数理解析研究所講究録 1233 巻 2001 年 90-94

90The dihedral coverings of the projective plane
東北学院大・教養

土橋 宏康 (Hiroyasu Tsuchihashi)

The Faculty of Liberal Arts,
Tohoku Gakuin University.

Introduction
Let

$r$

be an odd integer greater than 2and let$D_{2r}=\langle\sigma, \tau|\sigma^{2}=\tau^{r}=(\sigma\tau)^{2}=1\rangle$.

are homogeneous polyThen the following fact is well-known(see [3] or [5]). If and$\deg
g_{1}=r\degg_{2}$
covering of,then there exists anomials of three variables with 2, where acovering is aramifying only along the curve defined by. Moreover, if(branched) Galois covering with the Galois group isomorphic to$\deg
g_{1}\degg_{2}$$(2,r)$
cusps there. In this note, wepoints, then has
normally atcrossesramifying only along an irreducible reduced
covering ofshow that if there exists acurve $C=(f)$ , then there exist homogeneous polynomials , and of three variables,
satisfying$g_{2}$$g_{1}$$D_{2r}$$C$$\mathrm{P}^{2}$$g_{1}^{2}-g_{2}^{r}=0$$D_{2r}$$(g_{1})$$D_{2t}$$C$$(\# 2)$$\mathrm{P}^{2}$$D_{2r}$$h$$fh^{2}=g_{1}^{2}-g_{2}^{f}$$g_{1}$$g_{2}$.

ofand
Conversely, we also show that if there exist homogeneous polynomials , ,ofcomponents(/)
noirreducible
equation,
ifcontainsabove
satisfying
thethree variables
covering(h) and if
crosses (g2) normally at at least one point, then there exists acovering
ramifying only along (/). As an application, we give an example of aof$(2,5)$
cusps.
ramifying along asextic with four
$f$

$h$

$g_{1}$$g_{2}$$D_{2r}$$(g_{1})$$D_{10}$$\mathrm{P}^{2}$

1Aversal dihedral covering
We define the action of the dihedral group$\sigma:\xi\mapsto\xi^{-1}$$D_{2\mathrm{r}}$

and

on

$\mathrm{P}^{1}$by$\tau:\xi\mapsto\rho_{f}\xi$,where $\rho_{t}=\exp(2\pi\sqrt{-1}/r)$ and 4is anon-homogeneous coordinate ofholomorphic map:$\varpi$$\mathrm{P}^{1}$. Then the$\mathrm{P}^{1}\ni\xi\mapsto\xi^{f}+\xi^{-f}\in \mathrm{P}^{1}$

be a-covering. This covering plays akey role in this note. Let :is acomplementofthebethevariety
andlet
aprojectivemapfromdominant rational$\nu$$D_{2\mathrm{r}}$$\mathrm{Y}$$\mathrm{Y}_{0}$$\mathrm{Y}arrow \mathrm{P}^{1}$

91
set of the indeterminacy of . Then we have the following(see \S 4 in [2]).
$\nu$

Proposition 1. If the fiber productcovering
is irreducible, then there exists a.
map :withof the restrictionof to and ,:-equivariant rational
and a$\mathrm{Y}_{0}\cross_{\mathrm{P}^{1}}\mathrm{P}^{1}$$D_{2r}$$Xarrow \mathrm{P}^{1}$$\varpi$

$\mu$

$\nu$

$\nu|Y_{0}$$Xarrow \mathrm{Y}$$\pi$$\mathrm{Y}_{0}$$\varpi$$D_{2r}$$\circ\mu=\nu\circ\pi$$X$$arrow\mu$$\mathrm{P}^{1}$$arrow\nu$$\mathrm{P}^{1}$

$\ni$

$\xi$

$\ni$

$\xi^{r}+\xi^{-r}$1$\mathrm{J}\varpi$$\pi\downarrow$$\mathrm{Y}$Conversely, any-covering of aprojective variety can be constructed in this way
provided that is odd.
$D_{2r}$$r$

covering :of aprojectiveTheorem 2. Let be an odd integer. For any-equivariant rational map :variety , there exist aand adominant rational
.
map :
with
$D_{2r}$

