On Zariski's pairs of m-th canonical discriminant curves

TL;DR

This paper constructs examples of Zariski pairs of plane cuspidal curves that are discriminant curves of morphisms from homeomorphic surfaces of general type, highlighting their geometric properties and differences.

Contribution

It provides explicit examples of Zariski pairs of discriminant curves arising from linear systems on homeomorphic surfaces of general type, expanding understanding of their geometric distinctions.

Findings

Examples of Zariski pairs with m-th canonical discriminant curves

Discriminant curves from linear systems on homeomorphic surfaces

Differences in geometric properties despite topological similarity

Abstract

In this note we give examples of Zariski's pairs $B_{1,m}, B_{2,m}$ ($m \in N$ and $m \geq 5$) of plane cuspidal curves such that (i) $B_{i,m}$ is the discriminant curve of a generic morphism $f_{i,m}:S_i \to P^2$, $i=1, 2$, (ii) $S_1$ and $S_2$ are homeomorphic surfaces of general type, (iii) $f_{i,m}$ is given by linear three-dimensional subsystem of the mth canonical class of $S_i$.