Dynkin graphs and combinations of singularities on some algebraic varieties(Combinatorial Aspects in Representation Theory and Geometry)

(1)

Dynkin graphs

and

combinations

of

singularities

on some

algebraic

varieties

都立大理卜部東介 (Tohsuke Urabe)

Inthis article Iwouldlike to explainmy recentresults

on

Dynkin graphs andglobal theory of singularities

on

algebraic varieties. We

assume

thatevery varietyis defined

over

the complexfield $C$

.

First, let

us

consider plane cubiccurves, thatis,

curves

ofdegree 3 in

the 2-dimensional projective space. Itis easy to give thefollowing

classi-fication.

cuspidal

(normal definingpolynomial)

$y^{2}=x^{3}+a\kappa+b$ $y^{2}=x^{3}+x^{2}$ $y^{2}=x^{3}$

$(4a^{3}+27b^{2}\neq 0)$

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

$op$ $+\infty$

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

$O$

(2)

$D_{4}$

The first

one

is the smooth

one.

Inspite that

we

consider

curves

over

the complexfield, above

we

havedrawnthepictures of real points. Because

a

smooth cubic

curve

over

thecomplex fieldis

a

2-dimensional manifold in

a

4-dimensional manifold $P^{2}(C)$, it is impossible to draw

correctpictures. Anyway, wehave 9types of cubic

curves.

The last two havemultiple components, and they

are

not worthcalling cubiccurves, if

we

treat them

as

figures.

Here Iexplain

an

importantconceptin singularity theory. We have

a

series ofsingularitieswiththe

same

names as

Dynkin graphs without

multiple edges.

$A_{k}$

:

$x^{k+1}+y^{2}(+z^{2})=0$ $D_{l};x^{l-1}+xy^{2}(+z^{2})=0$

$E_{6}:x^{3}+y^{4}(+z^{2})=0$ $E_{7}$

:

$x^{3}y+y^{3}(+z^{2})=0$ $E_{8};x^{3}+y^{5}(+z^{2})=0$

When

we

consider singularities oncurves, we abbreviate the aboveterms

$+z^{2}$

.

When

we

considersurface singularities, we add the above _{$terms+z^{2}$}

.

If

a

singularity is definedby

one

oftheseequations under

a

suitablelocal coordinate,

we

callthe singularity

one

oftype$A,$$D$

or

$E$

.

Forexample, if

a curve

singularity isdefined by

a

powerseries of2 variables $x,$$y$with cubic termsin thebeginning part, and ifthe

sum

of

cubic terms defines

a

homogeneouspolynomial without

a

multipleroot,

then

we

can

make this

power

series into $x^{3}+xy^{2}$ by

a

suitable coordinate

change. Therefore, thesingularity is of type $D_{4}$

.

The above 7-th cubic

curve

has

a

unique singularity andit is oftype

$D_{4}$

.

We draw

a

Dynkin gr.aphoftype $D_{4}$ beneath the 7-th

curve.

By the

same

method

we can

associate

a

Dynkingraph(possiblywithmultiple

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

.

Now, perhaps you

can

notice that classification of cubic

curves

corre-spondsto subgraphs of $D_{4}.7$types of cubic

curves

have one-to-one

correspondences with7 kinds of subgraphs of $D_{4}$

.

This is not

an

acciden-talcoincide. We

can

give theoretical explanation. Moreover,

we

can

observe similarfactsfor

curves

andsurfaces oflow degree in

a

projective

space.

For suchobjects there exists

a

common

law dominating possible

appearance

ofsingularities.

By my study

so

farthe basic frame ofthe

common

theory

can

be explained

as

follows. For

a

given class of objects,

a

basic Dynkin graph

can

be determined, and

a

certainoperation by which

we

can

make

a new

Dynkin graph from

a

given Dynkin graph is defined. The set of all

Dynkin graphsmadeby the operationfrom the basic graphcoincideswith thesetof possible combinations ofsingularities.

Aclass ofobjects

basic Dynkingraph $arrow^{operation}$ {Dynkin graphsmade from the basicone}

I

1

possible combinationsofsingularities Needless to say, in the

case

of cubiccurves, thebasic graph $=D_{4}$, the

operation$=taking$

a

subgraph, andthe set ofgraphsmadefrom the basic

one

consists of 7 graphs. $r$

Bythe study

so

far, I know thatthe following geometrical objects

are

dominated by theory described by the above frame.