$r$

$\nu$

$\mathrm{Y}arrow \mathrm{P}^{1}$

Proof. Let$f_{0}$$Xarrow \mathrm{Y}$$\pi$$Xarrow \mathrm{P}^{1}$$D_{2r}$$\mathrm{Y}$$\mu$$\varpi\circ\mu=\nu\circ\pi$

be arational function on $X$ and let$f_{1}= \sum_{i=0}^{r-1}\frac{(\tau^{i})^{*}f_{0}}{\rho_{r}^{is}}$,$f= \frac{f_{1}}{\sigma^{*}f_{1}}$where $\rho_{r}=\exp(2\pi\sqrt{-1}/r)$ and $s=(r+1)/2$ . Then, because
and$f$map-equivariant:
arational
functiondefinesconstant for asuitable . Hence the rational map :.
dominant andand $\sigma^{*}f=f^{-1}$ . Hence. Therefore, the rational. Moreover, is notdefined by $f^{r}+f^{-r}$ , is$\tau^{*}f1=\rho_{r}^{s}f1$$\tau^{*}f=\rho_{r}f$$\tau^{*}\sigma^{*}f1=\sigma^{*}(\tau^{-1})^{*}f1=\rho_{r}^{-s}\sigma^{*}f_{1}$$D_{2r}$$\rho_{r}^{2s}=\rho_{r}$$Xarrow \mathrm{P}$$\mu$$f_{0}$$\nu$$\mathrm{Y}arrow \mathrm{P}^{1}$$f$$\square$$\varpi\circ\mu=\nu\circ\pi$

Remark. It is well-known that there exist $G$-coverings
also for the groups,. However, the above theorem does notisomorphic toandwith even ,coverings(see[1]).
hold for these$\mathrm{P}^{1}arrow \mathrm{P}^{1}$$G$$D_{2r}$$r$$A_{4}$$S_{4}$$A_{5}$

2Dihedral coverings ramifying along irreducible curves
We keep the notations in the previous section.
Proposition 3. Let $C=(f)$ be an irreducible reduced curve on
defined by ahomogeneous polynomial . If there exists a D2r-c0vering/K :of ramifying only
along $C$ , then there exist homogeneous polynomials ,,and satisfying$by\pm(4g_{1}^{2}/g_{2}^{r}-2)$
map
and is induced $bom$.
and the rational
defined$\mathrm{P}^{2}$$\mathrm{Y}arrow \mathrm{P}^{2}$

$f$

$h$

$\pi$

$g_{1}$$g_{2}$$\mathrm{P}^{2}$$fh^{2}=g_{1}^{2}-g_{2}^{r}$$\mathrm{P}^{2}arrow \mathrm{P}^{1}$$\varpi$along $C$ is equal to 2, because the double covering, has order
ramifies along
and any element in
not contained in
2. On the other hand,ramifies at 2, -2 andwith the ramification index 2, 2andProof. The ramification index of$\pi$$C$$\mathrm{Y}/\langle\tau\ranglearrow \mathrm{P}^{2}$$\varpi$$D_{2r}$$\infty$$\langle\tau\rangle$

92, respectively. Hence there exists adominant rational map:
such that$\nu(C)=2$ or -2, by Theorem 2. Let:
be the biholomorphic map definedby $(\xi+2)/4$ or $(-\xi+2)/4$ , accordingly as $\nu(C)=2$ or -2. Then $(\psi\circ\nu)(C)=1$ .
There exist homogeneous polynomials and such thatand that$\psi^{-1}(0)=\mp
2$ with theis defined by. Since ramifies alongand
ramification index and 2, respectively, and does not ramify along, there exist homogeneous polynomials and withandand$(\psi\circ\nu)^{-1}(1)=(g_{1}^{2}-g_{2}^{f})\supset
C=(f)$. Since
and ramifies only along (/), there
exists ahomogeneous polynomial with $fh^{2}=g_{1}^{2}-g_{2}^{r}$ .
$r$$\mathrm{P}^{2}arrow \mathrm{P}^{1}$$\nu$$\mathrm{P}^{1}arrow \mathrm{P}^{1}$$\psi$$\tilde{g}_{1}$$\tilde{g}_{1}/\tilde{g}_{2}$$\deg\tilde{g}_{1}=\deg\tilde{g}_{2}$$\tilde{g}_{2}$$\psi$$\circ\nu$$\psi^{-1}(\infty)=\mathrm{o}\mathrm{o}$$\varpi$$r$$(\psi\circ\nu)^{-1}(\infty)=(\mathrm{g}2)$$\pi$$(\psi\circ\nu)^{-1}(0)=(\tilde{g}_{1})$$g_{1}$$\tilde{g}_{2}=g_{2}^{f}$$\tilde{g}_{1}=g_{1}^{2}$$g_{2}$

