On the global theory of singularities(Analytic varieties and singularities)

RecendyI

am

considerhg

on

the global theory of shgularities.

Perhaps whatIhavejust statedreminds

you

ofthePl\"ucker_{fornula for plane irreducible}

curves.

As

you

know, itis

an

equality connecthgintegersdefmed

as

local invariantsof

shgularities and htegers defmed

as

globalinvariants ofvarieties such

as genus,

degree, and classnumber. It restrictspossible combhations of singularities. Thedata obtained from

an

actual combhation of shgularities satisfy the Plucker formula. However, given global

hvariantsand

a

combhation of local singularities satisfyingit,there doesnotnecessarilyexista plane

curve

with such data. Forexample,

a

planesexticrational

curve

with unique shgularityof

type $A_{20}$ (theshgularity locaUy defined by $x^{21}+y^{2}=0$)

never

exists,though the data satisfy

thePl\"uckerfornula:

(Inthis

case

$g=0,$$d=6$ _and

$\delta_{X}=1og=\frac{(d-1)(d-2)}{foranA_{20}2}-\sum\delta_{X}- siu1a1$

Our pointofviewisdifferent from this. We

use

graphs for the description instead of integers. (By

a

graph

we mean

a

fmiteone-dimensional complex with

some

additional

structure.)Moreover,

we

aim togive the

necessary

andsufficient conditionfor thedescription ofpossible combhations of singularities. The following figure explah$s$

our

basic framework.

TheBasic graph

$+Conditions$ $|$ $|$

givhgresmctions

Here

we

consider

a

typical exampletoexplaintheabove figure. We consider the

case

of cubic

curves

hthetwo-dimensionalprojective

space

$\mathbb{P}^{2}=\mathbb{P}^{2}(C)$

over

the complex field C.

(Below

we

always

assume

thatthe ground field is the complex field C.) Itis

easy

togivethe classificationof plane cubiccuives. We have the followhg

9

items. (Below

we

draw the figure

of thesetofreal pohts,i.e.thehtersectionof the

curve

in $\mathbb{P}^{2}(C)$ and $\mathbb{P}^{2}(\mathbb{R})$,because

we

cannotdraw the actual figure of thesetof complexpoints.) Thegraphs beneath the figures

are

$A_{1}$ $A_{2}$

$O$ $\mapsto$

$2A_{1}$ $A_{3}$ _{$3A_{1}$}

$O$

O O O $O$

mathematics.)Theconcept ofDynkin graphs iswell-knownbecause it plays the keyroleinthe

classificationtheory of semi-simple Lie

groups.

Thelocal defmingequations $f(x,y)=0$ of thesesingularities of dimension 1

are as

follows:

$A_{k}:x^{k+1}+y^{2}=0$ $(k=1,2,3,\cdots)$

$D_{t}:x^{\ell-1}+xy^{2}=0$ $(t=4,5,6.\cdots)$

$E_{6}:x^{3}+y^{4}=0$

$E_{7}:x^{3}+xy^{3}=0$

$E_{8}$

:

$x^{3}+y^{5}=0$

(Theabove

are

equationsof

curve

singularities. When

we

considersurface singularities

we

add

a

tern $z^{2}$ totheaboverespective equationand

we

consider the singularity defmed by

$f(x,y)+z^{2}=0$

.

_{Forexample thesurface singularity oftype} $A_{k}$ is defined by $x^{k+1}+y^{2}+z^{2}=0.)$

The above seventh cubic

curve

has aunique singularity and it isof type $D_{4}$. Wedraw

a

Dynkin graph oftype $D_{4}$ beneaththe seventh

curve.

By the

same

method

we

can

associate

a

Dynkin graph (possiblywith severalcomponents) toeach cubic

curve.

We have theempty

graph, $A_{1},$$A_{2},2A_{1},$$A_{3},3A_{1}$ and $D_{4}$

.

Here perhaps

you

can

noticethatthe classification of cubic

curves

corresponds tosubgraphs of $D_{4}.7$ typesof cubic

curves

haveone-to-onecorrespondences with7kinds ofsubgraphs of

$D_{4}$

.

Inthe

case

of plane cubic

curves

thebasic graphisthe Dynkin graph oftype $D_{4}$

.

The

operationistopick

a

subgraph. There is

no

condition givingrestrictions. Thesetofallgraph obtained from $D_{4}$ under this situationis equaltothesetofpossible combinations of

singularities.

As suggested by the above example of cubiccurves, the meaning of

our

basicframework aboveis

as

follows.

First

we

set$up_{\zeta}$

an

appropriate

range

of objects

we

treat.For example

we

consider

one

ofthe

followings([5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [17], [18]):

1. AU plane cubic

curves

without multiplecomponents.

2. Allplane quardc

curves

without multiplecomponents.

3.

Allplanesextic

curves

with only ADE singularities

as

singularities.

