zbMATH — the first resource for mathematics

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.
14H37 Automorphisms of curves
14Q05 Computational aspects of algebraic curves
Macaulay2; Maxima
Full Text: Link
[1] Dixmier, Jacques.On the projective invariants of quartic plane curves.Adv. in Math.64(1987), no. 3, 279-304.MR888630,Zbl 0668.14006, doi:10.1016/00018708(87)90010-7.636 · Zbl 0668.14006
[2] Dolgachev, Igor V.Classical algebraic geometry. A modern view.Cambridge University Press, Cambridge, 2012. xii+639 pp. ISBN: 978-1-107-01765-8.MR2964027, Zbl 1252.14001, doi:10.1017/CBO9781139084437.637 · Zbl 1252.14001
[3] Elsenhans, Andreas-Stephan.Explicit computations of invariants of plane quartic curves.J. Symbolic Comput.68(2015), part 2, 109-115.MR3283857,Zbl 1360.13017, doi:10.1016/j.jsc.2014.09.006.636 · Zbl 1360.13017
[4] Girard, Martine; Kohel, David R.Classification of genus 3 curves in special strata of the moduli space.Algorithmic number theory, 346-360, Lecture Notes in Comput. Sci., 4076.Springer, Berlin, 2006.MR2282935,Zbl 1143.14304, arXiv:math/0603555, doi:10.1007/11792086.639 · Zbl 1143.14304
[5] Grayson, Daniel R.; Stillman, Michael E.Macaulay2, a software system for research in algebraic geometry.https://faculty.math.illinois.edu/Macaulay2/. 637
[6] Helton, J. William; Vinnikov, Victor. Linear matrix inequality representation of sets.Comm. Pure Appl. Math.60(2007), no. 5, 654-674.MR2292953,Zbl 1116.15016, arXiv:math/0306180, doi:10.1002/cpa.20155.636 · Zbl 1116.15016
[7] Henn Peter G.Die Automorphismengruppen der algebraischen Funktionenk¨orper vom Geschlecht 3. Inagural-dissertation, Heidelberg, 1976.639
[8] Hesse, Otto.Uber Determinanten und ihre Anwendung in der Geometrie, ins-¨ besondere auf Curven vierter Ordnung.J. Reine Angew. Math.49(1855), 273-264. MR1578915,Zbl 049.1314cj, doi:10.1515/crll.1855.49.243.636
[9] Hesse,Otto.Uber die Doppeltangenten der Curven vierter Ordnung.¨J. ReineAngew.Math.49(1855),279-332.MR1578918,ERAM049.1317cj, doi:10.1515/crll.1855.49.279.636
[10] Holzapfel, Rolf-Peter.The ball and some Hilbert problems. Lectures in Mathematics ETH Z¨urich.Birkh¨auser Verlag, Basel, 1995. viii+160 pp. ISBN: 3-7643-28355.MR1350073,Zbl 0905.14013, doi:10.1007/978-3-0348-9051-9.637 · Zbl 0905.14013
[11] Jacobi, Carl G. J.Beweis des Satzes daß eine Curve ntenGrades im Allgeimeinen 1/2n(n−2)(n2−9) Doppeltangenten hat.J. Reine Angew. Math.40(1850), 237-260. MR1578697,Zbl 040.1107cj, doi:10.1515/crll.1850.40.237.636
[12] Kantor, S.Theorie der eindeutigen periodischen Transformationen in der Ebene.J. Reine Angew. Math.114(1895), 50-108.MR1580369, doi:10.1515/crll.1895.114.50. 637 · JFM 26.0769.04
[13] Kuribayashi, Akikazu; Komiya, Kaname.On Weierstrass points and automorphisms of curves of genus three.Algebraic geometry(Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), 253-299, Lecture Notes in Math., 732.Springer, Berlin, 1979.MR555703,Zbl 0494.14012, doi:10.1007/BFb0066649.639 · Zbl 0494.14012
[14] Maxima Development Team. Maxima manual.http://maxima.sourceforge.net/ docs/manual/maxima.html.
[15] Mumford, David.Theta characteristics of an algebraic curve.Ann. Sci. ´Ecole Norm. Sup.(4)4(1971), 181-192.MR292836,Zbl 0216.05904, doi:10.24033/asens.1209.638 · Zbl 0216.05904
[16] Ohno, Toshiaki.Application of representation theory of SL(3) to invariant elements of ternary quartics.Bull. Fac. Sci. Eng. Chuo Univ.43(2000), 7-16.MR1818572. 636
[17] Picard, Emile.Sur des fonctions de deux variables ind´ependantes analogues aux fonctions modulaires.Acta Math.2(1883), no. 1, 114-135.MR1554595,Zbl 15.0432.01, doi:10.1007/BF02612158.637
[18] Picard, Emile.Sur les formes quadratiques ternaires ind´efinies ´a ind´etermin´ees conjugu´ees et sur les fonctions hyperfuchsiennes correspondantes.Acta Math.5 (1884), no. 1, 121-182.MR1554651,JFM 16.0385.01, doi:10.1007/bf02421555.637
[19] Plaumann, Daniel; Sturmfels, Bernd; Vinzant, Cynthia.Quartic curves and their bitangents.J. Symbolic Comput.46(2011), no. 6, 712-733.MR2781949,Zbl 1214.14049,arXiv:1008.4104, doi:10.1016/j.jsc.2011.01.007.637,638 · Zbl 1214.14049
[20] Plaumann,Daniel;Sturmfels,Bernd;Vinzant,Cynthia.Computing linear matrix representations of Helton-Vinnikov curves.Mathematical methods in systems, optimization, and control, 259-277, Oper. Theory Adv. Appl., 222.Birkh¨auser/Springer Basel AG, Basel, 2012.MR2962788,Zbl 1328.14093, arXiv:1011.6057, doi:10.1007/978-3-0348-0411-0 19.638,649 · Zbl 1328.14093
[21] Popolitov, Aleksandr; Shakirov, Shamil.On undulation invariants of plane curves.Michigan Math. J.64(2015), no. 1, 143-153.MR3326583,Zbl 1349.14154, arXiv:1208.5775, doi:10.1307/mmj/1427203288.643,644 · Zbl 1349.14154
[22] Shioda, Tetsuji.On the graded ring of invariants of binary octavics.Amer. J. Math. 89(1967), 1022-1046.MR220738,Zbl 0188.53304, doi:10.2307/2373415.636 · Zbl 0188.53304
[23] Steiner, Jakob.Eigenschaften der Curven vierten Grades r¨ucksichtlich ihrer Doppeltangenten.J. Reine Angew. Math.49(1855), 265-272.MR1578916,ERAM 049.1315cj, doi:10.1515/crll.1855.49.265.636
[24] Vinnikov, Victor.Complete description of determinantal representations of smooth irreducible curves.Linear Algebra Appl.125(1989),103-140.MR1024486,Zbl 0704.14041, doi:10.1016/0024-3795(89)90035-9.636 · Zbl 0704.14041
[25] Vinnikov,Victor.Self-adjoint determinantal representations of real plane curves.Math. Ann.296(1993), no. 3, 453-479.MR1225986,Zbl 0789.14029, doi:10.1007/BF01445115.636 · Zbl 0789.14029
[26] Wiman, Anders.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.