$l$ plane

curves

cubic, quartic, _-, sextic

6

space surfaces cubic, quartic

$C$ deformationfibers ofasingularity

rational doublepoints $A_{k},$ $D_{l},$$E_{6},$ $E_{7},$$E_{8}$

simple elliptic singularities $P_{8},$$X_{9},$ $J_{10}$

9 of 14 triangle singularities

6quadrilateral singularities (singularitiesofrectangles)

Heretheconcept of

a

deformation fiber isto beexplained. Let

(4)

the origin. Let $g(x,y,z)$ be

an

arbitrarypowerseries. We consider the zero-locus defined in

a

ballwith

a

sufficiently smallradius $\epsilon>0$ with the

center atthe originbythefollowing equality. $f(x,y,z)+tg(x,y,z)=0$

Here $t$ is

a

complex numberwith $0<|t|<\delta$, where $\delta$ is

a

sufficiently

smallpositivenumber compared with $\epsilon$

.

This locus iscalled

a

deformation fiberofthe singularity definedby $f$

.

We

can

showthat the

combination ofsingularities onit is independentfrom the choice of $t$

.

Now,

we

have

a

classification list ofhypersurface singularitiesdue to

Arnold. (Amold [1].) Itshouldbe remarked that the singularities in the above item “

Ct

_{deformation fibers...”}

appear

in the beginning _part_{of his} list.

So far

we

considered singularities. Next

we

would like to consider ellipticsurfaces,because they

are

also relatedto Dynkin graphs of type$A$,

$D$

or

$E$

.

A compactcomplex surface $X$ with

a

morphism $\Phi$ to

a curve

$C,$ $\Phi:Xarrow C$ is called

an

elliptic surface, if the inverse image $\Phi^{-1}(c)$ is

aconnected smooth

curve

ofgenus 1 for

a

general point $c\in C$

.

Inthis article because of

a

technical reason,

we

assume moreover

that there is

a

section of $\Phi$, thatis,

a

morphism $s:Carrow X$ such that the

composition $\Phi s$ is the identity

on

$C$

.

Aninverse image of

a

point

on

$C$ iscalled

a

fiber inthis

case.

Possi-ble singular fibers in

an

elliptic surface

are

classifiedbyKodaira.

(Kodaira [2].)

(1) An irreducible fiberis

one

ofthefollowing.

(1.1)

an

elliptic

curve

$=a$

curve

ofgenus 1

(1.2)

a

rational

curve

with

an

$A_{1}$ singular point (It is isomorphicto

a

plane nodalcubiccurve.)

(1.3)

a

rational

curve

with

an

$A_{2}$ singular point(It is isomorphic to a

plane cuspidal cubic curve.)

(2)Areducible fiber is

a

union ofsmooth rational

curves.

The graph of intersectionof these smooth rational

curves

(Inthis graph the setof

ver-ticeshas one-to-one correspondence withthe set of smooth rational

curves.

If two rational

curves are

disjoint, then the correspondingtwo

ver-tices

are

notconnected. If theyhave intersection-number 1, then the correspondingtwo vertices

are

connected by a single edge. If they have intersection-number2, then the corresponding vertices

are

connectedby

a

bold edge. Anintersection-number $\geq 3$

never

appears.) coincides with

an

extendedDynkingraphoftype $A_{k},$ $D_{l},$ $E_{6},$$E_{7}$

or

$E_{8}$

.

Therefore

we

can

associate

a

Dynkin graphto eachsingular fiber in

a

natural

manner.

(We associate theempty graphto

an

irreducible fiber.

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cor-respondingextendedDynkingraph.) Under this correspondence

we

can

apply the basic frame explained above also to describe possible combina-tions of singularfibers

on

elliptic surfaces. So far

we

have obtained results to thisdirectionfor rationalelliptic surfaces and K3 elliptic

sur-faceswith

a

singular fiberof type $D_{4}$

.

8

elliptic surfaces

rational elliptic surfaces

K3 elliptic surfaces with

a

$D_{4}$-fiber.

We divide the above mentioned objects into 3 gradesI, II, and III. Inthe firstgradethe operationis the simplest

one

–taking

a

subgraph.

Cubic

curves

and rational doublepoints fallinto this grade. Thebasic graph for

a

rational double point coincides with the graph of the

name

of the singularity.

I. operation$=taking$

a

subgraph

In the second gradearathercomplicated operationcalled an elemen-tary transformation is introduced. In anelementary transformation, first

we

replace everycomponentto the associated extended Dynkin graph. In the second step

we

choose

a

propersubgraph.

