Structure of stable degeneration of K3 surfaces into pairs of rational elliptic surfaces
HJE
Structure of stable degeneration of K3 surfaces into pairs of rational elliptic surfaces
0 11 Oho , Tsukuba, Ibaraki 3050801 , Japan
1 KEK Theory Center, Institute of Particle and Nuclear Studies , KEK
Ftheory/heterotic duality is formulated in the stable degeneration limit of a bration on the Ftheory side. In this note, we analyze the structure of the stable
Di erential and Algebraic Geometry; FTheory; Gauge Symmetry; Super

K3
degeneration limit.
We discuss whether stable degeneration exists for pairs of rational
elliptic surfaces. We demonstrate that, when two rational elliptic surfaces have an
identical complex structure, stable degeneration always exists. We provide an equation that
systematically describes the stable degeneration of a K3 surface into a pair of isomorphic
rational elliptic surfaces. When two rational elliptic surfaces have di erent complex
structures, whether their sum glued along a smooth
ber admits deformation to a K3 surface
can be determined by studying the structure of the K3 lattice. We investigate the lattice
theoretic condition to determine whether a deformation to a K3 surface exists for pairs of
extremal rational elliptic surfaces. In addition, we discuss the con gurations of singular
bers under stable degeneration.
The sum of two isomorphic rational elliptic surfaces glued together admits a
deformation to a K3 surface, the singular bers of which are twice that of the rational elliptic
surface. For special situations, singular bers of the resulting K3 surface collide and they
are enhanced to a
ber of another type. Some K3 surfaces become attractive in these
situations. We determine the complex structures and the Weierstrass forms of these
attractive K3 surfaces. We also deduce the gauge groups in Ftheory compacti cations on
these attractive K3 surfaces times a K3. E6, E7, E8, SU(5), and SO(10) gauge groups arise
in these compacti cations.
string Vacua
2
Stable degeneration of K3 to a pair of isomorphic rational elliptic sur
Equation for the degeneration of K3 to a pair of isomorphic rational elliptic
surfaces
Extremal rational elliptic surfaces
Example of generic deformation using an extremal rational elliptic surface
Attractive K3 surfaces as a special deformation of two extremal rational
elliptic surfaces glued together
2.4.1
Complex structures of attractive K3 surfaces, and gauge groups in
2.4.2
2.4.3
Ftheory compacti cations
Weierstrass equation
Anomaly cancellation condition
3
Stable degeneration of K3 to a pair of nonisomorphic rational elliptic
1 Introduction
faces
2.1
2.2
2.3
2.4
surfaces
3.1
3.2
4
Conclusion
Lattice condition for stable degeneration limit
Pairs of extremal rational elliptic surfaces
1
Introduction
The Ftheory approach to particle physics model building has several advantages. It
naturally realizes SU(5) grand uni ed theories with matter in spinor representations of SO(10).
In contrast to Dbrane models, there is no di culty in generating uptype Yukawa
couplings. Furthermore, it can evade the problem of weakly coupled heterotic string theory
addressed in [1]. Recent studies on Ftheory model building [2{5] have emphasized the use
of local models. However, in order to address the issue of gravity such as in ation, a global
model of compacti cation need to be considered eventually. In particular, many insights
can be gained by the duality between heterotic string and Ftheory [6{10], which states the
equivalence between the former compacti ed on an elliptically
bered CY nfold and the
latter on a K3 bered CY (n + 1)fold in the stable degeneration limit [10, 11].1 The aim
of the present paper is to develop a systematic study of the process of stable degeneration
of a K3 surface for a pair of rational elliptic surfaces.
1For recent discussion of the stable degeneration limit of Ftheory and Ftheory/heterotic duality, see,
for example, [12{16].
{ 1 {
K3 surface stably degenerates into two rational elliptic surfaces in two distinct ways:
i) K3 surface splits into two rational elliptic surfaces with an identical complex structure
ii) K3 surface splits into two rational elliptic surfaces with di erent complex structures.
We discuss these cases separately, in section 2 and section 3, respectively.
We demonstrate in section 2.1 that, when a pair of rational elliptic surfaces are
isomorphic, stable degeneration can be described by a systematic equation. We analyze the
geometry of stable degeneration of the rst kind i) using this equation. We determine that
given any pair of isomorphic rational elliptic surfaces, there is some K3 surface that stably
degenerates into the pair.
However, it is considerably di cult to describe stable degeneration by an equation
when two rational elliptic surfaces have di erent complex structures. Moreover, a pair
of nonisomorphic rational elliptic surfaces glued together along a smooth
ber does not
necessarily admit a deformation to a K3 surface. Complex structures and con gurations of
singular bers are classi ed for a speci c class of rational elliptic surfaces, called extremal
rational elliptic surfaces. We focus on the extremal rational elliptic surfaces to analyze
the stable degeneration of the second kind ii). We determine whether stable degeneration
exists for pairs of these rational elliptic surfaces. We use the lattice theoretic approach to
analyze this process.
We also study the con guration of singular bers under the stable degeneration limit.
In Ftheory, nonAbelian gauge symmetries on the 7branes are in correspondence with the
types of singular
bers. Therefore, analyzing the con gurations of singular
bers under
the stable degeneration limit is of physical interest.
