A mirror duality for families of $K3$ surfaces associated to bimodular singularities

0.2 MB·

by  Makiko Mase

K
A mirror duality for families of 3 surfaces
associated to bimodular singularities
Makiko Mase
7
1 Key Words: K3 surfaces, toric varieties, Picard lattices
0
AMS MSC2010: 14J28 14M25 14C22
2

b
e Abstract
F
Ebeling and Ploog [4] studied a duality of bimodular singularities
1
which is part of the Berglund–Hu¨bsch mirror symmetry. Mase and Ueda

[7]showedthatthisdualityleadstoapolytopemirrorsymmetryoffami-
]
G liesof K3surfaces. Wediscussin thisarticle howthissymmetryextends
toa symmetry between lattices.
A
.
h
1 Introduction
t
a
m
BimodularsingularitiesclassifiedbyArnold[1]haveadualitystudiedbyEbeling
[ andPloog[4]analogoustoArnold’sstrangedualityforunimodularsingularities.

1 Namely, a pair ((B,f), (B′,f′)) of singularities B, B′ in C3 together with ap-
v propriatedefiningpolynomialsf, f′ isdualifthematricesAf, Af′ ofexponents
7 of f and f′ are transpose to each other. Moreover, in some cases, such poly-
0
nomials are compactified as anticanonical members of 3-dimensional weighted
1
projectivespaceswhosegeneralmembersareGorensteinK3. Thestrangedual-
0
ityforunimodularsingularitiesisrelatedwiththepolytopemirrorsymmetryfor
0
. families of K3 surfaces that are obtained by compactifying the singularities by
2
Kobayashi [6] in a certain sense. In the study of bimodular singularities, Mase
0
7 and Ueda [7] extend the duality by Ebeling and Ploog to a polytope mirror
1 symmetry of families of K3 surfaces. More precisely, the following statement is
: shown :
v
i
X Theorem [7] Let ((B, f), (B′, f′)) be a dual pair in the sense of [4] of sin-
r gularities B and B′ together with their defining polynomials f and f′ that are
a
respectively compactified into polynomials F and F′ as in [4]. Then, there exists
a reflexive polytope ∆ such that ∆F ⊂ ∆ and ∆F′ ⊂ ∆∗. Here, ∆F and ∆F′
are respectively the Newton polytopes of F and of F′, and ∆∗ is the polar dual
to ∆. (cid:4)
Inthisarticle,weconsiderwhetherornotitispossibletoextendtheduality
obtainedin[7]furthertothelatticemirrorsymmetryoffamiliesofK3surfaces.
More precisely, our problem is stated as follows:
Problem Let∆beareflexivepolytopeasin[7]. Doesthereexistgeneralmem-
bers S ∈ F∆ and S′ ∈ F∆∗ such that an isometry T(S) ≃ Pic(S′)⊕U holds
e e
1
? Here, F∆ and F∆∗ are families of K3 surfaces associated to the polytopes ∆
and ∆∗, S denotes the minimal model of S, Pic(S) and T(S) are respectively
the Picard and transcendental lattice of S.
e e e
The problem is answered in Theoreme3.2 together with an explicit descrip-
tion in Proposition 3.1 of the Picard lattices Pic(∆) and Pic(∆∗) defined in
section3,withranksρ(∆)andρ(∆∗),oftheminimalmodelofappropriategen-
eralmembersinthefamilies. Themainresultofthisarticleissummarizedhere.
In the sequel,the names of singularitiesfollow Arnold[1], and singularitiesin a
same row of Table 1 are dual to each other in the sense of [4].
Proposition3.1andTheorem3.2 Let ∆ be the reflexive polytope obtained in
