数理解析研究所講究録1182 巻 2001 年 25-42

25

NOTE ON CLASSIFICATION OF SEXTICS
DUC TAI PHO
ABSTRACT. We classify the local and global singularities of sextics which are tame torus
curves of type $(2, 3)$ and we also show the degenerations among tllese classes. As an application, two Zariski pairs are found.

INTRODUCTION
The classification of complex algebraic plane curves is a classical problem. For a survey forthe classification of curves of degree less than 6, see [N]. The classification of curves of degree
6 is still not completed, though there are some partial answers for instance in a series papers
by Urabe on sextics which has only simple singularities (see [U] and its references), and thenby Yang extended Urabe’s result in [Y].
According to [Li], singular plane curves of interest are given by(a) a curve which appears as a branch of generic projections of surfaces,(this(b) generic plane sections of discriminants of linear systems on projective space
includes dual curves),(c) a curve defined by explicit equation.
Deformation theory is also useful to prove the existence of curves which can be degenerated
into curves of type (a), (b), (c).
In this paper, we mainly use the method (c) for a special class of sextics, which is so called
torus curves of type $(2,3)$ , and actually these curves have also non-simple singularities, bythat reason we can not use the methods of Urabe or of Yang.
Let $C=\{(X;Y;Z)\in \mathrm{P}^{2}; F(X;\mathrm{Y};Z)=0\}$ be a complex projective plane curve and let$(x, y)$ be the affine coordinates given by $x=X/Z,$ $y=\mathrm{Y}/Z$ on. For simply
we also keep the notation $C$ in the affine chart , where its defining equation $f(x, y)$ is given
by $f(x, y):=F(x, y, 1)$ . Then $C$ is called torus curve of type $(p, q)$ (or $(p, q)$ -torus curve forof degree and respectively
shortly) if we can write
for some polynomialsin. In this paper, we consider reduced $(2,3)$ -torus curve. A sextic $C=\{f_{3}^{2}+f_{2}^{3}=0\}$
of type $(2,3)$ is called tame if its singularities are sitting only at the intersection of conic, whereare respectively defined by $f_{2}=0$ and $f_{3}=0$ .
and cubic
andIn the first section, we assume that conic and cubic passing through the origin of the affine
chart , and $(C, O)$ is an isolated singularity. We will classify all the local singularities at theorigin (in the sense of topological equivalence) by considering the geometrical relation of conic
plays an important role.
and cubic. For that purpose the intersection number
The result is contained in Theorem 1.
In the second section, we study the possible configurations of singularities on tame torus
curves of type $(2,3)$ using the local result obtained in the first section. The main result is
Theorem 2.
In the third section we study the spaces with fixed configurations and the possible degenerations among them. Actually these degenerations will be used in the investigation of topological$\mathrm{P}^{2}$$\mathbb{C}^{2}:=\mathrm{P}^{2}-\{Z=0\}$$\mathbb{C}^{2}$$f=f_{p}^{q}+f_{q}^{p}$$f_{p},$$f_{q}$$p$$q$$\mathbb{C}[x, y]$$C_{2}$$C_{3}$$C_{2}$$C_{3}$$\mathbb{C}^{2}$$\iota=\mathrm{I}(C_{2}, C_{3;}O)$

2000 Mathematics Subject Classification. Primary$14\mathrm{H}20$; Secondary$14\mathrm{J}10,14\mathrm{H}10$.

26, whereproblem, namely, the computation of the fundamental group of the complement$C$$(2,3)$[OP].
in. The details will be given
is a tame torus curve of type
Finally in the last section, we give two new examples of weak Zariski pairs. In each pair,both curves are torus curves of type $(2,3)$ .
We omit several proofs in this paper, the full version is given in [P].
Acknowledgement. The author expresses his deepest appreciation to Professor Mutsuo Oka
for generous help and inspiring guidance.
$\mathrm{P}^{2}\backslash C$

1. LOCAL CLASSIFICATION PROBLEM

1.1. Some classes of singularities. We use the following standard notations for “simple”
singularities, which have the normal form:$A_{n}$$\{$$D_{n}$: $x^{2}+y^{n+1}=0(n\geq 1)$: $x^{2}y+y^{n-1}=0(n\geq 4)$: $x^{3}+xy^{3}=0,$: $x^{3}+y^{4}=0,$$E_{7}$$E_{6}$$E_{8}$: $x^{3}+y^{5}=0$

Furtherlnore we define notations of some other (topological equivalence classes of) singularities
which we use later.
type): $x^{p}+y^{q}=0$ (: $x^{p}+y^{q}+x^{2}y^{2}=0$: $y^{4}+x^{3}y^{2}+x^{7}=0$(1.1)$\mathrm{B}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{s}\mathrm{k}\mathrm{o}\mathrm{r}\mathrm{n}- \mathrm{P}\mathrm{h}\mathrm{a}\ln$$B_{p,q}$$C_{p,q}$$\{$$D_{4,7}$$Sp_{1}$$Sp_{2}$: $(y^{2}-x^{3})^{2}+(xy)^{3}=0$: $(y^{2}-x^{3})^{2}-y^{6}=0$

Note the symmetry $B_{p,q}=B_{q,p},$ $C_{p,q}=C_{q,p}$ and the identities $B_{2,p}=A_{p-1},$ $B_{3,4}=E_{6}$ .
Suppose we have a germ $(C, O)$ of a plane curve. Recall three local invariants: $\mu(C, O)$ is
the Milnor llumber at $O,$ $\delta(C, O)$ is the -invariant which is the maximal llulnber of nodes
in a generic deformation of $C$ , and $r(C, O)$ is the nunlber of the analytic branches. It is
will be used for thewell-known that $\delta=(\mu+r-1)/2$ (see [M]). The invariant triple
later arguments.
$\delta$$(\mu, r, \delta)$

Definition 1.1. Suppose that we have two germs of plane curve singularities $(C,p)$ and$(C’,p’)$ . We say that $(C,p)$ and $(C’,p’)$ are topologically equivalent if there exists a local$U’$ such that $\phi(p)=p’$ andhomeomorphism of the respective ambient neighborhoods$(C,p)\sim(C’,p’)$$\phi(C\cap U)=C’\cap U’$ . We denote this equivalence relation as. When $p=p’=O$ ,let and $f’$ are defining polynomials of $C$ and $C’$ , then we also write $f\sim f’$ .
$U,$$\phi$$f$

For a reduced plane curve germ $(C,p)$ , the topological equivalence type (or shortly topological type) of the germ $(C,p)$ is determined by the following discrete characteristics (for] : the Puiseux pairs of its irreducible components and their linking
instance [Z2, L\^el,numbers. Alternatively, it is known that the embedded resolution graph of $(C,p)$ and themultiplicities of the total transforms of $(C,p)$ at infinitely near points (including ) deterlnine
the topological type.
$\mathrm{B}\mathrm{K}$

$)$

$p$

and $f_{2}(x, y),$ $f_{3}(x, y)$ are the1.2. Setting. Hereafter, $(x, y)$ is the affine coordinates ofby $f_{2}(x, y)^{3}+f_{3}(x, y)^{2}=0$ .
.
sexticisdefinedThe
andofpolynomial
affine defining
and it is an isolated
andis an intersection point of
We assulne that the origin
have noand the cubic
singularity of . This implies, in particular, that the conic
common component (such a line or the conic itself). We will classify the topological types ofand thethe germ $(C, O)$ , using the geometry of the respective singularities of the conic
and their lnutual position.
cubic$\mathbb{C}^{2}$$C_{2}$$O$$C$$C_{3}$$C$$C_{3}$$C_{2}$$C_{2}$$C_{3}$$C_{2}$$C_{3}$

27

1.3. Strategy of classification. We frequently use the fact that the topological type of anon-degenerate germ depends only on its Newton boundary. See [ , O2] for more details.
In our classification, most of the cases can be transformed into non-degenerate forms after
several changes of coordinates, and then the Newton principal part (NPP for short) gives us
andwhich can not be
a normal form. However there are two special cases, namelytransformed into non-degenerate form. For these cases, the topological type can be read offfrom the explicit resolution of the respective singularity.
From now on, we use $(u, v)$ for local coordinates (i.e. in the sense that we use an analyticchange of coordinates), and $(x, y)$ for global coordinates (i.e. affine coordinates). To determine the local type of singularities, it is convenient to use suitable changes of local analytic
coordinates. However to see the existence of a sextic of $(2,3)$ -torus type, it is usually betterto keep the affine coordinates.
We recall a following lemma [O3, lemma 4.3].
$\mathrm{K}$$Sp_{2}$$Sp_{1}$

Lemma 1.2. Assume that $c_{f}=\{(u, v)\in \mathbb{C}^{2};f(u, v)=0\}$ a germ of a smooth curve at
the origin. Let $C_{g}=\{(u, v)\in \mathbb{C}^{2}; g(u, v)=0\}$ be another germ of a curve at the origin.
Let be the multiplicity of at the origin and let $g_{d}(u, v)$ be the homogeneous polynomial of
be positive integers such that $p<dq$ .
degree , which defines the tangent cone of . Let
Consider the germ of a plane curve $C=\{(u, v)\in \mathbb{C}^{2}; f(u, v)^{p}-g(u, v)^{q}=0\}$ . Assume thateach irreducible component of $g_{d}(u, v)=0$ intersects transversely withat the origin. Then
$(C,
O)$
is
coincides with that
topological type ofand the tangential direction at the(1.1).
,
Brieskom-Pham
introduced
insingularity
where
is
the
$f=0$
of
$d$