4.

All

space

cubic surfaces with onlyisolated singularities.

5. All

space

quartic surfaceswithonly ADEsingularities

as

singularities.

7. Allcomplete intersectionswithbidegree$(2, 3)$inthe fourdimensional projective space.

Inthe seventh

case

we

assume

that allsingularities

are

ADEsingularities.

For the

range

of objects

we

consider,thebasicgraph is determined. For simpler

cases

the

basic graph is unique. Somebasic graphscan

appear

in the complicated

cases.

Nexttheoperation

on

graphs is defined. By the operation

we can

makenew graphs from

a

given graph. Atpresent

we

use

three kindsofoperations-topick

a

subgraph,

an

elementary

transformation, and

a

tie transformation. These threeoperationshavemanychoicesontheway oftheir

process,

and

we

can

make considerablymanykmds of graphs from

a

given

one.

In addition

we

have

some

conditionsgiving restrictions

on

the

process

ofoperations. Underthe above conditions

we

considerthesetof allgraphs obtained starting from

one

of

the basic graphs. Amemberof thissetisnotnecessarily connected. Thetypeof

a

connected

componentcorrespondstothetypeof

a

singularity, and thenumberofconnectedcomponents

of eachtypeindicates the number ofcorrespondingsingularities. Inthisway

a

graph is translated into the meaning of

a

combinationofsingularities, and thesetofobtained graphs coincides with thesetof possible combinations of singularities.

Sofar

we

have explainedthesuperficialpartof

our

theory. Indeed thegreater partishidden

inthe background. First,

as

the bestweapon, we

use

thetheoryofperiodsof algebraicvarieties. By this

we

translatethe problem intothe language ofintegral symmerric bilinear forms. Itis known that interesting

groups

are

associatedwithbilinear forms. So using the

group

theory, in particular usingthetheoryofreflectiongroups, wetranslate it furtherintothelanguage of

graphs.

The above is

our

philosophy. In the

case

of sufficientlylowdimension,lowdegreeandlow

codimension,thisphilosophy worksmiraculouslywell andwe

can

getthe splendid description ofpossible combinations of singularities. Infact thispartis the discovery of English

mathematician Timmsatthebeginning of thiscentury[6]. _{English mathematician Du}Valwrote

another

paper

giving interpretationof Timms’ result from hisview pointlittle later[1].

We

can

imaginethat

we

are

lookingatonly

a

smallpart, and thatthereexists ahidden

generalprinciple in the background, since Timms’ resultis

very

beautiful. Therefore we

can

try tosearch howfar

our

philosophy

can

be applied,andifpossible

we

wouldbe abletofindout

the hidden general principle. This is

our

objective. Today

we

have theory of periods of K3

surfaces, _{fully developed theory of integral bilinear}forms, the theory ofreflection

groups

includingWeyl

groups

andthelike. Using these theories

we

can

go

to

a

considerably deep point.

However,the world of actual algebraicvarietiesismuch

more

complicated than what

we

expect. Thereforesometimesthevertical equality in the first figuredoesnotexactly holdand

a

fewexceptions

appear.

Thisis

a

really strangephenomenon actuallyoccurring,thoughwehave

defined absolutelycorrectbasic graphs,operations, restrictionconditions etc. Our theory may have

some

defects,

or

there

may

be

a

faint break in the complete law because of the wisdom of the god.

Besides, atpresent,onlyADE singularities

can

correspondtoconnectedgraphs. More complicated singularities have appeared in the objects of

our

study,but I donotknowhow

we

treatthemexactly. We have

some

evidences showingthatsimilar coIrespondences

can

be

pointscorrespondingtoactual algebraicvarieties is enough for

us.

Perhapstogive

a necessary

andsufficient condition is

an

extremelydifficultproblem.)

Atpresent

on one

handI

am

making steadyefforts totake complicatedworsesingularities in

our

theoryandeffortstodevelope the theorytootheraspects,and

on

theotherhandI expect that I

can

fmd bychance

a

keytothehiddengeneral principlefromadifferentview point(for

example

a

viewpointoftherepresentationtheory of Liegroups).

Intheabove explanation of

our

philosophywehaveomitted

some

factstohelp reader’s understanding. Here

we

supplement them.

Themostimpomantpoint isthatbytheframeworkin the first figurewe

can

treat notonly combinations of singularities of global algebraicvarieties but also otherobjects. We

can

treat

the following twoitems bythe presenttheory of

ours:

A. Possible combinations of singular fibers of elliptic surfaces.

B. Possible combinations of singularities oflocalobjects such

as

deformation fibers in the

semi-universal deformation family of

a

fixedisolatedsingularity.

We explainthe item A. Let $\Phi:Xarrow \mathbb{P}^{1}$ be

an

elliptic surface

over

$\mathbb{P}^{1}$.

