Inose quartic surfaces & type II degeneration
Shigeru MUKAI
4/5/23(W)Arithmetic
A1

Even unimodular lattice of signature (1, 17)
II 1 = U+ E 8+ E 8
U= 0 1 , E 8= II 017810,,A2

II 1 1 + <-2>
,7Geometry
G1Aut (Inose)& type II degeneration of K3 surfaces'G2

Bir-Aut (Inose ) & type II degeneration of hol. symp. 4-folds[2 ]§1 Lattice II 1 7 as Pic (Inose),12

H (K3, Z) = II 3,19<3'9 II 1,17
さ 2-parameter family of K3 surfaces whose Picard latticecontains II 1 7,

1

Explicitly given by Inose(1978) as quartic surface
S G : y2 z+ ・・・ = 0ac[P3,Simpler realization is354

2

y = x - at x + t (t + 1-2bt),
2~elliptic K3 surface with 2 E8 -fibers at t = 0,and a section

We assume a,b are very general and Picard number ρ = 18.

sa.es
sectionBy vinberg's computation,these 19 (-2) classes define
a fundamental domain in
the positive cone.

Geometric implication(1) No other (-2)P 's other thanwhat we see.
××.

(2) Nef cone is polyhedral with 19 facets.
The Π-diagram is the Coxeter diagram.

(3) Aut(S) is finite.

a.fr功ぜ) が癉氎○_-.-Roman、、a.

曜

3§2 II1、 17

as type II K3 lattice ΛIKulikov model of type II degeneration of K3

ratio

N = log P \=0
N =0d

Rational

elliptic ruled
One diﬃculty of (compact) moduli construction of polarized K3 (X, L) isthat naive{RDP}U {type II}/ isom
is not Hausdorﬀ by the following reason. (Another reason caused byflop is discussed in other lectures and omitted here.)(T) Assume that the central fiber is reducible,
say, A B. Then tensoring line bundle
Restrictchanges the distribution of polarizations.

で を、っon>

LAA
DBLB,

↓ ・ 0剕
( LA ( FD )、LB は D)V。

0D=A。 B

cubicExample

Hdegeneration

ぱっと、

ansT Upつ4planeT Upcが12-pt blow-up to makefamily d-semi-stable']/yuponP0'T⑤ た (T)deg

ans et Dll
3=

0で "4l・deg

デPiーー> ど「D40Which one among two (or more) polarized type II K3 surfaces
should we take when (partially) compactifing moduli?
My proposal in 1989: Choose it following Vinberg's fundamental
domain (or diagram) since II1 1 7 is the Picard lattice of the standard
type II K3 surface.
、KII! Be
=がPUBGa'B B
and the fundamental domain coincides with the new cone.
=Standardization of type II K3's to make moduli Hausdorﬀ.
stdThe next page shows Π-diagram in Λ五 , the Picard lattice of K3互 .

IThus for Example above, our choice is the union T P, not arational surface with an elliptic singularity of type E .

5DE Ph

ポ-:lB

KC !

'BisだB'e、2

どPi、WhereK 3五S=(82--0)B

8=-B'ないz 。

一げ ) ⊕ Pi
Pie ( Bl、E、。

( Bfが ).

Before moving to §3, recall somebasics of holomorphic symplectic manifold of K3-type

K3

K3

X

S=

[ 2]

S

吟ならres.RS
Pie X
S
Cswith Beauvilleform=Pieに)>

s+Z S、

later
言 of exe.div.at
as=(f)

ニ・2

7た1§3 Bir-Aut ( Inose )Inose quartic

Sq= y zt + ・・・= 02<3D

3is (U+2E 8)-K3 surface embedded into [P by H = w + w ':ポがA 、、。

。

。

。

ハの。

く」

トに ひいRankSaw

7"たcubic-plane, i.e., T UP>
degeneration in Example

A "-RDP & E' -RDP & 2 lines
An
E6Image(α 0 )Image(αi )This quartic polarization plays a key role in the proof of the following:
た]

Theorem Bir-Aut (Inose ) is finite.
Proof: Similar to the Inose surface case after replacing nef cone,1(-2) P , Vinberg's diagram by movable cone, Q-eﬀective (-2)
eﬀective divisor, KLS-diagram, respectively.
KLS diagram = Π-diagram + 5 vertices
24=19+5Vertices are (-2) classes in the Picard lattice of Inose[ 1.
2Basis is written, using double Bl g[P model, on the nextpage.
2

8

δ

δδ

9Remark KLS stands for Kondo-Looijenga-Scattone. Kondo uses
the diagram to compute the automorphism group of a K3 surface.
Others, including the recent sophisticated construction by AlexeevEngel-Thompson, use for type II degeneration of K3's of degree 2.
Proof (cont'd) (*) Show all classes are Q-eﬀective.
1. 19 (-2) classes in sub-Π-diagram comes from (-2)P on Inosesurface. Easy to prove their eﬀectiveness.
2. Two comes from the Lagrangian fibration of Inose , induced
from the elliptic fibration of Inose . Also easy.
3. δ' and δ". Here Inose quartic is crucial. From a pair of points
p,q, one obtain new pair p',q' in the following way.
Thus one gets an involution of[2 ]Inose , which is called Beauville inv.

d.

iP

δ' and δ" are the image of δ by1
Beauville inv's. Since δ is Qeﬀective
so are δ' and δ". Note that both δ' andδ" are of the form 2(Inose pol) - 3δ.

P''q.

ElBy (*), cone of movable divisors Mov(Inose ) is finite polyhedral.
[ ]Hence Bir-Aut(Inose ) is finite.
QED2

[2]

§4 Type II degeneration of holomorphic symplectic 4-fold of K3 -type
We still do'nt have a good theory like Kulikov's about this. I just pose
Problem: Find a standardization ofハ, similar to K3, replacing Vinberg's
Π by KLS-diagram.
3

Seems diﬃcult to answer since geometric meaning of S - symmetryof KLS-diagram is still unclear.

RReferences§1 Inose, H.: Defining equations of singular K3 surfaces and a notion ofisogeny, Int'l Symp. on Algebraic Geometry, Kyoto, 1977,
Kinokuniya, Tokyo, pp. 495-502 (1978).
Shioda, T.: A note on K3 surfaces and sphere packings, Proc. Japan
Acad., 76(2000), 68-72.
Vinberg, E.B.: Some arithmetic in Lobacevskii spaces, in "Discrete
subgroups of Lie groups and applications to moduli", Oxford Univ.
Press, 1975, pp.323-348.
へ1§2 Kulikov, V.: Degenerations of K3 surfaces and Enrique's surfaces,Izv. Acad. Nauk SSSR 41(1977), 1008-1042.
Friedman, R.: A new proof of global Torelli theorem for K3 surfaces,
Ann. Math. 120(1984), 237-269.
Mukai, S.: On the moduli space of K3 surfaces (in Japanese), Proc.
Algebraic Geometry mini-symposium, U. Tokyo, 1989, pp. 94-124.
§3 Kondo, S.: Algebraic K3 surfaces with finite automorphism groups,
Nagoya Math. J. 116(1989), 1-15.
Scattone, F.: On the compactification of moduli spaces of K3surfaces, Memoirs AMS 70(1987), #374.
Alexeev, V., Engel, P and A. Thompson: Stable pair compactificationof moduli of K3 surfaces of degree 2, arXiv1903.09742.
§4 Kollar,J., Laza, R.,
Sacca, G. and C. Voisin: Remarks onof
degenerations of hyper-Kahler manifolds, Ann. Inst. Fourier 68(2018), 2837-2882.
st