Anti-Zariski pairs

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 $C^{\prime }$ and $C^{\prime \prime }$ belong to different components of the equisingular family. We exhibit examples of anti-Zariski pairs and discuss related issues.

Following comments of Enrique Artal Bartolo, we added new references and updated information on various types of equivalence of plane complex curves