1Igusa 3-fold and Enriques surfaces

4/4/23(T)Shigeru MUKAI

Abstract: This quartic was first found as the projective dual of Segre's
10-nodal cubic, the moduli of 6 points on the projective line. It wasre-discovered as moduli of p.p.a.s’s by Igusa(1962). I explain its newinterpretation (Contemp. Math., 2012) as the moduli of Enriquessurfaces of certain root type.

Two modular interpretations of Igusa (+ Steiner)
2-dim'l analogue of× ( 2)い=h=/ア ( 2×≡にhy / t ( 2に≡。

P、

EO , Iixyp1、30,B}

xA

A moduli of genus 2 curves with (full) level 2 str.
B
Enrique's surface of HG-type」

Igusa as moduliof curves

A

Bssー

1Periody = f (x)".

i]periods

Ω幽)

Sa Ω:

=,g(C)=2axtetF C)
SsΩ:εC2(i=1,2)

Jac

Abel

C"

4=/ 24 semcted ) 夂( by periods

matrixpeniod〜2( 12 z )hya /ε{tz=Imア{()
yf2Picdc=03dI=1onigin③= CSetckeccptfn2。

Z 70)/ 14 り hy
Sp (ry/p□ に)}{ (品 ) 14い《 a=)

h ±(.

=mraz

≡

P ( 2 ) /C ( u )≈Sp ( 42 ) ≡∞.
,Igusa (1964) embedding by 10 even theta "constants"
5 liuearrlsY厶hs /p ( )
2

9→

2

先(→p4c.

p( Om ( 224 ) mieven

Image is a quartic 3-fold whose singular locus isunion of 15 lines.
ノ,Eachsurface04m(22 =Im

cuts a (double) quadricfor even m.

shy ,m2periodmap.

34Igusa

Mcoverins〜UQRmkFact:Qu

Mz is the complement of=R ' ≡×Pproduct abelian surfaces

t*tt

X.

parametrizes

Ez = JaC E . VE 2E、 ×

the Jacobean of stable
curve of compact type.

*.

,tth* unigonalx^degeneration

Xa

3 Bielliptic curves and Steiner surfaces.

.
Edet involution σ such thatbielliitic.

CsC/σ is ellipticσ≥

2w

-',

",

woys?

permatctin typet

Weierstrass points

く > { wi,…

wo CBl,0is a line section ofUpiP;a complete quadrangle.

2in IP .

( ≤ : *) 々 4

4This classical theorem gives
a morphism.

IP2,*smoduli ofbielliptic
curvesonto one of 15 components inIgusa.
2,*

Image of IP is a Steiner(Roman) surface.
tetrahedronxyzt = 0

Klein's4-group3

Non-normal quartic surface
singular along (line) (line) (line).

Cp

Fact: Igusa quartic has 15 linear involutions σ with
Fix σ Steiner surfaces
e.

Key

5for passing from A to B

T。 ( 2 )"{=2( 凸) ≡.

index 8U品ア [2)mdzhy .
k↓ CzcFact:}

Xiis again Igusa
quartic.

3modulioohy [,2)CC , G )/GC JacCC ) ( 22Gopel, #ie .

,=G

4

Weslpaining G

Igusa quartic has a selfmorphism of degree 8.

≡ 0Igusa as moduli of Enrique's surfaces

B

3

Motivation

-dime
〜

Kummersurfece

JaY 1±Jacδr

C

E 、 × E2c{ Ksmfaes }
als.

-:

Abeliansmfacediml19?
〉*Euniqus{ sunfeces }10-dim'l family

GQ. Find X
Answer by M.-Ohashi(2013):
K should be Hutchinson-Gopel (HG) type.
.

mini-history
Kummer(1864) Found 3-dim'l family of quartic
surfaces with 16 nodes (16 is maximal possible)
Borchardt(1877) Uniformized them by abeliansurfaces, or h.e. functions.
Kummer's equation is equivalent to Gopel's
one found in 1847.