The outline of this study is as follows. In section 2, we discuss the stable degeneration
limit where a K3 surface degenerates into two isomorphic rational elliptic surfaces. We
provide an equation that systematically describes this process. We determine that any
pair of isomorphic rational elliptic surfaces glued together deforms to a K3 surface, which
is a double cover of P2 rami ed over a sextic curve. Moreover, we discuss the con gurations
of singular
bers under the degeneration. Furthermore, we review the properties of the
extremal rational elliptic surfaces. We discuss some examples of stable degeneration using
an extremal rational elliptic surface. The sum of two isomorphic rational elliptic surfaces
glued together admits a deformation to a K3 surface, the singular bers of which are twice
that of the rational elliptic surface. For special situations, two bers of the same type of
the resulting K3 surface collide, and they are enhanced to a
ber of another type. These
situations can be considered as special cases of stable degeneration. For such situations,
some K3 surfaces become attractive. In section 2.4, we determine the complex structures
of these attractive K3 surfaces and their Weierstrass forms.
We also deduce the non
Abelian gauge symmetries that form on the 7branes in Ftheory compacti cations on these
attractive K3 surfaces times a K3. E6, E7, E8, SU(5), and SO(10) gauge groups arise in
these models. In section 3, we investigate the stable degeneration limit where a K3 surface
degenerates into two nonisomorphic rational elliptic surfaces. Whether such a degeneration
exists can be determined by studying the lattice structure of the second integral cohomology
{ 2 {
surfaces
liptic surfaces
group H2(S; Z) of a K3 surface S. We obtain the lattice condition under which two
nonisomorphic rational elliptic surfaces glued together admits a deformation to a K3 surface.
We determine whether a deformation to a K3 surface exists for pairs of extremal rational
elliptic surfaces. We study the con guration of singular
bers under stable degeneration
using the lattice theoretic argument. We state the concluding remarks in section 4.
2
Stable degeneration of K3 to a pair of isomorphic rational elliptic
2.1
Equation for the degeneration of K3 to a pair of isomorphic rational
elwith a section. The base space of a rational elliptic surface is isomorphic to P1 by Luroth's
theorem, and it is known that every rational elliptic surface with a section is the blowup
of P2 in the nine base points of a cubic pencil [17]. The Picard number of a rational elliptic
surface is 10, and the rank of the MordellWeil group ranges from 0 to 8:
0
rk MW
8:
(2.1)
A generic rational elliptic surface has MordellWeil rank 8. Rational elliptic surfaces with
MordellWeil rank 0 are called extremal rational elliptic surfaces. For extremal rational
elliptic surfaces, the rank of the singularity type is 8, which is the highest for a rational
elliptic surface.
We will review the properties of extremal rational elliptic surfaces in
section 2.2. In section 3, we focus on extremal rational elliptic surfaces to discuss the
stable degeneration of a K3 surface into a pair of nonisomorphic rational elliptic surfaces.
When two rational elliptic surfaces have isomorphic smooth elliptic bers, we can glue
two rational elliptic surfaces in the following fashion: we choose a point in base P1 of each
rational elliptic surface over which the
ber is smooth and isomorphic, and we glue two
rational elliptic surfaces by identifying the isomorphic smooth bers over the chosen points.
As described in section 3, when a certain lattice theoretic condition is satis ed, the sum of
two (not necessarily isomorphic) rational elliptic surfaces glued together can be deformed
to a K3 surface.
In this section, we particularly consider the case wherein two rational elliptic surfaces
that are glued together are isomorphic. For this particular case, we explicitly provide an
equation that describes the stable degeneration of a K3 surface into the pair of isomorphic
{ 3 {
rational elliptic surfaces glued together. As stated above, a rational elliptic surface X is
the blowup of P2 in the base points of a cubic pencil; we denote the cubic pencil of a
rational elliptic surface X as f . The double cover of P2 rami ed along a degree 6 curve
is, in general, a K3 surface. We particularly consider double covers of P2 rami ed over a
degree 6 curve given by the following form of equations:
where g is a polynomial of degree 3 and g is the cubic pencil of the same type as the pencil
f , but the ratio of the coe cients of the pencil g is generally di erent from the ratio of the
coe cients of the pencil f . For this situation, K3 surface (2.2) is elliptically
bered. To
HJEP03(218)45
be explicit, when the cubic pencil f is given by
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
Each of the above two equations in (2.5) describes a rational elliptic surface given by cubic
pencil f ; therefore, when a cubic polynomial g goes to cubic pencil f , the K3 surface (2.2)
splits into two isomorphic rational elliptic surfaces, each given by the cubic pencil f . This
is the stable degeneration limit of the K3 surface (2.2) splitting into two copies of rational
elliptic surfaces X. Thus, we conclude that two isomorphic rational elliptic surfaces glued
along an isomorphic smooth
ber always admit a deformation to a K3 surface.