[7]. For the following transpose-dual pairs, the polar duality extends to a lattice
mirror symmetry between the families F∆ and F∆∗, where the Picard lattices
−4 1
are given in Table 1. Here we use the notation C6 := .
8 1 −2
(cid:18) (cid:19)
Singularity Pic(∆) ρ(∆) ρ(∆∗) Pic(∆∗) Singularity
Q U ⊕E ⊕E 16 4 U ⊕A E
12 6 8 2 18
Z U ⊕E ⊕E 17 3 U ⊕A E
1,0 7 8 1 19
E U ⊕E⊕2 18 2 U E
20 8 20
Q U ⊕A ⊕E 16 4 U ⊕C6 Z
2,0 6 8 8 17
E U ⊕E ⊕E 17 3 U ⊕A Z
25 7 8 1 19
Q U ⊕E ⊕E 16 4 U ⊕A E
18 6 8 2 30
Table 1: Picard lattices for lattice mirror symmetric pairs(cid:4)
Section2istodefine thepolytope-andlattice-mirrortheoriesinsubsection
2.3, andto define the transposeduality following[4]in subsection2.4, basedon
a brief introductionto lattice theory in subsection2.1, andof toric geometryin
subsection 2.2, where several formulas and results are stated without proof.
The main theorem of this article is stated in section 3 following auxiliary
results. The facts introduced in the previous section are used in their proof.
Denote by ∆ the reflexive polytope obtained in [7] for a singularity B. As
B
is seen in Table 1, there are isometric Picard lattices Pic(∆ ) ≃ Pic(∆ ),
Q12 Q18
and Pic(∆ )≃Pic(∆ ). We consider and affirmatively answer in Proposi-
Z1,0 E25
tion 4.1 the following question as an application in section 4:
Problem Are the families F (resp.F ) and F (resp.F ) es-
∆Q12 ∆Z1,0 ∆Q18 ∆E25
sentially the same in the sense that general members in these families are bira-
tionally equivalent ?
Acknowledgement
The author thanks to the referee for his helpful comments particularly about
the proof of the key Proposition 3.1 in the original manuscript.
2
2 Preliminary
We start with having a consensus as to Gorenstein K3 and K3 surfaces.
Definition 2.1 A compact complex connected 2-dimensional algebraic variety
S with at most ADE singularities is called Gorenstein K3 if (i) K ∼ 0; and
S
(ii) H1(S, O ) = 0. If a Gorenstein K3 surface S is nonsingular, S is simply
S
called a K3 surface. (cid:4)
2.1 Brief lattice theory
Alattice isa non-degeneratefinitely-generatedfree Z-module witha symmetric
bilinear formcalled an intersectionpairing. The discriminant group of a lattice
LisdefinedbyA :=L∗/L,whichisfinitely-generatedandabelian,whereL∗ :=
L
Hom(L, Z) is dual to L. It is known that the order |A | of the discriminant
L
groupis equal to the determinant of any intersection matrix of L. Let us recall
a standard lattice theory by Nikulin [8]:
Corollary 2.1 (Corollary 1.13.5-(1) [8]) If an even lattice L of signature
(t , t ) satisfies t ≥1, t ≥1, and t +t ≥3+lengthA , then, there exists
+ − + − + − L
a lattice T such that L≃U ⊕T, where U is the hyperbolic lattice of rank 2. (cid:4)
Inparticular,ifanevenlatticeLisofrkL>12,t ≥1,andt ≥1,then,there
+ −
exists a lattice T such that L≃U ⊕T.
SupposeLisasublatticeofalatticeL′ withinclusionι:L֒→L′. Denoteby
L⊥ theorthogonalcomplementofLinL′. Theembedding ιiscalledprimitive,
L′
and L called a primitive sublattice of L′ if the finite abelian group L′/ι(L) is
