an:07146807
Zbl 1439.13020
Lercier, Reynald; Ritzenthaler, Christophe; Sijsling, Jeroen
Reconstructing plane quartics from their invariants
EN
Discrete Comput. Geom. 63, No. 1, 73-113 (2020).
00443553
2020
j
13A50 14L24 14H10 14H25
plane quartic curves; invariant theory; Dixmier-Ohno invariants; moduli spaces; reconstruction
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 \textit{J. Dixmier} [Adv. Math. 64, 279--304 (1987; Zbl 0668.14006)] and by Ohno (unpublished, see also \textit{A.-S. Elsenhans} [J. Symb. Comput. 68, Part 2, 109--115 (2015; Zbl 1360.13017)] and \textit{M. Girard} and \textit{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 \textit{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 \url{https://github.com/JRSijsling/quartic\_reconstruction/}.
Peter Schenzel (Halle)
Zbl 0668.14006; Zbl 1360.13017; Zbl 1143.14304; Zbl 0752.14027; Zbl 1344.11049