Anelementary transformation

1.

a

Dynkin$grapharrow an$ extendedDynkin graph.

2. Choose

a

propersubgraph ofthe extended graph. In the second grade

as

the operationwe

use

elementary transformations repeatedtwice:

II. operation $=elementary$transformations repeatedtwice Quarticcurves, cubic surfaces, simple elliptic singularities and rational elliptic surfaces fallinto this grade. The basicgraphis

one

of type$E$

.

Consider quartic

curves

as

anexample. Now, 4 lines intersecting at

one

point is areducible quarticcurve., However, the singularityonit is not

oftype$A,$ $D$

or

$E$

.

We have toexclude this

case.

Excluding only this case, the set of possible combinations of singularitiescoincides with the set of Dynkin graphs which

can

be madefrom

a

basic graph $E_{7}$ by

ele-mentary transformationsrepeated twice.

{plane quartic

curves

without

$\{*\}$

a

multiplecomponent}

$\cup$

$E_{7}-$

{$possible$combinations ofsingularities}

elementary transformation

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In

an

elementary transformation starting from $E_{7}$

we

can erase a

vertex

as

in the figure below.

Under this choice

we

getthe graph $D_{6}+A_{1}$

.

We

can

repeat

an

elementary transformation

once

more.

In the second transformation vertices

can

be erased

as

in the figure below.

The remaining classes fallinto the third grade. In the third gradethe thirdoperation called

a

tie transformationhas to beintroduced, and the number of the basic graphs is notnecessarily 1.

III. operation $=elementary$ transformation&tie transformation repeated2 times

(Fourkinds ofcombinations, i.e., “elementary“ twice, “tie” twice,

“elememtary” after“tie”, and “tie” after“elementary”

are

all

permitted.)

basic graphs–possibly 2

or more

specialvertices (blackvertices) have tobe erased until the final stage.

For example forquartic surfaceswehave 9basic graphs. For sextic

curves

we

have 4basic graphs.

Here I explainthe

case

of

one

of6 quadrilateral singularities called

$J_{3,0}$

.

$J_{3,0}$ is definedby the followingequality.

$J_{3,0}:x^{3}+ax^{2}y^{3}+y^{9}+bxy^{7}+z^{2}=0$ $(4a^{3}+27\neq 0)$

Weconsider only deformation fibers with$ADE$ singularities only. Inthis

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The set ofallpossible combinations of singularitiesminus $3A_{3}+2A_{2}$

coincides with the setof Dynkin graphs with onlycomponentsof type$A$, $D$

or

$E$ made from $E_{8}+F_{4}$ by2 kinds of transformationsrepeated2 times.

Here

we

need to give

some

explanation

on

the Dynkin graph $F_{4}$,

becauseithas

a

double edge and

an arrow.

Now,

every

vertex in

a

Dynkin graph$COlTesponds$to

a

vectorcalled

a

rootin

an

Euclideanspace. Ifthe graphhas

an arrow

with

a

multiple edge, it indicates that the length of roots atboth ends

are

different. Thereforefor $F_{4}$ two vertices correspond

to shorter roots. In fact the ratio oflengthis

as

in the following.

Because length is different,

we can

replacethese vertices of shorterroots byblackvertices. In

our

theory

we use

the expressionincluding black

vertices forthe graph $F_{4}$

.

$F_{4}$ : $\ovalbox{\tt\small REJECT}$

HereIexplain the concept of tie transformations usingthis

case.

Atthe firststep of

a

tie transformation

we

make

every

component to the corresponding extendedDynkingraph, and

moreover

weattachthe corresponding coefficient ofthemaximal root to each vertexof the extended graph.

At the second step

we

choosetwo subsets $A$ and $B$ ofvertices

satis-fying th$e$followingthree conditions_$<a>,$ $<b>and<c>$

.

Thus the basic graph $E_{8}+F_{4}$ becomes the graphlike the following.

The conditions:

$<a>A\cap B=\emptyset$

$<b>$ Let $\overline{G}_{0}$ be

an

arbitrarycomponentofthe extendedgraph. Let _$N$ be

the

sum

ofintegers attached to $B\cap\tilde{G}_{0}$. Let

$n_{1},$ $n_{2},$$\cdots,$$n_{l}$.be all the

attached integers to $A\cap\tilde{G}_{0}$

.

Then, G.C._{$D.(N, n_{1}, n_{2}, \cdots, n_{l})=1$}

.

