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Reconstructing plane quartics from their invariants. (English) Zbl 1439.13020
Let \(R_{3,n}\) denote the ring of ternary forms of degree \(n\) over the complex field \(\mathbb{C}\). Consider the action of \(\operatorname{SL} _3(\mathbb{C})\) on the ring \(R_{3,n}\). Explicit generators are known for \(n \leq 4\). While the cases \(n \leq 3\) are classically known, the case of \(n = 4\) was shown by J. Dixmier [Adv. Math. 64, 279–304 (1987; Zbl 0668.14006)] and by Ohno (unpublished, see also A.-S. Elsenhans [J. Symb. Comput. 68, Part 2, 109–115 (2015; Zbl 1360.13017)] and M. Girard and D. R. Kohel [Lect. Notes Comput. Sci. 4076, 346–360 (2006; Zbl 1143.14304)]). By the work of these authors it follows that the ring \(\mathbb{C}[R_{3,4}]^{\operatorname{SL} _3(\mathbb{C})}\) is generated by 13 elements, the so-called Dixmier-Ohno invariants of ternary quartics. The main result of the present paper is an explicit method that, given a generic tuple of Dixmier-Ohno invariants, reconstructs a corresponding plane quartic curve. The main technical tool is a method of J.-F. Mestre [Prog. Math. 94, 313–334 (1991; Zbl 0752.14027)], see also the authors in [Open Book Ser. 1, 463–486 (2013; Zbl 1344.11049)]. A Magma package of the authors for reconstructing plane quartics from Dixmier-Ohno invariants is available under https://github.com/JRSijsling/quartic_reconstruction/.
MSC:
13A50 Actions of groups on commutative rings; invariant theory
14L24 Geometric invariant theory
14H10 Families, moduli of curves (algebraic)
14H25 Arithmetic ground fields for curves
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