We only consider rational elliptic surfaces with a global section in this note; therefore,
they admit transformation into the Weierstrass form. The coe cients of the Weierstrass
form depend on the coordinate of the base P1. We denote the homogeneous coordinate
of the base P
1 as [u : v]. In terms of the coordinate [u : v], the stable degeneration of a
K3 surface (2.2) splitting into two isomorphic rational elliptic surfaces is described by the
following equation:
2 = u2 + 2kuv + v2:
k in equation (2.6) denotes a parameter of deformation. k varies along the deformation,
and when k assumes the values
equation (2.6) splits into linear factors. This occurs when a K3 surface splits into two
isomorphic rational elliptic surfaces.
where a; b are coe cients of the pencil f , we choose the cubic pencil g as follows:
c; d are coe cients of the pencil g, and the ratio [c : d] is generally di erent from the ratio
[a : b]. For the limit at which polynomial g goes to cubic pencil f (i.e. for the limit at
which ratio [c : d] goes to the ratio [a : b]), equation (2.2) is split into the following two
equations:
= f
=
f:
2 = f g;
f = a h1 + b h2;
g = c h1 + d h2:
k =
1;
{ 4 {
We observe from equation (2.6) that the K3 surface (2.2) is the quadratic base change
of the rational elliptic surface (2.5) into which the K3 surface splits in the stable
degeneration limit, when g is the cubic pencil of the same type as the pencil f . In other words,
the Weierstrass equation of the K3 surface is obtained when some appropriate quadratic
equations are substituted into variables u; v in the coe cients of the Weierstrass form of a
rational elliptic surface. Thus, the generic K3 surface (2.2) that results from the
deformation of two isomorphic rational elliptic surfaces (2.5) glued together has twice the number
of singular bers as a rational elliptic surface.
For special situations, singular bers of the same type of the K3 surface, that is
obtained as the quadratic base change of a rational elliptic surface, collide and they are
HJEP03(218)45
enhanced to a ber of another type. We discuss these situations in section 2.4.
The aforementioned argument applies to the deformation of every pair of two
isomorphic rational elliptic surfaces with a global section glued along smooth ber to a K3 surface.
2.2
Extremal rational elliptic surfaces
We summarize the properties of extremal rational elliptic surfaces. In section 2.3, we
discuss the quadratic base change of extremal rational elliptic surfaces that rami es only
over smooth
bers. In section 2.4, we discuss the limits at which singular
bers of the
same type collide in the quadratic base change of extremal rational elliptic surfaces. We
discuss the structures of attractive K3 surfaces that result from the quadratic base change
of extremal rational elliptic surfaces in section 2.4.
Extremal rational elliptic surfaces have the singularity type of rank 8, and the
MordellWeil groups only have torsion parts. Singular
ber types of the extremal rational
elliptic surfaces are classi ed, and the complex structure of an extremal rational elliptic
surface is uniquely determined by the ber type, except for the surfaces with the ber type
[I0 ; I0 ] [18]. Jinvariant of bers of an extremal rational elliptic surface with the ber type
[I0 ; I0 ] is constant over the base P1, and the complex structure of a rational elliptic surface
with the ber type [I0 ; I0 ] depends on the value of j. The complex structure of an extremal
rational elliptic surface with the ber type [I0 ; I0 ] is xed when the jinvariant of the ber
is chosen.
Provided these facts, in this note, we denote an extremal rational elliptic surface using
its ber type as the subscript. For example, an extremal rational elliptic surface with the
ber type [IV; IV ] is denoted as X[IV; IV ]. We simply use n to represent the In
ber, and
m to represent the Im
ber. Therefore, an extremal rational elliptic surface with singular
bers of type IV , I3, and I1 is denoted as X[IV ; 3;1]. We denote a surface with the ber
type [I0 ; I0 ] as X[0 ; 0 ](j), because the complex structure of such a surface depends on
the jinvariant of bers. We list the con gurations of singular ber types of the extremal
rational elliptic surfaces in table 1. The Weierstrass forms of the extremal rational elliptic
surfaces were also derived in [18]. We include the Weierstrass forms of the extremal rational
elliptic surfaces in table 1.
The cubic pencils for all the extremal rational elliptic surfaces, except the surfaces
X[II; II ], X[III; III ], X[IV; IV ], X[0 ; 0 ](j), were obtained in [19]. The cubic pencil for the
{ 5 {
Extremal rational
elliptic surface
III , III
IV , IV
I0 , I0
II I1 I1
III I2 I1
IV I3 I1
I4 I1 I1
I2 I2 I2
I1 I4 I1
I9 I1 I1 I1
I8 I2 I1 I1
I6 I3 I2 I1
I5 I5 I1 I1
3u4
uv3
2u5v
v5(u v)
v4(16u2
72uv + 54v2)
uv3(2u2
9v2)
(u v)3(u3 + v3)
u(u 2v)3(2u2
9v2)
2(u6 + 36u3v3 + 216v6)
2u6 + 12u4v2 + 15u2v4
2v6
2u6 + 12u5v + 12u4v2
14u3v3
+3u2v4
6uv5 + 2v6
2(u6
18u5v + 75u4v2
+75u2v4 + 18uv5 + v6)
2u6
3u4v2
3u2v4 + 2v6
2(u6 + 20u3v3
8v6)
2 = x3 + a4x + a6. [u : v] is the homogeneous coordinate on the base P1. For surface
4s3 + 27t2 6= 0. jinvariant of bers of surface X[0 ; 0 ](j) is a function of s; t.
