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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.

Normal forms of smooth plane quartics and their restrictions