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Unexpected curves arising from special line arrangements

  • Michela Di Marca
  • Grzegorz Malara
  • Alessandro OnetoEmail author
Article
  • 23 Downloads

Abstract

In a recent paper, Cook et al. (Compos Math 154:2150–2194, 2018) used the splitting type of a line arrangement in the projective plane to study the number of conditions imposed by a general fat point of multiplicity j on the linear system of curves of degree \(j+1\) passing through the configuration of points dual to the given arrangement. If the number of conditions is less than the expected, they said that the configuration of points admits unexpected curves. In this paper, we characterize supersolvable line arrangements whose dual configuration admits unexpected curves and we provide other infinite families of line arrangements with this property.

Keywords

Fat points Line arrangements Linear systems Splitting types 

Mathematics Subject Classification

14N20 (primary) 13D02 14C20 14N05 05E40 14F05 (secondary) 

Notes

Acknowledgements

This project started during the “2017 Pragmatic Summer School: Powers of ideals and ideals of powers” which was held at the University of Catania, Italy (June 19th–7th, 2017). We are grateful to the organizers (Alfio Ragusa, Elena Guardo, Francesco Russo and Giuseppe Zappalá) and the teachers (Brian Harbourne, Adam Van Tuyl, Enrico Carlini and Tài Hà) of the school. In particular, we want to thank Brian Harbourne for suggesting and supervising this project and for useful comments on an early version of this paper. We also want to thank Michael Cuntz for sharing with us a database of crystallographic simplicial arrangements. Finally, we thank the anonymous referees for inspiring remarks and valuable comments on the original draft of the paper. The first author was partially supported by the “National Group for Algebraic and Geometric Structure, and their Applications” (GNSAGA-INdAM). The second author was partially supported by National Science Centre, Poland, grant 2016/21/N/ST1/01491. The third author was partially supported by the Aromath team of INRIA Sophia Antipolis Méditerranée (France) during the Pragmatic Summer School and acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445).

References

  1. 1.
    Akesseh, S.: Ideal containments under flat extensions and interpolation on linear systems in \(\mathbb{P}^2\). Doctoral Thesis, University of Nebraska–Lincoln, Lincoln (2017)Google Scholar
  2. 2.
    Anzis, B., Tohăneanu, Ş.: On the geometry of real or complex supersolvable line arrangements. J. Combin. Theory Ser. A 140, 76–96 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ciliberto, C., Miranda, R.: The Segre and Harbourne-Hirschowitz conjectures. In: Applications of Algebraic Geometry to Coding Theory, Physics and Computation, pp. 37–51. Springer, Dordrecht (2001)Google Scholar
  4. 4.
    Cook II, D., Harbourne, B., Migliore, J., Nagel, U.: Line arrangements and configurations of points with an unusual geometric property. Compos. Math. 154, 2150–2194 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cuntz, M.: Simplicial arrangements with up to \(27\) lines. Discrete Comput. Geom. 48, 682–701 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 4-1-0—A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2017)
  7. 7.
    Di Gennaro, R., Ilardi, G., Vallès, J.: Singular hypersurfaces characterizing the Lefschetz properties. J. Lond. Math. Soc. 89(1), 194–212 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dimca, A., Sticlaru, G.: On supersolvable and nearly supersolvable line arrangements, J. Algebraic Combin.  https://doi.org/10.1007/s10801-018-0859-6 (2018)
  9. 9.
    Farnik, Ł., Galuppi, F., Sodomaco, L., Trok, B.: On the unique unexpected quartic in \(\mathbb{P}^2\). Preprint arXiv:1803.02746
  10. 10.
    Faenzi, D., Vallès, J.: Logarithmic bundles and line arrangements, an approach via the standard construction. J. Lond. Math. Soc. 90(2), 675–694 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Geramita, A.V., Orecchia, F.: On the Cohen-Macaulay type of s-lines in \({\mathbb{A}}^n+1\). J. Algebra 70(1), 116–140 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gimigliano, A.: On linear systems of plane curves. Ph. D. Thesis, Queens University, Kingston, ON (1987)Google Scholar
  13. 13.
    Green, B., Tao, T.: On sets defining few ordinary lines. Discrete Comput. Geom. 50, 409–468 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grayson, D., Stillman, M.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
  15. 15.
    Grothendieck, A.: Sur la Classification des Fibres Holomorphes sur la Sphere de Riemann. Amer. J. Math. 79(1), 121–38 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Grünbaum, B.: A catalogue of simplicial arrangements in the real projective plane. Ars Math. Contemp. 2, 1–25 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Harbourne, B.: The geometry of rational surfaces and Hilbert functions of points in the plane. Can. Math. Soc. Conf. Proc. 6, 95–111 (1986)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Harbourne, B.: Asymptotics of linear systems, with connections to line arrangements. In: 2016 MiniPAGES Conference and Proceedings of Warsaw, Poland, (BCSim-2016-s02). arXiv:1705.09946 (To Appear)
  19. 19.
    Hirschowitz, A.: Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles génériques. J. Reine Angew. Math. 397, 208–213 (1989)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Nagata, M.: On rational surfaces. II. Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 33, 271–293 (1960)Google Scholar
  21. 21.
    Orlik, P., Terao, H.: Arrangement of Hyperplanes. Grundlehren der Mathematischen Wissenschaften, vol. 300. Springer, Berlin (1992)Google Scholar
  22. 22.
    Schenck, H.: Hyperplane arrangements: computations and conjectures. In: Advanced Studies in Pure Mathematics, vol. 62. Mathematical Society of Japan, Tokyo (2012)Google Scholar
  23. 23.
    Segre, B.: Alcune questioni su insiemi finiti di punti in Geometria Algebrica. Atti del Convegno Internaz. di Geom. Alg, Torino (1961)zbMATHGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di GenovaGenoaItaly
  2. 2.Department of MathematicsPedagogical University of CracowKrakówPoland
  3. 3.Barcelona Graduate School of Mathematics (BGSMath) and Department of MathematicsUniversitat Politècnica de CatalunyaBarcelonaSpain

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