Abstract
We show that all the possible pairs of integers occur as exponents for free or nearly free irreducible plane curves and line arrangements, by producing only two types of simple families of examples. The topology of the complements of these curves and line arrangements is also discussed, and many of them are shown not to be \(K(\pi ,1)\) spaces.
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Artal Bartolo, E., Cogolludo-Agustín, J.I., Matei, D.: Quasi-projectivity, Artin-Tits groups, and pencil maps. Topology of algebraic varieties and singularities, 113–136, Contemp. Math., 538, Amer. Math. Soc., Providence, RI, (2011)
Artal Bartolo, E., Gorrochategui, L., Luengo, I., Melle-Hernández, A.: On some conjectures about free and nearly free divisors, In: Singularities and Computer Algebra, Festschrift for Gert-Martin Greuel on the Occasion of his 70th Birthday, pp. 1–19, Springer (2017)
Damon, J.N.: Higher Multiplicities and Almost Free Divisors and Complete Intersections, vol. 589. Memoirs A.M.S, Providence (1996)
Dimca, A.: Singularities and Topology of Hpersurfaces, vol. Universitext. Springer, New York (1992)
Dimca, A.: Freeness versus maximal global Tjurina number for plane curves, to appear in Math. Proc. Cambridge Phil. Soc
Dimca, A.: Curve arrangements, pencils, and Jacobian syzygies. arXiv:1601.00607, to appear in Michigan Math. J
Dimca, A., Sernesi, E.: Syzygies and logarithmic vector fields along plane curves. Journal de l’École polytechnique-Mathématiques 1, 247–267 (2014)
Dimca, A., Sticlaru, G.: Free divisors and rational cuspidal plane curves. arXiv:1504.01242v4, to appear in Math. Res. Lett
Dimca, A., Sticlaru, G.: Nearly free divisors and rational cuspidal curves. arXiv:1505.00666v3
du Pleseis, A.A., Wall, C.T.C.: Application of the theory of the discriminant to highly singular plane curves. Math. Proc. Camb. Philos. Soc. 126, 259–266 (1999)
du Plessis, A.A., Wall, C.T.C.: Curves in \(P^2(\mathbb{C})\) with 1-dimensional symmetry. Rev. Mat. Complut. 12, 117–132 (1999)
Falk, M.: \( K(\pi,1)\) arrangements. Topology 34(1), 141–154 (1995)
Falk, M., Randell, R.: On the homotopy theory of arrangements. II. Arrangements-Tokyo 1998. Adv. Stud. Pure Math. 27, Kinokuniya, Tokyo, 93–125 (2000)
Hirzebruch, F.: Arrangements of lines and algebraic surfaces, In: Arithmetic and Geometry. Progress in Mathematics, II (36), pp. 113-140. Birkhäuser, Boston (1983)
Jambu, M., Terao, H.: Free arrangements of hyperplanes and supersolvable lattices. Adv. Math. 52, 248–258 (1984)
Mond, D.: Differential forms on free and almost free divisors. Proc. Lond. Math. Soc. 81, 587–617 (2000)
Orlik, P., Terao, H.: Arrangements of Hyperplanes. Springer, Berlin (1992)
Randell, R.: Lattice-isotopic arrangements are topologically isomorphic. Proc. Am. Math. Soc. 107, 555–559 (1989)
Saito, K.: Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27(2), 265–291 (1980)
Simis, A., Tohaneanu, S.O.: Homology of homogeneous divisors. Israel J. Math. 200, 449–487 (2014)
Yoshinaga, M.: Freeness of hyperplane arrangements and related topics. Annales de la Faculté des Sciences de Toulouse 23(2), 483–512 (2014)
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Alexandru Dimca: Partially supported by Institut Universitaire de France.
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Dimca, A., Sticlaru, G. On the exponents of free and nearly free projective plane curves. Rev Mat Complut 30, 259–268 (2017). https://doi.org/10.1007/s13163-017-0228-3
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DOI: https://doi.org/10.1007/s13163-017-0228-3