$g$

$d$

$C_{g}$$p,$ $q$$c_{f}$$B_{p,dq}$$\mathrm{o}r\dot{\mathrm{v}}gin$$B_{p,dq}$

For $(2, 3)$ -torus curve, we apply the above lemma with $(p, q)=(2,3)$ or $(p, q)=(3,2)$ andobtain the following result:Corollary 1.3. With the transverse iniersection condition as above, we have:(i) If $(C_{3}, O)$ is smooth, then $(C, O)$ is of type.
for $\iota=1,2,$(ii) If $(C_{3}, O)$ is singular and $(C_{2}, O)$ is smooth, then $(C, O)$ is of type$2,3,$.
$6$$A_{3\iota-1}$$\ldots,$$B_{3,2\iota}$, for$\iota=$$6$$\ldots,$

1.4. Classification steps. We will divide the situation into 3 cases by the multiplicityof the cubic
at the origin. The simplest case is the case when $(C_{3}, O)$ is smooth.
Case $\mathrm{A}:m_{3}=1$ . In the corollary 1.3 (i), we have to assume that the irreducible components. The claim is also true without thisintersects transversely withof the tangent cone ofcondition.
$m_{3}$$C_{3}$$C_{2}$$C_{3}$

Proposition 1.4. If $(C_{3}, O)$ is smooth, and it intersects the conic$(C, O)$ is of type.

$C_{2}$

with multiplicity$\iota_{f}$then$A_{3\iota-1}$

Proof. Up to an analytic change of coordinates, we may assume that $f_{3}=u$ and the conic has

the following form $f_{2}=cv^{\iota}+uh$ , where$w(u)=3\iota$ and $w(v)=2$ , we can see.
implies $(C, O)\sim A_{3\iota-1}$ , for $\iota=1,$$h\in \mathbb{C}\{u, v\},$ $c\in \mathbb{C}\{v\}$and$c(\mathrm{O})\neq 0$. Putting weights$f=u^{2}+(cv^{\iota}+uh)^{3}=u^{3}+c(0)^{3}v^{3\iota}+\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{e}\mathrm{r}$terms. This$6$$\square$$..,$is ( ( mod $2)+1,$ $[k/2]$ ) for
We recall here that the invariant triple
whereis the greatest integer less than or equal to .
$(\mu, r, \delta)$$[\alpha]$

$k$

$k,$

$A_{k}$

singularity,$\alpha$, suchand a conic
Proposition 1.5. For any $1\leq\iota\leq 6$ there enists a smooth cubic$A_{2}arrow
A_{5}arrow$$(C,O)$
family:degeneration
.
we
have
a
Furthermore
type
is of
that the$C_{3}$$C_{2}$$A_{3\iota-1}$$A_{8}arrow A_{11}arrow A_{14}arrow A_{17}$Proof. See in [P].

$\square$28

Remark 1.6. One can compute the Newton principal part by hand, but usually it is a boring
computation. We make a Maple package SCURVE1 , it can be used for compute invariants ofsingular plane curves, such that intersection number, Milnor number, NPP, toric modification,
etc.
The case $m_{3}=1$ is done by Proposition 1.5.
. We divide Case into two subcases by .
Case
(B-I) $(C_{2}, O)$ is smooth $(m_{2}=1)$ : there are 8 cases, indicated in the following figures, where, and the straight lines are affine lines. We remark here
the dotted lines denote the cubic
may not irreducible globally, however in the cases I-4 and I-8 it must be irreducible
thathas a tangent line which is a component of ).
(because
In the pictures I-3 and I-4, the fat dotted line denotes a line with multiplicity 2.
$\mathrm{B}:m_{3}=2$$\mathrm{B}$$m_{2}$$C_{3}$$C_{2}$$C_{3}$$C_{2}$/$\mathrm{C}2$$–.\mathrm{I}- 9$

FIGURE 1. Case B-I $(m_{3}=2, m_{2}=1)$ .

of the lemma 1.2, and$d=2,\mathrm{s}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{w}\mathrm{e}\mathrm{o}\mathrm{b}\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{n}(C,O)\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{f}B_{3,4}\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{s},\iota=2\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{b}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{g}\mathrm{u}\mathrm{r}\mathrm{e},\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{s}\mathrm{I}- \mathrm{l},\mathrm{I}- 3,\mathrm{I}- 5,\mathrm{I}-.7\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{i}^{r}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}.\mathrm{s}$The next case I-2 is the most interesting case in this classification, and we consider it at
the end of this part.
I-4. We assume $f_{2}=y-x^{2}$ and $f_{3}=y^{2}h$ , where is a linear term and $c:=h(\mathrm{O}, 0)\neq 0$ . We.
have $\iota=4$ . Putting $y_{1}=y-x^{2}$ , in the new coordinates $(x, y_{1})$ , we have$(C,O)\sim
B_{3,8}$.
Thus
is smooth at , we can take a local system of coordinates $(u, v)$ so that $f_{2}=v$ , andI-6. As. Putting weights $w(u)=1$ and $w(v)=2$ , we obtain, where$\iota=3$ and. We can easily see that this polynomial is non-degenerate forbeing two
. Thus $(C, O)\sim B_{3,6}$ . This computation applies also for the case ofanylines.
consists of a line and a conic . We may assume that $f_{2}=y-x^{2}$ andI-8. The cubic$f_{3}=y(y+ax^{2}+bxy+cy^{2})$ , whereor is non-zero. Putting $t_{1}=a+1,$ $t_{2}=b,$ $t_{3}=c$ ,$f_{3}(x, x^{2})=t_{3}x^{6}+t_{2}x^{5}+t_{1}x^{4}$ . Henceis equal to 4, 5 or 6$y=x^{2}$we getand substitute$C_{2;}O)=2,3$ or 4). By this setting we have:(respectively$h$$\mathrm{N}\mathrm{P}\mathrm{P}(f)=y_{1}^{3}+c^{2}x^{8}$$O$$C_{2}$$\mathrm{N}\mathrm{P}\mathrm{P}(f_{3})=\alpha v^{2}-\beta u^{3}$$\alpha,$$\beta\neq 0$$\mathrm{N}\mathrm{P}\mathrm{P}(f)=v^{3}+\beta^{2}u^{6}$$C_{2}$$\beta\neq 0$$C_{3}$$C_{2}’$$\ell$$b$$a$$\iota$$\mathrm{I}(C_{2}’,$

Proposition 1.7. Under the above situation the $ge7m(C, O)$ can be of type
$B_{3,8}arrow B_{3,10}arrow B_{3,12}$ .
$4,5,6$ . Furthermore there is a degeneration family:

Proof. See in [P].
$//\mathrm{w}\mathrm{w}\mathrm{w}$. comp. metro-u.

, for$\iota=$$\square$

1You can get SCURVE by a request
http:$B_{3,2\iota}$$\mathrm{e}$$\mathrm{a}\mathrm{c}.\mathrm{j}\mathrm{p}/\sim \mathrm{p}\mathrm{d}\mathrm{t}\mathrm{a}\mathrm{i}/\mathrm{s}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{v}\mathrm{e}/$-mail to the author or download from the web-page:29

Now we come back to the case I-2.
is defined by $f_{2}=b_{01}y+b_{20^{X^{2}}}+b_{11}xy+b_{02}y^{2}$ and the cubic
We may assume the conic$f_{3}(x,y)=y(a_{11}x+a_{02}y)+a_{30}x^{3}+a_{21}x^{2}y+a_{12}xy^{2}+a_{03}y^{3}$, whereis defined by.
and $a_{11}\neq 0$ . Because $C_{2}\cap\{y=0\}=C_{3}\cap\{y=0\}=\{O\}$ , therefore${\rmRes}_{y}(f)=x^{3}\varphi(x)$andWe have$C_{2}$$b_{01}\neq 0$$C_{3}$$\iota=\mathrm{v}\mathrm{a}1_{x=0}{\rm Res}_{y}(f)$(1.2)$\varphi(0)=b_{01}(b_{01}a_{30}-a_{11}b_{20})(b_{02}a_{02}-a_{03}b_{01})$

Thus $\iota=3$ iff none of these factors is zero.
First we consider the case $\iota=3$ . Putting weights $w(x)=1,$ $w(y)=2$ , by computation
we have NPP $(f;(x, y))$ is a weighted holnogeneous polynomial of degree 6. So that $f(x, y)$ is
non-degenerate if and only if$(27a_{30}b_{01}^{3}-27a_{11}b_{01}^{2}b_{20}-4a_{11}^{3})(a_{30}b_{01}-a_{11}b_{20})\neq 0$

Thus $f\sim B_{3,6}$ for a generic case. Note that the term $(a_{30}b_{01}-a_{11}b_{20})$ is non-zero by (1.2).
Thus to see further degeneration, we solve$(27a_{30}b_{01}^{3}-27a_{11}b_{01}^{2}b_{20}-4a_{11}^{3})=0$.
gets a multiply factor. Namely,inand then$y_{1}=y-(9b_{01}^{2}b_{20}+4a_{11}^{2})x^{2}/(9b_{01}^{3})$$(x,y_{1})$.
Inwith
Thus we need to take the new coordinates
is non-zero. In fact, this coefficient is
as long as the coefficient of
this case we getgiven by$(9b_{01}^{3}y+9b_{01}^{2}b_{20}x^{2}+4a_{11}^{2}x^{2})^{2}$$\mathrm{N}\mathrm{P}\mathrm{P}(f;(x, y))$$a_{30}$$x^{7}$$C_{3,7}$$c(4a_{11}^{2}+9b_{20}b_{01}^{2})(9a_{02}b_{20}b_{01}^{2}-9a_{21}b_{01}^{3}+9a_{11}b_{11}b_{01}^{2}+4a_{02}a_{11}^{2})$where is a non-zero constant.
Thus there are two exceptional cases:$c$$(\iota 3- 1)4a_{11}^{2}+9b_{20}b_{01}^{2}=0$or$(\iota 3- 2)9a_{02}b_{20}b_{01}^{2}-9a_{21}b_{01}^{3}+9a_{11}b_{11}b_{01}^{2}+4a_{02}a_{11}^{2}=0$.

