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<p>In this paper, we construct a Poincaré–type polynomial for any reduced plane curve. This polynomial is then used to decode the homological properties of plus-one generated curves. The construction presented herein offers a generalization of the results obtained in [<CitationRef CitationID="CR9">9</CitationRef>] for free plane curves. In accordance with the general result that has been established, combinatorial avatars of our Poincaré-type polynomial are presented for particular classes of reduced plane curves with quasi–homogeneous singularities. This phenomenon is exemplified by specific cubic-line arrangements and quartic-line arrangements.</p>

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On a Poincaré-type polynomial for plus-one generated plane curves

  • 期刊论文

  • 发表日期：

    2026年1月24日

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  • Bartosz Jarosławski

摘要

In this paper, we construct a Poincaré–type polynomial for any reduced plane curve. This polynomial is then used to decode the homological properties of plus-one generated curves. The construction presented herein offers a generalization of the results obtained in [9] for free plane curves. In accordance with the general result that has been established, combinatorial avatars of our Poincaré-type polynomial are presented for particular classes of reduced plane curves with quasi–homogeneous singularities. This phenomenon is exemplified by specific cubic-line arrangements and quartic-line arrangements.