torsion-free. In other words, if there is no overlattice that is an intermediate
lattice between L and L′ of rank equal to the rank of L. Note that if a direct
sumL ⊕L oflatticesisasublatticeofL′,then(L ⊕L )⊥ ≃(L )⊥ ⊕(L )⊥.
1 2 1 2 L′ 1 L′ 2 L′
For a K3 surface S, it is known that H2(S, Z) with the intersectionpairing
isisometrictotheK3 latticeΛ thatisevenunimodularofrank22andsigna-
K3
ture (3,19), being U⊕3⊕E⊕2, where E is the negative-definite even unimod-
8 8
ular lattice of rank 8. There is a standard exact sequence 0 → H1(S, O∗) →c1
S
H2(S, Z) → 0 of cohomologies so H1(S, O∗) is inherited a lattice structure
S
from H2(S, Z). Define the Picard lattice Pic(S) of a K3 surface S as the group
c (H1,1(S))∩H2(S, Z) with the lattice structure. The rank of Pic(S) is called
1
the Picard number, denoted by ρ(S). The Picard lattice is hyperbolic since a
K3 surface is complex and algebraic, and is known to be a primitive sublattice
of Λ under a marking H2(S, Z)→∼ Λ .
K3 K3
If an evenhyperbolic lattice L of rank rkL≤20 has |A | being square-free,
L
L is a primitive sublattice of Λ . Indeed, if so, for any lattice L ⊂ L′′ ⊂
K3
ΛK3 the general relation |AL| = [L′′ : L]2|AL′′| implies [L′′ : L] = 1 thus
L′′ ≃ L. Hence there is no overlattice of L. Moreover, by surjectivity of the
period mapping [2], there exists a K3 surface S such that Pic(S) ≃ L. Let
M ⊂ Λ be a hyperbolic sublattice. A K3 surface is M-polarised [3] if there
K3
exists a marking φ such that all divisors in φ−1(Cpol) are ample, where Cpol
M M
is the positive cone in MR minus d∈∆M Hd, ∆M = {d ∈ M|d.d = −2}, and
H ={x∈P(Λ )|x.d=0 for all d∈∆ }.
d K3 M
S
Nishiyama [9] gives the orthogonal complements of primitive sublattices of
type ADE of E in possible cases.
8
3
Lemma 2.1 (Lemma 4.3 [9]) There exist primitive embeddings of lattices of
type ADE into E with orthogonal complements given as follows. All the nota-
8
−4 1
tion follows Bourbaki except C6 := .
8 1 −2
(cid:18) (cid:19)
(A )⊥ ≃E (A )⊥ ≃E (A )⊥ ≃D (A )⊥ ≃A
(A1)E⊥8 ≃A7⊕A (A2)E⊥8 ≃C66 (A3)E⊥8 ≃(−58) 4 E8 4
(D5)E⊥8 ≃D1 2 (D6)E⊥8 ≃A8 (D7)E⊥8 ≃A⊕2 (D )⊥ ≃(−4)
(E4)⊥E8≃A4 (E5)⊥E8≃A3 6 E8 1 7 E8 (cid:4)
6 E8 2 7 E8 1
2.2 Brief toric geometry
Here we summarize toric divisors and ∆-regularity. Let M be a rank-3 lattice
with the standard basis {e , e , e }, N be its dual, and (, ) : M ×N → Z be
1 2 3
the natural pairing. From now on, a polytope means a 3-dimensional convex
hull of finitely-many points in Z3 embedded into R3, namely, integral, and the
origin is the only lattice point in the interior of it.
LetP∆ bethetoricvarietydefinedbyapolytope∆inM⊗ZR,towhichone
can associate a fan Σ whose one-dimensional cones, called one-simplices, are
∆
generatedbyprimitivelatticevectorseachofwhoseend-pointisanintersection
point of N and an edge of its polar dual ∆∗ defined by
∆∗ :={y ∈N ⊗R|(x, y)≥−1 for all x∈∆}.
LetP be the toricresolutionofsingularitiesinP . A toric divisor is adivisor
∆ ∆
admittingthetorusaction,identifiedwiththeclosureofthetorus-actionorbitof
