Anti-Zariski pairs

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 $C^{\mathrm{\prime }},C^{\mathrm{\prime }\mathrm{\prime }}\subset {\text{PP}}^2$ form an {\it anti-Zariski pair}, if $({\text{PP}}^2,C^{\mathrm{\prime }})$ and $({\text{PP}}^2,C^{\mathrm{\prime }\mathrm{\prime }})$ are homeomorhic, but $C^{\mathrm{\prime }}$ and $C^{\mathrm{\prime }\mathrm{\prime }}$ belong to different components of the equisingular family. We exhibit examples of anti-Zariski pairs and discuss related issues.