# On the number of moduli of plane sextics with six cusps

## TL;DR

This paper proves that the moduli space of irreducible sextic plane curves with six cusps has exactly seven moduli, matching the maximum possible, thus clarifying the structure of these algebraic curves.

## Contribution

It establishes that both irreducible components of the variety of sextic curves with six cusps have exactly seven moduli, confirming the maximal dimension of their moduli space.

## Key findings

- Both components have exactly seven moduli.
- The moduli map is dominant onto a 7-dimensional space.
- The result confirms the maximal number of moduli for these curves.

## Abstract

Let S be the variety of irreducible sextics with six cusps as singularities. Let W be one of irreducible components of W. Denoting by M_4 the space of moduli of smooth curves of genus 4, the moduli map of W is the rational map from W to M_4 sending the general point of W, corresponding to a plane curve D, to the point of M_4 parametrizing the normalization curve of D. The number of moduli of W is, by definition the dimension of the image of W with respect to the moduli map. We know that this number is at most equal to seven. In this paper we prove that both irreducible components of S have number of moduli exactly equal to seven.