X[0 ; 0 ](j), s; t in coe cients a4; a6 of the Weierstrass form are complex numbers, s; t 2 C, with
extremal rational elliptic surface X[IV; IV ] is given by
a yz(y + z) + b x3:
[x : y : z] represents the homogeneous coordinates on P2, and [a : b] represents the
homogeneous coordinate on P1. The cubic pencils of extremal rational elliptic surfaces are listed
in table 2.
2.3
Example of generic deformation using an extremal rational elliptic surface
We discuss an example of results in section 2.1 using an extremal rational elliptic surface.
We consider the surface X[IV; IV ]. As stated in section 2.2, the cubic pencil of surface
X[IV; IV ] is given by
(2.8)
(2.9)
f = a yz(y + z) + b x3:
{ 6 {
a f(2x + z)(xz + y2) + x3g + b x(xz + y2)
a (x + y)(xy + z2) + b x2y
a xf(x + y)z + y2g + b (x + y)z2
a (x2y + y2z + z2x
3xyz) + b xyz
a fx(xz
y2) + y2zg + b (y + 2z)(xz
y2 + z2)
a z(x2 + xy + xz + y2) + b ( x2z + xyz + y3)
a yz(x + y + z) + b x(x + y)(x + z)
a (x
y)(xy
z2) + b xy(x + y
2z)
a (x3 + y3 + z3) + b xyz
(2.10)
(2.11)
g to be the pencil of the same type as the cubic pencil f :
g = c yz(y + z) + d x3:
We x c; d to be nonzero constants. We choose c; d so that the ratio [c : d] is generally
di erent from the ratio [a : b] of the coe cients a; b in the pencil f . For the cubic pencil
f (2.9) and cubic pencil g (2.10), equation
generically provides a K3 surface with two type IV
bers and two type IV
bers. For the
limit at which ratio [c : d] goes to [a : b], the cubic pencil g coincides with the cubic pencil
f . In this stable degeneration limit, equation (2.11) is split into linear factors; accordingly,
the K3 surface (2.11) splits into two extremal rational elliptic surfaces X[IV; IV ].
We stated in section 2.1 that gluing together two isomorphic rational elliptic surfaces,
each given by cubic pencil f , to obtain a K3 surface (2.2) is equivalent to the quadratic
base change of a rational elliptic surface over P1, when the cubic pencil g is chosen to be
the same type as the pencil f . We consider, as an example, the case wherein two extremal
rational elliptic surfaces X[IV; IV ] are glued together to form a K3 surface, to demonstrate
this explicitly. We discuss the generic situation in which the quadratic base change rami es
2 = f g
{ 7 {
over smooth bers. As given in table 1, the Weierstrass form of X[IV; IV ] is given by
Gluing together two X[IV; IV ]s and deforming the resulting surface to a K3 surface is
equivalent to substituting the following quadratic equations into variables u; v:
u =
v =
1u~2 + 2u~v~ + 3v~2 =
4u~2 + 5u~v~ + 6v~2 =
1(u~
4(u~
1v~)(u~
3v~)(u~
2v~)
4v~):
i, i = 1;
; 6, are some constants, and the quadratic terms are split into linear factors
on the right extreme hand sides in equation (2.13). (In (2.13), we assume that 1 6=
and 3 6= 4.) The resulting K3 surface has the following Weierstrass form:
this kind, in which singular bers of the same type collide in the quadratic base change of
an extremal rational elliptic surface, will be discussed in section 2.4.
2.4
Attractive K3 surfaces as a special deformation of two extremal rational
elliptic surfaces glued together
2.4.1
Complex structures of attractive K3 surfaces, and gauge groups in Ftheory compacti cations
In section 2.1, we mainly discussed the quadratic base change whose rami cation occurs
over smooth
bers.
A smooth
ber remains smooth after the quadratic base change.
When the quadratic base change is unrami ed over a singular ber, we obtain two copies
of that ber.
As we saw in section 2.1, the quadratic base change of a rational elliptic surface to
obtain a K3 surface is the reverse of a process in which a K3 surface splits into two
isomorphic rational elliptic surfaces under stable degeneration. A quadratic base change
generically rami es only over smooth
bers; we obtain two copies of the singular ber for
each singular
ber after the quadratic base change. Thus, the resulting K3 surface has
twice as many singular
bers as the rational elliptic surface for such a generic situation.
We consider the special situation of the quadratic base change, where singular bers of the
same type collide and they are enhanced to a singular ber of another type. We particularly
consider the examples in which the resulting K3 surfaces after the quadratic base change
are enhanced to attractive K3 surfaces.2 We show in table 3 the resulting ber type after
the collision of two singular bers of the same type [20].
2Following the convention of the term in [21], we refer to a K3 surface with the Picard number 20 as an
attractive K3 surface in this study.
{ 8 {
Original ber type of
a pair of identical singular bers
Resulting ber type
In
II
III
IV
I2n
IV
I
0
IV
We consider stable degeneration in which two isomorphic extremal rational elliptic
surfaces X[II; II ] are glued together. When two type II bers collide in the quadratic base
change, we nd from table 3 that the resulting ber has type IV . The resulting K3 surface
has singular
bers of types II , II , and IV . The corresponding ADE type is E2
8
A2.