($X$ is

a

smooth

compactcomplex surface. For

any

generalpoint $x\in \mathbb{P}^{1}$ the inverse image _{$\Phi^{-1}(x)$} is

a curve

ofgenus 1.) We

assume

that $\Phi$ has

no

multiple fibers. It is known that the Eulernumber

$e(X)$ of $X$ ispositiveanda multiple of12.

When $e(X)=12,$$X$ is

a

rationalsurface, and

we can

describepossible combinations of

singular fibers using

our

framework. Inthis

case

the basic graph is the Dynkingraph of type

$E_{8}$ and

we

can

apply elementarytransformations twice

as

the operation.Thereis

no

resrriction

condition. We

can

associatetheresulting Dynkin graphswith possiblecombinations of singularfibers ([2], [7]).

When $e(X)=24,$ $X$ is

a

K3 surface. Also in this

case we

can

develope

our

theory. Some

partial results have already been obtained([16]). Ithink that

we can

completethetheory in this

case

in future.

As forthe item $B$,

we

knowthatin the

cases

correspondingtothe following singularities

possiblecombinationsofsingularitiesofdeformation fibers

can

bedescribed usingour framework:

1. ADE singularities.

2. 3kinds ofhypersurface simple elliptic singularities([2], [4], [7]).

3. Apartof

cusp

singularities([3]).

4. Apartof14kinds of hypersurface triangle singularities. 5. 6kindsof hypersurface quadrilateral singularities ([16]).

References

[1] DuVal,P.: On Isolated Singularities which donotAffecttheCondition of Adjunction I,

$\Pi$,III. Proc. Cambridge Philos. Soc. 30, 453-465,

483-491

(1934)

[2] Looijenga,E.: On the Semi-universal Deformation of

a

Simple Elliptic Hypersurface Singularity,

n.

Topology 17,

23-40

(1978)

[3] Looijenga,E.:Rational Surfaces with

an

Anti-canonical Cycle. Ann. ofMath. (2) 114,

267-322

(1981)

[4] Saito,K.: Einfach ElliptischeSingularitaten. Invent. Math. 23,

289-325

(1974)

[5] Tang, Li-Zhong: Rational Double Points

on a

NomlalOctic K-3 Surface.Thesis, Fudan

University, China(1991)

[6] Timms, G.: The Nodal Cubic Surfaces andtheSurfaces from which They

are

Derived by

Projection. Proc.Roy. Soc. Ser. A 119, 213-348 (1928)

[7] Urabe,Tohsuke: On Singularities

on

Degenerate Del Pezzo Surfaces ofDegree 1,2.

Proc. Symp. PureMath. 40Part 2, 587-591 (1983)

[8] Urabe,Tohsuke: On QuarticSurfaces andSextic Curves with Certain Singularities.

Proc. JapanAcad.

59

Ser. A No.9,

434-437

(1983)

[9] Urabe,Tohsuke: OnQuarticSurfacesandSextic CurveswithSingularities of Type $E_{8}$,

$T_{2.3.7},$ $E_{2}$ . Publ. RIMS. Kyoto Univ. 20, 1185-1245 (1984)

[10] Urabe,Tohsuke: Dynkin Graphs andCombinations ofSingularities

on

Quartic Surfaces.

Proc.Japan Acad. 61 Ser.A No. 8,

266-269

(1985)

[11] Urabe,Tohsuke: Singularities inaCertain Class ofQuartic Surfaces andSextic Curves andDynkin Graphs. Canad. Math. Soc. Conf. Proc. 6,

477-497

(1986)

[12] Urabe,Tohsuke: ElementaryTransformations of Dynkin Graphs and Singularities

on

QuarticSurfaces.Invent. Math. 87,

549-572

(1987)

[13] Urabe,Tohsuke: Combinations of Rational Singularities

on

PlaneSextic Curveswith the Sum of MilnorNumbers Less than Sixteen. Banach Center Publications 20, Singularities,

429-456:

PWN-Polish Scientific Publishers,Warsaw(1988)

[14] Urabe, Tohsuke: Dynkin Graphs andCombinations of Singularities

on

Plane Sextic Curves. Contemporary Math.90, Singularities (R.Randeled.),American Math. Soc., Providence,Rhode Island

295-316

(1989)

[15] Urabe,Tohsuke:Tie Transformations of Dynkin Graphs and Singularities

on

Quartic

Surfaces. Invent. Math. 100,

207-230

(1990)

[16] Urabe, Tohsuke:Dynkin Graphs andQuadrilateral Singularities. Manuscript of

a

Book

(1990)

[17] Wang, Ming: Rational Double Points

on

SexticK3 Surfaces. Thesis, Fudan University, China(1991)

[18] Yang,Jin-Gen: Rational DoublePoints

on a

NormalQuintic K3 Surface. Preprint, Institute of Mathematics Fudan University, China(1991)

Tokyo 192-03,Japan