On Zariski Multiplets of Branch Curves from Surfaces Isogenous to a Product

TL;DR

This paper establishes an asymptotic bound on the number of Zariski multiplets of certain plane curves with nodes and cusps, derived from branched covers of surfaces isogenous to a product, with implications for their moduli space.

Contribution

It provides the first asymptotic bounds on Zariski multiplets for these specific singular curves, linking their enumeration to the moduli of surfaces isogenous to a product.

Findings

Number of Zariski multiplets grows subexponentially with degree.

Constructs Zariski multiplets from branched covers of surfaces isogenous to a product.

Uses moduli space properties to estimate the cardinality of Zariski multiplets.

Abstract

In this paper we give an asymptotic bound of the cardinality of Zariski multiples of particular plane singular curves. These curves have only nodes and cusps as singularities and are obtained as branched curves of ramified covering of the plane by surfaces isogenous to a product of curves with group $(\mathbb{Z}/2\mathbb{Z})^k$. The knowledge of the moduli space of these surfaces will enable us to produce Zariski multiplets whose number grows subexponentialy in function of their degree.