数理解析研究所講究録
第 1843 巻 2013 年 163-165

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Geometric Properties of Plane Quartics
甲南大学知能情報学部

高橋正 *Tadashi Takahashi
Department of Intelligence and Informatics,Konan University
Abstract: It is well-known that smooth plane quartic curves in the two dimensional complex projective
space are (non-hyperelliptic) curves of genus three. And the local property of singular plane quartic
curves is also well known. We consider about the parameters of these defining equations.

1Introduction

be a 2-dimensional complex projective space with the coordinate $[x, y, z]$ and let
be a homogeneous polynomial of variables
with degree 4 in . We con$:=\{(x, y, z)|f_{4}(x, y, z)=0\}$ . We callsider the set
complex projective plane quartics(in short: plane quartics). Nonhyperelliptic curves of genus 3 are non-singular plane quartics.
Let
be the variety of moduli of curves of genus . Then
has dimension which is lessthan or equal to 6. Let $f(x, y, z)$ be a homogeneous polynomial of variables $x,y,$ with degree
4 in . The analytic set defined by $f(x, y, z)=0$ has a singular point at the origin inThe analytic set is a smooth plane quartic in
if it has only isolated singular point at theLet$\mathbb{P}^{2}$$f_{4}(x,y, z)$$\mathbb{P}^{2}$$x,$ $y,$ $z$$V_{4}$$V_{4}$$M_{g}$$M_{3}$$g$$z$$\mathbb{C}^{3}$$\mathbb{C}^{3}.$$\mathbb{P}^{2}$origin in2$\mathbb{C}^{3}.$Normal forms of smooth plane quartics

For the defining equations of smooth plane quartics in$\mathbb{P}^{2}$, the following theorem holds.

Theorem 2.1. For the defining equations of smooth quartics with the coordinate $[x, y, z],$there exist the following two normal forms.
Type I: $x^{3}z+(y^{3}+pyz^{2}+qz^{3})x+ry^{4}+sy^{3}z+ty^{2}z^{2}+uyz^{3}+vz^{4}=0,$
Type II: $x^{3}z+(py^{2}+qyz+rz^{2})xz+y^{4}+sy^{2}z^{2}+tyz^{3}+uz^{4}=0,$
where $(p, q, r, s,t, u, v)\in \mathbb{C}^{7}$ are parametric coefficients.
We can rewrite any defining equation of smooth plane quartic into the one of above
normal forms. This accords for a result of Shioda’s Propositions 1 and 2 ([1]). And we got therestrictions of parameters for the above Type I, Type ([2], [3]). As the result, the following
theorem holds.
$\Pi$

Theorem 2.2. Let

be a smooth plane quartic in a 2-dimensional complex projective spacewith the coordinate
Then we can take the multiplicity of $C$ and -axis at [1,0,0] suchthat it is equal to 3 or 4 by replacing a suitable coordinate.
$C$$[x, y, z]$’takahasi@ikonan-u.ac.jp

$y$

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3

Normal forms of singular plane quartics

By a same method as the rewriting of the $sm\infty th$ plane quartics, we can rewrite the defiming
equations.
equation of the singular quartic to one of the defiming$fo\mathbb{I}ow\dot{m}g$

Type I: $x^{2}yz+(y^{3}+a_{1}y^{2}z+z^{3})x+(a2y^{3}+a3y^{2}z+a_{4}yz^{2}+a5^{Z^{3})_{Z}}=0,$
Type II: $x^{2}yz+a_{1}xz^{3}+y^{4}+a_{2}y^{3}z+a_{3}y^{2}z^{2}+a_{4}yz^{3}+a_{5}z^{4}=0,$

Type III: $z(x^{2}y+a_{1}xz^{2}+y^{3}+a_{2}y^{2}z+aayz^{2}+a_{4}z^{3})=0,$
Type IV: $z(x^{2}y+a_{1}xz^{2}+a_{2}y^{2}z+a_{3}yz^{2}+a_{4}z^{3})=0,$
Type V: $x^{2}z^{2}+(y^{2}+2a_{1}z^{2})xy+(a_{2}y^{3}+a_{3}y^{2}z+a_{4}yz^{2}+a_{5}z^{3})z=0,$Type VI: $x^{2}z^{2}+2a_{1}xy^{2}z+y^{4}+a2y^{2}z^{2}+a3yz^{3}+a4z^{4}=0,$
Type VII: $x^{2}z^{2}+2xy^{2}z+y^{4}+y^{3}z+a_{1}yz^{3}+a_{2}z^{4}=0,$
Type VIII: $x^{2}z^{2}+2xy^{2}z+y^{4}+y^{2}z^{2}+a_{1}z^{4}=0,$
Type IX: $x^{2}z^{2}+2xy^{2}z+(y^{3}+z^{3})y=0,$Type X: $x^{2}z^{2}+2xy^{2}z+y^{4}+a_{1}z^{4}=0,$
Type XI: $z(x^{2}z+xy^{2}+a_{1}y^{2}z+a_{2}yz^{2}+a_{3}z^{3})=0,$
Type XII: $z(x^{2}z+y^{3}+a_{1}yz^{2}+a_{2}z^{3})=0,$
Type XIII: $xy^{2}z+xz^{3}+y^{4}+a_{1}yz^{3}+a_{2}z^{4}=0,$
Type XIV: $z(xy^{2}+xz^{2}+a_{1}yz^{2}+a_{2}z^{3})=0,$Type XV: $xyz^{2}+y^{4}+a_{1}y^{3}z+a_{2}z^{4}=0,$
Type XVI: $z(xyz+y^{3}+a_{1}z^{3})=0,$
Type XVII: $z^{2}(xy+a_{1}z^{2})=0,$
Type XVIII: $xz^{3}+(y^{2}+a_{1}z^{2})y^{2}=0,$
Type XIX: $z(xz^{2}+y^{3})=0,$
Type XX: $z^{2}(xz+y^{2})=0,$
Type XXI: $z(y^{3}+a_{1}yz^{2}+a_{2}z^{3})=0,$
Type XXII: $(y^{2}+z^{2})z^{2}=0,$
Type XXIII: $y^{2}z^{2}=0,$
Type XXIV: $yz^{3}=0,$
Type XXV: $z^{4}=0.$We obtain a following table as a classification of singular point at [1,0,0] on irreducible
plane quartics.

We can obtain the restrictions for the parameters of these defining equations by using the165
method of smooth cases. And we want to know those restrictions and their relations. It is
our next purpose.

References[1] T.Shioda, Plane quartics and Mordell-Weil lattices of type
Universitatis Sancti Pauli 42, no. 1, 61?79, 1993.

$E_{7}$, Commentarii Mathematici[2] T. Takahashi, A Note on the Application of Elimination Ideal, Proc. of the Intemational
Conference on Computational Sciences and its Apphcations ICCSA, pp. 66-71, 2011.
[3] T. Takahashi, A Note on the Application of Elimination Ideal II. Mathematical Sciences
International Research Journal, Vol. 1, No. 1, pp. 21-48, 2012.