aonef-simplex. LetDivT(P∆)bethesetofalltoricdivisorsDi =orb(R≥0vi), i=
1,...,s on P∆, where vi is a primitive lattice vector, then, −KPf = si=1 Di.
By a standard exact sequfence and a commutative diagram [10] ∆
P
f
0 → M → DivT(P∆) → Pic(P∆) → 0
↓ ↓
M → s ZfD → A (Pf) → 0
i=1 i 2 ∆
there is a system of linear equatLions among toric divisors
f
s
(e , v )D =0, j =1,2,3. (1)
j i i
i=1
X
Thus the solution set of the linear system is generated by (s − 3) elements
corresponding to divisors which generate the Picard group Pic(P ) of P .
∆ ∆
Definition 2.2 A polytope is reflexive if its polar dual is also integral. (cid:4)
f f
The importance of that we consider reflexive polytopes is by the following:
Theorem 2.1 c.f. [?] The following conditions are equivalent:
(1) A polytope ∆ is reflexive.
(2) The toric 3-fold P is Fano, in particular, general anticanonical members
∆
of P are Gorenstein K3. (cid:4)
∆
4
Forareflexivepolytope∆,denotebyF thefamilyofhypersurfacesparametrised
∆
by the complete anticanonicallinear system of P . Note that general members
∆
in F are Gorenstein K3 due to the previous theorem so that they are bira-
∆
tionally equivalent to K3 surfaces by the existence of crepant resolution. Thus
we call the family F a family of K3 surfaces.
∆
We recall from §3 of [?] the notion of ∆-regularity.
Definition 2.3 Let F be a Laurent polynomial defining a hypersurface Z ,
F
whose Newton polytope is a polytope ∆. The hypersurface Z is called ∆-
F
regular if for every face Γ of ∆, the corresponding affine stratum Z of Z is
F,Γ F
either empty or a smooth subvariety of codimension 1 in the torus T that is
Γ
contained in the affine variety associated to Γ. (cid:4)
Itisshownin[?]that∆-regularityisageneralcondition,andsingularitiesof
all ∆-regular members are simultaneously resolved by a toric desingularization
of P . From now on, suppose a polytope ∆ is reflexive and S is a ∆-regular
∆
member whose minimal model S is obtained by a toric resolution.
Definition 2.4 For a restriction r : P → S, let r : H1,1(P ) → H1,1(S) be
e ∆ ∗ ∆
theinducedmapping. DefinealatticeL (S):=r (H1,1(P ))∩H2(S, Z)ofthe
D ∗ ∆
intersection of the image of r and Hf2(S, Ze), and its orthofgonal compleement
∗
L (S):=L (S)⊥ in H2(S, Z). (cid:4) e f e
0 D H2(Se,Z)
e
Iet is knowne[6]that ρ(S) anderkL0(S) only depend onthe number oflattice
points in edges of ∆ and ∆∗. Thus we define the Picard number ρ(∆):=ρ(S),
andthe rankrkL :=rkeL (S)associaetedto ∆. Moreprecisely,denote by Γ∗
0,∆ 0
in ∆∗ the dual face to a face Γ of ∆, and l∗(Γ) is the number of lattice poinets
in the interior of Γ, and ∆[1] ethe set of edges in ∆. Let s be the number of
one-simplices of Σ . Then
∆
rkL0,∆ = l∗(Γ)l∗(Γ∗)=rkL0,∆∗, (2)
Γ∈X∆[1]
ρ(∆) = s−3+rkL , (3)
0,∆
ρ(∆)+ρ(∆∗) = 20+rkL . (4)
0,∆
If l∗(Γ∗)=n and l∗(Γ)=m for an edge Γ of ∆, there is a singularityof type
Γ Γ
A with multiplicity m +1 on an affine variety associated to Γ.
nΓ+1 Γ
As we will see later, we only need formulas when rkL = 0 for the inter-
0,∆
