Number of moduli of irreducible families of plane curves with nodes and cusps

Let $\s\subset\mathbb P^{\frac{n(n+3)}{2}}$ be the family of irreducible plane curves of degree $n$ with $d$ nodes and $k$ cusps as singularities. Let $\Sigma\subset\s$ be an irreducible component. We consider the natural rational map $$ \Pi_{\Sigma}:\Sigma\dashrightarrow \mathcal M_g, $$ from $\Sigma$ to the moduli space of curves of genus $g=\pa-d-k$. We define the \textit{number of moduli of $\Sigma$} as the dimension $dim(\Pi_{\Sigma}(\Sigma))$. If $\Sigma$ has the expected dimension equal to $3n+g-1-k$, then \begin{equation}\label{dis} dim(\Pi_{\Sigma}(\Sigma))\leq min(dim(\m_g), dim(\m_g)+\rho-k), \end{equation} where $\rho:=\rho(2,g,n)=3n-2g-6$ is the Brill-Neother number of the linear series of degree $n$ and dimension $2$ on a smooth curve of genus $g$. We say that $\Sigma$ has the expected number of moduli if the equality holds in \eqref{dis}. In this paper we construct examples of families of irreducible plane curves with nodes and cusps as singularities having expected number of moduli and with non-positive Brill-Noether number.