It turns out that the two cases give completely different geometries.
, putting $b_{20}=-c_{20}^{2}$ , thus the equivalence condition is $a_{11}=$In the first case
. Let assume $a_{11}=3b_{01}c_{20}/2$ , we get
are non-zero constants. One canwith(see Case B-II-5 for a detailcheck that this form is topological equivalent to
explanation). Thus $(C, O)\sim C_{3,8}$ . We also obtain same result for the case $a_{11}=-3b_{01^{C}20}/2$ .
member of this family is not irreducible but it is
An important observation is that thequintic.
andaa union of a line
, i.e. $a_{21}=(9b_{01}^{2}b_{20}a_{02}+9b_{01}^{2}a_{11}b_{11}+4a_{02}a_{11}^{2})/(9b_{01}^{3})$ . We
Next we consider the caselook at the Newton principal part of with respect to the coordinates $(x, y_{1})$ . It has two
faces, $AB$ and $BC$ where $A=(0,3),$ $B=(2,2)$ and $C=(8,0)$ . The face function with respectof the face function with respect to $BC$
to $AB$ is non-degenerate, while the discriminallt inis$(\iota 3- 1)$$\pm 3b_{01^{C_{20}}}/2$$\mathrm{N}\mathrm{P}\mathrm{P}(f;(x, y_{1}))=c_{1}x^{5}y_{1}+c_{2}x^{2}y_{1}^{2}+c_{3}y_{1}^{3}$$c_{1},$$c_{2},$$c_{3}$$x^{8}+x^{2}y_{1}^{2}+y_{1}^{3}$$gene\mathit{7}\dot{\mathrm{V}}C$$(\iota 3- 2)$$f$$y_{1}$$d=cx^{10}(4a_{11}^{2}+9b_{01}^{2}b_{20})^{2}(-9b_{11}^{2}b_{01}^{2}-16a_{11}a_{12}b_{01}+16b_{02}a_{11}^{2}),$$(c\neq 0)$, so the generic caseis already considered in the casetake place if the last factor of , say , is non-zero. In that case we have $(C, O)\sim C_{3,8}$ .
a generic member of this class isin caseIn colnparison with the previous class ofirreducible.
Finally if $d_{2}=0$ , it turlls out that $(C, O)\sim C_{3,9}$ . And it does not degenerate any further (as).
long as $(C_{2}, O)$ is smooth and $(C_{3}, O)$ is. We remark
andSummary: In the case $\iota=3$ there are 4 possibilities
that in this case, the intersection lnultiplity is not enough to determine the topology.

Since the factor$(\iota 3- 1)$$4a_{11}^{2}+9b_{01}^{2}b_{20}$$d$$d_{2}$$C_{3,8}$$(\iota 3- 1)$$A_{1}$${}_{7}C_{3,8}$${}_{6}C_{3},,$$B_{3},,$$C_{3,9}$30

is necessarily a smooth conic.
. By Bezout theorem,
Next we consider the caseThus we can assume that $f_{2}(x, y)=y-x^{2}$ . We have $a_{30}=-a_{11}$ and$\iota\geq 4$4$\iota=\{$ 56$C_{2}$

iff
iff
iff

$a_{02}+a_{21}\neq 0$$a_{02}+a_{21}=0,$ $a_{12}\neq 0$$a_{02}+a_{21}=a_{12}=0,$ $a_{03}\neq 0$

Let consider the case $\iota=4$ . We take the coordinate change $(x, y_{1})$ with $y_{1}=y-x^{2}$ . The-Newton principal part is given by
NPP $(f;(x, y_{1}))=(a_{02}+a_{21})^{2}x^{8}+2a_{11}(a_{02}+a_{21})x^{5}y_{1}+a_{11}^{2}x^{2}y_{1}^{2}+y_{1}^{3}$
It is degenerate for the weight $P={}^{t}(1,3)$ , where $f_{P}=x^{2}(x^{3}a_{21}+x^{3}a_{02}+a_{11}y_{1})^{2}$ .
the coordinate change $(x, y_{2})$ with $y_{2}=x^{3}a_{21}+x^{3}a_{02}+a_{11}y_{1}$ . We have$\mathrm{T}\mathrm{a}\mathrm{k}\mathrm{i}_{1\mathrm{l}}\mathrm{g}$

NPP $(f;(x, y_{2}))=-(a_{02}+a_{21})^{3}x^{9}/a_{11}^{3}+x^{2}y_{2}^{2}+y_{2}^{3}/a_{11}^{3}$
Hence for genericmethod, we get(i.e. $a_{02}+a_{21}\neq 0$ ) we have $(C, O)\sim C_{3,9}$ . Continuing by the samefor $\iota=5$ and 6 respectively.
and$a_{ij’}s$$C_{3,12}$$C_{3,15}$, such that the $(C, O)$ is of typeand a conic
Proposition 1.8. There exists a cubic(for $k=3,4,5$ ). Furthermore there are the following(for $k=7,8,9$) anddegeneration families:
with the same $\iota=3$ .
(i)(ii) $B_{3,6}arrow C_{3,9}arrow C_{3,12}arrow C_{3,15}$ , with : $3arrow 4arrow 5arrow 6$ .
$C_{2}$$C_{3}$$B_{3,6},$$C_{3,3k}$$C_{3,k}$$B_{3,6}arrow C_{3,7}arrow C_{3,8}arrow C_{3_{2}9\mathrm{z}}$$\iota$Proof. See in [P].

$\square$with
Remark 1.9. (a) There are two moduli components for the singularity classes$\iota=3$ . The generic curve in the component corresponding to the casedecomposes into
for the singularity in this moduli.
a line and a quintic, later we will use the notation, with different intersection(b) There are two components for the moduli space of the classfor the case $\iota=4$ .
numbers. Later we will distinguish by the new notation(c) There are more degenerations for the singularities if we admit to increase or if we. For example, we can degenerate
andadmit the degenerations for the singularities of$C_{3,9}arrow
B_{3,12}$$C_{3,8}arrow
B_{3,10}$by increasing the intersection$C_{3,7}arrow B_{3,8}arrow B_{3,10}arrow B_{3,12},$and. For the explicit construction, see \S 3.
degenerates also into
number .
$C_{3,8}$$(\iota 3- 1)$$C_{3,8}^{\#}$$C_{3,9}$$C_{3,9}^{\mathfrak{h}}$$\iota$$C_{2}$$B_{4,6}$$B_{3,8}$$\iota$$C_{3}$

is singular $(m_{2}=2)$ . There are 9 cases, indicated in Figure 2, where thestraight lines are again affine lines.
are 2 distinct lines, but 2 lines
Attention. In the cases II-1,5,8,9 though in the pictures oflnay also coincide.
II-1. Up to an analytic change of coordinates we may assume $f_{3}=uv$ , and $f_{2}=(a_{1}u+b_{1}v+$
for $i=1,2$ . Here$h_{1})(a_{2}u+b_{2}v+h_{2})$ , where
are non-zero constants, and$\iota=4$. Putting weights $w(u)=w(v)=1$ , we get. We haveis the maximal ideal in$f\sim u^{6}+v^{6}+u^{2}v^{2}$ . Thus $(C, O)\sim C_{6,6}$ .
transversely
II-2. Since conic consists of 2 distinct lines, say and , where intersects. Therefore
andand is a tangent cone direction of , hence$\iota=5$$f_{2}=(a_{1}u+b_{1}v+h_{1})(a_{2}u^{2}+b_{2}v+h_{Q})$. And weso that
we may assume $f_{3}=uv$ , andobtain $(C, O)\sim C_{6,9}$ .
II-3,4. In these case, the illtersection number $\iota=6$ . By the same way, we get respectively$(\mathrm{B}-|\mathrm{I}\mathrm{I})(C_{2}, O)$$C_{2}$$a_{i},$$h_{i}\in \mathfrak{m}^{2}$$b_{i}$$\mathfrak{m}$$\mathbb{C}\{u, v\}$$\ell_{1}$$C_{3}$$p_{2}$${}_{9}C_{6,12}$$C_{9},,$.

