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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''⊂PP2 form an {\it anti-Zariski pair}, if (PP2,C') and (PP2,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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Cite

BibTeXRIS
BibTeX
@misc{ren2026antizariski,
  title = {{Anti-Zariski pairs}},
  author = {Peng Ren and Eugenii Shustin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.20268},
  eprint = {2606.20268},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.20268},
}
RIS
TY  - UNPB
TI  - Anti-Zariski pairs
AU  - Peng Ren
AU  - Eugenii Shustin
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2606.20268
UR  - https://arxiv.org/abs/2606.20268
AB  - 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''⊂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.
ER  -