Jun 17, 1991 - such that $g(P_{\nu})$ is bounded and $||P_{\nu}||arrow\infty$ . We apply ...... of coordinates $\Phi(u, g)=(U, G)$ where $U=u,$ $G=g-\gamma ...
J. Math. Soc. Japan Vol. 44, No. 3, 1992
Symmetric plane curves with nodes and cusps Dedicated to Professor Heisuke Hironaka on his 60th birthday By Mutsuo OKA (Received
June 17,
1991)
\S 1. Introduction. In [Z1], Zariski considered the family of projective curves of degree 6 with 6 cusps on a conic. This family is defined by: $f(X, Y, Z)=f_{2}(X, Y, Z)^{3}+$ $f_{3}(X, Y, Z)^{2}=0$ where is a homogeneous polynomial of degree $i=2,3$ . He showed that the fundamental group $\pi_{1}(P^{2}-C)$ is isomorphic to the free product for a generic member of this family. He also proved that the fundamental group of the complement of a curve of degree 6 with 6 cusps which are not on . In fact, we will show in \S 5 that this a conic is not isomorphic to fundamental group is abelian. Zariski also studied a curve of degree 4 with 3 cusps as a degeneration of the first family in [Z1] and he claims that the complement of such a curve has a non-commutative finite fundamental group of order 12. We will reprove this assertion (\S 3 Theorem (3.12)). The purpose of this note is to construct systematically plane curves with nodes and cusps which are defined by symmetric polynomials $f(x, y)$ . A symmetric polynomial $f(x, y)$ can be written as a polynomial $h(u, v)$ where $u=x+y$ and $v=xy$ . In this expression, the degree of in is half of the original degree and the calculation of the fundamental group becomes comparatively easy. Let $p:C^{2}arrow C^{2}$ be the two-fold branched covering defined by $p(x, y)=(u, v)$ . The branching locus is the discriminant variety $D=\{u^{2}-4v=0\}$ . Let $C=\{h(u, v)=0\}$ . Under a certain condition, the homomorphism and $\pi_{1}(C^{2}-C)$ is an isomorphism (Theorem (2.3), \S 2). Symmetric polynomials give enough models for the cuspidal curves with small degree. In fact, we will give examples of symmetric plane curves of the following type and we will compute their fundamental groups. (1) Symmetric curve of degree 4 with 3 cusps (Theorem (3.12), \S 3). (2) Symmetric curve of degree 5, with 4 cusps (Theorem (3.14), \S 3). (3) Symmetric curve of degree 6, with conical 6 cusps (Theorem (4.5), \S 4). (4) Symmetric curve of degree 6, with non-conical 6 cusps (Theorem (5.8), $i,$
$f_{i}$
$Z_{2}*Z_{3}$
$Z_{2}*Z_{3}$
$h$
$\tilde{C}=p^{-1}(C)$
\S 5).
$v$
$p_{*}:$
$\pi_{1}(C^{2}-\tilde{C})arrow$
376
M. OKA
The fundamental groups of the complement of the above examples (2), (4) are abelian. We also discuss the degenerations of the above curves. For example, we will show that the symmetric curves of degree 5 with 4 cusps can be degenerated to a curve with 5 cusps (\S 3). The conical 6 cuspidal curve of degree 6 can be degenerated to a curve with nodes and $6+s$ cusps for any $d+s\leqq 3$ (\S 4). We will also prove that the moduli space of curves of with degree 4 with 3 cusps is an irreducible surface. All the curves which we treat in this paper except in \S 6 are defined over the real numbers and the essential information can be obtained from their real graphs. In \S 6, we will give explicit examples of a maximal nodal curve and a cuspidal curve of degree which has cusps. The three cuspidal curves of degree 4 is also treated asymptotically in [D.L]. I would like to thank Professors M. Namba and H. Tokunaga for pointing out an error in the first version and Professor A. Dimca for the information about [D-L]. $d$
$d,$
$s$
$n$
$n^{2}/4$
\S 2. Symmetric covering. Let $p:C^{2}arrow C^{2}$ be the two-fold covering mapping defined by $p(x, y)=(u, v)$ where $u=x+y,$ $v=xy$ . This is branched along the discriminant variety: $D=$ $\{(u, v);g(u, v)=0\}$ where $g(u, v)=u^{2}-4v$ . As and are elementary sym$p:C^{2}arrow C^{2}$ as the symmetric covering. Hereafter metric polynomials, we refer $\deg u=1,$ $\deg v=2$ unless otherwise stated. symmetric weight: we consider the Thus $g(u, v)$ is a weighted homogeneous polynomial of degree 2 under the symmetric weight. Let $h(u, v)$ be a reduced polynomial of degree (under the symmetric weight) and let $C=\{(u, v)\in C^{2} ; h(u, v)=0\}$ . We denote the inverse image $p^{-1}(C)$ of by . The defining equation of is $p^{*}h(x, y)=h(x+y, xy)$ $=0$ . Note that $p^{*}h(x, y)$ is a polynomial of degree in and . We say that is symmetrically regular at infinity if $u$
$v$
$n$
$\tilde{C}$
$\tilde{C}$
$C$
$n$
$x$
$y$
$C$
$\{(u, v)\in C^{2} ; h_{n}(u, v)=g(u, v)=0\}=\emptyset$
$(R_{\infty})$
where is the weighted homogeneous part of degree of . The geometric meaning of is the following. First, under the condition , the com$D=\{X-Y=0\}$ pactification of and the line in do not intersect at infinity $i.e.$ , on the infinite line $Z=0$ . Secondly, $h_{n}$
$n$
$(R_{\infty})$
$h$
$(R_{\infty})$
$\tilde{C}$
$P^{2}$
LEMMA (2.1). Assume that is symmetrically regular at infinity. Let : $Carrow C$ $g(u, v)=u^{2}-4v$ be the restriction of the function to C. Then the number , multiplicity, counting the the is constant fiber of for $c\in C$. $C$
$g_{C}$
$g_{\overline{c}^{1}}(c)$
$\nu=1,2,$ PROOF. Assume the contrary. Then there is a sequence of such that is bounded and . We apply the Curve Selection $P_{\nu},$
$C$
$g(P_{\nu})$
$||P_{\nu}||arrow\infty$
$\cdots$
Symmetric plane curves with nodes and
cusps
377
to find a real analytic curve $(u(t), v(t)),$ $0