$\ell_{2}$$\mathrm{I}(\ell_{1}, C_{3;}O)=2$$\ell_{1}$$C_{3}$$\mathrm{I}(\ell_{2}, C_{3;}O)=3$31

FIGURE 2. Case B-II $(m_{3}=m_{2}=2)$ .
Proposition 1.10. Under the situations II-I\sim II-4, the type of $(C, O)$ can beand. Furthermore there are the following degeneration families:$C_{6,6},$$C_{6,9},$$C_{9,9}$$C_{6,12}$$C_{6,6}arrow C_{6,9}arrow C_{9,9}$$C_{6,9}arrow C_{6,12}$

Proof. See in [P].

$\square$In the next three cases II-5\sim II-7, the cubic
has a cusp, and by a lirear change of coordinates
we may assume that
, i.e. $f_{3}=y^{2}-x^{3}+h$ where is the higher term,
and $f_{2}=(a_{1}x+b_{1}y)(a_{2}x+b_{2y})$ .
$a_{2}\neq 0$ , the intersection number $\iota=4$ , we may
II-5. This caseassume $a_{1}=1$ . We have
, and its discriminant in. Hence
if $1+a_{2}^{3}\neq 0$ , we have $(C, O)\sim B_{4,6}$ .
If $1+a_{2}^{3}=0$ , then NPP $(f)=3a_{1}^{2}(a_{1}b_{2}+b_{1})x^{5}y-2x^{3}y^{2}+y^{4}$ , the isolated singularity condition
requires the term $3a_{1}^{2}(a_{1}b_{2}+b_{1})x^{5}y$ does not vanish. Thus $f\sim x^{5}y+x^{3}y^{2}+y^{4}$ . So we get$(C, O)\sim D_{4,7}$ (because $x^{5}y+x^{3}y^{2}+y^{4}\sim x^{7}+x^{3}y^{2}+y^{4}$ ).
We relnark that the intersection multiplity is not enough to determine the topology of$(C, O)$ like in Case B-I-2.
II-6. One branch of the conic is tangent to $(C_{3}, O)$ , we may assume that
$C_{3}$$\mathrm{N}\mathrm{P}\mathrm{P}(f_{3})=y^{2}-x^{3}$$h$$a_{1},$$\mathrm{N}\mathrm{P}\mathrm{P}(f)=(1+a_{2}^{3})x^{6}-2x^{3}y^{2}+y^{4}$$f_{2}(x, y)=axy$ ,$x\mathrm{i}\mathrm{s}-46656(1+a_{2}^{3})^{2}a_{2}^{9}y^{20}$$f_{3}(x, y)=y^{2}-x^{3}+b_{1}yx^{2}+b_{2^{X}}y^{2}+b_{3y^{3}}$In this case $\iota=5$ . This is a degenerate singularity, since. Taking acanonical toric modification, we find that $\pi_{1}^{*}f=0$ has again a $(2,3)$ -cusp and:it has a simultaneous resolution by one more toric modification for any$i=1,2,3$ .
The tower of the weight vectors in the sense of [AO] is given by $\{^{t}(2,3),{}^{t}(2,3)\}$ . Thus the
topology of $(C, O)$ does not depend on the parameters $a\neq 0$ and. Moreover $(C, O)$
is locally irreducible and its Puiseux pairs is $P(C, O)=\{(2,3), (2,9)\}$ by [O1]. By taking$a=1,$ $b_{1}=b_{2}=b_{3}=0$ , we get $f=(y^{2}-x^{3})^{2}+(xy)^{3}$ . Thus $(C, O)\sim Sp_{1}$ . The resolution
diagram ofis given in Figure 3. The Milnor number is 18 by [A2] or [AO].
II-7. The conic is a double line and it coincides with the tangent direction of $(C_{3}, O)$ . We
assume that $f_{2}(x, y)=-a^{2}y^{2},$ $a\neq 0$ and $f_{3}(x, y)$ is as above. The intersection number$\iota=6$ . By the same method ascase, we obtain this singularity is equivalent toandthe resolution graph is given in Figure 4. We see thathas 2 irreducible components$f_{3}\pm a^{3}y^{3}=0$ , they have same Puiseux pair $\{(2,3)\}$ and theirlinking number is 9. Thus$\mu(Sp_{2})=21$ and $\delta(Sp_{2})=11$ .
$\mathrm{N}\mathrm{P}\mathrm{P}(f)=(y^{2}-x^{3})^{2}$$\pi_{1}$$X_{1}arrow \mathbb{C}^{2}$$b_{i}\in \mathbb{C},$$b_{1},$$b_{2},$$b_{3}$$Sp_{1}$$Sp_{1}$$Sp_{2}$$Sp_{2}$

Proposition 1.11. There are following degeneration families:(i) $B_{4,6}arrow D_{4,7}$ in Case II-5.

32

FIGURE 3. Resolution graph of$Sp_{1}$.

FIGURE 4. Resolution graph of$Sp_{2}$.

(ii) $B_{4,6}arrow Sp_{1}arrow Sp_{2}$ corresponding to.

$\mathrm{I}\mathrm{I}- 5arrow \mathrm{I}\mathrm{I}- 6arrow \mathrm{I}\mathrm{I}- 7$

Proof. The proof is obvious ffom the explanations in Cases II-5,6,7.

$\square$II-8. Assuming $f_{3}=y(y-x^{2})$ and $f_{2}=a(x+by)(x+cy)$ , where is non-zero. We have
. Thus $(C, O)\sim B_{4,6}$ .
$f_{3}=x^{2}(ax+by+k)$
and $f_{2}=(cx+y)(dx+y)$ , where is non-zero. We haveII-9. Assuming$(C,O)\sim
B_{4,6}$.
. ThusFinally we consider singular cubic with multiplicity 3 $(m_{3}=3)$ .
. Similarly, we also divide 2 cases by the multiplicity of the conic.
Case(C-I) Conic.
is smooth at . Obviously we haveC-I-1. $\iota=3$ , applying the corollary 1.3, we get $(C, O)\sim B_{3,6}$ .
Now we consider the case
. Then is irreducible, as $(C_{2}, O)$ is assumed to be smooth.
Therefore we may assume that $f_{2}=y-x^{2}$ and $f_{3}=(a_{1}x+b_{1}y)(a_{2}x+b_{2}y)(a_{3}x+b_{3y})$ . We
have is equal to the lowest degree in of $f_{3}(x, x^{2})=x^{3}(a_{1}+b_{1^{X}})(a_{2}+b_{2^{X}})(a_{3}+b_{3^{X}})$ .
we may assume $a_{1}=0$ andare none-zero. Putting
C-I-2. $\iota=4$ , by symmetry of$y_{1}=y-x^{2}$ , then
. Since $a_{1}=0$ thentermsmust be non-zero. Thus $(C, O)\sim B_{3,8}$ .
Similarly, we obtain: result is $(C, O)\sim B_{3,10}$ .
C-I-3. $\iota=5$ if $a_{1}=a_{2}=0$ and$\iota=6$
if $a_{1}=a_{2}=a_{3}=0$ : result is $(C, O)\sim B_{3,12}$ .
C-I-4.
(C-II) Conicis singular. We have $\iota=6$ . Since both conic and cubic are products of linear
terms, then is a homogeneous polynomial of degree 6. Since
is an isolated singularity,then should be non-degenerate and it is a product of linear terms. Thus $(C, O)\sim B_{6,6}$ .
$a$$\mathrm{N}\mathrm{P}\mathrm{P}(f)=y^{4}+a^{3}x^{6}$$k$$\mathrm{N}\mathrm{P}\mathrm{P}(f)=y^{6}+k^{2}x^{4}$$\mathrm{C}:m_{3}=3$$C_{2}$$O$$\iota\geq 3$$\iota\geq 4$$C_{2}$$x$$\iota$$a_{i}’s$$f(x, y_{1})=y_{1}^{3}+b_{1}^{2}a_{2}^{2}a_{3}^{2}x^{8}+\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{e}\mathrm{r}$$a_{2},$ $a_{3}$$\sim y_{1}^{3}+b_{1}^{2}a_{2}^{2}a_{3}^{2}x^{8}$$b_{1}$$a_{3}\neq 0$$C_{2}$

$O$

$f$

$f$

1.5. Statement of the result on the local classification. Now we can state the result inthe local classification:
Theorem 1 (Local Classification). Let $C=\{f=f_{2}^{3}+f_{3}^{2}=0\}$ is a tame $(2,3)$ -torus curve.
Put $C_{i}=\{f_{i}=0\}$ for $i=2,3$ . The topological type of the germ $(C, O)$ can be read off fromandas follows:
1. Cubic $(C_{3}, O)$ is smooth: $A_{3\iota-1}(\iota=1, \ldots, 6)$ .
2. Cubic is not smooth at (i.e., $m_{3}\geq 2$ ).
$C_{2}$$C_{3}$$O$

