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Research articles

ScienceAsia (): 26-33 |doi: 10.2306/scienceasia1513-1874...026

Normal forms of smooth plane quartics and their restrictions


ABSTRACT:     It is well known that smooth plane quartic curves in a two-dimensional complex projective space are curves of genus three and that the dimension of the parameters of the defining equation is less than seven. We show a process for obtaining the normal forms and their restrictions. For a homogeneous 4th-degree polynomial over the complex numbers, the vanishing set ℂ of the homogeneous polynomial in the complex projective plane ℙ2 is a curve of genus three, and such curves depend on six-dimensional parameters. By using the Gröbner basis of the elimination ideal, we show the restrictions on smooth plane quartics.

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Department˙of˙Intelligence˙and˙Informatics, Konan˙University, 8-9-1˙Okamoto, Higashinada, Kobe, Japan

* Corresponding author, E-mail: takahasi@konan-u.ac.jp

Received 31 Aug 2014, Accepted 20 Jul 2016