This is an extremal K3 surface. An extremal K3 surface is an attractive elliptic K3 surface
with a global section, with the MordellWeil group of rank 0. The extremal K3 surface with
ADE type E2
8
A2 is discussed in [22]. We next consider the gluing of two isomorphic
extremal rational elliptic surfaces X[III; III ], in which two type III
bers collide. The
resulting K3 surface has type III , III , and I0
bers. The corresponding ADE type is
D4. This is also an extremal K3 surface. This K3 surface is discussed in [23] as the
Jacobian
bration of some K3 genusone bration without a section. For the gluing of two
isomorphic extremal rational elliptic surfaces X[IV; IV ], when two type IV
bers collide,
the resulting
ber has type IV . Therefore, the resulting K3 surface has three type IV
singular bers. The corresponding ADE type is E3. This is an extremal K3 surface, and
6
this K3 surface is discussed in [24] as the Jacobian
bration of an attractive K3 genusone
bration without a section.
For the gluing of two isomorphic extremal rational elliptic surfaces X[II ; 1;1], we
consider the situation in which two pairs of type I1 bers collide in the quadratic base change.
The resulting
II , and two
bers have type I2; the resulting K3 surface has two singular
bers of type
bers of type I2. When we consider the limit of the quadratic base change
of extremal rational elliptic surface X[III ; 2;1] at which two type I2
bers and two type I1
bers collide, the resulting K3 surface has bers of types III , III , I4, and I2. The limit
of quadratic base change of extremal rational elliptic surface X[IV ; 3;1], at which two bers
of type I3 and two
bers of type I1 collide, gives K3 surface with singular bers of types
IV , IV , I6, and I2.
For the gluing of two extremal rational elliptic surfaces X[4 ; 1;1], when we consider the
limit of the quadratic base change at which two pairs of type I1
bers collide, the resulting
K3 surface has two singular bers of type I4 and two singular bers of type I2. For extremal
rational elliptic surface X[2 ; 2;2], when we consider the limit of the quadratic base change
at which two pairs of type I2
bers collide, the resulting K3 surface has two type I2
bers
and two type I4
bers. For the gluing of two extremal rational elliptic surface X[1 ; 4;1],
when we consider the limit of the quadratic base change at which two type I4 bers and
two type I1 bers collide, the resulting K3 surface has type I1 , I1 , I8, and I2 bers.
{ 9 {
HJEP03(218)45
For extremal rational elliptic surface X[5;5;1;1], when we consider the limit of the
quadratic base change at which two
bers of type I5 and two
bers of type I1 collide,
the resulting K3 surface has 1 type I10 ber, 2 type I5 bers, 1 type I2 ber, and 2 type
I1 bers.
From ADE types of the 10 K3 surfaces that we obtained above as the quadratic
base change of extremal rational elliptic surfaces, we conclude that they are extremal K3
surfaces. ADE types of the extremal K3 surfaces and the corresponding complex structures
were classi ed in [25]. MordellWeil groups of extremal K3 surfaces were also derived in [25].
Using table 2 in [25], we can deduce the complex structures of the 10 extremal K3 surfaces
from their ADE types.
The complex structure of an attractive K3 surface S is speci ed by the transcendental
lattice T (S), which is de ned to be the orthogonal complement of the NeronSeveri lattice in
the K3 lattice H2(S; Z) [
26
]. Therefore, we represent the complex structure of an attractive
K3 surface by the intersection matrix of the transcendental lattice in this study. For an
attractive K3 surface, the transcendental lattice is a 2
2 integral, symmetric, positive
de nite even lattice. See section 4 in [24] for a review of the correspondence of the complex
structures of attractive K3 surfaces and the transcendental lattice.
We list the
ber types, corresponding ADE types, the complex structures and the
MordellWeil groups of the 10 extremal K3 surfaces, that we obtained above as
deformation of two isomorphic extremal rational elliptic surfaces glued together, in table 4. The
global structures of the nonAbelian gauge symmetries3 that arise on the 7branes in
Ftheory compacti cations on the 10 extremal K3 surfaces times a K3 surface are also shown
in table 4. Extremal K3 surface has the MordellWeil rank 0, therefore these Ftheory
compacti cations do not have U(1) gauge eld.
2.4.2
Weierstrass equation
We obtained 10 extremal K3 surfaces in section 2.4.1 as the limit of the quadratic base
change4 of extremal rational elliptic surfaces, at which singular bers of the same type
collide. Therefore, by substituting appropriate quadratic polynomials into u; v in the
Weierstrass forms of extremal rational elliptic surfaces, listed in table 1 in section 2.2, we can
deduce the Weierstrass forms of the 10 extremal K3 surfaces, obtained in section 2.4.1.
As an example, extremal K3 surface whose transcendental lattice has the intersection
matrix
60 02 , with ADE type E2
6
A5
A1, is obtained via the quadratic base change of
extremal rational elliptic surface X[IV ; 3;1], in which two
bers of type I3 and two bers
of type I1 collide. The Weierstrass form of extremal rational elliptic surface X[IV ; 3;1] is
given by
y2 = x3 + v3(24u
27v)x + v4(16u2
72uv + 54v2):
The discriminant of the Weierstrass form (2.15) is [18]
on the 7branes.
3See [
8, 27
] for the correspondence of the types of the singular bers and the nonAbelian gauge groups
4The relationship of K3 surfaces and rational elliptic surfaces via base change is also discussed in [28].