33(a) Conic $(C_{2}, O)$ is smooth.
(i) $(C_{3}, O)$ is
and $\iota=2:E_{6}=B_{3,4}$ .
(ii) $(C_{3}, O)$ is
and $(C_{2}, O)$ is tangent to one of the branch:
for $\iota=3$ , and$\iota=4,5,6$
for.
$\iota=2,$(iii) $(C_{3}, O)$ is not:.
(b) Conic $(C_{2}, O)$ is:(i) $(C_{3}, O)$ is:$\iota=4$, no common tangential cone,
for
for $\iota=5$ , one common tangential cone,
for $\iota=6$ , two common tangential cones.
(ii) $(C_{3}, O)$ is eitheror:, for $\iota=4$ , no common tangential cone,or
if $\iota=5$ and $(C_{3}, O)=A_{2}$ and the tangential cones coincide.
(c)is a line with multiplicity 2:(i) $(C_{3}, O)$ is a:for $\iota=4$ , no common tangential cone,
if $\iota=6$ , a colnmon tangential cone.
(ii) $(C_{3}, O)$ is eitheror:$\iota=4$, no common tangential cone,
for
for $\iota=6,$ $(C_{3}, O)$ is, the same tangential cone.
(d) Cubic
consists of three lines passing through , andconsists of two lines
passing through $O:B_{6,6}(\iota=6)$ .
$A_{1}$$A_{1}$${}_{6}C_{3,7},$$B_{3},,$$C_{3,8},$$C_{3,9}$$C_{3,3\iota-3}$$A_{1}$$6$$B_{3,2\iota},$$\ldots,$$A_{1}$$A_{1}$$C_{6,6}$$C_{6,9}$$C_{9,9}$$B_{4,6}$$A_{2}$$A_{3}$$A_{2}$$A_{3}$$D_{4,7}$$Sp_{1}$$C_{2}$$A_{1}$$B_{4,6}$$C_{6,12}$$B_{4,6},$ $D_{4,7}$$Sp_{2}$$A_{2}$$O$$C_{3}$$C_{2}$

Remark 1.12. When $(C_{3}, O)$ is a node (i.e.,),
can be reduced (either a line and aconic meeting transversely ator three lines where two of them are passing through ).
$(C_{3}, O)$ is
if and only if
consists of a line and a conic which are tangent at O. $(C_{2}, O)$
has a node if it consists of two lines.
$A_{1}$$C_{3}$$O$$A_{3}$$O$$C_{3}$

Theorem 1-D (Local Degenerations). Under the notation in Theorem 1, we have following
degellerations:
1. $(C_{3}, O)$ is smooth: $A_{2}arrow A_{5}arrow A_{8}arrow A_{11}arrow A_{14}arrow A_{18}$
2. $(C_{3}, O)$ is
and $(C_{2}, O)$ is slnooth:$A_{1}$(a) $B_{3,4}arrow B_{3,6}arrow C_{3,7}arrow C_{3,8}arrow C_{3,9}$(b) $B_{3,4}arrow B_{3,6}arrow C_{3,9}^{\#}arrow C_{3,12}arrow C_{3,15}$$(C_{2}, O)$ is smooth:
3. $(C_{3}, O)$ is$B_{3,4}arrow B_{3,8}arrow B_{3,10}arrow B_{3,12}$or$(C_{2}, O)$ is
4. $(C_{3}, O)$ is: $B_{4,6}arrow D_{4,7},$ $B_{4,6}arrow Sp_{1}arrow Sp_{2}$ .
$(C_{2}, O)$ is
5. $(C_{3}, O)$ isor: $C_{6,6}arrow C_{6,9}arrow C_{9,9},$ $C_{6,9}arrow C_{6,12}$ .
$A_{2}$$A_{3},$$A_{1},$$A_{2}$

The$\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{i}_{1\mathrm{l}}\mathrm{g}$$A_{1}$$A_{3},$$A_{1}$

is well known.

Proposition 1.13. Let $C$ be a reduced curve of degree in
defined by $F(X, Y, Z)$ . Then
$0\leq t\leq 1$
there is a family
curvesdegreesuchthat $C_{t}\cong C$ for $t\neq 0$ and
for
of
of
where
is the class of lines meeting at .
$d$

$d$

$C_{t}$$C_{0}\cong B_{d,d}$$B_{d,d}$$\mathrm{P}^{2}$

$d$

$O$

Proof. We follow the method in [OS].

We may assume that the line at infinity $Z=0$
transversely. Take $F(X, Y, Z, t):=F(X/t, Y/t, Z)t^{d}$ and define the family$\{F(X, Y, Z, t)=0\}$ .

meets

$C$

$C_{t}\square =$Remark 1.14. For the sake of the global study of sextics, we distinguish
also
alld
though they are locally topologically equivalent.
$C_{3,9}$$C_{3,9}^{\mathfrak{h}}$$C_{3,8}$

and$C_{3,8}^{\#}$

, and

34

Table

Table
$\iota$

1
2
3
4
5

6

$m_{3}=1$$\mathrm{A}’$$\mathrm{A}$: Local classification: The local degeneration series

35

2. GLOBAL CLASSIFICATIONIn this section, we consider the possible combination of the local singularities of reduced
tame $(2,3)$ -torus curves, using the local classification obtained in Section 1.
Assume that $C$ is an irreducible curve of degree and denote the set of singularities of $C$. We recall the genus formulaby$d$$\Sigma(C)$

or

(2.1)

$g= \frac{(d-1)(d-2)}{2}-\sum_{P\in\Sigma(C)}\delta(C, P)\geq 0$(2.2)$\chi(C’)=3d-d^{2}+\sum_{P\in\Sigma(C)}\mu’(C, P)\leq 2$

where $C’$ is the normalization of $C,$ $\chi(C’)$ is the topological Euler characteristic, $\mu’(C, P)$ $:=$$\mu(C, P)+r(C, P)-1$ and $r(C, P)$ is the number of analytic branches at $P$ (see [BK]). We call$\mu’(C, P)$ the normalized Milnor
defect of $C$ at $P$ . When is a tame $(2,3)$ -torus curve, thenis given by $C_{2}\cap C_{3}=:\{P_{1}, P_{2}, \ldots, P_{n}\}$ . Therefore (2.1) is equivalent
its singularities set
to the following$C$$\Sigma(C)$(2.3)$\sum_{i=1}^{n}\delta(C, P_{i})\leq(6-1)(6-2)/2=10$, we call
DenoteBezout theorem, we have$i_{k}:=\mathrm{I}(f_{2}, f_{3;}P_{k})$$(i_{1}, \ldots, i_{n})$

(2.4)

.

is an -vector and$\sum_{i=1}^{n}i_{k}=6$

$i$

$n$

the length of -vector. By$\mathrm{i}$.

is reducible curve with $r(C)$ irreducible components, the inequality (2.1) does not
Whenhold and (2.2) has to be replaced by the following.
$C$

(2.5)

$\chi(C’)=3d-d^{2}+\sum_{P\in\Sigma(C)}\mu’(C, P)\leq 2r(C)$Our strategy for the global classification is the following steps:
Step 1. Consider every possible -vector, which satisfies (2.4).
Step 2. List up every possible combination of the local singularities having prescribed i-vector.
Step 3. Prove or disprove the existence of a reduced tame $(2,3)$ -torus curve with the configurations. In this stage, if the inequality (2.2) is not satisfied, we have to look for the reduced
curves with the given configuration.
For the later discussion, we recall$\mathrm{i}$Lemma 2.1. The possible topological types of a singularity on an irreducible quartic curve
.
and
are
$A_{1},$$\ldots,$$A_{6},$$D_{4},$$D_{5}$$E_{6}$

Proof. See for instance in [W]$\square$From the above strategy we can enumerate singularities as follows:

Theorem 2 (Global Classification). The configuration of singularities of a tame torus curves
is for the cases of reducibleof type $(2, 3)$ is given by Table , where the notation
.
$C$
$C$

$\mathrm{B}$$\{\Sigma(C)\}^{\star}$

36

Table

$\mathrm{B}$: Global classification

Proof. We will prove the assertion by three steps (corresponds to 3 steps in the strategy) as
follows.
Step: The -vector of length 6 (1, 1, 1, 1, 1, 1) is obviously given under the genericsituation where
andintersect transversely.
The -vector with $n=5$ is given by (1, 1, 1, 1, 2) and the possible configurations are
and.
The -vectors with $n=4$ is either (1, 1, 1, 3) or (1, 1, 2, 2). The possible configurations are(1, 1, 1, 3):, where$\{2A_{2},2E_{6}\}$(1, 1, 2, 2): $\{2A_{2},2A_{5}\},$The -vectors with $n=3$ are (1,1,4), (1,2,3), (2,2,2) and the possible configurations are(1, 1, 4):, where $\sigma=A_{11},$(1, 2, 3):, where(2, 2, 2):$1+\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{p}2$$C_{2}$$\mathrm{i}$$C_{3}$$\mathrm{i}$$\{A_{5},4A_{2}\}$$\{E_{6},4A_{2}\}$$\mathrm{i}$$\{3A_{2}, \sigma\}$${}_{8}C_{3}^{\#},’ {}_{8}C_{3,9}$$\sigma=A_{8},$ $B_{3},,$${}_{7}C_{3},,$${}_{6}C_{3},,$$\{2A_{2}, A_{5}, E_{6}\},$$\mathrm{i}$$\{2A_{2}, \sigma\}$$\{A_{2}, A_{5}, \sigma\},$$C_{3,9}^{\mathfrak{h}},$$\{A_{2}, E_{6}, \sigma\}$${}_{8}C_{6,6},$ $B_{4,6},$ $D_{4,7}$$B_{3},,$${}_{8}C_{3}^{\#},’ {}_{8}C_{3,9}$$\sigma=A_{8},$ $B_{3},,$${}_{7}C_{3},,$${}_{6}C_{3},,$$\{3A_{5}\},$ $\{3E_{6}\},$ $\{2A_{5}, E_{6}\},$ $\{A_{5},2E_{6}\}$

