcendental lattices are

· 38 0

0 2

¸

(for real),

· 8 ±2

±2 10

¸

(for non-real).

Example

The simple sextics with R_B =A₁₉ form one lattice type. The eq- uisingular family has two connected components, and both are real.

Artal et al. showed that they are conjugate by Gal(Q(√ 5)/Q).

Their transcendental lattices are

· 2 0 0 20

¸ .

We would like to know whether these two are∼_top or not.

Example

The simple sextics with R_B = A₁₄ +A₄ +A₁ form one lattice type. The equisingular family has six non-real connected compo- nents. Their transcendental lattices are all isomorphic to

· 10 0 0 30

¸ .

Summary

The “zoology” of simple sextics is interesting.