数理解析研究所講究録1182 巻 2001 年 80-82

80

Problem of Fenchel on the complex projective plane andof the 3rd braid group$\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}^{\perp}\iota \mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$Takanori MATSUNO
Department of Liberal Arts and Sciences
Osaka Prefectural College of Technology

1Abstractthe complex projective plane. Let $C=$
We denote by$\{(X_{0} : X_{1} : X_{2})\in P^{2}|X_{2}=0\}$ be the line of, which we call line. Let
be a curve ofare tangent at.
and$(0:0:1)$
is a rational curve of degree 3 with a cusp at
at infinity.
$D=e_{1}C+e_{2}L_{\infty}$. We consider thebe positive integers greater than 1. Put
1 : : . Letof the 3rd braid
folowing problem md give here a partial answer by constructing$\{(X_{0} :$\mathrm{P}^{2}$$\mathrm{P}^{2}$

X_{1} :

X_{2})\in \mathrm{P}^{2}|X_{2}X_{0}^{2}-X_{1}^{3}=0\}$$\mathrm{P}^{2}$$L_{\infty}=$

$C$

$C$

$($$0)$$0$$e_{1},$$L_{\infty}$$e_{2}$$\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{t}\dot{\mathrm{i}}\mathrm{o}\mathrm{n}\mathrm{s}$group.

Fenchel’s Problem Give a condition on the pairwhich branchae at .
covering :$Xarrow \mathrm{P}^{2}$

$D$

$\pi$

2

for the existence of a finite Galois$(e_{1},e_{2})$Elementary facts$\delta|\alpha\beta\alpha\beta\alpha\beta=\delta^{-1}>\mathrm{t}\mathrm{h}\mathrm{e}3\mathrm{r}\mathrm{d}\mathrm{b}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{d}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}.\mathrm{T}\mathrm{h}\mathrm{i}\mathrm{s}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{W}\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}p_{0}\in \mathrm{P}^{2}-\{C\bigcup_{=}L_{\infty}\}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{i}\mathrm{t}.\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}1\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}\pi_{1}(\mathrm{P}^{2}-\{C\cup L_{\infty}\},p_{0})$$\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{m}o\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{o}<\alpha,\beta,$. This isomorphism is given by. We identify(resp.
once
direction
which rounds counterclockwise, resp. ) with a closed path in$C$). Let be the smallest normalof (resp.
of , resp.
ofaround non-singular points. There is a finite Galois covering
which contains
andsubgroup ofof finite
which branches at $D$ if and only if there is a normal subgroup $K$ of$J\subset
K$) and(modthen,(1)Ifsatisfies
theconditions:following
whichindex with
$K$ .
asuch
(mod ). However it is difficult to look forthen(2)
be a finite group generated by two elements $A,$ , which satisfy the relation $ABA=$Let$BAB,A^{e_{1}}=B^{e_{1}}=1,$ $(ABA)^{e_{2}}=1$ . Obviously
and are conjugate to each other. If there is.
a finite group as above, we have a surjective homomorphism :which branches at $D$ .
Then the kernel of corresponds to a finite Galois covering :Put $Q=ABA$ . It is easy to see:$<\gamma,$ $\delta|\gamma^{3}=\delta^{2}>$$\delta\mapsto(\alpha\beta\alpha)^{-1}$$\alpha$$\mathrm{P}^{2}-\{C\cup L_{\infty}\}$$\delta$$\beta$$\gammarightarrow(\alpha\beta)^{-1},$$P_{\alpha}$$C$$\pi_{1}(\mathrm{P}^{2}-\{C\cup L_{\infty}\},p_{0})$$J$$L_{\infty}$$P_{\delta}$$P\rho$$\alpha^{e_{1}}$$\delta^{\mathrm{e}_{2}}$$\pi_{1}(\mathrm{P}^{2}-\{C\cup L_{\infty}\},p_{0})$$k\equiv 0$$\alpha^{k}\in K$$l\equiv 0$$\mathrm{I}\mathrm{f}\delta^{l}\in K$

$B$

$G$

$A$

$B$

$G$

$\Phi$

a 2.

Since$Q^{2}$$\pi$isabelian-, then$.\cdot 1.I,fG$$\pi_{1}(\mathrm{P}^{2}-\{C\cup L_{\infty}\},p_{0})arrow G$$Xarrow \mathrm{p}^{2}$$\Phi$$\mathrm{L}\mathrm{e}\mathrm{m}\mathrm{m}\backslash$$e_{1}$$e_{2}$$G$is a cyclic gmup.

is an element of the center of ,$G$

Lemma 2.2 If the order of$Q$

is odd, then$G$

is abelian ($G$is a cyclic$gro\prime up$).

Oence we have:

Theorem 2.1 If$e_{2}$

bividly we have:is odd, then any covering$\pi:Xarrow \mathrm{P}^{2}$

which branches at $D$ is cyclic.

81For given $odSnum$ der(modifexists
which bmnches $a.tD$ .

$\mathrm{P}_{\wedge}^{\mathrm{r}}\mathrm{o}\mathrm{p}\mathrm{s}_{\wedge}\urcorner \mathrm{t}\mathrm{i}\mathrm{o}\mathrm{m}2.1$$e_{2}\equiv \mathrm{C}$$e_{2_{f}}$.T

ken$t\hslash ere$

It is$\mathit{3}_{\grave{j}}pu\mathrm{f}e_{1}=e_{2}/2.,$.

$erw\prime sde?$$O_{J}^{+}\dot{j}$}put $e_{1}=e_{2}$ .