37

The -vectors with $n=2$ are $(1,5),(2,4),(3,3)$ and the possible configurations are$(1, 5)$ :, where $\sigma=A_{14},$$\sigma=A_{11},$$(2, 4)$ :where $\tau=A_{5},$$(3, 3)$ :where $\xi=A_{8},$For the case $n=1$ , we have the following obvious possibility.
(6) : $\{A_{17}\}.’\{C_{3,15}\},$
Step 3. Existence and Non-existence.
$n=6$ : The configuration
is given by generic conic and cubic which intersect transversely
and this case was first studied by Zariski, [Z1].
The configurationoris given by the generic member of the moduli
of the singularity
ordiscussed in \S 1.
$n=4$ : Any configurations
where $\sigma=A_{8},$are obtained by a,generic curve in the moduli of singularity at the origin. This holds also forbut we notice that a generic member of this class consists of a line and a quintic, though itsatisfies the inequality (2.2).
The configuration $\{2A_{2},2A_{5}\}$ is given by a smooth conic
and a smooth cubic
which
are tangent at two points and have two other transverse intersections. The configurations$\{2A_{2}, A_{5}, E_{6}\}$ and $\{2A_{2},2E_{6}\}$ are obtained by the similar device.
The existence of these
configuratiolls also shown by explicit equations, in Proposition 3.1 of Section 3.
,
For (1,1,4), the existence of the configurations
where $\sigma=A_{11},$is proved as the above. Similarly the configuration
exists and an example isgiven by $f_{2}=(x+y)(-x+2y),$ $f_{3}=y^{2}-x^{3}$ . In this case $C$ has two irreducible colnponents,
a lineand a quintic F5. Note thathas a singularityatand 2
singularities.
Now we consider the configuration with -vector (2,2,2).
The configuration
is given for instance by $f_{2}=(x-1)^{2}+y^{2}-1$ and $f_{3}=x(y^{2}-1)$ .
Note thatconsists of three simple tangent points. The configuration
is given
by $f_{2}=(x-1)^{2}+y^{2}-1$ and $f_{3}=(x-1)(x^{2}-y^{2})$ . This case was studied by Oka [O4]. The
existence of the configurations $\{2A_{2}, E_{6}\},$ $\{A_{5},2E_{6}\}$ is shown in Proposition 3.1.
Now we consider the configurations with -vector (1,2,3). Examples of the configurati’ons
andwith $\sigma=A_{8},$are given in Proposition 3.1.
The configuration
exists and an example is given by $f_{2}=y-x^{2},$ $f_{3}=$$(-23x^{3}-15yx^{2}+27xy-16y^{3}+27y^{2})/27$ . In this case, $C$ has two irreducible components,
a conic
which is defined by $5x^{2}-16xy-16y^{2}-9y=0$ and a quartic
which definedby $40x^{4}-10yx^{3}-117yx^{2}+51y^{2}x^{2}+18y^{2}x-16xy^{3}+81y^{2}+16y^{4}-63y^{3}=0$. Note that$(C, A)$ is
where $(C, O)$ isand
has two singularities,
at andat $B=(1,1)$ .
exists, by taking $f_{2}=y-x^{2},$ $f_{3}=2y^{3}+(3/2)yx-$
The configuration$(7/2)xy^{2}+yx^{2}-x^{3}$ . In this case,is also has two irreducible components: they are a line$\Gamma_{1}=\{y=0\}$ and a quintic. The line is a tangent direction
with multiplicity 5 at
, and they have no other intersection point except . Moreover,has 3 singularities, an
at $(-1/4,1/16)$ .
at , anat $(1, 1)$ and an
exists, by taking $f_{2}=y^{2}+y(x-1/4)-x^{2},$ $f_{3}=$The configuration$y(-(3/8)x+(9/16)y)-x^{3}+(19/8)x^{2}y-(63/32)y^{3}’$ . Againis a union of a line and aquintic.
exist byreason as the cases
The configurations with respect to -vector $(1,5)$(1,1,4), (1,1,1,3) or (1,1,1,1,2).
$\mathrm{i}$$\{A_{2}, \sigma\}$$C_{3,12},$ $B_{3,10},$ $C_{6,9},$ $Sp_{1}$$C_{3,9}^{\#},$$E_{6}$$\{\tau, \sigma\}$$\{\tau, \xi\}$${}_{8}C_{6,6},$ $B_{4,6},$ $D_{4,7}$$B_{3},,$${}_{8}C_{3}^{\#},’ {}_{8}C_{3,9}$${}_{7}C_{3},,$${}_{6}C_{3},,$$B_{3},,$$\tau,$$\{C_{9,9}\},$ $\{B_{3,12}\},$ $\{C_{6,12}\},$ $\{Sp_{2}\},$ $\{B_{6,6}\}$$\{6A_{2}\}$$\{4A_{2}, \sigma\},$$\underline{n=5.}$$A_{5}$$\sigma=A_{5}$$E_{6}$$E_{6}$$\{3A_{2}, \sigma\}$${}_{8}C_{3,9}$${}_{6}C_{3},,$${}_{7}C_{3},,$$B_{3},,$$\{3A_{2}, C_{3,8}^{\#}\}^{\star}$$\sigma$$C_{2}$$C_{3}$$\{2A_{2}, \sigma\}$$\underline{n=3.}$$C_{3,9}^{\#},$${}_{8}C_{6,6}$$B_{3},,$$\{2A_{2}, D_{4,7}\}^{\star}$$B_{4,6}$$\Gamma_{1}\cap\Gamma_{5}=\{O\},$$\Gamma_{1}$$E_{7}$$\Gamma_{5}$$O$$A_{2}$$\mathrm{i}$$\{3A_{5}\}$$C_{2}\cap C_{3}$$\{3E_{6}\}$$\mathrm{i}$$\{A_{2}, A_{5}, \sigma\}$$\{A_{2}, E_{6}, \sigma\}$${}_{6}C_{3,7}$$B_{3},,$$\{A_{2}, A_{5}, C_{3,8}\}^{\star}$$\Gamma_{2}$$\Gamma_{4}$$\Gamma_{2}\cap\Gamma_{4}=\{O, A\}$

$A_{5}$$C_{3,8},$$A_{3}$$\Gamma_{4}$$A_{2}$$\{A_{2}, A_{5}, C_{3,8}^{\#}\}^{\star}$

$C$

$O$

$A_{3}$$\dot{\mathrm{o}}\mathrm{f}\Gamma_{5}$$\Gamma_{1}$$\Gamma_{5}$

$O$

$O$

$A_{5}$$\Gamma_{5}$$A_{2}$$\{A_{2}, E_{6}, C_{3,8}^{\#}\}^{\star}$$C$$\acute{\mathrm{d}}\mathrm{o}$$\underline{n=2.}$$\mathrm{i}$simil.ar$O$

38

Now we consider the configuration with -vector $(2,4)$ . Examples of the configurations
are given in Proposition 3.1.
with $\sigma=A_{11},$andThere are 3 more configurations for -vector is $(2,4)$ , in these cases the conic is a line withmultiplicity 2. Hence at least has two factors. Explicit examples are given as follows.
$\{A_{5}, B_{4,6}\}^{\star}:f_{2}(x, y)=x^{2}$ and $f_{3}(x, y)=y^{2}-x^{3}$ . $C$ has two cubic components, defined by$y^{2}-(1\pm\sqrt{-1})x^{3}=0$ . The singularities ofat $(0,1)$ .
and anatare a$\{A_{5}, D_{4,7}\}^{\star}:f_{2}=-(x-y)^{2}$ and $f_{3}=y^{2}-x^{3}$ . In this case, $C$ has 3 irreducible components,
at and an
and a cuspidal cubic F3. The singularities of are aa line , a conic
intersects F3 at and $(1, 1)$ with respective intersection multiplicity 3 and 1,
at $(1, 1)$ .
at .
andis tangent to bothwhile
$f_{3}=(y-1)(x^{2}-y^{2})$$f_{2}=-x^{2}$
has 2 irreducible. In this case,and:at and ancomponents, both of them are nodal cubics. The singularities of are aat $(0,1)$ .
are givenand-vector is $(3,3)$ : Examples of the configurationsin Proposition 3.1.
is given by $f_{2}=y-x^{2},$ $f_{3}=-x^{3}+x^{2}y+(9y+16y^{2})x/6+$
An example ofis a tangent$y^{2}+32y^{3}/27$ . In this case,and a quintic , whereis a union of a line
and anatareaof. The singularities
at , anddirection ofat $(1,1)$ .
is given by $f_{3}=x(x-2)y,$ $f_{2}=y^{2}+x^{2}-2x$ . In this case, $C$ is
An example ofeach other other at two points and, and they ,a union of three smooth conics$A=(2,0)$ with multiplicity 2.
The existence in these cases is obvious from the local classification. The explicit equations for irreducible curves are given in Proposition 3.1. Explicit examples for the reducible
cases are given as following.
,
consists of three smooth conics: Let $f_{2}=y-x^{2},$ $f_{3}=y^{3}$ . In this case,4.
multiplicity
withwhich are tangent to each other at: Let $f_{2}=-y^{2},$ $f_{3}=x(y-x^{2})$ . In this case, $f=(x^{3}-xy-y^{3})(x^{3}-xy+y^{3})$ , thus
$C$ is a union of two nodal cubics.
: Let $f_{2}=-y^{2},$ $f_{3}=y^{2}-x^{3}$ . In this case, $f=(x^{3}-y^{2}-y^{3})(x^{3}-y^{2}+y^{3})$ , thus
is a union of two cuspidal cubics.
: Let $f_{2}=y^{2},$ $f_{3}=x^{3}$ . Thus $C$ consists of 6 concurrent lines.
To complete the proof of Theorem 2, it suffices to show the non-existence of the relnaining
configurations.
$\mathrm{i}$$\{E_{6}, \sigma\}$$\{A_{5}, \sigma\}$$C_{3,9}^{\mathfrak{h}},$$B_{3,8}$$\mathrm{i}$