(2.15)
(2.16)
Singular bers
Complex Str. MordellWeil group
II II IV
III III I0
D4
E3
6
E2 A2
8 1
III III I4 I2
E2 A3 A1
7
groups that form on the 7branes.
Type IV
ber is at [u : v] = [1 : 0], type I3 ber is at [u : v] = [0 : 1] and type I1 ber is
at [u : v] = [1 : 1]. We consider the following substitutions for u; v in the Weierstrass form
of extremal rational elliptic surface X[IV ; 3;1]:
u = u~
2
v = 2u~v~
2
v~ :
This gives the limit of the quadratic base change at which two bers of type I3 collide at
[u : v] = [0 : 1] and two bers of type I1 collide at [u : v] = [1 : 1]. The resulting equation
y2 = x3 + v3(2u
v)3(24u2 + 27v2
54uv)x
+ v4(2u
v)4(16u4
144u3v + 288u2v2
gives the Weierstrass form of extremal K3 surface with transcendental lattice 60 02 , with
A1. The discriminant of the Weierstrass form (2.18) is given by
We con rm from equations (2.18) and (2.19) that extremal K3 surface (2.18) has two type
IV
bers, at [u : v] = [1 : 0]; [1 : 2], type I6 ber at [u : v] = [0 : 1] and type I2 ber at
[u : v] = [1 : 1].
(2.18)
(2.19)
E2
8
D82
D62
A3
A5
A1
A1
A1
f
0
0
12u6
11+25p5 (2uv v2)
18u10
11+25p5 (2uv v2)
We show the Weierstrass forms of the 10 extremal K3 surfaces, which we obtained in
section 2.4.1, in table 5. The discriminants of the Weierstrass forms in table 5 are listed
in table 6. The Weierstrass forms of extremal K3 surfaces with ADE types E2
8
A21 are discussed in [22]. The Weierstrass forms of extremal K3 surfaces with ADE
A2 and
types E63 and E2
7
D4 are discussed in [24] and [23], respectively.
2.4.3
Anomaly cancellation condition
We consider Ftheory compacti cations on the extremal K3 surfaces obtained in
section 2.4.1 times a K3 surface. The resulting theory is a fourdimensional theory with
N = 2, without a fourform
ux. The anomaly cancellation condition determines the form
of the discriminant locus to be 24 K3 surfaces, counted with multiplicity; there are 24
7branes wrapped on the K3 surfaces.5
We show the correspondence of the numbers of
7branes and the
ber types in table 7. The Euler numbers of the singular
bers can be
found in [29]. The Euler number of ber type can be considered as the number of the
associated 7branes. We con rm from tables 4 and 7 that there are in fact 24 7branes in
5See, for example, [24] for discussion.
HJEP03(218)45
E2
8
E2
7
E2
8
D82
D62
A2
D4
A2
1
A2
1
A2
3
of the Weierstrass forms in table 5 are shown. We suppressed the
irrelevant constant factors of the discriminants.
Fiber type # of 7branes (Euler number)
Ftheory compacti cations on the 10 extremal K3 surfaces times a K3 surface. Therefore,
we conclude that the anomaly cancellation condition is satis ed for these compacti cations.
By turning on fourform
ux [30{34], Ftheory compacti cation on K3 times K3 gives
fourdimensional theory with N = 1 supersymmetry.
We con rm from table 1 in [35]
and table 2 in [13] that for Ftheory compacti cations on the 10 extremal K3 surfaces,
obtained in section 2.4.1, times some appropriate attractive K3 surface, the tadpole [31]
can be cancelled. See [23, 24] for the details.
surfaces
In this section, we discuss the stable degeneration where a K3 surface degenerates into two
nonisomorphic rational elliptic surfaces. It is considerably di cult to provide a general
equation to describe this kind of stable degeneration. Therefore, instead of providing an
equation to describe the process, we use a lattice theoretic approach to determine whether
stable degeneration exists for pairs of nonisomorphic rational elliptic surfaces. For the sake
of brevity, in this section, we simply say \a pair of rational elliptic surfaces" to indicate
a pair of nonisomorphic rational elliptic surfaces. We also discuss the con gurations of
singular bers under stable degeneration. In section 3.1, we will discuss a lattice theoretic
condition for the existence of stable degeneration for pairs of rational elliptic surfaces.
Applying the lattice condition, in section 3.2, we demonstrate that stable degeneration
exists for pairs of extremal rational elliptic surfaces.
Elliptic ber of a rational elliptic surface generally has the moduli of dimension 1 over
the base P1. Therefore, given a pair of rational elliptic surfaces, there is a pair of isomorphic
smooth elliptic
bers, and the pair of rational elliptic surfaces can be glued along the
isomorphic smooth
bers. However, when elliptic ber of a rational elliptic surface has the
constant moduli over the base P1, such gluing is not necessarily possible. For the three
extremal rational elliptic surfaces X[II; II ], X[III; III ], X[IV; IV ], the complex structure of
elliptic bers is constant over the base P .
1 6 We do not consider the three extremal rational
elliptic surfaces X[II; II ], X[III; III ], X[IV; IV ] in this section.
