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The 21 reducible polars of Klein’s quartic. (English) Zbl 1460.14071
The paper under review deals with the study of arrangements of lines, with an eye towards extension to higher degree curves, with particular reference to the Klein quartic curve \(C\) and its rich geometry. The curve \(C\) is defined by the homogeneous equation \(x^3y + y^3z + z^3x=0\) in the complex projective plane, and its relevant properties derive from the fact that its automorphism group has the maximal possible order, according to Hurzwitz’s bound. Moreover, it is realized by a subgroup \(G \subset \mathrm{PGL}(3, \mathbb C)\). There exists a unique arrangement of \(21\) lines invariant under the action of \(G\), discovered by Klein, which is particularly symmetric and extremal in many senses. The authors study an arrangement of lines and conics closely related to it. Every sufficiently general smooth plane quartic has 21 reducible polars and, in the case of \(C\), each of them splits as the union of one of the lines in Klein’s arrangement plus a smooth conic. The arrangement \(\mathcal K\) the authors study is that consisting of these 21 lines and 21 conics, taken together. All singularities of \(\mathcal K\) are described, in particular their position with respect to relevant curves associated with \(C\), as well as their number according to the various multiplicities. Further specific properties of \(\mathcal K\) are discussed in connection with problems in the study of arrangements of plane curves addressed in recent years by several authors.

MSC:
14H50 Plane and space curves
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14J26 Rational and ruled surfaces
14C20 Divisors, linear systems, invertible sheaves
14N05 Projective techniques in algebraic geometry
14E05 Rational and birational maps
13A15 Ideals and multiplicative ideal theory in commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
Software:
SINGULAR
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[1] Abe, T.; Dimca, A., Splitting Types of Bundles of Logarithmic Vector Fields along Plane Curves (2018) · Zbl 1394.14020
[2] Adler, A., The Eightfold Way. Mathematical Sciences Research Institute Publications, 35, 221-285 (1999), Cambridge: Cambridge University Press, Cambridge
[3] Akesseh, S., Ideal Containments under Flat Extensions.”, J. Algebra, 492, 44-51 (2017) · Zbl 1408.13053
[4] Bauer, T.; Di Rocco, S.; Harbourne, B.; Huizenga, J.; Lundman, A.; Pokora, P.; Szemberg, T., Bounded Negativity and Arrangements of Lines.”, Int. Math. Res. Not. IMRN, 19, 9456-9471 (2015) · Zbl 1330.14007
[5] Bauer, T.; Di Rocco, S.; Harbourne, B.; Huizenga, J.; Seceleanu, A.; Szemberg, T., Negative Curves on Symmetric Blowups of the Projective Plane, Resurgences, and Waldschmidt Constants, International Mathematics Research Notices (2018)
[6] Bertini, E., Le tangenti multiple della Cayleyana di una quartica piana generale, Torino Atti, 32, 32-33 (1896) · JFM 27.0490.05
[7] [Casas-Alvero 14] Casas-Alvero, E.. Analytic projective geometry. EMS Textbooks in Mathematics. Zürich: European Mathematical Society (EMS), 2014. doi:. · Zbl 1292.51002
[8] [Decker et al. 18] Decker, W., Greuel, G.-M., Pfister, G., and Schönemann, H.. Singular 4-1-1 — A computer algebra system for polynomial computations. Available online (http://www.singular.uni-kl.de), 2018.
