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arXiv · 2606.20268

Anti-Zariski pairs

Peng Ren·Eugenii Shustin
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Abstract

In 1929, O. Zariski found a pair of complex plane algebraic curves of the same degree and with the same collection of singularities, but embedded into the plane in a topologically different way. Accordingly, such curves belong to different components of the equisingular family. This phenomenon has been intensively studied till now. In this note, we propose a different insight on this subject: Two curves $C',C''\subset\PP^2$ form an {\it anti-Zariski pair}, if $(\PP^2,C')$ and $(\PP^2,C'')$ are homeomorhic, but $C'$ and $C''$ belong to different components of the equisingular family. We exhibit examples of anti-Zariski pairs and discuss related issues.

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Peng Ren, Eugenii Shustin. 2026-06-18. Anti-Zariski pairs. https://arxiv.org/abs/2606.20268

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