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A new hierarchy for complex plane curves

Part of: Local theory Singularities Commutative algebra: Homological methods Curves

Published online by Cambridge University Press:  12 November 2025

Piotr Pokora Open the ORCID record for Piotr Pokora [Opens in a new window] ,
Alexandru Dimca  and
Takuro Abe
Piotr Pokora*
Affiliation:
Department of Mathematics, University of the National Education Commission Krakow , Poland
Alexandru Dimca
Affiliation:
Université Côte d’Azur , France e-mail: Alexandru.Dimca@univ-cotedazur.fr
Takuro Abe
Affiliation:
Department of Mathematics, Rikkyo University , Japan e-mail: abetaku@rikkyo.ac.jp
*
e-mail: piotrpkr@gmail.com piotr.pokora@uken.krakow.pl
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Abstract

We define the type of a plane curve as the initial degree of the corresponding Bourbaki ideal. Then, we show that this invariant behaves well with respect to the union of curves. Curves of type $0$ are precisely the free curves, while curves of type $1$ are the plus-one generated curves. In this article, we first show that line arrangements and conic-line arrangements can exhibit all the theoretically possible types. In the second part, we study the properties of the curves of type $2$ and construct families of line arrangements and conic-line arrangements of this type.

Keywords

Jacobian modulefree curveplus-one generated curveline arrangementsconic-line arrangements

MSC classification

Primary: 14H50: Plane and space curves
Secondary: 14B05: Singularities 13D02: Syzygies, resolutions, complexes 32S22: Relations with arrangements of hyperplanes

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Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 25
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

T.A. is partially supported by JSPS KAKENHI Grant no. JP23K17298 and Grant no. JP23K20788. A.D. is partially supported by the project “Singularities and Applications” – CF 132/31.07.2023 funded by the European Union – NextGenerationEU – through Romania’s National Recovery and Resilience Plan. P.P. is supported by the National Science Centre (Poland) Sonata Bis Grant {2023/50/E/ST1/00025}. For the purpose of Open Access, the author has applied a CC-BY public copyright license to any Author Accepted Manuscript (AAM) version arising from this submission.

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