$\pi$

$h$

$\square$Remark. For any homogeneous polynomial of even degree there exist homogeneous$l^{r-3}f+$
polynomials ,and satisfying. For example,$\ldots+f^{(r-1)/2}$ ,and $g_{2}=l^{2}-f$ satisfy the equality
for any homogeneous polynomial 1with $\deg l=\deg f/2$ , because $(l\pm\sqrt{f})^{r}=$. However, then
$f$

$h$

$g_{1}$$fh^{2}=g_{1}^{2}-g_{2}^{\mathrm{r}}$$g_{2}$$h=(\begin{array}{l}r1\end{array})$$l^{r-1}+(\begin{array}{l}r3\end{array})$$g_{1}=l^{f}+(\begin{array}{l}f2\end{array})l^{r-2}f+\cdots+(\begin{array}{l}ft-\mathrm{l}\end{array})lf^{(\mathrm{r}-1)/2}$$g_{2}^{f}=g_{1}^{2}-fh^{2}$$g_{1}\pm\sqrt{f}h$$4 \frac{g_{1}^{2}}{g_{2}^{r}}-2=(2\frac{l^{2}+f}{g_{2}})^{f}+c_{2}(2\frac{l^{2}+f}{g_{2}})^{\mathrm{r}-2}+\cdots+c_{\mathrm{r}-1}(2\frac{l^{2}+f}{g_{2}})$where$c_{\dot{1}}$,are the integers determined by the equation$\xi^{f}+\xi^{-r}=(\xi+\xi^{-1})^{t}+c_{2}(\xi+\xi^{-1})^{\mathrm{r}-2}+\cdots+c_{\mathrm{r}-1}(\xi+\xi^{-1})$.

Hence the rational map, is equal to the composite
defined by$2(l^{2}+f)/g_{2}$of the rational map :defined byand the holomorphic
map:
induced ffom . Therefore, the fiber productof therestriction of toand , is reducible, whereis the complement of the set of points
of indeterminacy of .
$\nu:\mathrm{P}^{2}arrow \mathrm{P}^{1}$$\iota \mathit{1}$$\varpi’$$4g_{1}^{2}/g_{2}^{f}-2$$\mathrm{P}^{2}arrow \mathrm{P}^{1}/\langle\sigma\rangle\simeq \mathrm{P}^{1}$$\mathrm{P}^{1}/\langle\sigma\ranglearrow \mathrm{P}^{1}$$\nu$$Z_{0}$$\varpi’\circ\nu’$$Z_{0}\cross_{\mathrm{P}^{1}}\mathrm{P}^{1}$$\varpi$$Z_{0}$$\varpi$$\nu$

Proposition 4. Let $C=(f)$ be areduced curve on
defined by ahomogeneouspolynomial . Assume that there exist homogeneous polynomials ,and satisfying$C$. If contains no irreducible components of the zero divisor (h) of andcrossesnormally at at least one point, then there exists a-covering of$C$ramifying only alorig .
$\mathrm{P}^{2}$$f$$g_{1}$$h$$g_{2}$$fh^{2}=g_{1}^{2}-g_{2}^{f}$$h$$(\# 2)$$(g_{1})$$D_{2r}$$\mathrm{P}^{2}$be the rational map defined by. Since is dominant,there exists a-covering of
ramifying only along , if and only if the fiber productof the restriction of toand , is irreducible, whereis the complementof the set of points of indeterminacy of .
Letcross (g2) normally at , let $W$ be asmall neighborhood of and let
be alinear equation with $(l)\cap W=\emptyset$ . Then $((g_{1}/l^{\deg(g_{1})})|W, (g_{2}/l^{\deg(g_{2})})|W)$ is alocal
coordinate system of $W$ and there exists a-covering : $Uarrow W$ which is expressed
as $(u,v)\mapsto((u^{f}+v^{f})/2,uv)$ by alocal coordinate system $(u,v)$ of , where is an open
neighborhood of the origin in. Let :be the meromorphic map defined byProof. Let :$\nu$$D_{2\mathrm{r}}$$Z_{0}\mathrm{x}_{\mathrm{P}^{1}}\mathrm{P}^{1}$$\mathrm{P}^{2}arrow \mathrm{P}^{1}$$4g_{1}^{2}/g_{2}^{r}-2$