3.1
Lattice condition for stable degeneration limit
We use the Torelli theorem for K3 surfaces [36] to deduce a lattice theoretic condition
that determines whether pairs of rational elliptic surfaces admit stable degeneration. The
Torelli theorem for K3 surfaces states that the geometry of a K3 surface is determined by
the structure of the K3 lattice
K3. The K3 lattice
K3 of a K3 surface S is the second
integral cohomology group H2(S; Z). The K3 lattice
K3 is the inde nite even unimodular
lattice of signature (3,19), and it is the direct sum of three copies of the hyperbolic plane
U and two copies of the E8 lattice:
K3 = U 3
E82:
(3.1)
In this note, we assume that a rational elliptic surface has a global section; thus, we
presume that the K3 surface obtained as a deformation of the sum of two rational elliptic
surfaces also admits a global section.7 An elliptic K3 surface having a section is equivalent
6The complex structure of elliptic bers of extremal rational elliptic surface X[0 ; 0 ](j) is also constant
over the base P1, but there is a degree of freedom in choosing jinvariant of an elliptic
ber. Therefore,
when jinvariant is appropriately chosen, X[0 ; 0 ](j) can be glued with another rational elliptic surface.
For this reason, we include extremal rational elliptic surface X[0 ; 0 ](j).
7In general, genusone bered K3 surfaces need not have a global section. For discussion of the geometry
of genusone
bered K3 surfaces without a section and string compacti cations on such spaces, see, for
example, [23, 24, 37]. For recent progress in Ftheory compacti cations on genusone brations without a
section, see, for example, also [38{48].
to the primitive embedding of the hyperbolic plane U into the K3 lattice
K3 [36, 49].
We denote the ADE types of two rational elliptic surfaces, X1 and X2, as R1 and R2,
respectively. Applying the argument in [50] to the boundary of the closure of K3 moduli,
from the Torelli theorem for K3 surfaces, we deduce that a K3 surface exists and it admits
stable degeneration into two rational elliptic surfaces X1 and X2, exactly when there is a
primitive embedding of the lattice U
R2 into the K3 lattice
K3:
R1
R1
U
R2
K3:
(3.2)
Whether lattice U
R1
R2 primitively embeds into the K3 lattice
K3 can be
determined by the criterion given in [
51
]. Some lattice theoretic terms are necessary to
state the criterion; we introduce some lattice theoretic terms rst.
By lattice, we indicate a
nite rank free Zmodule with a nondegenerate integral
symmetric bilinear form. The lattice L is said to be even, when for every element x of L,
x2 = x x is even. The discriminant of lattice L, disc L, is the determinant of an intersection
matrix (ei ej )ij for a basis feig of the lattice L. The lattice L is said to be unimodular
when its discriminant is
1. U denotes the hyperbolic plane. The hyperbolic plane U is
the even unimodular lattice of signature (1,1). E8 denotes the even unimodular lattice of
signature (0,8). E8 and U are unique up to the isometries of lattice. When the lattice
L1 embeds into the lattice L2, the embedding L1
L2 is said to be primitive when the
quotient L2=L1 is free as a Zmodule. The dual lattice of lattice L is the lattice Hom(L; Z),
and is denoted as L . The quotient GL := L =L is a
nite Abelian group, and this group
is called the discriminant group. When L is an even lattice, the map qL : GL ! Q=2Z,
qL(x) = x
2 mod 2Z
(3.3)
de nes a nondegenerate quadratic form of the discriminant group GL; form qL is called
the discriminant form.
When A is a nite Abelian group, its length l(A) is de ned to be the minimum number
of elements required to generate group A.
Further, we state the criterion of primitive lattice embedding.
Criterion (C) [51].
M is an even lattice of signature (m+; m ), G is the discriminant
group of M and q is the discriminant form of M . Then, M primitively embeds into some
even unimodular lattice of signature (l+; l ) when all of the three conditions i){iii) are
satis ed:
i) l+ + l
rk M > l(G).
ii) l+
iii) l+
m+
l
0, l
m
0 (mod 8).
3.2
Pairs of extremal rational elliptic surfaces
Applying criterion (C), we discuss whether the lattice U
R1
R2 of pairs of extremal
rational elliptic surfaces, X1 q X2, primitively embeds into the K3 lattice
K3. This
Discriminant group
Z=2Z
Z=2Z
0
Z=2Z
Z=3Z
Z=2Z
Z=4Z
Z=3Z
Z=4Z
Z=6Z
Z=5Z
Z=2Z
Z=2Z
determines whether stable degeneration exists for pairs of extremal rational elliptic surfaces.
An even unimodular lattice of signature (3,19) is unique up to the isometries of lattice [52].
Therefore, when the lattice U
R1
R2 admits a primitive embedding into some even
unimodular lattice of signature (3,19), it primitively embeds into the K3 lattice
K3.
As stated in section 2.2, the complex structures and singular ber types of the
extremal rational elliptic surfaces were classi ed. For an extremal rational elliptic surface,
the discriminant group and the MordellWeil group are identical [53]. The MordellWeil
groups of the rational elliptic surfaces were computed in [
54
]. We list the discriminant
groups and ADE types of extremal rational elliptic surfaces in table 8.