section numbers of the divisors {r D } in H2(S, Z) given as follows.
∗ i
r D2 =r D .r D = 2l∗(vi∗)−2 eif vi is a vertex of ∆∗, (5)
∗ i ∗ i ∗ i
(−2 otherwise.
If end-points of v and v are on the edge Γ∗ of ∆∗, then
i j ij
1 if v and v are next to each other,
i j
r∗Di.r∗Dj = l∗(Γij)+1 if l∗(Γ∗ij)=0, and vi, vj are both vertices,(6)
0 otherwise.
ThePicardlatticesofthe minimalmodels ofany∆-regularmembers,which
are generated by components of restricted toric divisors, are isometric. Define
5
the Picard lattice Pic(∆) of the family F as the Picard lattice of the minimal
∆
model of a ∆-regular member with rank ρ(∆). The orthogonal complement
T(∆)=Pic(∆)⊥ is called the transcendental lattice of F .
ΛK3 ∆
2.3 Mirrors
We define the polytope- and lattice-mirror theories.
2.3.1 Polytope Mirror
We focus on polytopes that “represent” the anticanonical members in a toric
variety as is seen in subsection 2.2.
Definition 2.5 A pair (∆ , ∆ ) of reflexive polytopes or a pair (F , F )
1 2 ∆1 ∆2
of families of K3 surfaces associated to ∆ and ∆ is called polytope mirror
1 2
symmetric if an isometry ∆ ≃∆∗ holds. (cid:4)
1 2
2.3.2 Lattice Mirror
For a K3 surface S, Pic(S) is the Picard lattice, and T(S) = Pic(S)⊥ is the
ΛK3
transcendentallattice. AmirrorforfamilyofM-polarisedK3surfacesisdefined
whenM isasublattice ofΛ ingeneral[3]. Here,wedealwiththemoststrict
K3
case, namely, mirror for K3 surfaces with Picard lattice as their polarisation.
Definition 2.6 (1) A pair (S, S′) of K3 surfaces is called lattice mirror sym-
metric if an isometry T(S)≃Pic(S′)⊕U holds.
(2) A pair (F, F′) of families whose general members are Gorenstein K3 sur-
faces is lattice mirror symmetric if there exist general members S ∈ F and
S′ ∈F′ pair of whose minimal models is lattice mirror symmetric. (cid:4)
Note thata lattice mirrorpair (S, S′)of K3surfacessatisfies,by definition,
rkPic(S′)+2=rkT(S)=22−rkPic(S), thus
ρ(S′)+ρ(S)=20. (7)
2.4 Bimodular singularities and the transpose duality
BeingclassifiedbyArnold[1]in1970’s,bimodularsingularitieshavetwospecific
classes: quadrilateralandexceptional. Quadrilateralbimodularsingularitiesare
6 in number with exceptional divisor of type I∗, whilst exceptional are 14 in
0
numberwithexceptionaldivisoroftypeII∗, III∗ orIV∗ inKodaira’snotation.
Anon-degeneratepolynomialf inthreevariablesiscalledinvertibleiff has
three terms f = 3j=1xa1jya2jza3j such that its matrix Af := (aij)1≤i,j≤3 of
exponents is invertible in GL (Q).
3
P
Definition 2.7 c.f. [4] Let B = (0,(f = 0)) and B′ = (0,(f′ = 0)) be germs
of singularities in C3. A pair (B,B′) of singularities is called transpose dual if
the following three conditions are satisfied.
(1) Defining polynomials f and f′ are invertible.
(2) MatricesAf andAf′ ofexponentsoff andf′ aretransposetoeachother.
6
(3) f (resp. f′) is compactified to a four-term polynomial F (resp. F′) in
|−KP(a)| (resp. |−KP(b)|), where P(a) (resp. P(b)) is the 3-dimensional
weightedprojectivespacewhosegeneralmembersareGorensteinK3with
weight a (resp. b) out of the list of 95 weights classified by [12][5][11]. (cid:4)
