Unexpected curves arising from special line arrangements

  • Michela Di Marca
  • Grzegorz Malara
  • Alessandro OnetoEmail author


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.


Fat points Line arrangements Linear systems Splitting types 

Mathematics Subject Classification

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



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).


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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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