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Invariants, bitangents, and matrix representations of plane quartics with 3-cyclic automorphisms. (English) Zbl 1445.14048
As the author says in the introduction of the paper, “the study of the geometry of plane quartics is one of the most beautiful achievements in classical algebraic geometry”. Dixmier invariants of a plane quartic [J. Dixmier, Adv. Math. 64, 279–304 (1987; Zbl 0668.14006)] are 7 algebraically independent invariants of degrees 3, 6, 9, 12, 15, 18 and 27 that, together with 6 more invariants, generate the ring of invariants of the quartic [T. Shioda, Am. J. Math. 89, 1022–1046 (1967; Zbl 0188.53304)]. On the other hand, a bitangent is a line that is tangent to a curve at two points. There are many studies on the existence and configurations of the 28 bitangents of a plane quartic. In this article, the author computes Dixmier invariants and bitangents of the plane quartics with 3, 6 or 9-cyclic automorphisms. The invariants are computed with Maxima and the algebraic conditions between the invariants with Macaulay2. He also finds that a quartic curve with 6-cyclic automorphism has 3 horizontal bitangents which form an asyzygetic triple (the six points determined by these three horizontal bitangents do not lie in a conic). Finally, the linear matrix representation problem for these quartics is discussed and a degree 6 equation of 1 variable solving the symbolic solution of the linear matrix representation problem for the curve with 6-cyclic automorphism is also given.
MSC:
14H37 Automorphisms of curves
14Q05 Computational aspects of algebraic curves
Software:
Macaulay2; Maxima
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References:
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