Condition (1) and (2) is said that they are Berglund–Hu¨bsch mirror sym-
metric. Ebeling-Ploog [4] show that there are 16 transpose-dual pairs among
quadrilateral and exceptional bimodular singularities, exceptional unimodular
singularities, and the singularities E , E , X , and Z .
25 30 2,0 2,0
3 The transpose dual and the lattice mirror
For a transpose-dual pair (B,B′) of defining polynomial f (resp. f′) being
compactified to a polynomial F (resp. F′), consider the Newton polytope ∆
F
(resp. ∆F′) of F (resp. F′) all of whose corresponding monomials are fixed by
an automorphism action on (F =0) (resp. (F′ =0)).
Remark 1 AcompactifiedmemberF tof doesnotalwaysdefineaGorenstein
K3 surface because ∆ may not be reflexive.
F
However,the Newton polytopes are extended to be reflexive and dual.
Theorem 3.1 [7] For each transpose-dual pair (B,B′), there exists a reflexive
polytope ∆ such that ∆F ⊂∆ and ∆F′ ⊂∆∗. (cid:4)
Computing Picardlattices is generally difficult, but it seems possible for ∆-
regular members by subsection 2.2. Let us reformulate our problem.
Problem Forapolytope∆obtainedin[7],isapair(S, S′)ofminimalmodels
of ∆-regular S ∈F∆ and ∆∗-regular S′ ∈F∆∗ lattice mirror symmetric ?
e e
First we study the rank rkL .
0,∆
Lemma 3.1 The list of rkL for the reflexive polytope ∆ obtained in [7] is
0,∆
given in Table 2.
transpose-dual pair rkL transpose-dual pair rkL
0,∆ 0,∆
(Q , E ) 0 (Z , Q ) 2
12 18 17 2,0
(Z , E ) 0 (U , U ) 2
1,0 19 1,0 1,0
(E , E ) 0 (U , U ) 2
20 20 16 16
(Q , Z ) 0 (Q , Z ) 2
2,0 17 17 2,0
(E , Z ) 0 (W , W ) 3
25 19 1,0 1,0
(Q , E ) 0 (W , S ) 5
18 30 17 1,0
(Z , Z ) 1 (W , W ) 6
1,0 1,0 18 18
(Z , J ) 2 (S , X ) 6
13 3,0 17 2,0
Table 2: rkL
0,∆
Proof. The assertion follows from direct and case-by-case computation by
formula (2) in subsection 2.2.
7
1. (Q , E ) The polytope ∆ is given in subsection 4.7 of [7]. There is
12 18
no contribution to rkL since l∗(Γ) or l∗(Γ∗) is zero for any edge Γ. Thus,
0,∆
rkL =0 by formula (2).
0,∆
Similarforthecases(Z , E ), (E , E ), (Q , Z ), (E , Z ), (Q , E ).
1,0 19 20 20 2,0 17 25 19 18 30
2. (Z , J ) The polytope ∆ is given in subsection 4.1 of [7]. The only con-
13 3,0
tributiontorkL isbyanedgeΓbetweenvertices(0, 0, 1)and(−2, −6, −9),
0,∆
whosedualΓ∗isbetween(8, −1, −1)and(−1, 2, −1)sol∗(Γ)=1andl∗(Γ∗)=
2. Thus, rkL =1×2=2 by formula (2).
0,∆
Similarforthecases(Z ,Z ), (Z ,Q ), (U ,U ), (U ,U ), (Q ,Z ),
1,0 1,0 17 2,0 1,0 1,0 16 16 17 2,0
(W ,W ), (W ,S ), (W ,W ), (S ,X ). (cid:3)
1,0 1,0 17 1,0 18 18 17 2,0
Corollary 3.1 No∆-and∆∗-regularmembersfortranspose-dualpairs(Z ,J ),
13 3,0
(Z ,Z ), (Z ,Q ), (U ,U ), (U ,U ), (Q ,Z ), (W ,W ), (W ,S ),
1,0 1,0 17 2,0 1,0 1,0 16 16 17 2,0 1,0 1,0 17 1,0
(W ,W ), (S ,X ) admit a lattice mirror symmetry.
18 18 17 2,0
Proof. For each ∆ associated to the presented pairs, by formula (4),
ρ(∆)+ρ(∆∗)=20+rkL >20
0,∆
since rkL >0 by Lemma 3.1. Thus, the equation (7) does not hold. There-
0,∆
fore, ∆- and ∆∗-regular members do not admit a lattice mirror symmetry. (cid:3)