$<c>7he$_{resulting graph after}_{the third} _step isa Dynkingraph.

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togetherwith edgesissuing from them. Next

we

draw

a new

white$ve$rtex

called $\theta$ andconnecteach vertex in $B$to $\theta$by

an

edge.

Under the choice of $A$ and $B$ in the above figure, we get the graph

$E_{7}+B_{6}$

.

This is

an

example of

a

tie transformation.

Now,

we

can

apply

a

transformation

once more.

Atthe second step of the secondtie transformation

we

getthefollowing graph under

a

choice of $A$ and $B$

.

Underthe above choice,

_we

get 2$E_{7}$

as

the resultofthe secondtie

transformation. Note that $2E_{7}$ contains

no

black vertex.

Here Iexplain the roleplayed by blackvertices. Ifwe startfrom

a

graph withablack vertex,we

can

make agraph with

a

black vertex after the second transformation. However, obviously such

a

graph does not correspondto

a

combination ofsingularities

on

a

surface. Thus in the thirdgrade

we

considerthe setof Dynkingraphs with only white vertex madefrom

a

basic graphby two transformation. This setcoincideswith theset of possible combination of singularities

on

surfaces inthegiven

class.

Therefore we

can

conclude that there is

a

deformation fiber of $J_{3,0}$

with $2E_{7}$ singularities.

{defornationfibers of $J_{3,0}$ with only$ADE$ singularities}

$u$

{possiblecombinations of singularities}-- $\{3A_{3}+2A_{2}\}$

$11$

{Dynkin$grap^{\text{ヒ}}hs$ made {Dynkingraphs with

$E_{8}+F_{4}arrow$ from $E_{8}+F_{4}$ } blackvertices} elementary transf.

2 times tie transf.

Here Iconclude the explanation ofmy theory. I do notexplain why these facts

can

hold.

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theory relatedtosimple algebraic

groups.

For the grade II and thegrade III

we

apply the theory of periods of algebraic surfaces. By periods

we

reduce

our

problemto the theory of integral bilinear forms. With the aid of the theory ofreflectiongroups

we can

show theabovefacts.

Itis slightly strange that forthe grade IIand III

no

Lie groups appearin thetheory at present, thoughthe grade I iscloselyrelatedto Lie groups. It is quite natural that

we

expect

a

relation betweenLie

groups

andthe grade II and III.

I hope that

someone

inthe world–perhaps

a

reader ofthis $article-$

can

find outthe relation to Lie groups.

References

[1] Arnold,V.:Localnormal forms of functions. Invent. Math. 35, 87-109 (1976) [2] Kodaira,K.:Oncompactanalytic surfaces II.Ann.of Math.77,563-626(1963) [3]Urabe,T.: On singularitiesondegenerate DelPezzo surfacesofdegree 1,2. Proc.

Symp. Pure Math. 40,(Part2)587-591 (1983)

[4]Urabe,T.: On quartic surfaces andsexticcurves withcertain singularities. Proc.

JapanAcad.,Ser. A 59,434-437(1983)

[5]Urabe, T.:On quartic surfaces andsexticcurves withsingularities oftype

$\tilde{E}_{8},$

$T_{2,3,7},$$E_{12}$.Publ.RIMS. Kyoto Univ. 20, 1185-1245 (1984)

[6]Urabe,T.: Dynkingraphs andcombinations of singularitiesonquartic surfaces. Proc. JapanAcad.,Ser.A61,266-269 (1985)

[7]Urabe,T.: Singulanities inacertainclassof quartic surfaces andsexticcurvesand

Dynkin graphs.Proc. 1984VancouverConf. Alg. Geom.,CMSConf. Proc. 6,

477-497 (1986)

[8]Urabe, T.:Classification of non-normal quartic surfaces. Tokyo J. Math. 9,265-295

(1986)

[9]Urabe,T.: Elementary

_{transformations.of}

Dynkin graphs and singularities on

quartic surfaces. Invent. Math. 87,549-572 (1987)

[10]Urabe, _{T.: Dynkin graphs and combinations of singularities}onplanesextic

curves.In:Randell,R. (ed.) Singularities.Proceedings,Univ. Iowa1986

(Contemporary Math.,vol. 90,pp.295-316)Providence,RhodeIsland: Amer.

Math. Soc. 1989

[11]Urabe,T.: Tietransformations of Dynkin graphsandsingularitiesonquartic surfaces. Invent. Math. 100,207-230 (1990)