$f$

$C$

$B_{4,6}$$O$$A_{5}$$C$$\Gamma_{2}$$\Gamma_{1}$$A_{5}$$O$$D_{4,7}$$O$$\Gamma_{2}$$\Gamma_{3}$$\Gamma_{2}$$\Gamma_{1}$

$O$

$C$

$\{A_{5}, C_{6,6}\}^{\star}$$C$$\{2A_{8}\},$$\mathrm{i}$$O$$C_{6,6}$$\{A_{8}, B_{3,6}\}$$A_{5}$$\{A_{8}, C_{3,7}\}$$\{A_{8}, C_{3,8}^{\#}\}^{\star}$$C$$\Gamma_{5}$$C$$\Sigma(\Gamma_{5})=\{A_{8}, A_{3}\}$$\mathit{0}$$\Gamma_{1}$$\Gamma_{5}$$\Gamma_{1}$$C_{3,8}^{\#}$$O$$A_{8}$$\{2B_{3,6}\}^{\star}$$\Gamma_{2}^{(i)}$$O$$\mathrm{i}_{1}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}$$\underline{n=1.\cdot}$$\Gamma_{2}^{(i)}$$C$$\{B_{3,12}\}^{\star}$$O$$\{C_{6,12}\}^{\star}$$C$$\{Sp_{2}\}^{\star}$$\{B_{6,6}\}^{\star}$$\square$-torus curves having a configuration of singularities inLemma 2.2. There are no tamethe following list.
,
and
(1)
$6\leq k,$ $l\leq 9,$ $(k, l)\neq(6,6)$ . Here we use the notation
any
(2)
for
for
is also included in these pairs.
for notation’s consistency, and$\sigma=C_{6,6},$
.
(3)
where
.
(4)
$(\mathit{2},\mathit{3})$$\{A_{2}, E_{6}, C_{3,8}\}^{\star},$$\{A_{2}, E_{6}, C_{3,9}\}^{\star}$$\{A_{2}, A_{5}, C_{3,9}\}^{\star}$$C_{3,6}$$\{C_{3},, {}_{k}C_{3,l}\}^{\star}$$B_{3,6}$$C_{3,8}^{\#}$$B_{4,6},$ $D_{4,7}$$\{E_{6}, \sigma\}^{\star}$$\{A_{8}, C_{3,8}\}^{\star},$ $\{A_{8}, C_{3,9}\}^{\star}$

Proof. (1) For a singular point $P\in C$ , recall that $\mu’(P):=\mu(P)+r(C, P)-1$ . We first. Note that $\chi(C’)=4$
and
consider the case$r(C)\geq2$anygeneric member of the
beforeobserved
fact,
have
we
.
In
Thus
for each of them.
theabove three configuof
Eachconsists of a conic and a quartic.
moduli of. In the case of
rations should be a degeneration of a family of$\{A_{2}, E_{6}, C_{3,8}\}^{\star},$$\{A_{2}, E_{6}, C_{3,9}\}^{\star}$$\{A_{2}, A_{5}, C_{3,9}\}^{\star}$$\{A_{2}, A_{5}, C_{3,8}\}^{\star}$$\{A_{2}, A_{5}, C_{3,8}\}^{\star}$$\{A_{2}, E_{6}, C_{3,8}\}^{\star}$

39

and
(respectively in the case of), a quartic can not contain singularities
and) by the inequality (2.2), which proves
simultaneously (resp.
andthe non-existence of the case (1).
(2) Now we consider the configurations
for any $6\leq k,$ $l\leq 9$ . Note that$\mu’(C_{3,\ell})=12,12,14,14$ respectively and the tangent cones are irreducible. Thus if
has any
two of them simultaneously, we need to have $r(C)\geq 3$ . Rrthermore if
oris included,$r(C)\geq 4$ , in which case,has at least two line components. If $C$ has two line components,
we have an obvious contradiction by the number of the tangential cone argument. Thus theremaining cases are$l=6,7$ and $C$ has either 3 conics or one line and twoother components of degree 2 and 3 respectively. Observe that
has irreducible singularity, which can not exist on a cubic or a conic. Thus the non-existence is proved except. This exceptional case exists as we have seen before.
(3) Assume that the singularity atisor. Then the conic
consists of
two different lines
or a line with multiplicity two by the local classification argument in\S 2. Note that in any case, the intersection multiplicity of a line (or the reduced line) inandat
is 2. Thus there exists a simple intersection outside of . In the first case, theother partner singularities are two, which is not the case (3). If
is a double line, theother partner singularity is
as we have seen in Corollary 1.3, which is also not in case (3).
(4) Assume thathas a configurationor. Then they must be adegeneration of the configuration of typewhose generic member consists of aconic and a quartic. Thus by the same argulnent as above, this is impossible.
$\{A_{2}, E_{6}, C_{3,9}\}^{\star}$$A_{2}$$\{A_{2},$ $A_{5},$$E_{6}$$A_{2}$$C_{3,9}\}^{\star}$$A_{6}$$\{C_{3},, {}_{k}C_{3,l}\}^{\star}$$C$$C_{3,8}$$C_{3,9}$$C$$\{C_{3},, {}_{k}C_{3,l}\},$ $k,$

$C_{3,7}$$A_{4}$$\{C_{3},, {}_{6}C_{3,6}\}^{\star}=\{B_{3,6}, B_{3,6}\}^{\star}$$O$$\ell_{1},$$C_{6,6},$$D_{4,7}$$B_{4,6}$$C_{2}$$\ell_{2}$$C_{2}$$C_{3}$$O$$O$$A_{2}$$C_{2}$$A_{5}$$C$$\{A_{8}, C_{3,8}\}^{\star}$$\{A_{8}, C_{3,9}\}^{\star}$$\{A_{2}, A_{5}, C_{3,8}\}^{\star}$$\square$I am grateful to Professor Oka for pointing out some missing cases in Table
the above non-computational proof of Lemma 2.2.

$\mathrm{B}$and showing

3. THE GLOBAL DEGENERATION
3.1. A certain degenerations. In this section we consider the degenerations of irreducible
tame torus curves of type $(2,3)$ , our aim is finding the list of all maximal curves. Here an
irreducible sextic $C$ of a torus type is called maximal if $C$ does not have any degeneration in
the space of irreducible sextics of torus type.
The degenerations between the curves in the table may come from the configurations inthe same level of -vectors in the table , and also from different levels of -vectors. We firstshow the following.
$\mathrm{B}$$\mathrm{B}$$\mathrm{i}$$\mathrm{i}$Proposition 3.1. In the same -vectors level, the degeneration families are given in Table$i$$\mathrm{B}’$.

Proof. See in [P].

$\square$

A configuration at the end of each degeneration family in the above proposition is calledsemi-maximal. Next we consider degenerations between the semi-maximal curves. For later
purpose, we first give the following degenerations, which will be used to prove Theorem 3.4.
Proposition 3.2. Among the semi-maximal curves and together
degeneration families:(1) $\{6A_{2}\}arrow\{4A_{2}, E_{6}\}arrow\{2A_{2},2E_{6}\}arrow\{3E_{6}\}arrow\{E_{6}, B_{3,8}\}$(2) $\{A_{2}, E_{6}, C_{3,7}\}arrow\{E_{6}, B_{3,8}\}$$\{6A_{2}\}_{f}$there are following(3). $\{2A_{2}, B_{4,6}\}arrow\{A_{2}, Sp_{1}\}$

Proof. See in [P].