Corollary 3.2 The restriction mapping r : Pic(P ) → Pic(S), for ∆-regular
∗ ∆
S ∈F issurjectiveforthetranspose-dualpairs(Q ,E ), (Z ,E ), (E ,E ),
∆ 12 18 1,0 19 20 20
(Q2,0,Z17), (E25,Z19), (Q18,E30). f e
Proof. By Lemma 3.1, rkL = 0 for each case. By definition, rkL is
0,∆ 0,∆
equal to the rank of the orthogonal complement of r (H1,1(P )) in H2(S, Z).
∗ ∆
Thus, that rkL =0 means that r is surjective. (cid:3)
0,∆ ∗
f e
By Corollary 3.1, we may only focus on the transpose-dual pairs appearing
in Corollary 3.2, whose statement means moreoverthat Pic(∆) is generated by
restricted toric divisors generating Pic(P ), and analogous to ∆∗.
∆
Let AL denote the discriminant grofup, qL the quadratic form, and discrL
the discriminantofalattice L. Ifp=discrLisprime,thenA ≃Z/pZ. Before
L
stating our main results, note a fact in Proposition 1.6.1 in [8]. Suppose that
latticesS andT areprimitivelyembeddedintotheK3latticeΛ . IfA ≃A
K3 S T
and q =−q , then, it is determined that the orthogonal complement S⊥ in
S T ΛK3
Λ is T. And q =−q if and only if discrS =−discrT.
K3 S T
Proposition 3.1 The Picard lattice Pic(∆∗) for ∆ in Corollary 3.2 is as in
Table 3, where singularities in a row are transpose-dual. In each case, one gets
discrPic(∆)=−discr(U ⊕Pic(∆∗)), and APic(∆) ≃AU⊕Pic(∆∗).
8
−4 1
Denote C6 := .
8 1 −2
(cid:18) (cid:19)
Singularity ρ(∆∗) Pic(∆∗) Singularity
Q 4 U ⊕A E
12 2 18
Z 3 U ⊕A E
1,0 1 19
E 2 U E
20 20
Q 4 U ⊕C6 Z
2,0 8 17
E 3 U ⊕A Z
25 1 19
Q 4 U ⊕A E
18 2 30
Table 3: Pic(∆∗) for ∆ in Corollary 3.2
Proof.
1.Q and E The polytope ∆ is given in subsection 4.7 of [7] and the asso-
12 18
ciated toric 3-fold has 19 toric divisors D corresponding to the one-simplices
i
generated by vectors
m =(1, 2, 2) m =(0, 1, 1) m =(−8, −11, −9)
1 2 3
m =(1, −2, 0) m =(1, 1, 0) m =(−4, −5, −4)
4 5 6
m =(−7, −10, −8) m =(−6, −9, −7) m =(−5, −8, −6)
7 8 9
m =(−4, −7, −5) m =(−3, −6, −4) m =(−2, −5, −3)
10 11 12
m =(−1, −4, −2) m =(0, −3, −1) m =(1, 0, 1)
13 14 15
m =(−5, −7, −6) m =(−2, −3, −3) m =(1, −1, 0)
16 17 18
m =(1, 0, 0)
19
and by solving the linear system (1) : 19 (e , m )D = 0 (j = 1,2,3), we
i=1 j i i
get linear relations among toric divisors
P
D ∼ −9D +3D −D −2D −3D −4D −5D −6D −7D −8D
1 4 5 7 8 9 10 11 12 13 14
−5D +D +2D −5D −D ,
15 16 17 18 19
D ∼ D +10D −2D +2D +3D +4D +5D +6D +7D +8D
2 3 4 5 7 8 9 10 11 12 13
+9D +5D −D +6D +2D ,
14 15 17 18 19
D ∼ −2D −2D +D −2D −2D −2D −2D −2D −2D −2D
6 3 4 5 7 8 9 10 11 12 13
−2D −D −D −D .
14 15 16 18
So the set {D |i 6= 1,2,6} of toric divisors is linearly independent. Let L be
i
the lattice generated by the set {r D |i 6= 1,2,6} of their restrictions to a ∆-
∗ i
regular member. We shall check that L is primitively embedded into the K3
latticetoshowthatLisindeedthePicardlatticeofthefamilyF . Bycomputer
∆
calculationwithformulas(5)and(6),thedeterminantofanintersectionmatrix
9
of L is −3 since this matrix is given by
−2 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0
0 −2 0 0 0 0 0 0 0 0 1 1 0 0 1 0
 
