On the number of moduli of plane sextics with six cusps

Let $\Sigma_{6,0}^6$ be the variety of irreducible sextics with six cusps as singularities. Let $\Sigma\subset\Sigma_{6,0}^6$ be one of irreducible components of $\Sigma_{6,0}^6$. Denoting by $\m_4$ the space of moduli of smooth curves of genus $4$, we consider the rational map $\Pi:\Sigma\dashrightarrow\m_4$ sending the general point $\ge$ of $\Sigma$, corresponding to a plane curve $\g\subset\p$, to the point of $\m_4$ parametrizing the normalization curve of $\g$. The number of moduli of $\Sigma$ is, by definition the dimension of $\Pi(\Sigma)$. We know that $dim(\Pi(\Sigma))\leq dim(\m_4)+\rho(2,4,6)-6=7$, where $\rho(2,4,6)$ is the Brill-Neother number of linear series of dimension $2$ and degree $6$ on a curve of genus $4$. We prove that both irreducible components of $\Sigma_{6,0}^6$ have number of moduli equal to seven.