$\pi:Xarrow \mathrm{p}^{2}$-knowe (see for example$\mathrm{w}\mathrm{e}^{1}\mathrm{J}$Lenama 2*3 For given
order 2 and of order 3 with
By putting$\hat{Q}=\hat{A}\hat{B}\hat{A}$and):integer$\mathrm{p}os_{\iota}^{=}i\mathrm{i}ve$$\hat{R}$$[_{\perp}^{\tau}]$$\hat{Q}\hat{R}$

of

$n$

ihere is a finte greup $G$ generated by two elements$\mathit{0}\tau dern$$\hat{R}=\hat{A}\hat{B}$.

of, we have:

Theorem 2.2 Ifisthen for any positive integer
which bmnches $aiD$ .
$\mathit{2}_{f}$$e_{2}$$\hat{Q}$$e_{1}$

greater than 1 there is a covering$\pi$:$Xarrow \mathrm{p}^{2}$

Let $D$ be as before and kt $D’=e_{1^{J}}C+e_{2^{J}}L_{\infty}$ . Let

put $D^{u}=e_{1’’}C+e_{2^{u}}L_{\infty}$ .
By constructing the fiber product, we have:$e_{j’’}$

be the LCM $<e_{j},$ $e_{j’}>(j=1,2)$ andPropsiticn 2.2 If there is a covering :which branches atwhich bmnches at:then there is a covering:$Xarrow \mathrm{P}^{2}$$\pi$$\pi’$$X’arrow \mathrm{P}^{2}$

3$D_{f}’$$\pi^{\prime J}$$D$;
and there is awhich branches at $D”$
$C\mathit{0}ver$$X”arrow \mathrm{p}^{2}$

Cyclic extension

group of letters. Let
theWe denote by
be a finite group generated by, whichtwo permutations
the relation. Then is a product of cycles oflength 2 with no common letters and is a product of cycles ofwith no common letters.
there is a
We may aesunle has the fohowing properties. (l)transitivity: For each lettersletter,permutaion of which maps to . (2)then it isa
permutation
fixesof: If a
the unit element of .
ofNow by showing examples, we give a method to construct a cyclic extension
by an element of its center.
$S_{n}$$\hat{G}\subset S_{\gamma}$$n$$s\mathrm{y}\mathrm{m}\mathrm{I}\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}$$\hat{Q},\hat{P}_{\vee}$$\hat{Q}^{2}=\hat{R}^{3}=1$$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\Phi$$\hat{Q}$$\hat{R}$$1\mathrm{e}\mathrm{n}_{\mathrm{o}}\sigma \mathrm{t}\mathrm{h}3$$\hat{G}$$x,$$\hat{G}$$y$$\hat{G}$

$x$

$y$

$\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{y}$.
$Q=$$\hat{G}$$G\subset S_{rq}$The case $f=3$ . Put $\hat{Q}=(a b)$ and $\hat{R}=(abc)$ . In this caseneed to assume is odd. Put$\hat{G}=S_{3}$$\hat{G}\subset S_{r}$and non-abelian. We$q$$R=($$\cdot$$a_{1}b_{1}$$c_{1}b_{1}$$a_{2}b_{2}$$c_{2}b_{2}$$a_{q}b_{q}$$c_{q}b_{q}$$a_{2}c_{1}$$a_{3}c_{2}$$\cdot$.

$c_{q-1}$$a_{1}c_{q}$).

Then$F=Q^{2}=R^{3}=(a_{1}\ldots a_{q})(b_{1}\ldots b_{q})(c_{1}\ldots c_{q})$and$A=R^{-1}Q=(a_{1}c_{p+1}a_{p+2^{C_{1}}}\ldots)$

where $q=2p+1$ . The order of isLet be a finite group generated by two
exact sequence:
way we have the$A$$2q\wedge\cdot$$G$$\mathrm{p}\mathrm{e}\mathrm{r}\iota \mathrm{n}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}Q$, R.

$F$

is a center of . In a natural$G$$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}$$1arrow<F>^{G}arrow Garrow\hat{G}arrow 1$

where $<F>^{G}$ is a subgroup of generated by $F$ . $<F>^{G}$ is a cyclic group of order . Then we. Hence we have:hisln :
can have a surjective$G$$\mathrm{h}\mathrm{o}\mathrm{r}_{-}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{r}_{1}\mathrm{D}$$q$$\Phi$$\pi_{1}(\mathrm{P}^{2}-\{C\cup L_{\infty}\},p_{0})arrow G$

82

Theorem 3.1$x_{jq}^{x}$is odd,

$tf\iota ent\dot{n}ere$is a find$e$-alois$c\sigma ve\dot{n}ng\pi$:$Xarrow \mathrm{P}^{2}$

which branches at$2qC+2qL_{\infty}$

and non-abelian. For. Put $\hat{Q}=(a b)$ and $\hat{R}=(bcd)$ . In this case
The case
have:
weway,the extension we need to assume the LCM $<6,$ $q>=1$ . In a similar$\hat{G}\subset S_{4}$$r=4_{\wedge}$

Theoren 3.2 If
at $4qC+2qL_{\infty}$$q$

is as above, then fhere is a finite Galois covering$\pi$:$Xarrow \mathrm{p}^{2}$

which bmnches

The case $r=12$ . Put $\hat{Q}=(aj)(bd)(ch)(el)(fi)(gk)$ and $\hat{R}=(abc)(def)(ghi)(jkl)$ .
and non-abelian. In a similar way, we have:
In this case$\hat{G}\subset S_{12}$Theorem 3.3 There is a finite Galois covering$\pi:Xarrow \mathrm{P}^{2}$

which bmnches at $3q(q-1)C+2qL_{\infty}$

References[1] R.H.Fox, On Fenchel’s conjecture about $F$ -groups, Mat. Tidsskrift, volB (1952) 61-65(1987) vol[2] M.Namba, Branched coverings and aIgebmic fimctions, Research Notes in Math.
161 Pitman-Longman