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1
 1 0 0 −2 1 0 0 0 0 0 0 0 0 0 0 0 
 
 0 0 0 1 −2 1 0 0 0 0 0 0 0 0 0 0 
 
 0 0 0 0 1 −2 1 0 0 0 0 0 0 0 0 0 
 
 0 0 0 0 0 1 −2 1 0 0 0 0 0 0 0 0 
 
 0 0 0 0 0 0 1 −2 1 0 0 0 0 0 0 0 
 ,
 0 0 0 0 0 0 0 1 −2 1 0 0 0 0 0 0 
 
 0 0 0 0 0 0 0 0 1 −2 1 0 0 0 0 0 
 
 0 1 0 0 0 0 0 0 0 1 −2 0 0 0 0 0 
 
 0 1 0 0 0 0 0 0 0 0 0 −2 0 0 0 0 
 
 1 0 0 0 0 0 0 0 0 0 0 0 −2 1 0 0 
 
 0 0 1 0 0 0 0 0 0 0 0 0 1 −2 0 0 
 
 0 1 0 0 0 0 0 0 0 0 0 0 0 0 −2 1 
 
 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 −2 
 
 
Since the discriminant of L is −3 that is square-free,there exists no overlattice
of L ; indeed, if H ⊂ Λ were an overlattice of L, then, by the standard
K3
relation −3 = [H : L]2discrH, we get [H : L] = 1 and discrH = −3 so that
L≃H. Hence, LisprimitivelyembeddedintotheK3lattice. Byconstruction,
L is indeed the Picard lattice of the family F .
∆
The dual polytope ∆∗ associates a toric 3-fold with 7 toric divisors D′ cor-
i
responding to the one-simplices generated by vectors
v =(1, 1, −2) v =(1, −2, 1) v =(2, −3, 2) v =(−1, 0, 1)
1 2 3 4
v =(−1, 0, 0) v =(1, 0, −1) v =(1, −1, 0)
5 6 7
and by solving the linear system (1) : 7 (e , m )D′ = 0 (j = 1,2,3), we
i=1 j i i
get linear relations among toric divisors
P
D′ ∼2D′ +3D′ +D′, D′ ∼3D′ +4D′ +D′ +2D′, D′ ∼D′.
1 2 3 7 4 2 3 6 7 5 3
Thus the set {D′|i 6= 1,4,5} of toric divisors is linearly independent. Let L′
i
be the lattice generated by the set {r D′|i 6= 1,4,5} of their restrictions to a
∗ i
∆∗-regularmember. By formulas(5) and(6), anintersectionmatrix associated
to L′ is given by
−2 1 0 1
1 0 0 0
 
0 0 −2 1
 1 0 1 −2 
 
 
that is equivalent to U ⊕A by re-taking the generators as
2
{r D′ +r D′, r D′, r D′, r D′ +r D′}.
∗ 2 ∗ 3 ∗ 3 ∗ 6 ∗ 3 ∗ 7
Since the discriminantofL′ is −3thatis square-free,there existsno overlattice
of L′ ; indeed, if H′ ⊂ Λ were an overlattice of L′, then, by the standard
K3
relation −3 = [H′ : L′]2discrH′, we get [H′ : L′] = 1 and discrH′ = −3 so
that H′ ≃ L′. Hence, L′ is primitively embedded into the K3 lattice. By
10