Hutchinson(1901) Found a new equation ofKummer quartic
1 p ? withkn ( c )εreference to Gopel subgroup

GC

CJac

which is invariant under standard Cremonatransformation
記
7 t ←of IP 3.
ーー

More precisely, we have(x :y::)2(C:)( )2.

方 話 当)::

( C,a)fielliptua≥=GplareBI(G={acε Jac3desevercte) laca }=)(a)caseOtherwise standard Cremona transformation induces
an involutionc )EG ε Ant ( km.

Quotient Km Gis free.

EGEnriquessurface

XE K3 surf. free inv.

S

[period of S]1

is called Enriques of HG type if EGEDl

bdd symmetricdomain of type IV0

well-defined modulo O π ( , 102)

the orthogonal group of diag[1, 1, -1, ..., -1]〜<Torelli type thm> S≡s'l⇐)0periods are the same(modulo OR (2 107 ),<surj. thm>

{periods of S}

品
enD'/Oπis the complement of a divisor(2,e10)Rmk(1)

Borcherds' Φ.
(2)
.

t.

ee

is the zero locus of(quasi-projectivity of moduli)parametrizes Coble surfaces X/EFixε={m8enodes }

,I ≤≤

m10.

×/ ε,

has m singular points of type (1, 1)/4.
2

Y&

3o

DyCs

natural embeddings

*l山
=b%

,

(2)

→oいD

which is geometrically interpreted as follows:
…

Th2x.

of 2 Steiner surfaces,eト) 4&is the union
H8.

The

complement of Ha UH8 is moduli of Enriquessurfaces of HG-type. (Root type D + A )G3CC , GIE

HGCp.
'i. bielliptic involution σ s.t.
< a 2,=< G 7 okn (c )9is the union of 2 conics.
Their strict transforms R 1 and R2
are disjoint on Km(C).

2、4Contract R's to 2 nodes andtake quotient by σ. Then one
obtains a Coble surface with two(1,1)/4 singular points (m=2).

de3( Conte )alm

In c(C , G )EHg くftp.liatimbyfI.ie
hasan"「をいが,.

,7

G = {1,2,3,4}

Je ChoEnd (ココ を EEI

which is findby EG(Coble surfaces with m=1).
References
1. Dolgachev, I.: Kummer surfaces: 200 years of study, Notices AMS,
2022, pp. 1527-1533.
2. Geer, van der G.: On the geometry of Siegel modular threefold, Math.
Ann. 260(1982), 317-350.
3. Hirzeburch, F.: The ring of Hilbert modular forms for real quadratic
fields of small discriminant, Lect. Notes in Math., 6(1976), 288-323.
4. Igusa, J.: On the Siegel modular forms of genus 2, Amer. J. Math.,
1964.
5. Mukai, S.: Lecture notes on K3 and Enriques surfaces in"Contribution to algebraic geometry", EMS Publ., 2011, pp. 389-405.
6. ---: Igusa quartics and Steiner surfaces, Contemp. Math. 584(2012).
7. --- and Ohashi, H.: Enriques surfaces of Hutchinson-Gopel type andMathieu automorphisms, Fields Inst. Commun. 67(2013), 429-454.
(The next page was used at the beginning of my 3rd talk on 5(W) toexplain type II & III boundaries.)self-morphism of Igusa/ c xC

422×C10≥
2〜MOc2.
oいty /.

,2em{ Evo }

p 12 )〜M2↳y。

.

T, ( 2 )complement divisor
Yo ( 2 )と××02 Steiner surfaces

(2 )

。

These are nterior divisor when regarded as moduli
of covering K3'sof ppAS's
True bdry =Ueiz
I ≤{CayleyRichmondconfiguration(15 -153 )True bdry = U 6 lines

cj ≤ 4

Remark Remaining 9 lines
parametrize Enrique's
surfaces with extra
automorphism.

= Sing Igusa4(15 lines)

type{type

15 intersection points
complement

N

tHreI
皿N02

N = log P0キoquasi-unipotent
monodromy P