[9] Dimca, A., On rational Cuspidal Plane Curves, and the Local Cohomology of Jacobian Rings, ArXiv e-prints (2017)
[10] Dimca, A., Freeness versus Maximal Global Tjurina Number for Plane Curves.”, Math. Proc. Camb. Philos. Soc, 163, 1, 161-172 (2017) · Zbl 1387.14080
[11] Dimca, A.; Sticlaru., G., Free and Nearly Free Curves vs. Rational Cuspidal Plane Curves.”, Publ. Res. Inst. Math. Sci, 54, 1, 163-179 (2018) · Zbl 1391.14057
[12] Dolgachev, I., Classical Algebraic Geometry (2012), Cambridge: Cambridge University Press, Cambridge · Zbl 1252.14001
[13] Dolgachev, I.; Kanev., V., Polar Covariants of Plane Cubics and Quartics.”, Adv. Math, 98, 2, 216-301 (1993) · Zbl 0791.14013
[14] Dumnicki, M.; Szemberg, T.; Tutaj-Gasińska, H., Counterexamples to the \(####\) Containment.”, J. Algebra, 393, 24-29 (2013) · Zbl 1297.14008
[15] Ein, L.; Lazarsfeld, R.; Smith, K. E., Uniform Bounds and Symbolic Powers on Smooth Varieties.”, Invent. Math, 144, 2, 241-252 (2001) · Zbl 1076.13501
[16] Elkies, N. D., The Eightfold Way, vol. 35 of Mathematical Sciences Research Institute Publications, The Klein Quartic in Number Theory, 51-101 (1999), Cambridge: Cambridge University Press, Cambridge
[17] Gerbaldi, F., Sui gruppi di sei coniche in involuzione, Torino Atti, 17, 566-580 (1882) · JFM 14.0537.02
[18] [Harbourne 17] Harbourne., B. “Asymptotics of Linear Systems, with Connections to Line Arrangements.” (2017), arXiv:1705.09946. · Zbl 1400.14028
[19] Harbourne, B.; Huneke, C., Are Symbolic Powers Highly Evolved?”, J. Ramanujan Math. Soc, 28A, 247-266 (2013) · Zbl 1296.13018
[20] Hochster, M.; Huneke, C., Comparison of Symbolic and Ordinary Powers of Ideals.”, Invent. Math, 147, 2, 349-369 (2002) · Zbl 1061.13005
[21] Hurwitz, A., Ueber algebraische Gebilde mit eindeutigen Transformationen in sich.”, Math. Ann, 41, 3, 403-442 (1892) · JFM 24.0380.02
[22] Jeurissen, R. H.; van Os, C. H.; Steenbrink, J. H. M., The Configuration of Bitangents of the Klein Curve.”, Discrete Math, 132, 1-3, 83-96 (1994) · Zbl 0808.05059
[23] Klein, F., Ueber die Transformation siebenter Ordnung der elliptischen Functionen.”, Math. Ann, 14, 3, 428-471 (1878) · JFM 11.0297.01
[24] Levy, S., The Eightfold Way, Vol. 35, Mathematical Sciences Research Institute Publications (1999), Cambridge: Cambridge University Press, Cambridge · Zbl 0941.00006
[25] Orlik, P.; Terao, H., Arrangements of hyperplanes, volume 300 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] (1992), Berlin: Springer-Verlag, Berlin · Zbl 0757.55001
[26] Pokora, P., Hirzebruch-Type Inequalities and Plane Curve Configurations.”, Internat. J. Math, 28, 2, 1750013 (2017) · Zbl 1375.14032
[27] Pokora., P., The Orbifold Langer-Miyaoka-Yau Inequality and Hirzebruch-Type Inequalities.”, Electron. Res. Announc. Math. Sci, 24, 21-27 (2017) · Zbl 1407.52032
[28] Pokora, P.; Tutaj-Gasińska, H., Harbourne Constants and Conic Configurations on the Projective Plane.”, Math. Nachr, 289, 7, 888-894 (2016) · Zbl 1343.14008
[29] Schenck, H.; Tohǎneanu, Ş. O., “Freeness of Conic-Line Arrangements in, Comment. Math. Helv, 84, 2, 235-258 (2009) · Zbl 1183.52014
[30] Steiner, J., Allgemeine Eigenschaften der algebraischen Curven.”, J. Reine Angew. Math, 47, 1-6 (1854) · ERAM 047.1255cj
[31] Szemberg, T.; Szpond., J., On the Containment Problem.”, Rend. Circ. Mat. Palermo (2), 66, 2, 233-245 (2017) · Zbl 1386.14045
[32] Terao., H., Generalized Exponents of a free Arrangement of Hyperplanes and Shepherd-Todd-Brieskorn Formula.”, Invent. Math, 63, 1, 159-179 (1981) · Zbl 0437.51002
[33] Valentiner, H., De endelige Transformationsgruppers Theori. Avec un résumé en français, Kjób. Skrift, 6, 64-235 (1889) · JFM 21.0135.01
[34] Wiman, A., Ueber eine einfache Gruppe von 360 ebenen Collineationen.”, Math. Ann, 47, 531-556 (1896) · JFM 27.0103.03
[35] Wiman, A., Zur Theorie der endlichen Gruppen von birationalen Transformationen in der Ebene.”, Math. Ann, 48, 1-2, 195-240 (1896) · JFM 30.0600.01
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