Let G denote the product group of the discriminant groups of two extremal rational
elliptic surfaces X1 and X2. We determine the pairs of extremal rational elliptic surfaces
that satisfy the condition i)
rk K3
rk U
R1
R2 > l(G):
in criterion (C). From table 8, we observe that the length of the discriminant group of an
extremal rational elliptic surface is either 1 or 2. For two extremal rational elliptic surfaces,
X1 and X2, the singular ber types R1 and R2 are rank 8 even lattices
(3.4)
(3.5)
rk R1 = rk R2 = 8
of signature (0,8). Thus, the lattice U
R2 has the signature (1,17). The di erence
of the rank of the K3 lattice
K3 and the rank of the lattice U
R1
R2 is
The length l(G) of the discriminant group G attains this bound only when the discriminant
groups of two extremal rational elliptic surfaces both have the length 2. The product group
G of the discriminant groups of extremal rational elliptic surfaces has the length 4 only for
three pairs8 of extremal rational elliptic surfaces:
X[4;4;2;2] q X[2 ; 2;2]; X[2 ; 2;2] q X[0 ; 0 ]; X[0 ; 0 ] q X[4;4;2;2]:
For all other pairs of extremal rational elliptic surfaces, the criterion (C) applies. Both
conditions ii) and iii) in criterion (C) are satis ed:
HJEP03(218)45
(3.6)
(3.7)
(3.8)
(3.9)
and
3
rational elliptic surfaces.
pairs9 of extremal rational elliptic surfaces.
From lattice embedding
Thus, we determine that the lattice U
R2 primitively embeds into the K3 lattice
K3
for all pairs of extremal rational elliptic surfaces, except the three pairs X[4;4;2;2] q X[2 ; 2;2],
X[2 ; 2;2] q X[0 ; 0 ] and X[0 ; 0 ] q X[4;4;2;2]. We conclude that the stable degeneration of a
K3 surface exists for all pairs of extremal rational elliptic surfaces except the three pairs.
The remaining three pairs, X[4;4;2;2] q X[2 ; 2;2], X[2 ; 2;2] q X[0 ; 0 ] and X[0 ; 0 ] q
X[4;4;2;2], have the ADE types D6
A4, D6
1
D42
A2, D42
1
A21, respectively.
ADE types of the singular bers of elliptic K3 surfaces with a global section were classi ed
in [
55
]. We conclude from table 1 in [
55
] that the lattice U
R1
R2 primitively embeds
into the K3 lattice
K3 for the three pairs of extremal rational elliptic surfaces. (They
correspond to No.2079, 2043, 2152 in table 1 in [
55
], respectively.) This demonstrates that
the stable degeneration of a K3 surface exists for the remaining three pairs of extremal
The aforementioned argument demonstrates that stable degeneration exists for all
U
R1
R2
K3;
(3.10)
we deduce that the ADE type of the singular
bers of the resulting K3 surface is the
sum of the ADE types of the singular bers of the two nonisomorphic extremal rational
elliptic surfaces.
8The product of discriminant group Z=3Z
Z=3Z and another discriminant group with length 2 has
length 2. For example, the product of Z=3Z Z=3Z with Z=2Z
Z=2Z is isomorphic to Z=6Z
Z=6Z, which
has the length 2.
9As we stated at the beginning of this section, the three extremal rational elliptic surfaces X[II; II ],
X[III; III ], X[IV; IV ] are not considered in this section.
degeneration limit.
In this study, we analyzed the stable degeneration of a K3 surface into two rational
elliptic surfaces.
We also discussed the con gurations of singular
bers under the stable
We demonstrated that gluing together two isomorphic rational elliptic surfaces and
deforming the resulting surface to a K3 surface is always possible. We provided an equation
to describe this kind of stable degeneration. The sum of two isomorphic rational elliptic
surfaces glued together admits a deformation to a K3 surface, the singular bers of which
are twice the singular
bers of the rational elliptic surface. For special cases, two bers
of the same type of the resulting K3 surface collide, and they are enhanced to a ber of
another type. Some K3 surfaces become attractive in these cases. We determined the
complex structures and the Weierstrass forms of these attractive K3 surfaces.
We also
deduced the gauge groups in Ftheory compacti cations on these attractive K3 surfaces
times a K3.
We also investigated the deformation of two nonisomorphic rational elliptic surfaces
glued together to a K3 surface, using the Torelli theorem of K3 surfaces.
We deduced
the lattice theoretic condition that must be satis ed to ensure that a deformation to a K3
surface exists for pairs of nonisomorphic rational elliptic surfaces. We con rmed that the
lattice condition is satis ed for all pairs of the extremal rational elliptic surfaces. Thus,
all such pairs of extremal rational elliptic surfaces glued together admit a deformation to
a K3 surface. This demonstrates that for any pair of extremal rational elliptic surfaces,
except the three extremal rational elliptic surfaces X[II; II ], X[III; III ], X[IV; IV ], there is
a K3 surface that stably degenerates into that pair. The ADE type of singular bers of the
resulting K3 surface is the sum of those of the two nonisomorphic extremal rational elliptic
surfaces glued together. The lattice condition discussed in this study can be extended to
general pairs of rational elliptic surfaces.
Acknowledgments
We would like to thank Shun'ya Mizoguchi and Shigeru Mukai for discussions. This work is
partially supported by GrantinAid for Scienti c Research #16K05337 from The Ministry
of Education, Culture, Sports, Science and Technology of Japan.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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