$\square$

40

Table
$\{A_{17}\}$$\mathrm{B}’$$\{A_{17}\}$$\{C_{3,15}\}$$arrow$: Global degenerations (with same i-vector)$arrow$$\{C_{9,9}\}$$———–,———————–\{A_{2},A_{14}\}arrow\{A_{2},C_{312}\}arrow\{A_{2},B_{3,10}\}$$\{A_{2}, A_{14}\}$$arrowarrow\{A_{2}, C_{6,9}\}$$arrow$$\{A_{2}, Sp_{1}\}$$\{A_{5}, A_{11}\}$$arrow$$\{A_{5}, C_{3,9}^{\#}\}$$arrow$$\{A_{5}, B_{3,8}\}$$(*)$$\{E_{6},A_{11}\}\downarrow$$\{E_{6},B_{3,8}\}\downarrow$$\{E_{6},C_{3,9}^{\#}\}\downarrow$$arrow$$arrow$$———–,———————–\{2A_{8}\}arrow\{A_{8},B_{36}\}arrow\{A_{8},C_{37)}\}$$\{2A_{2}, A_{11}\}$$arrow$$\{2A_{2}, C_{3,9}^{\mathfrak{h}}\}$$arrow$$\{2A_{2}, B_{3,8}\}$$\searrow$$\{A_{2}, A_{5}, A_{8}\}$$arrow$$\downarrow$$\{2A_{2}, C_{6,6}\}$$arrow$$\{2A_{2}, B_{4,6}\}$$\{A_{2}, A_{5}, B_{3,6}\}$$arrow$$\{A_{2}, A_{5}, C_{3,7}\}$$(**)$$\downarrow$$\downarrow$$\downarrow$$\{A_{2}, E_{6}, A_{8}\}$$arrow$$\{A_{2}, E_{6}, B_{3,6}\}$$arrow$$\{A_{2}, E_{6}, C_{3,7}\}$$\{3A_{5}\}$$arrow$$\{2A_{5}, E_{6}\}$$arrow$$\{A_{5},2E_{6}\}$$\{3E_{6}\}$$arrow$$\{3A_{2}, A_{8}\}arrow\{3A_{2}, B_{3,6}\}arrow\{3A_{2}, C_{3,7}\}arrow\{3A_{2}, C_{3,8}\}arrow\{3A_{2}, C_{3,9}\}$$\{2A_{2},2A_{5}\}arrow\{2A_{2}, A_{5}, E_{6}\}arrow\{2A_{2},2E_{6}\}$$\{4A_{2}, A_{5}\}$$arrow$$\{4A_{2}, E_{6}\}$3.2. Maximal sextics. First we remark the following.
such that
Lemma 3.3. Assume that we have an analytic family of plane curves
has only isolated singularities in a fixed open neighborhood of the origin for any and is
are the singular points of
the unique singularity of . We assume thatwhich converges to for $t=0$ . Then we have(1) If $\nu\geq 2_{f}\sum_{i=1}^{\nu}\mu(C_{t}, P_{t,i})>\mu(C_{0}, O)$ .
(2) If $\nu=1,$ $\mu(C_{0}, O)\geq\mu(C_{t}, P_{t,1})$ for $t\neq 0$ and the equality holds if and only if $(C_{0}, O)$is equivalent tofor $t\neq 0$ .
$(|t|\leq\epsilon)$$U$$C_{0}$$P_{t,1},$$\ldots,$$C_{t}$$t$$O$$C_{t}\cap U$$P_{t,\nu}$$O$$(C_{t}, P_{t,1})$

Proof. The first assertions follows from the vanishing theorem of Lefschetz number of themonodromy$([\mathrm{A}1], [\mathrm{L}\hat{\mathrm{e}}2])$

and the second assertion is due to$([\mathrm{L}\hat{\mathrm{e}}2],[\mathrm{L}\mathrm{R}])$.

$\square$Using Propositions 3.1 and 3.2 we obtain the following theorem.
Theorem 3.4. The maximal sextics of torus type has the following configurations.
.
(1) $n=1:\{C_{3,15}\},$.
(2) $n=2:\{A_{2}, B_{3,10}\},$ $\{A_{2}, Sp_{1}\},$(3) $n=4:\{3A_{2}, C_{3,9}\}$ .
$\{C_{9,9}\}$$\{E_{6}, B_{3,8}\},$ $\{A_{8}, C_{3,7}\}$. By Lelnma 3.3,, the assertion
.
for
For
increasing
the sum of Milnor numbers is strictly
is obvious as they have $\mu=19$ and no other place to degenerate. Now we consider the

Proof. Assume that there is a family of degeneration$C_{t}arrow C_{0}$$tarrow \mathrm{O}$for$tarrow \mathrm{O}$$\{C_{3,15}\},$ $\{C_{9,9}\}$41

configurations $\{A_{2}, B_{3,10}\},$. The total Milnor numbers are 20 for eachof them and they are obviously maximal. For, the total Milnor number is 19 andthe possibility of the degeneration is to one of. If this is
the case, there must be degenerations of each of the two singularities to the corresponding
singularities in the above configurations. Then
has to degenerate into eitheror, and thenor
has no partner to degenerate.
It remains to show that the configuration $\{3A_{2}, C_{3,9}\}$ can not degenerate to any configuration in (1) or (2). By the total Milnor number argument, the possibility is to $\{A_{2}, B_{3,10}\},$. Assume that $\{3A_{2}, C_{3,9}\}arrow\{E_{6}, B_{3,8}\}$ . As $2A_{2}arrow E_{6}$ but $3A_{2}\neq+E_{6}$ , we needorto have a degeneration $A_{2}+C_{3,9}arrow B_{3,8}$ which is ridiculous as the Milnor number is decreasing. The impossibly of the degeneration $\{3A_{2}, C_{3,9}\}arrow\{A_{2}, B_{3,10}\}$ or
can beproved by computation, see in [P].
$\{A_{2}, Sp_{1}\},$$\{E_{6}, B_{3,8}\}$$\{A_{8}, C_{3,7}\}$$\{A_{2}, B_{3_{\gamma}10}\},$$\{A_{2}, Sp_{1}\},$$\{E_{6}, B_{3,8}\}$$Sp_{1}$$C_{3,7}$$B_{3,10}$$A_{8}$$B_{3,8}$$\{A_{2}, Sp_{1}\}$$\{E_{6}, B_{3,8}\}$$\{A_{2}, Sp_{1}\}$$\square$Remark 3.5. The impossibility of the above degeneration can be shown by looking the dual
curves. Suppose that we have a degenerating family
with
has $\{3A_{2}, C_{3,9}\}$ for $t\neq 0$ andhas the configuration. We assume also that the singularities
andareat the origin, with $y=0$ as the tangent cone. Then we first notice that the dual curve ofand. Thishas degree 6that the corresponding dual curve can be considered
as an analytic family of sextics. Secondly we can see that the dual singularity ofis againat $(0,1,0)$ andhas further three
singularities also. On the other hand, the dual
, [O4]. Let be the line supporting the tangent cone of
singularity ofisby$L\capC_{0}=\{O\}$at the singularity. Then. This implies that in the dual curve , it has, we have the degeneration
singularity at $(0,1,0)$ . Then in the family.
This isas the Milnor nulnber is decreasing. The detail will be studied in our next
paper [OP].
$C_{t}$$C_{t}$$\{A_{2}, Sp_{1}\}$$C_{0}$$Sp_{1}$$C_{3,9}$$C_{t}$$([\mathrm{O}4])$$C_{0}$$\mathrm{i}_{1}\mathrm{n}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{s}$$C_{3,9}$$C_{3,9}$$A_{2}$$C_{t}^{*}$$Sp_{1}$$A_{8}$$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\ln 14$$L$$Sp_{1}$$C_{0}$$C_{0}^{*}$$A_{8}$$C_{3,9}arrow A_{8}$$C_{t}^{*}arrow C_{0}^{*}$$\mathrm{i}_{1}\mathrm{n}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}$4. NEW ZARISKI PAIRS
We first modify thethe original one.

$\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}_{11}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$of a Zariski pair given in [Ar]. Our definition is weaker than

Definition 4.1. A pair of plane curves $(C, C’)$ is called a weak Zariski pair if they have same
degree and $\Sigma(C)\sim\Sigma(C’)$ , but
is not homeomorphic to.
$\mathrm{P}^{2}-C$$\mathrm{P}^{2}-C’$

We do not ask the respective tubular neighborhood of and $C’$ are homeolnorphic. The
notation $\Sigma(C)\sim\Sigma(C’)$ means that there is a bijection : $\Sigma(C)arrow\Sigma(C’)$ such that $(C, \xi)$ is
equivalent to $(C’, \phi(\xi))$ for any $\xi\in\Sigma(C)$ .
As a corollary of
we have two new Zariski pairs with respect to the following
configurationsa).
andb) $\{3A_{2}, C_{3,8}\}$ and.
$C$$\phi$$\mathrm{t}\mathrm{o}_{\mathrm{I}})\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{c}\mathrm{a}1$$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\ln 2$$\{A_{2}, A_{5}, C_{3,8}\}^{\star}$$\{A_{2}, A_{5}, C_{3,8}^{\#}\}^{\star}$$\{3A_{2}, C_{3,8}^{\#}\}^{\star}$In the first pair,is a union of a conic and a quartic, whileisa ullion of a line and a quintic.
In the second pair, $\{3A_{2}, C_{3,8}\}$ is irreducible, while
is reducible, a union of aline and a quintic.
The explicit equations of the above pairs and the discussions about geometry of these curves
are given in the previous section.
$\{A_{2}, A_{5}, C_{3,8}^{\#}\}^{\star}$$\{A_{2}, A_{5}, C_{3,8}\}^{\star}$$\{3A_{2}, C_{3,8}^{\#}\}^{\star}$

42

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$\mathrm{v}\mathrm{o}\mathrm{l}87$

METROPOLITAN UNIVERSITY, MINAMI-OHSAWA 1-1, HACHIOJIDEPARTMENT OF MATHEMATICS, TOI, TOKYO 192-0397, JAPAN-mail address: pdtai@comp.metro-u.ac.jp$<\mathrm{v}\mathrm{o}$$\mathrm{S}\mathrm{H}\mathrm{I}$$E$