$\nu$

$\nu$

$C$

$\mathrm{P}^{2}$$Z_{0}$$Z_{0}$$\varpi$$\nu$$(g_{1})$

$p$

$p$

$D_{2t}$

$\pi$

$U$

$\mathrm{C}^{2}$$\mu$$Uarrow \mathrm{P}^{1}$

$U$

$l$

93$u/v$. Then$\mu$is$D_{2r}$-equivariant and$\varpi$$\circ\mu^{=\nu}|\psi\circ\pi$.
$u$$U\ni$$arrow\mu$$(u,v)$$=$$\in \mathrm{P}^{1}$

$\xi$

$v$
$\mathrm{J}\varpi$$\pi\downarrow$$W$a$( \frac{u^{r}+v^{f}}{2}, uv)$$=(x,y)$$arrow\nu_{|W}$$4 \frac{x^{2}}{y^{r}}\backslash -2$$\xi^{f}+\xi^{-r}$$\in \mathrm{P}^{1}$.
Xpl
is irreducible. Therefore, also is$C$ ,induced from and , ramifies only along because-covering of$\nu^{-1}(\infty)=r(g_{2})$and $\nu^{-1}(2)=(f)+2(h)$ .
,Hence
The$Z_{0}$$(W\backslash \{\mathrm{p}\})\cross_{\mathrm{P}^{1}}\mathrm{P}^{1}$$2(g_{1})$$=$$\mathrm{P}^{2}$$D_{2r}$$\nu$$\mathrm{P}^{1}$$\varpi$$\nu^{-1}(-2)=\square$3An example
be homogeneous polynomials of degree 1, 1and 2, respectively, with, where$(l)\cap(m)\cap(h)=\emptyset$ . Let $g_{1}=l^{5}-5l^{3}h+6mh^{2}$ and let $g_{2}=l^{2}-2h$ . Then
$f=-15l^{6}+12l^{5}m+80l^{4}h-60l^{3}mh-80l^{2}h^{2}+36m^{2}h^{2}+32h^{3}$ . Assume that $(l-m)$ crosses
also does, because $g_{1}=lg_{2}(l^{2}-3h)-6(l-m)h^{2}$(02) normally at two points. Then
ramifying along-covering ofand $(l-m)\cap(g_{2})\cap(h)=\emptyset$ . Hence there exists a
the sextic curve (/). We easily see that if (I) crosses (h) normally, then (/) has two $(2, 5)$.
as well as at $(l-m)\cdot$
cusps at (/)

Let , $m$ and$l$$h$$g_{1}^{2}-g_{2}^{5}=h^{2}f$$(g_{1})$$D_{10}$$\cdot(h)$$\mathrm{P}^{2}$$(g_{2})$References[1] J. Buhler and Z. Reichstein: On the essential dimension of afinite group, Compositio
Math. 106(1997), 159-179[2] M. Namba, On finite Galois coverings of projective manifolds, J. of Math. Soc. Japan41 (3), 1989, 391-403[3] M. Oka, Some plane curves whose complements have non-abelian fundamentalgroups, Math. Ann. 218(1975), 55- 6

94[4] M. Oka, On the fundamental group of the c complement ofcertain plane curves, J.
Math. Soc. Japan 30(1978), 579-597[5] H. Tokunaga, Dihedral coverings branched along maximizing sextics, Math. Ann.
308(1997), 633-64